Analysis of PDEs (math.AP)

Abstract: The main purpose of this paper is to study weak solutions of time-fractional of porous medium equation with nonlocal pressure: \[ \partial^\alpha_t u=\operatorname{div}\left( |u|^{m}\nabla (-\Delta)^{-s} u\right) \,\, \text{in } \mathbb{R}^N\times (0,T) \,, \] with $m\geq 1$, $N\geq 2$, $\frac{1}{2}\leq s<1$, and $\alpha\in(0,1)$. We first prove an existence of weak solutions to the equation with initial data in $L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ (possibly mixed sign). After that, we establish the $L^q-L^\infty$ decay estimate of weak solutions: \[ \|u(t)\|_{L^\infty(\mathbb{R}^N)} \leq C t^{-\frac{\alpha}{q(1-\lambda_0)+ m}} \|u_0\|_{L^{q}(\mathbb{R}^N)}^{\frac{q(1-\lambda_0)}{q(1-\lambda_0) + m}} ,\quad \text{for } t\in(0,\infty), \] with $\lambda_0=\frac{N-2(1-s)}{N}$.

Abstract: This article deals with the multidimensional Borg-Levinson theorem for perturbed bi-harmonic operator. More precisely, in a bounded smooth domain of $\R^n$, with $n \geq 2$, we prove the stability of the first and zero order coefficients of the bi-harmonic operator from some asymptotic behavior of the boundary spectral data of the corresponding bi-harmonic operator i.e., the Dirichlet eigenvalues and the Neumann trace on the boundary of the associated eigenfunctions.

Abstract: We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system with perturbations and two parameters $\lambda_1$ and $\lambda_2$ on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-$(p,q)$ conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-$(p,q)$ conditions which is weaker than the well-known Ambrosetti-Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters $\lambda_1$ and $\lambda_2$.

Abstract: This paper explores the nonuniqueness of solutions to the $L_p$ chord Minkowski problem for negative $p.$ The $L_p$ chord Minkowski problem was recently posed by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the $L_p$ chord measure of a convex body, and it includes the chord Minkowski problem and the $L_p$ Minkowski problem.

Abstract: We consider a branched transport type problem which describes the magnetic flux through type-I superconductors in a regime of very weak applied fields. At the boundary of the sample, deviation of the magnetization from being uniform is penalized through a negative Sobolev norm. It was conjectured by S. Conti, F. Otto and S. Serfaty that as a result, the trace of the magnetization on the boundary should be a measure of Hausdorff dimension $8/5$. We prove that this conjecture is equivalent to the proof of local energy bounds with an optimal exponent. We then obtain local bounds which are however not optimal. These yield improved lower bounds on the dimension of the irrigated measure but unfortunately does not improve on the trivial upper bound. In order to illustrate the dependence of this dimension on the choice of penalization, we consider in the last part of the paper a toy model where the boundary energy is given by a Wasserstein distance to Lebesgue. In this case minimizers are finite graphs and thus the trace is atomic.

Abstract: We prove the existence of time-periodic solutions to non-linear massive Klein-Gordon equations in Anti-de Sitter as well as their orbital stability over exponentially long times for certain values of the mass corresponding to completely resonant spectrum. We analyse the resonant system in the Fourier space by relying in particular on Zeilberger's algorithm which allows for a systematic way to derive recurrence formulae for the Fourier coefficients. We also show that the derivation and analysis of the Fourier systems easily extends to other semi-linear wave equations such as the co-rotational wave map equation.

Abstract: The focus of our paper is to investigate the possibility of a minimizer for the Thomas-Fermi-Dirac-von Weizs\"{a}cker model on the lattice graph $\mathbb{Z}^{3}$. The model is described by the following functional: \begin{equation*} E(\varphi)=\sum_{y\in\mathbb{Z}^{3}}\left(|\nabla\varphi(y)|^2+ (\varphi(y))^{\frac{10}{3}}-(\varphi(y))^{\frac{8}{3}}\right)+ \sum_{x,y\in\mathbb{Z}^{3}\atop ~\ y\neq x\hfill}\frac{{\varphi}^2(x){\varphi}^2(y)}{|x-y|}, \end{equation*} with the additional constraint that $\sum\limits_{y\in\mathbb{Z}^{3}} {\varphi}^2(y)=m$ is sufficiently small. We also prove the nonexistence of a minimizer provided the mass $m$ is adequately large. Furthermore, we extend our analysis to a subset $\Omega \subset \mathbb{Z}^{3}$ and prove the nonexistence of a minimizer for the following functional: \begin{equation*} E(\Omega)=|\partial\Omega|+\sum_{x,y\in\Omega\atop ~y\neq x\hfill}\frac{1}{|x-y|}, \end{equation*} under the constraint that $|\Omega|=V$ is sufficiently large.

Abstract: We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on the waveguide manifold $\mathbb{R}^3\times\mathbb{T}$ and establish almost sure scattering for random initial data, where no symmetry conditions are imposed and the result is available for arbitrarily rough data $f\in H^s$ with $s\in\mathbb{R}$. The main new ingredient is a layer-by-layer refinement of the newly established randomization introduced by Shen-Soffer-Wu \cite{ShenSofferWu21}, which enables us to also obtain strongly smoothing effect from the randomization for the forcing term along the periodic direction. It is worth noting that such smoothing effect generally can not hold for purely compact manifolds, which is on the contrary available for the present model thanks to the mixed type nature of the underlying domain. As a byproduct, by assuming that the initial data are periodically trivial, we also obtain the almost sure scattering for the defocusing cubic NLS on $\mathbb{R}^3$ which parallels the ones by Camps \cite{Camps} and Shen-Soffer-Wu \cite{Shen2022}. To our knowledge, the paper also gives the first almost sure well-posedness result for NLS on product spaces.

Abstract: We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. The problem can be formulated as the comparison of the entropy solutions in $BV\cap L^{1}$ between the two initial-boundary value problems for the compressible full Euler equations with parameter $\tau>0$ and the hypersonic small-disturbance equations with curved characteristic boundaries. We establish the $L^1$--convergence estimate of these two solutions with the optimal convergence rate, which justifies Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the $L^1$--stability of the entropy solutions under the smallness assumptions on the total variation of both the initial data and the tangential slope of the wedge boundary. Based on the $L^1$--stability properties of the approximate solutions to the scaled equations with parameter $\tau$, a uniform Lipschtiz continuous map with respect to the initial data and the wedge boundary is obtained. Next, we compare the solutions given by the Riemann solvers of the two systems by taking the boundary perturbations into account case by case. Then, for a given fixed hypersonic similarity parameter, as the Mach number tends to infinity, we establish the desired $L^1$--convergence estimate with the optimal convergence rate. Finally, we show the optimality of the convergence rate by investigating a special solution.

Abstract: We prove a restriction type estimate for sub-Laplacians on arbitrary two-step stratified Lie groups. Despite being slightly weaker than previously known estimates for the subclass of Heisenberg type groups, these estimates surprisingly turn out to be sufficient to prove an $L^p$-spectral multiplier theorem with sharp regularity condition $s>d\left(1/p-1/2\right)$ for sub-Laplacians on M\'etivier groups, up to some exceptional class in dimension $d=15$.