Damping of kinetic transport equation with diffuse boundary condition

混合双曲-动理学系统与稠密喷雾中的朗道阻尼

Landau damping in mixed hyperbolic-kinetic systems and thick sprays

本文研究稠密喷雾模型中 Vlasov 方程与正压可压缩 Euler 方程耦合的朗道阻尼现象，证明声波因与近速粒子相互作用而被阻尼或放大，并揭示该机制在双曲守恒律与 Vlasov 方程耦合时自然出现。

Statistical mechanics of Landau damping

Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions however approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by Vlasov equation. On the other hand, density and kinetic energy density, which are integrals of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when Vlasov equation is reduced to the evolution equations for density and kinetic energy…

有界区域中动理学Fokker-Planck方程的正则性

Regularity of kinetic Fokker-Planck equations in bounded domains

本研究针对有界区域中带流入、漫反射、镜面反射等边界条件且系数有界可测的动理学Fokker-Planck方程，证明了解的存在性、唯一性及正则性。

A study of Landau damping with random initial inputs

For the Vlasov-Poisson equation with random uncertain initial data, we prove that the Landau damping solution given by the deterministic counterpart (Caglioti and Maffei, {\it J. Stat. Phys.}, 92:301-323, 1998) depends smoothly on the random variable if the time asymptotic profile does, under the smoothness and smallness assumptions similar to the deterministic case. The main idea is to generalize the deterministic contraction argument to more complicated function spaces to estimate derivatives in space, velocity and random variables. This result suggests that the random space regularity can p…

Invariant density & time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

This paper deals with collisionless transport equations in bounded open domains $Ω\subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial Ω$, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $% L^{1}(Ω\times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $% \left( U_{\ma…

Dynamics of Riemann waves with sharp measure-controlled damping

This paper is concerned with locally damped semilinear wave equations defined on compact Riemannian manifolds with boundary. We present a construction of measure-controlled damping regions which are sharp in the sense that their summed interior and boundary measures are arbitrarily small. The construction of this class of open sets is purely geometric and allows us to prove a new observability inequality in terms of potential energy rather than the usual one with kinetic energy. A unique continuation property is also proved. Then, in three-dimension spaces, we establish the existence of finite…