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Analysis of PDEs (math.AP)

Mon, 22 May 2023

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1.Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, III: singular coefficients

Authors:Claudia Garetto, Bolys Sabitbek

Abstract: In this paper we continue the analysis of non-diagonalisable hyperbolic systems initiated in \cite{GarJRuz, GarJRuz2}. Here we assume that the system has discontinuous coefficients or more in general distributional coefficients. Well-posedness is proven in the very weak sense for systems with singularities with respect to the space variable or the time variable. Consistency with the classical theory is proven in the case of smooth coefficients.

2.pointwise boundary $\bm{{C}^{1,α}}$ Estimates for some degenerate fully nonlinear elliptic equations on $\bm{C^{1,α}}$ Domains

Authors:Xuemei Li, Dongsheng Li

Abstract: In this paper, we establish pointwise boundary ${{C}^{1,\alpha}}$ estimates for viscosity solutions of some degenerate fully nonlinear elliptic equations on ${C^{1,\alpha}}$ domains. Instead of straightening out the boundary, we utilize the perturbation and compactness techniques.

3.Standing Waves for Schrödinger Equations with Kato-Rellich potentials

Authors:Aleksander Ćwiszewski, Piotr Kokocki

Abstract: We show the existence of standing waves for the nonlinear Schr\"{o}dinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman-Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski.

4.Analytical approximations in short times of exact operational solutions to reaction diffusion problems on bounded intervals

Authors:Anani Kwassi

Abstract: This paper aims at obtaining, by means of integral transforms, analytical approximations in short times of solutions to boundary value problems for the one-dimensional reaction-diffusion equation with constant coefficients. The general form of the equation is considered on a bounded generic interval and the three classical types of boundary conditions, i.e., Dirichlet as well as Neumann and mixed boundary conditions are considered in a unified way. The Fourier and Laplace integral transforms are successively applied and an exact solution is obtained in the Laplace domain. This operational solution is proven to be the accurate Laplace transform of the infinite series obtained by the Fourier decomposition method and presented in the literature as solutions to this type of problem. On the basis of this unified operational solution, four cases are distinguished where innovative formulas expressing consistent analytical approximations in short time limits are derived with respect to the behavior of the solution at the boundaries. Compared to the infinite series solutions, the analytical approximations may open new perspectives and applications, among which can be noted the improvement of numerical efficiency in simulations of one-dimensional moving boundary problems, such as in Stefan models.

5.The fundamental eigenfrequency is simple in the two-dimensional sloshing problem

Authors:Nikolay Kuznetsov

Abstract: The two-dimensional sloshing problem is considered; it describes the transversal free oscillations of water in an open, infinitely long canal of uniform cross-section. It is proved that the fundamental eigenfrequency is simple, whereas the corresponding velocity potential has only one nodal line connecting the free surface and the bottom; its harmonic conjugate (stream function) does not change sign under the proper choice of the additive constant.

6.Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics

Authors:Deng-Shan Wang, Peng Yan

Abstract: The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. So far, the full analysis of this problem remains open. In this work, the long-time asymptotics for the defocusing nonlinear Schr\"{o}dinger equation with general step-like initial data is investigated by Whitham modulation theory and Riemann-Hilbert formulation. The Whitham modulation theory shows that there are six cases for the initial discontinuity problem according to the orders of the Riemann invariants. The leading-order terms and the corresponding error estimates for each region of the six cases are formulated by Deift-Zhou nonlinear steepest method for oscillatory Riemann-Hilbert problems. It is demonstrated that the long-time asymptotic solutions match very well with the results from Whitham modulation theory and the numerical simulations.

7.On the Study of the Klein-Gordon Equation in the Dunkl Setting

Authors:Mohamed Gaidi, Mounir Bedhiafi

Abstract: In Dunkl theory on $\mathbb{R}^{n}$ which generalizes classical Fourier analysis, we study the solution of the Klein-Gordon-equation defined by: \begin{eqnarray} \nonumber \partial_{t}^{2}u-\Delta_{k}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \ \ \partial_{t}u(x,0)=f(x) \end{eqnarray} with \ $m > 0$ \ and \ $\partial_{t}^{2}u$ \ is the second derivative of the solution $u$ with respect to $t$ and $\Delta_{k}u$ is the Dunkl Laplacian with respect to $x$ where $f$ and $g$ the two functions in $\mathcal{S}(\mathbb{R}^{n})$ which surround the initial conditions. We obtain an integral representation for its solution which we gives some properties. As a specific result, we studied the associated energies to the Dunkl-Klein-Gordon equation.

8.On the weak Harnack inequality for unbounded non-negative super-solutions of degenerate double-phase parabolic equations

Authors:Mariia Savchenko, Igor Skrypnik, Yevgeniia Yevgenieva

Abstract: In the case $q> p\dfrac{n+2}{n}$, we give a proof of the weak Harnack inequality for non-negative super-solutions of degenerate double-phase parabolic equations under the additional assumption that $u\in L^{s}_{loc}(\Omega_{T})$ with some $s >p\dfrac{n+2}{n}$.

9.Excursus on modulation spaces via metaplectic operators and related time-frequency representations

Authors:Elena Cordero, Gianluca Giacchi

Abstract: Modulation spaces were originally introduced by Feichtinger in 1983. Since the 2000s there have been thousands of contributions using them as correct framework; they range from PDEs, pseudodifferential operators, quantum mechanics, signal analysis. This justifies a deep study of such spaces and the related Wiener ones. Recently, metaplectic Wigner distributions, which contain as special examples the $\tau$-Wigner distributions, the ambiguity function and the Short-time Fourier transform, have proved to characterize modulation spaces, under suitable assumptions. We investigate the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. We add a new result on this topic and conclude with an exhaustive vision of these characterizations. Similar results hold for the Wiener amalgam ones.

10.Non-uniqueness for the compressible Euler-Maxwell equations

Authors:Shunkai Mao, Peng Qu

Abstract: We consider the Cauchy problem for the isentropic compressible Euler-Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth equilibrium-charged ion density, we could construct infinitely many $\alpha$-H\"older continuous entropy solutions emanating from the same initial data for $\alpha<\frac{1}{7}$. Especially, the electromagnetic field belongs to the H\"older class $C^{1,\alpha}$. Furthermore, we provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme. Due to the constrain of the Maxwell equations, we propose a method of Mikado potential and construct new building blocks.

11.Korteweg-de Vries waves in peridynamical media

Authors:Michael Herrmann, Katia Kleine

Abstract: We consider a one-dimensional peridynamical medium and show the existence of solitary waves with small amplitudes and long wavelength. Our proof uses nonlinear Bochner integral operators and characterizes their asymptotic properties in a singular scaling limit.