Kinetic Fokker-Planck and Landau Equations with Specular Reflection Boundary Condition

TL;DR

This paper proves the existence, regularity, and uniqueness of weak solutions to the kinetic Fokker-Planck and linearized Landau equations with specular reflection boundary conditions in general domains.

Contribution

It introduces a method of reflection and $S_p$ estimates to establish regularity and uniqueness of solutions for these kinetic equations with boundary conditions.

Findings

Existence of finite energy weak solutions near Maxwellian.

Regularity in kinetic Sobolev and anisotropic Hölder spaces.

Uniqueness of weak solutions.

Abstract

We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linearized Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the $S_p$ estimate previously established by the first and the last author, we prove regularity in the kinetic Sobolev spaces $S_p$ and anisotropic H\"older spaces for such weak solutions. Such $S_p$ regularity leads to the uniqueness of weak solutions.