Analysis of PDEs (math.AP)

Abstract: Diffusive limit of the Vlasov-Poisson-Boltzmann system with cutoff soft potentials $-3<\gamma<0$ in the perturbative framework around global Maxwellian still remains open. By introducing a new weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay, we solve this problem for the full range of cutoff potentials $-3<\gamma\leq 1$. The core of this approach lies in the interplay between the velocity weighted $H_{x,v}^2$ energy estimate with time decay and the time-velocity weighted $W_{x,v}^{2,\infty}$ estimate with time decay for the Vlasov-Poisson-Boltzmann system, which leads to the uniform estimate with respect to the Knudsen number $\varepsilon\in (0,1]$ globally in time. As a result, global strong solution is constructed and incompressible Navier-Stokes-Fourier-Poisson limit is rigorously justified for both hard and soft potentials. Meanwhile, this uniform estimate with respect to $\varepsilon\in (0,1]$ also yields optimal $L^2$ time decay rate and $L^\infty$ time decay rate for the Vlasov-Poisson-Boltzmann system and its incompressible Navier-Stokes-Fourier-Poisson limit. This newly introduced weighted $H_{x,v}^2$-$W_{x,v}^{2, \infty}$ approach with time decay is flexible and robust, as it can deal with both optimal time decay problems and hydrodynamic limit problems in a unified framework for the Boltzmann equation as well as the Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials. It is also expected to shed some light on the more challenging hydrodynamic limit of the Landau equation and the Vlasov-Poisson-Landau system.

Abstract: We consider a general curvature equation $F(\kappa)=G(X,\nu(X))$, where $\kappa$ is the principal curvature of the hypersurface $M$ with position vector $X$. It includes the classical prescribed curvature measures problem and area measures problem. However, Guan-Ren-Wang \cite{GRW} proved that the $C^2$ estimate fails usually for general function $F$. Thus, in this paper, we pose some additional conditions of $G$ to get existence results by a suitably designed parabolic flow. In particular, if $F=\sigma_{k}^\frac{1}{k}$ for $\forall 1\le k\le n-1$, the existence result has been derived in the famous work \cite{GLL} with $G=\psi(\frac{X}{|X|})\langle X,\nu\rangle^{\frac1k}{|X|^{-\frac{n+1}{k}}}$. This result will be generalized to $G=\psi(\frac{X}{|X|})\langle X,\nu\rangle^\frac{{1-p}}{k}|X|^\frac{{q-k-1}}{k}$ with $p>q$ for arbitrary $k$ by a suitable auxiliary function. The uniqueness of the solutions in some cases is also studied.

Abstract: In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{cases} u -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f & \text{in}\;\Omega, \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary. In particular we explore the regularizing effect given by the absorption term in order to get a unique solutions for data $f$ merely belonging to $L^1(\Omega)$ and with no smallness assumptions. We also prove a sharp boundedness result for solutions for data in $L^{N}(\Omega)$.

Abstract: In this article we study the problem of nonlinear Landau damping for the Vlasov-Poisson equations on the torus. As our main result we show that for perturbations initially of size $\epsilon>0$ and time intervals $(0,\epsilon^{-N})$ one obtains nonlinear stability in regularity classes larger than Gevrey $3$, uniformly in $\epsilon$. As a complementary result we construct families of Sobolev regular initial data which exhibit nonlinear Landau damping. Our proof is based on the methods of Grenier, Nguyen and Rodnianski.