Analysis of PDEs (math.AP)

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}.

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.

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.

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 $\mathbb{X}$.

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.

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.

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.