arXiv daily: Analysis of PDEs

arXiv daily: Analysis of PDEs (math.AP)

1.Smooth transonic flows with nonzero vorticity to a quasi two dimensional steady Euler flow model

Authors:Shangkun Weng, Zhouping Xin

Abstract: This paper concerns studies on smooth transonic flows with nonzero vorticity in De Laval nozzles for a quasi two dimensional steady Euler flow model which is a generalization of the classical quasi one dimensional model. First, the existence and uniqueness of smooth transonic flows to the quasi one-dimensional model, which start from a subsonic state at the entrance and accelerate to reach a sonic state at the throat and then become supersonic are proved by a reduction of degeneracy of the velocity near the sonic point and the implicit function theorem. These flows can have positive or zero acceleration at their sonic points and the degeneracy types near the sonic point are classified precisely. We then establish the structural stability of the smooth one dimensional transonic flow with positive acceleration at the sonic point for the quasi two dimensional steady Euler flow model under small perturbations of suitable boundary conditions, which yields the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity and positive acceleration to the quasi two dimensional model. The positive acceleration of the one dimensional transonic solutions plays an important role in searching for an appropriate multiplier for the linearized second order mixed type equations. A deformation-curl decomposition for the quasi two dimensional model is utilized to deal with the transonic flows with nonzero vorticity.

2.Multi-parameter perturbations for the space periodic heat equation

Authors:Matteo Dalla Riva, Paolo Luzzini, Riccardo Molinarolo, Paolo Musolino

Abstract: This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of the results from the first part. Specifically, we consider a transmission problem for the heat equation in a periodic two-phase composite material and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and conductivity parameters. Finally, in the last part of the paper, we fix all parameters except for the contrast parameter and outline a strategy to deduce an explicit expansion of the solution using a Neumann-type series.

3.Strict Faber-Krahn type inequality for the mixed local-nonlocal operator under polarization

Authors:K. Ashok Kumar, Nirjan Biswas

Abstract: Let $\Omega \subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class $\mathcal{C}^{1,\beta }$ for some $\beta \in (0,1)$. For $p\in (1, \infty )$ and $s\in (0,1)$, let $\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local-nonlocal operator $-\Delta _p+(-\Delta _p)^s$ in $\Omega $ with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber-Krahn type inequality for $\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of $\Lambda _{p}^s(\cdot )$ over annular domains, and characterize the uniqueness of the balls in the classical Faber-Krahn inequality for $-\Delta _p+(-\Delta _p)^s$.

4.Null-controllability for weakly dissipative heat-like equations

Authors:Paul Alphonse UMPA-ENSL, Armand Koenig IMB

Abstract: We study the null-controllability properties of heat-like equations posed on the whole Euclidean space $\mathbb R^n$. These evolution equations are associated with Fourier multipliers of the form $\rho(\vert D_x\vert)$, where $\rho\colon[0,+\infty)\rightarrow\mathbb C$ is a measurable function such that $\Re\rho$ is bounded from below. We consider the ``weakly dissipative'' case, a typical example of which is given by the fractional heat equations associated with the multipliers $\rho(\xi) = \xi^s$ in the regime $s\in(0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $\rho$, in particular assuming that $\rho(\xi) = o(\xi)$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $\rho$.Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.

5.An elliptic problem involving critical Choquard and singular discontinuous nonlinearity

Authors:Gurdev C. Anthal, Jacques Giacomoni, Konijeti Sreenadh

Abstract: The present article investigates the existence, multiplicity and regularity of weak solutions of problems involving a combination of critical Hartree type nonlinearity along with singular and discontinuous nonlinearity. By applying variational methods and using the notion of generalized gradients for Lipschitz continuous functional, we obtain the existence and the multiplicity of weak solutions for some suitable range of $\lambda$ and $\gamma$. Finally by studying the $L^\infty$-estimates and boundary behavior of weak solutions, we prove their H\"{o}lder and Sobolev regularity.

6.Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field I: Schrödinger equation

Authors:Haoran Wang, Fang Zhang, Junyong Zhang

Abstract: This is the first of a series of papers in which we investigate the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In this first starting paper, we prove the local-in-time dispersive estimates and Strichartz estimates for Schr\"odinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The key ingredient is the construction of Schr\"odinger propagator, we provide two methods to construct the propagator. The first one is combined the strategies of \cite{FFFP1} and \cite{GYZZ22, FZZ22}, and the second one is based on the Schulman-Sunada formula in sprit of \cite{stov, stov1} in which the heat kernel has been studied. In future papers, we will continue investigating this quantum model for wave with one or multiple Aharonov-Bohm solenoids in a uniform magnetic field.

7.Uniqueness of blowup at singular points for superconductivity problem

Authors:Lili Du, Xu Tang, Cong Wang

Abstract: In this paper, we prove that the uniqueness of blowup at the maximum point of coincidence set of the superconductivity problem, mainly based on the Weiss-type and Monneau-type monotonicity formulas, and the proof of the main results in this paper is inspired the recent paper \cite{CFL22} by Chen-Feng-Li.

8.Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field II: wave equation

Authors:Haoran Wang, Fang Zhang, Junyong Zhang

Abstract: This is the second of a series of papers in which we investigate the decay estimates for dispersive equations with Aharonov-Bohm solenoids in a uniform magnetic field. In our first starting paper \cite{WZZ}, we have studied the Strichartz estimates for Schr\"odinger equation with one Aharonov-Bohm solenoid in a uniform magnetic field. The wave equation in this setting becomes more delicate since a difficulty is raised from the square root of the eigenvalue of the Schr\"odinger operator $H_{\alpha, B_0}$ so that we cannot directly construct the half-wave propagator. An independent interesting result concerning the Gaussian upper bounds of the heat kernel is proved by using two different methods. The first one is based on establishing Davies-Gaffney inequality in this setting and the second one is straightforward to construct the heat kernel (which efficiently captures the magnetic effects) based on the Schulman-Sunada formula. As byproducts, we prove optimal bounds for the heat kernel and show the Bernstein inequality and the square function inequality for Schr\"odinger operator with one Aharonov-Bohm solenoid in a uniform magnetic field.

9.Generalized Newton-Busemann Law For Two-Dimensional Steady Hypersonic-limit Euler Flows Passing Ramps With Skin-Frictions

Authors:Aifang Qu, Xueying Su, Hairong Yuan

Abstract: By considering Radon measure solutions for boundary value problems of stationary non-isentropic compressible Euler equations on hypersonic-limit flows passing ramps with frictions on their boundaries, we construct solutions with density containing Dirac measures supported on the boundaries of the ramps, which represent the infinite-thin shock layers under different assumptions on the skin-frictions. We thus derive corresponding generalizations of the celebrated Newton-Busemann law in hypersonic aerodynamics for distributions of drags/lifts on ramps.

10.A mean field problem approach for the double curvature prescription problem

Authors:Luca Battaglia, Rafael López-Soriano

Abstract: In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions: $$\left\{\begin{array}{ll} -\Delta u+2K_g=2Ke^u&\text{in }\Sigma\\ \partial_\nu u+2h_g=2he^\frac u2&\text{on }\partial\Sigma. \end{array}\right.$$ We provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques.

11.Energy decay of some multi-term nonlocal-in-time Moore--Gibson--Thompson equations

Authors:Mostafa Meliani, Belkacem Said-Houari

Abstract: This paper aims at exploring the long-term behavior of some nonlocal high-order-in-time wave equations. These equations, which have come to be known as Moore--Gibson--Thompson equations, arise in the context of acoustic wave propagation when taking into account thermal relaxation mechanisms in complex media such as human tissue. While the long-term behavior of linear local-in-time acoustic equations is well understood, their nonlocal counterparts still retain many mysteries in that regard. We establish here a set of assumptions which ensures exponential decay of the energy of the system. These assumptions are then shown to be verified by a large class of rapidly decaying memory kernels. Under weaker assumptions on the kernel we show that one may still obtain that the energy vanishes but without a rate of convergence. We furthermore refine previous results on the local well-posedness of the studied equation and establish a necessary initial-data compatibility condition for the solvability of the problem.

12.Scattered wavefield in the stochastic homogenization regime

Authors:Josselin Garnier CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France, Laure Giovangigli POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France, Quentin Goepfert CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France, Pierre Millien Institut Langevin, ESPCI Paris, PSL University, CNRS, 1 rue Jussieu, F-75005 Paris, France

Abstract: In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments.

13.The heat equation with the $L^p$ primitive integral

Authors:Erik Talvila

Abstract: For each $1\leq p<\infty$ a Banach space of integrable Schwartz distributions is defined by taking the distributional derivative of all functions in $L^p({\mathbb R})$. Such distributions can be integrated when multiplied by a function that is the integral of a function in $L^q({\mathbb R})$, where $q$ is the conjugate exponent of $p$. The heat equation on the real line is solved in this space of distributions. The initial data is taken to be the distributional derivative of an $L^p({\mathbb R})$ function. The solutions are shown to be smooth functions. Initial conditions are taken on in norm. Sharp estimates of solutions are obtained and a uniqueness theorem is proved.

1.Well-posedness for an hyperbolic-hyperbolic-elliptic system describing cold plasmas

Authors:Diego Alonso-Orán, Rafael Granero-Belinchón

Abstract: In this short note, we provide the well-posedness for an hyperbolic-hyperbolic-elliptic system of PDEs describing the motion of collision free-plasma in magnetic fields. The proof combines a pointwise estimate together with a bootstrap type of argument for the elliptic part of the system.

2.Long-time instability of planar Poiseuille-type flow in compressible fluid

Authors:Andrew Yang, Zhu Zhang

Abstract: It is well-known that at the high Reynolds number, the linearized Navier-Stokes equations around the inviscid stable shear profile admit growing mode solutions due to the destabilizing effect of the viscosity. This phenomenon, called Tollmien-Schlichting instability, has been rigorously justified by Grenier-Guo-Nguyen [Adv. Math. 292 (2016); Duke J. Math. 165 (2016)] for Poiseuille flows and boundary layers in the incompressible fluid. To reveal this intrinsic instability mechanism in the compressible setting, in this paper, we study the long-time instability of the Poiseuille flow in a channel. Note that this instability arises in a low-frequency regime instead of a high-frequency regime for the Prandtl boundary layer. The proof is based on the quasi-compressible-Stokes iteration introduced by Yang-Zhang in [50] and subtle analysis of the dispersion relation for the instability. Note that we do not require symmetric conditions on the background shear flow or perturbations.

3.Artificial boundary conditions for random ellitpic systems with correlated coefficient field

Authors:Nicolas Clozeau, Lihan Wang

Abstract: We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.

4.Extremal functions for a fractional Morrey inequality: Symmetry properties and limit at infinity

Authors:Alireza Tavakoli

Abstract: In a series of articles, Ryan Hynd and Francis Seuffert have studied extremal functions for the Morrey inequality. Building upon their work, we study the extremals of a Morrey-type inequality for fractional Sobolev spaces. We verify a few of the results in the spirit of Hynd and Seuffert concerning the symmetry of extremals and their limit at infinity.

5.Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

Authors:Lucrezia Cossetti, Luca Fanelli, David Krejcirik

Abstract: We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.

6.Qualitative properties of the fourth-order hyperbolic equations

Authors:K. Buryachenko

Abstract: We investigate the qualitative properties of the weak solutions to the boundary value problems for the hyperbolic fourth-order linear equations with constant coefficients in the plane bounded domain convex with respect to characteristics. The main question is to prove the analogue of maximum principle, solvability and uniqueness results for the weak solutions of initial and boundary value problems in the case of weak regularities of initial data from $L^2.$

7.Planar loops with prescribed curvature via Hardy's inequality

Authors:Gabriele Cora, Roberta Musina

Abstract: We investigate the existence of closed planar loops with prescribed curvature. Our approach is variational, and relies on a Hardy type inequality and its associated functional space.

8.A model for the approximation of vortex rings by almost rigid bodies

Authors:David Meyer

Abstract: We consider a model that approximates vortex rings in the axisymmetric 3D Euler equation by the movement of almost rigid bodies described by Newtonian mechanics. We assume that the bodies have a circular cross-section and that the fluid is irrotational and interacts with the bodies through the pressure exerted at the boundary. We show that this kind of system can be described through an ODE in the positions of the bodies and that in the limit, where the bodies shrink to massless filaments, the system converges to an ODE system similar to the point vortex system. In particular, we can show that in a suitable set-up, the bodies perform a leapfrogging motion.

9.Stabilization of 2D Navier-Stokes equations by means of actuators with locally supported vorticity

Authors:Sérgio S. Rodrigues, Dagmawi A. Seifu

Abstract: Exponential stabilization to time-dependent trajectories for the incompressible Navier-Stokes equations is achieved with explicit feedback controls. The fluid is contained in two-dimensional spatial domains and the control force is, at each time instant, a linear combination of a finite number of given actuators. Each actuator has its vorticity supported in a small subdomain. The velocity field is subject to Lions boundary conditions. Simulations are presented showing the stabilizing performance of the proposed feedback. The results also apply to a class of observer design problems.

1.Free boundary regularity for tumor growth with nutrients and diffusion

Authors:Carson Collins, Matt Jacobs, Inwon Kim

Abstract: In this paper, we study a tumor growth model where the growth is driven by nutrient availability and the tumor expands according to Darcy's law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the hitting time T, which records the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that T is C^{\alpha}, where \alpha depends only on the dimension, which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in spacetime except on a set of Hausdorff dimension at most $d-\alpha$. On the set of regular points, we further show that the tumor patch is locally $C^{1,\alpha}$ in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.

2.Some attempts on $L^{2}$ boundedness for 1-D wave equations with time variable coeffecients

Authors:Ryo Ikehata

Abstract: We consider the $L^2$-boundedness of the solution itself of the Cauchy problem for wave equations with time-dependent wave speeds. We treat it in the one-dimensional Euclidean space. To study these, we adopt a simple multiplier method by using a special property equiped with the one dimensional space.

3.Domains of dependence for subelliptic wave equations and unique continuation for square roots of H{ö}rmander's operators

Authors:Nicolas Burq LMO, Claude Zuily LMO

Abstract: We prove the sharp domain of dependence property for solutions to subelliptic wave equations for sums of squares of vector fields satisfying H{\"o}rmander bracket condition. We deduce a unique continuation property for the square root of subelliptic Laplace operators under an additional analyticity condition.

4.Determination of quasilinear terms from restricted data and point measurements

Authors:Yavar Kian

Abstract: We study the inverse problem of determining uniquely and stably quasilinear terms appearing in an elliptic equation from boundary excitations and measurements associated with the solutions of the corresponding equation. More precisely, we consider the determination of quasilinear terms depending simultaneously on the solution and the gradient of the solution of the elliptic equation from measurements of the flux restricted to some fixed and finite number of points located at the boundary of the domain generated by Dirichlet data lying on a finite dimensional space. Our Dirichlet data will be explicitly given by affine functions taking values in $\mathbb R$. We prove our results by considering a new approach based on explicit asymptotic properties of solutions of these class of nonlinear elliptic equations with respect to a small parameter imposed at the boundary of the domain.

5.Linear and quasilinear evolution equations in the context of weighted $L_p$-spaces

Authors:Mathias Wilke

Abstract: In 2004, the article "Maximal regularity for evolution equations in weighted $L_p$-spaces" by J. Pr\"{u}ss and G. Simonett has been published in Archiv der Mathematik. We provide a survey of the main results of that article and outline some applications to semilinear and quasilinear parabolic evolution equations which illustrate their power.

6.Explicit formula for the Gamma-convergence homogenized quadratic curvature energy in isotropic Cosserat shell models

Authors:Maryam Mohammadi Saem, Emilian Bulgariu, Ionel-Dumitrel Ghiba, Patrizio Neff

Abstract: We show how to explicitly compute the homogenized curvature energy appearing in the isotropic $\Gamma$-limit for flat and for curved initial configuration Cosserat shell models, when a parental three-dimensional minimization problem on $\Omega \subset \mathbb{R}^3$ for a Cosserat energy based on the second order dislocation density tensor $\alpha:=\overline{R} ^T {\rm Curl}\,\overline{R} \in \mathbb{R}^{3\times 3}$, $\overline{R}\in {\rm SO}(3)$ is used.

7.Determination of Lower Order Perturbations of a Polyharmonic Operator in Two Dimensions

Authors:Rajat Bansal, Venkateswaran P. Krishnan, Rahul Raju Pattar

Abstract: We study an inverse boundary value problem for a polyharmonic operator in two dimensions. We show that the Cauchy data uniquely determine all the anisotropic perturbations of orders at most $m-1$ and several perturbations of orders $m$ to $2m-2$ under some restriction. The uniqueness proof relies on the $\bar{\partial}$-techniques and the method of stationary phase.

8.On Morrey's inequality in Sobolev-Slobodeckiĭ spaces

Authors:Lorenzo Brasco, Francesca Prinari, Firoj Sk

Abstract: We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodecki\u{\i} spaces on the whole $\mathbb{R}^N$. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some regularity estimates. We also analyze the sharp asymptotic behaviour of this constant as we reach the borderline case $s\,p=N$, where the inequality fails. This can be done by means of a new elementary proof of the Morrey inequality, which combines: a local fractional Poincar\'e inequality for punctured balls, the definition of capacity of a point and Hardy's inequality for the punctured space. Finally, we compute the limit of the sharp Morrey constant for $s\nearrow 1$, as well as its limit for $p\nearrow \infty$. We obtain convergence of extremals, as well.

9.Quantitative unique continuation property for solutions to a bi-Laplacian equation with a potential

Authors:H. Liu, L. Tian, X. Yang

Abstract: In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} \Delta^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than \begin{equation} C\left(\|V\|^{\frac{1}{4}}_{L^{\infty}}+\|\nabla V\|_{L^{\infty}}+1\right). \end{equation} Our key argument is to lift the original equation to that with a positive potential, then decompose the resulted fourth-order equation into a special system of two second-order equations. Based on the special system, we define a variant frequency function with weights and derive its almost monotonicity to establishing some doubling inequalities with explicit dependence on the Sobolev norm of the potential function.

10.On the quasilinear Schrödinger equations on tori

Authors:Felice Iandoli

Abstract: We improve the result by Feola and Iandoli [J. de Math. Pures et App., 157:243-281, 2022], showing that quasilinear Hamiltonian Schr\"odinger type equations are well posed on $H^s(\mathbb{T}^d)$ if $s>d/2+3$. We exploit the sharp paradifferential calculus on $\mathbb{T}^d$ introduced by Berti, Maspero and Murgante [J. Dynam. and Differential Equations, 33 (3): 1475-1513, 2021].

11.Dynamics and spreading speeds of a nonlocal diffusion model with advection and free boundaries

Authors:Chengcheng Cheng

Abstract: In this paper, we investigate a Fisher-KPP nonlocal diffusion model incorporating the effect of advection and free boundaries, aiming to explore the propagation dynamics of the nonlocal diffusion-advection model. Considering the effects of the advection, the existence, uniqueness, and regularity of the global solution are obtained. We introduce the principal eigenvalue of the nonlocal operator with the advection term and discuss the asymptotic properties influencing the long-time behaviors of the solution for this model. Moreover, we give several sufficient conditions determining the occurrences of spreading or vanishing and obtain the spreading-vanishing dichotomy. Most of all, applying the semi-wave solution and constructing the upper and the lower solution, we give an explicit description of the finite asymptotic spreading speeds for the double free boundaries on the effects of the nonlocal diffusion and advection compared with the corresponding problem without an advection term.

12.Spreading speeds of a nonlocal diffusion model with free boundaries in the time almost periodic media

Authors:Chengcheng Cheng, Rong Yuan

Abstract: In this paper, we mainly investigate the spreading dynamics of a nonlocal diffusion KPP model with free boundaries which is firstly explored in time almost periodic media. As the spreading occurs, the long-run dynamics are obtained. Especially, when the threshold condition for the kernel function is satisfied, applying the novel positive time almost periodic function, we accurately express the unique asymptotic spreading speed of the free boundary problem.

13.Non Linear Hyperbolic-Parabolic Systems with Dirichlet Boundary Conditions

Authors:Rinaldo M. Colombo, Elena Rossi

Abstract: We prove the well posedness of a class of non linear and non local mixed hyperbolic-parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on data and parameters are also provided. These equations appear in models devoted to population dynamics or to epidemiology, for instance.

14.Invariant Gibbs measures for $(1+1)$-dimensional wave maps into Lie groups

Authors:Bjoern Bringmann

Abstract: We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Lie group. The corresponding Gibbs measure is given by a Brownian motion on the Lie group, which plays a central role in stochastic geometry. Our main theorem is the almost sure global well-posedness and invariance of the Gibbs measure for the wave maps equation. It is the first result of this kind for any geometric wave equation. Our argument relies on a novel finite-dimensional approximation of the wave maps equation which involves the so-called Killing renormalization. The main part of this article then addresses the global convergence of our approximation and the almost invariance of the Gibbs measure under the corresponding flow. The proof of global convergence requires a carefully crafted Ansatz which includes modulated linear waves, modulated bilinear waves, and mixed modulated objects. The interactions between the different objects in our Ansatz are analyzed using an intricate combination of analytic, geometric, and probabilistic ingredients. In particular, geometric aspects of the wave maps equation are utilized via orthogonality, which has previously been used in the deterministic theory of wave maps at critical regularity. The proof of almost invariance of the Gibbs measure under our approximation relies on conservative structures, which are a new framework for the approximation of Hamiltonian equations, and delicate estimates of the energy increment.

15.Suppression of lift-up effect in the 3D Boussinesq equations around a stably stratified Couette flow

Authors:Michele Coti Zelati, Augusto Del Zotto

Abstract: In this paper, we establish linear enhanced dissipation results for the three-dimensional Boussinesq equations around a stably stratified Couette flow, in the viscous and thermally diffusive setting. The dissipation rates are faster compared to those observed in the homogeneous Navier-Stokes equations, in light of the interplay between velocity and temperature, driven by buoyant forces. Our approach involves introducing a change of variables grounded in a Fourier space symmetrization framework. This change elucidates the energy structure inherent in the system. Specifically, we handle non-streaks modes through an augmented energy functional, while streaks modes are amenable to explicit solutions. This explicit treatment reveals the oscillatory nature of shear modes, providing the elimination of the well-known three-dimensional instability mechanism known as the ``lift-up effect''.

1.Sharp Decay of the Fisher Information for Degenerate Fokker-Planck Equations

Authors:Anton Arnold, Amit Einav, Tobias Wöhrer

Abstract: The goal of this work is to find the sharp rate of convergence to equilibrium under the quadratic Fisher information functional for solutions to Fokker-Planck equations governed by a constant drift term and a constant, yet possibly degenerate, diffusion matrix. A key ingredient in our investigation is a recent work of Arnold, Signorello, and Schmeiser, where the $L^2$-propagator norm of such Fokker-Planck equations was shown to be identical to the propagator norm of a finite dimensional ODE which is determined by matrices that are intimately connected to those appearing in the associated Fokker-Planck equations.

2.A Dirichlet inclusion problem on Finsler manifolds

Authors:Ágnes Mester, Károly Szilák

Abstract: In this paper we study a Dirichlet-type differential inclusion involving the Finsler-Laplace operator on a complete Finsler manifold. Depending on the positive $\lambda$ parameter of the inclusion, we establish non-existence, as well as existence and multiplicity results by applying non-smooth variational methods. The main difficulties are given by the problem's highly nonlinear nature due to the general Finslerian setting, as well as the nonsmooth context.

3.Large time behaviour of the 2D thermally non-diffusive Boussinesq equations with Navier-slip boundary conditions

Authors:Fabian Bleitner, Elizabeth Carlson, Camilla Nobili

Abstract: The goal of this paper is to study the large-time bahaviour of a buoyancy driven fluid without thermal diffusion and Navier-slip boundary conditions in a bounded domain with Lipschitz-continuous second derivatives. After showing global well-posedness and regularity of classical solutions, we study their large-time asymptotics. Specifically we prove that, in suitable norms, the solutions converge to the hydrostatic equilibrium. Moreover, we prove linear stability for the hydrostatic equilibrium when the temperature is an increasing affine function of the height, i.e. the temperature is vertically stably stratified. This work is inspired by results in [Doe+18] for free-slip boundary conditions.

4.Maximizers of nonlocal interactions of Wasserstein type

Authors:Almut Burchard, Davide Carazzato, Ihsan Topaloglu

Abstract: We characterize the maximizers of a functional involving the minimization of the Wasserstein distance between equal volume sets. This functional appears as a repulsive interaction term in some models describing biological membranes. We combine a symmetrization-by-reflection technique with the uniqueness of optimal transport plans to prove that balls are the only maximizers. Further, in one dimension, we provide a sharp quantitative version of this maximality result.

5.On the Cauchy problem for $p$-evolution equations with variable coefficients: a necessary condition for Gevrey well-posedness

Authors:Alexandre Arias Junior, Alessia Ascanelli, Marco Cappiello

Abstract: In this paper we consider a class of $p$-evolution equations of arbitrary order with variable coefficients depending on time and space variables $(t,x)$. We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces.

6.Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations

Authors:Mihaela Ifrim, Ben Pineau, Daniel Tataru, Mitchell A. Taylor

Abstract: We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.

1.Stochastic Cahn-Hilliard and conserved Allen-Cahn equations with logarithmic potential and conservative noise

Authors:Andrea Di Primio, Maurizio Grasselli, Luca Scarpa

Abstract: We investigate the Cahn-Hilliard and the conserved Allen-Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure that the order parameter takes its values in the physical range and, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. For the Cahn-Hilliard equation, existence and uniqueness of probabilistically-strong solutions is shown up to the three-dimensional case. For the conserved Allen-Cahn equation, under a restriction on the noise magnitude, existence of martingale solutions is proved even in dimension three, while existence and uniqueness of probabilistically-strong solutions holds in dimension one. The analysis is carried out by studying the Cahn-Hilliard/conserved Allen-Cahn equations jointly, that is a linear combination of both the equations, which has an independent interest.

2.Suppression of Chemotactic Blowup by Strong Buoyancy in Stokes-Boussinesq Flow with Cold Boundary

Authors:Zhongtian Hu, Alexander Kiselev

Abstract: In this paper, we show that the Keller-Segel equation equipped with zero Dirichlet Boundary condition and actively coupled to a Stokes-Boussinesq flow is globally well-posed provided that the coupling is sufficiently large. We will in fact show that the dynamics is quenched after certain time. In particular, such active coupling is blowup-suppressing in the sense that it enforces global regularity for some initial data leading to a finite-time singularity when the flow is absent.

3.Duality Arguments in the Analysis of a Viscoelastic Contact Problem

Authors:Piotr Bartman, Anna Ochal, Mircea Sofonea

Abstract: We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations.

4.Nonlinear Stability of Static Néel Walls in Ferromagnetic Thin Films

Authors:A. Capella, C. Melcher, L. Morales, R. G. Plaza

Abstract: In this paper, the nonlinear (orbital) stability of static 180^\circ N\'eel walls in ferromagnetic films, under the reduced wave-type dynamics for the in-plane magnetization proposed by Capella, Melcher and Otto [CMO07], is established. It is proved that the spectrum of the linearized operator around the static N\'eel wall lies in the stable complex half plane with non-positive real part. This information is used to show that small perturbations of the static N\'eel wall converge to a translated orbit belonging to the manifold generated by the static wall.

5.Laplacian with singular drift in a critical borderline case

Authors:Damir Kinzebulatov

Abstract: We develop a strong well-posedness theory for parabolic diffusion equation with singular drift, in the case when the singularities of the drift reach critical magnitude.

6.Ground state solutions for quasilinear Schrodinger type equation involving anisotropic p-laplacian

Authors:Kaushik Bal, Sanjit Biswas

Abstract: This paper is concerned with the existence of a nonnegative ground state solution of the following quasilinear Schr\"{o}dinger equation \begin{equation*} \begin{split} -\Delta_{H,p}u+V(x)|u|^{p-2}u-\Delta_{H,p}(|u|^{2\alpha}) |u|^{2\alpha-2}u=\lambda |u|^{q-1}u \text{ in }\;R^n;\; u\in W^{1,p}(\;R^n)\cap L^\infty(\;R^N) \end{split} \end{equation*} where $N\geq2$; $(\alpha,p)\in D_N=\{(x,y)\in \;R^2 : 2xy\geq y+1,\; y\geq2x,\; y<N\}$ and $\lambda>0$ is a parameter. The operator $\Delta_{H,p}$ is the reversible Finsler p-Laplacian operator with the function $H$ being the Minkowski norm on $\;R^N$. Under certain conditions on $V$, we establish the existence of a non-trivial non-negative bounded ground state solution of the above equation.

7.Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications

Authors:Shihshu Walter Wei

Abstract: In this paper we discuss convexity, its average principle, an extrinsic average variational method in the Calculus of Variations, an average method in Partial Differential Equations, a link of convexity to $p$-subharmonicity, subsolutions to the $p$-Laplace equation, uniqueness, existence, isometric immersions in multiple settings. In particular, we show that a convex function on $\mathbb{R}^n$ is a $p$-subharmonic function, for every $p > 1$, and a $C^2$ convex function on a Riemannian manifold is a $p$-subharmonic function $f$, for every $p > 1\, .$ We also show that a $C^2$ convex function which is a submersion on a Riemannian manifold is a $p$-subharmonic function, for every $p \ge 1\, .$ This result is sharp. As further applications, via function growth estimates in $p$-harmonic geometry, we prove that every $p$-balanced nonnegative $C^2$ convex function on a complete noncompact Riemannian manifold is constant for $p > 1$. In particular, every $L^q$, nonnegative, convex function of class $C^2$ on a complete noncompact Riemannian manifold is constant for $q > p -1 > 0\, .$

1.Liouville equations on complete surfaces with nonnegative Gauss curvature

Authors:Xiaohan Cai, Mijia Lai

Abstract: We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases: on the one end, if the solution decays not too fast, then $(M,g)$ must be isometric to the standard Euclidean plane; on the other end, if $(M,g)$ is isometric to the flat cylinder $\mathbb{S}^1\times \mathbb{R}$, then solutions must decay linearly and are completely classified.

2.Description of Chemical Systems by means of Response Functions

Authors:E. Franco, B. Kepka, J. J. L. Velázquez

Abstract: In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

3.Guided modes in a hexagonal periodic graph like domain

Authors:Bérangère Delourme LAGA, Sonia Fliss POEMS

Abstract: This paper deals with the existence of guided waves and edge states in particular two-dimensional media obtained by perturbing a reference periodic medium with honeycomb symmetry. This reference medium is a thin periodic domain (the thickness is denoted $\delta$ > 0) with an hexagonal structure, which is close to an honeycomb quantum graph. In a first step, we show the existence of Dirac points (conical crossings) at arbitrarily large frequencies if $\delta$ is chosen small enough. We then perturbe the domain by cutting the perfectly periodic medium along the so-called zigzag direction, and we consider either Dirichlet or Neumann boundary conditions on the cut edge. In the two cases, we prove the existence of edges modes as well as their robustness with respect to some perturbations, namely the location of the cut and the thickness of the perturbed edge. In particular, we show that different locations of the cut lead to almost-non dispersive edge states, the number of locations increasing with the frequency. All the results are obtained via asymptotic analysis and semi-explicit computations done on the limit quantum graph. Numerical simulations illustrate the theoretical results.

4.Optimal quantitative stability for a Serrin-type problem in convex cones

Authors:Filomena Pacella, Giorgio Poggesi, Alberto Roncoroni

Abstract: We consider a Serrin-type problem in convex cones in the Euclidean space and motivated by recent rigidity results we study the quantitative stability issue for this problem. In particular, we prove both sharp Lipschitz estimates for an $L^2-$pseudodistance and estimates in terms of the Hausdorff distance.

5.Null-controllability for a fourth order parabolic equation under general boundary conditions

Authors:Emmanuel Wend-Benedo Zongo, Luc Robbiano

Abstract: In this paper, we consider a fourth order inner-controlled parabolic equation on an open bounded subset of $R^d$, or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskii-Sapiro condition. We derive a spectral inequality for the solution of the parabolic system that yields a null-controllability result. The spectral inequality is a consequence of an interpolation inequality obtained via a Carleman inequality for the bi-Laplace operator under the considered boundary conditions.

6.Wiener type regularity for non-linear integro-differential equations

Authors:Shaoguang Shi, Guanglan Wang, Zhichun Zhai

Abstract: The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional $p-$Laplace equation. In doing so, with the help of tools from potential analysis, such as fractional relative Sobolev capacities, Wiener type integrals, Wolff potentials, $(\alpha,p)-$barriers, and $(\alpha,p)-$balayages, we first prove the characterizations of the fractional thinness and the Perron boundary regularity. Then, we establish a Wiener test and a generalized fractional Wiener criterion. Furthermore, we also prove the continuity of the fractional superharmonic function, the fractional resolutivity, a connection between $(\alpha,p)-$potentials and $(\alpha,p)-$Perron solutions, and the existence of a capacitary function for an arbitrary condenser.

1.Global solvability for semi-discrete Kirchhoff equation

Authors:Fumihiko Hirosawa

Abstract: In this paper, we consider the global solvability and energy conservation for initial value problem of nonlinear semi-discrete wave equation of Kirchhoff type, which is a discretized model of Kirchhoff equation.

2.On the trace theorem to Volterra-type equations with local or non-local derivatives

Authors:Jae-Hwan Choi, Jin Bong Lee, Jinsol Seo, Kwan Woo

Abstract: This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $A_p$ weighted $L_p$ spaces. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.

3.A survey on the boundary behavior of the double layer potential in Schauder spaces in the frame of an abstract approach

Authors:M. Lanza de Cristoforis

Abstract: We provide a summary of the continuity properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in H\"{o}lder and Schauder spaces on the boundary of a bounded open subset of ${\mathbb{R}}^n$. The purpose is two-fold. On one hand we try present in a single paper all the known continuity results on the topic with the best known exponents in a H\"{o}lder and Schauder space setting and on the other hand we show that many of the properties we present can be deduced by applying results that hold in an abstract setting of metric spaces with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by Garc\'{\i}a-Cuerva and Gatto in the frame of H\"{o}lder spaces and later by the author.

4.Subsonic steady-states for bipolar hydrodynamic model for semiconductors

Authors:Siying Li, Ming Mei, Kaijun Zhang, Guojing Zhang

Abstract: In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case.

5.Strong solutions for the Navier-Stokes-Voigt equations with non-negative density

Authors:Hermenegildo Borges de Oliveira, Khonatbek Khompysh, Aidos Ganizhanuly Shakir

Abstract: The aim of this work is to study the Navier-Stokes-Voigt equations that govern flows with non-negative density of incompressible fluids with elastic properties. For the associated non-linear initial-and boundary-value problem, we prove the global-in-time existence of strong solutions (velocity, density and pressure). We also establish some other regularity properties of these solutions and find the conditions that guarantee the uniqueness of velocity and density. The main novelty of this work is the hypothesis that, in some subdomain of space, there may be a vacuum at the initial moment, that is, the possibility of the initial density vanishing in some part of the space domain.

6.Continuity estimates for doubly degenerate parabolic equations with lower order terms via nonlinear potentials

Authors:Qifan Li

Abstract: This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. A model equation is \begin{equation*}\partial_t u-\operatorname{div}(u^{m-1}|Du|^{p-2}Du)=f\qquad \text{in}\quad\Omega\times(-T,0)\subset \mathbb{R}^{n+1}.\end{equation*} Here, we consider the case $m>1$ and $2<p<n$. We establish a continuity estimate for $u$ in terms of elliptic Riesz potentials of the right-hand side of the equation.

7.On the Sobolev Stability Threshold for the 2D MHD Equations with Horizontal Magnetic Dissipation

Authors:Niklas Knobel, Christian Zillinger

Abstract: In this article we consider the stability threshold of the 2D magnetohydrodynamics (MHD) equations near a combination of Couette flow and large constant magnetic field. We study the partial dissipation regime with full viscous and only horizontal magnetic dissipation. In particular, we show that this regime behaves qualitatively differently than both the fully dissipative and the non-resistive setting.

8.On the inversion of the momenta ray transform of symmetric tensors in the plane

Authors:David Omogbhe, Kamran Sadiq, Alexandru Tamasan

Abstract: We present a reconstruction method which stably recovers some sufficiently smooth, real valued, symmetric tensor fields compactly supported in the Euclidean plane, from knowledge of their non/attenuated momenta ray transform. The reconstruction method extends Bukhgeim's $A$-analytic theory from an equation to a system.

9.Linearized Analysis of Adiabatic Oscillations of Rotating Gaseous Stars

Authors:Tetu Makino

Abstract: We study adiabatic oscillations of rotating self-gravitating gaseous stars in mathematically rigorous manner. The internal motion of the star is supposed to be governed by the Euler-Poisson equations with rotation of constant angular velocity under the equation of state of the ideal gas. The motion is supposed to be adiabatic, but not to be barotropic in general. This causes a free boundary problem to gas-vacuum interface. Existence of solutions to the linearized equation in the Lagrange coordinates of the perturbations around a fixed stationary solution, the eigenvalue problem with concept of quadratic pencil of operators, and the stability problem with a new concept of stability introduced in this article are discussed.

10.On blow-up conditions for nonlinear higher order evolution inequalities

Authors:A. A. Kon'kov, A. E. Shishkov

Abstract: We obtain exact conditions for global weak solutions of the problem $$ \left\{ \begin{aligned} & u_t - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1}, & u (x, 0) = u_0 (x) \ge 0, \end{aligned} \right. $$ to be identically zero, where ${\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty)$, $m, n \ge 1$. In so doing, we assume that $u_0 \in L_{1, loc} ({\mathbb R}^n)$ and $a_\alpha$ and $f$ are some functions.

1.Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model

Authors:Siran Li, Jianing Yang

Abstract: In this paper, a novel observation is made on a one-dimensional compressible Navier--Stokes model for the dynamic combustion of a reacting mixture of $\gamma$-law gases ($\gamma>1$) with discontinuous Arrhenius reaction rate function, on both bounded and unbounded domains. We show that the mass fraction of the reactant (denoted as $Z$) satisfies a weighted gradient estimate $Z_y/ \sqrt{Z} \in L^\infty_t L^2_y$, provided that at time zero the density is Lipschitz continuous and bounded strictly away from zero and infinity. Consequently, the graph of $Z$ cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any instant, and the entropy of $Z$ is bounded from above. The key ingredient of the proof is a new estimate based on the Fisher information, first exploited by [2, 7] with applications to PDEs in chemorepulsion and thermoelasticity. Along the way, we also establish a Lipschitz estimate for the density.

2.BMO-type functionals, total variation, and $Γ$-convergence

Authors:Panu Lahti, Quoc-Hung Nguyen

Abstract: We study the BMO-type functional $\kappa_{\varepsilon}(f,\mathbb R^n)$, which can be used to characterize BV functions $f\in BV(\mathbb R^n)$. The $\Gamma$-limit of this functional, taken with respect to $L^1_{\mathrm{loc}}$-convergence, is known to be $\tfrac 14 |Df|(\mathbb R^n)$. We show that the $\Gamma$-limit with respect to $L^{\infty}_{\mathrm{loc}}$-convergence is \[ \tfrac 14 |D^a f|(\mathbb R^n)+\tfrac 14 |D^c f|(\mathbb R^n)+\tfrac 12 |D^j f|(\mathbb R^n), \] which agrees with the ``pointwise'' limit in the case of SBV functions.

3.Highest Cusped Waves for the Fractional KdV Equations

Authors:Joel Dahne

Abstract: In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations $f_t + f f_x = |D|^{\alpha} f_x$ for all $\alpha \in (-1,0)$ and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach.

4.Abstract multiplicity results for $(p,q)$-Laplace equations with two parameters

Authors:Vladimir Bobkov, Mieko Tanaka

Abstract: We investigate the existence and multiplicity of abstract weak solutions of the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and $\alpha,\beta \in \mathbb{R}$. We determine three generally different ranges of parameters $\alpha$ and $\beta$ for which the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy. As auxiliary results, which are also of independent interest, we provide alternative characterizations of variational eigenvalues of the $q$-Laplacian using narrower and larger constraint sets than in the standard minimax definition.

5.A Class of Initial-Boundary Value Problems Governed by Pseudo-Parabolic Weighted Total Variation Flows

Authors:Toyohiko Aiki, Daiki Mizuno, Ken Shirakawa

Abstract: In this paper, we consider a class of initial-boundary value problems governed by pseudo-parabolic total variation flows. The principal characteristic of our problem lies in the velocity term of the diffusion flux, a feature that can bring about stronger regularity than what is found in standard parabolic PDEs. Meanwhile, our total variation flow contains singular diffusion, and this singularity may lead to a degeneration of the regularity of solution. The objective of this paper is to clarify the power balance between these conflicting effects. Consequently, we will present mathematical results concerning the well-posedness and regularity of the solution in the Main Theorems of this paper.

6.Hydrodynamic limit and Newtonian limit from the relativistic Boltzmann equation to the classical Euler equations

Authors:Yong Wang, Changguo Xiao

Abstract: The hydrodynamic limit and Newtonian limit are important in the relativistic kinetic theory. We justify rigorously the validity of the two independent limits from the special relativistic Boltzmann equation to the classical Euler equations without assuming any dependence between the Knudsen number $\varepsilon$ and the light speed $\mathfrak{c}$. The convergence rates are also obtained. This is achieved by Hilbert expansion of relativistic Boltzmann equation. New difficulties arise when tacking the uniform in $\mathfrak{c}$ and $\varepsilon$ estimates for the Hilbert expansion, which have been overcome by establishing some uniform-in-$\mathfrak{c}$ estimate for relativistic Boltzmann operators.

7.Asymptotics and geometric flows for a class of nonlocal curvatures

Authors:Wojciech Cygan, Tomasz Grzywny, Julia Lenczewska

Abstract: We consider a family of nonlocal curvatures determined through a kernel which is symmetric and bounded from above by a radial and radially non-increasing profile. It turns out that such definition encompasses various variants of nonlocal curvatures that have already appeared in the literature, including fractional curvature and anisotropic fractional curvature. The main task undertaken in the article is to study the limit behaviour of the introduced nonlocal curvatures under an appropriate limiting procedure. This enables us to recover known asymptotic results e.g. for fractional curvature, but also for anisotropic fractional curvature where we identify the limit object as a curvature being the first variation of the related anisotropic perimeter. Our other goal is to prove existence, uniqueness and stability of viscosity solutions to the corresponding level-set parabolic Cauchy problem formulated in terms of the investigated nonlocal curvature.

8.An exceptional property of the one-dimensional Bianchi-Egnell inequality

Authors:Tobias König

Abstract: In this paper, for $d \geq 1$ and $s \in (0,\frac{d}{2})$, we study the Bianchi-Egnell quotient \[ \mathcal Q(f) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B} \frac{\|(-\Delta)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{\frac{2d}{d-2s}}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal B)^2}, \qquad f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B, \] where $S_{d,s}$ is the best Sobolev constant and $\mathcal B$ is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when $d = 1$, there is a neighborhood of $\mathcal B$ on which the quotient $\mathcal Q(f)$ is larger than the lowest value attainable by sequences converging to $\mathcal B$. This behavior is surprising because it is contrary to the situation in dimension $d \geq 2$ described recently in \cite{Koenig}. This leads us to conjecture that for $d = 1$, $\mathcal Q(f)$ has no minimizer on $\dot{H}^s(\mathbb R^d) \setminus \mathcal B$, which again would be contrary to the situation in $d \geq 2$. As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every $d \geq 1$. For $d \geq 2$, this family yields an alternative proof of the main result of \cite{Koenig}. For $d =1$ we make some numerical observations which support the conjecture stated above.

9.Strichartz estimates for the $(k,a)$-generalized Laguerre operators

Authors:Kouichi Taira, Hiroyoshi Tamori

Abstract: In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}(-|x|^{2-a}\Delta_k+|x|^a)$ which were introduced by Ben Sa\"{\i}d-Kobayashi-{\0}rsted, and for the operators $|x|^{2-a}\Delta_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $\Delta_k$ and $a$ denotes a positive real number satisfying certain conditions. The cases $a=1,2$ were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.

10.Point sources identification problems with pointwise overdetermination

Authors:Sergey Pyatkov, Lyubov Neustroeva

Abstract: This article is devoted to inverse problems of recovering point sources in mathematical models of heat and mass transfer. The main attention is paid to well-posedness questions of these inverse problems with pointwise overdetermination conditions. We present conditions for existence and uniqueness of solutions to the problem, display non-uniqueness examples, and, in model situations, we give estimates on the number of measurements that allow completely identify sources and their locations. The results rely on asymptotic representations of Green functions of the corresponding elliptic problems with a parameter. They can be used in constructing new numerical algorithms for determining a solution.

11.On fractional and classical hyperbolic obstacle-type problems

Authors:Pedro Miguel Campos, José Francisco Rodrigues

Abstract: We consider weak solutions for the obstacle-type viscoelastic ($\nu>0$) and very weak solutions for the obstacle inviscid ($\nu=0$) Dirichlet problems for the heterogeneous and anisotropic wave equation in a fractional framework based on the Riesz fractional gradient $D^s$ ($0<s<1$). We use weak solutions of the viscous problem to obtain very weak solutions of the inviscid problem when $\nu\searrow 0$. We prove that the weak and very weak solutions of those problems in the fractional setting converge as $s\nearrow 1$ to a weak solution and to a very weak solution, respectively, of the correspondent problems in the classical framework.

1.The logarithmic Schr{ö}dinger equation with spatial white noise on the full space

Authors:Quentin Chauleur LPP, Paradyse, Antoine Mouzard LPENSL, IRMAR, MINGUS

Abstract: We solve the Schr{\"o}dinger equation with logarithmic nonlinearity and multiplicative spatial white noise on R d with d $\le$ 2. Because of the nonlinearity, the regularity structures and the paracontrolled calculus can not be used. To solve the equation, we rely on an exponential transform that has proven useful in the context of other singular SPDEs.

2.Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems

Authors:Sunghan Kim, Kaj Nyström

Abstract: We prove new optimal $C^{1,\alpha}$ regularity results for obstacle problems involving evolutionary $p$-Laplace type operators in the degenerate regime $p > 2$. Our main results include the optimal regularity improvement at free boundary points in intrinsic backward $p$-paraboloids, up to the critical exponent, $\alpha \leq 2/(p-2)$, and the optimal regularity across the free boundaries in the full cylinders up to a universal threshold. Moreover, we provide an intrinsic criterion by which the optimal regularity improvement at free boundaries can be extended to the entire cylinders. An important feature of our analysis is that we do not impose any assumption on the time derivative of the obstacle. Our results are formulated in function spaces associated to what we refer to as higher order or $C^{1,\alpha}$ intrinsic interpolative geometries.

3.Asymptotic stability of a wide class of stationary solutions for the Hartree and Schrödinger equations for infinitely many particles

Authors:Sonae Hadama

Abstract: We consider the Hartree and Schr\"{o}dinger equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators, and they have infinitely many stationary solutions. In this paper, we prove the asymptotic stability of a wide class of stationary solutions. We emphasize that our result includes Fermi gas at zero temperature. This is one of the most important steady states from the physics point of view; however, its asymptotic stability has been left open after Lewin and Sabin first formulated this stability problem and gave significant results in their seminal work [arXiv:1310.0604].

4.On localization of eigenfunctions of the magnetic Laplacian

Authors:Jeffrey S. Ovall, Hadrian Quan, Robyn Reid, Stefan Steinerberger

Abstract: Let $\Omega \subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $ H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:\Omega \rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) \psi = \lambda \psi$ that are highly localized in a small region of $\Omega$. The main goal of this paper is to show that if $|\psi|$ assumes its maximum in $x_0 \in \Omega$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrt{\lambda}-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where $\left|\mbox{curl} A \right|$ is small. The result is illustrated with numerical examples.

5.Stability of the Wulff shape with respect to anisotropic curvature functionals

Authors:Julian Scheuer, Xuwen Zhang

Abstract: For a function $f$ which foliates a one-sided neighbourhood of a closed hypersurface $M$, we give an estimate of the distance of $M$ to a Wulff shape in terms of the $L^{p}$-norm of the traceless $F$-Hessian of $f$, where $F$ is the support function of the Wulff shape. This theorem is applied to prove quantitative stability results for the anisotropic Heintze-Karcher inequality, the anisotropic Alexandrov problem, as well as for the anisotropic overdetermined boundary value problem of Serrin-type.

6.On an age-structured juvenile-adult model with harvesting pulse in moving and heterogeneous environment

Authors:Haiyan Xu, Zhigui Lin, Huaiping Zhu

Abstract: This paper concerns an age-structured juvenile-adult model with harvesting pulse and moving boundaries in a heterogeneous environment, in which the moving boundaries describe the natural expanding front of species and human periodic pulse intervention is carried on the adults. The principal eigenvalue is firstly defined and its properties involving the intensity of harvesting and length of habitat sizes are analysed. Then the criteria to determine whether the species spread or vanish is discussed, and some relevant sufficient conditions characterized by pulse are established. Our results reveal that the co-extinction or coexistence of species is influenced by internal expanding capacities from species itself and external harvesting pulse from human intervention, in which the intensity and timing of harvesting play key roles. The final numerical approximations indicate that the larger the harvesting rate and the shorter the harvesting period, the worse the survival of all individuals due to the cooperation among juveniles and adults, and such harvesting pulse can even alter the situation of species, from persistence to extinction. In addition, expanding capacities also affect or alter the outcomes of spreading-vanishing

7.A new $p$-harmonic map flow with Struwe monotonicity

Authors:Erik Hupp, Michał Miśkiewicz

Abstract: We construct and analyze solutions to a regularized homogeneous $p$-harmonic map flow equation for general $p \geq 2$. The homogeneous version of the problem is new and features a monotonicity formula extending the one found by Struwe for $p = 2$; such a formula is not available for the nonhomogeneous equation. The construction itself is via a Ginzburg-Landau-type approximation \`a la Chen-Struwe, employing tools such as a Bochner-type formula and an $\varepsilon$-regularity theorem. We similarly obtain strong subsequential convergence of the approximations away from a concentration set with parabolic codimension at least $p$. However, the quasilinear and non-divergence nature of the equation presents new obstacles that do not appear in the classical case $p = 2$, namely uniform-time existence for the approximating problem, and thus our basic existence result is stated conditionally.

8.On $L_{p}$- theory for integro-differential operators with spatially dependent coefficients

Authors:Sutawas Janreung, Tatpon Siripraparat, Chukiat Saksurakan

Abstract: The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by L\'evy measures with O-regularly varying profile. The coefficients are assumed to be bounded and H\"older continuous in the spatial variable. Our results can cover interesting classes of L\'evy measures that go beyond those comparable to $dy/\left|y\right|^{d+\alpha}$.

9.Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms

Authors:Francesco Balducci, Francescantonio Oliva, Francesco Petitta

Abstract: In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial \Omega$,} \ \end{cases} \end{equation*} in a domain $\Omega \subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(\Omega)$.

10.Global Well-Posedness of Displacement Monotone Degenerate Mean Field Games Master Equations

Authors:Mohit Bansil, Alpár R. Mészáros, Chenchen Mou

Abstract: In this manuscript we construct global in time classical solutions to mean field games master equations in the lack of idiosyncratic noise in the individual agents' dynamics. These include both deterministic models and dynamics driven solely by a Brownian common noise. We consider a general class of non-separable Hamiltonians and final data functions that are supposed to be displacement monotone. Our main results unify and generalize in particular some of the well-posedness results on displacement monotone master equations obtained recently by Gangbo--M\'esz\'aros and Gangbo--M\'esz\'aros--Mou--Zhang.

11.Improved low regularity theory for gravity-capillary waves

Authors:Albert Ai

Abstract: This article concerns the Cauchy problem for the gravity-capillary water waves system in general dimensions. We establish local well-posedness for initial data in $H^s$, with $s > \frac{d}{2} + 2 - \mu$, with $\mu = \frac{3}{14}$ and $\mu = \frac37$ in the cases $d = 1$ and $d \geq 2$ respectively. This represents an improvement over the the state-of-the-art low regularity theory in $d \geq 2$ dimensions.

1.Subordinated Bessel heat kernels

Authors:Krzysztof Bogdan, Konstantin Merz

Abstract: We prove new bounds for Bessel heat kernels and Bessel heat kernels subordinated by stable subordinators. In particular, we provide a 3G inequality in the subordinated case.

2.Finite-dimensional leading order dynamics for the fast diffusion equation near extinction

Authors:Beomjun Choi, Christian Seis

Abstract: The fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near the vanishing solution to any prescribed convergence rate.

3.Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system

Authors:Bernard Di Martino, Chourouk El Hassanieh, Edwige Godlewski, Julien Guillod, Jacques Sainte-Marie

Abstract: By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. This new system can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multi-layer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.

4.A note on Rubio de Francia's extrapolation in tent spaces and applications

Authors:José María Martell, Pierre Portal

Abstract: The Rubio de Francia extrapolation theorem is a very powerful result which states that in order to show that certain operators satisfy weighted norm inequalities with Muckenhoupt weights it suffices to see that the corresponding inequalities hold for some fixed exponent, for instance $p=2$. In this paper we extend this result and show that this extrapolation principle allows one to obtain weighted estimates in tent spaces. From our extrapolation result we automatically derive new estimates (and reprove some other) concerning Calder\'on-Zygmund operators, operators associated with the Kato conjecture, or fractional operators.

5.Multi-scale techniques and homogenization for viscoelastic non-simple materials at large strains

Authors:Markus Gahn

Abstract: In this paper we present the homogenization for nonlinear viscoelastic second-grade non-simple perforated materials at large strain in the quasistatic setting. The reference domain $\Omega_{\varepsilon}$ is periodically perforated and is depending on the scaling parameter $\varepsilon$ which describes the ratio between the size of the whole domain and the small periodic perforations. The mechanical energy depends on the gradient and also the second gradient of the deformation, and also respects positivity of the determinant of the deformation gradient. For the viscous stresses we assume dynamic frame indifference and is therefore depending of the rate of the Cauchy-stress tensor. For the derivation of the homogenized model for $\varepsilon \to 0$ we use the method of two-scale convergence. For this uniform a priori estimates with respect to $\varepsilon$ are necessary. The most crucial part is to estimate the rate of the deformation gradient. Due to the time-dependent frame indifference of the viscous term, we only get coercivity with respect to the rate of the Cauchy-stress tensor. To overcome this problem we derive a Korn inequality for non-constant coefficients on the perforated domain. The crucial point is to verify that the constant in this inequality, which is usually depending on the domain, can be chosen independently of the parameter $\varepsilon$. Further, we construct an extension operator for second order Sobolev spaces on perforated domains with operator norm independent of $\varepsilon$.

6.A comparison principle for semilinear Hamilton-Jacobi-Bellman equations in the Wasserstein space

Authors:Samuel Daudin, Benjamin Seeger

Abstract: The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the $2$-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the $1$-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.

7.On a Lack of Stability of Parametrized BV Solutions to Rate-Independent Systems with Non-Convex Energies and Discontinuous Loads

Authors:Merlin Andreia, Christian Meyer

Abstract: We consider a rate-independent system with nonconvex energy under discontinuous external loading. The underlying space is finite dimensional and the loads are functions in $BV([0,T];\mathbb{R}^d)$. We investigate the stability of various solution concepts w.r.t. a sequence of loads converging weakly$*$ in $BV([0,T];\mathbb{R}^d)$ with a particular emphasis on the so-called normalized, $\mathfrak{p}$-parametrized balanced viscosity solutions. By means of two counterexamples, it is shown that common solution concepts are not stable w.r.t. weak$*$ convergence of loads in the sense that a limit of a sequence of solutions associated with these loads need not be a solution corresponding to the load in the limit. We moreover introduce a new solution concept, which is stable in this sense, but our examples show that this concept necessarily allows "solutions" that are physically meaningless.

8.Enhanced dissipation and blow-up suppression for the three dimensional Keller-Segel equation with a non-shear incompressible flow

Authors:Binbin Shi, Weike Wang

Abstract: In this paper, we consider the Cauchy problem for the three dimensional parabolic-elliptic Keller-Segel equation with a large non-shear incompressible flow. Without advection, there exist solution with arbitrarily mass which blow up in finite time. Firstly, we introduce a three dimensional non-shear incompressible flow and study the enhanced dissipation of such flows by resolvent estimate method. Next, we show that the enhanced dissipation of such flow can suppress blow-up of solution to three dimensional parabolic-elliptic Keller-Segel equation and establish global classical solution with large initial data.

9.Linearized partial data Calderón problem for Biharmonic operators

Authors:Divyansh Agrawal, Ravi Shankar Jaiswal, Suman Kumar Sahoo

Abstract: We consider a linearized partial data Calder\'on problem for biharmonic operators extending the analogous result for harmonic operators. We construct special solutions and utilize Segal-Bargmann transform to recover lower order perturbations.

10.Well-posedness and Low Mach Number Limit of the Free Boundary Problem for the Euler--Fourier System

Authors:Xumin Gu, Yanjin Wang

Abstract: We consider the free boundary problem for the Euler--Fourier system that describes the motion of compressible, inviscid and heat-conducting fluids. The effect of surface tension is neglected and there is no heat flux across the free boundary. We prove the local well-posedness of the problem in Lagrangian coordinates under the Taylor sign condition. The solution is produced as the limit of solutions to a sequence of tangentially-smoothed approximate problems, where the so-called corrector is crucially introduced beforehand in the temperature equation so that the approximate initial data satisfying the corresponding compatibility conditions can be constructed. To overcome the strong coupling effect between the Euler part and the Fourier part in solving the linearized approximate problem, the temperature equation is further regularized by a pseudo-parabolic equation. Moreover, we prove the uniform estimates with respect to the Mach number of the solutions to the free-boundary Euler--Fourier system with large temperature variations, which allow us to justify the convergence towards the free-boundary inviscid low Mach number limit system by the strong compactness argument.

11.Obstructions to topological relaxation for generic magnetic fields

Authors:Alberto Enciso, Daniel Peralta-Salas

Abstract: For any axisymmetric toroidal domain $\Omega \subset \mathbf{R}^3$ we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) equilibrium in $\Omega$. Each vector field in this set is Morse-Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS equilibrium. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.

12.Illposedness for dispersive equations: Degenerate dispersion and Takeuchi--Mizohata condition

Authors:In-Jee Jeong, Sung-Jin Oh

Abstract: We provide a unified viewpoint on two illposedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi--Mizohata condition. Our approach is based on a robust energy- and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydynamics. Concretely, the main results in this paper concern strong illposedness of the Cauchy problem (e.g., non-existence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schr\"odinger- and KdV-type equations, including the Hunter--Smothers equation, $K(m, n)$ models of Rosenau--Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of combination of two effects: degenerate dispersion -- which is a property of the principal term in the presence of degenerating coefficients -- and the evolution of the amplitude governed by the Takeuchi--Mizohata condition -- which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical $L^{2}$-illposedness result by Mizohata for linear variable-coefficient Schr\"odinger equations with failed Takeuchi--Mizohata condition.

13.The Feynman-Lagerstrom criterion for boundary layers

Authors:Theodore D. Drivas, Sameer Iyer, Trinh T. Nguyen

Abstract: We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a \textit{necessary} condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice -- known to Batchelor (1956) and Wood (1957) -- is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.

1.On a set of some recent contributions to energy equality for the Navier-Stokes equations

Authors:Hugo Beirão da Veiga, Jiaqi Yang

Abstract: In these notes we want, in addition to presenting some new results, to both clean up and refine some reflections on a couple of articles published on paper a few years ago, 2019-20. These papers concerned integral sufficient conditions on $u\,,$ $\n u\,,$ and mixed, to guarantee the equality of the energy, EE in the sequel, for solutions of the Navier-Stokes equations under the classical non-slip boundary condition. Concerning the $\n u$ case a crucial role was enjoyed by a previous well known Berselli and Chiodaroli's pioneering 2019 work on the subject. The above three papers are the main sources of these notes. References will be mostly concentrated on their direct relation to the above papers at the time of pubblication. More recent results will be not stated throughout the article. However, in the last section, the reader will be suitably sent to the more recent bibliography. Below, we also turn back to the innovative interpretation of some main parameters which allowed to overcome their apparent incongruence. Non-Newtonian fluids were also considered in our 2019 paper, maybe for the first time in the above Berselli-Chiodaroli's particular $\n u\,$ context. However we will stick mostly to the Newtonian case since in the end we come to the conclusion that there are not particular additional obstacles to extend the present results from Newtonian to non-Newtonian fluids. Hence we avoid to go further in this direction.

2.On the Number of Normalized Ground State Solutions for a class of Elliptic Equations with general nonlinearities and potentials

Authors:Hichem Hajaiej, Eliot Pacherie, Linjie Song

Abstract: We provide a precise description of the set of normalized ground state solutions (NGSS) for the class of elliptic equations: $$ -\Delta u - \lambda u + V (| x |) u - f (| x |, u) = 0,\quad\text{in}\quad \mathbb{R}^n,\ n\geq 1. $$ In particular, we show that under suitable assumptions on $V$ and $f$, the NGSS is unique for all the masses except for at most a finite number. Moreover, we prove that when unique, the NGSS $u_c$ is a smooth function of the mass $c.$ Our method is as follow: using the NGSS for a given mass $c$, we construct an exhaustive list of potential candidates to the minimization problem for masses close to $c$, and we develop a strategy how to pick the right one. In particular, if there is a unique NGSS for a given mass $c_0,$ then this uniqueness property is inherited for all the masses $c$ close to $c_0.$ Our method is general and applies to other equations provided that some key properties hold true.

3.s-stability for W^{s,n/s}-harmonic maps in homotopy groups

Authors:Katarzyna Mazowiecka, Armin Schikorra

Abstract: We study $s$-dependence for minimizing $W^{s,n/s}$-harmonic maps $u\colon \mathbb{S}^n \to \mathbb{S}^\ell$ in homotopy classes. Sacks--Uhlenbeck theory shows that, for each $s$, minimizers exist in a generating subset of $\pi_{n}(\S^\ell)$. We show that this generating subset can be chosen locally constant in $s$. We also show that as $s$ varies the minimal $W^{s,n/s}$-energy in each homotopy class changes continuously. In particular, we provide progress to a question raised by Mironescu and Brezis--Mironescu.

4.The regularity theory for the Mumford-Shah functional on the plane

Authors:Camillo De Lellis, Matteo Focardi

Abstract: The aim of these notes is to give a complete self-contained account of the current state of the art in the regularity for planar minimizers and critical points of the Mumford-Shah functional.

1.Generic properties in free boundary problems

Authors:Xavier Fernández-Real, Hui Yu

Abstract: In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regularity of free boundaries for minimizers of the one-phase Alt-Caffarelli and Alt-Phillips functionals, for a monotone family of boundary data $\{\varphi_t\}_{t\in(-1,1)}$. More precisely, we show that for a co-countable subset of $\{\varphi_t\}_{t\in(-1,1)}$, minimizers have smooth free boundaries in $\mathbb{R}^5$ for the Alt-Caffarelli and in $\mathbb{R}^3$ for the Alt-Phillips functional. In general dimensions, we show that the singular set is one dimension smaller than expected for almost every boundary datum in $\{\varphi_t\}_{t\in(-1,1)}$.

2.Improved Lerey inequality and Trudinger-Moser type inequality involving the Leray potential

Authors:Huyuan Chen, Yihong Du, Feng Zhou

Abstract: We obtain three types of results in this paper. Firstly we improve Leray's inequality by providing several types of reminder terms, secondly we introduce several Hilbert spaces based on these improved Leray inequalities and discuss their embedding properties, thirdly we obtain some Trudinger-Moser type inequalities in the unit ball of R2 associated with the norms of these Hilbert spaces where the Leray potential is used. Our approach is based on analysis of radially symmetric functions.

3.Modified scattering for nonlinear Schrödinger equations with long-range potentials

Authors:Masaki Kawamoto, Haruya Mizutani

Abstract: We study the final state problem for the nonlinear Schr\"{o}dinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data. The class of potential includes a repulsive long-range potential with a short-range perturbation, especially the positive Coulomb potential in two and three space dimensions. The asymptotic profile is constructed by combining Yafaev's type linear modifier [41] associated with the long-range part of the potential and the nonlinear modifier introduced by Ozawa [32]. Finally, we also show that one can replace Yafaev's type modifier by Dollard's type modifier under a slightly stronger decay assumption on the long-range potential. This is the first positive result on the modified scattering for the nonlinear Schr\"{o}dinger equation in the case when both of the nonlinear term and the linear potential are of long-range type.

4.Capacity of infinite graphs over non-Archimedean ordered fields

Authors:Florian Fischer, Matthias Keller, Anna Muranova, Noema Nicolussi

Abstract: In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator.

5.Intrinsic Harnack's inequality for a general nonlinear parabolic equation in non-divergence form

Authors:Tapio Kurkinen, Jarkko Siltakoski

Abstract: We prove the intrinsic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.

6.Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori

Authors:Filippo Giuliani

Abstract: In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (Comm. Math. Phys., 2014) for the square case, to obtain the $H^s$-instability ($s>1$) of the elliptic equilibrium $u=0$. We also provide the existence of solutions $u(t)$ with arbitrarily small $L^2$ norm which achieve a prescribed growth, say $\| u(T)\|_{H^s}\geq K \| u(0)\|_{H^s}, K\gg 1$, within a time $T$ satisfying polynomial estimates, namely $0<T\le K^c$ for some $c>0$.

1.Global existence of spherically symmetry solutions for isothermal Euler-Poisson system outside a ball

Authors:Lingjun Liu

Abstract: In this paper, we consider an isothermal Euler-Poisson system with self-gravitational force, modeling a compact star such as strange quark star. We prove that there exists a global entropy solution with spherically symmetry outside a ball, through the fractional Lax-Friedrichs scheme and the theory of compensated compactness.

2.Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables

Authors:Michele Dolce

Abstract: We consider a 2D incompressible and electrically conducting fluid in the domain $\mathbb{T}\times\mathbb{R}$. The aim is to quantify stability properties of the Couette flow $(y,0)$ with a constant homogenous magnetic field $(\beta,0)$ when $|\beta|>1/2$. The focus lies on the regime with small fluid viscosity $\nu$, magnetic resistivity $\mu$ and we assume that the magnetic Prandtl number satisfies $\mu^2\lesssim\mathrm{Pr}_{\mathrm{m}}=\nu/\mu\leq 1$. We establish that small perturbations around this steady state remain close to it, provided their size is of order $\varepsilon\ll\nu^{2/3}$ in $H^N$ with $N$ large enough. Additionally, the vorticity and current density experience a transient growth of order $\nu^{-1/3}$ while converging exponentially fast to an $x$-independent state after a time-scale of order $\nu^{-1/3}$. The growth is driven by an inviscid mechanism, while the subsequent exponential decay results from the interplay between transport and diffusion, leading to the dissipation enhancement. A key argument to prove these results is to reformulate the system in terms of symmetric variables, inspired by the study of inhomogeneous fluid, to effectively characterize the system's dynamic behavior.

3.The incompressible Navier-Stokes-Fourier-Maxwell system limits of the Vlasov-Maxwell-Boltzmann system for soft potentials: the noncutoff cases and cutoff cases

Authors:Ning Jiang, Yuanjie Lei

Abstract: We obtain the global-in-time and uniform in Knudsen number $\epsilon$ energy estimate for the cutoff and non-cutoff scaled Vlasov-Maxwell-Boltzmann system for the soft potential. For the non-cutoff soft potential cases, our analysis relies heavily on additional dissipative mechanisms with respect to velocity, which are brought about by the strong angular singularity hypothesis, i.e. $\frac12\leq s<1$. In the case of cutoff cases, our proof relies on two new kinds of weight functions and complex construction of energy functions, and here we ask $\gamma\geq-1$. As a consequence, we justify the incompressible Navier-Stokes-Fourier-Maxwell equations with Ohm's law limit.

4.Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear Schr\" odinger equation

Authors:Chenmin Sun, Nikolay Tzvetkov

Abstract: We consider the $3d$ energy critical nonlinear Schr\" odinger equation with data distributed according to the Gaussian measure with covariance operator $(1-\Delta)^{-s}$, where $\Delta$ is the Laplace operator and $s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $1d$ to higher dimensions.

5.Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces

Authors:Marius Beceanu, Michael Goldberg

Abstract: We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $R^3$ to the case of perturbed Hamiltonians of the form $H=-\Delta+V$, where $V$ is a scalar real-valued potential. Results include sharp endpoint bounds for Mihlin multipliers, confirming a conjecture made in [BeGo3] about intertwining operators, a characterization of the twisted Hardy spaces that correspond to these perturbed Hamiltonians, upgrading previous Strichartz estimates from [BeGo2] and [BeGo3], and maximum principles.

6.Uniqueness of the 2D Euler equation on rough domains

Authors:Siddhant Agrawal, Andrea R. Nahmod

Abstract: We consider the 2D incompressible Euler equation on a bounded simply connected domain $\Omega$. We give sufficient conditions on the domain $\Omega$ so that for all initial vorticity $\omega_0 \in L^{\infty}(\Omega)$ the weak solutions are unique. Our sufficient condition is slightly more general than the condition that $\Omega$ is a $C^{1,\alpha}$ domain for some $\alpha>0$, with its boundary belonging to $H^{3/2}(\mathbb{S}^1)$. As a corollary we prove uniqueness for $C^{1,\alpha}$ domains for $\alpha >1/2$ and for convex domains which are also $C^{1,\alpha}$ domains for some $\alpha >0$. Previously uniqueness for general initial vorticity in $L^{\infty}(\Omega)$ was only known for $C^{1,1}$ domains with possibly a finite number of acute angled corners. The fundamental barrier to proving uniqueness below the $C^{1,1}$ regularity is the fact that for less regular domains, the velocity near the boundary is no longer log-Lipschitz. We overcome this barrier by defining a new change of variable which we then use to define a novel energy functional.

7.On the regularity problem for parabolic operators and the role of half-time derivative

Authors:Martin Dindoš

Abstract: In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$ where $\mathcal O\subset\mathbb R^n$ is a uniform domain (it satisfies both corkscrew and Harnack chain conditions) and has uniformly $n-1$ rectifiable boundary. Let $u$ be a solution of such PDE in $\Omega$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}(\nabla u)$ belongs to $L^p(\partial\Omega)$ for some $p>1$. Furthermore, assume that for $u|_{\partial\Omega}=f$ we have that $D^{1/2}_tf\in L^p(\partial\Omega)$. Then both $\tilde{N}(D^{1/2}_t u)$ and $\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\partial\Omega)$, where $D^{1/2}_t$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.

1.Fractional boundary Hardy inequality for the critical case

Authors:Adimurthi, Prosenjit Roy, Vivek Sahu

Abstract: We establish fractional boundary Hardy-type inequality for the critical cases. In 2004, Dyda established the inequalities for the subcritical regime, whereas we establish more general inequalities (Caffarelli-Kohn-Nirenberg type) for the critical cases.

2.Nonlocal critical growth elliptic problems with jumping nonlinearities

Authors:Giovanni Molica Bisci, Kanishka Perera, Raffaella Servadei, Caterina Sportelli

Abstract: In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in presence of jumping nonlinearities. In the main results of the paper we prove the existence of a nontrivial solution for the problem under consideration, using variational and topological methods and applying a new linking theorems recently got by Perera and Sportelli in [10]. The existence results provided in this paper can be seen as the nonlocal counterpart of the ones obtained in [10] in the context of the Laplacian equations. In the nonlocal framework the arguments used in the classical setting have to be refined. Indeed the presence of the fractional Laplacian operator gives rise to some additional difficulties, that we are able to overcome proving new regularity results for weak solutions of nonlocal problems, which are of independent interest.

3.The massless and the non-relativistic limit for the cubic Dirac equation

Authors:Timothy Candy, Sebastian Herr

Abstract: Massive and massless Dirac equations with Lorentz-covariant cubic nonlinearities are considered in spatial dimension $d=2,3$. Global well-posedness of the Cauchy problem for small initial data in scale-invariant Sobolev spaces and scattering of solutions is proved by a new approach which uses bilinear Fourier restriction estimates and atomic function spaces. Furthermore, global uniform convergence results, both in the massless and in the non-relativistic limit, are proved at optimal regularity. In both regimes, these are the first results which imply convergence of scattering states and wave operators.

4.On Phase Boundaries in Relativistic Korteweg Fluids

Authors:Heinrich Freistuhler

Abstract: This is the first of several planned papers that study the existence and local-in-time persistence of phase fronts in the Lorentz invariant Euler equations for gases of van der Waals type, aiming at transferring earlier results of Slemrod and of Benzoni-Gavage and collaborators to the context of the theory of relativity. While the later papers will examine more general admissibility criteria, this one uses and extends the author's Lorentz invariant formulation of Korteweg's theory of capillarity, establishing a family of phase fronts that have a regular heteroclinic profile with respect to the associated Euler-Korteweg equations.

5.Finite time singularities to the 3D incompressible Euler equations for solutions in $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,α}\cap L^2$

Authors:Diego Córdoba, Luis Martínez-Zoroa, Fan Zheng

Abstract: We introduce a novel mechanism that reveals finite time singularities within the 1D De Gregorio model and the 3D incompressible Euler equations. Remarkably, we do not construct our blow up using self-similar coordinates, but build it from infinitely many regions with vorticity, separated by vortex-free regions in between. It yields solutions of the 3D incompressible Euler equations in $\mathbb{R}^3\times [-T,0]$ such that the velocity is in the space $C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2$ for times $t\in (-T,0)$ and is not $C^1$ at time 0.

6.The Snapshot Problem for the Wave equation

Authors:Fulton Gonzalez, Tomoyuki Kakehi, Jens Christensen, Jue Wang

Abstract: By definition, a wave is a $C^\infty$ solution $u(x,t)$ of the wave equation on $\mathbb R^n$, and a snapshot of the wave $u$ at time $t$ is the function $u_t$ on $\mathbb R^n$ given by $u_t(x)=u(x,t)$. We show that there are infinitely many waves with given $C^\infty$ snapshots $f_0$ and $f_1$ at times $t=0$ and $t=1$ respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave $u$ to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to shifted wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the shifted wave equation on the $n$-sphere.

7.The blow-up rate for a loglog non-scaling invariant semilinear wave equation

Authors:Tristan Roy, Hatem Zaag

Abstract: We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.

8.[Existence of multiple solutions for a Schröndiger logarithmic equation

Authors:Claudianor O. Alves, Ismael S. da Silva

Abstract: This paper concerns the existence of multiple solutions for a Schr\"odinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\varepsilon^2\Delta u + V(x)u & =u\log u^2,\;\;\mbox{in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P_\varepsilon)} \end{equation} where $V:\mathbb{R}^N\longrightarrow \mathbb{R}$ is a continuous function that satisfies some technical conditions and $\varepsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_\varepsilon)$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.

1.Quantitative stability for overdetermined nonlocal problems with parallel surfaces and investigation of the stability exponents

Authors:Serena Dipierro, Giorgio Poggesi, Jack Thompson, Enrico Valdinoci

Abstract: In this article, we analyze the stability of the parallel surface problem for semilinear equations driven by the fractional Laplacian. We prove a quantitative stability result that goes beyond that previously obtained in [Cir+23]. Moreover, we discuss in detail several techniques and challenges in obtaining the optimal exponent in this stability result. In particular, this includes an upper bound on the exponent via an explicit computation involving a family of ellipsoids. We also sharply investigate a technique that was proposed in [Cir+18] to obtain the optimal stability exponent in the quantitative estimate for the nonlocal Alexandrov's soap bubble theorem, obtaining accurate estimates to be compared with a new, explicit example.

2.Well-posedness of a Nonlinear Acoustics -- Structure Interaction Model

Authors:Barbara Kaltenbacher, Amjad Tuffaha

Abstract: We establish local-in-time and global in time well-posedness for small data, for a coupled system of nonlinear acoustic structure interactions. The model consists of the nonlinear Westervelt equation on a bounded domain with non homogeneous boundary conditions, coupled with a 4th order linear equation defined on a lower dimensional interface occupying part of the boundary of the domain, with transmission boundary conditions matching acoustic velocities and acoustic pressures. While the well-posedness of the Westervelt model has been well studied in the literature, there has been no works on the literature on the coupled structure acoustic interaction model involving the Westervelt equation. Another contribution of this work, is a novel variational weak formulation of the linearized system and a consideration of various boundary conditions.

3.Thermocapillary Thin Films: Periodic Steady States and Film Rupture

Authors:Gabriele Brüll, Bastian Hilder, Jonas Jansen

Abstract: We study stationary, periodic solutions to the thermocapillary thin-film model \begin{equation*} \partial_t h + \partial_x \Bigl(h^3(\partial_x^3 h - g\partial_x h) + M\frac{h^2}{(1+h)^2}\partial_xh\Bigr) = 0,\quad t>0,\ x\in \mathbb{R}, \end{equation*} which can be derived from the B\'enard-Marangoni problem via a lubrication approximation. When the Marangoni number $M$ increases beyond a critical value $M^*$, the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

4.A collision result for both non-Newtonian and heat conducting Newtonian compressible fluids

Authors:Šárka Nečasová, Florian Oschmann

Abstract: We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature depending viscosities.

5.Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels

Authors:Xavier Fernández-Real, Xavier Ros-Oton

Abstract: We study integro-differential elliptic equations (of order $2s$) with variable coefficients, and prove the natural and most general Schauder-type estimates that can hold in this setting, both in divergence and non-divergence form. Furthermore, we also establish H\"older estimates for general elliptic equations with no regularity assumption on $x$, including for the first time operators like $\sum_{i=1}^n(-\partial^2_{\textbf{v}_i(x)})^s$, provided that the coefficients have ``small oscillation''.

6.Quantitative global well-posedness of Boltzmann-Bose-Einstein equation and incompressible Navier-Stokes-Fourier limit

Authors:Ling-Bing He, Ning Jiang, Yu-long Zhou

Abstract: In the diffusive scaling and in the whole space, we prove the global well-posedness of the scaled Boltzmann-Bose-Einstein (briefly, BBE) equation with high temperature in the low regularity space $H^2_xL^2$. In particular, we quantify the fluctuation around the Bose-Einstein equilibrium $\mathcal{M}_{\lambda,T}(v)$ with respect to the parameters $\lambda$ and temperature $T$. Furthermore, the estimate for the diffusively scaled BBE equation is uniform to the Knudsen number $\epsilon$. As a consequence, we rigorously justify the hydrodynamic limit to the incompressible Navier-Stokes-Fourier equations. This is the first rigorous fluid limit result for BBE.

1.Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs

Authors:Shanshan Wang, Mamadou Diagne, Miroslav Krstić

Abstract: Deep neural networks that approximate nonlinear function-to-function mappings, i.e., operators, which are called DeepONet, have been demonstrated in recent articles to be capable of encoding entire PDE control methodologies, such as backstepping, so that, for each new functional coefficient of a PDE plant, the backstepping gains are obtained through a simple function evaluation. These initial results have been limited to single PDEs from a given class, approximating the solutions of only single-PDE operators for the gain kernels. In this paper we expand this framework to the approximation of multiple (cascaded) nonlinear operators. Multiple operators arise in the control of PDE systems from distinct PDE classes, such as the system in this paper: a reaction-diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle, both of which are bilinear in their input functions and not explicitly solvable. For the delay-compensated PDE backstepping controller, which employs the learned control operator, namely, the approximated gain kernel, we guarantee exponential stability in the $L^2$ norm of the plant state and the $H^1$ norm of the input delay state. Simulations illustrate the contributed theory.

2.Smooth Subsonic and Transonic Flows with Nonzero Angular Velocity and Vorticity to steady Euler-Poisson system in a Concentric Cylinder

Authors:Weng Shangkun, Yang Wengang, Na Zhang

Abstract: In this paper, both smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder are studied. We first establish the existence of cylindrically symmetric smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder. On one hand, we investigate the structural stability of smooth cylindrically symmetric subsonic flows under three-dimensional perturbations on the inner and outer cylinders. On the other hand, the structural stability of smooth transonic flows under the axi-symmetric perturbations are examined. There is no any restrictions on the background subsonic and transonic solutions. A deformation-curl-Poisson decomposition to the steady Euler-Poisson system is utilized in our work to deal with the hyperbolic-elliptic mixed structure in subsonic region. It should be emphasized that there is a special structure of the steady Euler-Poisson system which yields a priori estimates and uniqueness of a second order elliptic system for the velocity potential and the electrostatic potential.

3.The Kuznetsov and Blackstock equations of nonlinear acoustics with nonlocal-in-time dissipation

Authors:B. Kaltenbacher, M. Meliani, V. Nikolić

Abstract: In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin--Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag-Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

4.Quantized Vortex Dynamics of the Nonlinear Schrödinger Equation with Wave Operator on the Torus

Authors:Yongxing Zhu

Abstract: We derive rigorously the reduced dynamical law for quantized vortex dynamics of the nonlinear Schr\"odinger equation with wave operator on the torus when the core size of vortex $\varepsilon \to 0$. It is proved that the reduced dynamical law of the nonlinear Schr\"odinger equation with wave operator is a mixed state of the vortex motion laws for the nonlinear wave equation and the nonlinear Schr\"odinger equation. We will also investigate the convergence of the reduced dynamical law of the nonlinear Schr\"odinger equation with wave operator to the vortex motion law of the nonlinear Schr\"odinger equation via numerical simulation.

5.Asymptotics of solving a singularly perturbed system of transport equations with fast and slow components in the critical case

Authors:Andrey Nesterov

Abstract: An asymptotic expansion with respect to a small parameter of the solution of the Cauchy problem is constructed for a system of three transfer equations, two of which are singularly perturbed by the degeneracy of the entire senior part of the transfer operator, and the third equation clearly does not contain a small parameter. The peculiarity of the problem is that it belongs to the so-called critical case: the solution of a degenerate problem is a one-parameter family. The asymptotic expansion of the solution under smooth initial conditions is constructed as the sum of the regular part and boundary functions.

6.The Total Variation-Wasserstein Problem

Authors:Antonin Chambolle CEREMADE, MOKAPLAN, Vincent Duval MOKAPLAN, Joao Miguel Machado CEREMADE, MOKAPLAN

Abstract: In this work we analyze the Total Variation-Wasserstein minimization problem. We propose an alternative form of deriving optimality conditions from the approach of Calier\&Poon'18, and as result obtain further regularity for the quantities involved. In the sequel we propose an algorithm to solve this problem alongside two numerical experiments.

7.Liouville theorems and Harnack inequalities for Allen-Cahn type equation

Authors:Zhihao Lu

Abstract: We first give a logarithmic gradient estimate for positive solutions of Allen-Cahn equation on Riemannian manifolds with Ricci curvature bounded below. As its natural corallary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane-Emden equation, static Whitehead-Newell equation and static Fisher-KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry-\'{E}mery curvature condition.

8.Hölder and Sobolev regularity of optimal transportation potentials with rough measures

Authors:Pierre-Emmanuel Jabin, Antoine Mellet

Abstract: We consider a Kantorovich potential associated to an optimal transportation problem between measures that are not necessarily absolutely continuous with respect to the Lebesgue measure, but are comparable to the Lebesgue measure when restricted to balls with radius greater than some $\delta>0$. Our main results extend the classical regularity theory of optimal transportation to this framework. In particular, we establish both H\"older and Sobolev regularity results for Kantorovich potentials up to some critical length scale depending on $\delta$. Our assumptions are very natural in the context of the numerical computation of optimal maps, which often involves approximating by sums of Dirac masses some measures that are absolutely continuous with densities bounded away from zero and infinity on their supports.

9.Inverse problem of recoverying a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations

Authors:Song-Ren Fu, Peng-Fei Yao, Yongyi Yu

Abstract: In this paper, we consider the inverse problem of recovering a time-dependent nonlinearity for a third order nonlinear acoustic equation, which is known as the Jordan-Moore-Gibson-Thompson equation (J-M-G-T equation for short). This third order in time equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained for the small initial and boundary data. By the higher order linearization to the nonlinear equation, and construction of complex geometric optics (CGO for short) solutions for the linearized equation, we derive the uniqueness of recovering the nonlinearity.

10.Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations

Authors:Gabriele Bruell, Long Pei

Abstract: We study the existence, regularity, and symmetry of periodic traveling solutions to a class of Gardner-Ostrovsky type equations, including the classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Gardner-Ostrovsky equation is a model for internal ocean waves of large amplitude. We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves.

1.Interior and boundary mixed norm derivative estimates for nonstationary Stokes equations

Authors:Hongjie Dong, Hyunwoo Kwon

Abstract: We obtain weighted mixed norm Sobolev estimates in the whole space for nonstationary Stokes equations in divergence and nondivergence form with variable viscosity coefficients that are merely measurable in time variable and have small mean oscillation in spatial variables in small cylinders. As an application, we prove interior mixed norm derivative estimates for solutions to both equations. We also discuss boundary mixed norm Hessian estimates for solutions to equations in nondivergence form under the Lions boundary conditions.

2.The Dichotomy Property in Stabilizability of $2\times2$ Linear Hyperbolic Systems

Authors:Xu Huang, Zhiqiang Wang, Shijie Zhou

Abstract: This paper is devoted to discuss the stabilizability of a class of $ 2 \times2 $ non-homogeneous hyperbolic systems. Motivated by the example in \cite[Page 197]{CB2016}, we analyze the influence of the interval length $L$ on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all $L>0$ or it possesses the dichotomy property: there exists a critical length $L_c>0$ such that the system is stabilizable for $L\in (0,L_c)$ but unstabilizable for $L\in [L_c,+\infty)$. In addition, for $L\in [L_c,+\infty)$, we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.

3.Damping for fractional wave equations and applications to water waves

Authors:Thomas Alazard, Jeremy L. Marzuola, Jian Wang

Abstract: Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.

1.Stability of planar shock wave for the 3-dimensional compressible Navier-Stokes-Poisson equations

Authors:Xiaochun Wu

Abstract: This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain $\Omega:=\mathbb{R}\times \mathbb{T}^2$ with $\mathbb{T}^2=(\mathbb{R}/\mathbb{Z})^2$. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below.

2.KdV limit for the Vlasov-Poisson-Landau system

Authors:Renjun Duan, Dongcheng Yang, Hongjun Yu

Abstract: Two fundamental models in plasma physics are given by the Vlasov-Poisson-Landau system and the compressible Euler-Poisson system which both capture the complex dynamics of plasmas under the self-consistent electric field interactions at the kinetic and fluid levels, respectively. Although there have been extensive studies on the long wave limit of the Euler-Poisson system towards Korteweg-de Vries equations, few results on this topic are known for the Vlasov-Poisson-Landau system due to the complexity of the system and its underlying multiscale feature. In this article, we derive and justify the Korteweg-de Vries equations from the Vlasov-Poisson-Landau system modelling the motion of ions under the Maxwell-Boltzmann relation. Specifically, under the Gardner-Morikawa transformation $$ (t,x,v)\to (\delta^{\frac{3}{2}}t,\delta^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t),v) $$ with $ \varepsilon^{\frac{2}{3}}\leq \delta\leq \varepsilon^{\frac{2}{5}}$ and $\varepsilon>0$ being the Knudsen number, we construct smooth solutions of the rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the Korteweg-de Vries equations as $\delta\to 0$. Moreover, the explicit rate of convergence in $\delta$ is also obtained. The proof is obtained by an appropriately chosen scaling and the intricate weighted energy method through the micro-macro decomposition around local Maxwellians.

3.Existence results for some nonlinear elliptic systems on graphs

Authors:Shoudong Man

Abstract: In this paper, several nonlinear elliptic systems are investigated on graphs. One type of the sobolev embedding theorem and a new version of the strong maximum principle are established. Then, by using the variational method, the existence of different types of solutions to some elliptic systems is confirmed. Such problems extend the existence results on closed Riemann surface to graphs and extend the existence results for one single equation on graphs [A. Grigor'yan, Y. Lin, Y. Yang, J. Differential Equations, 2016] to nonlinear elliptic systems on graphs. Such problems can also be viewed as one type of discrete version of the elliptic systems on Euclidean space and Riemannian manifold.

4.On Dirac equations with Hartree type nonlinearity in modulation spaces

Authors:Seongyeon Kim, Hyeongjin Lee, Ihyeok Seo

Abstract: We obtain well-posedness for Dirac equations with a Hartree-type nonlinearity derived by decoupling the Dirac-Klein-Gordon system. We extend the function space of initial data, enabling us to handle initial data that were not addressed in previous studies.

5.Usable boundary for visibility-based surveillance-evasion games

Authors:Carlos Esteve-Yagüe, Richard Tsai

Abstract: We consider a surveillance-evasion game in an environment with obstacles. In such an environment, a mobile pursuer seeks to maintain the visibility with a mobile evader, who tries to get occluded from the pursuer in the shortest time possible. In this two-player zero-sum game setting, we study the discontinuities of the value of the game near the boundary of the target set (the non-visibility region). In particular, we describe the transition between the usable part of the boundary of the target (where the value vanishes) and the non-usable part (where the value is positive). We show that the value enjoys a different behaviour depending on the regularity of the obstacles involved in the game. Namely, we prove that the boundary profile is continuous for the case of smooth obstacles, and that it exhibits a jump discontinuity when the obstacle contains corners. Moreover, we prove that, in the latter case, there is a semi-permeable barrier emanating from the interface between the usable and the non-usable part of the boundary of the target set.

1.A dynamic Green's function for the homogeneous viscoelastic and isotropic half-space

Authors:Tsviatko V. Rangelov, Petia S. Dineva, George D. Manolis

Abstract: A dynamic 3D Green's function for the homogeneous, isotropic and viscoelastic (of the Zener type) half-space is derived in a closed form. The results obtained here can be used as either stand-alone solutions for simple problems or in conjunction with a boundary integral equation formulations to account for complex boundary conditions. In the later case, mesh-reducing boundary element formulations can be constructed as an alternative method for numerical implementation purposes.

2.On location of maximum of gradient of torsion function

Authors:Qinfeng Li, Ruofei yao

Abstract: It has been a widely belief that for a planar convex domain with two axes of symmetry, the location of maximal norm of gradient of torsion function is related to curvature and contact points of largest inscribed circle. We show that this is not quite true in general. Actually, we derive the formula for the location of maximal norm of gradient of torsion function on nearly ball domains in $\mathbb{R}^n$, which does not in general relate to curvature or contact points of largest inscribed ball. As a consequence, we explicitly construct several examples from which some open questions in Saint Venant elasticity theory are also solved. We also prove that for a rectangular domain, the maximum of the norm of gradient of torsion function exactly occurs at the centers of the faces of largest $(n-1)$-volume.

3.Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: application to the nonlinearly damped p-system

Authors:Timothée Crin-Barat, Ling-Yun Shou, Enrique Zuazua

Abstract: A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform that we avoid here, developing the "physical space version" of the hyperbolic hypocoercivity approach introduced by Beauchard and Zuazua, to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows to recover the natural heat-like time-decay of solutions under sharp rank conditions, without employing Fourier analysis or $L^1$ assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped $p$-system.

4.A new understanding of grazing limit

Authors:Tong Yang, Yu-Long Zhou

Abstract: The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this paper, we will provide a new understanding by simply applying a natural scaling on the Boltzmann operator without angular cutoff. The proof is based on a new well-posedness theory on the Boltzmann equation without angular cutoff in the regime with optimal ranges of parameters so that the grazing limit can be justified directly for any $\gamma>-5$ that includes the Coulomb potential corresponding to $\gamma=-3$. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two components. The first one converges to the Landau operator when the singular parameter $s$ of interaction angle tends to $1^{-}$ and the second one vanishes in this limit.

5.The Cauchy-Dirichlet Problem for the Fast Diffusion Equation on Bounded Domains

Authors:Matteo Bonforte, Alessio Figalli

Abstract: The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc. In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.

6.Well-posedness and stability for a class of fourth-order nonlinear parabolic equations

Authors:Xinye Li, Christof Melcher

Abstract: In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(\xi) = \pm \lvert \xi \rvert^2 \xi$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.

7.The probabilistic scaling paradigm

Authors:Yu Deng, Andrea R. Nahmod, Haitian Yue

Abstract: In this note we further discuss the probabilistic scaling introduced by the authors in [21, 22]. In particular we do a case study comparing the stochastic heat equation, the nonlinear wave equation and the nonlinear Schrodinger equation.

8.Existence of Solutions to $L_p$-Gaussian Minkowski problem

Authors:Shengyu Tang

Abstract: In this paper, we discuss the $L_p$-Gaussian Minkowski problem with small volume condition in $R^n$, which implies that there are at least two symmetric solutions for the $L_p$-Gaussian Minkowski problem without volume limit when $1\leq p<n$.

9.Improved algebraic lower bound for the radius of spatial analyticity for the generalized KdV equation

Authors:Mikaela Baldasso, Mahenda Panthee

Abstract: We consider the initial value problema (IVP) for the generalized Korteweg-de Vries (gKdV) equation \begin{equation} \begin{cases} \partial_tu+\partial_x^3u+\mu u^k\partial_xu=0, \,\;\; x\in \mathbb{R}, \, t \in \mathbb{R},\\ u(x,0)=u_0(x), \end{cases} \end{equation} where $u(x,\,t)$ is a real valued function, $u_0(x)$ is a real analytic function, $\mu=\pm 1$ and $k\geq 4$. We prove that if the initial data $u_0$ has radius of analyticity $\sigma_0$, then there exists $T_0>0$ such that the radius of spatial analyticity of the solution remains the same in the time interval $[-T_0, \, T_0]$. In the defocusing case, for $k\geq 4$ even, we prove that when the local solution extends globally in time, then for any $T\geq T_0$, the radius of analyticity cannot decay faster than $cT^{-\left(\frac{2k}{k+4}+\epsilon\right)}$, $\epsilon>0$ arbitrarily small and $c>0$ a constant. The result of this work improves the one obtained by Bona et al. in [ J. L. Bona, Z. Gruji\'c, H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincar\'e, 22 (2005) 783--797].

1.Existence of solutions for a poly-Laplacian system involving concave-convex nonlinearity on locally finite graphs

Authors:Ping Yang, Xingyong Zhang

Abstract: We investigate the existence of two nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearity and parameters with Dirichlet boundary value condition on the locally finite graph. By using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one non-semi-trivial solution of positive energy and one non-semi-trivial solution of negative energy, respectively. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result in [10] which is based on the fibering maps and the method of Nehari manifold, we obtain the existence of ground state solution to the single equation corresponding to poly-Laplacian system. Especially, we present the concrete range of parameters in all of results.

2.Sharp results for spherical metric on flat tori with conical angle 6$π$ at two symmetric points

Authors:Ting-Jung Kuo

Abstract: A conjecture about the existence or nonexistence of solutions to the curvature equation (1.1) defined on a rectangle torus $E_{\tau},$ $\tau\in i\mathbb{R}_{>0}$ with four conical singularties at its symmetric points is proposed in [3]. See Conjecture 1. For the purposes to understand this problem, in this paper, we study the following equation: \[ \Delta u+e^{u}=8\pi(\delta_{0}+\delta_{\frac{\omega_{k}}{2}})\text{in}E_{\tau}\,\tau\in\mathbb{H}\, \label{a} \] where $\frac{\omega_{k}}{2}$ is one of the half periods of $E_{\tau}$, i.e., the case $(m_{0},m_{1},$ $m_{2},m_{3})$ $=(1,1,0,0)$, $(1,0,1,0)$, $(1,0,0,1)$ for $k=1,2,3,$ respectively. Among others, we prove that the existence of \textit{non-even family of solutions} (see the definition in Section 1 ) is related to the existence of solutions for the equation with single conical singularity: \[ \Delta u+e^{u}=8\pi\delta_{0}\text{ in }E_{\tau}\text{, }\tau\in \mathbb{H}\text{.} \] Consequently, equation (0.1) does not have any non-even family of solutions for all $k=1,2,3$. As an application, we completely understand the solution structure of the equation (0.1) for rectangle torus and give a confirmative answer for this conjecture in this three cases. See Theorem 1.3.

3.Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms

Authors:Herbert Koch, Dongxiao Yu

Abstract: We study the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the B\"acklund transform which reduces the study of the asymptotic stability of the kinks to the study of the asymptotic decay of solutions near zero. Our results consist of two parts. First, we present a different proof of the local asymptotic stability result in arXiv:2009.04260. In its proof, we apply a result obtained by the inverse scattering method on the local decay of the solutions with sufficiently small and localized initial data. Moreover, we prove an $L^\infty$-type asymptotic stability result which is similar to that in arXiv:2106.09605; the main difference is that we remove the assumptions on the spatial symmetry of the perturbations. In its proof, we apply a result obtained by the method of testing by wave packets on the pointwise decay of the solutions with small and localized data.

4.Decay rates for mild solutions of QGE with critical fractional dissipation in $L^2(\mathbb{R}^2)$

Authors:Jamel Benameur

Abstract: In \cite{MRSC1} the authors proved some asymptotic results for the global solution of critical Quasi-geostrophic equation with a condition on the decay of $\widehat{\theta_0}$ near at zero. In this paper, we prove that this condition is not necessary. Fourier analysis and standard techniques are used.

5.Local well-posedness for incompressible neo-Hookean Elastic equations in almost critical Sobolev spaces

Authors:Huali Zhang

Abstract: Inspired by a pineeor work of Andersson-Kapitanski \cite{AK}, we prove the local well-posedness of Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to $H^{\frac{n+2}{2}+}(\mathbb{R}^n) \times H^{\frac{n}{2}+}(\mathbb{R}^n)$ ($n=2,3$), where $\frac{n+2}{2}$ and $\frac{n}{2}$ is respectively a scaling-invariant exponent for deformation and velocity in Sobolev spaces. Our new observation relies on two folds: a reduction to a second-order wave-elliptic system of deformation and velocity; a "wave-map type" null form intrinsic in this coupled system. In particular the wave nature with "wave-map type" null form allows us to prove a bilinear estimate of Klainerman-Machedon type for nonlinear terms. So we can lower $\frac12$-order regularity in 3D and $\frac34$-order regularity in 2D for well-posedness compared with \cite{AK}. Moreover, as an application, we also prove a new local well-posedness result for the ideal incompressible magnetohydrodynamic (MHD) system in the presence of a strong magnetic field.

6.Norm inflation for the viscous nonlinear wave equation

Authors:Pierre de Roubin, Mamoru Okamoto

Abstract: In this article, we study the ill-posedness of the viscous nonlinear wave equation for any polynomial nonlinearity in negative Sobolev spaces. In particular, we prove a norm inflation result above the scaling critical regularity in some cases. We also show failure of $C^k$-continuity, for $k$ the power of the nonlinearity, up to some regularity threshold.

7.Low Mach number limit for non-isentropic magnetohydrodynamic equations with ill-prepared data and zero magnetic diffusivity in bounded domains

Authors:Yaobin Ou, Lu Yang

Abstract: In this article, we verify the low Mach number limit of strong solutions to the non-isentropic compressible magnetohydrodynamic equations with zero magnetic diffusivity and ill-prepared initial data in three-dimensional bounded domains, when the density and the temperature vary around constant states. Invoking a new weighted energy functional, we establish the uniform estimates with respect to the Mach number, especially for the spatial derivatives of high order. Due to the vorticity-slip boundary condition of the velocity, we decompose the uniform estimates into the part for the fast variables and the other one for the slow variables. In particular, the weighted estimates of highest-order spatial derivatives of the fast variables are crucial for the uniform bounds. Finally, the low Mach number limit is justified by the strong convergence of the density and the temperature, the divergence-free component of the velocity, and the weak convergence of other variables. The methods in this paper can be applied to singular limits of general hydrodynamic equations of hyperbolic-parabolic type, including the full Navier-Stokes equations.

8.Inverse problems for nonlinear progressive waves

Authors:Yan Jiang, Hongyu Liu, Tianhao Ni, Kai Zhang

Abstract: We propose and study several inverse problems associated with the nonlinear progressive waves that arise in infrasonic inversions. The nonlinear progressive equation (NPE) is of a quasilinear form $\partial_t^2 u=\Delta f(x, u)$ with $f(x, u)=c_1(x) u+c_2(x) u^n$, $n\geq 2$, and can be derived from the hyperbolic system of conservation laws associated with the Euler equations. We establish unique identifiability results in determining $f(x, u)$ as well as the associated initial data by the boundary measurement. Our analysis relies on high-order linearisation and construction of proper Gaussian beam solutions for the underlying wave equations. In addition to its theoretical interest, we connect our study to applications of practical importance in infrasound waveform inversion.

9.On the low regularity phase space of the Benjamin-Ono equation

Authors:Patrick Gérard, Peter Topalov

Abstract: In this paper we prove that the Benjamin-Ono equation is globally in time $C^0$-well-posed in the Hilbert space $H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ of periodic distributions in $H^{-1/2}(\mathbb{T},\mathbb{R})$ with $\sqrt{\log}$-weights. The space $H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R})$ can thus be considered as a maximal low regularity phase space for the Benjamin-Ono equation corresponding to the scale $H^s(\mathbb{T},\mathbb{R})$, $s>-1/2$.

10.Abstract multiplicity theorems and applications to critical growth problems

Authors:Kanishka Perera

Abstract: We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian and $(p,q)$-Laplacian type problems. We show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter $\lambda > 0$. In particular, the number of solutions goes to infinity as $\lambda \to \infty$. Moreover, we give an explicit lower bound on $\lambda$ in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues constructed using the ${\mathbb Z}_2$-cohomological index. This is a consequence of the fact that our abstract multiplicity results make essential use of the piercing property of the cohomological index, which is not shared by the genus.

11.Degenerate Stability of the Caffarelli-Kohn-Nirenberg Inequality along the Felli-Schneider Curve

Authors:Rupert L. Frank, Jonas W. Peteranderl

Abstract: We show that the Caffarelli-Kohn-Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli-Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi-Egnell strategy, the heart of our proof is verifying a `secondary non-degeneracy condition'. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain.

1.Transient asymptotics of the modified Camassa-Holm equation

Authors:Taiyang Xu, Yiling Yang, Lun Zhang

Abstract: We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlev\'e-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem.

2.Inversions of two wave-forward operators with variable coefficients

Authors:Sunghwan Moon, Ihyeok Seo

Abstract: As the most successful example of a hybrid tomographic technique, photoacoustic tomography is based on generating acoustic waves inside an object of interest by stimulating electromagnetic waves. This acoustic wave is measured outside the object and converted into a diagnostic image. One mathematical problem is determining the initial function from the measured data. The initial function describes the spatial distribution of energy absorption, and the acoustic wave satisfies the wave equation with variable speed. In this article, we consider two types of problems: inverse problem with Robin boundary condition and inverse problem with Dirichlet boundary condition. We define two wave-forward operators that assign the solution of the wave equation based on the initial function to a given function and provide their inversions.

3.Hardy-Littlewood-Sobolev inequalities with partial variable weight on the upper half space and related inequalities

Authors:Jingbo Dou, Jingjing Ma

Abstract: In this paper, we establish a class of Hardy-Littlewood-Sobolev inequality with partial variable weight functions on the upper half space using a weighted Hardy type inequality. Overcoming the impact of weighted functions, the existence of extremal functions is proved via the concentration compactness principle, whereas Riesz rearrangement inequality is not available. Moreover, the cylindrical symmetry with respect to $t$-axis and the explicit forms on the boundary of all nonnegative extremal functions are discussed via the method of moving planes and method of moving spheres, as well as, regularity results are obtained by the regularity lift lemma and bootstrap technique. As applications, we obtain some weighted Sobolev inequalities with partial variable weight function for Laplacian and fractional Laplacian.

4.Remarks about the mean value property and some weighted Poincaré-type inequalities

Authors:Giorgio Poggesi

Abstract: We start providing a quantitative stability theorem for the rigidity of an overdetermined problem involving harmonic functions in a punctured domain. Our approach is inspired by and based on the proof of rigidity established by Enciso and Peralta-Salas, and reveals essential differences with respect to the stability results obtained in the literature for the classical overdetermined Serrin problem. Secondly, we provide new weighted Poincar\'e-type inequalities for vector fields. These are crucial tools for the study of the quantitative stability issue initiated by the author concerning a class of rigidity results involving mixed boundary value problems. Finally, we provide a mean value-type property and an associated weighted Poincar\'e-type inequality for harmonic functions in cones. A duality relation between this new mean value property and a partially overdetermined boundary value problem is discussed, providing an extension of a classical result due to Payne and Schaefer.

5.On solutions of an ill-posed Stefan problem

Authors:Evgeny Yu. Panov

Abstract: We study multi-phase Stefan problem with increasing Riemann initial data and with generally negative latent specific heats for the phase transitions. We propose the variational formulation of self-similar solutions, which allows to find precise conditions for existence and uniqueness of the solution.

6.Existence and Multiplicity of Solutions for Fractional $p$-Laplacian Equation Involving Critical Concave-convex Nonlinearities

Authors:Weimin Zhang

Abstract: We investigate the following fractional $p$-Laplacian equation \[ \begin{cases} \begin{aligned} (-\Delta)_p^s u&=\lambda |u|^{q-2}u+|u|^{p_s^*-2}u &&\text{in}~\Omega,\\ u &=0 &&\text{in}~ \mathbb{R}^n\setminus\Omega, \end{aligned} \end{cases} \] where $s\in (0,1)$, $p>q>1$, $n>sp$, $\lambda>0$, $p_s^*=\frac{np}{n-sp}$ and $\Omega$ is a bounded domain (with $C^{1, 1}$ boundary). Firstly, we get a dichotomy result for the existence of positive solution with respect to $\lambda$. For $p\ge 2$, $p-1<q<p$, $n>\frac{sp(q+1)}{q+1-p}$, we provide two positive solutions for small $\lambda$. Finally, without sign constraint, for $\lambda$ sufficiently small, we show the existence of infinitely many solutions.

7.Cancellation properties and unconditional well-posedness for the fifth order KdV type equations with periodic boundary condition

Authors:Takamori Kato, Kotaro Tsugawa

Abstract: We consider the fifth order KdV type equations and prove the unconditional well-posedness in $H^s(\mathbb{T})$ for $s \ge 1$. The main idea is to employ the normal form reduction and a kinds of cancellation properties to deal with the derivative losses.

8.Domain branching in micromagnetism: scaling law for the global and local energies

Authors:Tobias Ried, Carlos Román

Abstract: We study the occurrence of domain branching in a class of $(d+1)$-dimensional sharp interface models featuring the competition between an interfacial energy and a non-local field energy. Our motivation comes from branching in uniaxial ferromagnets corresponding to $d=2$, but our result also covers twinning in shape-memory alloys near an austenite-twinned-martensite interface (corresponding to $d=1$, thereby recovering a result of Conti [Comm. Pure Appl. Math. 53 (2000), 1448-1474.<1448::AID-CPA6>3.0.CO;2-C ]). We prove that the energy density of a minimising configuration in a large cuboid domain $Q_{L,T}=[-L,L]^d\times [0,T]$ scales like $T^{-\frac{2}{3}}$ (irrespective of the dimension $d$) if $L\gg T^{\frac{2}{3}}$. While this already provides a lot of insight into the nature of minimisers, it does not characterise their behaviour close to the top and bottom boundaries of the sample, i.e. in the region where the branching is concentrated. More significantly, we show that minimisers exhibit a self-similar behaviour near the top and bottom boundaries in a statistical sense through local energy bounds: for any minimiser in $Q_{L,T}$, the energy density in a small cuboid $Q_{\ell,t}$ centred at the top or bottom boundaries of the sample, with side lengths $\ell \gg t^{\frac{2}{3}}$, satisfies the same scaling law, that is, it is of order $t^{-\frac{2}{3}}$.

9.The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in $1D$ and multi-dimensions

Authors:Leonardo Abbrescia, Jared Speck

Abstract: In this article, we provide notes that complement the lectures on the relativistic Euler equations and shocks that were given by the second author at the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime, which was hosted by the Erwin Schrodinger International Institute for Mathematics and Physics in Vienna in February, 2022. We set the stage by introducing a standard first-order formulation of the relativistic Euler equations and providing a brief overview of local well-posedness in Sobolev spaces. Then, using Riemann invariants, we provide the first detailed construction of a localized subset of the maximal globally hyperbolic developments of an open set of initially smooth, shock-forming isentropic solutions in 1D, with a focus on describing the singular boundary and the Cauchy horizon that emerges from the singularity. Next, we provide an overview of the new second-order formulation of the 3D relativistic Euler equations derived in [41], its rich geometric and analytic structures, their implications for the mathematical theory of shock waves, and their connection to the setup we use in our 1D analysis of shocks. We then highlight some key prior results on the study of shock formation and related problems. Furthermore, we provide an overview of how the formulation of the flow derived in [41] can be used to study shock formation in multiple spatial dimensions. Finally, we discuss various open problems tied to shocks.

10.Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

Authors:L. Galo-Mendoza, M. López-García

Abstract: In this paper we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm-Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier-Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.

1.Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

Authors:Shengbing Deng, Xingliang Tian

Abstract: This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{equation*} \mathrm{div}(|x|^{\alpha}\nabla(\mathrm{div}(|x|^\alpha\nabla u)))=u^{2^{**}_{\alpha}-1},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 2$, $\frac{4-N}{2}<\alpha<2$ and $2^{**}_{\alpha}=\frac{2N}{N-4+2\alpha}$. It is obvious that the solutions of above equation are invariant under the scaling $\lambda^{\frac{N-4+2\alpha}{2}}u(\lambda x)$ while they are not invariant under translation when $\alpha\neq 0$. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if $\alpha$ satisfies $(2-\alpha)(2N-2+\alpha)\neq4k(N-2+k)$ for all $k\in\mathbb{N}^+$ the radial solution is non-degenerate, otherwise there exist new solutions to the linearized problem that ``replace'' the ones due to the translations invariance. As applications, firstly we investigate the remainder terms of some inequalities related to above equation. Then when $N\geq 5$ and $0<\alpha<2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality \begin{equation*} \int_{\mathbb{R}^N} |\mathrm{div}(|x|^\alpha\nabla u)|^2 \mathrm{d}x \geq C \left(\int_{\mathbb{R}^N}|u|^{2^{**}_{\alpha}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{\alpha}}},\quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N), \end{equation*} and in this case we consider a prescribed perturbation problem by using Lyapunov-Schmidt reduction.

2.Hölder continuity of functions in the fractional Sobolev spaces: 1-dimensional case

Authors:Yan Rybalko

Abstract: This paper deals with the embedding of the Sobolev spaces of fractional order into the space of H\"older continuous functions. More precisely, we show that the function $f\in H^s(\mathbb{R})$ with $\frac{1}{2}<s<1$ is H\"older continuous with the exponent $s-\frac{1}{2}$. Our result is an improvement of the Sobolev embedding theorem in the one dimensional case, which states that every such a function $f$ is continuous. The H\"older exponent $s-\frac{1}{2}$ is consistent with the Morrey's inequality, which yields that $f\in H^1(\mathbb{R})$ is H\"older continuous with the exponent $\frac{1}{2}$.

3.Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type

Authors:Carlo Alberto Antonini, Matteo Cozzi

Abstract: We address some regularity issues for mixed local-nonlocal quasilinear operators modeled upon the sum of a $p$-Laplacian and of a fractional $(s, q)$-Laplacian. Under suitable assumptions on the right-hand sides and the outer data, we show that weak solutions of the Dirichlet problem are $C^{1, \theta}$-regular up to the boundary. In addition, we establish a Hopf type lemma for positive supersolutions. Both results hold assuming the boundary of the reference domain to be merely of class $C^{1, \alpha}$, while for the regularity result we also require that $p > s q$.

4.The Mullins-Sekerka problem via the method of potentials

Authors:Joachim Escher, Anca-Voichita Matioc, Bogdan-Vasile Matioc

Abstract: It is shown that the two-dimensional Mullins-Sekerka problem is well-posed in all subcritical Sobolev spaces $H^r(\mathbb{R})$ with $r\in(3/2,2).$ This is the first result where this issue is established in an unbounded geometry. The novelty of our approach is the use of potential theory to formulate the model as an evolution problem with nonlinearities expressed by singular integral operators.

5.Transformation of a variational problem in the Euclidean space to that of a tuple of functions on several regions

Authors:Sohei Ashida

Abstract: We obtain a method to transform an minimization problem of the quadratic form corresponding to the Schr\"odinger operator in the Euclidean space to a variational problem of a tuple of functions defined on several regions. The method is based on a characterization of elements of the orthogonal complement of $H^1_0(\Omega)$ in $H^1(\Omega)$ as weak solutions to an elliptic partial differential equation on a region $\Omega$ with a bounded boundary.

6.Approximation of (some) FPUT lattices by KdV Equations

Authors:Joshua A. McGinnis, J. Douglas Wright

Abstract: We consider a Fermi-Pasta-Ulam-Tsingou lattice with randomly varying coefficients. We discover a relatively simple condition which when placed on the nature of the randomness allows us to prove that small amplitude/long wavelength solutions are almost surely rigorously approximated by solutions of Korteweg-de Vries equations for very long times. The key ideas combine energy estimates with homogenization theory and the technical proof requires a novel application of autoregressive processes.

7.Multiplicity of solutions for a class of nonhomogeneous quasilinear elliptic system with locally symmetric condition in $\mathbb{R}^N$

Authors:Cuiling Liu, Xingyong Zhang, Liben Wang

Abstract: This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole-space $\mathbb{R}^N$. By a variant of Clark's theorem without the global symmetric condition and a Moser's iteration technique, we obtain the existence of multiple solutions when the nonlinear term satisfies some growth conditions only in a circle with center 0 and the perturbation term is any continuous function with a small parameter and no any growth hypothesis.

8.Nonlinear asymptotic stability of compressible vortex sheets with viscosity effects

Authors:Feimin Huang, Zhouping Xin, Lingda Xu, Qian Yuan

Abstract: This paper concerns the stabilizing effect of viscosity on the vortex sheets. It is found that although a vortex sheet is not a time-asymptotic attractor for the compressible Navier-Stokes equations, a viscous wave that approximates the vortex sheet on any finite time interval can be constructed explicitly, which is shown to be time-asymptotically stable in the $ L^\infty $-space with small perturbations, regardless of the amplitude of the vortex sheet. The result shows that the viscosity has a strong stabilizing effect on the vortex sheets, which are generally unstable for the ideal compressible Euler equations even for short time [26,8,1]. The proof is based on the $ L^2 $-energy method.In particular, the asymptotic stability of the vortex sheet under small spatially periodic perturbations is proved by studying the dynamics of these spatial oscillations. The first key point in our analysis is to construct an ansatz to cancel these oscillations. Then using the Galilean transformation, we are able to find a shift function of the vortex sheet such that an anti-derivative technique works, which plays an important role in the energy estimates. Moreover, by introducing a new variable and using the intrinsic properties of the vortex sheet, we can achieve the optimal decay rates to the viscous wave.

9.Well-posedness and global attractor for wave equation with nonlinear damping and super-cubic nonlinearity

Authors:Cuncai Liu, Fengjuan Meng, Chang Zhang

Abstract: In the paper, we study the semilinear wave equation involving the nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. Under the wider ranges of exponents of $g$ and $f$, the well-posedness of the weak solution is achieved by establishing a priori space-time estimates. Then, the existence of the global attractor in the naturally phase space $H^1_0(\Omega)\times L^2(\Omega)$ is obtained. Moreover, we prove that the global attrator is regular, that is, the global attractor is a bounded subset of $(H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega)$.

10.Global Well-posedness for The Fourth-order Nonlinear Schrödinger Equations on $\mathbb{R}^{2}$

Authors:Engin Başakoğlu, Barış Yeşiloğlu, Oğuz Yılmaz

Abstract: We study the global well-posedness of the two-dimensional defocusing fourth-order Schr\"odinger initial value problem with power type nonlinearities $\vert u\vert^{2k}u$ where $k$ is a positive integer. By using the $I$-method, we prove that global well-posedness is satisfied in the Sobolev spaces $H^{s}(\mathbb{R}^{2})$ for $2-\frac{3}{4k}<s<2$.

11.Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains

Authors:Simon Labrunie, Hassan Mohsen, Victor Nistor

Abstract: We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ of elliptic second order differential operators on a domain $U_0 \subset \RR^m$ whose coefficients depend on the space variable $x \in U_0$ and on $\omega \in \Omega,$ a probability space. We allow the coefficients $a_{ij}$ of $P_\omega$ to have jumps over a fixed interface \Gamma \subset U_0$ (independent of $\omega \in \Omega$). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution $u_\omega$ to the equation $P_\omega u_\omega = f$ with transmission and mixed boundary conditions (we consider ``sign-changing'' problems as well). In particular, we show that, if $f$ and the coefficients $a_{ij}$ are smooth enough and follow a log-normal-type distribution, then the map $\Omega \ni \omega \to \|u_\omega\|_{H^{k+1}(U_0)}$ is in $L^p(\Omega)$, for all $1 \le p < \infty$. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.

12.Analysis on noncompact manifolds and Index Theory: Fredholm conditions and Pseudodifferential operators

Authors:Ivan Beschastnyi, Catarina Carvalho, Victor Nistor, Yu Qiao

Abstract: We provide Fredholm conditions for compatible differential operators on certain Lie manifolds (that is, on certain possibly non-compact manifolds with nice ends). We discuss in more detail the case of manifolds with cylindrical, hyperbolic, and Euclidean ends, which are all covered by particular instances of our results. We also discuss applications to Schr\"odinger operators with singularities of the form r^{-2\gamma}$, $\gamma \in \RR_+$.

1.Inverse problem for the subdiffusion equation with non-local in time condition

Authors:Ravshan Ashurov, Marjona Shakarova

Abstract: In the Hilbert space $H$, the inverse problem of determining the right-hand side of the abstract subdiffusion equation with the fractional Caputo derivative is considered. For the forward problem, a non-local in time condition $u(0)=u(T)$ is taken. The right-hand side of the equation has the form $fg(t)$, and the unknown element is $f\in H$. If function $g(t)$ does not change sign, then under a over-determination condition $ u (t_0)= \psi $, $t_0\in (0, T)$, it is proved that the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution for some sign-changing functions $g(t)$. For such functions $g(t)$, under certain conditions on this function, one can achieve well-posedness of the problem by choosing $t_0$. And for some $g(t)$, for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied and in this case there is no uniqueness. All the results obtained are also new for the classical diffusion equations.

2.Existence and Multiplicity of Normalized Solutions for Dirac Equations with non-autonomous nonlinearities

Authors:Anouar Bahrouni, Qi Guo, Hichem Hajaiej, Yuanyang Yu

Abstract: In this paper, we study the following nonlinear Dirac equations \begin{align*} \begin{cases} -i\sum\limits_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u, \displaystyle \int_{\mathbb{R}^3} |u|^2dx=a^2, \end{cases} \end{align*} where $u: \mathbb{R}^{3}\rightarrow \mathbb{C}^{4}$, $m>0$ is the mass of the Dirac particle, $\omega\in \mathbb{R}$ arises as a Lagrange multiplier, $\partial_k=\frac{\partial}{\partial x_k}$, $\alpha_1,\alpha_2,\alpha_3$ are $4\times 4$ Pauli-Dirac matrices, $a>0$ is a prescribed constant, and $f(x,\cdot)$ has several physical interpretations that will be discussed in the Introduction. Under general assumptions on the nonlinearity $f$, we prove the existence of $L^2$-normalized solutions for the above nonlinear Dirac equations by using perturbation methods in combination with Lyapunov-Schmidt reduction. We also show the multiplicity of these normalized solutions thanks to the multiplicity theorem of Ljusternik-Schnirelmann. Moreover, we obtain bifurcation results of this problem.

3.A limiting case in partial regularity for quasiconvex functionals

Authors:Mirco Piccinini

Abstract: Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $p$-growth of the type $$ w \mapsto \int \left[F(Dw)-f\cdot w\right]dx $$ feature almost everywhere $\mbox{BMO}$-regular gradient provided that $f$ belongs to the borderline Marcinkiewicz space $L(n,\infty)$.

4.On higher integrability for $p(x)$-Laplacian equations with drift

Authors:Jingya Chen, Bin Guo, Baisheng Yan

Abstract: In this paper, we study the higher integrability for the gradient of weak solutions of $p(x)$-Laplacians equation with drift terms. We prove a version of generalized Gehring's lemma under some weaker condition on the modulus of continuity of variable exponent $p(x)$ and present a modified version of Sobolev-Poincar\'{e} inequality with such an exponent. When $p(x)>2$ we derive the reverse H\"older inequality with a proper dependence on the drift and force terms and establish a specific high integrability result. Our condition on the exponent $p(x)$ is more specific and weaker than the known conditions and our results extend some results on the $p(x)$-Laplacian equations without drift terms.

5.On a nonlocal two-phase flow with convective heat transfer

Authors:Šárka Ňecasová, John Sebastian H. Simon

Abstract: We study a system describing the dynamics of a two-phase flow of incompressible viscous fluids influenced by the convective heat transfer of Caginalp-type. The separation of the fluids is expressed by the order parameter which is of diffuse interface and is known as the Cahn-Hilliard model. We shall consider a nonlocal version of the Cahn-Hilliard model which replaces the gradient term in the free energy functional into a spatial convolution operator acting on the order parameter and incorporate with it a potential that is assumed to satisfy an arbitrary polynomial growth. The order parameter is influenced by the fluid velocity by means of convection, the temperature affects the interface via a modification of the Landau-Ginzburg free energy. The fluid is governed by the Navier--Stokes equations which is affected by the order parameter and the temperature by virtue of the capillarity between the two fluids. The temperature on the other hand satisfies a parabolic equation that considers latent heat due to phase transition and is influenced by the fluid via convection. The goal of this paper is to prove the global existence of weak solutions and show that, for an appropriate choice of sequence of convolutional kernels, the solutions of the nonlocal system converges to its local version.

6.On solvability in the small of higher order elliptic equations in Orlicz-Sobolev spaces

Authors:Javad A. Asadzade

Abstract: In this article, we consider a higher-order elliptic equation with nonsmooth coefficients with respect to Orlicz spaces on the domain $\Omega\subset\mathbb{R}^{n}$. The separable subspace of this space is distinguished in which infinitely differentiable and compactly supported functions are dense; Sobolev spaces generated by these subspaces are determined. We demonstrate the local solvability of the equation in Orlicz-Sobolev spaces under specific restrictions on the coefficients of the equation and the Boyd indices of the Orlicz space. This result strengthens the previously known classical $L_{p}$ analog.

7.Time periodic solutions of completely resonant Klein-Gordon equations on $\mathbb{S}^3$

Authors:Massimiliano Berti, Beatrice Langella, Diego Silimbani

Abstract: We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, regarded as toy models in General Relativity. The solutions are obtained by a variational Lyapunov- Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on $\mathbb{S}^3$.

1.Approximation of a solution to the stationary Navier-Stokes equations in a curved thin domain by a solution to thin-film limit equations

Authors:Tatsu-Hiko Miura

Abstract: We consider the stationary Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under the slip boundary conditions. Our aim is to show that a solution to the bulk equations is approximated by a solution to limit equations on the surface appearing in the thin-film limit of the bulk equations. To this end, we take the average of the bulk solution in the thin direction and estimate the difference of the averaged bulk solution and the surface solution. Then we combine an obtained difference estimate on the surface with an estimate for the difference of the bulk solution and its average to get a difference estimate for the bulk and surface solutions in the thin domain, which shows that the bulk solution is approximated by the surface one when the thickness of the thin domain is sufficiently small.

2.On a diffusion equation with rupture

Authors:Yoshikazu Giga, Yuki Ueda

Abstract: We propose a model to describe an evolution of a bubble cluster with rupture. In a special case, the equation is reduced to a single parabolic equation with evaporation for the thickness of a liquid layer covering bubbles. We postulate that a bubble collapses if this liquid layer becomes thin. We call this collapse a rupture. We prove for our model that there is a periodic-in-time solution if the place of rupture occurs only in the largest bubble. Numerical tests indicate that there may not exist a periodic solution if such an assumption is violated.

3.Existence of a local strong solution to the beam-polymeric fluid interaction system

Authors:Dominic Breit, Prince Romeo Mensash

Abstract: We construct a unique local strong solution to the finitely extensible nonlinear elastic (FENE) dumbbell model of Warner-type for an incompressible polymer fluid (described by the Navier-Stokes-Fokker-Planck equations) interacting with a flexible elastic shell. The latter occupies the flexible boundary of the polymer fluid domain and is modeled by a beam equation coupled through kinematic boundary conditions and the balance of forces. A main step in our approach is the proof of local well-posedness for just the solvent-structure system in higher-order topologies which is of independent interest. Different from most of the previous results in the literature, the reference spatial domain is an arbitrary smooth subset of $\mathbb{R}^3$, rather than a flat one. That is, we cover viscoelastic shells rather than elastic plates. Our result also supplements the existing literature on the Navier-Stokes-Fokker-Planck equations posed on a fixed bounded domain.

4.Existence and Uniqueness of Solution to Unsteady Darcy-Brinkman Problem with Korteweg Stress for Modelling Miscible Porous Media Flow

Authors:Sahil Kundu Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, India, Surya Narayan Maharana Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, India, Manoranjan Mishra Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, India

Abstract: The work investigates a model in the first half that combines a convection-diffusion equation for solute concentration with an unsteady Darcy-Brinkman equation for the flow field, including the Kortweg stress. Such models are used to describe flows in porous mediums such as fractured karst reservoirs, mineral wool, industrial foam, coastal mud, etc. The system of equations has Neumann boundary conditions for the solute concentration and no-flow conditions for the velocity field, and the well-posedness of the model is discussed for a wide range of initial data. In the second half, the study extends the model to include reactive solute transport under a forced flow field and precipitated flow where permeability varies with solute concentration. The existence and uniqueness of weak solutions are analyzed for these cases. Specifically, the study proves the existence and uniqueness of weak solutions for precipitated flow without reaction or body force, but the techniques used can be extended to the reactive precipitated and forced flow cases.

5.Exponential mixing for the white-forced complex Ginzburg--Landau equation in the whole space

Authors:Vahagn Nersesyan, Meng Zhao

Abstract: In the last two decades, there has been a significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In the present paper, we consider the damped complex Ginzburg--Landau equation on the real line driven by a white-in-time noise. Under the assumption that the noise is sufficiently non-degenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foia\c{s}--Prodi estimate to the case of the real line and special space-time weighted estimates which help to handle the behavior of solutions at infinity.

6.A Becker-Döring type model for cell polarization

Authors:Lorena Pohl Universität Bonn, Germany, Barbara Niethammer Universität Bonn, Germany

Abstract: We propose a model for cell polarization based on the Becker-D\"oring equations with the first coagulation coefficient equal to zero. We show convergence to equilibrium for power-law coagulation and fragmentation rates and obtain a loss of mass in the limit $t \rightarrow \infty$ depending on the initial mass and the relative strengths of the coagulation and fragmentation processes. In the case of linear rates, we further show that large clusters evolve in a self-similar manner at large times by comparing limits of appropriately rescaled solutions in different spaces.

7.On the Lawson-Osserman conjecture

Authors:Jonas Hirsch, Connor Mooney, Riccardo Tione

Abstract: We prove that if $u : B_1 \subset \mathbb{R}^2 \rightarrow \mathbb{R}^n$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case.

8.Generalized curvature for the optimal transport problem induced by a Tonelli Lagrangian

Authors:Yuchuan Yang

Abstract: We propose a generalized curvature that is motivated by the optimal transport problem on $\mathbb{R}^d$ with cost induced by a Tonelli Lagrangian $L$. We show that non-negativity of the generalized curvature implies displacement convexity of the generalized entropy functional on the $L-$Wasserstein space along $C^2$ displacement interpolants.

9.Concave solutions to Finsler $p$-Laplace type equations

Authors:Sunra Mosconi, Giuseppe Riey, Marco Squassina

Abstract: We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions describing the anisotropies to be not even.

10.Nonlocal problems with local boundary conditions II: Green's identities and regularity of solutions

Authors:James M. Scott, Qiang Du

Abstract: We study nonlocal integral equations on bounded domains with finite-range nonlocal interactions that are localized at the boundary. We establish a Green's identity for the nonlocal operator that recovers the classical boundary integral, which, along with the variational analysis established previously, leads to the well-posedness of these nonlocal problems with various types of classical local boundary conditions. We continue our analysis via boundary-localized convolutions, using them to analyze the Euler-Lagrange equations, which permits us to establish global regularity properties and classical Sobolev convergence to their classical local counterparts.

11.Local solvability and dilation-critical singularities of supercritical fractional heat equations

Authors:Yohei Fujishima, Kotaro Hisa, Kazuhiro Ishige, Robert Laister

Abstract: We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

1.Diffusive limit of Boltzmann Equation in exterior Domain

Authors:Junhwa Jung

Abstract: The study of flows over an obstacle is one of the fundamental problems in fluids. In this paper we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain. To overcome the well-known difficulty of the lack of Poincare's inequality, we develop a new $L^2-L^6$ splitting for dissipative hydrodynamic part Pf for nonlinear closure.

2.Uniform Decaying Property of Solutions for Anisotropic Viscoelastic Systems

Authors:Maarten V. de Hoop, Ching-Lung Lin, Gen Nakamura

Abstract: The paper concerns about the uniform decaying property (abbreviated by UDP) of solutions for an anisotropic viscoelastic system in the form of integrodifferential system (abbreviated by VID system) with mixed type boundary condition. The mixed type condition consists of the homogeneous displacement boundary condition and a homogeneous traction boundary condition or with a dissipation. By using a dissipative structure of this system, we will prove the UDP in a unified way for the two cases, which are, when the time derivative of relaxation tensor decays with polynomial order and it decays with exponential order.

3.Local second order regularity of solutions to elliptic Orlicz-Laplace equation

Authors:Arttu Karppinen, Saara Sarsa

Abstract: We consider Orlicz--Laplace equation $-div(\frac{\varphi'(|\nabla u|)}{|\nabla u|}\nabla u)=f$ where $\varphi$ is an Orlicz function and either $f=0$ or $f\in L^\infty$. We prove local second order regularity results for the weak solutions $u$ of the Orlicz--Laplace equation. More precisely, we show that if $\psi$ is another Orlicz function that is close to $\varphi$ in a suitable sense, then $\frac{\psi'(|\nabla u|)}{|\nabla u|}\nabla u\in W^{1,2}_{loc}$. This work contributes to the building up of quantitative second order Sobolev regularity for solutions of nonlinear equations.

4.$p$-Laplacian operator with potential in generalized Morrey Spaces

Authors:René Erlin Castillo, Héctor Camilo Chaparro

Abstract: We study some basic properties of generalized Morrey spaces $\mathcal{M}^{p,\phi}(\R^{d})$. Also, the problem $-\mbox{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u=0$ in $\Omega$, where $\Omega$ is a bounded open set in $\R^d$, and potential $V$ is assumed to be not equivalent to zero and lies in $\mathcal{M}^{p,\phi}(\Omega)$, is studied. Finally, we establish the strong unique continuation for the $p$-Laplace operator in the case $V\in\mathcal{M}^{p,\phi}(\R^d)$.

1.Reconstruction of the initial data from the solutions of damped wave equations

Authors:Seongyeon Kim, Sunghwan Moon, Ihyeok Seo

Abstract: In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation and show that the initial velocity from the solution on the unit sphere. This inverse problem is related to Photoacoustic Tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.

2.Notes on Overdetermined Singular Problems

Authors:Francesco Esposito, Berardino Sciunzi, Nicola Soave

Abstract: We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.

3.On traveling waves and global existence for a nonlinear Schrödinger system with three waves interaction

Authors:Yuan Li

Abstract: In this paper, we consider three components system of nonlinear Schr\"odinger equations related to the Raman amplification in a plasma. By using variational method, a new result on the existence of traveling wave solutions are obtained under the non-mass resonance condition. We also study the new global existence result for oscillating data. Both of our results essentially due to the absence of Galilean symmetry in the system.

4.Nonexistence of multi-dimensional solitary waves for the Euler-Poisson system

Authors:Junsik Bae, Daisuke Kawagoe

Abstract: We study the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler-Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler-Poisson system for ions and electrons.

5.Prandtl Boundary Layers in An Infinitely Long Convergent Channel

Authors:Chen Gao, Zhouping Xin

Abstract: This paper concerns the large Reynold number limits and asymptotic behaviors of solutions to the 2D steady Navier-Stokes equations in an infinitely long convergent channel. It is shown that for a general convergent infinitely long nozzle whose boundary curves satisfy curvature-decreasing and any given finite negative mass flux, the Prandtl's viscous boundary layer theory holds in the sense that there exists a Navier-Stokes flow with no-slip boundary condition for small viscosity, which is approximated uniformly by the superposition of an Euler flow and a Prandtl flow. Moreover, the singular asymptotic behaviors of the solution to the Navier-Stokes equations near the vertex of the nozzle and at infinity are determined by the given mass flux, which is also important for the constructions of the Prandtl approximation solution due to the possible singularities at the vertex and non-compactness of the nozzle. One of the key ingredients in our analysis is that the curvature-decreasing condition on boundary curves of the convergent nozzle ensures that the limiting inviscid flow is pressure favourable and plays crucial roles in both the Prandtl expansion and the stability analysis.

6.Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

Authors:M. Fasondini, J. R. King, J. A. C. Weideman

Abstract: Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified space domain. Blow up in finite time is caused by these singularities eventually reaching the real axis. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales on which blow up is approached. It is shown that an ordinary differential equation with quadratic nonlinearity plays a central role in the asymptotic analysis. This equation is studied in detail, including its numerical computation on multiple Riemann sheets, and the far-field solutions are shown to be given at leading order by a Weierstrass elliptic function.

7.Rigorous Derivation of Discrete Fracture Models for Darcy Flow in the Limit of Vanishing Aperture

Authors:Maximilian Hörl, Christian Rohde

Abstract: We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying hydraulic conductivity matrices in both bulk and fractures. The width-to-length ratio of a fracture is of the order of a small parameter $\varepsilon$ and the ratio $K_\mathrm{f}^\star / K_\mathrm{b}^\star$ of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with $\varepsilon^\alpha$ for a parameter $\alpha \in \mathbb{R}$. The fracture geometry is parameterized by aperture functions on a submanifold of codimension one. Given a fracture, we derive the limit models as $\varepsilon \rightarrow 0$. Depending on the value of $\alpha$, we obtain five different limit models as $\varepsilon \rightarrow 0$, for which we present rigorous convergence results.

8.On a Space Fractional Stefan problem of Dirichlet type with Caputo flux

Authors:S. D. Roscani, K. Ryszewska, L. D. Venturato

Abstract: We study a space-fractional Stefan problem with the Dirichlet boundary conditions. It is a model that describes superdiffusive phenomena. Our main result is the existence of the unique classical solution to this problem. In the proof we apply evolution operators theory and the Schauder fixed point theorem. It appears that studying fractional Stefan problem with Dirichlet boundary conditions requires a substantial modifications of the approach in comparison with the existing results for problems with different kinds of boundary conditions.

9.On counterexamples to unique continuation for critically singular wave equations

Authors:Simon Guisset, Arick Shao

Abstract: We consider wave equations with a critically singular potential $\xi \cdot \sigma^{-2}$ diverging as an inverse square at a hypersurface $\sigma = 0$. Our aim is to construct counterexamples to unique continuation from $\sigma = 0$ for this equation, provided there exists a family of null geodesics trapped near $\sigma = 0$. This extends the classical geometric optics construction of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of $\sigma = 0$ - by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence.

10.Maximum principle for the weak solutions of the Cauchy problem for the fourth-order hyperbolic equations

Authors:Kateryna Buryachenko

Abstract: We investigate the maximum principle for the weak solutions to the Cauchy problem for the hyperbolic fourth-order linear equations with constant complex coefficients in the plane bounded domain

1.Improvement of the general theory for one dimensional nonlinear wave equations related to the combined effect

Authors:Shu Takamatsu

Abstract: We focus on the general theory to the Cauchy problem for one dimensional nonlinear wave equations with small initial data. In the general theory, we aim to obtain the lower bound estimate of the lifespan of classical solution. In this paper, we improve it in some case related to the combined effect, which was expected complete more than 30 years ago.

2.A priori estimates for higher-order fractional Laplace equations

Authors:Yugao Ouyang, Meiqing Xu, Ran Zhuo

Abstract: In this paper, we establish a priori estimates for the positive solutions to a higher-order fractional Laplace equation on a bounded domain by a blowing-up and rescaling argument. To overcome the technical difficulty due to the high-order and fractional order mixed operators, we divide the high-order fractional Laplacian equation into a system, and provide uniform estimates for each equation in the system. Finding a proper scaling parameter for the domain is the crux of rescaling argument to the above system, and the new idea is introduced in the rescaling proof, which may hopefully be applied to many other system problems. In order to derive a contradiction in the blowing-up proof, combining the moving planes method and suitable Kelvin transform, we prove a key Liouville-type theorem under a weaker regularity assumption in a half space.

3.A priori estimates for anti-symmetric solutions to a fractional Laplacian equation in a bounded domain

Authors:Chenkai Liu, Shaodong Wang, Ran Zhuo

Abstract: In this paper, we obtain a priori estimates for the set of anti-symmetric solutions to a fractional Laplacian equation in a bounded domain using a blowing-up and rescaling argument. In order to establish a contradiction to possible blow-ups, we apply a certain variation of the moving planes method in order to prove a monotonicity result for the limit equation after rescaling.

4.Flat blow-up solutions for the complex Ginzburg Landau equation

Authors:Giao Ky Duong, Nejla Nouaili, Hatem Zaag

Abstract: In this paper, we consider the complex Ginzburg Landau equation $$ \partial_t u = (1 + i \beta ) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u \text{ where } \beta, \delta, \alpha \in \mathbb R. $$ The study aims to investigate the finite time blowup phenomenon. In particular, for fixed $ \beta\in \mathbb R$, the existence of finite time blowup solutions for an arbitrary large $|\delta|$ is still unknown. Especially, Popp, Stiller, Kuznetsov, and Kramer formally conjectured in 1998 that there is no blowup (collapse) in such a case. In this work, considered as a breakthrough, we give a counter example to this conjecture. We show the existence of blowup solutions in one dimension with $\delta $ arbitrarily given and $\beta =0$. The novelty is based on two main contributions: an investigation of a new blowup scaling (flat blowup regime) and a suitable modulation.

5.Compact embeddings for weighted fractional Sobolev spaces and applications to Nonlinear Schrödinger Equations

Authors:Federico Bernini, Sergio Rolando, Simone Secchi

Abstract: The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation $$ (-\Delta)^su + V(|x|)u = K(|x|)f(u), \quad \text{ in } \mathbb{R}^N, $$ where the two measurable functions $K > 0$ and $V \geq 0$ could vanish at infinity.

6.Non-standard Sobolev scales and the mapping properties of the X-ray transform on manifolds with strictly convex boundary

Authors:François Monard

Abstract: This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These questions are motivated by classical inverse problems questions (e.g. range characterization, stability estimates, mapping properties on Hilbert scales), and more recently by uncertainty quantification and operator learning questions.

7.Well-posedness for the extended Schrödinger-Benjamin-Ono system

Authors:Felipe Linares, Argenis Mendez, Didier Pilod

Abstract: In this work we prove that the initial value problem associated to the Schr\"odinger-Benjamin-Ono type system \begin{equation*} \left\{ \begin{array}{ll} \mathrm{i}\partial_{t}u+ \partial_{x}^{2} u= uv+ \beta u|u|^{2}, \partial_{t}v-\mathcal{H}_{x}\partial_{x}^{2}v+ \rho v\partial_{x}v=\partial_{x}\left(|u|^{2}\right) u(x,0)=u_{0}(x), \quad v(x,0)=v_{0}(x), \end{array} \right. \end{equation*} with $\beta,\rho \in \mathbb{R}$ is locally well-posed for initial data $(u_{0},v_{0})\in H^{s+\frac12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for $s>\frac54$. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold $s=\frac32$, we employ a refined Strichartz estimate introduced in the Benjamin-Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.

8.Parabolic equations with non-standard growth and measure or integrable data

Authors:Miroslav Bulíček, Jakub Woźnicki

Abstract: We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition, controlled by the exponent $p$ depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-H\"{o}lder continuity of the variable exponent~$p$ with respect to the spatial variable. On the other hand, we do not assume any smoothness of~$p$ with respect to the time variable.

1.Part I: Rebuttal to "Uniform stabilization for the Timoshenko beam by a locally distributed damping"

Authors:Fatiha Alabau-Boussouira

Abstract: A paper, entitled "Uniform stabilization for the Timoshenko beam by a locally distributed damping" was published in 2003, in the journal Electronic Journal of Differential Equations. Its title concerns exclusively its Section 3, devoted to the case of equal speeds of propagation and to its main theorem, namely Theorem 3.1. It states that the solutions of the Timoshenko system (see (1.3) in [1]) decays exponentially when the damping coefficient b is locally distributed. The proof of Theorem 3.1 is crucially based on Lemma 3.6, which states the existence of a strict Lyapunov function along which the solutions of (1.3) decay when the speeds of propagation are equal. This rebuttal shows the major gap and flaws in the proof of Lemma 3.6, which invalidate the proofs of Lemma 3.6 and Theorem 3.1. Lemma 3.6 is stated at the top of page 12. The main part of its proof is given in the pages 12 and 13. In the last eight lines of page 13, eight inequalities are requested to hold together for the proof of Lemma 3.6. They don't appear in the statements of Lemma 3.6. The subsequent flaws come from the evidence that several of them are contradictory either between them or with claims in the title of the article. We also point in this rebuttal other flaws, or gaps in the proofs of Theorem 2.2 related to strong stability and non uniform stability for the case of distinct speeds of propagation. In [3], we correct and complete the proof of strong stability. We also correct, set up the missing functional frames, fill the gaps in the proof of non uniform stability in the cases of different speeds of propagation, and complete a missing argument in the proof of Theorem A in [4] (see Remark 4.3), the result of Theorem A being used in the paper [1] on which this rebuttal is mainly devoted.

2.Maximal regularity of Stokes problem with dynamic boundary condition -- Hilbert setting

Authors:Tomáš Bárta, Paige Davis, Petr Kaplický

Abstract: For the evolutionary Stokes problem with dynamic boundary condition we show maximal regularity of weak solutions in time. Due to the characteriation of $R$-sectorial operators on Hilbert spaces, the proof reduces to finding the correct functional analytic setting and proving that an operator is sectorial, i.e. generates an analytic semigroup.

3.Part II On strong and non uniform stability of locally damped Timoshenko beam: Mathematical corrections to the proof of Theorem 2.2 in the publication referenced as [1] in the bibliography

Authors:Fatiha Alabau-Boussouira

Abstract: In part I of the rebuttal (see [2] to the article [1] entitled "Uniform stabilization for the Timoshenko beam by a locally distributed damping" published in 2003, in the journal Electronic Journal of Differential Equations, we prove that Lemma 3.6 and Theorem 3.1 are unproved due to major flaws (contradictory assumptions). We also show that Theorem 2.2 and its proofs of strong stability, and non uniform stability in the case of different speeds of propagation, contain several incorrect arguments and several gaps (including missing functional frames). In this part II, we give the precise missing functional frames, fill the gaps and correct several parts contained in the proof of Theorem 2.2 in [1]. We also complete a missing argument (see Remark 4.23 and Remark 3.2) in the proof of Theorem A in [5] used by [1]. For this we state and prove Proposition 4.4 (see also Proposition 4.6 for a general formulation in Banach spaces). We also give the correct formulations, and proofs of strong stability and non uniform stability (in case of different speeds of propagation) for Timoshenko beams.

4.Blow-up for semilinear wave equations with damping and potential in high dimensional Schwarzschild spacetime

Authors:Mengliang Liu, Mengyun Liu

Abstract: In this work, we study the blow up results to power-type semilinear wave equation in the high dimensional Schwarzschild spacetime, with damping and potential terms. We can obtain the upper bound estimates of lifespan without the assumption that the support of the initial date should be far away from the black hole.

5.Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions

Authors:Armand Bernou

Abstract: We study the degenerate linear Boltzmann equation inside a bounded domain with the Maxwell and the Cercignani-Lampis boundary conditions, two generalizations of the diffuse reflection, with variable temperature. This includes a model of relaxation towards a space-dependent steady state. For both boundary conditions, we prove for the first time the existence of a steady state and a rate of convergence towards it without assumptions on the temperature variations. Our results for the Cercignani-Lampis boundary condition make also no hypotheses on the accommodation coefficients. The proven rate is exponential when a control condition on the degeneracy of the collision operator is satisfied, and only polynomial when this assumption is not met, in line with our previous results regarding the free-transport equation. We also provide a precise description of the different convergence rates, including lower bounds, when the steady state is bounded. Our method yields constructive constants.

6.Obstacle problems for nonlocal operators with singular kernels

Authors:Xavier Ros-Oton, Marvin Weidner

Abstract: In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in contrast to all previous known results, we are able to treat nonlocal operators whose kernels are not necessarily pointwise comparable to the one of the fractional Laplacian. Such operators might be very anisotropic in the sense that they "do not see" certain directions at all, or might have substantial oscillatory behavior, causing the nonlocal Harnack inequality to fail.

7.A strict maximum principle for nonlocal minimal surfaces

Authors:Serena Dipierro, Ovidiu Savin, Enrico Valdinoci

Abstract: In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious, since the surfaces may touch at an irregular point, therefore a suitable blow-up analysis must be combined with a bespoke regularity theory to obtain this result. For the classical case, an analogous result was proved by Leon Simon. Our proof also relies on a Harnack Inequality for nonlocal minimal surfaces that has been recently introduced by Xavier Cabr\'e and Matteo Cozzi and which can be seen as a fractional counterpart of a classical result by Enrico Bombieri and Enrico Giusti. In our setting, an additional difficulty comes from the analysis of the corresponding nonlocal integral equation on a hypersurface, which presents a remainder whose sign and fine properties need to be carefully addressed.

8.Pushed and pulled fronts in a logistic Keller-Segel model with chemorepulsion

Authors:Montie Avery, Matt Holzer, Arnd Scheel

Abstract: We analyze spatial spreading in a population model with logistic growth and chemorepulsion. In a parameter range of short-range chemo-diffusion, we use geometric singular perturbation theory and functional-analytic farfield-core decompositions to identify spreading speeds with marginally stable front profiles. In particular, we identify a sharp boundary between between linearly determined, pulled propagation, and nonlinearly determined, pushed propagation, induced by the chemorepulsion. The results are motivated by recent work on singular limits in this regime using PDE methods.

9.Infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domain: An Introduction

Authors:Birgit Jacob, Hans Zwart

Abstract: We provide an introduction to infinite-dimensional port-Hamiltonian systems. As this research field is quite rich, we restrict ourselves to the class of infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domainand we will focus on topics such as Dirac structures, well-posedness, stability and stabilizability, Riesz-bases and dissipativity. We combine the abstract operator theoretic approach with the more physical approach based on Hamiltonians. This enables us to derive easy verifiable conditions for well-posedness and stability.

10.Blow up dynamics for the 3D energy-critical Nonlinear Schrödinger equation

Authors:Tobias Schmid

Abstract: We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i \alpha(t)}\lambda(t)^{\frac{1}{2}}W(\lambda(t) x) + \eta(t, x )$, $ t \in[0, T)$, $ x \in \mathbb{R}^3$, where $ W( x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}}$ is the ground state solution, $\lambda(t) = (T-t)^{- \frac12 - \nu} $ for suitable $ \nu > 0 $, $ \alpha(t) = \alpha_0 \log(T - t)$ and $ T= T(\nu, \alpha_0) > 0 $. Further $ \|\eta(t) - \eta_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1)$ as $ t \to T^-$ for some $ \eta_T \in \dot{H}^{1} \cap~ \dot{H}^2$.

1.Study of fractional semipositone problems on $\mathbb{R}^N$

Authors:Nirjan Biswas

Abstract: Let $s \in (0,1)$ and $N \geq 2$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-\Delta)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in C(\mathbb{R})$ is negative on $\mathbb{R}^{-}$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near the origin. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^{\infty}(\mathbb{R}^N)$.

2.Regularity of oscillatory integral operators

Authors:Anders Israelsson, Tobias Mattsson, Wolfgang Staubach

Abstract: In this paper, we establish the global boundedness of oscillatory integral operators on Besov-Lipschitz and Triebel-Lizorkin spaces, with amplitudes in general $S^m_{\rho,\delta}(\mathbb{R}^n)$-classes and non-degenerate phase functions in the class $\textart F^k$. Our results hold for a wide range of parameters $0\leq\rho\leq1$, $0\leq\delta<1$, $0<p\leq\infty$, $0<q\leq\infty$ and $k>0$. We also provide a sufficient condition for the boundedness of operators with amplitudes in the forbidden class $S^m_{1,1}(\mathbb{R}^n)$ in Triebel-Lizorkin spaces.

3.Propagation of chaos and hydrodynamic description for topological models

Authors:Dario Benedetto Sapienza University of Rome, Thierry Paul LJLL, Stefano Rossi Sapienza University of Rome

Abstract: In this work, we study the deterministic Cucker-Smale model with topological interaction.Focusing on the solutions of the corresponding Liouville equation, we show that propagation of chaos holds.Moreover, by looking at the monokinetic solutions, we also obtain a rigorous derivation of the hydrodynamic description given by a pressureless Euler-type system.

4.New Lipschitz estimates and long-time asymptotic behavior for porous medium and fast diffusion equations

Authors:Noemi David MMCS, ICJ, Filippo Santambrogio MMCS, ICJ

Abstract: We obtain new estimates for the solution of both the porous medium and the fast diffusion equations by studying the evolution of suitable Lipschitz norms. Our results include instantaneous regularization for all positive times, long-time decay rates of the norms which are sharp and independent of the initial support, and new convergence results to the Barenblatt profile. Moreover, we address nonlinear diffusion equations including quadratic or bounded potentials as well. In the slow diffusion case, our strategy requires exponents close enough to 1, while in the fast diffusion case, our results cover any exponent for which the problem is well-posed and mass-preserving in the whole space.

5.Stability Analysis for a Class of Heterogeneous Catalysis Models

Authors:Christian Gesse, Matthias Köhne, Jürgen Saal

Abstract: We prove stability for a class of heterogeneous catalysis models in the $L_p$-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.

6.Measure Data for a General Class of Nonlinear Elliptic Problems

Authors:Mohammed El Ansari, Youssef Akdim, Soumia Lalaoui Rhali

Abstract: We consider nonlinear elliptic inclusion having a measure in the right-hand side of the type $\beta(u)-div a(x,Du)\ni \mu$ in $\Omega$ a bounded domain in $\mathbb{R}^{N},$ with $\beta$ is a maximal monotone graph in $\mathbb{R}^2$ and $a(x,Du)$ is a Leray-Lions type operator. We study a suitable notion of solution for this kind of problem. The functional setting involves anisotropic Sobolev spaces.

7.Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall

Authors:Jiawei Wang, Junyan Zhang

Abstract: We prove the incompressible limit of compressible ideal magnetohydrodynamic (MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space $H^m~(m\geq 2)$, while the incompressible problem is still well-posed in $H^m$. The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight $\varepsilon^2$. Thus, the energy functional should be defined by using the anisotropic Sobolev space $H_*^{2m}$. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is parallel to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD such as current-vortex sheets.

8.Conservation, convergence, and computation for evolving heterogeneous elastic wires

Authors:Anna Dall'Acqua, Gaspard Jankowiak, Leonie Langer, Fabian Rupp

Abstract: The elastic energy of a bending-resistant interface depends both on its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham-Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal $L^2$-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.

9.Struwe's Global Compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group

Authors:Giampiero Palatucci, Mirco Piccinini, Letizia Temperini

Abstract: We investigate some of the effects of the lack of compactness in the critical Folland-Stein-Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi's $\Gamma$-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe (Math. Z. 1984).

10.Regularity results for quasiminima of a class of double phase problems

Authors:Antonella Nastasi, Cintia Pacchiano Camacho

Abstract: We prove boundedness, H\"older continuity, Harnack inequality results for local quasiminima to elliptic double phase problems of $p$-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach and they are based on the De Giorgi method, a careful phase analysis and estimates in the intrinsic geometries.

11.Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation

Authors:Xing Cheng, Chang-Yu Guo, Yunrui Zheng

Abstract: In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schr\"odinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Sch\"ordinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schr\"odinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$.

12.Well-posedness and Long-time Behavior of a Bulk-surface Coupled Cahn-Hilliard-diffusion System with Singular Potential for Lipid Raft Formation

Authors:Hao Wu, Shengqin Xu

Abstract: We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term $q$. For the system with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term $q$. Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term $q$. For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as $t\to +\infty$. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.

13.One inverse source problem generated by the Dunkl operator

Authors:Bayan Bekbolat, Niyaz Tokmagambetov

Abstract: The aim of this paper is to study time-fractional pseudo-parabolic type equations generated by the Dunkl operator. The forward problem is considered and its well-posedness is established. In particular, a prior estimates are obtained in the Sobolev type spaces and, explicit formulas for solutions of the problems are derived. Here we also deal with the left-sided Caputo fractional time derivative. As an application, we investigate an inverse source problem. Existence and uniqueness of the solution is proved. Moreover, we show that a solution pair is continuously depending on the initial and additional data, finalizing with a numerical test.

14.Blow-up and lifespan estimate for the generalized tricomi equation with the scale-invariant damping and time derivative nonlinearity on exterior domain

Authors:Makram Hamouda, Mohamed Ali Hamza, Bouthaina Yousfi

Abstract: The article is devoted to investigating the initial boundary value problem for the damped wave equation in the scale-invariant case with time-dependent speed of propagation on the exterior domain. By presenting suitable multipliers and applying the test-function technique, we study the blow-up and the lifespan of the solutions to the problem with derivative-type nonlinearity $ \d u_{tt}-t^{2m}\Delta u+\frac{\mu}{t}u_t=|u_t|^p, \quad \mbox{in}\ \Omega^{c}\times[1,\infty), $ that we associate with appropriate small initial data.

1.Free boundary regularity and support propagation in mean field games and optimal transport

Authors:Pierre Cardaliaguet, Sebastian Munoz, Alessio Porretta

Abstract: We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys $C^{1,1}$ regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be H\"older continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

2.Fick's las selects the Neumann boundary condition

Authors:Danielle Hilhorst, Seung-Min Kang, Ho-Youn Kim, Yong-Jung Kim

Abstract: We study the appearance of a boundary condition along an interface between two regions, one with constant diffusivity $1$ and the other with diffusivity $\eps>0$, when $\eps\to0$. In particular, we take Fick's diffusion law in a context of reaction-diffusion equation with bistable nonlinearity and show that the limit of the reaction-diffusion equation satisfies the homogeneous Neumann boundary condition along the interface. This problem is developed as an application of heterogeneous diffusion laws to study the geometry effect of domain.

3.A regularity theory for parabolic equations with anisotropic non-local operators in $L_{q}(L_{p})$ spaces

Authors:Jae-Hwan Choi, Jaehoon Kang, Daehan Park

Abstract: In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $$ -\sum_{i=1}^{\ell}a_{i}(t)(-\Delta_{x_i})^{\alpha_i/2}. $$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calder\'on-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.

4.Mixing in incompressible flows: transport, dissipation, and their interplay

Authors:Michele Coti Zelati, Gianluca Crippa, Gautam Iyer, Anna L. Mazzucato

Abstract: In this survey, we address mixing from the point of view of partial differential equations, motivated by applications that arise in fluid dynamics. We give an account of optimal mixing, loss of regularity for transport equations, enhanced dissipation, and anomalous dissipation.

5.The asymptotic stability of solitons in the focusing Hirota equation on the line

Authors:Ruihong Ma, Engui Fan

Abstract: In this paper, the $\overline\partial$-steepest descent method and B\"acklund transformation are used to study the asymptotic stability of solitons for the following Cauchy problem of focusing Hirota equation \begin{align}\nonumber & iq_t+\alpha(2|q|^2q+q_{xx})+i\beta(q_{xxx}+6|q|^2q_x)=0, \\\nonumber & q(x,0)=q_0(x), \end{align} where $q_0 \in H^1(\mathbb{R})\,\cap\,L^{2,s}(\mathbb{R}),s\in(\frac{1}{2},1] .$ We first express the solution of the Cauchy problem in term of the solution of a Riemann-Hilbert (RH) problem. Then the RH problem is further decomposed into pure radiation solution and solitons solution,which are solved by using $\overline\partial$-techniques and B\"acklund transformation respectively. As a directly consequence, we obtain the asymptotic stability of solitons for the Hirota equation.

6.Anomalous smoothing effect on the incompressible Navier-Stokes-Fourier limit from Boltzmann with periodic velocity

Authors:Zhongyang Gu, Xin Hu, Tsuyoshi Yoneda

Abstract: Adding a nontrivial term composed from a microstructure, we prove the existence for global-in-time weak solutions to an incompressible 3D Navier-Stokes-Fourier system, whose enstrophy is bounded for all the time. The main idea is employing the hydrodynamic limit of the Boltzmann equation with periodic velocity and a specially designed collision operator.

7.Multi-frequency averaging and uniform accuracy towards numerical approximations for a Bloch model

Authors:Brigitte Bidégaray-Fesquet EDP, Clément Jourdana EDP, Léopold Trémant TONUS, IRMA

Abstract: We are interested in numerically solving a transitional model derived from the Bloch model. The Bloch equation describes the time evolution of the density matrix of a quantum system forced by an electromagnetic wave. In a high frequency and low amplitude regime, it asymptotically reduces to a non-stiff rate equation. As a middle ground, the transitional model governs the diagonal part of the density matrix. It fits in a general setting of linear problems with a high-frequency quasi-periodic forcing and an exponentially decaying forcing. The numerical resolution of such problems is challenging. Adapting high-order averaging techniques to this setting, we separate the slow (rate) dynamics from the fast (oscillatory and decay) dynamics to derive a new micro-macro problem. We derive estimates for the size of the micro part of the decomposition, and of its time derivatives, showing that this new problem is non-stiff. As such, we may solve this micro-macro problem with uniform accuracy using standard numerical schemes. To validate this approach, we present numerical results first on a toy problem and then on the transitional Bloch model.

8.Decay estimates for a class of semigroups related to self-adjoint operators on metric measure spaces

Authors:Guoxia Feng, Manli Song, Huoxiong Wu

Abstract: Assume that $(X,d,\mu)$ is a metric space endowed with a non-negative Borel measure $\mu$ satisfying the doubling condition and the additional condition that $\mu(B(x,r))\gtrsim r^n$ for any $x\in X, \,r>0$ and some $n\geq1$. Let $L$ be a non-negative self-adjoint operator on $L^2(X,\mu)$. We assume that $e^{-tL}$ satisfies a Gaussian upper bound and the Schr\"odinger operator $e^{itL}$ satisfies an $L^1\to L^\infty$ decay estimate of the form \begin{equation*} \|e^{itL}\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}. \end{equation*} Then for a general class of dispersive semigroup $e^{it\phi(L)}$, where $\phi: \mathbb{R}^+ \to \mathbb{R}$ is smooth, we establish a similar $L^1\to L^\infty$ decay estimate by a suitable subordination formula connecting it with the Schr\"odinger operator $e^{itL}$. As applications, we derive new Strichartz estimates for several dispersive equations related to Hermite operators, twisted Laplacians and Laguerre operators.

9.A quantitative version of the Gidas-Ni-Nirenberg Theorem

Authors:Giulio Ciraolo, Matteo Cozzi, Matteo Perugini, Luigi Pollastro

Abstract: A celebrated result by Gidas-Ni-Nirenberg asserts that classical solutions to semilinear equations~$- \Delta u = f(u)$ in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small perturbations of this equation and study its quantitative stability counterpart.

10.Diffusion laws select boundary conditions

Authors:Jaywan Chung, Seungmin Kang, Ho-Youn Kim, Yong-Jung Kim

Abstract: The choice of boundary condition makes an essential difference in the solution structure of diffusion equations. The Dirichlet and Neumann boundary conditions and their combination have been the most used, but their legitimacy has been disputed. We show that the diffusion laws may select boundary conditions by themselves, and through this, we clarify the meaning of boundary conditions. To do that we extend the domain with a boundary into the whole space by giving a small diffusivity $\eps>0$ outside the domain. Then, we show that the boundary condition turns out to be Neumann or Dirichlet as $\eps\to0$ depending on the choice of a heterogeneous diffusion law. These boundary conditions are interpreted in terms of a microscopic-scale random walk model.

11.The Non-cutoff Boltzmann Equation in Convex Domains

Authors:Dingqun Deng

Abstract: The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann equation is a long-standing open problem. In this paper, we study the stability and long-time dynamics of the Boltzmann equation near global Maxwellian without angular cutoff assumption in a convex domain $\Omega$ with physical boundary conditions: inflow and Maxwell-reflection (including diffuse-reflection) boundary conditions. When the domain $\Omega$ is bounded, we obtain the global stability in time, which has an exponential decay rate for the inflow boundary for both hard and soft potentials, and for the Maxwell-reflection boundary for hard potentials. The crucial method is to extend the boundary problem in a convex domain to the whole space, followed by the De Giorgi iteration and the $L^2$--$L^\infty$ method. We believe that the current work will have a significant impact on the generation of robust applications for the kinetic equations in bounded domains.

12.Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere

Authors:Alden Waters, Yan-Long Fang

Abstract: The goal of this article is to establish general principles for high frequency dispersive estimates for the $p$-form Laplacian with relative boundary conditions on co-closed forms. In dimension $3$ for the case $p=1$, we show that the propagator corresponding to Maxwell's equations on compactly supported co-closed forms satisfies the same dispersive estimates as in $\mathbb{R}^3$ for the corresponding wave equation in the exterior of a ball -- but only for certain polarizations. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave equation and this rate is not expected to hold in general. The Dirichlet Laplacian wave equation $L^1-L^{\infty}$ scattering rate does not hold true for the $1-$form Laplacian with relative boundary conditions by itself in the exterior of a sphere, or any smooth obstacle for that matter. We also do not expect it to hold in general for Maxwell's equations because of the presence of $L^2$ harmonic $1$ forms.

13.Nonlinear Gagliardo-Nirenberg inequality and a priori estimates for nonlinear elliptic eigenvalue problems

Authors:Agnieszka Kałamajska, Dalimil Peša, Tomáš Roskovec

Abstract: We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||\mathcal{T}_{H}(u(x))|}\right)^{2}h(u(x))\, {\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{\rm loc}(\Omega)$ is nonnegative, $P$ is a uniformly elliptic operator in nondivergent form, ${\cal T}_{H}(\cdot )$ is certain transformation of the nonnegative continuous function $h(\cdot)$, and $\Theta$ is the boundary term which depends on boundary values of $u$ and $\nabla u$, which holds under some additional assumptions. We apply such inequalities to obtain a priori estimates for solutions of nonlinear eigenvalue problems like $Pu=f(x)\tau (u)$, where $f\in L^1(\Omega)$, and provide several examples dealing with $\tau(\cdot)$ being power, power-logarithmic or exponential function. Our results are also linked with several issues from the probability and potential theory like Douglas formulae and representation of harmonic functions.

14.Regularity for the Timoshenko system with fractional damping

Authors:Fredy Maglorio Sobrado SuÁrez

Abstract: We study the Regularity of the Timoshenko system with two fractional dampings $ (-\Delta)^\tau u_t\quad $ and $ \quad (-\Delta)^\sigma \psi_t$, both of the parameters. $\; ( \tau, \sigma)$ vary in the interval $\; [0,1]$. We note that ( $\; \tau=0$ or $\sigma=0$ ) and ( $ \tau=1$ or $\sigma=1$ ) the dampings are called frictional and viscous respectively. Our main contribution is to show that the corresponding semigroup $S(t)=e^{\mathcal{B}t}$, is analytic for. $(\tau,\sigma)\in R_A:=[\frac{1}{2},1]\times[ \frac{1}{2},1]$ and determine the Gevrey's class $\nu>\dfrac{1}{\phi}$, where $$\phi=\left\{\begin{array}{ccc} \frac{2\sigma}{\sigma+1} &{\rm for} & \tau\leq \sigma,\\\\ \frac{2\tau}{\tau+1} &{\rm for} & \sigma\leq \tau. \end{array}\right.$$ and $(\tau,\sigma)\in R_{CG}:= (0,1)^2$.

15.An operator-asymptotic approach to periodic homogenization applied to equations of linearized elasticity

Authors:Yi-Sheng Lim, Josip Žubrinić

Abstract: We explain an operator-asymptotic approach to homogenization for periodic composite media. This approach was developed by Cherednichenko and Vel\v{c}i\'c in [Cherednichenko and Vel\v{c}i\'c (2022) Sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic properties. J. London Math. Soc.] in the context of thin elastic plates, and here we demonstrate the approach under the simpler setting of equations of linearized elasticity. As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher order $L^2\to L^2$ norm-resolvent estimates. The correctors for the $L^2\to H^1$, and higher order $L^2\to L^2$ results are constructed from boundary value problems that arise during the asymptotic procedure, and the first-order corrector is shown to coincide with classical formulae.

1.Decay estimates for Beam equations with potential in dimension three

Authors:Miao Chen, Ping Li, Avy Soffer, Xiaohua Yao

Abstract: This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{t t}+\big(\Delta^2+V\big)u=0, \,\ u(0, x)=f(x),\ u_{t}(0, x)=g(x)$$ in dimension three, where $V$ is a real-valued and decaying potential on $\R^3$. Assume that zero is a regular point of $H:= \Delta^2+V $, we first prove the following optimal time decay estimates of the solution operators \begin{equation*} \big\|\cos (t\sqrt{H})P_{ac}(H)\big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{3}{2}}\ \ \hbox{and} \ \ \Big\|\frac{\sin(t\sqrt{H})}{\sqrt{H}} P_{a c}(H)\Big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{1}{2}}. \end{equation*} Moreover, if zero is a resonance of $H$, then time decay of the solution operators above also are considered. It is noticed that the first kind resonance does not effect the decay rates for the propagator operators $\cos(t\sqrt{H})$ and $\frac{\sin(t\sqrt{H})}{\sqrt{H}}$, but their decay will be dramatically changed for the second and third resonance types.

2.Self-regulated biological transportation structures with general entropy dissipations, part I: the 1D case

Authors:Clarissa Astuto, Jan Haskovec, Peter Markowich, Simone Portaro

Abstract: We study self-regulating processes modeling biological transportation networks as presented in \cite{portaro2023}. In particular, we focus on the 1D setting for Dirichlet and Neumann boundary conditions. We prove an existence and uniqueness result under the assumption of positivity of the diffusivity $D$. We explore systematically various scenarios and gain insights into the behavior of $D$ and its impact on the studied system. This involves analyzing the system with a signed measure distribution of sources and sinks. Finally, we perform several numerical tests in which the solution $D$ touches zero, confirming the previous hints of local existence in particular cases.

3.On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

Authors:Kotaro Hisa, Mizuki Kojima

Abstract: We discuss the existence and nonexistence of nonnegative local and global-in-time solutions of the time-fractional problem \[ \partial_t^\alpha u -\Delta u = u^p,\quad t>0,\,\,\, x\in{\bf R}^N, \qquad u(0) = \mu \quad \mbox{in}\quad {\bf R}^N, \] where $N\geq1$, $0<\alpha<1$, $p>1$, and $\mu$ is a nonnegative Radon measure on ${\bf R}^N$. Here, $\partial_t^\alpha$ is the Caputo derivative of order $\alpha$. The corresponding usual equation $\partial_tu-\Delta u=u^p$ may not be globally or locally-in-time solvable, under certain critical situations. In contrast, the solvability of the time-fractional equation is guaranteed, under such situations. In this paper, we deduce necessary and sufficient conditions on the initial data $\mu$ for the solvability of this equation. As application, we describe the collapse of the global and local-in-time solvability for the time-fractional equation as $\alpha \to1-0$.

4.On asymptotic stability on a center hypersurface at the solition for even solutions of the NLKG when $2\ge p> \frac{5}{3}$

Authors:Scipio Cuccagna, Masaya Maeda, Federico Murgante, Stefano Scrobogna

Abstract: We extend the result M. Kowalczyk, Y. Martel, C. Mu\~noz, JEMS 2022, on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power Nonlinear Klein Gordon Equation with $p>3$, to the case $2\ge p> \frac{5}{3}$.

5.Analysis of a dilute polymer model with a time-fractional derivative

Authors:Marvin Fritz, Endre Süli, Barbara Wohlmuth

Abstract: We investigate the well-posedness of a coupled Navier-Stokes-Fokker-Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modelled by a stochastic process exhibiting power-law waiting time, in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modelled by a finitely extensible nonlinear elastic (FENE) dumbbell model, and the drag term in the Fokker--Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order $\alpha \in (\tfrac12,1)$, and derive an energy inequality satisfied by weak solutions.

6.Well-posedness and simulation of weak solutions to the time-fractional Fokker-Planck equation with general forcing

Authors:Marvin Fritz

Abstract: In this paper, we investigate the well-posedness of weak solutions to the time-fractional Fokker-Planck equation. Its dynamics is governed by anomalous diffusion, and we consider the most general case of space-time dependent forces. Consequently, the fractional derivatives appear on the right-hand side of the equation, and they cannot be brought to the left-hand side, which would have been preferable from an analytical perspective. For showing the model's well-posedness, we derive an energy inequality by considering nonstandard and novel testing methods that involve a series of convolutions and integrations. We close the estimate by a Henry-Gronwall-type inequality. Lastly, we propose a numerical algorithm based on a nonuniform L1 scheme and present some simulation results for various forces.

7.Simultaneous determination of initial value and source term for time-fractional wave-diffusion equations

Authors:Paola Loreti, Daniela Sforza, Masahiro Yamamoto

Abstract: We consider initial boundary value problems for time fractional diffusion-wave equations: $$ d_t^{\alpha} u = -Au + \mu(t)f(x) $$ in a bounded domain where $\mu(t)f(x)$ describes a source and $\alpha \in (0,1) \cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $\mu(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $\Omega$, $\mu$ in $(0,T)$, and initial values of $u$ by data $u\vert_{\omega\times (T_1,T_2)}$, provided that the order $\alpha$ does not belong to a countably infinite set in $(0,1) \cup (1,2)$ which is characterized by $\mu$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.

8.Convergence to equilibrium for a degenerate McKean-Vlasov Equation

Authors:Manh Hong Duong, Amit Einav

Abstract: In this work we study the convergence to equilibrium for a (potentially) degenerate nonlinear and nonlocal McKean-Vlasov equation. We show that the solution to this equation is related to the solution of a linear degenerate and/or defective Fokker-Planck equation and employ recent sharp convergence results to obtain an easily computable (and many times sharp) rates of convergence to equilibrium for the equation in question.

9.Nonlinear fractional equations in the Heisenberg group

Authors:Giampiero Palatucci, Mirco Piccinini

Abstract: We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order $(s,p)$, with summability exponent $p \in (1,\infty)$ and differentiability exponent $s\in (0,1)$, whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

10.Global Compactness, subcritical approximation of the Sobolev quotient, and a related concentration result in the Heisenberg group

Authors:Giampiero Palatucci, Mirco Piccinini, Letizia Temperini

Abstract: We investigate some effects of the lack of compactness in the critical Sobolev embedding in the Heisenberg group.

11.On the kinetic description of the objective molecular dynamics

Authors:Richard D. James, Kunlun Qi, Li Wang

Abstract: In this paper, we develop a multiscale hierarchy framework for objective molecular dynamics (OMD), a reduced order molecular dynamics with a certain symmetry, that connects it to the statistical kinetic equation, and the macroscopic hydrodynamic model. In the mesoscopic regime, we exploit two interaction scalings that lead, respectively, to either a mean-field type or to a Boltzmann type equation. It turns out that, under the special symmetry of OMD, the mean-field scaling results in vastly simplified dynamics that extinguishes the underlying molecular interaction rule, whereas the Boltzmann scaling yields a meaningful reduced model called the homo-energetic Boltzmann equation. At the macroscopic level, we derive the corresponding Euler and Navier-Stokes systems by conducting a detailed asymptotic analysis. The symmetry again significantly reduces the complexity of the resulting hydrodynamic systems.

12.Remarks on the linear wave equation

Authors:John M. Ball

Abstract: We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.

1.Massive Thirring Model: Inverse Scattering and Soliton Resolution

Authors:Cheng He, Jiaqi Liu, Changzheng Qu

Abstract: In this paper the long-time dynamics of the massive Thirring model is investigated. Firstly the nonlinear steepest descent method for Riemann-Hilbert problem is explored to obtain the soliton resolution of the solutions to the massive Thirring model whose initial data belong to some weighted-Sobolev spaces. Secondly, the asymptotic stability of multi-solitons follow as a corollary. The main difficulty in studying the massive Thirring model through inverse scattering is that the corresponding Lax pair has singularities at the origin and infinity. We overcome this difficulty by making use of two transforms that separate the singularities.

2.On a mathematical model for cancer invasion with repellent pH-taxis and nonlocal intraspecific interaction

Authors:Maria Eckardt, Christina Surulescu

Abstract: Starting from a mesoscopic description of cell migration and intraspecific interactions we obtain by upscaling an effective reaction-difusion-taxis equation for the cell population density involving spatial nonlocalities in the source term and biasing its motility and growth behavior according to environmental acidity. We prove global existence, uniqueness, and boundedness of a nonnegative solution to a simplified version of the coupled system describing cell and acidity dynamics. A 1D study of pattern formation is performed. Numerical simulations illustrate the qualitative behavior of solutions.

3.Initial-boundary value problem for 2D temperature-dependent tropical climate model

Authors:Jitao Liu, Yunxiao Zhao, Shasha Wang

Abstract: It is well known that the tropical climate model is an important model to describe the interaction of large scale flow fields and precipitation in the tropical atmosphere. In this paper, we address the issue of global well-posedness for 2D temperature-dependent tropical climate model in a smooth bounded domain. Through classical energy estimates and De Giorgi-Nash-Moser iteration method, we obtain the global existence and uniqueness of strong solution in classical energy spaces. Compared with Cauchy problem, we establish more delicate a priori estimates with exponential decay rates. To the best of our knowledge, this is the first result concerning the global well-posedness for the initial-boundary value problem in 2D tropical climate model.

4.Long-Timescale Soliton Dynamics in the Korteweg-de Vries Equation with Multiplicative Translation-Invariant Noise

Authors:Rik W. S. Westdorp, Hermen Jan Hupkes

Abstract: This paper studies the behavior of solitons in the Korteweg-de Vries equation under the influence of multiplicative noise. We introduce stochastic processes that track the amplitude and position of solitons based on a rescaled frame formulation and stability properties of the soliton family. We furthermore construct tractable approximations to the stochastic soliton amplitude and position which reveal their leading-order drift. We find that the statistical properties predicted by our method agree well with numerical evidence.

5.Gradient Riesz potential estimates for a general class of measure data quasilinear systems

Authors:Iwona Chlebicka, Minhyun Kim, Marvin Weidner

Abstract: We study the gradient regularity of solutions to measure data elliptic systems with Uhlenbeck-type structure and Orlicz growth. For any bounded Borel measure, pointwise estimates for the gradient of solutions are provided in terms of the truncated Riesz potential. This allows us to show a precise transfer of regularity from data to solutions on various scales.

6.The Cheeger constant as limit of Sobolev-type constants

Authors:Grey Ercole

Abstract: Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}:u\in W_{0}^{1,p}(\Omega)\text{ \ and \ }\int_{\Omega}\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} . \] We prove that if $\lim_{p\rightarrow1^{+}}q(p)=1,$ then $\lim_{p\rightarrow 1^{+}}\lambda_{p,q(p)}=h(\Omega)$, where $h(\Omega)$ denotes the Cheeger constant of $\Omega.$ Moreover, we study the behavior of the minimizers $u_{p,q(p)}$ as $p\rightarrow1^{+}.$ Our results extend those obained by Kawohl and Fridman (2003) for $q(p)=p.$

7.Schrödinger-Maxwell equations driven by mixed local-nonlocal operators

Authors:Nicolò Cangiotti, Maicol Caponi, Alberto Maione, Enzo Vitillaro

Abstract: In this paper we prove existence of solutions to Schr\"odinger-Maxwell type systems involving mixed local-nonlocal operators. Two different models are considered: classical Schr\"odinger-Maxwell equations and Schr\"odinger-Maxwell equations with a coercive potential, and the main novelty is that the nonlocal part of the operator is allowed to be nonpositive definite according to a real parameter. We then provide a range of parameter values to ensure the existence of solitary standing waves, obtained as Mountain Pass critical points for the associated energy functionals.

1.Long-time asymptotics and the radiation condition for linear evolution equations on the half-line with time-periodic boundary conditions

Authors:Yifeng Mao, Dionyssios Mantzavinos, Mark A. Hoefer

Abstract: The large time $t$ asymptotics for scalar, constant coefficient,linear, third order, dispersive equations are obtained for asymptotically time-periodic Dirichlet boundary data and zero initial data on the half-line modeling a wavemaker acting upon an initially quiescent medium. The asymptotic Dirichlet-to-Neumann (D-N) map is constructed by expanding upon the recently developed $Q$-equation method. The D-N map is proven to be unique if and only if the radiation condition that selects the unique wavenumber branch of the dispersion relation for a sinusoidal, time-dependent boundary condition holds: (i) for frequencies in a finite interval, the wavenumber is real and corresponds to positive group velocity, (ii) for frequencies outside the interval, the wavenumber is complex with positive imaginary part. For fixed spatial location $x$, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time-periodic wave. Uniform-in-$x$ asymptotic solutions for the physical cases of the linearized Korteweg-de Vries and Benjamin-Bona-Mahony (BBM) equations are obtained via integral asymptotics. The linearized BBM asymptotics are found to quantitatively agree with viscous core-annular fluid experiments.

2.A new approach for stability analysis of 1-D wave equation with time delay

Authors:Shijie Zhou, Hongyinping Feng, Zhiqiang Wang

Abstract: In our manuscript, we develop a new approach for stability analysis of one-dimensional wave equation with time delay. The major contribution of our work is to develop a new method for spectral analysis. We derive sufficient and necessary conditions for the feedback gain and time delay which guarantee the exponential stability of the closed-loop system. Comparing with similar conditions developed in the past literatures, we discuss all the situation when the time delay is positive, including when it is irrational. We prove that the exponential stability can be achieved if and only if the time delay is an even number. We also get the general formula term of the stability region of the coupling gain for different even multiples of time delay, and from this we easily obtain the shrink of the stability region as time delay increases. In addition, we explore the impact of slight perturbations in time delay on high frequency robustness.

3.Competing effects in fourth-order aggregation-diffusion equations

Authors:José Antonio Carrillo, Antonio Esposito, Carles Falcó, Alejandro Fernández-Jiménez

Abstract: We give sharp conditions for global in time existence of gradient flow solutions to a Cahn-Hilliard-type equation, with backwards second order degenerate diffusion, in any dimension and for general initial data. Our equation is the 2-Wasserstein gradient flow of a free energy with two competing effects: the Dirichlet energy and the power-law internal energy. Homogeneity of the functionals reveals critical regimes that we analyse. Sharp conditions for global in time solutions, constructed via the minimising movement scheme, also known as JKO scheme, are obtained. Furthermore, we study a system of two Cahn-Hilliard-type equations exhibiting an analogous gradient flow structure.

4.Boundedness through nonlocal dampening effects in a fully parabolic chemotaxis model with sub and superquadratic growth

Authors:Yutaro Chiyo, Fatma Ga mze D Düzgün, Silvia Frassu, Giuseppe Viglialoro

Abstract: This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough.

5.Partial regularity for degenerate parabolic systems with general growth via caloric approximations

Authors:Jihoon Ok, Giovanni Scilla, Bianca Stroffolini

Abstract: We establish a partial regularity result for solutions of parabolic systems with general $\varphi$-growth, where $\varphi$ is an Orlicz function. In this setting we can develop a unified approach that is independent of the degeneracy of system and relies on two caloric approximation results: the $\varphi$-caloric approximation, which was introduced in Diening, Schwarzacher, Stroffolini and Verde (2017) (arXiv:1606.01706), and an improved version of the \mathcal{A}-caloric approximation, which we prove without using the classical compactness method.

6.Asymptotic approach to singular solutions for the CR Yamabe equation, and a conjecture by H. Brezis and L. A. Peletier in the Heisenberg group

Authors:Giampiero Palatucci, Mirco Piccinini

Abstract: We investigate some effects of the lack of compactness in the critical Sobolev embedding by proving that a famous conjecture of Brezis & Peletier (Essays in honor of Ennio De Giorgi -- Progr. Differ. Equ. Appl. 1989) does still hold in the Heisenberg framework: optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point which can be localized via the Green function associated to the involved domain and in clear accordance with the underlying sub-Riemannian geometry -- and consequently a new suitable definition of domains geometrical regular near their characteristic set is given. In order to achieve the aforementioned result, we need to combine proper estimates and tools to attack the related CR Yamabe equation (Jerison & Lee, J. Diff. Geom. 1987) with novel feasible ingredients in PDEs and Calculus of Variations which also aim to constitute general independent results in the Heisenberg framework, as e.g. a fine asymptotic control of the optimal functions via the Jerison & Lee extremals realizing the equality in the critical Sobolev inequality (J. Amer. Math. Soc. 1988).

1.Diffusive Limit of the Vlasov-Poisson-Boltzmann System for the Full Range of Cutoff Potentials

Authors:Weijun Wu, Fujun Zhou, Yongsheng Li

Abstract: Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff soft potentials $-3<\gamma<0$ in the perturbative framework around global Maxwellian still remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay, we solve this problem for the full range of cutoff potentials $-3<\gamma\leq 1$. The core of this approach lies in the interplay between the velocity weighted $H_{x,v}^2$ energy estimate with time decay and the time-velocity weighted $W_{x,v}^{2,\infty}$ estimate with time decay for the Vlasov-Poisson-Boltzmann system, which leads to the uniform estimate with respect to the Knudsen number $\varepsilon\in (0,1]$ globally in time. As a result, global strong solution is constructed and incompressible Navier-Stokes-Fourier-Poisson limit is rigorously justified for both hard and soft potentials. Meanwhile, this uniform estimate with respect to $\varepsilon\in (0,1]$ also yields optimal $L^2$ time decay rate and $L^\infty$ time decay rate for the Vlasov-Poisson-Boltzmann system and its incompressible Navier-Stokes-Fourier-Poisson limit. This newly introduced weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay is flexible and robust, as it can deal with both optimal time decay problems and hydrodynamic limit problems in a unified framework for the Boltzmann equation as well as the Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials. It is also expected to shed some light on the more challenging hydrodynamic limit of the Landau equation and the Vlasov-Poisson-Landau system.

2.A flow method for curvature equations

Authors:Shanwei Ding, Guanghan Li

Abstract: We consider a general curvature equation $F(\kappa)=G(X,\nu(X))$, where $\kappa$ is the principal curvature of the hypersurface $M$ with position vector $X$. It includes the classical prescribed curvature measures problem and area measures problem. However, Guan-Ren-Wang \cite{GRW} proved that the $C^2$ estimate fails usually for general function $F$. Thus, in this paper, we pose some additional conditions of $G$ to get existence results by a suitably designed parabolic flow. In particular, if $F=\sigma_{k}^\frac{1}{k}$ for $\forall 1\le k\le n-1$, the existence result has been derived in the famous work \cite{GLL} with $G=\psi(\frac{X}{|X|})\langle X,\nu\rangle^{\frac1k}{|X|^{-\frac{n+1}{k}}}$. This result will be generalized to $G=\psi(\frac{X}{|X|})\langle X,\nu\rangle^\frac{{1-p}}{k}|X|^\frac{{q-k-1}}{k}$ with $p>q$ for arbitrary $k$ by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.

3.The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

Authors:Francescantonio Oliva, Francesco Petitta, Sergio Segura de León

Abstract: In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{cases} u -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f & \text{in}\;\Omega, \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary. In particular we explore the regularizing effect given by the absorption term in order to get a unique solutions for data $f$ merely belonging to $L^1(\Omega)$ and with no smallness assumptions. We also prove a sharp boundedness result for solutions for data in $L^{N}(\Omega)$.

4.On nonlinear Landau damping and Gevrey regularity

Authors:Christian Zillinger

Abstract: In this article we study the problem of nonlinear Landau damping for the Vlasov-Poisson equations on the torus. As our main result we show that for perturbations initially of size $\epsilon>0$ and time intervals $(0,\epsilon^{-N})$ one obtains nonlinear stability in regularity classes larger than Gevrey $3$, uniformly in $\epsilon$. As a complementary result we construct families of Sobolev regular initial data which exhibit nonlinear Landau damping. Our proof is based on the methods of Grenier, Nguyen and Rodnianski.

1.KPP transition fronts in a one-dimensional two-patch habitat

Authors:François Hamel I2M, Mingmin Zhang IMT

Abstract: This paper is concerned with the existence of transition fronts for a one-dimensional twopatch model with KPP reaction terms. Density and flux conditions are imposed at the interface between the two patches. We first construct a pair of suitable super-and subsolutions by making full use of information of the leading edges of two KPP fronts and gluing them through the interface conditions. Then, an entire solution obtained thanks to a limiting argument is shown to be a transition front moving from one patch to the other one. This propagating solution admits asymptotic past and future speeds, and it connects two different fronts, each associated with one of the two patches. The paper thus provides the first example of a transition front for a KPP-type two-patch model with interface conditions.

2.$L^2$-growth property for wave equations with higher derivative terms

Authors:Ryo Ikehata, Xiaoyan Li

Abstract: We consider the Cauchy problems in the whole space for wave equations with higher derivative terms. We derive sharp growth estimates of the $L^2$-norm of the solution itself in the case of the space 1, 2 dimensions. By imposing the weighted $L^1$-initial velocity, we can get the lower and upper bound estimates of the solution itself. In three or more dimensions, we observe that the $L^2$-growth behavior of the solution never occurs in the ($L^2 \cap L^1$)-framework of the initial data.

3.Global weak solutions of an initial-boundary value problem on a half-line for the higher order nonlinear Schrödinger equation

Authors:Andrei V. Faminskii

Abstract: An initial-boundary value problem with one boundary condition is considered for the higher order nonlinear Schr\"odinger equation. It is assumed that either the boundary condition is homogeneous or the nonlinearity in the equation is quadratic. Results on existence, uniqueness and continuous dependence on input data of global weak solutions are obtained.

4.Homogenization of non-autonomous evolution problems for convolution type operators in random media

Authors:Andrey Piatnitski, Elena Zhizhina

Abstract: We study homogenization problem for non-autonomous parabolic equations of the form $\partial_t u=L(t)u$ with an integral convolution type operator $L(t)$ that has a non-symmetric jump kernel which is periodic in spatial variables and stationary random in time. We show that the homogenized equation is a SPDE with a finite dimensional multiplicative noise.

5.The gyrokinetic limit for the Plasma-Charge model in $\mathbb{R}^2$

Authors:Jingpeng Wu

Abstract: In this article, we investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system with point charges. We show that the solution converges to a measure-valued solution of the Euler equation with a defect measure, which extends the results in [Miot, Nonlinearity, 2019] to the case of multi-point charges and removes the small condition $\sup_{0<\varepsilon<1}\|f_{\varepsilon}^0\|_{L^1}<1$.

6.Weak solutions to gradient flows of functionals with inhomogeneous growth in metric spaces

Authors:Wojciech Górny

Abstract: We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solutions to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their existence and uniqueness and show that they are also variational solutions; in particular, this is an existence result for variational solutions. Then, we apply this technique in the case of a gradient flow of a functional with inhomogeneous growth.

7.Asymptotic behavior and life-span estimates for the damped inhomogeneous nonlinear Schrödinger equation

Authors:Lassaad Aloui, Sirine Jbari, Slim Tayachi

Abstract: We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schr\"odinger equation $ i\partial_tu+\Delta u+\mu|x|^{-b}|u|^{\alpha}u+iau=0$, $\mu \in\mathbb{C} $, $b>0$, $a \in \mathbb{C}$ such that $\Re \textit{e}(a) \geq 0$, $\alpha>0$. We establish lower and upper bound estimates of the life-span. In particular for $a\geq 0$, we obtain explicit values $a_*,\; a^*$ such that if $a<a_*$ then blow up occurs, while for $a>a^*,$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$

8.Existence and uniqueness of solutions to some anisotropic elliptic equations with a singular convection term

Authors:Giuseppina di Blasio, Filomena Feo, Gabriella Zecca

Abstract: We prove the existence and uniqueness of weak solutions to a class of anisotropic elliptic equations with coefficients of convection term belonging to some suitable Marcinkiewicz spaces. Some useful a priori estimates and regularity results are also derived.

9.Relaxation of one-dimensional nonlocal supremal functionals in the Sobolev setting

Authors:Andrea Torricelli, Elvira Zappale

Abstract: We provide necessary and sufficient conditions on the density $W:\mathbb R^d\times\mathbb R ^d\to\mathbb R$ in order to ensure the sequential weak* lower semicontinuity of the functional $J: W^{1,\infty}(I;\mathbb R^d)\to \mathbb R$, defined as \begin{align*} J(u):=ess\,sup_{I\times I}W(u'(x), u'(y)), \end{align*} when $I$ is an open and bounded interval of $\mathbb R$. We also show that, when $d=1$, the lower semicontinuous envelope of $I$ in general can be obtained by replacing $W$ by its separately level convex envelope.

10.Matrix displacement convexity along density flows

Authors:Yair Shenfeld

Abstract: A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schr\"odinger equations. Matrix displacement convexity is stronger than the classical notions of displacement convexity, and its verification (formal and rigorous) relies on matrix differential inequalities along the density flows. The matrical nature of these differential inequalities upgrades dimensional functional inequalities to their intrinsic dimensional counterparts, thus improving on many classical results. Applications include turnpike properties, evolution variational inequalities, and entropy growth bounds, which capture the behavior of the density flows along different directions in space.

11.Finite- and Infinite-Time Cluster Formation for Alignment Dynamics on the Real Line

Authors:Trevor M. Leslie, Changhui Tan

Abstract: We show that the locations where finite- and infinite-time clustering occurs for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated to a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated to a scalar balance law formulation of the system.

12.On the Poincaré inequality on open sets in $\mathbb{R}^n$

Authors:A. -K. Gallagher

Abstract: We show that the Poincar\'{e} inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian.

1.Exact Global Control of Small Divisors in Rational Normal Form

Authors:Jianjun Liu, Duihui Xiang

Abstract: Rational normal form is a powerful tool to deal with Hamiltonian partial differential equations without external parameters. In this paper, we build rational normal form with exact global control of small divisors. As an application to nonlinear Schr\"{o}dinger equations in Gevrey spaces, we prove sub-exponentially long time stability results for generic small initial data.

2.Optimal regularity of the thin obstacle problem by an epiperimetric inequality

Authors:Matteo Carducci

Abstract: The key point to prove the optimal $C^{1,\frac12}$ regularity of the thin obstacle problem is that the frequency at a point of the free boundary $x_0\in\Gamma(u)$, say $N^{x_0}(0^+,u)$, satisfies the lower bound $N^{x_0}(0^+,u)\ge\frac32$. In this paper we show an alternative method to prove this estimate, using an epiperimetric inequality for negative energies $W_\frac32$. It allows to say that there are not $\lambda-$homogeneous global solutions with $\lambda\in (1,\frac32)$, and by this frequancy gap, we obtain the desired lower bound, thus a new self contained proof of the optimal regularity.

3.Thin Film Equations with Nonlinear Deterministic and Stochastic Perturbations

Authors:Oleksiy Kapustyan, Olha Martynyuk, Oleksandr Misiats, Oleksandr Stanzhytskyi

Abstract: In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovych noise, as well as nonlinear colored Wiener noise. By means of Trotter-Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.

4.On the stability of a double porous elastic system with visco-porous dampings

Authors:Ahmed Keddi, Aicha Nemsi, Abdelfeteh Fareh

Abstract: In this paper we consider a one dimensional elastic system with double porosity structure and with frictional damping in both porous equations. We introduce two stability numbers $\chi_{0}$ and $\chi_{1}$ and prove that the solution of the system decays exponentially provided that $\chi_{0}=0$ and $\chi_{1}\neq0.$ Otherwise, we prove the lack of exponential decay. Our results improve the results of \cite{Bazarra} and \cite{Nemsi}.

5.On the Fisher infinitesimal model without variability

Authors:Cécile Taing LMA-Poitiers, Amic Frouvelle CEREMADE

Abstract: We study the long-time behavior of solutions to a model of sexual populations structured in phenotypes. The model features a nonlinear integral reproduction operator derived from the Fisher infinitesimal operator and a trait-dependent selection term. The reproduction operator describes here the inheritance of the mean parental traits to the offspring without variability. We show that, under assumptions on the growth of the selection rate, Dirac masses are stable around phenotypes for which the difference between the selection rate and its minimum value is less than 1 2. Moreover, we prove the convergence in some Fourier-based distance of the centered and rescaled solution to a stationary profile under some conditions on the initial moments of the solution.

6.From the Brunn-Minkowski inequality to a class of generalized Poincaré-type inequality for torsional rigidity

Authors:Niufa Fang, Jinrong Hu, Leina Zhao

Abstract: In this paper, we put forward an argument which leads from the Brunn-Minkowski inequality to a class of Poincar\'{e}-type inequality for torsional rigidity on the boundary of a convex body of class $C^{2}_{+}$ in $\rnnn$.

7.The zero dispersion limit for the Benjamin--Ono equation on the line

Authors:Patrick Gérard

Abstract: We identify the zero dispersion limit of a solution of the Benjamin--Ono equation on the line corresponding to every initial datum in $L^2(\R)\cap L^\infty(\R )$. We infer a maximum principle and a local smoothing property for this limit. The proof is based on an explicit formula for the Benjamin--Ono equation and on the combination of calculations in the special case of rational initial data, with approximation arguments. We also investigate the special case of an initial datum equal to the characteristic function of a finite interval, and prove the lack of semigroup property for this zero dispersion limit.

8.On the exact boundary controllability of semilinear wave equations

Authors:Sue Claret, Jérôme Lemoine, Arnaud Münch

Abstract: We address the exact boundary controllability of the semilinear wave equation $\partial_{tt}y-\Delta y + f(y)=0$ posed over a bounded domain $\Omega$ of $\mathbb{R}^d$. Assuming that $f$ is continuous and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f(r)\vert /(\vert r\vert \ln^p\vert r\vert)\leq \beta$ for some $\beta$ small enough and some $p\in [0,3/2)$, we apply the Schauder fixed point theorem to prove the uniform controllability for initial data in $L^2(\Omega)\times H^{-1}(\Omega)$. Then, assuming that $f$ is in $\mathcal{C}^1(\mathbb{R})$ and satisfies the condition $\limsup_{\vert r\vert\to \infty} \vert f^\prime(r)\vert/\ln^p\vert r\vert\leq \beta$, we apply the Banach fixed point theorem and exhibit a strongly convergent sequence to a state-control pair for the semilinear equation.

9.Graph Limit for Interacting Particle Systems on Weighted Random Graphs

Authors:Nathalie Ayi SU, LJLL, ANGE, MAMBA, Nastassia Pouradier Duteil SU, MAMBA, LJLL

Abstract: In this article, we study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We prove that as the number of particles tends to infinity, the finite-dimensional particle system converges in probability to the solution of a deterministic graph-limit equation, in which the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also study interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We show that these systems converge to the same graph-limit equation, in which the interaction is prescribed by a constant-in-time graphon.

10.Schwartz regularity of differential operators on the cylinder

Authors:André Pedroso Kowacs

Abstract: This article presents an investigation of global properties of a class of differential operators on $\T^1\times\R$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus and partial Fourier transform in Euclidean space. By examining the behavior of the mixed Fourier coefficients, we obtain necessary and sufficient conditions for the Schwartz global hypoellipticity of this class of differential operators, as well as conditions for the Schwartz global solvability of said operators.

11.On the transverse stability of smooth solitary waves in a two-dimensional Camassa-Holm equation

Authors:Anna Geyer, Yue Liu, Dmitry E. Pelinovsky

Abstract: We consider the propagation of smooth solitary waves in a two-dimensional generalization of the Camassa--Holm equation. We show that transverse perturbations to one-dimensional solitary waves behave similarly to the KP-II theory. This conclusion follows from our two main results: (i) the double eigenvalue of the linearized equations related to the translational symmetry breaks under a transverse perturbation into a pair of the asymptotically stable resonances and (ii) small-amplitude solitary waves are linearly stable with respect to transverse perturbations.

12.Estimates on the Neumann and Steklov principal eigenvalues of collapsing domains

Authors:Paolo Acampora, Vincenzo Amato, Emanuele Cristoforoni

Abstract: We investigate the relationship between the Neumann and Steklov principal eigenvalues emerging from the study of collapsing convex domains in $\mathbb{R}^2$. Such a relationship allows us to give a partial proof of a conjecture concerning estimates of the ratio of the former to the latter: we show that thinning triangles maximize the ratio among convex thinning sets, while thinning rectangles minimize the ratio among convex thinning with some symmetry property.

13.The best approximation of a given function in $L^2$-norm by Lipschitz functions with gradient constraint

Authors:Stefano Buccheri, Tommaso Leonori, Julio D. Rossi

Abstract: The starting point of this paper is the study of the asymptotic behavior, as $p\to\infty$, of the following minimization problem $$ \min\left\{\frac1{p}\int|\nabla v|^{p}+\frac12\int(v-f)^2 \,, \quad \ v\in W^{1,p} (\Omega)\right\}. $$ We show that the limit problem provides the best approximation, in the $L^2$-norm, of the datum $f$ among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover such approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the $(N-1)$-Hausdorff measure of the jump set of the function.

14.On the strong maximum principle for fully nonlinear parabolic equations of second order

Authors:Alessandro Goffi

Abstract: We provide a proof of strong maximum and minimum principles for fully nonlinear uniformly parabolic equations of second order. The approach is of parabolic nature, slightly differs from the earlier one proposed by L. Nirenberg and does not exploit the parabolic Harnack inequality.

15.Quantitative and qualitative properties for Hamilton-Jacobi PDEs via the nonlinear adjoint method

Authors:Fabio Camilli, Alessandro Goffi, Cristian Mendico

Abstract: We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from $L^p$-rates of convergence for the vanishing viscosity approximation and homogenization to regularizing effects for the Cauchy problem in the whole Euclidean space and Liouville-type theorems. Our approach is based on duality techniques \`a la Evans and a careful study of advection-diffusion equations. The optimality of the results is discussed by several examples.

16.Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime

Authors:Giovanni Di Fratta, Valeriy Slastikov, Arghir Zarnescu

Abstract: The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields $H^1(S,T)$, where $S$ and $T$ are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface $T$ is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.

1.Dimensional Reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals

Authors:Giacomo Canevari, Antonio Segatti

Abstract: In this paper we discuss the behavior of the Oseen-Frank model for nematic liquid crystals in the limit of vanishing thickness. More precisely, in a thin slab~$\Omega\times (0,h)$ with~$\Omega\subset \mathbb{R}^2$ and $h>0$ we consider the one-constant approximation of the Oseen-Frank model for nematic liquid crystals. We impose Dirichlet boundary conditions on the lateral boundary and weak anchoring conditions on the top and bottom faces of the cylinder~$\Omega\times (0,h)$. The Dirichlet datum has the form $(g,0)$, where $g\colon\partial\Omega\to \mathbb{S}^1$ has non-zero winding number. Under appropriate conditions on the scaling, in the limit as~$h\to 0$ we obtain a behavior that is similar to the one observed in the asymptotic analysis of the two-dimensional Ginzburg-Landau functional. More precisely, we rigorously prove the emergence of a finite number of defect points in $\Omega$ having topological charges that sum to the degree of the boundary datum. Moreover, the position of these points is governed by a Renormalized Energy, as in the seminal results of Bethuel, Brezis and H\'elein.

2.Approximating a continuously stratified hydrostatic system by the multi-layer shallow water system

Authors:Mahieddine Adim

Abstract: In this article we consider the multi-layer shallow water system for the propagation of gravity waves in density-stratified flows, with additional terms introduced by the oceanographers Gent and McWilliams in order to take into account large-scale isopycnal diffusivity induced by small-scale unresolved eddies. We establish a bridge between the multi-layer shallow water system and the corresponding system for continuously stratified flows, that is the incompressible Euler equations with eddy-induced diffusivity under the hydrostatic approximation. Specifically we prove that, under an assumption of stable stratification, sufficiently regular solutions to the incompressible Euler equations can be approximated by solutions to multi-layer shallow water systems as the number of layers, $N$, increases. Moreover, we provide a convergence rate of order $1/N^2$. A key ingredient in the proof is a stability estimate for the multi-layer system which relies on suitable energy estimates mimicking the ones recently established by Bianchini and Duch\^ene on the continuously stratified system. This requires to compile a dictionary that translates continuous operations (differentiation, integration, etc.) into corresponding discrete operations.

3.Bifurcation domains from any high eigenvalue for an overdetermined elliptic problem

Authors:Guowei Dai, Yong Zhang

Abstract: Let $\lambda_k$ be the $k$-th eigenvalue of the zero-Dirichlet Laplacian on the unit ball for $k\in \mathbb{N^+}$. We prove the existence of $k$ smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a sign-changing solution with changing the sign by $k-1$ times. This nonsymmetric sign-changing solution can be seen as the perturbation of the eigenfunction corresponding to $\lambda_k$ with $k\geq2$. The main contribution of the paper is to provide some counterexamples to the Berenstein conjecture on unbounded domain.

4.The five gradients inequality on differentiable manifolds

Authors:S. Di Marino, S. Murro, E. Radici

Abstract: The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds with Ricci curvature bounded from below.

5.The Global well-posedness for Klein-Gordon-Hartree equation in modulation spaces

Authors:Divyang G. Bhimani

Abstract: Modulation spaces have received considerable interest recently as it is the natural function spaces to consider low regularity Cauchy data for several nonlinear evolution equations. We establish global well-posedness for 3D Klein-Gordon-Hartree equation $$u_{tt}-\Delta u+u + ( |\cdot|^{-\gamma} \ast |u|^2)u=0$$ with initial data in modulation spaces $M^{p, p'}_1 \times M^{p,p}$ for $p\in \left(2, \frac{54 }{27-2\gamma} \right),$ $2<\gamma<3.$ We implement Bourgain's high-low frequency decomposition method to establish global well-posedness, which was earlier used for classical Klein-Gordon equation. This is the first result on low regularity for Klein-Gordon-Hartree equation with large initial data in modulation spaces (which do not coincide with Sobolev spaces).

6.The lack of exponential stability of a Bresse system subjected only to two dampings

Authors:Virginie Régnier, Waël Youssef

Abstract: In this paper, we study the indirect boundary stabilization of a Bresse system with only two dissipation laws. This system, which models the dynamics of a beam, is a hyperbolic system with three wave speeds. We study the asymptotic behaviour of the eigenvalues and of the eigenvectors of the underlying operator in the case of three distinct wave velocities which is not physically relevant. Since the imaginary axis is proved to be an asymptote for one family of eigenvalues, the stability can not be exponential. Of course, this paper is only interesting from a mathematical point of view.

7.Global well-posedness and optimal time decay rates of solutions to the pressureless Euler-Navier-Stokes system

Authors:Feimin Huang, Houzhi Tang, Weiyuan Zou

Abstract: In this paper, we present a new framework for the global well-posedness and large-time behavior of a two-phase flow system, which consists of the pressureless Euler equations and incompressible Navier-Stokes equations coupled through the drag force. To overcome the difficulties arising from the absence of the pressure term in the Euler equations, we establish the time decay estimates of the high-order derivative of the velocity to obtain uniform estimates of the fluid density. The upper bound decay rates are obtained by designing a new functional and the lower bound decay rates are achieved by selecting specific initial data. Moreover, the upper bound decay rates are the same order as the lower one. Therefore, the time decay rates are optimal. When the fluid density in the pressureless Euler flow vanishes, the system is reduced into an incompressible Navier-Stokes flow. In this case, our works coincide with the classical results by Schonbek \cite{M.S3} [JAMS,1991], which can be regarded as a generalization from a single fluid model to the two-phase fluid one.

8.Improved Spectral Cluster Bounds for Orthonormal Systems

Authors:Tianyi Ren, An Zhang

Abstract: We improve Frank-Sabin's work concerning the spectral cluster bounds for orthonormal systems at $p=\infty$, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from $[\lambda^{2}, (\lambda+1)^{2})$ to $[\lambda^{2}, (\lambda+\epsilon(\lambda))^{2})$, where $\epsilon(\lambda)$ is a function of $\lambda$ that goes to $0$ as $\lambda$ goes to $\infty$. In achieving this, we invoke the method developed by Bourgain-Shao-Sogge-Yao.

9.Generic Singularities for 2D Pressureless Flow

Authors:Alberto Bressan, Geng Chen, Shoujun Huang

Abstract: In this paper, we consider the Cauchy problem for pressureless gases in two space dimensions with generic smooth initial data (density and velocity). These equations give rise to singular curves, where the mass has positive density w.r.t.~1-dimensional Hausdorff measure. We observe that the system of equations describing these singular curves is not hyperbolic. For analytic data, local solutions are constructed using a version of the Cauchy-Kovalevskaya theorem. We then study the interaction of two singular curves, in generic position. Finally, for a generic initial velocity field, we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed.

10.Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

Authors:Lucia Scardia, Konstantinos Zemas, Caterina Ida Zeppieri

Abstract: In this paper we study the convergence of integral functionals with $q$-growth in a randomly perforated domain of $\mathbb R^n$, with $1<q<n$. Under the assumption that the perforations are small balls whose centres and radii are generated by a \emph{stationary short-range marked point process}, we obtain in the critical-scaling limit an averaged analogue of the nonlinear capacitary term obtained by Ansini and Braides in the deterministic periodic case \cite{Ansini-Braides}. In analogy to the random setting introduced by Giunti, H\"ofer, and Vel\'azquez \cite{Giunti-Hofer-Velasquez} to study the Poisson equation, we only require that the random radii have finite $(n-q)$-moment. This assumption on the one hand ensures that the expectation of the nonlinear $q$-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

11.On the existence, regularity and uniqueness of $L^p$-solutions to the steady-state 3D Boussinesq system in the whole space

Authors:Oscar Jarrin

Abstract: We consider the steady-state Boussinesq system in the whole three-dimensional space, with the action of external forces and the gravitational acceleration. First, for $3<p\leq +\infty$ we prove the existence of weak $L^p$-solutions. Moreover, within the framework of a slightly modified system, we discuss the possibly non-existence of $L^p-$solutions for $1\leq p \leq 3$. Then, we use the more general setting of the $L^{p,\infty}-$spaces to show that weak solutions and their derivatives are H\"older continuous functions, where the maximum gain of regularity is determined by the initial regularity of the external forces and the gravitational acceleration. As a bi-product, we get a new regularity criterion for the steady-state Navier-Stokes equations. Furthermore, in the particular homogeneous case when the external forces are equal to zero; and for a range of values of the parameter $p$, we show that weak solutions are not only smooth enough, but also they are identical to the trivial (zero) solution. This result is of independent interest, and it is also known as the Liouville-type problem for the steady-state Boussinesq system.

12.Weighted analytic regularity for the integral fractional Laplacian in polyhedra

Authors:Markus Faustmann, Carlo Marcati, Jens Markus Melenk, Christoph Schwab

Abstract: We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polytopal three-dimensional domains and with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides control of higher order derivatives.

13.Macroscopic estimate of the linear Boltzmann and Landau equations with Specular reflection boundary

Authors:Hongxu Chen, Chanwoo Kim

Abstract: In this short note, we prove an $L^6$-control of the macroscopic part of the linear Boltzmann and Landau equations. This result is an extension of the test function method of Esposito-Guo-Kim-Marra~\cite{EGKM}\cite{EGKM2} to the specular reflection boundary condition, in which we crucially used the Korn's inequality and the system of symmetric Poisson equations.

1.Remark on the ill-posedness for KdV-Burgers equation in Fourier amalgam spaces

Authors:Divyang G. Bhimani, Saikatul Haque

Abstract: We have established (a weak form of) ill-posedness for the KdV-Burgers equation on a real line in Fourier amalgam spaces $\widehat{w}_s^{p,q}$ with $s<-1$. The particular case $p=q=2$ recovers the result of L. Molinet and F. Ribaud [Int. Math. Res. Not., (2002), pp. 1979-2005]. The result is new even in Fourier Lebesgue space $\mathcal{F}L_s^q$ which corresponds to the case $p=q(\neq 2)$ and in modulation space $M_s^{2,q}$ which corresponds to the case $p=2,q\neq 2$.

2.Local data inverse problem for the polyharmonic operator with anisotropic perturbations

Authors:Sombuddha Bhattacharyya, Pranav Kumar

Abstract: In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

3.Global well-posedness of quadratic and subquadratic half wave Schr{ö}dinger equations

Authors:Xi Chen LMO

Abstract: We consider the following $p$ order nonlinear half wave Schr{\"o}dinger equations$$\left(i \partial\_{t}+\partial\_{x }^2-\left|D\_{y}\right|\right) u=\pm|u|^{p-1} u$$on the plane $\mathbb{R}^2$ with $1<p\leq 2$. This equation is considered as a toy model motivated by the study of solutions to weakly dispersive equations. In particular, the global well-posedness of this equation is a difficult problem due to the anisotropic property of the equation, with one direction corresponding to the half-wave operator, which is not dispersive. In this paper, we prove the global well-posedness of this equation in $L\_x^2 H\_y^s(\mathbb{R}^2) \cap H\_x^1 L\_y^2(\mathbb{R}^2)$($\frac{1}{2}\leq s \leq 1$), which is the first global well-posedness result of nonlinear half wave Schr{\"o}dinger equations. With the global well-posedness in the energy space for the focusing equation and the study on the solitary wave in [1], we complete the proof of the stability of the set of ground states. Moreover, we consider the half wave Schr{\"o}dinger equations on $\mathbb{R}\_{x}\times\mathbb{T}\_{y}$, which can also be called the wave guide Schr{\"o}dinger equations on $\mathbb{R}\_{x}\times\mathbb{T}\_{y}$. Using a similar approach in the analysis of the Cauchy problem of half wave Schr{\"o}dinger equations on $\mathbb{R}^2$, we can also deduce the global well-posedness of $p$ ($1<p\leq2$) order wave guide Schr{\"o}dinger equations in $L\_x^2 H\_y^s(\mathbb{R}\times\mathbb{T}) \cap H\_x^1 L\_y^2(\mathbb{R}\times\mathbb{T})$ with $\frac{1}{2}\leq s \leq 1$. With the global well-posedness in the energy space for the focusing wave guide Schr{\"o}dinger equations and the study on the ground states in [2], we complete the proof of the orbital stability of the ground states with small frequencies.

4.A nonlocal Gray-Scott model: well-posedness and diffusive limit

Authors:Philippe Laurençot LAMA, Christoph Walker

Abstract: Well-posedness in $L_\infty$ of the nonlocal Gray-Scott model is studied for integrable kernels, along with the stability of the semi-trivial spatially homogeneous steady state. In addition, it is shown that the solutions to the nonlocal Gray-Scott system converge to those to the classical Gray-Scott system in the diffusive limit.

5.On the topological size of the class of Leray solutions with algebraic decay

Authors:Lorenzo Brandolese UCBL, Cilon F Perusato UFPE, Paulo R Zingano UFRGS

Abstract: In 1987, Michael Wiegner in his seminal paper [17] provided an important result regarding the energy decay of Leray solutions $\boldsymbol u(\cdot,t)$ to the incompressible Navier-Stokes in $\mathbb{R}^{n}$: if the associated Stokes flows had their $\hspace{-0.020cm}L^{2}\hspace{-0.050cm}$ norms bounded by $O(1 + t)^{-\;\!\alpha} $ for some $ 0 < \alpha \leq (n+2)/4 $, then the same would be true of $ \|\hspace{+0.020cm} \boldsymbol u(\cdot,t) \hspace{+0.020cm} \|_{L^{2}(\mathbb{R}^{n})} $. The converse also holds, as shown by Z.Skal\'ak [15] and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier-Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two-sided bounds of the form $(1+t)^{-\alpha}\lesssim \| \boldsymbol u(\cdot,t)\|_{L^2(\mathbb{R}^n)} \lesssim (1+t)^{-\alpha}$.

6.Existence and stability of nonmonotone hydraulic shocks for the Saint Venant equations of inclined thin-film flow

Authors:Grégory Faye IMT, L. Miguel Rodrigues IRMAR, Zhao Yang AMSS, Kevin Zumbrun

Abstract: Extending work of Yang-Zumbrun for the hydrodynamically stable case of Froude number F < 2, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin-film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams are all rigorously obtained by mathematical analysis and explicit calculation.

7.On competing (p,q) -Laplacian Drichlet problem with unbounded weight

Authors:Josef Diblik, Marek Galewski, Igor Kossowski, Dumitru Motreanu

Abstract: We investigate the existence of generalized solutions to coercive competing system driven by the (p,q) -Laplacian with unbounded perturbation corresponding to the leading term in the differential operator and with convection depending on the gradient. Some abstract principle leading to the existence of generalized solutions is also derived basing on the Galerkin scheme.

8.Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups

Authors:Fares Essebei, Gianmarco Giovannardi, Simone Verzellesi

Abstract: In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.

9.Inversion of the Momenta Doppler Transform in two dimensions

Authors:Hiroshi Fujiwara, David Omogbhe, Kamran Sadiq, Alexandru Tamasan

Abstract: We introduce an analytic method which stably reconstructs both components of a (sufficiently) smooth, real valued, vector field compactly supported in the plane from knowledge of its Doppler transform and its first moment Doppler transform. The method of proof is constructive. Numerical inversion results indicate robustness of the method.

10.Decay at infinity for solutions to some fractional parabolic equations

Authors:Agnid Banerjee, Abhishek Ghosh

Abstract: For $s \in [1/2, 1)$, let $u$ solve $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^{n} \times [-T, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb R^n \times [-T, 0])} < \infty$. We show that if for some $0< c< T$ and $\epsilon>0$ $$\frac{1}{c} \int_{[-c,0]} u^2(x, t) dt \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb R^n,$$ then $u \equiv 0$ in $\mathbb R^{n} \times [-T, 0]$.

11.Rellich inequalities via Riccati pairs on model space forms

Authors:Sándor Kajántó

Abstract: We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kaj\'ant\'o, Krist\'aly, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.

12.Structured Population Models on Polish Spaces: A unified approach including Graphs, Riemannian Manifolds and Measure Spaces

Authors:Christian Düll, Piotr Gwiazda, Anna Marciniak-Czochra, Jakub Skrzeczkowski

Abstract: We provide well-posedness theory of a nonlinear structured population model on an abstract metric space which is only assumed to be separable and complete. To this end, we leverage the structure of the space of nonnegative Radon measures under the dual bounded Lipschitz distance (flat metric) which can be seen as a generalization of Wasserstein distance to nonconservative problems. Motivated by applications, the formulation of models on fairly general metric spaces allows us to consider processes on infinite-dimensional state spaces or on graphs combining discrete and continuous structures.

1.On long-time asymptotics to the nonlocal Lakshmanan -Porsezian-Daniel equation with step-like initial data

Authors:Wen-yu zhou, Shou-Fu Tian, Xiao-fan Zhang

Abstract: In this work, the nonlinear steepest descent method is employed to study the long-time asymptotics of the integrable nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation with a step-like initial data: $q_{0}(x)\rightarrow0$ as $x\rightarrow-\infty$ and $q_{0}(x)\rightarrow A$ as $x\rightarrow+\infty$, where $A$ is an arbitrary positive constant. Firstly, we develop a matrix Riemann-Hilbert (RH) problem to represent the Cauchy problem of LPD equation. To remove the influence of singularities in this RH problem, we introduce the Blaschke-Potapov (BP) factor, then the original RH problem can be transformed into a regular RH problem which can be solved by the parabolic cylinder functions. Besides, under the nonlocal condition with symmetries $x\rightarrow-x$ and $t\rightarrow t$, we give the asymptotic analyses at $x>0$ and $x<0$, respectively. Finally, we derive the long-time asymptotics of the solution $q(x,t)$ corresponding to the complex case of three stationary phase points generated by phase function.

2.A non-local traffic flow model for 1-to-1 junctions with buffer

Authors:F. A. Chiarello, J. Friedrich, S. GÖttlich

Abstract: In this paper, we introduce a non-local PDE-ODE traffic model devoted to the description of a 1-to-1 junction with buffer. We present a numerical method to approximate solutions and show a maximum principle which is uniform in the non-local interaction range. Further, we exploit the limit models as the support of the kernel tends to zero and to infinity. We compare them with other already existing models for traffic and production flow and present numerical examples.

3.The Helmholtz decomposition of a $BMO$ type vector field in general unbounded domains

Authors:Yoshikazu Giga, Zhongyang Gu

Abstract: We consider a space of $L^2$ vector fields with bounded mean oscillation whose ``normal'' component to the boundary is well-controlled. In the case when the dimension $n \geq 3$, we establish its Helmholtz decomposition for arbitrary uniformly $C^3$ domain in $\mathbf{R}^n$.

4.Linear stability of elastic 2-line solitons for the KP-II equation

Authors:Tetsu Mizumachi

Abstract: The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using the Darboux transformations, we study linear stability of 2-line solitons whose line solitons interact elastically each other. Time evolution of resonant continuous eigenfunctions is described by a damped wave equation in the transverse variable which is supposed to be a linear approximation of the local phase shifts of modulating line solitons.

5.Unique Determination of a Planar Screen in Electromagentic Inverse Scattering

Authors:Sadia Sadique, Petri Ola, Lassi Päivärint

Abstract: We show that the far--field pattern of a scattered electromagnetic field corresponding to a single incoming plane--wave uniquely determines a bounded supraconductive planar screen. This generalises a previous scalar result of Bl\aa sten, P\"aiv\"arinta and Sadique.

1.Properties of periodic Dirac--Fock functional and minimizers

Authors:Isabelle Catto, Long Meng

Abstract: Existence of minimizers for the Dirac--Fock model in crystals was recently proved by Paturel and S\'er\'e and the authors \cite{crystals} by a retraction technique due to S\'er\'e \cite{Ser09}. In this paper, inspired by Ghimenti and Lewin's result \cite{ghimenti2009properties} for the periodic Hartree--Fock model, we prove that the Fermi level of any periodic Dirac--Fock minimizer is either empty or totally filled when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. Here $c$ is the speed of light, $\alpha$ is the fine structure constant, and $C_{\rm cri}$ is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for $C_{\rm cri}$. Our result implies that any minimizer of the periodic Dirac--Fock model is a projector when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. In particular, the non-relativistic regime (i.e., $c\gg1$) and the weak coupling regime (i.e., $0<\alpha\ll1$) are covered. The proof is based on a delicate study of a second-order expansion of the periodic Dirac--Fock functional composed with the retraction used in \cite{crystals}.

2.Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion

Authors:Renata O. Figueira, Mahendra Panthee

Abstract: We consider the initial value problems (IVPs) for the modified Korteweg-de Vries (mKdV) equation \begin{equation*} \label{mKdV} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u+\mu u^2\partial_xu =0, \quad x\in\mathbb{R},\; t\in \mathbb{R} , \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and $\mu=\pm 1$, and the cubic nonlinear Schr\"odinger equation with third order dispersion (tNLS equation in short) \begin{equation*} \label{t-NLS} \left\{\begin{array}{l} \partial_t v+i\alpha \partial_x^2v+\beta \partial_x^3v+i\gamma |v|^2v = 0, \quad x\in\mathbb{R},\; t\in\mathbb{R} , \\ v(x,0) = v_0(x), \end{array}\right. \end{equation*} where $\alpha, \beta$ and $\gamma$ are real constants and $v$ is a complex valued function. In both problems, the initial data $u_0$ and $v_0$ are analytic on $\mathbb{R}$ and have uniform radius of analyticity $\sigma_0$ in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same $\sigma_0$ till some lifespan $0<T_0\le 1$. We also consider the evolution of the radius of spatial analyticity $\sigma(t)$ when the local solution extends globally in time and prove that for any time $T\ge T_0$ it is bounded from below by $c T^{-\frac43}$, for the mKdV equation in the defocusing case ($\mu = -1$) and by $c T^{-(4+\varepsilon)}$, $\varepsilon>0$, for the tNLS equation. The result for the mKdV equation improves the one obtained in [ J. L. Bona, Z. Gruji\'c and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincar\'e 22 (2005) 783--797] and, as far as we know, the result for the tNLS equation is the new one.

3.Dissipation in Onsager's critical classes and energy conservation in $BV\cap L^\infty$ with and without boundary

Authors:Luigi De Rosa, Marco Inversi

Abstract: This paper is concerned with the incompressible Euler equations. In Onsager's critical classes we provide explicit formulas for the Duchon-Robert measure in terms of the regularization kernel and a family of vector-valued measures $\{\mu_z\}_z$, having some H\"older regularity with respect to the direction $z\in B_1$. Then, we prove energy conservation for $L^\infty_{x,t}\cap L^1_t BV_x$ solutions, in both the absence or presence of a physical boundary. This result generalises the previously known case of Vortex Sheets, showing that energy conservation follows from the structure of $L^\infty\cap BV$ incompressible vector fields rather than the flow having "organized singularities". The interior energy conservation features the use of Ambrosio's anisotropic optimization of the convolution kernel and it differs from the usual energy conservation arguments by heavily relying on the incompressibility of the vector field. In particular the same argument fails to apply to solutions to the Burgers equation, coherently with compressible shocks having non-trivial entropy production. To run the boundary analysis we introduce a notion "normal Lebesgue trace" for general vector fields, very reminiscent of the one for $BV$ functions. We show that having such a null normal trace is basically equivalent to have vanishing boundary energy flux. This goes beyond the previous approaches, laying down a setup which apply to every Lipschitz bounded domain. Allowing any Lipschitz boundary introduces several technicalities to the proof, with a quite geometrical/measure-theoretical flavour.

4.Pointwise convergence to initial data for some evolution equations on symmetric spaces

Authors:Tommaso Bruno, Effie Papageorgiou

Abstract: Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the fractional heat equation, and the Caffarelli--Silvestre extension problem associated with $\mathscr{L}$, and in each of these cases we characterize the weights $v$ on $\mathbb{X}$ for which the solution converges pointwise a.e. to the initial data when the latter is in $L^{p}(v)$, $1\leq p < \infty$. As a tool, we also establish vector-valued weak type $(1,1)$ and $L^{p}$ estimates ($1<p<\infty$) for the local Hardy--Littlewood maximal function on $\mathbb{X}$.

5.Local and Global Results for Shape optimization problems with weighted source

Authors:Qinfeng Li, Hang Yang

Abstract: We consider shape optimization problems of maximizing the averaged heat under various boundary conditions. Assuming that the heat source is radial, we obtain several local stability and global optimality results on ball shape. As a byproduct of stability analysis, we show that Talenti type pointwise comparison result is no longer true under Robin conditions even if the domain is a smooth small perturbation of a ball.

6.Stationary equilibria and their stability in a Kuramoto MFG with strong interaction

Authors:Annalisa Cesaroni, Marco Cirant

Abstract: Recently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of rational interacting oscillators. The MFG model exhibits several stationary equilibria, but the characterization of these equilibria and their ability to capture dynamic equilibria in long time remains largely open. In this paper, we demonstrate that, up to a phase translation, there are only two possible stationary equilibria: the incoherent equilibrium and the self-organizing equilibrium, given that the interaction parameter is sufficiently large. Furthermore, we present some local stability properties of the self-organizing equilibrium.

7.Weak solutions to the heat conducting compressible self-gravitating flows in time-dependent domains

Authors:Kuntal Bhandari, Bingkang Huang, Šárka Nečasová

Abstract: In this paper, we consider a flow of heat-conducting self-gravitating compressible fluid in a time-dependent domain. The flow is governed by the 3-D Navier-Stokes-Fourier-Poisson equations where the velocity is supposed to fulfill the full-slip boundary condition and the temperature on the boundary is given by a non-homogeneous Dirichlet condition. We establish the global-in-time weak solution to the system. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, to accommodate the non-homogeneous boundary heat flux, we introduce the concept of {\em ballistic energy} in this work.

1.Vanishing viscosity limits of compressible viscoelastic equations in half space

Authors:Xumin Gu, Dehua Wang, Feng Xie

Abstract: In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, these results also provide two different kinds of suitable boundary conditions for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the vanishing viscosity limit method. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper.

2.Global Existence and Aggregation of Chemotaxis-fluid Systems in dimension two

Authors:Fanze Kong, Chen-Chih Lai, Juncheng Wei

Abstract: To describe the cellular self-aggregation phenomenon, some strongly coupled PDEs named as Patlak--Keller--Segel (PKS) systems were proposed in 1970s. Since PKS systems possess relatively simple structures but admit rich dynamics, plenty of scholars have studied them and obtained many significant results. However, the cells or bacteria in general direct their movement in liquid. As a consequence, it seems more realistic to consider the influence of ambient fluid flow on the chemotactic mechanism. Motivated by this, we consider the chemotaxis-fluid model proposed by He et al. (SIAM J. Math. Anal., Vol. 53, No. 3, 2021) in the two-dimensional bounded domain. It is well-known that the PKS system admits the critical mass phenomenon in 2D and for the whole space $\mathbb R^2$, He et al. also showed there exists the same phenomenon in the chemotaxis-fluid system. In this paper, we first study the global well-posedness of two-dimensional chemotaxis-fluid model in the bounded domain and prove the solution exists globally with the subcritical mass. Then concerning the critical mass case, we construct the boundary spot steady states rigorously via the inner-outer gluing method. While studying the concentration phenomenon with the critical mass, we develop the global $W^{2,p}$ theory of the stationary Stokes operator in 2D.

3.On the energy and helicity conservation of the incompressible Euler equations

Authors:Yanqing Wang, Wei Wei, Gnag Wu, Yulin Ye

Abstract: In this paper, we are concerned with the minimal regularity of weak solutions implying the law of balance for both energy and helicity in the incompressible Euler equations. In the spirit of recent works due to Berselli [5] and Berselli-Georgiadis [6], it is shown that the energy of weak solutions is invariant if $v\in L^{p}(0,T;B^{\frac1p}_{\frac{2p}{p-1},c(\mathbb{N})} )$ with $1<p\leq3$ and the helicity is conserved if $v\in L^{p}(0,T;B^{\frac2p}_{\frac{2p}{p-1},c(\mathbb{N})} )$ with $2<p\leq3 $ for both the periodic domain and the whole space, which generalizes the classical work of Cheskidov-Constantin-Friedlander-Shvydkoy in [10]. This indicates the role of the time integrability, spatial integrability and differential regularity of the velocity in the conserved quantities of weak solutions of the ideal fluid.

4.Global controllability and stabilization of the wave maps equation from a circle to a sphere

Authors:Jean-Michel Coron, Joachim Krieger, Shengquan Xiang

Abstract: Continuing the investigations started in the recent work [Krieger-Xiang, 2022] on semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$, where {\it semi-global} refers to the $2\pi$-energy bound, we prove global exact controllability of the same system for $k>1$ and show that the $2\pi$-energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in [Krieger-Xiang, 2022] is sharp. Lastly, the global exact controllability for $\mathbb{S}^1$-target within minimum time is discussed.

5.Stability results for a hierarchical size-structured population model with distributed delay

Authors:Dandan Hu, József Z. Farkas, Gang Huang

Abstract: In this paper we investigate a structured population model with distributed delay. Our model incorporates two different types of nonlinearities. Specifically we assume that individual growth and mortality are affected by scramble competition, while fertility is affected by contest competition. In particular, we assume that there is a hierarchical structure in the population, which affects mating success. The dynamical behavior of the model is analysed via linearisation by means of semigroup and spectral methods. In particular, we introduce a reproduction function and use it to derive linear stability criteria for our model. Further we present numerical simulations to underpin the stability results we obtained.

6.Boundedness of solutions to singular anisotropic elliptic equations

Authors:Barbara Brandolini, Florica Corina Cirstea

Abstract: We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$-\Delta_{\overrightarrow{p}}u+\Phi_0(u,\nabla u)=\Psi(u,\nabla u) +f $$ on a bounded open subset $\Omega\subset \mathbb R^N$ $(N\geq 2)$, where $ \Delta_{\overrightarrow{p}}u=\sum_{j=1}^N \partial_j (|\partial_j u|^{p_j-2}\partial_j u)$ and $\Phi_0(u,\nabla u)=\left(\mathfrak{a}_0+\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}\right)|u|^{m-2}u$, with $\mathfrak{a}_0>0$, $m,p_j>1$, $\mathfrak{a}_j\geq 0$ for $1\leq j\leq N$ and $N/p=\sum_{k=1}^N (1/p_k)>1$. We assume that $f \in L^r(\Omega)$ with $r>N/p$. The feature of this study is the inclusion of a possibly singular gradient-dependent term $\Psi(u,\nabla u)=\sum_{j=1}^N |u|^{\theta_j-2}u\, |\partial_j u|^{q_j}$, where $\theta_j>0$ and $0\leq q_j<p_j$ for $1\leq j\leq N$. The existence of such weak solutions is contained in a recent paper by the authors.

7.Optimization of the principal eigenvalue of the Neumann Laplacian with indefinite weight and monotonicity of minimizers in cylinders

Authors:Claudia Anedda, Fabrizio Cuccu

Abstract: Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$ in $\Omega$ with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with homogeneous Neumann boundary conditions. We study weak* continuity, convexity and G\^ateaux differentiability of the map $m\mapsto1/\lambda_1(m)$, where $\lambda_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ the class of rearrangements of a fixed weight $m_0$, under the assumptions that $m_0$ is positive on a set of positive Lebesgue measure and $\int_\Omega m\,dx<0$, we prove the existence and a characterization of minimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, we show that, if $\Omega$ is a cylinder, then every minimizer is monotone with respect to the direction of the generatrix. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats for a population to survive.

8.Global Wellposedness of a Class of Weakly Hyperbolic Cauchy Problems with Variable Multiplicities on $\mathbb{R}^d$

Authors:Sandro Coriasco, Giovanni Girardi, N. Uday Kiran

Abstract: We study a class of weakly hyperbolic Cauchy problems on $\mathbb{R}^d$, involving linear operators with characteristics of variable multiplicities, whose coefficients are unbounded in the space variable. The behaviour in the time variable is governed by a suitable shape function. We develop a parameter-dependent symbolic calculus, corresponding to an appropriate subdivision of the phase space. By means of such calculus, a parametrix can be constructed, in terms of (generalized) Fourier integral operators naturally associated with the employed symbol class. Further, employing the parametrix, we prove $\mathscr{S}(\mathbb{R}^{d})$-wellposedness and give results about the global decay and regularity of the solution, within a scale of weighted Sobolev space.

1.A resonant Lyapunov centre theorem with an application to doubly periodic travelling hydroelastic waves

Authors:Rami Ahmad, Mark David Groves, Dag Nilsson

Abstract: We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate $1:1$ or $1:-1$ semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a "trivial" eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application we show how Kirchg\"{a}ssner's spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (hydroelastic waves). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a $1:1$ or $1:-1$ semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.

2.Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial data

Authors:Dongjin Park

Abstract: We consider the defocusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation (INLS) $iu_t + \Delta u = |x|^{-b}|u|^{k}u$ in $\mathbb{R} \times \mathbb{R}^{n}$ where $n \geq 3$, $0<b<\min(2, n/2)$, and $k=(4-2b)/(n-2)$. We show that for every spherically symmetric initial data $\phi \in H^1(\mathbb{R}^n)$, or preferably $\dot{H}^1(\mathbb{R}^n)$, the solution is globally well-posed and scatters for every such $n$ and $b$ except for $n=4$ with $1\leq b<2$ and $n=5$ with $1/2\leq b\leq 5/4$. We mainly apply the arguments of Tao (2005), but inspired by the work of Aloui and Tayachi (2021), we utilize Lorentz spaces to define spacetime norms. This method is distinct from the widespread concentration compactness principle and establishes a quantitative bound for the solution's spacetime norm. The bound has an exponential form $C\exp(CE[\phi]^C)$ in terms of the energy $E[\phi]$, similar to Tao's work.

3.Semi-linear parabolic equations on homogeneous Lie groups arising from mean field games

Authors:Paola Mannucci, Claudio Marchi, Cristian Mendico

Abstract: The existence and the uniqueness of solutions to some semilinear parabolic equations on homogeneous Lie groups, namely, the Fokker-Planck equation and the Hamilton-Jacobi equation, are addressed. The anisotropic geometry of the state space plays a crucial role in our analysis and creates several issues that need to be overcome. Indeed, the ellipticity directions span, at any point, subspaces of dimension strictly less than the dimension of the state space. Finally, the above results are used to obtain the short-time existence of classical solutions to the mean field games system defined on an homogenous Lie group.

4.Energy stability for a class of semilinear elliptic problems

Authors:Danilo Gregorin Afonso, Alessandro Iacopetti, Filomena Pacella

Abstract: In this paper, we consider semilinear elliptic problems in a bounded domain $\Omega$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $\Omega$ inside $\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.

5.Normalized bound state solutions of fractional Schrödinger equations with general potential

Authors:Xin Bao, Ying Lv, Zeng-Qi Ou

Abstract: In this paper, we study a class of fractional Schr\"{o}dinger equation \begin{equation} \label{eq0} \left\{ \begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\ u\in H^{s}(\mathbb{R}^{N}), \end{aligned} \right. \end{equation} where $N>2s$, $s\in(0,1)$ and $p\in(2,2+4s/N), c>0$. $a(x)\in C(\mathbb{R}^{N},\mathbb{R})$ is a positive potential function. By using Fixed Point Theorem of Brouwer, barycenter function and variational method, we obtain the existence of normalized bound solutions for the problem.

1.Weak solutions to the Hall-MHD equations whose singular sets in time have Hausdorff dimension strictly less than 1

Authors:Yi Peng, Huaqiao Wang

Abstract: In this paper, we focus on the three-dimensional hyper viscous and resistive Hall-MHD equations on the torus, where the viscous and resistive exponent $\alpha\in [\rho, 5/4)$ with a fixed constant $\rho\in (1,5/4)$. We prove the non-uniqueness of a class of weak solutions to the Hall-MHD equations, which have bounded kinetic energy and are smooth in time outside a set whose Hausdorff dimension strictly less than 1. The proof is based on the construction of the non-Leray-Hopf weak solutions via a convex integration scheme.

2.Normalized solutions for a fractional Choquard-type equation with exponential critical growth in $\mathbb{R}$

Authors:Wenjing Chen, Qian Sun, Zexi Wang

Abstract: In this paper, we study the following fractional Choquard-type equation with prescribed mass \begin{align*} \begin{cases} (-\Delta)^{1/2}u=\lambda u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}, \displaystyle\int_{\mathbb{R}}|u|^2 \mathrm{d}x=a^2, \end{cases} \end{align*} where $(-\Delta)^{1/2}$ denotes the $1/2$-Laplacian operator, $a>0$, $\lambda\in \mathbb{R}$, $I_\mu(x)=\frac{{1}}{{|x|^\mu}}$ with $\mu\in(0,1)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Trudinger-Moser inequality. By using a minimax principle based on the homotopy stable family, we obtain that there is at least one normalized ground state solution to the above equation.

3.Well-posedness of the three-dimensional heat conductive compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

Authors:Qin Duan, Zhouping Xin, Shengguo Zhu

Abstract: For the degenerate viscous and heat conductive compressible fluids, the momentum equations and the energy equation are degenerate both in the time evolution and spatial dissipation structures when vacuum appears, and then the physical entropy S behaves singularly, which make it challenging to study the corresponding well-posedness of regular solutions with high order regularities of S near the vacuum. In this paper, when the coefficients of viscosities and heat conductivity depend on the absolute temperature {\theta} in a power law ({\theta}^{\nu} with {\nu}>0) of Chapman-Enskog, by some elaborate analysis of the intrinsic degenerate-singular structures of the full compressible Navier-Stokes equations (CNS), we identify a class of initial data admitting a local-in-time regular solution with far field vacuum to the Cauchy problem of the three-dimensional (3-D) CNS in terms of the mass density {\rho}, velocity u and S. Furthermore, it is shown that within its life span of such a regular solution, u stays in an inhomogeneous Sobolev space, i.e., u\in H^3(R^3), S has uniformly finite lower and upper bounds in R^3, and the laws of conservation of total mass, momentum and total energy are all satisfied. The key idea for proving the existence is to introduce an enlarged system by considering some new variables, which includes a singular parabolic system for u, and one degenerate-singular parabolic equation for S. It is worth pointing out that this reformulation can transfer part of the degeneracies of the full CNS to some singular source terms, and then one can carry out a series of singular or degenerate weighted energy estimates carefully designed for this reformulated system, which provides successfully an effective propagation mechanism of S's high order regularities along with the time.

4.Global regularity for the 2D micropolar Rayleigh-Bénard convection system with velocity zero dissipation and temperature critical dissipation

Authors:Baoquan Yuan, Changhao Li

Abstract: This paper studies the global regularity problem for the 2D micropolar Rayleigh-B\'{e}nard convection system with velocity zero dissipation, micro-rotation velocity Laplace dissipation and temperature critical dissipation. By introducing a combined quantity and using the technique of Littlewood-Paley decomposition, we establish the global regularity result of solutions to this system.

5.Landscape of wave localisation at low frequencies

Authors:Bryn Davies, Yiqi Lou

Abstract: High-contrast scattering problems are special among classical wave systems as they allow for strong wave localisation at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterisation in terms of the generalised capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalised capacitance matrix and use it to predict the positions of three-dimensional wave localisation in random and non-periodic systems of subwavelength resonators.

6.The cubic Szegő equation on the real line: explicit formula and well-posedness on the Hardy class

Authors:Patrick Gérard, Alexander Pushnitski

Abstract: We establish an explicit formula for the solution of the the cubic Szeg\H{o} equation on the real line. Using this formula, we prove that the evolution flow of this equation can be continuously extended to the whole Hardy class $H^2$ on the real line.

7.Anomalous Dissipation for the d-dimensional Navier-Stokes Equations

Authors:Jinlu Li, Yanghai Yu, Weipeng Zhu

Abstract: The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier--Stokes equations in the whole space: \begin{equation*} \begin{cases} \partial_tu^\varepsilon+u^\varepsilon\cdot \nabla u^\varepsilon-\varepsilon\Delta u^\varepsilon+\nabla p^\varepsilon=0,\\ \mathrm{div}\ u^\varepsilon=0. \end{cases} \end{equation*} We aim to presenting a simple rigorous examples of initial data which generates the corresponding solutions of the Navier--Stokes equations do exhibit anomalous dissipation. Precisely speaking, we show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero.

8.Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain: linearized system

Authors:Jean Cauvin-Vila, Virginie Ehrlacher, Amaury Hayat

Abstract: We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time.

9.Upper bounds for the relaxed area of $\mathbb S^1$-valued Sobolev maps and its countably subadditive interior envelope

Authors:Giovanni Bellettini, Riccardo Scala, Giuseppe Scianna

Abstract: Given a bounded open connected Lipschitz set $\Omega \subset \mathbb R^2$, we show that the relaxed Cartesian area functional $\overline{\mathcal A}(u,\Omega)$ of a map $u\in W^{1,1}(\Omega;\mathbb S^1)$ is finite, and provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture [17] adapted to $W^{1,1}(\Omega;\mathbb S^1)$, on the largest countably subadditive set function $\overline {\overline{\mathcal A}}(u, \cdot)$ smaller than or equal to $\overline{\mathcal A}(u,\cdot)$.

1.Parabolic-elliptic Keller-Segel's system

Authors:Valentin Lemarié

Abstract: We study on the whole space R d the compressible Euler system with damping coupled to the Poisson equation when the damping coefficient tends towards infinity. We first prove a result of global existence for the Euler-Poisson system in the case where the damping is large enough, then, in a second step, we rigorously justify the passage to the limit to the parabolic-elliptic Keller-Segel after performing a diffusive rescaling, and get an explicit convergence rate. The overall study is carried out in 'critical' Besov spaces, in the spirit of the recent survey [16] by R. Danchin devoted to partially dissipative systems.

2.Stationary solutions and large time asymptotics to a cross-diffusion-Cahn-Hilliard system

Authors:Jean Cauvin-Vila ENPC, MATHERIALS, Virginie Ehrlacher ENPC, MATHERIALS, Greta Marino, Jan-Frederik Pietschmann

Abstract: We study some properties of a multi-species degenerate Ginzburg-Landau energy and its relation to a cross-diffusion Cahn-Hilliard system. The model is motivated by multicomponent mixtures where crossdiffusion effects between the different species are taken into account, and where only one species does separate from the others. Using a comparison argument, we obtain strict bounds on the minimizers from which we can derive first-order optimality conditions, revealing a link with the single-species energy, and providing enough regularity to qualify the minimizers as stationary solutions of the evolution system. We also discuss convexity properties of the energy as well as long time asymptotics of the time-dependent problem. Lastly, we introduce a structure-preserving finite volume scheme for the time-dependent problem and present several numerical experiments in one and two spatial dimensions.

3.Construction of minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$

Authors:André de Laire, Philippe Gravejat, Didier Smets

Abstract: As a sequel to our previous analysis in [9] arXiv:2202.09411 on the Gross-Pitaevskii equation on the product space $\mathbb{R} \times \mathbb{T}$, we construct a branch of finite energy travelling waves as minimizers of the Ginzburg-Landau energy at fixed momentum. We deduce that minimizers are precisely the planar dark solitons when the length of the transverse direction is less than a critical value, and that they are genuinely two-dimensional solutions otherwise. The proof of the existence of minimizers is based on the compactness of minimizing sequences, relying on a new symmetrization argument that is well-suited to the periodic setting.

4.Global well-posedness of smooth solutions to the Landau-Lifshitz-Slonczewski equation

Authors:Chenlu Zhang, Huaqiao Wang

Abstract: In this paper, we mainly consider the global solvability of smooth solutions for the Cauchy problem of the three-dimensional Landau-Lifshitz-Slonczewski equation in the Morrey space. We derive the covariant complex Ginzburg-Landau equation by using moving frames to address the nonlinear parts. Applying the semigroup estimates and energy methods, we extend local classical solutions to global solutions and prove the boundedness of $\|\nabla\boldsymbol{m}\|_{L^{\infty}(\mathbb{R}^{3})}$, where $\boldsymbol{m}$ is the magnetic intensity. Moreover, we obtain a global weak solution by using an approximation result and improve the regularity of the obtained solution by the regularity theory. Finally, we establish the existence and uniqueness of global smooth solutions under some conditions on $\nabla\boldsymbol{m}_{0}$ and the density of the spin-polarized current.

5.Optimal control of the 2D constrained Navier-Stokes equations

Authors:Sangram Satpathi

Abstract: We study the 2D Navier-Stokes equations within the framework of a constraint that ensures energy conservation throughout the solution. By employing the Galerkin approximation method, we demonstrate the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on the torus $\mathbb{T}^2$. Moreover, we investigate the linearized system associated with the 2D-constrained Navier-Stokes equations, exploring its existence and uniqueness. Subsequently, we establish the Lipschitz continuity and Fr$\'{e}$chet differentiability properties of the solution mapping. Finally, employing the formal Lagrange method, we prove the first-order necessary optimality conditions.

6.Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers

Authors:Onur Baysal, Alemdar Hasanov, Alexandre Kawano

Abstract: In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation $\rho(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to the clamped boundary conditions $u(0,t)=u_x(0,t)=0$ at $x=0$, and the boundary conditions $\left(-r(x)u_{xx}\right)_{x=\ell}=k_r u_x(\ell,t)+k_a u_{xt}(\ell,t)$, $\left(-\left(r(x)u_{xx}\right)_{x}\right )_{x=\ell}$$=- k_d u(\ell,t)-k_v u_{t}(\ell,t)$ at $x=\ell$, is analyzed. The boundary conditions at $x=\ell$ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient $\mu(x) \ge 0$, and the boundary springs $k_r,k_d \ge 0$ and dampers $k_a,k_v \ge 0$. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.

7.Taylor's expansions of Riesz convolution and the fractional Laplacians with respect to the order

Authors:Huyuan Chen

Abstract: We build the n-th order Taylor expansion for Riesz operators and fractional Laplacian with respect to the order.

8.Surfaces in which every point sounds the same

Authors:Feng Wang, Emmett L. Wyman, Yakun Xi

Abstract: We address a maximally structured case of the question, "Can you hear your location on a manifold," posed in arXiv:2304.04659 for dimension $2$. In short, we show that if a compact surface without boundary sounds the same at every point, then the surface has a transitive action by the isometry group. In the process, we show that you can hear your location on Klein bottles and that you can hear the lengths and multiplicities of looping geodesics on compact hyperbolic quotients.

9.A coupled rate-dependent/rate-independent system for adhesive contact in Kirchhoff-Love plates

Authors:Giovanna Bonfanti, Elisa Davoli, Riccarda Rossi

Abstract: We perform a dimension reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in the setting of visco-elastodynamic plates. We work with a weak solvability notion inspired by the theory of (purely) rate-independent processes, and accordingly term the related solutions `Semistable Energetic'. For Semistable Energetic solutions, the momentum balance holds in a variational sense, whereas the flow rule for the adhesion parameter is replaced by a semi-stability condition coupled with an energy-dissipation inequality. Prior to addressing the dimension reduction analysis, we show that Semistable Energetic solutions to the three-dimensional damped adhesive contact model converge, as the viscosity term tends to zero, to three-dimensional Semistable Energetic solutions for the undamped corresponding system. We then perform a dimension reduction analysis, both in the case of a vanishing viscosity tensor (leading, in the limit, to an undamped model), and in the complementary setting in which the damping is assumed to go to infinity as the thickness of the plate tends to zero. In both regimes, the presence of adhesive contact yields a nontrivial coupling of the in-plane and out-of-plane contributions. In the undamped scenario we obtain in the limit an energy-dissipation inequality and a semistability condition. In the damped case, instead, we achieve convergence to an enhanced notion of solution, fulfilling an energy-dissipation balance.

1.Global smooth solution for the 3D generalized tropical climate model with partial viscosity and damping

Authors:Hui Liu, Hongjun Gao, Chengfeng Sun

Abstract: The three-dimensional generalized tropical climate model with partial viscosity and damping is considered in this paper. Global well-posedness of solutions of the three-dimensional generalized tropical climate model with partial viscosity and damping is proved for $\alpha\geq\frac{3}{2}$ and $\beta\geq4$. Global smooth solution of the three-dimensional generalized tropical climate model with partial viscosity and damping is proved in $H^s(\mathbb R^3)$ $(s>2)$ for $\alpha\geq\frac{3}{2}$ and $4\leq\beta\leq5$.

2.On the thermoelastic coupling of anisotropic laminates

Authors:Paolo Vannucci

Abstract: The analysis of the mathematical and mechanical properties of thermoelastic coupling tensors in anisotropic laminates is the topic of this paper. Some theoretical results concerning the compliance tensors are shown and their mechanical consequences analyzed. Moreover, the case of thermally stable laminates, important for practical applications, is also considered. The study is carried out in the framework of the polar method, a mathematical formalism particularly well suited for the analysis of planar anisotropic problems, introduced by Prof. G. Verchery in 1979.

3.On the Hang-Yang conjecture for GJMS equations on $\mathbb S^n$

Authors:Ali Hyder, Quôc Anh Ngô

Abstract: This work concerns a Liouville type result for positive, smooth solution $v$ to the following higher-order equation \[ {\mathbf P}^{2m}_n (v) = \frac{n-2m}2 Q_n^{2m} (\varepsilon v+v^{-\alpha} ) \] on $\mathbb S^n$ with $m \geq 2$, $3 \leq n < 2m $, $0<\alpha \leq (2m+n)/(2m-n)$, and $\varepsilon >0$. Here $ {\mathbf P}^{2m}_n$ is the GJMS operator of order $2m$ on $\mathbb S^n$ and $Q_n^{2m} =(2/(n-2m)) {\mathbf P}^{2m}_n (1)$ is constant. We show that if $\varepsilon >0$ is small and $0<\alpha \leq (2m+n)/(2m-n)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon =0$ and $0<\alpha < (2m+n)/(2m-n)$. In the special case $n=3$, $m=2$, and $\alpha=7$, such Liouville type result was recently conjectured by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020). As a by-product, we obtain the sharp (subcritical and critical) Sobolev inequalities \[ \Big( \int_{\mathbb S^n} v^{1-\alpha} d\mu_{\mathbb S^n} \Big)^{\frac {2}{\alpha -1}} \int_{\mathbb S^n} v {\mathbf P}^{2m}_n (v) d\mu_{\mathbb S^n} \geq \frac{\Gamma (n/2 + m)}{\Gamma (n/2 - m )} | \mathbb S^n|^\frac{\alpha + 1}{\alpha - 1} \] for the GJMS operator $ {\mathbf P}^{2m}_n$ on $\mathbb S^n$ under the conditions $n \geq 3$, $n=2m-1$, and $\alpha \in(0,1) \cup (1, 2n+1]$. A log-Sobolev type inequality, as the limiting case $\alpha=1$, is also presented.

1.Three results on the Energy conservation for the 3D Euler equations

Authors:Luigi C. Berselli, Stefanos Georgiadis

Abstract: We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient, recovering some results which were known, up to now, only for the Navier-Stokes equations and for weak solutions of the Leray-Hopf type. Finally, we make some remarks on the Onsager singularity problem, identifying conditions which allow to pass to the limit from solutions of the Navier-Stokes equations to solution of the Euler ones, producing weak solutions which are energy conserving.

2.The Brezis-Nirenberg problem in 4D

Authors:Angela Pistoia, Serena Rocci

Abstract: The problem \begin{equation} \label{bn} -\Delta u=|u|^{4\over n-2}u+\lambda V u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega \end{equation} where $\Omega$ is a bounded regular domain in $\mathbb R^n$, $\lambda\in \mathbb R$ and $V\in C^0(\overline \Omega),$ that was introduced by Brezis and Nirenberg in their famous paper, where they address the existence of positive solutions in the autonomous case, i.e. the potential $V$ is constant. Since then, a huge amount of work has been done. In the following we will make a brief history highlighting the results which are much closer to the problem we wish to study in the present paper.

3.Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

Authors:Vishnu Raveendran, Ida de Bonis, Emilio N. M. Cirillo, Adrian Muntean

Abstract: We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

4.Perturbation method for second order strongly elliptic systems of PDEs with constant coefficients

Authors:Astamur Bagapsh

Abstract: The classical Dirichlet problem for a second-order strongly elliptic system with constant coefficients in a Jordan domain is considered. We show that the solution of the problem can be represented as a functional series in powers of the parameter, which determines the deviation of the system operator from the Laplacian. This series converges uniformly in the closure of the region under the assumption that the boundary of the region and the boundary function satisfy the sufficient regularity conditions: the trace of a conformal mapping of the domain onto a circle composed with the boundary function belongs to the Holder class with exponent greater than 1/2.

5.Lipschitz stability for determination of states and inverse source problem for the mean field game equations

Authors:Oleg Imanuvilov, Hongyu Liu, Masahiro Yamamoto

Abstract: In a bounded domain $\Omega \subset \mathbb{R}^d$ over time interval $(0,T)$, we consider mean field game equations whose principal coefficients depend on the time and state variables with a general Hamiltonian. We attach the non-zero Robin boundary condition. We first prove the Lipschitz stability in $\Omega \times (\varepsilon, T-\varepsilon)$ with given $\varepsilon>0$ for the determination of the solutions by Dirichlet data on arbitrarily chosen subboundary of $\partial\Omega$. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time.

6.On the time behavior of a porous thermoelastic system with only thermal dissipation given by Gurtin-Pipkin law

Authors:Afaf Ahmima, Abdelfeteh Fareh

Abstract: In the present paper we consider a porous thermoelastic system with only one dissipative mechanism generated by the heat conductivity modelled by the Gurtin-Pipkin thermal law. By the use of a semigroup approach and the Lumer-Phillips theorem we prove the existence of a unique solution. We introduce a stability number $\chi_g$ depends on the coefficients of the system, and establish an exponential stability result provided that $\chi_g=0$. Otherwise, if $\chi_g\ne 0$, we prove the lack of exponential decay. Our result improves and generalizes the previous results in the literature obtained for Fourier's and Cattaneo's laws of thermal conductivity.

7.Global solutions versus finite time blow-up for the fast diffusion equation with spatially inhomogeneous source

Authors:Razvan Gabriel Iagar, Ariel Sánchez

Abstract: Solutions in self-similar form, either global in time or presenting finite time blow-up, to the fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,\infty)$, $N\geq1$, are classified with respect to the exponents $m\in(0,1)$, $\sigma\in(-2,\infty)$ and $p>\max\{1,p_L(\sigma)\}$, where $$ p_L(\sigma)=1+\frac{\sigma(1-m)}{2}. $$ In the supercritical range $m_c=(N-2)_{+}/N<m<1$, global solutions are classified with respect to their tail behavior as $|x|\to\infty$, proving that a specific tail behavior $$ u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty $$ exists exactly for $p\in(p_F(\sigma),p_s(\sigma))$, where $$ p_F(\sigma)=m+\frac{\sigma+2}{N}, \qquad p_s(\sigma)=\left\{\begin{array}{ll}\frac{m(N+2\sigma+2)}{N-2}, & N\geq3,\\\infty, & N\in\{1,2\}, \end{array}\right. $$ are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any $\sigma\in(-2,0)$ and $p>p_L(\sigma)$, but do not exist for any $\sigma\geq0$ and $p\in(p_F(\sigma),p_s(\sigma))$. In the subcritical range $0<m<m_c$, $N\geq3$, we introduce a new transformation between radially symmetric solutions to the equation, which can be understood as a kind of symmetry of the solutions with respect to the critical exponents $m_s=(N-2)/(N+2)$ and $p_s(\sigma)$, and we employ this symmetry to classify both global and blow-up self-similar solutions. We stress that all these results are new also in the homogeneous case $\sigma=0$.

1.Global existence and weak-strong uniqueness for chemotaxis compressible Navier-Stokes equations modeling vascular network formation

Authors:Xiaokai Huo, Ansgar Jüngel

Abstract: A model of vascular network formation is analyzed in a bounded domain, consisting of the compressible Navier-Stokes equations for the density of the endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, which triggers the migration of the endothelial cells and the blood vessel formation. The coupling of the equations is realized by the chemotaxis force in the momentum balance equation. The global existence of finite energy weak solutions is shown for adiabatic pressure coefficients $\gamma>8/5$. The solutions satisfy a relative energy inequality, which allows for the proof of the weak--strong uniqueness property.

2.Finite-time blowup for an Euler and hypodissipative Navier-Stokes model equation on a restricted constraint space

Authors:Evan Miller

Abstract: In this paper, we introduce the $\mathcal{M}$-restricted Euler and hypodissipative Navier-Stokes equations. These equations are analogous to the Euler equation and hypodissipative Navier-Stokes equation, respectively, but with the Helmholtz projection replaced by a projection onto a more restrictive constraint space. The nonlinear term arising from the self-advection of velocity is otherwise unchanged. We prove finite time-blowup when the dissipation is weak enough, by making use of a permutation symmetric Ansatz that allows for a dyadic energy cascade of the type found in the Friedlander-Katz-Pavlovi\'{c} dyadic Euler/Navier-Stokes model equation. The $\mathcal{M}$-restricted Euler and hypodissipative Navier-Stokes equations respect both the energy equality and the identity for enstrophy growth for the full Euler and hypodissipative Navier-Stokes equations.

3.Trend to equilibrium for run and tumble equations with non-uniform tumbling kernels

Authors:Josephine Evans, Havva Yoldas

Abstract: We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels.

4.Phase-field topology optimization with periodic microstructure

Authors:Stefano Almi, Ulisse Stefanelli

Abstract: Progresses in additive manufacturing technologies allow the realization of finely graded microstructured materials with tunable mechanical properties. This paves the way to a wealth of innovative applications, calling for the combined design of the macroscopic mechanical piece and its underlying microstructure. In this context, we investigate a topology optimization problem for an elastic medium featuring a periodic microstructure. The optimization problem is variationally formulated as a bilevel minimization of phase-field type. By resorting to Gamma-convergence techniques, we characterize the homogenized problem and investigate the corresponding sharp-interface limit. First-order optimality conditions are derived, both at the homogenized phase-field and at the sharp-interface level.

5.Sharp Sobolev inequalities on Riemannian manifolds with ${\sf Ric}\geq 0$ via Optimal Mass Transportation

Authors:Alexandru Kristály

Abstract: In their seminal work, Cordero-Erausquin, Nazaret and Villani [Adv. Math., 2004] proved sharp Sobolev inequalities in Euclidean spaces via Optimal Mass Transportation, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. In this paper we affirmatively answer their question for Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Mass Transportation with quadratic distance cost, sharp $L^p$-Sobolev and $L^p$-logarithmic Sobolev inequalities (both for $p>1$ and $p=1$) are established, where the optimal constants contain the asymptotic volume growth arising from precise asymptotic properties of the Talentian and Gaussian bubbles. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia [Compos. Math., 2004] (and subsequent results) concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support certain Sobolev inequalities.

6.Embedded corrector problems for homogenization in linear elasticity

Authors:Virginie Ehrlacher, Frederic Legoll, Benjamin Stamm, Shuyang Xiang

Abstract: In this article, we extend the study of embedded corrector problems, that we have previously introduced in the context of the homogenization of scalar diffusive equations, to the context of homogenized elastic properties of materials. This extension is not trivial and requires mathematical arguments specific to the elasticity case. Starting from a linear elasticity model with highly-oscillatory coefficients, we introduce several effective approximations of the homogenized tensor. These approximations are based on the solution to an embedded corrector problem, where a finite-size domain made of the linear elastic heterogeneous material is embedded in a linear elastic homogeneous infinite medium, the constant elasticity tensor of which has to be appropriately determined. The approximations we provide are proven to converge to the homogenized elasticity tensor when the size of the embedded domain tends to infinity. Some particular attention is devoted to the case of isotropic materials.

7.Spreading, flattening and logarithmic lag for reaction-diffusion equations in R^N: old and new results

Authors:François Hamel I2M, Luca Rossi Sapienza University of Rome, CAMS

Abstract: This paper is concerned with the large-time dynamics of bounded solutions of reaction-diffusion equations with bounded or unbounded initial support in R N. We start with a survey of some old and recent results on the spreading speeds of the solutions and their asymptotic local one-dimensional symmetry. We then derive some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. Lastly, we reclaim some known results about the logarithmic lag between the position of the solutions and that of planar or spherical fronts expanding with minimal speed, for almost-planar or compactly supported initial conditions. We then prove some new logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. These estimates entail in particular that the same lag as for compactly supported initial data holds true for a class of unbounded initial supports. The paper also contains some related conjectures and open problems.

8.Solitary waves for dispersive equations with Coifman-Meyer nonlinearities

Authors:Johanna Ulvedal Marstrander

Abstract: Using a modified version of Weinstein's argument for constrained minimization in nonlinear dispersive equations, we prove existence of solitary waves in fully nonlocally nonlinear equations, as long as the linear multiplier is of positive and slightly higher order than the Coifman-Meyer nonlinear multiplier. It is therefore the relative order of the linear term over the nonlinear one that determines the method and existence for these types of equations. In analogy to KdV-type equations and water waves in the capillary regime, smooth solutions of all amplitudes can be found. We consider two structural types of symmetric Coifman-Meyer symbols $n(\xi-\eta,\eta)$, and show that cyclical symmetry is necessary for the existence of a functional formulation. Estimates for the solution and wave speed are given as the solutions tend to the bifurcation point of solitary waves.

9.Gaussian estimates vs. elliptic regularity on open sets

Authors:Tim Böhnlein, Simone Ciani, Moritz Egert

Abstract: Given an elliptic operator $L= - \mathrm{div} (A \nabla \cdot)$ subject to mixed boundary conditions on an open subset of $\mathbb{R}^d$, we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, H\"older continuity of $L$-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet we prove consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

10.Complete metrics with constant fractional higher order $Q$-curvature on the punctured sphere

Authors:João Henrique Andrade, Juncheng Wei, Zikai Ye

Abstract: This manuscript is devoted to constructing complete metrics with constant higher fractional curvature on punctured spheres with finitely many isolated singularities. Analytically, this problem is reduced to constructing singular solutions for a conformally invariant integro-differential equation that generalizes the critical GJMS problem. Our proof follows the earlier construction in Ao {\it et al.} \cite{MR3694645}, based on a gluing method, which we briefly describe. Our main contribution is to provide a unified approach for fractional and higher order cases. This method relies on proving Fredholm properties for the linearized operator around a suitably chosen approximate solution. The main challenge in our approach is that the solutions to the related blow-up limit problem near isolated singularities need to be fully classified; hence we are not allowed to use a simplified ODE method. To overcome this issue, we approximate solutions near each isolated singularity by a family of half-bubble tower solutions. Then, we reduce our problem to solving an (infinite-dimensional) Toda-type system arising from the interaction between the bubble towers at each isolated singularity. Finally, we prove that this system's solvability is equivalent to the existence of a balanced configuration.

1.Lipschitz potential estimates for diffusion with jumps

Authors:Nirjan Biswas, Harsh Prasad

Abstract: For $p \in (1, \infty)$ and $s \in (0,1)$, we consider the following mixed local-nonlocal equation $$ - \Delta_p u + (-\Delta_p)^s u = f \; \text{in} \; \Omega,$$ where $\Omega \subset \mathbb{R}^d$ is a bounded domain and the function $f \in L_{loc}^1(\Omega)$. Depending on the dimension $d$, we prove gradient potential estimates of weak solutions for the entire ranges of $p$ and $s$. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $p=2$.

2.Minimization of the buckling load of a clamped plate with perimeter constraint

Authors:Michele Carriero, Simone Cito, Antonio Leaci

Abstract: We look for minimizers of the buckling load problem with perimeter constraint in any dimension. In dimension 2, we show that the minimizing plates are convex; in higher dimension, by passing through a weaker formulation of the problem, we show that any optimal set is open and connected. For higher eigenvalues, we prove that minimizers exist among convex sets with prescribed perimeter.

3.Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds

Authors:Csaba Farkas, Sándor Kajántó, Alexandru Kristály

Abstract: The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from Cheng and Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in Krist\'aly [Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional Cartan-Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

4.Large deformations in terms of stretch and rotation and global solution to the quasi-stationary problem

Authors:Abramo Agosti, Pierluigi Colli, Michel Frémond

Abstract: In this paper we derive a new model for visco-elasticity with large deformations where the independent variables are the stretch and the rotation tensors which intervene with second gradients terms accounting for physical properties in the principle of virtual power. Another basic feature of our model is that there is conditional compatibility, entering the model as kinematic constraint and depending on the magnitude of an internal force associated to dislocations. Moreover, due to the kinematic constraint, the virtual velocities depend on the solutions of the problem. As a consequence, the variational formulation of the problem and the related mathematical analysis are neither standard nor straightforward. We adopt the strategy to invert the kinematic constraints through Green propagators, obtaining a system of integro-differential coupled equations. As a first mathematical step, we develop the analysis of the model in a simplified setting, i.e. considering the quasi-stationary version of the full system where we neglect inertia. In this context, we prove the existence of a global in time strong solution in three space dimensions for the system, employing techniques from PDEs and convex analysis, thus obtaining a novel contribution in the field of three dimensional finite visco-elasticity described in terms of the stretch and rotation variables. We also study a limit problem, letting the magnitude of the internal force associated to dislocations tend to zero, in which case the deformation becomes incompatible and the equations takes the form of a coupled system of PDEs. For the limit problem we obtain global existence, uniqueness and continuous dependence from data in three space dimensions.

5.Variational eigenvalues of quasilinear subelliptic equations

Authors:Mukhtar Karazym

Abstract: We find variational eigenvalues of quasilinear subelliptic equations by the Lusternik-Schnirelmann theory.

6.Global second-order estimates in anisotropic elliptic problems

Authors:Carlo Alberto Antonini, Andrea Cianchi, Giulio Ciraolo, Alberto Farina, Vladimir Maz'ya

Abstract: We deal with boundary value problems for second-order nonlinear elliptic equations in divergence form, which emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. Integrands with non polynomial growth are included in our discussion. The $W^{1,2}$-regularity of the stress-field associated with solutions, namely the nonlinear expression of the gradient subject to the divergence operator, is established under the weakest possible assumption that the datum on the right-hand side of the equation is a merely $L^2$-function. Global regularity estimates are offered in domains enjoying minimal assumptions on the boundary. They depend on the weak curvatures of the boundary via either their degree of integrability or an isocapacitary inequality. By contrast, none of these assumptions is needed in the case of convex domains. An explicit estimate for the constants appearing in the relevant estimates is exhibited in terms of the Lipschitz characteristic of the domains, when their boundary is endowed with H\"older continuous curvatures.

7.Sobolev spaces and trace theorems for time-fractional evolution equations

Authors:Doyoon Kim, Kwan Woo

Abstract: We establish trace and extension theorems for evolutionary equations with the Caputo fractional derivatives in (weighted) $L_p$ spaces. To achieve this, we identify weighted Sobolev and Besov spaces with mixed norms that accommodate solution spaces and their initial values well-suited for equations involving time-fractional derivatives. Our analysis encompasses both time-fractional sub-diffusion and super-diffusion equations. We also provide observations on the initial behavior of solutions to time-fractional equations.

8.Trend to equilibrium for flows with random diffusion

Authors:Shrey Aryan, Matthew Rosenzweig, Gigliola Staffilani

Abstract: Motivated by the possibility of noise to cure equations of finite-time blowup, recent work arXiv:2109.09892 by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak-Keller-Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from arXiv:2109.09892 in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow\infty$.

9.Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension

Authors:Lawrence Frolov, Samuel Leigh, A. Shadi Tahvildar-Zadeh

Abstract: In this paper we prove that the static solution of the Cauchy problem for a massless real scalar field that is sourced by a point charge in $1+1$ dimensions is asymptotically stable under perturbation by compactly-supported incoming radiation. This behavior is due to the process of back-reaction. Taking the approach of Kiessling, we rigorously derive the expression for the force on the particle from the principle of total energy-momentum conservation. We provide a simple, closed form for the self-force resulting from back-reaction, and show that it is restorative, i.e. proportional to negative velocity, and causes the charge to return to rest after the radiation passes through. We establish these results by studying the joint evolution problem for the particle-scalar field system, and proving its global well-posedness and the claimed asymptotic behavior.

10.Invertibility criteria for the biharmonic single-layer potential

Authors:Alexandre Munnier

Abstract: While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems.

11.Some remarks on the solution of the cell growth equation

Authors:Adolf Mirotin

Abstract: The analytical solution to the initial-boundary value problem for the cell growth equation was given in the paper Zaidi A. A., Van Brunt B., Wake G.C., Solutions to an advanced functional partial differential equation of the pantograph type, Proc. R. Soc. A 471: 20140947 (2015). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups.

1.On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type

Authors:Qian Lei, Chi Seng Pun

Abstract: This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium strategies and the associated value functions for time-inconsistent stochastic control problems. Specifically, we consider nonlocality in both time and space, which allows for modelling of the stochastic control problems with initial-time-and-state dependent objective functionals. We leverage the method of continuity to show the global well-posedness within our proposed Banach space with our established Schauder prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization method and Banach's fixed point arguments to show the local well-posedness of the nonlocal fully nonlinear case, while the global well-posedness is attainable provided that a very sharp a-priori estimate is available. On top of the well-posedness results, we also provide a probabilistic representation of the solutions to the nonlocal fully nonlinear PDEs and an estimate on the difference between the value functions of sophisticated and na\"{i}ve controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable.

2.Global existence and optimal time decay rate to one-dimensional two-phase flow model

Authors:Xushan Huang, Yi Wang

Abstract: We investigate the global existence and optimal time decay rate of solution to the one dimensional (1D) two-phase flow described by compressible Euler equations coupled with compressible Navier-Stokes equations through the relaxation drag force on the momentum equations. First, we prove the global existence of strong solution to 1D Euler-Navier-Stokes system by using the standard continuity argument for small $H^{1}$ data while the second order derivative can be large. Then we derive the optimal time decay rate to the equilibrium state $(\rho_*, 0, n_*, 0)$. Compared with multi-dimensional case, it is much hard to get time decay rate by direct spectrum method due to a slower convergence rate of the fundamental solution in 1D case. To overcome this main difficulty, we need to first carry out time-weighted energy estimates for higher order derivatives, and based on these time-weighted estimates, we can close a priori assumptions and get the optimal time decay rate by spectrum analysis method. Moreover, due to non-conserved form and insufficient decay rate of the coupled drag force terms between the two-phase flows, we essentially need to use momentum variables $(m= \rho u, M=n\omega)$, not velocity variables $(u, \omega)$ in the spectrum analysis, to fully cancel out those non-conserved and insufficiently decay drag force terms.

1.Local and global regularity for the Stokes and Navier-Stokes equations with the localized boundary data in the half-space

Authors:Kyungkeun Kang, Chanhong Min

Abstract: We study the Stokes system with the localized boundary data in the half-space. We are concerned with the local regularity of its solution near the boundary away from the support of the given boundary data which are product forms of each spatial variable and the temporal variable. We first show that if the boundary data are smooth in time, the corresponding solutions are also smooth in space and time near the boundary, even if the boundary data are only spatially integrable. Secondly, if the normal component of the boundary data is absent, we are able to construct a solution such that its second normal derivatives of the tangential components become singular near the boundary. Perturbation argument enables us to construct solutions of the Navier-Stokes equations with similar singular behaviors near the boundary in the half-space as the case of Stokes system. Lastly, we provide specific types of the localized boundary data to obtain the pointwise bounds of the solutions up to second derivatives. It turns out that such solutions are globally strong, and the second normal derivatives are, however, unbounded near the boundary. These results can be compared to the previous works in which only the normal component is present. In fact, the temporally non-smooth tangential boundary data can also cause spatially singular behaviors near the boundary, although such behaviors are milder than those caused by the normal boundary data of the same type.

1.Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation

Authors:Lucas Huysmans, Edriss S. Titi

Abstract: We consider the transport equation of a passive scalar $f(x,t)\in\mathbb{R}$ along a divergence-free vector field $u(x,t)\in\mathbb{R}^2$, given by $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$; and the associated advection-diffusion equation of $f$ along $u$, with positive viscosity/diffusivity parameter $\nu>0$, given by $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -\nu\Delta f = 0$. We demonstrate failure of the vanishing viscosity limit of advection-diffusion to select unique solutions, or to select entropy-admissible solutions, to transport along $u$. First, we construct a bounded divergence-free vector field $u$ which has, for each (non-constant) initial datum, two weak solutions to the transport equation. Moreover, we show that both these solutions are renormalised weak solutions, and are obtained as strong limits of a subsequence of the vanishing viscosity limit of the corresponding advection-diffusion equation. Second, we construct a second bounded divergence-free vector field $u$ admitting, for any initial datum, a weak solution to the transport equation which is perfectly mixed to its spatial average, and after a delay, unmixes to its initial state. Moreover, we show that this entropy-inadmissible unmixing is the unique weak vanishing viscosity limit of the corresponding advection-diffusion equation.

2.Deconvolutional determination of the nonlinearity in a semilinear wave equation

Authors:Nicholas Hu, Rowan Killip, Monica Visan

Abstract: We demonstrate that in three space dimensions, the scattering behaviour of semilinear wave equations with quintic-type nonlinearities uniquely determines the nonlinearity. The nonlinearity is permitted to depend on both space and time.

3.Observability of the Schr{ö}dinger equation with subquadratic confining potential in the Euclidean space

Authors:Antoine Prouff LMO

Abstract: We consider the Schr{\"o}dinger equation in $\mathbf{R}^d$, $d \ge 1$, with a confining potential growing at most quadratically. Our main theorem characterizes open sets from which observability holds, provided they are sufficiently regular in a certain sense. The observability condition involves the Hamiltonian flow associated with the Schr{\"o}dinger operator under consideration. It is obtained using semiclassical analysis techniques. It allows to provide with an accurate estimation of the optimal observation time. We illustrate this result with several examples. In the case of two-dimensional harmonic potentials, focusing on conical or rotation-invariant observation sets, we express our observability condition in terms of arithmetical properties of the characteristic frequencies of the oscillator.

4.Blow-up vs. global existence for a Fujita-type Heat exchanger system

Authors:Samuel Tréton UNIROUEN

Abstract: We analyze a reaction-diffusion system on $\mathbb{R}^{N}$ which models the dispersal of individuals between two exchanging environments for its diffusive component and incorporates a Fujita-type growth for its reactive component. The originality of this model lies in the coupling of the equations through diffusion, which, to the best of our knowledge, has not been studied in Fujita-type problems. We first consider the underlying diffusive problem, demonstrating that the solutions converge exponentially fast towards those of two uncoupled equations, featuring a dispersive operator that is somehow a combination of Laplacians. By subsequently adding Fujita-type reaction terms to recover the entire system, we identify the critical exponent that separates systematic blow-up from possible global existence.

5.On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval

Authors:Tadeusz Kulczycki, Jacek Wszoła

Abstract: We study the structure of solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval $D$ with parameter $\lambda > 0$. In particular, we show that there exists a constant $\lambda_{\alpha,D} > 0$ (called the Bernoulli constant) such that the problem has no solution for $\lambda \in (0,\lambda_{\alpha,D})$, at least one solution for $\lambda = \lambda_{\alpha,D}$ and at least two solutions for $\lambda > \lambda_{\alpha,D}$. We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for $\alpha = 1$ that there exist solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval which are not minimizers of the corresponding variational problem.

6.Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities

Authors:Hiroyuki Hirayama, Masahiro Ikeda

Abstract: We consider the Cauchy problem of the system of nonlinear Schr\"odinger equations with derivative nonlinearlity. This system was introduced by Colin-Colin (2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin-Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for $1$-dimension.

7.Blow-up of solutions for relaxed compressible Navier-Stokes equations

Authors:Yuxi Hu, Reinhard Racke

Abstract: We present a blow-up result for large data for relaxed compressible Navier-Stokes models avoiding the possibility of reaching the boundary of hyperbolicity. Thus a previous result is improved and further examples are given illustrating possible effects of a relaxation and contrasting the classical compressible Navier-Stokes equations without relaxation where solutions for large data exist globally.

8.On a chemotaxis-hapotaxis model with nonlinear diffusion modelling multiple sclerosis

Authors:S. Fagioli, E. Radici, L. Romagnoli

Abstract: We investigated existence of global weak solutions for a system of chemotaxis-hapotaxis type with nonlinear degenerate diffusion, arising in modelling Multiple Sclerosis disease. The model consists of three equations describing the evolution of macrophages ($m$), cytokine ($c$) and apoptotic oligodendrocytes ($d$). The main novelty in our work is the presence of a nonlinear diffusivity $D(m)$, which results to be more appropriate from the modelling point of view. Under suitable assumptions and for sufficiently regular initial data, adapting the strategy in [30,44], we show the existence of global bounded solutions for the model analysed.

9.A fractional Hopf Lemma for sign-changing solutions

Authors:Serena Dipierro, Nicola Soave, Enrico Valdinoci

Abstract: In this paper we prove some results on the boundary behavior of solutions to fractional elliptic problems. Firstly, we establish a Hopf Lemma for solutions to some integro-differential equations. The main novelty of our result is that we do not assume any global condition on the sign of the solutions. Secondly, we show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary. We provide concrete examples to show that the results obtained are sharp.

10.Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. II

Authors:Kirill Cherednichenko, Alexander Kiselev, Igor Velčić, Josip Žubrinić

Abstract: For the system of equations of linear elasticity with periodic coefficients displaying high contrast, in the regime of resonant scaling between the material properties of the ``soft" part of the medium and the spatial period, we construct an asymptotic approximation exhibiting time-dispersive properties of the medium and prove associated order-sharp error estimates in the sense of the $L^2\to L^2$ operator norm. The analytic framework presented can be viewed as an advanced version of the one presented in [Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties.\,I. Commun. Math. Phys. 375, 1833--1884], where the corresponding setup of a scalar PDEs was addressed.

11.Multiple normalized solutions to a logarithmic Schrödinger equation via Lusternik-Schnirelmann category

Authors:Claudianor O. Alves, Chao Ji

Abstract: In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2, \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}\epsilon^N, \end{aligned} \right. \end{align*} where $a, \epsilon>0, \lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and $V: \mathbb{R}^N \rightarrow[-1, \infty)$ is a continuous function. Our analysis demonstrates that the number of normalized solutions of the equation is associated with the topology of the set where the potential function $V$ attains its minimum value. To prove the main result, we employ minimization techniques and use the Lusternik-Schnirelmann category. Additionally, we introduce a new function space where the energy functional associated with the problem is of class $C^1$.

12.Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves

Authors:Mariana Haragus, Mathew A. Johnson, Wesley R. Perkins, Björn de Rijk

Abstract: We study the nonlinear dynamics of perturbed, spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is (orbitally) asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential decay rates of perturbations depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in $N$. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This in turn yields an improved nonuniform subharmonic stability result providing an $N$-independent ball of initial perturbations which eventually exhibit exponential decay at an $N$-dependent rate. Finally, we argue that our results connect in the limit $N \to \infty$ to previously established stability results against localized perturbations, thereby unifying existing theories.

13.Variational integrals on Hessian spaces: partial regularity for critical points

Authors:Arunima Bhattacharya, Anna Skorobogatova

Abstract: We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $\mathbb{R}^n$, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. We show that for smooth convex functionals, a $W^{2,\infty}$ critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most $n-p_0$, for some $p_0 \in (2,3)$. We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.

14.Waves in cosmological background with static Schwarzschild radius in the expanding universe

Authors:Karen Yagdjian

Abstract: In this paper we prove the existence of global in time small data solutions of semilinear Klein-Gordon equations in the space-time with a static Schwarzschild radius in the expanding universe.

15.A lower bound for the weighted-Hardy constant for domains satisfying a uniform exterior cone condition

Authors:Ujjal Das, Yehuda Pinchover

Abstract: We consider weighted Hardy inequalities involving the distance function to the boundary of a domain in the $N$-dimensional Euclidean space with nonempty boundary. We give a lower bound for the corresponding best Hardy constant for a domain satisfying a uniform exterior cone condition. This lower bound depends on the aperture of the corresponding infinite circular cone.

16.Recovering coefficients in a system of semilinear Helmholtz equations from internal data

Authors:Kui Ren, Nathan Soedjak

Abstract: We study an inverse problem for a coupled system of semilinear Helmholtz equations where we are interested in reconstructing multiple coefficients in the system from internal data measured in applications such as thermoacoustic imaging. We derive results on the uniqueness and stability of the inverse problem in the case of small boundary data based on the technique of first- and higher-order linearization. Numerical simulations are provided to illustrate the quality of reconstructions that can be expected from noisy data.

1.Graph-to-local limit for a multi-species nonlocal cross-interaction system

Authors:Antonio Esposito, Georg Heinze, Jan-Frederik Pietschmann, André Schlichting

Abstract: In this note we continue the study of nonlocal interaction dynamics on a sequence of infinite graphs, extending the results of [Esposito et. al 2023+] to an arbitrary number of species. Our analysis relies on the observation that the graph dynamics form a gradient flow with respect to a non-symmetric Finslerian gradient structure. Keeping the nonlocal interaction energy fixed, while localising the graph structure, we are able to prove evolutionary {\Gamma}-convergence to an Otto-Wassertein-type gradient flow with a tensor-weighted, yet symmetric, inner product. As a byproduct this implies the existence of solutions to the multi-species non-local (cross-)interacation system on the tensor-weighted Euclidean space

2.Large-time asymptotics for degenerate cross-diffusion population models with volume filling

Authors:Xiuqing Chen, Ansgar Jüngel, Xi Lin, Ling Liu

Abstract: The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including "iterated degenerate" functions.

3.Stability of a one-dimensional full viscous quantum hydrodynamic system

Authors:Xiaoying Han, Yuming Qin, Wenlong Sun

Abstract: A full viscous quantum hydrodynamic system for particle density, current density, energy density and electrostatic potential coupled with a Poisson equation in one dimensional bounded intervals is studied. First, the existence and uniqueness of a steady-state solution to the quantum hydrodynamic system is established. Then, utilizing the fact that the third order perturbation term has an appropriate sign, the local-in-time existence of the solution is investigated by introducing a fourth order viscous regularization and using the entropy dissipation method. In the end, the exponential stability of the steady-state solution is shown by constructing a uniform a-priori estimate.

4.Variational principles in quaternionic analysis with applications to the stationary MHD equations

Authors:Paula Cerejeiras, Uwe Kaehler, Rolf Soeren Krausshar

Abstract: In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder's fixed point theorem. The advantage of the approach using Schauder's fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These consideration will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

5.Characterization of solutions of a generalized Helmholtz problem

Authors:Daniel Hauer, David Lee

Abstract: In this article, we classify all distributional solutions of $f(-\Delta)u=f(1)u$ where $f$ is a non-constant Bernstein function. Specifically, we show that the Fourier transform of $u$ is a single-layer distribution on the unit sphere. Examples of such operators include $(-\Delta)^\sigma$ (for $\sigma \in (0,1]$), $\log(1-\Delta)$ and $(-\Delta)^\frac{1}{2}\text{tanh}((-\Delta)^\frac{1}{2})$.

6.Stability of the quermassintegral inequalities in hyperbolic space

Authors:Prachi Sahjwani, Julian Scheuer

Abstract: For the quermassintegral inequalities of horospherically convex hypersurfaces in the $(n+1)$-dimensional hyperbolic space, where $n\geq 2$, we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in the quermassintegral inequality. The exponent of the deficit is explicitly given and does not depend on the dimension. The estimate is valid in the class of domains with upper and lower bound on the inradius and an upper bound on a curvature quotient. This is achieved by some new initial value independent curvature estimates for locally constrained flows of inverse type.

7.Upscaling and Effective Behavior for Two-Phase Porous-Medium Flow using a Diffuse Interface Model

Authors:Mathis Kelm, Carina Bringedal, Bernd Flemisch

Abstract: We investigate two-phase flow in porous media and derive a two-scale model, which incorporates pore-scale phase distribution and surface tension into the effective behavior at the larger Darcy scale. The free-boundary problem at the pore scale is modeled using a diffuse interface approach in the form of a coupled Allen-Cahn Navier-Stokes system with an additional momentum flux due to surface tension forces. Using periodic homogenization and formal asymptotic expansions, a two-scale model with cell problems for phase evolution and velocity contributions is derived. We investigate the computed effective parameters and their relation to the saturation for different fluid distributions, in comparison to commonly used relative permeability saturation curves. The two-scale model yields non-monotone relations for relative permeability and saturation. The strong dependence on local fluid distribution and effects captured by the cell problems highlights the importance of incorporating pore-scale information into the macro-scale equations.

8.Asymptotic limits of the principal spectrum point of a nonlocal dispersal cooperative system and application to a two-stage structured population model

Authors:Maria A. Onyido, Rachidi B. Salako, Markjoe O. Uba, Cyril I. Udeani

Abstract: This work examines the limits of the principal spectrum point, $\lambda_p$, of a nonlocal dispersal cooperative system with respect to the dispersal rates. In particular, we provide precise information on the sign of $\lambda_p$ as one of the dispersal rates is : (i) small while the other dispersal rate is arbitrary, and (ii) large while the other is either also large or fixed. We then apply our results to study the effects of dispersal rates on a two-stage structured nonlocal dispersal population model whose linearized system at the trivial solution results in a nonlocal dispersal cooperative system. The asymptotic profiles of the steady-state solutions with respect to the dispersal rates of the two-stage nonlocal dispersal population model are also obtained. Some biological interpretations of our results are discussed.

9.Autonomous and asymptotically quasiconvex functionals with general growth conditions

Authors:Francesca Angrisani

Abstract: We obtain local regularity for minimizers of autonomous vectorial integrals of the Calculus of Variations, assuming $\psi$-growth hypothesis and imposing $\varphi$ - quasiconvexity assumptions only in asymptotic sense, both in the sub-quadratic and the super-quadratic case. In particular we obtain $C^{1,\alpha}$ regularity at points $x_0$ such that $Du$ is large enough around $x_0$ and clearly Lipschitz regularity on a dense set. \\ The results hold for all couple of Young functions $(\varphi,\psi)$ with $\Delta_2$ condition.

10.Continuous Data Assimilation for the 3D and Higher-Dimensional Navier--Stokes equations with Higher-Order Fractional Diffusion

Authors:Adam Larios, Collin Victor

Abstract: We study the use of the Azouani-Olson-Titi (AOT) continuous data assimilation algorithm to recover solutions of the Navier--Stokes equations modified to have higher-order fractional diffusion. The fractional diffusion case is of particular interest, as it is known to be globally well-posed for sufficiently large diffusion exponent $\alpha$. In this work, we prove that the assimilation equations are globally well-posed, and we demonstrate that the solutions produced by the AOT algorithm exhibit exponential convergence in time to the reference solution, given a sufficiently high spatial resolution of observations and a sufficiently large nudging parameter. We also note that the results hold in spatial dimensions $d$ where $2\leq d\leq 8$, so long as $\alpha\geq \frac12 +\frac{d}{4}$. Though the cases $3<d\leq8$ are likely only a mathematical curiosity, we include them as they cause no additional difficulty in the proof. Note that we show in a companion paper the $d=2$ case allows for $\alpha<1$.

1.Nodal solutions with synchronous sign changing components and Constant sign solutions for singular Gierer-Meinhardt type system

Authors:Abdelkrim Moussaoui

Abstract: We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant-sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray-Schauder topological degree involving perturbation argument.

2.Semiconvexity estimates for nonlinear integro-differential equations

Authors:Xavier Ros-Oton, Clara Torres-Latorre, Marvin Weidner

Abstract: In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro-differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabr\'e-Dipierro-Valdinoci [CDV22]. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.

3.The aggregation-diffusion equation with the intermediate exponent

Authors:Shen Bian, Jiale Bu

Abstract: We consider a Keller-Segel model with non-linear porous medium type diffusion and nonlocal attractive power law interaction, focusing on potentials that are less singular than Newtonian interaction. Here, the nonlinear diffusion is chosen to be $\frac{2d}{d+2s}<m<2-\frac{2s}{d}$ in which case the steady states are compactly supported. We analyse under which conditions on the initial data the regime that attractive forces are stronger than diffusion occurs and classify the global existence and finite time blow-up of solutions. It is shown that there is a threshold value which is characterized by the optimal constant of a variant of Hardy-Littlewood-Sobolev inequality such that the solution will exist globally if the initial data is below the threshold, while the solution blows up in finite time when the initial data is above the threshold.

4.Phenotype divergence and cooperation in isogenic multicellularity and in cancer

Authors:Frank Alvarez CEREMADE, INSA Toulouse, Jean Clairambault MAMBA, LJLL

Abstract: We discuss the mathematical modelling of two of the main mechanisms which pushed forward the emergence of multicellularity: phenotype divergence in cell differentiation, and between-cell cooperation. In line with the atavistic theory of cancer, this disease being specific of multicellular animals, we set special emphasis on how both mechanisms appear to be reversed, however not totally impaired, rather hijacked, in tumour cell populations. Two settings are considered: the completely innovating, tinkering, situation of the emergence of multicellularity in the evolution of species, which we assume to be constrained by external pressure on the cell populations, and the completely planned-in the body plan-situation of the physiological construction of a developing multicellular animal from the zygote, or of bet hedging in tumours, assumed to be of clonal formation, although the body plan is largely-but not completely-lost in its constituting cells. We show how cancer impacts these two settings and we sketch mathematical models for them. We present here our contribution to the question at stake with a background from biology, from mathematics, and from philosophy of science.

5.Quantitative stability of a nonlocal Sobolev inequality

Authors:Paolo Piccione, Minbo Yang, Shuneng Zhao

Abstract: In this paper, we study the quantitative stability of the nonlocal Soblev inequality \begin{equation*} S_{HL}\left(\int_{\mathbb{R}^N}\big(|x|^{-\mu} \ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}} dx\right)^{\frac{1}{2_{\mu}^{\ast}}}\leq\int_{\mathbb{R}^N}|\nabla u|^2 dx , \quad \forall~u\in \mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation*} where $2_{\mu}^{\ast}=\frac{2N-\mu}{N-2}$ and $S_{HL}$ is a positive constant depending only on $N$ and $\mu$. For $N\geq3$, and $0<\mu<N$, it is well-known that, up to translation and scaling, the nonlocal Soblev inequality has a unique extremal function $W[\xi,\lambda]$ which is positive and radially symmetric. We first prove a result of quantitative stability of the nonlocal Soblev inequality with the level of gradients. Secondly, we also establish the stability of profile decomposition to the Euler-Lagrange equation of the above inequality for nonnegative functions. Finally we study the stability of the nonlocal Soblev inequality \begin{equation*} \Big\|\nabla u-\sum_{i=1}^{\kappa}\nabla W[\xi_i,\lambda_i]\Big\|_{L^2}\leq C\Big\|\Delta u+\left(\frac{1}{|x|^{\mu}}\ast |u|^{2_{\mu}^{\ast}}\right)|u|^{2_{\mu}^{\ast}-2}u\Big\|_{(\mathcal{D}^{1,2}(\mathbb{R}^N))^{-1}} \end{equation*} with the parameter region $\kappa\geq2$, $3\leq N<6-\mu$, $\mu\in(0,N)$ satisfying $0<\mu\leq4$, or dimension $N\geq3$ and $\kappa=1$, $\mu\in(0,N)$ satisfying $0<\mu\leq4$.

6.On the spectrum of sets made of cores and tubes

Authors:Francesca Bianchi, Lorenzo Brasco, Roberto Ognibene

Abstract: We analyze the spectral properties of a particular class of unbounded open sets. These are made of a central bounded ``core'', with finitely many unbounded tubes attached to it. We adopt an elementary and purely variational point of view, studying the compactness (or the defect of compactness) of level sets of the relevant constrained Dirichlet integral. As a byproduct of our argument, we also get exponential decay at infinity of variational eigenfunctions. Our analysis includes as a particular case a planar set (sometimes called ``bookcover''), already encountered in the literature on curved quantum waveguides. J. Hersch suggested that this set could provide the sharp constant in the {\it Makai-Hayman inequality} for the bottom of the spectrum of the Dirichlet-Laplacian of planar simply connected sets. We disprove this fact, by means of a singular perturbation technique.

7.On a class of generalised solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids: existence and macroscopic closure

Authors:Tomasz Dębiec, Endre Süli

Abstract: We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain, coupled to a Fokker-Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent. The micro-macro interaction is reflected by the presence of a drag term in the Fokker-Planck equation and the divergence of a polymeric extra-stress tensor in the Navier-Stokes balance of momentum equation. We introduce the concept of generalised dissipative solution - a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation. This defect measure accounts for the lack of compactness in the polymeric extra-stress tensor. We prove global existence of generalised dissipative solutions satisfying additionally an energy inequality for the macroscopic deformation tensor. Using this inequality, we establish a conditional regularity result: any generalised dissipative solution with a sufficiently regular velocity field is a weak solution to the Hookean dumbbell model. Additionally, in two space dimensions we provide a rigorous derivation of the macroscopic closure of the Hookean model and discuss its relationship with the Oldroyd-B model with stress diffusion. Finally, we improve an earlier result by Barrett and S\"{u}li by establishing the global existence of weak solutions for a larger class of initial data.

8.A fractional Willmore-type energy functional -- subcritical observations

Authors:Simon Blatt, Giovanni Giacomin, Julian Scheuer, Armin Schikorra

Abstract: We investigate surfaces with bounded L^p-norm of the fractional mean curvature, a quantity we shall refer to as fractional Willmore-type functional. In the subcritical case and under convexity assumptions we show how this Willmore-functional controls local parametrization, and conclude as consequences lower Ahlfors-regularity, a weak Michael-Simon type inequality, and an application to stability.

9.L2 to Lp bounds for spectral projectors on thin intervals in Riemannian manifolds

Authors:Pierre Germain

Abstract: Given a Riemannian manifold endowed with its Laplace-Beltrami operator, consider the associated spectral projector on a thin interval. As an operator from L2 to Lp, what is its operator norm? For a window of size 1, this question is fully answered by a celebrated theorem of Sogge, which applies to any manifold. For smaller windows, the global geometry of the manifold comes into play, and connections to a number of mathematical fields (such as Differential Geometry, Combinatorics, Number Theory) appear, but the problem remains mostly open. The aim of this article is to review known results, focusing on cases exhibiting symmetry and emphasizing harmonic analytic rather than microlocal methods.

10.On the global bifurcation diagram of the equation $-Δ u=μ|x|^{2α}e^u$ in dimension two

Authors:Daniele Bartolucci, Aleks Jevnikar, Ruijun Wu

Abstract: The aim of this note is to present the first qualitative global bifurcation diagram of the equation $-\Delta u=\mu|x|^{2\alpha}e^u$. To this end, we introduce the notion of domains of first/second kind for singular mean field equations and base our approach on a suitable spectral analysis. In particular, we treat also non-radial solutions and non-symmetric domains and show that the shape of the branch of solutions still resembles the well-known one of the model regular radial case on the disk. Some work is devoted also to the asymptotic profile for $\mu\to-\infty$.

11.A Note on $L^1-$contractive property of the solutions of the scalar conservation laws through the method by Lax-Oleĭnik

Authors:Abhishek Adimurthi

Abstract: In this note, we study the $L^1-$contractive property of the solutions the scalar conservation laws, got by the method of Lax-{O}le\u{\i}nik. First, it is proved when f is merely convex and the initial data is in $L^{\infty}(\mathbb{R})$. And then, it is shown for the case when the initial data is in $L^1(\mathbb{R})$ with the convex flux having super-linear growth. Finally, the $L^1-$contractive property is shown for the scalar conservation laws with the initial data in $L^1(\mathbb{R})$ and the flux is "semi-super-linear". This entire note does not assume any results mentioned through the approach by Kruzkov.

12.Boundedness, Ultracontractive Bounds and Optimal Evolution of the Support for Doubly Nonlinear Anisotropic Diffusion

Authors:Simone Ciani, Vincenzo Vespri, Matias Vestberg

Abstract: We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is \begin{align*} \partial_t \big(|u|^{\alpha -1}u \big) - \sum^N_{i=1} \partial_i \big( |\partial_i u|^{p_i - 2} \partial_i u \big) = 0, \end{align*} where $\alpha \in (0,1)$ and $p_i \in (1, \infty)$. We obtain super and ultracontractive bounds, and global boundedness in space for solutions to the Cauchy problem with initial data in $L^{\alpha+1}(\mathbb{R}^N)$, and show that the mass is nonincreasing over time. As a consequence, compactly supported evolution is shown for optimal exponents. We introduce a seemingly new paradigm, by showing that Caccioppoli estimates, local boundedness and semicontinuity are consequences of the membership to a suitable energy class. This membership is proved by first establishing the continuity of the map $t \mapsto |u|^{\alpha-1}u(\cdot,t) \in L^{1+1/\alpha}_{loc}(\Omega)$ permitting us to use a suitable mollified weak formulation along with an appropriate test function.

1.Partial Data Inverse Problems for the Nonlinear Schrödinger Equation

Authors:Ru-Yu Lai, Xuezhu Lu, Ting Zhou

Abstract: In this paper we prove the uniqueness and stability in determining a time-dependent nonlinear coefficient $\beta(t, x)$ in the Schr\"odinger equation $(i\partial_t + \Delta + q(t, x))u + \beta u^2 = 0$, from the boundary Dirichlet-to-Neumann (DN) map. In particular, we are interested in the partial data problem, in which the DN-map is measured on a proper subset of the boundary. We show two results: a local uniqueness of the coefficient at the points where certain type of geometric optics (GO) solutions can reach; and a stability estimate based on the unique continuation property for the linear equation.

2.Regularity theory for nonlocal obstacle problems with critical and subcritical scaling

Authors:Alessio Figalli, Xavier Ros-Oton, Joaquim Serra

Abstract: Despite significant recent advances in the regularity theory for obstacle problems with integro-differential operators, some fundamental questions remained open. On the one hand, there was a lack of understanding of parabolic problems with critical scaling, such as the obstacle problem for $\partial_t+\sqrt{-\Delta}$. No regularity result for free boundaries was known for parabolic problems with such scaling. On the other hand, optimal regularity estimates for solutions (to both parabolic and elliptic problems) relied strongly on monotonicity formulas and, therefore, were known only in some specific cases. In this paper, we present a novel and unified approach to answer these open questions and, at the same time, to treat very general operators, recovering as particular cases most previously known regularity results on nonlocal obstacle problems.

3.Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces

Authors:Ophélie Cuvillier, Francesco Fanelli, Elena Salguero

Abstract: In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain $\mathbb{T}^d$, for space dimensions $d=2,3$. We admit the average turbulent kinetic energy $k$ to vanish in part of the domain, \textsl{i.e.} we consider the case $k \geq 0$; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces $H^s$, for any $s>1+d/2$. We expect this regularity to be optimal, due to the degeneracy of the system when $k \approx 0$. We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the non-linear terms involved in the computations.

4.Pad{é}-type High-Order Absorbing Boundary Condition for a Coupled Hydrodynamic Wave Model with Surface Tension Effect

Authors:Olivier Wilk M2N

Abstract: This paper presents a specific way to develop High-Order Absorbing Boundary Conditions (HABC) of the Pad{\'e} family on a Coupled Hydrodynamic Wave Model (CHWM) especially with surface tension effect (small scales in space). Inspired by the Neumann-Kelvin model, the CHWM is composed by a fluid - basin model to allow to use multiple objects below the surface coupled to a free surface model with a small added mass surface term. With the surface tension effect, we introduce new coefficients (similar as Higdon coefficients) on each HABC (for the surface model and the basin model) to ensure the continuity of the two HABC at the interface between the coupled models. So, we propose a useful specific compatibility condition, and a strong reduction of the Pad{\'e} approximation in particular in the water case.

5.A note on the Long-Time behaviour of Stochastic McKean-Vlasov Equations with common noise

Authors:Raphael Maillet CEREMADE

Abstract: This paper presents an investigation into the long-term behaviour of solutions to a nonlinear stochastic McKean-Vlasov equation with common noise. The equation arises naturally in themean-field limit of systems composed of interacting particles subject to both idiosyncratic and common noise. Initially, we demonstrate that the addition of common diffusion in each particle's dynamics does not disrupt the established stability results observed in the absence of common noise.However, our main objective is to understand how the presence of common noise can restore theuniqueness of equilibria. Specifically, in a non-convex landscape, we establish uniqueness and convergence towards equilibria under two specific conditions: (1) when the dimension of the ambientspace equals 1, and (2) in the absence of idiosyncratic noise in the system.

6.Approximation and Homogenization of Thermoelastic wave model

Authors:Salem Nafiri

Abstract: This paper deals with the approximation and homogenization of thermoelastic wave model. First, we study the homogenization problem of a weakly coupled thermoelastic wave model with rapidly varying coefficients, using a semigroup approach, two-scale convergence method and some variational techniques. We show that the limit semigroup can be obtained by using a weak version of the Trotter Kato convergence Theorem. Secondly, we consider the approximation of two thermoelastic wave model, one with exponential decay and the other one with polynomial decay. the numerical experiments indicate that the two discrete systems show different behavior of the spectra. Moreover, their discrete energies inherit the same behavior of the continuous ones. Finally we show numerically how the smoothness of data can impact the rate of decay of the energy associated the weakly coupled thermoelastic wave model.

1.Finite time blow-up of non-radial solutions for some inhomogeneous Schrödinger equations

Authors:Ruobing Bai, Tarek Saanouni

Abstract: This work studies the inhomogeneous Schr\"odinger equation $$ i\partial_t u-\mathcal{K}_{s,\lambda}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, $s\in\{1,2\}$, $N>2s$ and $\lambda>-\frac{(N-2)^2}{4}$. The linear Schr\"odinger operator reads $\mathcal{K}_{s,\lambda}:= (-\Delta)^s +(2-s)\frac{\lambda}{|x|^2}$ and the focusing source term is local or non-local $$F(x,u)\in\{|x|^{-2\tau}|u|^{2(q-1)}u,|x|^{-\tau}|u|^{p-2}(J_\alpha *|\cdot|^{-\tau}|u|^p)u\}.$$ The Riesz potential is $J_\alpha(x)=C_{N,\alpha}|x|^{-(N-\alpha)}$, for certain $0<\alpha<N$. The singular decaying term $|x|^{-2\tau}$, for some $\tau>0$ gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely $1+\frac{2(1-\tau)}N<q<1+\frac{2(1-\tau)}{N-2s}$ and $1+\frac{2-2\tau+\alpha}{N}<p<1+\frac{2-2\tau+\alpha}{N-2s}$. The purpose is to prove the finite time blow-up of solutions with datum in the energy space, non necessarily radial or with finite variance. The assumption on the data is expressed in terms of non-conserved quantities. This is weaker than the ground state threshold standard condition. The blow-up under the ground threshold or with negative energy are consequences. The proof is based on Morawetz estimates and a non-global ordinary differential inequality.

2.Weighted estimates and large time behavior of small amplitude solutions to the semilinear heat equation

Authors:Ryunosuke Kusaba, Tohru Ozawa

Abstract: We present a new method to obtain weighted $L^{1}$-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in $\mathbb{R}^{n}$, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions.

3.Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system

Authors:Yoshikazu Giga, Ayato Kubo, Hirotoshi Kuroda, Jun Okamoto, Koya Sakakibara, Masaaki Uesaka

Abstract: This paper is concerned with a singular limit of the Kobayashi-Warren-Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica-Mortola functional in a one-dimensional setting.

4.On $p$-Dirac Equation on Compact Spin Manifolds

Authors:Lei Xian, Xu Yang

Abstract: By using the Ljusternik-Schnirelman principle, we establish the existence of a nondecreasing sequence of nonnegative eigenvalues for the p-Dirac operator on compact spin manifold. Using the biorthogonal system theory on separable Banach space and some critical point theorems, we prove the existence and multiplicity of solutions to p-superlinear and p-sublinear nonlinear p-Dirac equations on compact spin manifold.

5.Growth of Sobolev norms and strong convergence for the discrete nonlinear Schr{ö}dinger equation

Authors:Quentin Chauleur LPP, Paradyse

Abstract: We show the strong convergence in arbitrary Sobolev norms of solutions of the discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the nonlinear Schr{\"o}dinger equation on the whole space. We restrict our attention to the one and two-dimensional case, with a set of parameters which implies global well-posedness for the continuous equation. Our proof relies on the use of bilinear estimates for the Shannon interpolation as well as the control of the growth of discrete Sobolev norms that we both prove.

6.The $2D$ nonlinear shallow water equations with a partially immersed obstacle

Authors:David Lannes IMB, Tatsuo Iguchi KEIO UNIVERSITY

Abstract: This article is devoted to the proof of the well-posedness of a model describing waves propagating in shallow water in horizontal dimension $d=2$ and in the presence of a fixed partially immersed object. We first show that this wave-interaction problem reduces to an initial boundary value problem for the nonlinear shallow water equations in an exterior domain, with boundary conditions that are fully nonlinear and nonlocal in space and time. This hyperbolic initial boundary value problem is characteristic, does not satisfy the constant rank assumption on the boundary matrix, and the boundary conditions do not satisfy any standard form of dissipativity. Our main result is the well-posedness of this system for irrotational data and at the quasilinear regularity threshold. In order to prove this, we introduce a new notion of weak dissipativity, that holds only after integration in time and space. This weak dissipativity allows high order energy estimates without derivative loss; the analysis is carried out for a class of linear non-characteristic hyperbolic systems, as well as for a class of characteristic systems that satisfy an algebraic structural property that allows us to define a generalized vorticity. We then show, using a change of {unknowns}, that {it} is possible to transform the linearized wave-interaction {problem} into a non-characteristic system, {which} satisfies this structural property and for which the boundary conditions are weakly dissipative. We can therefore use our general analysis to derive linear, and then nonlinear, a priori energy estimates. Existence for the linearized problem is obtained by a regularization procedure that makes the problem non-characteristic and strictly dissipative, and by the approximation of the data by more regular data satisfying higher order compatibility conditions for the regularized problem. Due to the fully nonlinear nature of the boundary conditions, it is also necessary to implement a quasilinearization procedure. Finally, we have to lower the standard requirements on the regularity of the coefficients of the operator in the linear estimates to be able to reach the quasilinear regularity threshold in the nonlinear well-posedness result.

7.Linearization and localization of nonconvex functionals motivated by nonlinear peridynamic models

Authors:Tadele Mengesha, James M. Scott

Abstract: We consider a class of nonconvex energy functionals that lies in the framework of the peridynamics model of continuum mechanics. The energy densities are functions of a nonlocal strain that describes deformation based on pairwise interaction of material points, and as such are nonconvex with respect to nonlocal deformation. We apply variational analysis to investigate the consistency of the effective behavior of these nonlocal nonconvex functionals with established classical and peridynamic models in two different regimes. In the regime of small displacement, we show the model can be effectively described by its linearization. To be precise, we rigorously derive what is commonly called the linearized bond-based peridynamic functional as a $\Gamma$-limit of nonlinear functionals. In the regime of vanishing nonlocality, the effective behavior the nonlocal nonconvex functionals is characterized by an integral representation, which is obtained via $\Gamma$-convergence with respect to the strong $L^p$ topology. We also prove various properties of the density of the localized quasiconvex functional such as frame-indifference and coercivity. We demonstrate that the density vanishes on matrices whose singular values are less than or equal to one. These results confirm that the localization, in the context of $\Gamma$-convergence, of peridynamic-type energy functionals exhibit behavior quite different from classical hyperelastic energy functionals.

8.On fractional quasilinear equations with elliptic degeneracy

Authors:Damião J. Araújo, Disson dos Prazeres, Erwin Topp

Abstract: In this work, we develop a systematic approach to study existence, multiplicity, and local gradient regularity estimates for solutions of nonlocal quasilinear equations with local gradient degeneracy. We develop an interactive geometric argument that interplays with uniqueness property for the corresponding homogeneous problem, in which, gradient H\"older regularity estimates are obtained. This machinery is intrinsically made for nonlocal scenarios since for the local ones solutions to the homogeneous problem are unique. We illustrate our results by exhibiting classes of exterior data in which multiplicity of solutions are observed, showing in parallel, relevant cases where uniqueness is verified.

9.Wrinkling of an elastic sheet floating on a liquid sphere

Authors:Peter Bella, Carlos Román

Abstract: A thin circular elastic sheet floating on a drop-like liquid substrate gets deformed due to incompatibility between the curved substrate and the planar sheet. We adopt a variational viewpoint by minimizing the non-convex membrane energy plus a higher-order convex bending energy. Being interested in thin sheets, we expand the minimum of the energy in terms of a small thickness $h$, and identify the first two terms of this expansion. The leading order term comes from a minimization of a family of one-dimensional ``relaxed'' problems, while for the next-order term we only identify its scaling law. This generalizes the previous work [P. Bella and R.V. Kohn. Wrikling of a thin circular sheet bonded to a spherical substrate, Philos. Trans. Roy. Soc. A, 375(2017). arXiv:1611.01781] to the physically relevant case of a liquid substrate.

10.Towards the optimality of the ball for the Rayleigh Conjecture concerning the clamped plate

Authors:Roméo Leylekian

Abstract: In 1995, Nadirashvili and subsequently Ashbaugh and Benguria proved the Rayleigh Conjecture concerning the first eigenvalue of the bilaplacian with clamped boundary conditions in dimension $2$ and $3$. Since then, the conjecture has remained open in dimension $d>3$. In this document, we contribute in answering the conjecture under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti's comparison principle, made possible after a fine study of the geometry of the eigenfunction's nodal domains.

11.Weak and parabolic solutions of advection-diffusion equations with rough velocity field

Authors:Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa

Abstract: We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

12.A proof of Guo-Wang's conjecture on the uniqueness of positive harmonic functions in the unit ball

Authors:Pingxin Gu, Haizhong Li

Abstract: Guo-Wang [Calc.Var.Partial Differential Equations,59(2020)] conjectured that for $1<q<\frac{n}{n-2}$ and $0<\lambda\leq \frac{1}{q-1}$, the positive solution $u\in C^{\infty}(\bar B)$ to the equation \[ \left\{ \begin{array}{ll} \Delta u=0 &in\ B^n,\\ u_{\nu}+\lambda u=u^q&on\ S^{n-1}, \end{array} \right. \] must be constant. In this paper, we give a proof of this conjecture.

13.Transmission problems: regularity theory, interfaces and beyond

Authors:Vincenzo Bianca, Edgard A. Pimentel, José Miguel Urbano

Abstract: Modelling diffusion processes in heterogeneous media requires addressing inherent discontinuities across interfaces, where specific conditions are to be met. These challenges fall under the purview of Mathematical Analysis as \emph{transmission problems}. We present a pa\-no\-ra\-ma of the theory of transmission problems, encompassing the seminal contributions from the 1950s and subsequent developments. Then we delve into the discussion of regularity issues, including recent advances matching the minimal regularity requirements of interfaces and the optimal regularity of the solutions. A discussion on free transmission problems closes the survey.

14.Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

Authors:Noah Stevenson, Ian Tice

Abstract: We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a generic phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form. In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

15.Fourier Analysis on $\mathbb{T}^m\times\mathbb{R}^n$ and Applications to Global Hypoellipticity

Authors:André Pedroso Kowacs

Abstract: This article presents a convenient approach to Fourier analysis for the investigation of functions and distributions defined in $\mathbb{T}^m \times \mathbb{R}^n$. Our approach involves the utilization of a mixed Fourier transform, incorporating both partial Fourier series on the torus for the initial variables and partial Fourier transform in Euclidean space for the remaining variables. By examining the behaviour of the mixed Fourier coefficients, we achieve a comprehensive characterization of the spaces of smooth functions and distributions in this context. Additionally, we apply our results to derive necessary and sufficient conditions for the global hypoellipticity of a class first order differential operators defined on $\mathbb{T}^m \times \mathbb{R}^n$, including all constant coefficient first order differential operators.

16.Global Properties for first order differential operators on $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$

Authors:André Pedroso Kowacs, Alexandre Kirilov, Wagner Augusto Almeida de Moraes

Abstract: In this paper, we study the global properties of a class of evolution-like differential operator with a 0-order perturbation defined on the product of $r+1$ tori and $s$ spheres $\mathbb{T}^{r+1}\times(\mathbb{S}^{3})^s$, with $r$ and $s$ non-negative integers. By varying the values of $r$ and $s$, we show that it is possible to recover results already known in the literature and present new results. The main tool used in this study is Fourier analysis, taken partially with respect to each copy of the torus and sphere. We obtain necessary and sufficient conditions related to Diophantine inequalities, change of sign and connectivity of level sets associated the operator's coefficients.

17.Mathematical foundations of the non-Hermitian skin effect

Authors:Habib Ammari, Silvio Barandun, Jinghao Cao, Bryn Davies, Erik Orvehed Hiltunen

Abstract: We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

1.Boundary Strichartz estimates and pointwise convergence for orthonormal systems

Authors:Neal Bez, Shinya Kinoshita, Shobu Shiraki

Abstract: We consider maximal estimates associated with fermionic systems. First we establish maximal estimates with respect to the spatial variable. These estimates are certain boundary cases of the many-body Strichartz estimates pioneered by Frank, Lewin, Lieb and Seiringer. We also prove new maximal-in-time estimates, thereby significantly extending work of Lee, Nakamura and the first author on Carleson's pointwise convergence problem for fermionic systems.

2.A note on Strichartz estimates for the wave equation with orthonormal initial data

Authors:Neal Bez, Shinya Kinoshita, Shobu Shiraki

Abstract: This note is concerned with Strichartz estimates for the wave equation and orthonormal families of initial data. We provide a survey of the known results and present what seems to be a reasonable conjecture regarding the cases which have been left open. We also provide some new results in the maximal-in-space boundary cases.

3.Norm inflation with infinite loss of regularity for the generalized improved Boussinesq equation

Authors:Pierre de Roubin

Abstract: In this paper, we study the ill-posedness issue for the generalized improved Boussinesq equation. In particular we prove there is norm inflation with infinite loss of regularity at general initial data in $\langle \nabla \rangle^{-s}\big(L^2 \cap L^\infty\big)(\mathbb{R})$ for any $s < 0$. This result is sharp in the $L^2$-based Sobolev scale in view of the well-posedness in $L^2(\mathbb{R}) \cap L^\infty(\mathbb{R})$. We also show that the same result applies to the multi-dimensional generalized improved Boussinesq equation. Finally, we extend our norm inflation result to Fourier-Lebesgue, modulation and Wiener amalgam spaces.

4.Finite-strain poro-visco-elasticity with degenerate mobility

Authors:Willem J. M. van Oosterhout, Matthias Liero

Abstract: A quasistatic nonlinear model for poro-visco-elastic solids at finite strains is considered in the Lagrangian frame using the concept of second-order nonsimple materials. The elastic stresses satisfy static frame-indifference, while the viscous stresses satisfy dynamic frame-indifference. The mechanical equation is coupled to a diffusion equation for a solvent or fluid content. The latter is pulled-back to the reference configuration. To treat the nonlinear dependence of the mobility tensor on the deformation gradient, the result by Healey & Kr\"omer is used to show that the determinant of the deformation gradient is bounded away from zero. Moreover, the focus is on the physically relevant case of degenerate mobilities. The existence of weak solutions is shown using a staggered time-incremental scheme and suitable energy-dissipation inequalities.

5.On Scaling Properties for a Class of Two-Well Problems for Higher Order Homogeneous Linear Differential Operators

Authors:Bogdan Raiţă, Angkana Rüland, Camillo Tissot, Antonio Tribuzio

Abstract: We study the scaling behaviour of a class of compatible two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in direction of the associated element in the wave cone. We complement the lower bound estimates by a detailed analysis of the two-well problem for generalized (tensor-valued) symmetrized derivatives with the help of the (tensor-valued) Saint-Venant compatibility conditions. In two spatial dimensions (but arbitrary tensor order $m\in \mathbb{N}$) we provide upper bound constructions matching the lower bound estimates. This illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone. The scaling for the symmetrized gradient from \cite{CC15} which was also discussed in \cite{RRT23} provides an example of this family of new scaling laws.

6.Blow-up result for a weakly coupled system of wave equations with a scale-invariant damping, mass term and time derivative nonlinearity

Authors:Mohamed Fahmi Ben Hassen, Makram Hamouda, Mohamed Ali Hamza

Abstract: We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms. The latter are specifically of great interest from both physical and mathematical points of view since they allow the speeds of propagation to be time-dependent ones. However, we assume in this work that both waves are propagating with the same speeds. Employing this fact together with other hypotheses on the aforementioned parameters (mass and damping coefficients), we obtain a new blow-up region for the system under consideration, and we show a lifespan estimate of the maximal existence time.

7.Nonlinear spectral problem for Hörmander vector fields

Authors:Mukhtar Karazym, Durvudkhan Suragan

Abstract: Based on variational methods, we study a nonlinear eigenvalue problem for a $p$-sub-Laplacian type quasilinear operator arising from smooth H\"ormander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction and show H\"older regularity of eigenfunctions. Moreover, we determine the best constant for the $L^{p}$-Poincar\'e inequality as a byproduct.

1.Viscous shocks and long-time behavior of scalar conservation laws

Authors:Thierry Gallay, Arnd Scheel

Abstract: We study the long-time behavior of scalar viscous conservation laws via the structure of $\omega$-limit sets. We show that $\omega$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $\omega$-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.

2.Evolution of crystalline thin films by evaporation and condensation in three dimensions

Authors:Paolo Piovano, Francesco Sapio

Abstract: The morphology of crystalline thin films evolving on flat rigid substrates by condensation of extra film atoms or by evaporation of their own atoms in the surrounding vapor is studied in the framework of the theory of Stress Driven Rearrangement Instabilities (SDRI). By following the SDRI literature both the elastic contributions due to the mismatch between the film and the substrate lattices at their theoretical (free-standing) elastic equilibrium, and a curvature perturbative regularization preventing the problem to be ill-posed due to the otherwise exhibited backward parabolicity, are added in the evolution equation. The resulting Cauchy problem under investigation consists in an anisotropic mean-curvature type flow of the fourth order on the film profiles, which are assumed to be parametrizable as graphs of functions measuring the film thicknesses, coupled with a quasistatic elastic problem in the film bulks. Periodic boundary conditions are considered. The results are twofold: the existence of a regular solution for a finite period of time and the stability for all times, of both Lyapunov and asymptotic type, of any configuration given by a flat film profile and the related elastic equilibrium. Such achievements represent both the generalization to three dimensions of a previous result in two dimensions for a similar Cauchy problem, and the complement of the analysis previously carried out in the literature for the symmetric situation in which the film evolution is not influenced by the evaporation-condensation process here considered, but it is entirely due to the volume preserving surface-diffusion process, which is instead here neglected. The method is based on minimizing movements, which allow to exploit the the gradient-flow structure of the evolution equation.

3.Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane-Emden equation in dimension two

Authors:Zhijie Chen, Zetao Cheng, Hanqing Zhao

Abstract: We study the Lane-Emden problem \[\begin{cases} -\Delta u_p =|u_p|^{p-1}u_p&\text{in}\quad \Omega, u_p=0 &\text{on}\quad\partial\Omega, \end{cases}\] where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. We obtain sharp estimates and non-degeneracy of low energy nodal solutions $u_p$ (i.e. nodal solutions satisfying $\lim_{p\to+\infty}p\int_{\Omega}|u_p|^{p+1}dx=16\pi e$). As applications, we prove that the comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$ holds automatically for least energy nodal solutions, which confirms a conjecture raised by (Grossi-Grumiau-Pacella, Ann.I.H. Poincare-AN, 30 (2013), 121-140) and (Grossi-Grumiau-Pacella, J.Math.Pures Appl. 101 (2014), 735-754).

4.Traveling Waves of the Vlasov--Poisson System

Authors:Masahiro Suzuki, Masahiro Takayama, Katherine Zhiyuan Zhang

Abstract: We consider the Vlasov--Poisson system describing a two-species plasma with spatial dimension $1$ and the velocity variable in $\mathbb{R}^n$. We find the necessary and sufficient conditions for the existence of solitary waves, shock waves, and wave trains of the system, respectively. To this end, we need to investigate the distribution of ions trapped by the electrostatic potential. Furthermore, we classify completely in all possible cases whether or not the traveling wave is unique. The uniqueness varies according to each traveling wave when we exclude the variant caused by translation. For the solitary wave, there are both cases that it is unique and nonunique. The shock wave is always unique. No wave train is unique.

5.Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains

Authors:Marian Stengl, Patrick Gelß, Stefan Klus, Sebastian Pokutta

Abstract: The Koopman--von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman--von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set's closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman--von Neumann framework to transport equations is highlighted.

6.Stokes waves at the critical depth are modulational unstable

Authors:Massimiliano Berti, Alberto Maspero, Paolo Ventura

Abstract: This paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves -- called Stokes waves -- at the critical Whitham-Benjamin depth $ \mathtt{h}_{\scriptscriptstyle WB} = 1.363... $ and nearby values. We prove that Stokes waves of small amplitude $ \mathcal{O}( \epsilon ) $ are, at the critical depth $ \mathtt{h}_{\scriptscriptstyle WB} $, linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^2 $, $ c > 0 $, depending on the amplitude of the wave. This problem was not rigorously solved in previous literature because the expansions degenerate at the critical depth. In order to resolve this degenerate case, and describe in a mathematically exhaustive way how the eigenvalues change their stable-to-unstable nature along this shallow-to-deep water transient, we Taylor expand the computations of arXiv:2204.00809v2 at a higher degree of accuracy, derived by the fourth order expansion of the Stokes waves. We prove that also in this transient regime a pair of unstable eigenvalues depict a closed figure "8", of smaller size than for $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} $, as the Floquet exponent varies.

7.Classification and stability of positive solutions to the NLS equation on the $\mathcal{T}$-metric graph

Authors:Francisco Agostinho, Simão Correia, Hugo Tavares

Abstract: Given $\lambda>0$ and $p>2$, we present a complete classification of the positive $H^1$-solutions of the equation $-u''+\lambda u=|u|^{p-2}u$ on the $\mathcal{T}$-metric graph (consisting of two unbounded edges and a terminal edge of length $\ell>0$, all joined together at a single vertex). This study implies, in particular, the uniqueness of action ground states and that, for $p\sim 6^-$, the notions of action and energy ground states do not coincide in general. In the $L^2$-supercritical case $p>6$, we prove that, for both $\lambda$ small and large, action ground states are orbitally unstable for the flow generated by the associated time-dependent NLS equation $i\partial_tu + \partial^2_{xx} u + |u|^{p-2}u=0$. Finally, we provide numerical evidence of the uniqueness of energy ground states for $p<6$ and of the existence of both stable and unstable action ground states for $p\sim6$.

8.Nonlinear asymptotic stability and transition threshold for 2D Taylor-Couette flows in Sobolev spaces

Authors:Xinliang An, Taoran He, Te Li

Abstract: In this paper, we investigate the stability of the 2-dimensional (2D) Taylor-Couette (TC) flow for the incompressible Navier-Stokes equations. The explicit form of velocity for 2D TC flow is given by $u=(Ar+\frac{B}{r})(-\sin \theta, \cos \theta)^T$ with $(r, \theta)\in [1, R]\times \mathbb{S}^1$ being an annulus and $A, B$ being constants. Here, $A, B$ encode the rotational effect and $R$ is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor-Couette flow is asymptotically stable, even at high Reynolds number ($Re\sim \nu^{-1}$), with a sharp exponential decay rate of $\exp(-\nu^{\frac13}|B|^{\frac23}R^{-2}t)$ as long as the initial perturbation is less than or equal to $\nu^\frac12 |B|^{\frac12}R^{-2}$ in Sobolev space. The powers of $\nu$ and $B$ in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor-Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient $B$. The larger the $|B|$, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor-Couette flows.

9.Eigenvalue for a problem involving the fractional (p,q)-Laplacian operator and nonlinearity with a singular and a supercritical Sobolev growth

Authors:A. L. A. de Araujo, Aldo H. S. Medeiros

Abstract: In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered involves supercritical Sobolev growth. Our approach is variational togheter with the sub- and supesolution methods, and in this way we can address a wide range of problems not yet contained in the literature. Even when $W^{s_1,p}_0(\Omega) \hookrightarrow L^{\infty}\left(\Omega\right)$ failing, we establish $\|u\|_{L^{\infty}\left(\Omega\right)} \leq C[u]_{s_1,p}$ (for some $C>0$ ), when $u$ is a solution.

1.An elliptic problem of the Prandtl-Batchelor type with a singularity

Authors:Debajyoti Choudhuri, Dušan D. Repovš

Abstract: We establish the existence of at least two solutions of the {\it Prandtl-Batchelor} like elliptic problem driven by a power nonlinearity and a singular term. The associated energy functional is nondifferentiable and hence the usual variational techniques do not work. We shall use a novel approach in tackling the associated energy functional by a sequence of $C^1$ functionals and a {\it cutoff function}. Our main tools are fundamental elliptic regularity theory and the mountain pass theorem.

2.Liouville-type results for some quasilinear anisotropic elliptic equations

Authors:Alberto Farina, Berardino Sciunzi, Domenico Vuono

Abstract: We prove some Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Our results cover the particular case of the pure Finsler p-Laplacian.

3.Spectral projectors on hyperbolic surfaces

Authors:Jean-Philippe Anker, Pierre Germain, Tristan Léger

Abstract: In this paper, we prove $L^2 \to L^p$ estimates, where $p>2$, for spectral projectors on a wide class of hyperbolic surfaces. More precisely, we consider projections in small spectral windows $[\lambda-\eta,\lambda+\eta]$ on geometrically finite hyperbolic surfaces of infinite volume. In the convex cocompact case, we obtain optimal bounds with respect to $\lambda$ and $\eta$, up to subpolynomial losses. The proof combines the resolvent bound of Bourgain-Dyatlov and improved estimates for the Schr\"odinger group (Strichartz and smoothing estimates) on hyperbolic surfaces.

4.Global dynamics of a predator-prey model with alarm-taxis

Authors:Songzhi Li, Kaiqiang Wang

Abstract: This paper concerns with the global dynamics of classical solutions to an important alarm-taxis ecosystem, which demonstrates the behaviors of prey that attract secondary predator when threatened by primary predator. And the secondary predator pursues the signal generated by the interaction of the prey and primary predator. However, it seems that the necessary gradient estimates for global existence cannot be obtained in critical case due to strong coupled structure. Thereby, we develop a new approach to estimate the gradient of prey and primary predator which takes advantage of slightly higher damping power. Then the boundedness of classical solutions in two dimension with Neumann boundary conditions can be established by energy estimates and semigroup theory. Moreover, by constructing Lyapunov functional, it is proved that the coexistence homogeneous steady states is asymptotically stability and the convergence rate is exponential under certain assumptions on the system coefficients.

5.Quantum optimal transport and weak topologies

Authors:Laurent Lafleche

Abstract: Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge--Kantorovich--Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term in the semiclassical approximation, which can be expressed using the Wigner--Yanase Skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data.

6.Global existence of 2D electron MHD near a steady state

Authors:Mimi Dai

Abstract: We study the electron magnetohydrodynamics (MHD) in two dimensional geometry, which has a rich family of steady states. In an anisotropic resistivity context, we show global in time existence of small smooth solution near a shear type steady state. Convergence rate of the solution to the steady state is also obtained.

1.The $τ_N$-configurations and polyconvex gradient flows

Authors:Baisheng Yan

Abstract: We study a generalization of $T_N$-configurations, called the $\tau_N$-configurations, for constructing certain irregular solutions of some nonlinear diffusion systems by the method of convex integration. We construct some polyconvex functions that support a parametrized family of $\tau_N$-configurations satisfying a general openness condition; this will guarantee the existence of nowhere-$C^1$ Lipschitz weak solutions to the initial boundary value problems of the polyconvex gradient flows. We elaborate on such constructions and the subsequent verification of the openness condition when the dimension is at least 4 to avoid some complicated calculations that cannot be done by hand but would otherwise be needed for dimensions 2 and 3.

1.On the critical regularity of nonlinearities for semilinear classical wave equations

Authors:Wenhui Chen, Michael Reissig

Abstract: In this paper, we consider the Cauchy problem for semilinear classical wave equations \begin{equation*} u_{tt}-\Delta u=|u|^{p_S(n)}\mu(|u|) \end{equation*} with the Strauss exponent $p_S(n)$ and a modulus of continuity $\mu=\mu(\tau)$, which provides an additional regularity of nonlinearities in $u=0$ comparing with the power nonlinearity $|u|^{p_S(n)}$. We obtain a sharp condition on $\mu$ as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted $L^{\infty}_tL^{\infty}_r$ estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply the critical regularity of source nonlinearities for semilinear classical wave equations.

2.A blow-up result for semilinear wave equations with modulus of continuity in derivative type nonlinearity

Authors:Wenhui Chen

Abstract: In this paper, we study blow-up of solutions to the Cauchy problem for semilinear classical wave equations with derivative type nonlinearity $|u_t|^{p_{\mathrm{Gla}}(n)}\mu(|u_t|)$ for $n\geqslant 2$, where $p_{\mathrm{Gla}}(n)=\frac{n+1}{n-1}$ denotes the Glassey exponent and $\mu(\tau)$ is a modulus of continuity. By applying iteration methods for a suitable time-dependent functional, and deriving additionally two times logarithms lower bounds for the functional, we obtain a new blow-up condition for the modulus of continuity satisfying \begin{equation*} \lim\limits_{\tau\to +0}\mu(\tau)\left(\log\frac{1}{\tau}\right)=C_{\mathrm{Gla}}\in(0,+\infty], \end{equation*} under suitable assumptions for initial data. This result shows that blow-up phenomena still occur for the critical exponent case $|u_t|^{p_{\mathrm{Gla}}(n)}$ with additional regularities $\mu(|u_t|)$ of the nonlinear term in $u_t=0$, which is weaker than any H\"older's modulus of continuity.

3.A Weakly Turbulent solution to the cubic Nonlinear Harmonic Oscillator on $\mathbb{R}^2$ perturbed by a real smooth potential decaying to zero at infinity

Authors:Maxine Chabert

Abstract: We build a smooth real potential $V(t,x)$ on $(t_0,+\infty)\times \mathbb{R}^2$ decaying to zero as $t\to \infty$ and a smooth solution to the associated perturbed cubic Nonlinear Harmonic Oscillator whose Sobolev norms blow up logarithmically as $t\to \infty$. Adapting the method of Faou and Raphael for the linear case, we modulate the solitons associated to the Nonlinear Harmonic Oscillator by time-dependent parameters solving a quasi-Hamiltonian dynamical system whose action grows up logarithmically, thus yielding logarithmic growth for the Sobolev norm of the solution.

4.Radial symmetry and Liouville theorem for master equations

Authors:Lingwei Ma, Yahong Guo, Zhenqiu Zhang

Abstract: This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation \begin{equation*} (\partial_t-\Delta)^s u(x,t) =f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb{R}, \end{equation*} subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in $B_1(0)$ for any $t\in \mathbb{R}$. Another one is to establish the Liouville theorem for homogeneous master equation \begin{equation*} (\partial_t-\Delta)^s u(x,t)=0 ,\,\, \mbox{in}\,\, \mathbb{R}^n\times\mathbb{R}, \end{equation*} which states that all bounded solutions must be constant. We propose a new methodology for a direct method of moving planes applicable to the fully fractional heat operator $(\partial_t-\Delta)^s$, and the proof of our main results based on this direct method involves the perturbation technique, limit argument as well as Fourier transform. This study opens up a way to investigate the geometric behavior of master equations, and provides valuable insights for establishing qualitative properties of solutions and even for deriving important Liouville theorems for other types of fractional order parabolic equations.

5.Nondivergence form degenerate linear parabolic equations on the upper half space

Authors:Hongjie Dong, Tuoc Phan, Hung Vinh Tran

Abstract: We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial {\mathbb R}^d_+$, where ${\mathbb R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in {\mathbb R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded positive definite matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. The divergence form equations in this setting were studied in [14]. Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations.

6.Emergence of Gaussian fields in noisy quantum chaotic dynamics

Authors:Maxime Ingremeau, Martin Vogel

Abstract: We study the long time Schr\"odinger evolution of Lagrangian states $f_h$ on a compact Riemannian manifold $(X,g)$ of negative sectional curvature. We consider two models of semiclassical random Schr\"odinger operators $P_h^\alpha=-h^2\Delta_g +h^\alpha Q_\omega$, $0<\alpha\leq 1$, where the semiclassical Laplace-Beltrami operator $-h^2\Delta_g$ on $X$ is subject to a small random perturbation $h^\alpha Q_\omega$ given by either a random potential or a random pseudo-differential operator. Here, the potential or the symbol of $Q_\omega$ is bounded, but oscillates and decorrelates at scale $h^{\beta}$, $0< \beta < \frac{1}{2}$. We prove a quantitative result that, under appropriate conditions on $\alpha,\beta$, in probability with respect to $\omega$ the long time propagation $$\mathrm{e}^{\frac{i}{h}t_h P_h^\alpha } f_h, \quad o(|\log h|)=t_h\to\infty, ~~h\to 0,$$ rescaled to the local scale of $h$ around a uniformly at random chosen point $x_0$ on $X$, converges in law to an isotropic stationary monochromatic Gaussian field -- the Berry Gaussian field. We also provide and $\omega$-almost sure version of this convergence along sufficiently fast decaying subsequences $h_j\to 0$.

7.Modified Scattering of Solutions to the Relativistic Vlasov-Maxwell System Inside the Light Cone

Authors:Stephen Pankavich, Jonathan Ben-Artzi

Abstract: We consider the relativistic Vlasov-Maxwell system in three dimensions and study the limiting asymptotic behavior as $t \to \infty$ of solutions launched by small, compactly supported initial data. In particular, we prove that such solutions scatter to a modification of the free-streaming asymptotic profile. More specifically, we show that the spatial average of the particle distribution function converges to a smooth, compactly-supported limit and establish the precise, self-similar asymptotic behavior of the electric and magnetic fields, as well as, the macroscopic densities and their derivatives in terms of this limiting function. Upon constructing the limiting fields, a modified $L^\infty$ scattering result for the particle distribution function along the associated trajectories of free transport corrected by the limiting Lorentz force is then obtained. When the plasma is non-neutral, our estimates are sharp up to a logarithmic correction. However, when the plasma is neutral, the limiting charge and current densities may vanish, which gives rise to decay rates that are faster than those attributed to the dispersive mechanisms in the system.

8.The relativistic Vlasov-Maxwell system: Local smooth solvability for weak topologies

Authors:Christophe Cheverry, Slim Ibrahim

Abstract: This article is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local smooth solvability for weak topologies (and its long time version for small data). This result is derived from a representation formula decoding how the momentum spreads, and showing that the domain of influence in momentum is controlled by mild information. We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In the end, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas. This is done by investigating the stability properties near a class of approximate solutions.

9.New results on controllability and stability for degenerate Euler-Bernoulli type equations

Authors:Alessandro Camasta, Genni Fragnelli

Abstract: In this paper we study the controllability and the stability for a degenerate beam equation in divergence form via the energy method. The equation is clamped at the left end and controlled by applying a shearing force or a damping at the right end.

1.Temporal periodic solutions to nonhomogeneous quasilinear hyperbolic equations driven by time-periodic boundary conditions

Authors:Xixi Fang, Peng Qu, Huimin Yu

Abstract: We consider the temporal periodic solutions to general nonhomogeneous quasilinear hyperbolic equations with a kind of weak diagonal dominant structure. Under the temporal periodic boundary conditions, the existence, stability and uniqueness of the time-periodic classical solutions are obtained.Moreover, the W2, regularity and stability around the time-periodic solutions are discussed. Our results reveal that the feedback boundary control with dissipative structure can stabilize the K-weakly diagonally dominant nonhomogeneous quasilinear hyperbolic system around the temporal periodic solution.

2.Homogenization of eigenvalues for problems with high-contrast inclusions

Authors:Xin Fu

Abstract: We study quantitative homogenization of the eigenvalues for elliptic systems with periodically distributed inclusions, where the conductivity of inclusions are strongly contrast to that of the matrix. We propose a quantitative version of periodic unfolding method, based on this and the recent results concerned on high-contrast homogenization, the convergence rates of eigenvalues are studied for any contrast $\delta \in (0,\infty)$.

3.Local-in-time well-posedness of the inhomogeneous anisotropic incompressible Navier-Stokes equations with far-field vacuum in two-dimensional whole space

Authors:Jincheng Gao, Lianyun Peng, Zheng-an Yao

Abstract: In this paper, we investigate the local-in-time well-posedness theory for the inhomogeneous incompressible Navier-Stokes equations with only horizontal dissipation in the two-dimensional whole space. Due to the lack of the vertical dissipative term and non-uniform positive lower bound of density, it is a highly challenging obstacle for us to establish well-posedenss theory because of the disappearance of estimate for vertical derivative of velocity. Thus, as we deal with the nonlinear term, our method is to control the vertical derivative of velocity by the horizontal one with the help of Hardy inequality. The weight, producing by Hardy inequality, can be absorbed by the good property of density and vorticity. Therefore, we find the appropriate function space to establish the local-in-time well-posedness theory for the inhomogeneous incompressible Navier-Stokes equations with only horizontal dissipation and far-field vacuum.

4.Numerical study of the Serre-Green-Naghdi equations in 2D

Authors:S. Gavrilyuk, C. Klein

Abstract: A numerical approach for the Serre-Green-Naghdi (SGN) equations in 2D based on a Fourier spectral method with a Krylov subspace technique is presented. The code is used to study the transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D waves independent of the second variable. The study of localised initial data as well as crossing 1D solitary waves does not give an indication of stable structures in SGN solutions localised in two spatial dimensions.

5.Liouville comparison theory for blowup of Euler-Arnold equations

Authors:Martin Bauer, Stephen C. Preston, Justin Valletta

Abstract: In this article we introduce a new blowup criterion for (generalized) Euler-Arnold equations on $\mathbb R^n$. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and comparison with the Liouville equation; in contrast to the usual comparison approach at a single point, we apply comparison in an infinite dimensional function space. We thereby show that the Jacobian of the Lagrangian flow map of the solution reaches zero in finite time, which corresponds to $C^1$-blowup of the velocity field solution. We demonstrate the applicability of our result by proving blowup of smooth solutions to some higher-order versions of the EPDiff equation in all dimensions $n\geq 3$. Previous results on blowup of higher dimensional EPDiff equations were only for versions where the geometric description corresponds to a Sobolev metric of order zero or one. In these situations the behavior does not depend on the dimension and thus already solutions to the one-dimensional version were exhibiting blowup. In the present paper blowup is proved even in situations where the one-dimensional equation has global solutions, such as the EPDiff equation corresponding to a Sobolev metric of order two.

6.Parabolic $α$-Riesz flows and limit cases $α\to 0^+$, $α\to d^-$

Authors:Lucia De Luca, Massimiliano Morini, Marcello Ponsiglione, Emanuele Spadaro

Abstract: In this paper we introduce the notion of parabolic $\alpha$-Riesz flow, for $\alpha\in(0,d)$, extending the notion of $s$-fractional heat flows to negative values of the parameter $s=-\frac{\alpha}{2}$. Then, we determine the limit behaviour of these gradient flows as $\alpha \to 0^+$ and $\alpha \to d^-$. To this end we provide a preliminary $\Gamma$-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly $\lambda$-convex functionals which guarantee that $\Gamma$-convergence commutes with the gradient flow structure.

1.Observability inequality, the interpolation inequality and the spectral inequality for the degenerate parabolic equation in R

Authors:Yuanhang Liu, Weijia Wu, Donghui Yang

Abstract: This paper investigates the interrelationships between the observability inequality, the H\"older-type interpolation inequality, and the spectral inequality for the degenerate parabolic equation in $\mathbb{R}$. We elucidate the distinctive properties of observable sets pertaining to the degenerate parabolic equation. Specifically, we establish that a measurable set in $\mathbb{R}$ fulfills the observability inequality when it exhibits $\gamma$-thickness at a scale $L$, where $\gamma>0$ and $L>0$.

2.Optimal constants of smoothing estimates for Dirac equations with radial data

Authors:Makoto Ikoma, Soichiro Suzuki

Abstract: Kato--Yajima smoothing estimates are one of the fundamental results in study of dispersive equations such as Schr\"odinger equations and Dirac equations. For $d$-dimensional Schr\"odinger-type equations ($d \geq 2$), optimal constants of smoothing estimates were obtained by Bez--Saito--Sugimoto (2017) via the so-called Funk--Hecke theorem. Recently Ikoma (2022) considered optimal constants for $d$-dimensional Dirac equations using a similar method, and it was revealed that determining optimal constants for Dirac equations is much harder than the case of Schr\"odinger-type equations. Indeed, Ikoma obtained the optimal constant in the case $d = 2$, but only upper bounds (which seem not optimal) were given in other dimensions. In this paper, we give optimal constants for $d$-dimensional Schr\"odinger-type and Dirac equations with radial initial data for any $d \geq 2$. In addition, we also give optimal constants for the one-dimensional Schr\"odinger-type and Dirac equations.

3.New minimax theorems for lower semicontinuous functions and applications

Authors:Claudianor O Alves, Giovanni Molica Bisci, Ismael S. da Silva

Abstract: In this paper we prove a version of the Fountain Theorem for a class of nonsmooth functionals that are sum of a $C^1$ functional and a convex lower semicontinuous functional, and also a version of a theorem due to Heinz for this class of functionals. The new abstract theorems will be used to prove the existence of infinitely many solutions for some elliptic problems whose the associated energy functional is of the above mentioned type. We study problems with logarithmic nonlinearity and a problem involving the 1-Laplacian operator.

4.Unique solvability of a rate independent damage model with fatigue

Authors:Livia Betz

Abstract: This paper investigates a rate independent damage model with fatigue. Its particular feature is that the dissipation depends on the history of the state. We establish an existence result for the original problem and for the control thereof. By imposing a nonrestrictive smoothness condition on the fatigue degradation map, we are able to derive a crucial a priori estimate. Based on this, we show uniqueness of solutions to the rate independent model. The a priori estimate also opens the door to future research on the topic of optimization, as it allows us to conclude an essential uniform Lipschitz property.

5.Existence and uniqueness of weak solutions to the Smoluchowski coagulation equation with source and sedimentation

Authors:Prasanta Kumar Barik, Asha K. Dond, Rakesh Kumar

Abstract: This article is devoted to a generalized version of Smoluchowski's coagulation equation. This model describes the time evolution of a system of aggregating particles under the effect of external input and output particles. We show that for a large class of coagulation kernels, output rates, and exponentially decaying input rates, there is a weak solution. Moreover, the solution satisfies the mass-conservation property for linear coagulation rate and an additional condition on input and output rates. The uniqueness of weak solutions is also established by applying additional restrictions on the rates.

6.Stationarity and Fredholm theory in subextremal Kerr-de Sitter spacetimes

Authors:Oliver Petersen, András Vasy

Abstract: In a recent paper, we proved that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the existing results. One central ingredient in the argument was a new definition of quasinormal modes, where a non-standard choice of stationary Killing vector field had to be used in order for the Fredholm theory to be applicable. In this paper, we show that there is in fact a variety of allowed choices of stationary Killing vector fields. In particular, the horizon Killing vector fields work for the analysis, in which case one of the corresponding ergoregions is completely removed.

1.Reduced energies for thin ferromagnetic films with perpendicular anisotropy

Authors:G. Di Fratta, C. B. Muratov, V. V. Slastikov

Abstract: We derive four reduced two-dimensional models that describe, at different spatial scales, the micromagnetics of ultrathin ferromagnetic materials of finite spatial extent featuring perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. Starting with a microscopic model that regularizes the stray field near the material's lateral edges, we carry out an asymptotic analysis of the energy by means of $\Gamma$-convergence. Depending on the scaling assumptions on the size of the material domain vs. the strength of dipolar interaction, we obtain a hierarchy of the limit energies that exhibit progressively stronger stray field effects of the material edges. These limit energies feature, respectively, a renormalization of the out-of-plane anisotropy, an additional local boundary penalty term forcing out-of-plane alignment of the magnetization at the edge, a pinned magnetization at the edge, and, finally, a pinned magnetization and an additional field-like term that blows up at the edge, as the sample's lateral size is increased. The pinning of the magnetization at the edge restores the topological protection and enables the existence of magnetic skyrmions in bounded samples.

2.Gibbs Dynamics for the Weakly Dispersive Nonlinear Schrödinger Equations

Authors:Rui Liang, Yuzhao Wang

Abstract: We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear fractional Schr\"odinger equation (FNLS) with initial data distributed via Gibbs measure. We construct global strong solutions with the flow property to FNLS on the support of the Gibbs measure in the full dispersive range, thus resolving a question proposed by Sun-Tzvetkov (2021). As a byproduct, we prove the invariance of the Gibbs measure and almost sure global well-posedness for FNLS.

3.Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups

Authors:Sekhar Ghosh, Vishvesh Kumar, Michael Ruzhansky

Abstract: In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional $p$-sublaplacian ($1<p<\infty$) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schr\"odinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676 (1982/1983)) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. (2023)). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for $p=2$ even in the context of the Heisenberg group.

4.Wave breaking in the unidirectional non-local wave model

Authors:Shaojie Yang, Jian Chen

Abstract: In this paper we study wave breaking in the unidirectional non-local wave model describing the motion of a collision-free plasma in a magnetic field. By analyzing the monotonicity and continuity properties of a system of the Riccati-type differential inequalities involving the extremal slopes of flows, we show a new sufficient condition on the initial data to exhibit wave breaking. Moreover, the estimates of life span and wave breaking rate are derived.

5.Phase transition for invariant measures of the focusing Schrödinger equation

Authors:Leonardo Tolomeo, Hendrik Weber

Abstract: In this paper, we consider the Gibbs measure for the focusing nonlinear Schr\"odinger equation on the one-dimensional torus, that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the measure exhibits a phase transition, depending on the size of the nonlinearity. This phase transition was originally conjectured on the basis of numerical simulation by Lebowitz, Rose and Speer (1988). Its existence is however striking in view of a series of negative results by McKean (1995) and Rider (2002).

6.On the Read-Shockley energy for grain boundaries in poly-crystals

Authors:Adriana Garroni, Martino Fortuna, Emanuele Spadaro

Abstract: In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearised elasticity and an ansazt on the distribution of incompatibilities of the lattice at the interface. In this paper we derive a sharp interface limiting functional starting from a nonlinear semidiscrete model for dislocations proposed by Lauteri--Luckhaus. Building upon their analysis we obtain, via $\Gamma$-convergence, an interfacial energy depending on the rotations of the grains and the relative orientation of the interface which agrees for small angle grain boundaries with the Read and Shockley logarithmic scaling.

7.Global harmonic analysis for $Φ^4_3$ on closed Riemannian manifolds

Authors:I. Bailleul, N. V. Dang, L. Ferdinand, T. D. Tô

Abstract: Following Parisi \& Wu's paradigm of stochastic quantization, we constructed in \cite{BDFT} a $\Phi^4$ measure on an arbitrary compact, boundaryless, Riemannian manifold as an invariant measure of a singular stochastic partial differential equation. The present work is a companion to \cite{BDFT}. We describe here in detail the harmonic and microlocal analysis tools that we used. We also introduce some new tools to treat the vectorial $\Phi^4_3$ model. This relies on a new Cole-Hopf transform involving random bundle maps. We do not aim here for the greatest generality; rather, we tried to keep our exposition relatively self-contained and pedagogical enough in the hope that the techniques we show can be used in other settings.

8.Normalized solutions to Schödinger equations with potential and inhomogeneous nonlinearities on large convex domains

Authors:Thomas Bartsch, Shijie Qi, Wenming Zou

Abstract: The paper addresses an open problem raised in [Bartsch, Molle, Rizzi, Verzini: Normalized solutions of mass supercritical Schr\"odinger equations with potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existence of normalized solutions to Schr\"odinger equations with potentials and inhomogeneous nonlinearities. We consider the problem \[ -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad \|u\|^2_2=\int|u|^2dx = \alpha, \] both on $\mathbb{R}^N$ as well as on domains $r\Omega$ where $\Omega\subset\mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponents satisfy $2<p<2+\frac4N<q<2^*=\frac{2N}{N-2}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Due to the presence of the potential a by now standard approach based on the Pohozaev identity cannot be used. We develop a robust method to study the existence of normalized solutions of nonlinear Schr\"odinger equations with potential and find conditions on $V$ so that normalized solutions exist. Our results are new even in the case $\beta=0$.

9.Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

Authors:Luigi Forcella, Oana Pocovnicu

Abstract: We consider a fractional nonlinear wave equations (fNLW) with a general power-type nonlinearity, on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We first construct the Gibbs measure associated with this equation by using the variational approach of Barashkov and Gubinelli. We then prove almost sure local well-posedness with respect to Gibbsian initial data, by exploiting the second order expansion. Finally, we extend solutions globally in time using Bourgain's invariant measure argument.

1.Continuity of a spatial gradient of a weak solution to a very singular parabolic equation involving anisotropic diffusivity

Authors:Shuntaro Tsubouchi

Abstract: We consider weak solutions to very singular parabolic equations involving both one-Laplace-type operators, which have anisotropic diffusivity, and $p$-Laplace-type operators with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$ denotes the space dimension. This equation becomes no longer uniformly parabolic near a facet, the place where a spatial gradient vanishes. The aim of this paper is to prove that spatial derivatives of weak solutions are continuous even across facets. This is possible by showing local a priori H\"{o}lder continuity of gradients suitably truncated near facets. To give a rigorous proof, we consider an approximating parabolic problems, and appeal to standard methods including De Giorgi's truncation and comparisons to Dirichlet heat flows.

2.On the Cauchy problem for the Fadaray tensor on globally hyperbolic manifolds with timelike boundary

Authors:Nicoló Drago, Nicolas Ginoux, Simone Murro

Abstract: We study the well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary. The existence of Green operators for the operator $\mathrm{d}+\delta$ and a suitable pre-symplectic structure on the space of solutions are discussed.

3.An analog of the Tricomi problem for a mixed type equation with Riemann-Liouville fractional derivative

Authors:Akmaljon Okboev Bakhromjonovich

Abstract: In this article, the Tricomi problem for a parabolic-hyperbolic type equation in a mixed domain is investigated. Riemann-Liouville fractional derivative participates in the parabolic part of the considerated equation, and the hyperbolic part consists of a degenerate hyperbolic equation of the second kind.

4.Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

Authors:Stuti Das

Abstract: We prove the local H\"older regularity of the weak solution of the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} for $0<s<1$, for all exponent $\alpha_0\in(0,1)$. Next we show that the gradient of the weak solution is also H\"older continuous for some $\alpha\in (0,1)$. Our approach is purely analytic and it is based on perturbation techniques.

5.Error estimates for the highly efficient and energy stable schemes for the 2D/3D two-phase MHD

Authors:Ke Zhang, Haiyan Su, Xinlong Feng

Abstract: In this paper, we mainly focus on the rigorous convergence analysis for two fully decoupled, unconditional energy stable methods of the two-phase magnetohydrodynamics (MHD) model, which described in our previous work \cite{2022Highly}. The two methods consist of semi-implicit stabilization method/invariant energy quadratization (IEQ) method \cite{2019EfficientCHEN, Yang2016Linear, Yang2017Efficient, 2019EfficientYANG} for the phase field system, the pressure projection correction method for the saddle point MHD system, the exquisite implicit-explicit treatments for nonlinear coupled terms, which leads to only require solving a sequence of small elliptic equations at each time step. As far as we know, it's the first time to establish the optimal convergence analysis of fully decoupled and unconditional energy stable methods for multi-physics nonlinear two-phase MHD model. In addition, several numerical examples are showed to test the accuracy and stability of the presented methods.

6.Learning zeros of Fokker-Planck operators

Authors:Pinak Mandal, Amit Apte

Abstract: In this paper we devise a deep learning algorithm to find non-trivial zeros of Fokker-Planck operators when the drift is non-solenoidal. We demonstrate the efficacy of our algorithm for problem dimensions ranging from 2 to 10. Our method scales linearly with dimension in memory usage. We show that this method produces better approximations compared to Monte Carlo methods, for the same overall sample sizes, even in low dimensions. Unlike the Monte Carlo methods, our method gives a functional form of the solution. We also demonstrate that the associated loss function is strongly correlated with the distance from the true solution, thus providing a strong numerical justification for the algorithm. Moreover, this relation seems to be linear asymptotically for small values of the loss function.

7.The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

Authors:Takiko Sasaki, Shu Takamatsu, Hiroyuki Takamura

Abstract: This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.

8.Upper bounds for the blow-up time of the 2-D parabolic-elliptic Patlak-Keller-Segel model of chemotaxis

Authors:Patrick Maheux

Abstract: In this paper, we obtain upper bounds for the critical time $T^*$ of the blow-up for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the mass is assumed. The result is expressed not only in terms of the supercritical mass $M> 8\pi$, but also in terms of the {\it shape} of the initial data.

9.A Strichartz estimate for quasiperiodic functions

Authors:Friedrich Klaus

Abstract: In this work we prove a Strichartz estimate for the Schr\"odinger equation in the quasiperiodic setting. We also show a lower bound on the number of resonant frequency interactions in this situation.

10.Weak solutions for steady, fully inhomogeneous generalized Navier-Stokes equations

Authors:Julius Jeßberger, Michael Růžička

Abstract: We consider the question of existence of weak solutions for the fully inhomogeneous, stationary generalized Navier-Stokes equations for homogeneous, shear-thinning fluids. For a shear rate exponent $p \in \big(\tfrac{2d}{d+1}, 2\big)$, previous results require either smallness of the norm or vanishing of the normal component of the boundary data. In this work, combining previous methods, we propose a new, more general smallness condition.

11.Spectral Closure for the Linear Boltzmann-BGK Equation

Authors:Florian Kogelbauer, Ilya Karlin

Abstract: We give an explicit description of the spectral closure for the three-dimensional linear Boltzmann-BGK equation in terms of the macroscopic fields, density, flow velocity and temperature. This results in a new linear fluid dynamics model which is valid for any relaxation time. The non-local exact fluid dynamics equations are compared to the Euler, Navier--Stokes and Burnett equations. Our results are based on a detailed spectral analysis of the linearized Boltzmann-BGK operator together with a suitable choice of spectral projection.

12.Convergence of solutions for a reaction-diffusion problem with fractional Laplacian

Authors:Jiaouhui Xu, Tomás Caraballo, José Valero

Abstract: A kind of nonlocal reaction-diffusion equations on an unbounded domain containing fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the solutions of the classical equation governed by the standard Laplacian, when the fractional parameter grows to 1. The existence of global attractors is investigated as well. The novelty of this paper is concerned with the convergence of solutions when the fractional parameter varies, which, as far as the authors are aware, seems to be the first result of this kind of problems in the literature.

13.Supercaloric functions for the porous medium equation in the fast diffusion case

Authors:Kristian Moring, Christoph Scheven

Abstract: We study a generalized class of supersolutions, so-called supercaloric functions to the porous medium equation in the fast diffusion case. Supercaloric functions are defined as lower semicontinuous functions obeying a parabolic comparison principle. We prove that bounded supercaloric functions are weak supersolutions. In the supercritical range, we show that unbounded supercaloric functions can be divided into two mutually exclusive classes dictated by the Barenblatt solution and the infinite point-source solution, and give several characterizations for these classes. Furthermore, we study the pointwise behavior of supercaloric functions and obtain connections between supercaloric functions and weak supersolutions.

14.On two notions of solutions to the obstacle problem for the singular porous medium equation

Authors:Kristian Moring, Christoph Scheven

Abstract: We show that two different notions of solutions to the obstacle problem for the porous medium equation, a potential theoretic notion and a notion based on a variational inequality, coincide for regular enough compactly supported obstacles.

1.On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

Authors:Angkana Rüland, Antonio Tribuzio

Abstract: We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in \cite{RT21} we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

2.Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator

Authors:Hongli Sun, Weijia Wu, Donghui Yang

Abstract: We consider the problems of extreming the first eigenvalue and the fundamental gap of a sub-elliptic operator with Dirichlet boundary condition, when the potential $V$ is subjected to a $p$-norm constraint. The existence results for weak solutions, compact embedding theorem and spectral theory for sub-elliptic equation are given. Moreover, we provide the specific characteristics of the corresponding optimal potential function.

3.Rigidity of inverse problems for nonlinear elliptic equations on manifolds

Authors:Ali Feizmohammadi, Yavar Kian, Lauri Oksanen

Abstract: We consider the inverse problem of determining coefficients appearing in semilinear elliptic equations stated on Riemannian manifolds with boundary given the knowledge of the associated Dirichlet-to-Neumann map. We begin with a negative answer to this problem. Owing to this obstruction, we consider a new formulation of our inverse problem in terms of a rigidity problem. Precisely, we consider cases where the Dirichlet-to-Neumann map of a semilinear equation coincides with the one of a linear equation and ask whether this implies that the equation must indeed be linear. We give a positive answer to this rigidity problem under some assumptions imposed to the Riemannian manifold and to the semilinear term under consideration.

4.The Pauli-Poisson equation and its semiclassical limit

Authors:Jakob Möller

Abstract: The Pauli-Poisson equation is a semi-relativistic model for charged spin-1/2-particles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in 3 space dimensions. It is a system of two magnetic Schr\"odinger equations for the two components of the Pauli 2-spinor, representing the two spin states of a fermion, coupled by the additional Stern-Gerlach term representing the interaction of magnetic field and spin. We study the global wellposedness in the energy space and the semiclassical limit of the Pauli-Poisson to the magnetic Vlasov-Poisson equation with Lorentz force and the semiclassical limit of the linear Pauli equation to the magnetic Vlasov equation with Lorentz force. We use Wigner transforms and a density matrix formulation for mixed states, extending the work of P. L. Lions & T. Paul as well as P. Markowich & N.J. Mauser on the semiclassical limit of the non-relativistic Schr\"odinger-Poisson equation.

5.A general support theorem for analytic double fibration transforms

Authors:Marco Mazzucchelli, Mikko Salo, Leo Tzou

Abstract: We develop a systematic approach for resolving the analytic wave front set for a class of integral geometry transforms appearing in various tomography problems. Combined with microlocal analytic continuation, this leads to uniqueness and support theorems for analytic integral transforms which are in the microlocal double fibration framework introduced by Guillemin. For the case of ray transforms, we show that the double fibration setup has a concrete interpretation in terms of curve families obtained by projecting integral curves of a fixed vector field on some fiber bundle down to the base. This setup includes geodesic X-ray type transforms, null bicharacteristic ray transforms and transforms related to real principal type systems. We also study transforms integrating over submanifolds of any codimension, and give geometric characterizations for the Bolker condition required for recovering singularities. Our approach is based on a general result related to recovering the analytic wave front set of a function from its transform given by a suitable analytic elliptic Fourier integral operator. This approach extends and unifies a number of previous works. We use wave packet transforms to extrapolate the geometric features of wave front set propagation for such operators when their canonical relation satisfies the Bolker condition.

6.Damped nonlinear Schrödinger equation with Stark effect

Authors:Yi Hu, Yongki Lee, Shijun Zheng

Abstract: We study the $L^2$-critical damped NLS with a Stark potential. We prove that the threshold for global existence and finite time blowup of this equation is given by $\|Q\|_2$, where $Q$ is the unique positive radial solution of $\Delta Q + |Q|^{4/d} Q = Q$ in $H^1(\mathbb{R}^d)$. Moreover, in any small neighborhood of $Q$, there exists an initial data $u_0$ above the ground state such that the solution flow admits the log-log blowup speed. This verifies the structural stability for the ``$\log$-$\log$ law'' associated to the NLS mechanism under the perturbation by a damping term and a Stark potential. The proof of our main theorem is based on the Avron-Herbst formula and the analogous result for the unperturbed damped NLS.

7.Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces

Authors:Miriam Buck, Stefano Modena

Abstract: We construct by convex integration examples of energy dissipating solutions to the 2D Euler equations on $\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$, for any $2/3<p<1$.

8.Continuity up to the boundary for obstacle problems to porous medium type equations

Authors:Kristian Moring, Leah Schätzler

Abstract: We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This result seems to be new even for signed solutions to the (obstacle free) Cauchy-Dirichlet problem to the singular porous medium equation, which is retrieved as a special case.

9.Normalized solutions to Schrödinger equations in the strongly sublinear regime

Authors:Jarosław Mederski, Jacopo Schino

Abstract: We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, with $N\ge2$. The behaviour of $g$ at the origin is allowed to be strongly sublinear, i.e., $\lim_{s\to0}g(s)/s = -\infty$, which includes the case \[ g(s) = \alpha s \ln s^2 + \mu |s|^{p-2} s \] with $\alpha > 0$ and $\mu \in \mathbb{R}$, $2 < p \le 2^*$ properly chosen. We consider a family of approximating problems that can be set in $H^1(\mathbb{R}^N)$ and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about $g$ that allow us to work in a suitable subspace of $H^1(\mathbb{R}^N)$, we prove the existence of infinitely many solutions.

10.Global solution and blow-up for the SKT model in Population Dynamics

Authors:Ichraf Belkhamsa, Messaoud Souilah

Abstract: In this paper, we prove the existence and uniqueness of the global solution to the reaction diffusion system SKT with homogeneous Newmann boundary conditions. We use the lower and upper solution method and its associated monotone iterations where the reaction functions are locally Lipschitz .We study the blowing-up property of the solution, we give a sufficient condition on the reaction parameters of the model to ensure the blow-up of the solution continuous functions spaces.

1.Quantitative spectral stability for Aharonov-Bohm operators with many coalescing poles

Authors:Veronica Felli, Benedetta Noris, Roberto Ognibene, Giovanni Siclari

Abstract: The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, we obtain an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed.

2.On stability and instability of the ground states for the focusing inhomogeneous NLS with inverse-square potential

Authors:JinMyong An, HakBom Mun, JinMyong Kim

Abstract: In this paper, we study the stability and instability of the ground states for the focusing inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential (for short, INLS$_c$ equation): \[iu_{t} +\Delta u+c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0}(x) \in H^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0<b<2$, $0<\sigma<\frac{4-2b}{d-2}$ and $c\neq 0$ be such that $c<c(d):=\left(\frac{d-2}{2}\right)^{2}$. In the mass-subcritical case $0<\sigma<\frac{4-2b}{d}$, we prove the stability of the set of ground states for the INLS$_{c}$ equation. In the mass-critical case $\sigma=\frac{4-2b}{d}$, we first prove that the solution of the INLS$_c$ equation with initial data $u_{0}$ satisfying $E(u_0)<0$ blows up in finite or infinite time. Using this fact, we then prove that the ground state standing waves are unstable by blow-up. In the intercritical case $\frac{4-2b}{d}<\sigma<\frac{4-2b}{d-2}$, we finally show the instability of ground state standing waves for the INLS$_c$ equation.

3.Vanishing of long time average p-enstrophy dissipation rate in the inviscid limit of the 2D damped Navier-Stokes equations

Authors:Raphael Wagner

Abstract: In 2007, Constantin and Ramos proved a result on the vanishing long time average enstrophy dissipation rate in the inviscid limit of the 2D damped Navier-Stokes equations. In this work, we prove a generalization of this for the p-enstrophy, sequences of distributions of initial data and sequences of strongly converging right-hand sides. We simplify their approach by working with invariant measures on the global attractors which can be characterized via bounded complete solution trajectories. Then, working on the level of trajectories allows us to directly employ some recent results on strong convergence of the vorticity in the inviscid limit.

4.Unique Continuation Properties from one time for hyperbolic Schrödinger equations

Authors:Juan Antonio Barceló, Biagio Cassano, Luca Fanelli

Abstract: In this paper, we investigate properties of unique continuation for hyperbolic Schr\"odinger equations with time-dependent complex-valued electric fields and time-independent real magnetic fields. We show that positive masses inside of a bounded region at a single time propagate outside the region and prove gaussian lower bounds for the solutions, provided a suitable average in space-time cylinders is taken.

5.Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions

Authors:Elisa Davoli, Lorenza D'Elisa, Jonas Ingmanns

Abstract: We perform a stochastic-homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere, and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii's predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.

6.Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions

Authors:Jeaheang Bang, Changfeng Gui, Hao Liu, Yun Wang, Chunjing Xie

Abstract: Solutions with scaling-invariant bounds such as self-similar solutions, play an important role in the understanding of the regularity and the asymptotic structure of solutions for the Navier-Stokes problem. In this paper, we prove that any steady solutions satisfying $|\bodysymbol{u}(x)|\leq C/|x|$ in $\mathbb{R}^n\setminus \{0\}, n \geq 4$, are trivial. Our main idea is to analyze the velocity field and the total head pressure via weighted energy estimates with suitable multipliers so that the proof is pretty elementary and short. These results not only give the Liouville-type theorem for steady solutions in higher dimensions with neither smallness nor self-similarity assumptions but also help remove the possible singularities of the solutions.

7.Design of Sturm global attractors 2: Time-reversible Chafee-Infante lattices of 3-nose meanders

Authors:Bernold Fiedler, Carlos Rocha

Abstract: This sequel continues our exploration arxiv:2302.12531 of a deceptively ``simple'' class of global attractors, called Sturm due to nodal properties. They arise for the semilinear scalar parabolic PDE \begin{equation}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) \tag{$*$} \end{equation} on the unit interval $0 < x<1$, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ODE boundary value problem of equilibrium solutions $u=v(x)$. Specifically, we address meanders with only three ``noses'', each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm of 1974, with cubic nonlinearity $f=f(u)$, features just two noses. We present, and fully prove, a precise description of global PDE connection graphs, graded by Morse index, for such gradient-like Morse-Smale systems \eqref{eq:*}. The directed edges denote PDE heteroclinic orbits $v_1 \leadsto v_2$ between equilibrium vertices $v_1, v_2$ of adjacent Morse index. The connection graphs can be described as a lattice-like structure of Chafee-Infante subgraphs. However, this simple description requires us to adjoin a single ``equilibrium'' vertex, formally, at Morse level -1. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graphs then also exhibit global time reversibility.

8.Discrete wave turbulence for the Benjamin-Bona-Mahony equation, Part I: oscillations for the correlations between the solutions and its initial datum

Authors:Anne-Sophie de Suzzoni

Abstract: We investigate different problems regarding wave turbulence for the Benjamin-Bona-Mahony (BBM) equation in the context of discrete turbulence regime. In the part I, we investigate the behaviour of the correlations between the solution to the BBM equation at latter times with the initial datum.

9.Anisotropic flows of Forchheimer-type in porous media and their steady states

Authors:Luan Hoang, Thinh Kieu

Abstract: We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. The first half of the paper is devoted to understanding the nonlinear structure of the anisotropic momentum equations. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. In the second half of the paper, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local H\"older continuous dependence on the forcing function and the boundary data.

10.Relativistic BGK model for gas mixtures

Authors:Byung-Hoon Hwang, Myeong-Su Lee, Seok-Bae Yun

Abstract: Unlike the case for classical particles, the literature on BGK type models for relativistic gas mixture is extremely limited. There are a few results %\cite{Kremer,Kremer3,KP} in which such relativistic BGK models for gas mixture are employed to compute transport coefficients. However, to the best knowledge of authors, relativistic BGK models for gas mixtures with complete presentation of the relaxation operators are missing in the literature. In this paper, we fill this gap by suggesting a BGK model for relativistic gas mixtures for which the existence of each equilibrium coefficients in the relaxation operator is rigorously guaranteed in a way that all the essential physical properties are satisfied such as the conservation laws, the H-theorem, the capturing of the correct equilibrium state, the indifferentiability principle, and the recovery of the classical BGK model in the Newtonian limit.

1.On the Defocusing Cubic Nonlinear Wave Equation on $\mathbb{H}^3$ with Radial Initial Data in $H^{\frac{1}{2}+δ} \times H^{-\frac{1}{2}+δ}$

Authors:Chutian Ma

Abstract: In this paper we prove global well-posedness and scattering for the defocusing cubic nonlinear wave equation in the hyperbolic space $\mathbb{H}^3$, under the assumption that the initial data is radial and lies in $H^{\frac{1}{2}+\delta}(\mathbb{H}^3)\times H^{-\frac{1}{2}+\delta}(\mathbb{H}^3)$

2.The $α$-SQG patch problem is illposed in $C^{2,β}$ and $W^{2,p}$

Authors:Alexander Kiselev, Xiaoyutao Luo

Abstract: We consider the patch problem for the $\alpha$-SQG system with the values $\alpha=0$ and $\alpha= \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint $C^{k,\beta}$ H\"older spaces, as well as in $W^{2,p},$ $1<p<\infty$ spaces. In stark contrast to the Euler case, we prove that for $0<\alpha< \frac{1}{2}$, the $\alpha$-SQG patch problem is strongly illposed in \emph{every} $C^{2,\beta} $ H\"older space with $\beta<1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for \emph{every} $W^{2,p}$ Sobolev space unless $p=2$.

3.Discrete-to-continuum limits of interacting particle systems in one dimension with collisions

Authors:Patrick van Meurs

Abstract: We study a class of interacting particle systems in which $n$ signed particles move on the real line. At close range particles with the same sign repel and particles with opposite sign attract each other. The repulsion and attraction are described by the same singular interaction force $f$. Particles of opposite sign may collide in finite time. Upon collision, pairs of colliding particles are removed from the system. In a recent paper by Peletier, Po\v{z}\'ar and the author, one particular particle system of this type was studied; that in which $f(x) = \frac1x$ is the Coulomb force. Global well-posedness of this particle system was shown and a discrete-to-continuum limit (i.e. $n \to \infty$) to a nonlocal PDE for the signed particle density was established. Both results rely on innovative use of techniques in ODE theory and viscosity solutions. In the present paper we extend these results to a large class of particle systems in order to cover many new applications. Motivated by these applications, we consider the presence of an external force $g$, consider interaction forces $f$ with a large range of singularities and allow $f$ to scale with $n$. To handle this class of $f$ we develop several new proof techniques in addition to those used for the Coulomb force.

4.Some remarks about the stationary Micropolar fluid equations: existence, regularity and uniqueness

Authors:Diego Chamorro LaMME, David Llerena LaMME, Gastón Vergara-Hermosilla LaMME

Abstract: We consider here the stationary Micropolar fluid equations which are a particular generalization of the usual Navier-Stokes system where the microrotations of the fluid particles must be taken into account. We thus obtain two coupled equations: one based mainly in the velocity field u and the other one based in the microrotation field $\omega$. We will study in this work some problems related to the existence of weak solutions as well as some regularity and uniqueness properties. Our main result establish, under some suitable decay at infinity conditions for the velocity field only, the uniqueness of the trivial solution.

5.About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations

Authors:Lorenzo Marino PAN, Stéphane Menozzi LaMME, Enrico Priola UNIPV

Abstract: We study here the effects of a time-dependent second order perturbation to a degenerate Ornstein-Uhlenbeck type operator whose diffusive part can be either local or non-local. More precisely, we establish that some estimates, such as the Schauder and Sobolev ones, already known for the non-perturbed operator still hold, and with the same constants, when we perturb the Ornstein-Uhlenbeck operator with second order diffusions with coefficients only depending on time in a measurable way. The aim of the current work is twofold: we weaken the assumptions required on the perturbation in the local case which has been considered already in [KP17] and we extend the approach presented therein to a wider class of degenerate Kolmogorov operators with non-local diffusive part of symmetric stable type.

6.Weyl Calculus on Graded Groups

Authors:Serena Federico, David Rottensteiner, Michael Ruzhansky

Abstract: The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on $\mathbb{R}^n$. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following [Doc. Math., 22, 1539--1592, 2017], using the H\"{o}rmander symbol classes $S^m_{\rho, \delta}(G)$ introduced in [Progress in Mathematics, 314. Birkh\"{a}user/Springer, 2016]. We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn-Nirenberg calculus, on $\mathbb{R}^n$ and on general graded groups [Progress in Mathematics, 314. Birkh\"{a}user/Springer, 2016]. Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\r{a}rding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups. In the particular case of the Heisenberg group $\mathbb{H}_n$, we are able to answer the fundamental questions of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on $\mathbb{H}_n$? And which are the criteria that determine it uniquely? The surprisingly simple but compelling answers raise the question whether what is true for $\mathbb{R}^n$ and $\mathbb{H}_n$ also extends to general graded groups, which we answer in the affirmative in this paper. Among other things, we discuss and investigate an analogue of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as the notion of the Poisson bracket for symbols in the setting of stratified groups, linking it to the symbolic properties of the commutators.

7.Stochastic optimal transport and Hamilton-Jacobi-Bellman equations on the set of probability measures

Authors:Charles Bertucci CMAP

Abstract: We introduce a stochastic version of the optimal transport problem. We provide an analysis by means of the study of the associated Hamilton-Jacobi-Bellman equation, which is set on the set of probability measures. We introduce a new definition of viscosity solutions of this equation, which yields general comparison principles, in particular for cases involving terms modeling stochasticity in the optimal control problem. We are then able to establish results of existence and uniqueness of viscosity solutions of the Hamilton-Jacobi-Bellman equation. These results rely on controllability results for stochastic optimal transport that we also establish.

8.Global convergence towards pushed travelling fronts for parabolic gradient systems

Authors:Ramon Oliver-Bonafoux, Emmanuel Risler

Abstract: This article addresses the issue of global convergence towards pushed travelling fronts for solutions of parabolic systems of the form \[ u_t = - \nabla V(u) + u_{xx} \,, \] where the potential $V$ is coercive at infinity. It is proved that, if an initial condition $x\mapsto u(x,t=0)$ approaches, rapidly enough, a critical point $e$ of $V$ to the right end of space, and if, for some speed $c_0$ greater than the linear spreading speed associated with $e$, the energy of this initial condition in a frame travelling at the speed $c_0$ is negative $\unicode{x2013}$ with symbols, \[ \int_{\mathbb{R}} e^{c_0 x}\left(\frac{1}{2} u_x(x,0)^2 + V\bigl(u(x,0)\bigr)- V(e)\right)\, dx < 0 \,, \] then the corresponding solution invades $e$ at a speed $c$ greater than $c_0$, and approaches, around the leading edge and as time goes to $+\infty$, profiles of pushed fronts (in most cases a single one) travelling at the speed $c$. A necessary and sufficient condition for the existence of pushed fronts invading a critical point at a speed greater than its linear spreading speed follows as a corollary. In the absence of maximum principle, the arguments are purely variational. The key ingredient is a Poincar\'e inequality showing that, in frames travelling at speeds exceeding the linear spreading speed, the variational landscape does not differ much from the case where the invaded equilibrium $e$ is stable. The proof is notably inspired by ideas and techniques introduced by Th. Gallay and R. Joly, and subsequently used by C. Luo, in the setting of nonlinear damped wave equations.

9.Asymptotic behavior of least energy nodal solutions for biharmonic Lane-Emden problems in dimension four

Authors:Zhijie Chen, Zetao Cheng, Hanqing Zhao

Abstract: In this paper, we study the asymptotic behavior of least energy nodal solutions $u_p(x)$ to the following fourth-order elliptic problem \[ \begin{cases} \Delta^2 u =|u|^{p-1}u \quad &\hbox{in}\;\Omega, \\ u=\frac{\partial u}{\partial \nu}=0 \ \ &\hbox{on}\;\partial\Omega, \end{cases} \] where $\Omega$ is a bounded $C^{4,\alpha}$ domain in $\mathbb{R}^4$ and $p>1$. Among other things, we show that up to a subsequence of $p\to+\infty$, $pu_p(x)\to 64\pi^2\sqrt{e}(G(x,x^+)-G(x,x^-))$, where $x^+\neq x^-\in \Omega$ and $G(x,y)$ is the corresponding Green function of $\Delta^2$. This generalize those results for $-\Delta u=|u|^{p-1}u$ in dimension two by (Grossi-Grumiau-Pacella, Ann.I.H.Poincar\'{e}-AN, 30 (2013), 121-140) to the biharmonic case, and also gives an alternative proof of Grossi-Grumiau-Pacella's results without assuming their comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$.

10.A remark on a conjecture on the symmetric Gaussian Problem

Authors:Nicola Fusco, Domenico Angelo La Manna

Abstract: In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a functional for certain value of the mass. We give a positive answer in dimension two while in higher dimension the situation is different. In fact, for small value of mass the ball centered at the origin is a local minimizer while for large values the ball is a maximizer among convex sets with uniform bound on the curvature.

11.Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli

Authors:Alexis Michelat, Tristan Rivière

Abstract: We obtain a nigh optimal estimate for the first eigenvalue of two natural weighted problems associated to the bilaplacian (and of a continuous family of fourth-order elliptic operators in dimension $2$) in degenerating annuli (that are central objects in bubble tree analysis) in all dimension. The estimate depends only on the conformal class of the annulus. We also show that in dimension $2$ and dimension $4$, the first eigenfunction (of the first problem) is never radial provided that the conformal class of the annulus is large enough. The other result is a weighted Poincar\'e-type inequality in annuli for those fourth-order operators. Applications to Morse theory are given.

1.Decay of extremals of Morrey's inequality

Authors:Ryan Hynd, Simon Larson, Erik Lindgren

Abstract: We study the decay (at infinity) of extremals of Morrey's inequality in $\mathbb{R}^n$. These are functions satisfying $$ \displaystyle \sup_{x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}= C(p,n)\|\nabla u\|_{L^p(\mathbb{R}^n)} , $$ where $p>n$ and $C(p,n)$ is the optimal constant in Morrey's inequality. We prove that if $n \geq 2$ then any extremal has a power decay of order $\beta$ for any $$ \beta<-\frac13+\frac{2}{3(p-1)}+\sqrt{\left(-\frac13+\frac{2}{3(p-1)}\right)^2+\frac13}. $$

2.Graph-to-local limit for the nonlocal interaction equation

Authors:Antonio Esposito, Georg Heinze, André Schlichting

Abstract: We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.

3.Boundary regularity of uniformly rotating vortex patch

Authors:Yuchen Wang, Maolin Zhou, Guanghui Zhang

Abstract: In this paper we consider the singularities on the boundary of limiting $V$-states of the 2-dim incompressible Euler equation. By setting up a Weiss-type monotoncity formula for a sign-changing unstable elliptic free boundary problem, we obtain the classification of singular points on the free boundary: the boundary of vortical domain would form either a right angle ($90^\circ$) or a cusp ($0^\circ$) near these points in the limiting sense. For the first alternative, we further prove the uniformly regularity of the free boundary near these isolated singular points.

4.Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions

Authors:Qionglei Chen, Zhiwen Zhao

Abstract: A high-contrast two-phase nonlinear composite material with adjacent inclusions of $m$-convex shapes is considered for $m>2$. The mathematical formulation consists of the insulated conductivity problem with $p$-Laplace operator in $\mathbb{R}^{d}$ for $p>1$ and $d\geq2$. The stress, which is the gradient of the solution, always blows up with respect to the distance $\varepsilon$ between two inclusions as $\varepsilon$ goes to zero. We first establish the pointwise upper bound on the gradient possessing the singularity of order $\varepsilon^{-\beta}$ with $\beta=(1-\alpha)/m$ for some $\alpha\geq0$, where $\alpha=0$ if $d=2$ and $\alpha>0$ if $d\geq3$. In particular, we give a quantitative description for the range of horizontal length of the narrow channel in the process of establishing the gradient estimates, which provides a clear understanding for the applied techniques and methods. For $d\geq2$, we further construct a supersolution to sharpen the upper bound with any $\beta>(d+m-2)/(m(p-1))$ when $p>d+m-1$. Finally, a subsolution is also constructed to show the almost optimality of the blow-up rate $\varepsilon^{-1/\max\{p-1,m\}}$ in the presence of curvilinear squares. This fact reveals a novel dichotomy phenomena that the singularity of the gradient is uniquely determined by one of the convexity parameter $m$ and the nonlinear exponent $p$ except for the critical case of $p=m+1$ in two dimensions.

5.Inverse problem of determining time-dependent leading coefficient in the time-fractional heat equation

Authors:Daurenbek Serikbaev, Michael Ruzhansky, Niyaz Tokmagambetov

Abstract: In this paper, we investigate direct and inverse problems for the time-fractional heat equation with a time-dependent diffusion coefficient for positive operators. First, we consider the direct problem, and the unique existence of the generalized solution is established. We also deduce some regularity results. Here, our proofs are based on the eigenfunction expansion method. Second, we consider the inverse problem of determining the diffusion coefficient. The well-posedness of this inverse problem is shown by reducing the problem to an operator equation for the diffusion coefficient.

6.Asymptotic stability in the critical space of 2D monotone shear flow in the viscous fluid

Authors:Hui Li, Weiren Zhao

Abstract: In this paper, we study the long-time behavior of the solutions to the two-dimensional incompressible free Navier Stokes equation (without forcing) with small viscosity $\nu$, when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold $\nu^{\frac{1}{2}}$ for perturbations in the critical space $H^{log}_xL^2_y$. Specifically, if the initial velocity $V_{in}$ and the corresponding vorticity $W_{in}$ are $\nu^{\frac{1}{2}}$-close to the shear flow $(b_{in}(y),0)$ in the critical space, i.e., $\|V_{in}-(b_{in}(y),0)\|_{L_{x,y}^2}+\|W_{in}-(-\partial_yb_{in})\|_{H^{log}_xL^2_y}\leq \epsilon \nu^{\frac{1}{2}}$, then the velocity $V(t)$ stay $\nu^{\frac{1}{2}}$-close to a shear flow $(b(t,y),0)$ that solves the free heat equation $(\partial_t-\nu\partial_{yy})b(t,y)=0$. We also prove the enhanced dissipation and inviscid damping, namely, the nonzero modes of vorticity and velocity decay in the following sense $\|W_{\neq}\|_{L^2}\lesssim \epsilon\nu^{\frac{1}{2}}e^{-c\nu^{\frac{1}{3}}t}$ and $\|V_{\neq}\|_{L^2_tL^2_{x,y}}\lesssim \epsilon\nu^{\frac{1}{2}}$. In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator $b(t,y)\partial_x-\partial_{yy}b(t,y)\partial_x\Delta^{-1}$, which could be useful in future studies.

7.Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow

Authors:Haigang Li, Longjuan Xu, Peihao Zhang

Abstract: The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.

8.Nonexistence results for semilinear elliptic equations on weighted graphs

Authors:Dario Daniele Monticelli, Fabio Punzo, Jacopo Somaglia

Abstract: We study semilinear elliptic inequalities with a potential on infinite graphs. Given a distance on the graph, we assume an upper bound on its Laplacian, and a growth condition on a suitable weighted volume of balls. Under such hypotheses, we prove that the problem does not admit any nonnegative nontrivial solution. We also show that our conditions are optimal.

9.Modica type estimates and curvature results for overdetermined elliptic problems

Authors:David Ruiz, Pieralberto Sicbaldi, Jing Wu

Abstract: In this paper, we establish a Modica type estimate on bounded solutions to the overdetermined elliptic problem \begin{equation*} \begin{cases} \Delta u+f(u) =0& \mbox{in $\Omega$, }\\ u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=c_0 &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^{n},n\geq 2$. As we will see, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. We will also discuss the equality case. From such estimates we will deduce information about the curvature of $\partial \Omega$ under a certain condition on $c_0$ and $f$. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.

10.Asymptotic Stability of Solitary Waves for One Dimensional Nonlinear Schrödinger Equations

Authors:Charles Collot, Pierre Germain

Abstract: We show global asymptotic stability of solitary waves of the nonlinear Schr\"odinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard scattering if it is higher order. We handle a general nonlinearity without any vanishing condition, requiring that the linearized operator around the solitary wave has neither nonzero eigenvalues, nor threshold resonances. Initial data are chosen in a neighborhood of the solitary waves in the natural space $H^1 \cap L^{2,1}$ (where the latter is the weighted $L^2$ space). The proof relies on the analysis of resonances as seen through the distorted Fourier transform, combined for the first time with modulation and renormalization techniques.

11.The qualitative behavior at a vortex point for the Chern-Simon-Higgs equation

Authors:Jiayu Li, Lei Liu

Abstract: In this paper, we study the qualitative behavior at a vortex blow-up point for Chern-Simon-Higgs equation. Roughly speaking, we will establish an energy identity at a each such point, i.e. the local mass is the sum of the bubbles. Moreover, we prove that either there is only one bubble which is a singular bubble or there are more than two bubbles which contains no singular bubble. Meanwhile, we prove that the energies of these bubbles must satisfy a quadratic polynomial which can be used to prove the simple blow-up property when the multiplicity is small. As is well known, for many Liouville type system, Pohozaev type identity is a quadratic polynomial corresponding to energies which can be used directly to compute the local mass at a blow-up point. The difficulty here is that, besides the energy's integration, there is a additional term in the Pohozaev type identity of Chern-Simon-Higgs equation. We need some more detailed and delicated analysis to deal with it.

12.Exact controllability of incompressible ideal magnetohydrodynamics in $2$D

Authors:Manuel Rissel

Abstract: This work examines the controllability of planar incompressible ideal magnetohydrodynamics (MHD). Interior controls are obtained for problems posed in doubly-connected regions; simply-connected configurations are driven by boundary controls. Up to now, only straight channels regulated at opposing walls have been studied. Hence, the present program adds to the literature an exploration of interior controllability, extends the known boundary controllability results, and contributes ideas for treating general domains. To transship obstacles stemming from the MHD coupling and the magnetic field topology, a divide-and-control strategy is proposed. This leads to a family of nonlinear velocity-controlled sub-problems which are solved using J.-M. Coron's return method. The latter is here developed based on a reference trajectory in the domain's first cohomology space.

13.Block-radial symmetry breaking for ground states of biharmonic NLS

Authors:Rainer Mandel, Diogo Oliveira e Silva

Abstract: We prove that the biharmonic NLS equation $\Delta^2 u +2\Delta u+(1+\varepsilon)u=|u|^{p-2}u$ in $\mathbb R^d$ has at least $k+1$ different solutions if $\varepsilon>0$ is small enough and $2<p<2_\star^k$, where $2_\star^k$ is an explicit critical exponent arising from the Fourier restriction theory of $O(d-k)\times O(k)$-symmetric functions. This extends the recent symmetry breaking result of Lenzmann-Weth and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of $k$. We further prove that, as $\varepsilon\to 0^+$, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.

14.On the inviscid limit connecting Brinkman's and Darcy's models of tissue growth with nonlinear pressure

Authors:Charles Elbar, Jakub Skrzeczkowski

Abstract: Several recent papers have addressed modelling of the tissue growth by the multi-phase models where the velocity is related to the pressure by one of the physical laws (Stoke's, Brinkman's or Darcy's). While each of these models has been extensively studied, not so much is known about the connection between them. In the recent paper (arXiv:2303.10620), assuming the linear form of the pressure, the Authors connected two multi-phase models by an inviscid limit: the viscoelastic one (of Brinkman's type) and the inviscid one (of Darcy's type). Here, we prove that the same is true for a nonlinear, power-law pressure. The new ingredient is that we use relation between the pressure $p$ and the Brinkman potential $W$ to deduce compactness in space of $p$ from the compactness in space of $W$.

15.Plasmons for the Hartree equations with Coulomb interaction

Authors:Toan T. Nguyen, Chanjin You

Abstract: In this work, we establish the existence and decay of {\em plasmons}, the quantum of Langmuir's oscillatory waves found in plasma physics, for the linearized Hartree equations describing an interacting gas of infinitely many fermions near general translation-invariant steady states, including compactly supported Fermi gases at zero temperature, in the whole space $\RR^d$ for $d\ge 2$. Notably, these plasmons exist precisely due to the long-range pair interaction between the particles. Next, we provide a survival threshold of spatial frequencies, below which the plasmons purely oscillate and disperse like a Klein-Gordon's wave, while at the threshold they are damped by {\em Landau damping}, the classical decaying mechanism due to their resonant interaction with the background fermions. The explicit rate of Landau damping is provided for general radial homogenous equilibria. Above the threshold, the density of the excited fermions is well approximated by that of the free gas dynamics and thus decays rapidly fast for each Fourier mode via {\em phase mixing}. Finally, pointwise bounds on the Green function and dispersive estimates on the density are established.

16.A class of fractional parabolic reaction-diffusion systems with control of total mass: theory and numerics

Authors:Maha Daoud, El-Haj Laamri, Azeddine Baalal

Abstract: In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems posed in a bounded domain of $\mathbb{R}^N$. The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide non-negativity of the solutions and uniform control of the total mass. The diffusion operators are of type $u_i\mapsto d_i(-\Delta)^s u_i$ where $0<s<1$. Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type $u_i\mapsto -d_i\Delta u_i$. On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case $s=1$.

1.Internal Schauder estimates for Hörmander type equations with Dini continuous source

Authors:Giovanna Citti, Bianca Stroffolini

Abstract: We study the regularity properties of a general second order H\"ormander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the H\"ormander condition, we consider the linear operator in $\R^{N}$, with $N>m+1$: \begin{equation*} \L u := \sum_{i, j= 1}^{m} a_{ij} X_{i}X_{j} u - X_0 u. \end{equation*} The vector field $X_0$ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\L u = f$ are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.

2.Global approximation for the cubic NLS with strong magnetic confinement

Authors:Jumpei Kawakami

Abstract: We consider nonlinear Schr\"{o}dinger equation with strong magnetic fields in 3D. This model was derived by R L. Frank, F. M\'{e}hats, C. Sparber in 2017. We prove modified scattering for small initial data and the existence of modified wave operator for small final data. To describe asymptotic behavior of the NLS we use the time-averaged model which was derived by the same authors as "the strong magnetic confinement limit" of the NLS. We construct asymptotic solutions which satisfy both asymptotic in time evolution and convergence in the strong magnetic confinement limit. We also analyze the error between the solution to the NLS and the time-averaged model for the same initial data and obtain global estimates.

3.Heat equations associated to harmonic oscillator with exponential nonlinearity

Authors:Divyang G. Bhimani, Mohamed Majdoub, Ramesh Manna

Abstract: We consider the Cauchy problem for heat equation with fractional harmonic oscillator and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global {weak-mild} solutions for small initial data and obtain decay estimates for large time in Lebesgue spaces. In particular, we show that the decay depends on the behavior of the nonlinearity near the origin. Finally, we show that for some non-negative initial data in the appropriate Orlicz space, there is no local non-negative classical solution.

4.Strong Persistence of a Class of Strongly Coupled Parabolic Systems of $m$ Equations

Authors:Dung Le

Abstract: We establish one of the most important assumptions of the strong persistence theory for dynamical systems associated to cross diffusion systems of $m$ equations ($m\ge2$): the stable sets of semi-trivial steady cannot intersect the interior of the positive cone of $C(\Omega,\mathbb{R}^m)$. Many examples will be provided to show the effects of the cross diffusion.

5.On The Weak Harnack Estimate For Nonlocal Equations

Authors:Harsh Prasad

Abstract: We prove a weak Harnack estimate for a class of doubly nonlinear nonlocal equations modelled on the nonlocal Trudinger equation \begin{align*} \partial_t(|u|^{p-2}u) + (-\Delta_p)^s u = 0 \end{align*} for $p\in (1,\infty)$ and $s \in (0,1)$. Our proof relies on expansion of positivity arguments developed by DiBenedetto, Gianazza and Vespri adapted to the nonlocal setup. Even in the linear case of the nonlocal heat equation and in the time-independent case of fractional $p-$Laplace equation, our approach provides an alternate route to Harnack estimates without using Moser iteration, log estimates or Krylov-Safanov covering arguments.

6.Normalized ground states for a biharmonic Choquard system in $\mathbb{R}^4$

Authors:Wenjing Chen, Zexi Wang

Abstract: In this paper, we study the existence of normalized ground state solutions for the following biharmonic Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \Delta^2v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad \displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $a,b>0$ are prescribed, $\lambda_1,\lambda_2\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential subcritical or critical growth in the sense of the Adams inequality. By using a minimax principle and analyzing the behavior of the ground state energy with respect to the prescribed mass, we obtain the existence of ground state solutions for the above problem.

7.Stefan problem with surface tension: uniqueness of physical solutions under radial symmetry

Authors:Yucheng Guo, Sergey Nadtochiy, Mykhaylo Shkolnikov

Abstract: We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently introduced in [NS23]. The paper at hand is devoted to the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. In the course of the proof, we establish a wide variety of regularity estimates for the free boundary and for the temperature function, of interest in their own right.

8.Sign changing bubble tower solutions to a slightly subcritical elliptic problem with non-power nonlinearity

Authors:Shengbing Deng, Fang Yu

Abstract: We study the following elliptic problem involving slightly subcritical non-power nonlinearity $$\left\{\begin{array}{lll} -\Delta u =\frac{|u|^{2^*-2}u}{[\ln(e+|u|)]^\epsilon}\ \ &{\rm in}\ \Omega, \\[2mm] u= 0 \ \ & {\rm on}\ \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, $n\geq 3$, $2^*=\frac{2n}{n-2}$ is the critical Sobolev exponent, $\epsilon>0$ is a small parameter. By the finite dimensional Lyapunov-Schmidt reduction method, we construct a sign changing bubble tower solution with the shape of a tower of bubbles as $\epsilon$ goes to zero.

1.Multiple positive solutions for a double phase system with singular nonlinearity

Authors:Zhanbing Bai, Yizhe Feng

Abstract: In this paper, we study a class of double phase systems which contain the singular and mixed nonlinear terms. Unlike the single equation, the mixed nonlinear terms make the problem more complicate. The geometry of the fibering mapping has multiple possibilities. To overcome the difficulties posed by the mixed nonlinear terms, we need to repeatedly construct concave functions, discuss different cases, and use the properties of concave functions and basic inequalities such as H\"{o}lder inequality, Poincar\'{e}'s inequality and Young's inequality. By the use of the Nehari manifold, the existence and multiplicity of positive solutions which have nonnegative energy are obtained.

2.Shock profiles for hydrodynamic models for fluid-particles flows in the flowing regime

Authors:Thierry Goudon COFFEE, LJAD, Pauline Lafitte MICS, FR3487, Corrado Mascia Sapienza University of Rome

Abstract: Starting from coupled fluid-kinetic equations for the modeling of laden flows, we derive relevant viscous corrections to be added to asymptotic hydrodynamic systems, by means of Chapman-Enskog expansions and analyse the shock profile structure for such limiting systems. Our main findings can be summarized as follows. Firstly, we consider simplified models, which are intended to reproduce the main difficulties and features of more intricate systems. However, while they are more easily accessible to analysis, such toy-models should be considered with caution since they might lose many important structural properties of the more realistic systems. Secondly, shock profiles can be identified also in such a case, which can be proven to be stable at least in the regime of small amplitude shocks. Last, but not least, regarding at the temperature of the mixture flow as a parameter of the problem, we show that the zero-temperature model admits viscous shock profiles. Numerical results indicate that a similar conclusion should apply in the regime of small positive temperatures.

3.On the exponential ergodicity of the McKean-Vlasov SDE depending on a polynomial interaction

Authors:Mohamed Alfaki Ag Aboubacrine Assadeck MATHSTIC, LAREMA

Abstract: In this paper, we study the long time behaviour of the Fokker-Planck and the kinetic Fokker-Planck equations with many body interaction, more precisely with interaction defined by U-statistics, whose macroscopic limits are often called McKean-Vlasov and Vlasov-Fokker-Planck equations respectively. In the continuity of the recent papers [63, [43],[42]] and [44, [74],[75]], we establish nonlinear functional inequalities for the limiting McKean-Vlasov SDEs related to our particle systems. In the first order case, our results rely on large deviations for U-statistics and a uniform logarithmic Sobolev inequality in the number of particles for the invariant measure of the particle system. In the kinetic case, we first prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 ($\mu$) with a rate of convergence which is explicitly computable and independent of the number of particles. In a second time, we quantitatively establish an exponential return to equilibrium in Wasserstein's W 2 --metric for the Vlasov-Fokker-Planck equation.

4.The Navier-Stokes equations in mixed-norm time-space parabolic Morrey spaces

Authors:Pierre Gilles Lemarié-Rieusset LaMME

Abstract: We discuss the Navier-Stokes equations with forces in the mixed norm time-space parabolic Morrey spaces of Krylov.

5.Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function

Authors:Muhammed Ali Mehmood

Abstract: We study the Aw-Rascle system in a one-dimensional domain with periodic boundary conditions, where the offset function is replaced by the gradient of the function $\rho_{n}^{\gamma}$, where $\gamma \to \infty$. The resulting system resembles the 1D pressureless compressible Navier-Stokes system with a vanishing viscosity coefficient in the momentum equation and can be used to model traffic and suspension flows. We first prove the existence of a unique global-in-time classical solution for $n$ fixed. Unlike the previous result for this system, we obtain global existence without needing to add any approximation terms to the system. This is by virtue of a $n-$uniform lower bound on the density which is attained by carrying out a maximum-principle argument on a suitable potential, $W_{n} = \rho_{n}^{-1}\partial_{x}w_{n}$. Then, we prove the convergence to a weak solution of a hybrid free-congested system as $n \to \infty$, which is known as the hard-congestion model.

6.A strong comparison principle for the generalized Dirichlet problem for Monge-Ampere

Authors:Brittany Froese Hamfeldt

Abstract: We prove a strong form of the comparison principle for the elliptic Monge-Ampere equation, with a Dirichlet boundary condition interpreted in the viscosity sense. This comparison principle is valid when the equation admits a Lipschitz continuous weak solution. The result is tight, as demonstrated by examples in which the strong comparison principle fails in the absence of Lipschitz continuity. This form of comparison principle closes a significant gap in the convergence analysis of many existing numerical methods for the Monge-Ampere equation. An important corollary is that any consistent, monotone, stable approximation of the Dirichlet problem for the Monge-Ampere equation will converge to the viscosity solution.

7.Generalized Monge-Ampère functionals and related variational problems

Authors:Freid Tong, Shing-Tung Yau

Abstract: In this paper, we introduce a family of real Monge-Amp\`ere functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated Dirichlet problems. In particular, we prove the existence of solutions for a nonlinear eigenvalue problem associated to this family of functionals.

8.Existence of a global attractor for the compressible Euler equation in a bounded interval

Authors:Yun-guang Lu, Okihiro Sawada, Naoki Tsuge

Abstract: In this paper, we are concerned with the one-dimensional initial boundary value problem for isentropic gas dynamics. Through the contribution of great researchers such as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay theory of solutions was established. They treated with the Cauchy problem and the corresponding initial data have the small total variation. On the other hand, the decay for initial data with large oscillation has been open for half a century. In addition, due to the reflection of shock waves at the boundaries, little is known for the decay of the boundary value problem on a bounded interval. Our goal is to prove the existence of a global attractor, which yields a decay of solutions for large data. To construct approximate solutions, we introduce a modified Godunov scheme.

1.Linear asymptotic stability of small-amplitude periodic waves of the generalized Korteweg--de Vries equations

Authors:Corentin Audiard, L. Miguel Rodrigues, Changzhen Sun

Abstract: In this note, we extend the detailed study of the linearized dynamics obtained for cnoidal waves of the Korteweg--de Vries equation in \cite{JFA-R} to small-amplitude periodic traveling waves of the generalized Korteweg-de Vries equations that are not subject to Benjamin--Feir instability. With the adapted notion of stability, this provides for such waves, global-in-time bounded stability in any Sobolev space, and asymptotic stability of dispersive type. When doing so, we actually prove that such results also hold for waves of arbitrary amplitude satisfying a form of spectral stability designated here as dispersive spectral stability.

2.Eigenvalue Variations of the Neumann Laplace Operator Due to Perturbed Boundary Conditions

Authors:Medet Nursultanov, William Trad, Justin Tzou, Leo Tzou

Abstract: This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold $(M,g,\partial M)$ under the singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive a sharp asymptotic of the perturbed eigenvalues, as the Dirichlet part shrinks to a point $x^*\in \partial M$, in terms of the spectral parameters of the unperturbed system. This asymptotic demonstrates the impact of the geometric properties of the manifold at a specific point $x^*$. Furthermore, it becomes evident that the shape of the Dirichlet region holds significance as it impacts the first terms of the asymptotic. A crucial part of this work is the construction of the singularity structure of the restricted Neumann Green's function which may be of independent interest. We employ a fusion of layer potential techniques and pseudo-differential operators during this work.

3.Existence and stability of weak solutions of the Vlasov--Poisson system in localized Yudovich spaces

Authors:Gianluca Crippa, Marco Inversi, Chiara Saffirio, Giorgio Stefani

Abstract: We consider the Vlasov--Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. In our first main theorem, we prove the uniqueness and the quantitative stability of Lagrangian solutions $f=f(t,x,v)$ whose associated spatial density $\rho_f=\rho_f(t,x)$ is potentially unbounded but belongs to suitable uniformly-localized Yudovich spaces. This requirement imposes a condition of slow growth on the function $p \mapsto \|\rho_f(t,\cdot)\|_{L^p}$ uniformly in time. Previous works by Loeper, Miot and Holding--Miot have addressed the cases of bounded spatial density, i.e., $\|\rho_f(t,\cdot)\|_{L^p} \lesssim 1$, and spatial density such that $\|\rho_f(t,\cdot)\|_{L^p} \sim p^{1/\alpha}$ for $\alpha\in[1,+\infty)$. Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov--Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov--Poisson systems.

4.Semi-classical observation sufficices for observability: wave and Schrödinger equations

Authors:Nicolas BURQ, Belhassen DEHMAN, Jérôme LE ROUSSEAU

Abstract: For the wave and the Schr\"odinger equations we show how observability can be deduced from the observability of solutions localized in frequency according to a dyadic scale.

5.Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow

Authors:Mihaela Ifrim, Daniel Tataru

Abstract: The first target of this article is the local well-posedness question for 1D quasilinear Schr\"odinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a \emph{sharp local well-posedness} result. The second goal of this article is to consider the long time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: ``\emph{Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions}''; the conjecture was initially proved for a well chosen semilinear model of Schr\"odinger type. Our work here establishes the above conjecture for 1D quasilinear Schr\"{o}dinger flows. Precisely, we show that if the problem has \emph{phase rotation symmetry} and is \emph{conservative and defocusing}, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows. Furthermore, we prove it at the minimal Sobolev regularity in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that \emph{small, $\epsilon$ size data yields long time solutions on the $\epsilon^{-8}$ time-scale}. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schr\"{o}dinger flows.

6.Phase space analysis of spectral multipliers for the twisted Laplacian

Authors:S. Ivan Trapasso

Abstract: We prove boundedness results on modulation and Wiener amalgam spaces concerning some spectral multipliers for the twisted Laplacian. Techniques of pseudo-differential calculus are inhibited due to the lack of global ellipticity of the special Hermite operator, therefore a phase space approach must rely on different pathways. In particular, we exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for spectral multipliers. We focus on a wide class of oscillating multipliers, including fractional powers of the twisted Laplacian and the corresponding dispersive flows of Schr\"odinger and wave type. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.

7.Persistence of solutions in a nonlocal predator-prey system with a shifting habitat

Authors:Min Zhao, Rong Yuan

Abstract: In this paper, we mainly study the propagation properties of a nonlocal dispersal predator-prey system in a shifting environment. It is known that Choi et al. [J. Differ. Equ. 302 (2021), pp. 807-853] studied the persistence or extinction of the prey and the predator separately in various moving frames. In particular, they achieved a complete picture in the local diffusion case. However, the question of the persistence of the prey and the predator in some intermediate moving frames in the nonlocal diffusion case is left open in Choi et al.'s paper. By using some prior estimates, the Arzela-Ascoli theorem and a diagonal extraction process, we can extend and improve the main results of Choi et al. to achieve a complete picture in the nonlocal diffusion case.

8.Sign-changing solutions to the slightly supercritical Lane-Emden system with Neumann boundary conditions

Authors:Qing Guo, Shuangjie Peng

Abstract: We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Delta u_2=|u_1|^{q_\epsilon-1}u_1, \ &in\ \Omega,\\ \partial_\nu u_1=\partial_\nu u_2=0,\ &on\ \partial\Omega \end{cases} \end{equation*} where $\Omega=B_1(0)$ is the unit ball in $\mathbb{R}^n$ ($n\geq4$) centered at the origin, $p_\epsilon=p+\alpha\epsilon, q_\epsilon=q+\beta\epsilon$ with $\alpha,\beta>0$ and $\frac1{p+1}+\frac1{q+1}=\frac{n-2}n$. We show the existence and multiplicity of concentrated solutions based on the Lyapunov-Schmidt reduction argument incorporating the zero-average condition by certain symmetries. It is worth noting that we simultaneously consider two cases: $p>\frac n{n-2}$ and $p<\frac n{n-2}$. The coupling mechanisms of the system are completely different in these different cases, leading to significant changes in the behavior of the solutions. The research challenges also vary. Currently, there are very few papers that take both ranges into account when considering solution construction. Therefore, this is also the main feature and new ingredient of our work.

9.Traveling Wave in a Ratio-dependent Holling-Tanner System with Nonlocal Diffusion and Strong Allee Effect

Authors:Hongliang Li, Min Zhao, Rong Yuan

Abstract: In this paper, a ratio-dependent Holling-Tanner system with nonlocal diffusion is taken into account, where the prey is subject to a strong Allee effect. To be special, by applying Schauder's fixed point theorem and iterative technique, we provide a general theory on the existence of traveling waves for such system. Then appropriate upper and lower solutions and a novel sequence, similar to squeeze method, are constructed to demonstrate the existence of traveling waves for c>c*. Moreover, the existence of traveling wave for c=c* is also established by spreading speed theory and comparison principle. Finally, the nonexistence of traveling waves for c<c* is investigated, and the minimal wave speed then is determined.

10.Analysis of Heterogeneous Vehicular Traffic: Using Proportional Densities

Authors:Nanyondo Josephine, Henry Kasumba

Abstract: An extended multi-class Aw-Rascle (AR) model with pressure term described as a function of area occupancy defined in form of proportional densities is presented. Two vehicle classes that is; cars and motorcycles are considered based on an assumption that proportions of these form total traffic density. Qualitative properties of the proposed equilibrium velocity is established. Conditions under which the proposed model is stable are determine by linear stability analysis. To compute numerical flux, the model is discretized by the original Roe decomposition scheme, where Roe matrix, averaged data variables and wave strengths are explicitly derived. The Roe matrix is shown to be hyperbolic, consistent and conservative. From the numerical results, the effect of motorcycles proportion on the flow of vehicle classes is determined. Results obtained remain within limits therefore, the proposed model is realistic.

11.Traveling Waves of Modified Leslie-Gower Predator-prey Systems

Authors:Hongliang Li, Min Zhao, Rong Yuan

Abstract: The spreading phenomena in modified Leslie-Gower reaction-diffusion predator-prey systems are the topic of this paper. We mainly study the existence of two different types of traveling waves. Be specific, with the aid of the upper and lower solutions method, we establish the existence of traveling wave connecting the prey-present state and the coexistence state or the prey-present state and the prey-free state by constructing different and appropriate Lyapunov functions. Moreover, for traveling wave connecting the prey-present state and the prey-free state, we gain more monotonicity information on wave profile based on the asymptotic behavior at negative infinite. Finally, our results are applied to modified Leslie-Gower system with Holling II type or Lotka-Volterra type, and then a novel Lyapunov function is constructed for the latter, which further enhances our results. Meanwhile, some numerical simulations are carried to support our results.

12.Long time well-posedness and full justification of a Whitham-Green-Naghdi system

Authors:Louis Emerald, Martin Oen Paulsen

Abstract: We establish the full justification of a "Whitham-Green-Naghdi" system modeling the propagation of surface gravity waves with bathymetry in the shallow water regime. It is an asymptotic model of the water waves equations with the same dispersion relation. The model under study is a nonlocal quasilinear symmetrizable hyperbolic system without surface tension. We prove the consistency of the general water waves equations with our system at the order of precision $O(\mu^2 (\varepsilon + \beta))$, where $\mu$ is the shallow water parameter, $\varepsilon$ the nonlinearity parameter, and $\beta$ the topography parameter. Then we prove the long time well-posedness on a time scale $O(\frac{1}{\max\{\varepsilon,\beta\}})$. Lastly, we show the convergence of the solutions of the Whitham-Green-Naghdi system to the ones of the water waves equations on the later time scale.

13.A sufficient condition on successful invasion by the predator

Authors:Hongliang Li, Min Zhao, Rong Yuan

Abstract: In this paper, we provide a sufficient condition on successful invasion by the predator. Specially, we obtain the persistence of traveling wave solutions of predator-prey system, in which the predator can survive without the predation of the prey. This proof heavily depends on comparison principle of scalar monostable equation, the rescaling method and phase-plane analysis.

14.Semiclassical resolvent bounds for short range $L^\infty$ potentials with singularities at the origin

Authors:Jacob Shapiro

Abstract: We consider, for $h, E > 0$, resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V - E$. The potential takes the form $V = V_L+ V_S$, where $V_L$ is a long range potential which is Lipschitz with respect to the radial variable, while $V_S = O(|x|^{-1} (\log |x|)^{-\rho})$ as $|x| \to \infty$ for some $\rho > 1$. Both $|V_L|$ and $|V_S|$ may behave like $|x|^{-\beta}$ as $|x| \to 0$, provided $0 \le \beta < 2(\sqrt{3} -1)$. We find that, as $h \to 0^+$, the resolvent bound is of the form $\exp(Ch^{-2} (\log(h^{-1}))^{1 + \rho})$ for some $C > 0$. The $h$-dependence of the bound improves if $V_S$ decays at a faster rate toward infinity.

15.A well-posed variational formulation of the Neumann boundary value problem for the biharmonic operator

Authors:Alberto Valli

Abstract: In this note we devise and analyze a well-posed variational formulation of the Neumann boundary value problem associated to the biharmonic operator $\Delta^2$.

16.Long-time behavior of the Stokes-transport system in a channel

Authors:Anne-Laure Dalibard, Julien Guillod, Antoine Leblond

Abstract: The coupling between the transport equation for the density and the Stokes equation is considered in a periodic channel. More precisely, the density is advected by pure transport by a velocity field given by the Stokes equation with source force coming from the gravity due to differences in the density. Dirichlet boundary conditions are taken for the velocity field on the bottom and top of the channel, and periodic conditions in the horizontal variable. We prove that the affine stratified density profile is stable under small perturbations in Sobolev spaces and prove convergence of the density to another limiting stratified density profile for large time with an explicit algebraic decay rate. Moreover, we are able to precisely identify the limiting profile as the decreasing vertical rearrangement of the initial density. Finally, we study boundary layers formation to precisely characterize the long-time behavior beyond the constant limiting profile and enlighten the optimal decay rate.

17.Inverse problems of identifying the time-dependent source coefficient for subelliptic heat equations

Authors:Mansur I. Ismailov, Tohru Ozawa, Durvudkhan Suragan

Abstract: We discuss inverse problems of determining the time-dependent source coefficient for a general class of subelliptic heat equations. We show that a single data at an observation point guarantees the existence of a (smooth) solution pair for the inverse problem. Moreover, additional data at the observation point implies an explicit formula for the time-dependent source coefficient. We also explore an inverse problem with nonlocal additional data, which seems a new approach even in the Laplacian case.

18.Boundary Layers for the Lane-Emden System with supercritical exponents

Authors:Qing Guo, Junyuan Liu, Shuangjie Peng

Abstract: We consider the following supercritical problem for the Lane-Emden system: \begin{equation}\label{eq00} \begin{cases} -\Delta u_1=|u_2|^{p-1}u_2\ &in\ D,\\ -\Delta u_2=|u_1|^{q-1}u_1 \ &in\ D,\\ u_1=u_2=0\ &on\ \partial D, \end{cases} \end{equation} where $D$ is a bounded smooth domain in $\mathbb R^N$, $N\geq4.$ What we mean by supercritical is that the exponent pair $(p,q)\in(1,\infty)\times(1,\infty)$ satisfies $\frac1{p+1}+\frac1{q+1}<\frac{N-2}N$. We prove that for some suitable domains $D\subset\mathbb R^N$, there exist positive solutions with layers concentrating along one or several $k$-dimensional sub-manifolds of $\partial D$ as $$\frac1{p+1}+\frac1{q+1}\rightarrow\frac{n-2}{n},\ \ \ \ \frac{n-2}{n}<\frac1{p+1}+\frac1{q+1}<\frac{N-2}N,$$ where $n:=N-k$ with $1\leq k\leq N-3$. By transforming the original problem into a lower $n$-dimensional weighted system, we carry out the reduction framework and apply the blow-up analysis. In this process, the properties of the ground state related to the limit problem make all the difference. The corresponding exponent pair $(p_0,q_0)$, which is the limit pair of $(p,q)$, is on the critical hyperbola $\frac n{p_0+1}+\frac n{q_0+1}=n-2$. It is well known that the range of the smaller exponent, say $p_0$, has a great influence on the solutions. It is worth noting that in this paper, we consider both two ranges, which is contained in $p_0>\frac n{n-2}$ and $p_0<\frac n{n-2}$ respectively. The coupling mechanism of the strongly indefinite problem in these two cases is totally different, which is the main feature and new ingredient here.

19.Anisotropic regularity for elliptic problems with Dirac measures as data

Authors:Ignacio Ojea

Abstract: We study the Possion problem with singular data given by a source supported on a one dimensional curve strictly contained in a three dimensional domain. We prove regularity results for the solution on isotropic and on anisotropic weighted spaces of Kondratiev type. Our technique is based on the study of a regularized problem. This allows us to exploit the local nature of the singularity. Our results hold with very few smoothness hypotheses on the domain and on the support of the data. We also discuss some extensions of our main results, including the two dimensional case, sources supported on closed curves and on polygonals.

1.Periodic Vlasov-Stokes' system: Existence and Uniqueness of strong solutions

Authors:Harsha Hutridurga, Krishan Kumar, Amiya K. Pani

Abstract: This paper deals with the Vlasov-Stokes' system in three dimensions with periodic boundary conditions in the spatial variable. We prove the existence of a unique strong solution to this two-phase model under the assumption that initial velocity moments of certain order are bounded. We use a fixed point argument to arrive at a global-in-time solution.

2.Globally analytical solutions of the compressible Oldroyd-B model without retardation

Authors:Xinghong Pan

Abstract: In this paper, we prove the global existence of analytical solutions to the compressible Oldroyd-B model without retardation near a non-vacuum equilibrium in ${\mathbb R}^n$ $(n=2,3)$. Zero retardation results in zero dissipation in the velocity equation, which is the main difficulty that prevents us to obtain the long time well-posedness of solutions. Through dedicated analysis, we find that the linearized equations of this model have damping effects, which ensures the global-in-time existence of small data solutions. However, the nonlinear quadratic terms have one more order derivative than the linear part and no good structure is discovered to overcome this derivative loss problem. So we can only build the result in the analytical energy space rather than Sobolev space with finite order derivatives.

3.Continuity of the double layer potential of a second order elliptic differential operator in Schauder spaces on the boundary

Authors:Massimo Lanza de Cristoforis

Abstract: We prove the validity of a regularizing property on the boundary of the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M.~G\"{u}nter, S.~Mikhlin, V.D.~Kupradze, T.G.~Gegelia, M.O.~Basheleishvili and T.V.~Bur\-chuladze, U.~Heinemann and extends the work of A.~Kirsch who has considered the case of the Helmholtz operator.

4.Existence and Stability of Random Transition Waves for Nonautonomous Fisher-KPP Equations with Nonlocal Diffusion

Authors:Min Zhao, Rong Yuan

Abstract: In this paper, we study the existence and stability of random transition waves for time heterogeneous Fisher-KPP Equations with nonlocal diffusion. More specifically, we consider general time heterogeneities both for the nonlocal diffusion kernel and the reaction term. We use the comparison principle of the scalar equation and the method of upper and lower solutions to investigate the existence of random transition wave solution when the wave speed is large enough. In addition, we show the stability of random transition fronts for non-autonomous Fisher-KPP equations with nonlocal diffusion.

5.On a class of elliptic equations with Critical Perturbations in the hyperbolic space

Authors:Debdip Ganguly, Diksha Gupta, K. Sreenadh

Abstract: We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1} \,\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, $$ where $\mathbb{B}^N$ denotes the hyperbolic space, $2<p<2^*:=\frac{2N}{N-2}$, if $N \geqslant 3; 2<p<+\infty$, if $N = 2,\;\lambda < \frac{(N-1)^2}{4}$, and $0< a\in L^\infty(\mathbb{B}^N).$ We first prove the existence of a positive radially symmetric ground-state solution for $a(x) \equiv 1.$ Next, we prove that for $a(x) \geq 1$, there exists a ground-state solution for $\varepsilon$ small. For proof, we employ ``conformal change of metric" which allows us to transform the original equation into a singular equation in a ball in $\mathbb R^N$. Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case $a(x) \leq 1$ is considered where we first show that there is no ground-state solution, and prove the existence of a \it bound-state solution \rm (high energy solution) for $\varepsilon$ small. We employ variational arguments in the spirit of Bahri-Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

6.On the focusing fractional nonlinear Schrödinger equation on the waveguide manifolds

Authors:Amin Esfahani, Hichem Hajaiej, Yongming Luo, Linjie Song

Abstract: In this paper, we consider the focusing fractional nonlinear Schr\"{o}dinger equation (FNLS) on the waveguide manifolds $\mathbb{R}^d\times\mathbb{T}^m$ both in the isotropic and anisotropic case. Under different conditions, we establish the existence and periodic dependence of the ground states of the focusing FNLS. In the intercritical regime, we also establish the large data scattering for the anisotropic focusing FNLS by appealing to the framework of semivirial vanishing geometry.

7.Sharp quantitative stability of the Möbius group among sphere-valued maps in arbitrary dimension

Authors:André Guerra, Xavier Lamy, Konstantinos Zemas

Abstract: In this work we prove a sharp quantitative form of Liouville's theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$ with degree $\pm 1$ are M\"obius transformations. In the case $n=3$ this estimate was first obtained by Bernand-Mantel, Muratov and Simon (Arch. Ration. Mech. Anal. 239(1):219-299, 2021), with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq 4$ requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the distance to M\"obius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.

8.On the Lavrentiev gap for convex, vectorial integral functionals

Authors:Lukas Koch, Matthias Ruf, Mathias Schäffner

Abstract: We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$ F: g+W_0^{1,1}(\Omega)^m\to\mathbb{R}\cup\{+\infty\},\qquad F(u)=\int_\Omega W(x,\mathrm{D} u)\,\mathrm{d}x, $$ where the boundary datum $g:\Omega\subset \mathbb{R}^d\to\mathbb{R}^m$ is sufficiently regular, $\xi\mapsto W(x,\xi)$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\leq d-1$, we assume $q$-growth from above with $q\leq \frac{(d-1)p}{d-1-p}$, while for $p>d-1$ we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the $x$-dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions.

9.Solutions to the nonlinear obstacle problem with compact contact sets

Authors:Simon Eberle, Hui Yu

Abstract: For the obstacle problem with a nonlinear operator, we characterize the space of global solutions with compact contact sets. This is achieved by constructing a bijection onto a class of quadratic polynomials describing the asymptotic behavior of solutions.

10.A clustering theorem in fractional Sobolev spaces

Authors:Fatma Gamza Düzgün, Antonio Iannizzotto, Vincenzo Vespri

Abstract: We prove a general clustering result for the fractional Sobolev space W^{s,p}. Then we show how corresponding results in W^{1,p} and BV, respectively, can be deduced as special cases.

1.Local existence of solutions and comparison principle for initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory

Authors:Alexander Gladkov

Abstract: We consider an initial value problem for a nonlinear parabolic equation with memory under nonlinear nonlocal boundary condition. In this paper we study classical solutions. We establish the existence of a local maximal solution. It is shown that under some conditions a supersolution is not less than a subsolution. We find conditions for the positiveness of solutions. As a consequence of the positiveness of solutions and the comparison principle of solutions, we prove the uniqueness theorem.

2.Corners and collapse: Some simple observations concerning critical masses and boundary blow-up in the fully parabolic Keller-Segel system

Authors:Mario Fuest, Johannes Lankeit

Abstract: Our main result shows that the mass $2\pi$ is critical for the minimal Keller-Segel system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - v + u, \end{cases} \end{align} considered in a quarter disc $\Omega = \{\,(x_1, x_2) \in \mathbb R : x_1 > 0, x_2 > 0, x_1^2 + x_2^2 < R^2\,\}$, $R > 0$, in the following sense: For all reasonably smooth nonnegative initial data $u_0, v_0$ with $\int_\Omega u_0 < 2\pi$, there exists a global classical solution to the Neumann initial boundary value problem associated to \eqref{prob:abstract}, while for all $m > 2 \pi$ there exist nonnegative initial data $u_0, v_0$ with $\int_\Omega u_0 = m$ so that the corresponding classical solution of this problem blows up in finite time. At the same time, this gives an example of boundary blow-up in \eqref{prob:abstract}. Up to now, precise values of critical masses had been observed in spaces of radially symmetric functions or for parabolic-elliptic simplifications of \eqref{prob:abstract} only.

3.Persistence, extinction and spreading properties of non-cooperative Fisher--KPP systems in space-time periodic media

Authors:Léo Girardin CNRS, ICJ

Abstract: This paper is concerned with asymptotic persistence, extinction and spreading properties for non-cooperative Fisher--KPP systems with space-time periodic coefficients. In a preceding paper, a family of generalized principal eigenvalues associated with an appropriate linear problem was studied. Here, a relation with semilinear systems is established. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time. In contrast with the scalar case, multiple space-time periodic uniformly positive entire solutions might coexist. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the two generalized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial data. Finally, with similar arguments, a Freidlin--G{\"a}rtner-type formulafor the asymptotic spreading speed of solutions with compactly supported initial data is established.

4.Existence of positive ground state solutions for the coupled Choquard system with potential

Authors:Jianqing Chen, Qian Zhang

Abstract: In this paper, we study the following coupled Choquard system in $\mathbb R^N$: $$\left\{\begin{align}&-\Delta u+A(x)u=\frac{2p}{p+q} \bigl(I_\alpha\ast |v|^q\bigr)|u|^{p-2}u,\\ &-\Delta v+B(x)v=\frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v,\\ &\ u(x)\to0\ \ \hbox{and}\ \ v(x)\to0\ \ \hbox{as}\ |x|\to\infty,\end{align}\right.$$ where $\alpha\in(0,N)$ and $\frac{N+\alpha}{N}<p,\ q<2_*^\alpha$, in which $2_*^\alpha$ denotes $\frac{N+\alpha}{N-2}$ if $N\geq 3$ and $2_*^\alpha := \infty$ if $N=1,\ 2$. The function $I_\alpha$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of positive ground state solution in the case of bounded potential and periodic potential respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\neq q$ and $u(x)\neq v(x)$. Moreover it seems to be the first existence result for the case of $p\neq q$.

5.The master equation for mean field game systems with fractional and nonlocal diffusions

Authors:Espen Robstad Jakobsen, Artur Rutkowski

Abstract: We prove existence and uniqueness of classical solutions of the master equation for mean field game (MFG) systems with fractional and nonlocal diffusions. We cover a large class of L\'evy diffusions of order greater than one, including purely nonlocal, local, and even mixed local-nonlocal operators. In the process we prove refined well-posedness results for the MFG systems, results that include the mixed local-nonlocal case. We also show various auxiliary results on viscous Hamilton-Jacobi equations, linear parabolic equations, and linear forward-backward systems that may be of independent interest. This includes a rigorous treatment of certain equations and systems with data and solutions in the duals of H\"older spaces $C^\gamma_b$ on the whole of $\mathbb{R}^d$. We do not assume existence of any moments for the initial distributions of players. In a future work we will use the results of this paper to prove the convergence of $N$-player games to mean field games as $N\to\infty$.

6.Strongly anisotropic Anzellotti pairings and their applications to the anisotropic $p$-Laplacian

Authors:Wojciech Górny

Abstract: In this paper, we study the parabolic and elliptic problems related to the anisotropic $p$-Laplacian operator in the case when it has linear growth on some of the coordinates. In order to define properly a notion of weak solutions and prove their existence, we first construct an anisotropic analogue of Anzellotti pairings and prove a weak Gauss-Green formula which relates the newly constructed pairing with a normal trace of a sufficiently regular vector field.

7.Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization

Authors:Herbert Egger, Jan Giesselmann

Abstract: We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem.

8.Navier-Stokes Modelling of Non-Newtonian Blood Flow in Cerebral Arterial Circulation and its Dynamic Impact on Electrical Conductivity in a Realistic Multi-Compartment Head Model

Authors:Maryam Samavaki, Arash Zarrin Nia, Santtu Söderholm, Sampsa Pursiainen

Abstract: Background and Objective: This study aims to evaluate the dynamic effect of non-Newtonian cerebral arterial circulation on electrical conductivity distribution (ECD) in a realistic multi-compartment head model. It addresses the importance and challenges associated with electrophysiological modalities, such as transcranial electrical stimulation, electro-magnetoencephalography, and electrical impedance tomography. Factors such as electrical conductivity's impact on forward modeling accuracy, complex vessel networks, data acquisition limitations (especially in MRI), and blood flow phenomena are considered. Methods: The Navier-Stokes equations (NSEs) govern the non-Newtonian flow model used in this study. The solver comprises two stages: the first solves the pressure field using a dynamical pressure-Poisson equation derived from NSEs, and the second updates the velocity field using Leray regularization and the pressure distribution from the first stage. The Carreau-Yasuda model establishes the connection between blood velocity and viscosity. Blood concentration in microvessels is approximated using Fick's law of diffusion, and conductivity mapping is obtained via Archie's law. The head model used corresponds to an open 7 Tesla MRI dataset, differentiating arterial vessels from other structures. Results: The results suggest the establishment of a dynamic model of cerebral blood flow for arterial and microcirculation. Blood pressure and conductivity distributions are obtained through numerically simulated pulse sequences, enabling approximation of blood concentration and conductivity within the brain. Conclusions: This model provides an approximation of dynamic blood flow and corresponding ECD in different brain regions. The advantage lies in its applicability with limited a priori information about blood flow and compatibility with arbitrary head models that distinguish arteries.

9.Convergence of infinitesimal generators and stability of convex monotone semigroups

Authors:Jonas Blessing, Michael Kupper, Max Nendel

Abstract: Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do not rely on the theory of viscosity solutions but use a recent comparison principle which uniquely determines the semigroup via its $\Gamma$-generator defined on the Lipschitz set and therefore resembles the classical analogue from the linear case. The framework also allows for discretizations both in time and space and covers a variety of applications. This includes Euler schemes and Yosida-type approximations for upper envelopes of families of linear semigroups, stability results and finite-difference schemes for convex HJB equations, Freidlin-Wentzell-type results and Markov chain approximations for a class of stochastic optimal control problems and continuous-time Markov processes with uncertain transition probabilities.

10.Enhanced profile estimates for ovals and translators

Authors:Kyeongsu Choi, Robert Haslhofer, Or Hershkovits

Abstract: We consider the profile function of ancient ovals and of noncollapsed translators. Recall that pioneering work of Angenent-Daskalopoulos-Sesum (JDG '19, Annals '20) gives a sharp $C^0$-estimate and a quadratic concavity estimate for the profile function of two-convex ancient ovals, which are crucial in their papers as well as a slew of subsequent papers on ancient solutions of mean curvature flow and Ricci flow. In this paper, we derive a sharp gradient estimate, which enhances their $C^0$-estimate, and a sharp Hessian estimate, which can be viewed as converse of their quadratic concavity estimate. Motivated by our forthcoming work on ancient noncollapsed flows in $\mathbb{R}^4$, we derive these estimates in the context of ancient ovals in $\mathbb{R}^3$ and noncollapsed translators in $\mathbb{R}^4$, though our methods seem to apply in other settings as well.

11.A sub-Riemannian maximum modulus theorem

Authors:Federico Buseghin, Nicolò Forcillo, Nicola Garofalo

Abstract: In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one.

12.Asymptotic stability of the fourth order $φ^4$ kink for general perturbations in the energy space

Authors:Christopher Maulén, Claudio Muñoz

Abstract: The Fourth order $\phi^4$ model generalizes the classical $\phi^4$ model of quantum field theory, sharing the same kink solution. It is also the dispersive counterpart of the well-known parabolic Cahn-Hilliard equation. Mathematically speaking, the kink is characterized by a fourth-order nonnegative linear operator with a simple kernel at the origin but no spectral gap. In this paper, we consider the kink of this theory, and prove orbital and asymptotic stability for any perturbation in the energy space.

13.Two-phase free boundary problems for a class of fully nonlinear double-divergence systems

Authors:Pêdra D. S. Andrade, Julio C. Correa-Hoyos

Abstract: In this article, we study a class of fully nonlinear double-divergence systems with free boundaries associated with a minimization problem. The variational structure of Hessian-dependent functional plays a fundamental role in proving the existence of the minimizers and then the existence of the solutions for the system. In addition, we establish gains of the integrability for the double-divergence equation. Consequently, we improve the regularity for the fully nonlinear equation in Sobolev and H\"older spaces.

1.Continuous dependence of the Cauchy problem for the inhomogeneous biharmonic NLS equation in Sobolev spaces

Authors:JinMyong An, YuIl Jo, JinMyong Kim

Abstract: In this paper, we study the continuous dependence of the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{o}dinger (IBNLS) equation \[iu_{t} +\Delta^{2} u=\lambda |x|^{-b}|u|^{\sigma}u,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] in the standard sense in $H^s$, i.e. in the sense that the local solution flow is continuous $H^s\to H^s$. Here $d\in \mathbb N$, $s>0$, $\lambda\in \mathbb R$ and $\sigma>0$. To arrive at this goal, we first obtain the estimates of the term $f(u)-f(v)$ in the fractional Sobolev spaces which generalize the similar results of An-Kim [5](2021) and Dinh [16](2018), where $f(u)$ is a nonlinear function that behaves like $\lambda |u|^{\sigma}u$ with $\lambda\in \mathbb R$. These estimates are then applied to obtain the standard continuous dependence result for IBNLS equation with $0<s <\min \{2+\frac{d}{2},\frac{3}{2}d\}$, $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$ and $0<\sigma< \sigma_{c}(s)$, where $\sigma_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $\sigma_{c}(s)=\infty$ if $s\ge \frac{d}{2}$. Our continuous dependence result generalizes that of Liu-Zhang [27](2021) by extending the validity of $s$ and $b$.

2.Superlinear elliptic equations with unbalanced growth and nonlinear boundary condition

Authors:Eleonora Amoroso, Ángel Crespo-Blanco, Patrizia Pucci, Patrick Winkert

Abstract: In this paper we first prove the existence of a new equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann problems, both of independent interest. Moreover, we study a variable exponent double phase problem with a nonlinear boundary condition and prove the existence of multiple solutions under very general assumptions on the nonlinearities. To be more precise, we get constant sign solutions (nonpositive and nonnegative) via a mountain-pass approach and a sign-changing solution by using an appropriate subset of the corresponding Nehari manifold along with the Brouwer degree and the Quantitative Deformation Lemma.

3.Viscosity solutions to uniformly elliptic complex equations

Authors:Wei Sun

Abstract: In this paper, we shall extend the definition of $\mathcal{C}$-subsolution condition and adapt the argument of Guo-Phong-Tong[18] to replace Alexandroff-Bakelman-Pucci estimate in complex cases. As an application, we shall define and study the viscosity solutions to uniformly elliptic complex equations and prove the H\"older regularity, following the argument for real equations. Our results show that the new method can improve the dependence in regularity and a priori estimates for complex elliptic equations.

4.Propagation of polarization sets for systems of MHD type

Authors:Rayhana Darwich

Abstract: Polarization sets were introduced by Dencker (1982) as a refinement of wavefront sets to the vector-valued case. He also clarified the propagation of polarization sets when the characteristic variety of the pseudodifferential system under study consists of two hypersurfaces intersecting tangentially (1992), or transversally (1995). In this paper, we consider the case of more than two intersecting characteristic hypersurfaces that are interesting transversally (and we give a note on the tangential case). Mainly, we consider two types of systems which we name "systems of generalized transverse type" and "systems of MHD type", and we show that we can get a result for the propagation of polarization set similar to Dencker's result for systems of transverse type. Furthermore, we give an application to the MHD equations.

5.Quantized Vortex Dynamics of the Nonlinear Wave Equation on the Torus

Authors:Yongxing Zhu

Abstract: We derive rigorously the reduced dynamical laws for quantized vortex dynamics of the nonlinear wave equation on the torus when the core size of vortex $\varepsilon\to 0$. It is proved that the reduced dynamical laws are second-order nonlinear ordinary differential equations which are driven by the renormalized energy on the torus, and the initial data of the reduced dynamical laws are determined by the positions of vortices and the momentum. We will also investigate the effect of the momentum on the vortex dynamics.

6.The 2D Onsager conjecture: a Newton-Nash iteration

Authors:Vikram Giri, Razvan-Octavian Radu

Abstract: For any $\gamma<1/3$, we construct a nontrivial weak solution $u$ to the two-dimensional, incompressible Euler equations, which has compact support in time and satisfies $u\in C^\gamma(\mathbb R_t \times \mathbb T^2_x)$. In particular, the constructed solution does not conserve energy and, thus, settles the flexible part of the Onsager conjecture in two dimensions. The proof involves combining the Nash iteration technique with a new linear Newton iteration.

7.A wavelet-inspired $L^3$-based convex integration framework for the Euler equations

Authors:Vikram Giri, Hyunju Kwon, Matthew Novack

Abstract: In this work, we develop a wavelet-inspired, $L^3$-based convex integration framework for constructing weak solutions to the three-dimensional incompressible Euler equations. The main innovations include a new multi-scale building block, which we call an intermittent Mikado bundle; a wavelet-inspired inductive set-up which includes assumptions on spatial and temporal support, in addition to $L^p$ and pointwise estimates for Eulerian and Lagrangian derivatives; and sharp decoupling lemmas, inverse divergence estimates, and space-frequency localization technology which is well-adapted to functions satisfying $L^p$ estimates for $p$ other than $1$, $2$, or $\infty$. We develop these tools in the context of the Euler-Reynolds system, enabling us to give both a new proof of the intermittent Onsager theorem from [32] in this paper, and a proof of the $L^3$-based strong Onsager conjecture in the companion paper [22].

8.The $L^3$-based strong Onsager theorem

Authors:Vikram Giri, Hyunju Kwon, Matthew Novack

Abstract: In this work, we prove the $L^3$-based strong Onsager conjecture for the three-dimensional Euler equations. Our main theorem states that there exist weak solutions which dissipate the total kinetic energy, satisfy the local energy inequality, and belong to $C^0_t (W^{\frac 13-, 3} \cap L^{\infty-})$. More precisely, for every $\beta<\frac 13$, we can construct such solutions in the space $C^0_t ( B^{\beta}_{3,\infty} \cap L^{\frac{1}{1-3\beta}} )$.

9.Diffusion enhancement and Taylor dispersion for rotationally symmetric flows in discs and pipes

Authors:Michele Coti Zelati, Michele Dolce, Chia-Chun Lo

Abstract: In this note, we study the long-time dynamics of passive scalars driven by rotationally symmetric flows. We focus on identifying precise conditions on the velocity field in order to prove enhanced dissipation and Taylor dispersion in three-dimensional infinite pipes. As a byproduct of our analysis, we obtain an enhanced decay for circular flows on a disc of arbitrary radius.

10.Local well-posedness of the higher order nonlinear Schrödinger equation on the half-line: single boundary condition case

Authors:Aykut Alkın, Dionyssios Mantzavinos, Türker Özsarı

Abstract: We establish local well-posedness for the higher-order nonlinear Schr\"odinger equation, formulated on the half-line. We consider the scenario of associated coefficients such that only one boundary condition is required, which is assumed to be Dirichlet type. Our functional framework centers around fractional Sobolev spaces. We treat both high regularity and low regularity solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of initial value problems. The linear analysis, which is the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method. In this connection, we note that the higher-order Schr\"odinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivative. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multi-term linear differential operator.

11.The anisotropic Cahn--Hilliard equation: regularity theory and strict separation properties

Authors:Harald Garcke, Patrik Knopf, Julia Wittmann

Abstract: The Cahn--Hilliard equation with anisotropic energy contributions frequently appears in many physical systems. Systematic analytical results for the case with the relevant logarithmic free energy have been missing so far. We close this gap and show existence, uniqueness, regularity, and separation properties of weak solutions to the anisotropic Cahn--Hilliard equation with logarithmic free energy. Since firstly, the equation becomes highly non-linear, and secondly, the relevant anisotropies are non-smooth, the analysis becomes quite involved. In particular, new regularity results for quasilinear elliptic equations of second order need to be shown.

12.Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum

Authors:Hind Al Baba, Bilal Al Taki, Amru Hussein

Abstract: We discuss in this short note the local-in-time strong well-posedness of the compressible Navier-Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, V\'{a}clav, and Ne\v{c}asova in [\doi{10.1007/s00208-021-02301-8}] can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in [\doi{10.1016/j.matpur.2003.11.004}] for compressible Navier-Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

13.Principal eigenvalues for Fully non linear singular or degenerate operators in punctured balls

Authors:Françoise Demengel

Abstract: This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions $( \bar\lambda_\gamma, u_\gamma)$ of the equation $$| \nabla u |^\alpha F( D^2 u_\gamma)+ \bar \lambda_\gamma {u_\gamma^{1+\alpha} \over r^\gamma} = 0\ {\rm in} \ B(0,1)\setminus \{0\}, \ u_\gamma = 0 \ {\rm on} \ \partial B(0,1)$$ where $u_\gamma>0$ in $B(0,1)$, $\alpha >-1$ and $\gamma >0$. We prove existence of radial solutions which are continuous on $\overline{ B(0,1)}$ in the case $\gamma <2+\alpha$, existence of unbounded solutions which do ot satisfy the boundary condition in the case $\gamma = 2+\alpha $ and a non existence result for $\gamma >2+\alpha$. We also give the explicit value of $\bar \lambda_{2+\alpha} $ in the case of Pucci's operators, which generalizes the Hardy--Sobolev constant for the Laplacian, and the previous results of Birindelli, Demengel and Leoni

14.Asymptotic behavior of Kawahara equation with memory effect

Authors:Roberto de A. Capistrano Filho, Boumediène Chentouf, Isadora Maria de Jesus

Abstract: In this work, we are interested in a detailed qualitative analysis of the Kawahara equation, a model that has numerous physical motivations such as magneto-acoustic waves in a cold plasma and gravity waves on the surface of a heavy liquid. First, we design a feedback law, which combines a damping control and a finite memory term. Then, it is shown that the energy associated with this system exponentially decays.

15.Well-posedness for a transmission problem connecting first and second-order operators

Authors:Héctor A. Chang-Lara

Abstract: We establish the existence and uniqueness of viscosity solutions within a domain $\Omega\subseteq\mathbb R^n$ for a class of equations governed by elliptic and eikonal type equations in disjoint regions. Our primary motivation stems from the Hamilton-Jacobi equation that arises in the context of a stochastic optimal control problem.

1.Existence and concentration of ground state solution to a nonlocal Schrödinger equation

Authors:Anmin Mao, Qian Zhang

Abstract: We study a class of Schr\"{o}dinger-Kirchhoff system involving critical exponent. We aim to find suitable conditions to assure the existence of a positive ground state solution of Nehari-Poho\u{z}aev type $u_{\varepsilon}$ with exponential decay at infinity for $\varepsilon$ and $ u_{\varepsilon}$ concentrates around a global minimum point of $ V$ as $ \varepsilon\rightarrow0^{+}.$ The nonlinear term includes the nonlinearity $f(u)\sim|u|^{p-1}u$ for the well-studied case $ p\in[3,5)$, and the less-studied case $p\in(2,3)$.

2.Normalized solutions to the biharmonic nonlinear Schrödinger equation with combined nonlinearities

Authors:Wenjing Chen, Zexi Wang

Abstract: In this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schr\"{o}dinger equation with combined nonlinearities \begin{equation*} \Delta^2u=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u,\quad \text {in $\mathbb{R}^N$} \end{equation*} having prescribed mass \begin{equation*} \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{equation*} where $N\geq2$, $\mu\in \mathbb{R}$, $a>0$, $2<q<p<\infty$ if $2\leq N\leq 4$, $2<q<p\leq 4^*$ if $N\geq 5$, and $4^*=\frac{2N}{N-4}$ is the Sobolev critical exponent and $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier. By using the Sobolev subcritical approximation method, we prove the second critical point of mountain pass type for the case $N\geq5$, $\mu>0$, $p=4^*$, and $2<q<2+\frac{8}{N}$. Moreover, we also consider the case $\mu=0$ and $\mu<0$.