Analysis of PDEs (math.AP)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $(\tau,\sigma)\in R_{CG}:= (0,1)^2$.

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.