Analysis of PDEs (math.AP)

Abstract: In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\infty}(\Omega;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in \Omega} W(x,y, \nabla u(x),\nabla u(y)),$$ where $\Omega$ is a bounded open subset of $\mathbb R^N$ and $W:\Omega \times \Omega \times \mathbb R^{d \times N}\times \mathbb R^{d \times N} \to \mathbb R$.

Abstract: In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the $p$-capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for $p\in(1,n)$. However, the smoothness and uniqueness of solutions are still open. Notice that the $p$-capacitary Orlicz-Minkowski problem can be converted equivalently to a Monge-Amp\`{e}re type equation in smooth case: \begin{align}\label{0.1} f\phi(h_K)|\nabla\Psi|^p=\tau G \end{align} for $p\in(1,n)$ and some constant $\tau>0$, where $f$ is a positive function defined on the unit sphere $\mathcal{S}^{n-1}$, $\phi$ is a continuous positive function defined in $(0,+\infty)$, and $G$ is the Gauss curvature. In this paper, we confirm the existence of smooth solutions to $p$-capacitary Orlicz-Minkowski problem with $p\in(1,n)$ for the first time by a class of inverse Gauss curvature flows, which converges smoothly to the solution of Equation (\ref{0.1}). Furthermore, we prove the uniqueness result for Equation (\ref{0.1}) in a special case.

Abstract: We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic nonlinearity), we convert the Cauchy problem into a system of ordinary differential equations in a Banach space. Using this approach, we are able to demonstrate local well-posedness in the Sobolev space $H^{s}$ where $s > 5/2$. We also establish the continuity properties for the data-to-solution map for a range of Sobolev spaces. Finally, we briefly explore the relationship between the two-component system and the bi-Hamiltonian AKNS hierarchy.

Abstract: In this paper, we prove the uniform nonlinear structural stability of Poiseuille flows with suitably large flux for the steady Navier-Stokes system in a two-dimensional strip with arbitrary period. Furthermore, the well-posedness theory for the Navier-Stokes system is also proved even when the $L^2$-norm of the external force is large. In particular, if the vertical velocity is suitably small where the smallness is independent of the flux, then Poiseuille flow is the unique solution of the steady Navier-Stokes system in the periodic strip. The key point is to establish uniform a priori estimates for the corresponding linearized problem via the boundary layer analysis, where we explore the particular features of odd and even stream functions. The analysis for the even stream function is new, which not only generalizes the previous study for the symmetric flows in \cite{Rabier1}, but also provides an explicit relation between the flux and period.

Abstract: This paper is concerned with a quasilinear Schr\"{o}dinger system in $\mathbb R^{N}$ $$\left\{\aligned &-\Delta u+A(x)u-\frac{1}{2}\triangle(u^{2})u=\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &-\Delta v+B(x)v-\frac{1}{2}\triangle(v^{2})v=\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ & u(x)\to 0\ \hbox{and}\quad v(x)\to 0\ \hbox{as}\ |x|\to \infty,\endaligned\right. $$ where $\alpha,\beta>1$ and $2<\alpha+\beta<\frac{4N}{N-2}$ ($N \geq 3$). $A(x)$ and $B(x)$ are two periodic functions. By minimization under a convenient constraint and concentration-compactness lemma, we prove the existence of ground states solution. Our result covers the case of $\alpha+\beta\in(2,4)$ which seems to be the first result for coupled quasilinear Schr\"{o}dinger system in the periodic situation.

Abstract: Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection-diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg-Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg-Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimisers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.

Abstract: We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and H\"older continuity of weak solutions to this problem. Our results substantially generalise analogous known results for the problem with single order fractional time derivative.

Abstract: In this paper, we study the initial-boundary value problem for the Poiseuille flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals in one space dimension. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of H\"older continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.

Abstract: In this paper, we prove the decay estimate of Maxwell-Higgs system on four dimensional Schwarzschild spacetimes. We show that if the field equations support a Morawetz type estimate supported around the trapped surface, the uniform decay properties in the entire exterior of the Schwarzschild black holes can be obtained by using Sobolev inequalities and energy estimates. Our results also consider various forms of physical potential such as the mass terms, $\phi^4$-theory, sine Gordon potential, and Toda potential.

Abstract: In this paper, a critical fourth-order Kirchhoff type elliptic equation with a subcritical perturbation is studied. The main feature of this problem is that it involves both a nonlocal coefficient and a critical term, which bring essential difficulty for the proof of the existence of weak solutions. When the dimension of the space is smaller than or equals to $7$, the existence of weak solution is obtained by combining the Mountain Pass Lemma with some delicate estimate on the Talenti's functions. When the dimension of the space is larger than or equals to $8$, the above argument no longer works. By introducing an appropriate truncation on the nonlocal coefficient, it is shown that the problem admits a nontrivial solution under appropriate conditions on the parameter.

Abstract: We present a rigorous approach and related techniques to construct global solutions of the 2-D Riemann problem with four-shock interactions for the Euler equations for potential flow. With the introduction of three critical angles: the vacuum critical angle from the compatibility conditions, the detachment angle, and the sonic angle, we clarify all configurations of the Riemann solutions for the interactions of two-forward and two-backward shocks, including the subsonic-subsonic reflection configuration that has not emerged in previous results. To achieve this, we first identify the three critical angles that determine the configurations, whose existence and uniqueness follow from our rigorous proof of the strict monotonicity of the steady detachment and sonic angles for 2-D steady potential flow with respect to the Mach number of the upstream state. Then we reformulate the 2-D Riemann problem into the shock reflection-diffraction problem with respect to a symmetric line, along with two independent incident angles and two sonic boundaries varying with the choice of incident angles. With these, the problem can be further reformulated as a free boundary problem for a second-order quasilinear equation of mixed elliptic-hyperbolic type. The difficulties arise from the degenerate ellipticity of the nonlinear equation near the sonic boundaries, the nonlinearity of the free boundary condition, the singularity of the solution near the corners of the domain, and the geometric properties of the free boundary. To the best of our knowledge, this is the first rigorous result for the 2-D Riemann problem with four-shock interactions for the Euler equations. The approach and techniques developed for the Riemann problem for four-wave interactions should be useful for solving other 2-D Riemann problems for more general Euler equations and related nonlinear hyperbolic systems of conservation laws.

Abstract: Differently from their classical counterpart, nonlocal minimal surfaces are known to present boundary discontinuities, by sticking at the boundary of smooth domains. It has been observed numerically by J. P. Borthagaray, W. Li, and R. H. Nochetto ``that stickiness is larger near the concave portions of the boundary than near the convex ones, and that it is absent in the corners of the square'', leading to the conjecture ``that there is a relation between the amount of stickiness on $\partial\Omega$ and the nonlocal mean curvature of $\partial\Omega$''. In this paper, we give a positive answer to this conjecture, by showing that the nonlocal minimal surfaces are continuous at convex corners of the domain boundary and discontinuous at concave corners. More generally, we show that boundary continuity for nonlocal minimal surfaces holds true at all points in which the domain is not better than $C^{1,s}$, with the singularity pointing outward, while, as pointed out by a concrete example, discontinuities may occur at all point in which the domain possesses an interior touching set of class $C^{1,\alpha}$ with $\alpha>s$.

Abstract: We study the nonlinear Schr\"odinger equation for the $s-$fractional $p-$Laplacian strongly coupled with the Poisson equation in dimension two and with $p=\frac2s$, which is the limiting case for the embedding of the fractional Sobolev space $W^{s,p}(\mathbb{R}^2)$. We prove existence of solutions by means of a variational approximating procedure for an auxiliary Choquard equation in which the uniformly approximated sign-changing logarithmic kernel competes with the exponential nonlinearity. Qualitative properties of solutions such as symmetry and decay are also established by exploiting a suitable moving planes technique.

Abstract: A one-dimensional cross-diffusion system modeling the transport of vesicles in neurites is analyzed. The equations are coupled via nonlinear Robin boundary conditions to ordinary differential equations for the number of vesicles in the reservoirs in the cell body and the growth cone at the end of the neurite. The existence of bounded weak solutions is proved by using the boundedness-by-entropy method. Numerical simulations show the dynamical behavior of the concentrations of anterograde and retrograde vesicles in the neurite.

Abstract: A modified Poisson-Nernst-Planck system in a bounded domain with mixed Dirichlet-Neumann boundary conditions is analyzed. It describes the concentrations of ions immersed in a polar solvent and the correlated electric potential due to the ion--solvent interaction. The concentrations solve cross-diffusion equations, which are thermodynamically consistent. The considered mixture is saturated, meaning that the sum of the ion and solvent concentrations is constant. The correlated electric potential depends nonlocally on the electric potential and solves the fourth-order Poisson-Fermi equation. The existence of global bounded weak solutions is proved by using the boundedness-by-entropy method. The novelty of the paper is the proof of the weak--strong uniqueness property. In contrast to the existence proof, we include the solvent concentration in the cross-diffusion system, leading to a diffusion matrix with nontrivial kernel. Then the proof is based on the relative entropy method for the extended cross-diffusion system and the positive definiteness of a related diffusion matrix on a subspace.

Abstract: Let $\Omega$ be an open subset of $\mathbb{R}^N.$ We identify various classes of Young functions $\Phi,\,\Psi$, and weight functions $g\in L^1_\text{loc}(\Omega)$ so that the following generalized weighted Sobolev inequality holds: \begin{equation*}\label{ineq:Orlicz} \Psi^{-1}\left(\int_{\Omega}|g(x)|\Psi( |u(x)| )dx \right)\leq C\Phi^{-1}\left(\int_{\Omega}\Phi(|\nabla u(x)|) dx \right),\,\,\,\forall\,u\in \mathcal{C}^1_c(\Omega), \end{equation*} for some $C>0$. As an application, we study the existence of non-negative solutions for certain nonlinear weighted eigenvalue problems.

Abstract: In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.

Abstract: The present note is devoted to the studies of the relation of the time-zero limits of Kaden's spirals and the 2D Euler equation. It is shown that the time-zero limits of Kaden's spirals satisfy inhomogeneous 2D Euler in a weak sense. As a corollary, the necessity of both, the decay of spherical averages around the origin of the spiral as well as the velocity matching condition, for the 2D Euler equation to hold in a weak sense, is shown. Finally, some preliminary results concerning the Kaden spirals are obtained.