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

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 Sp estimate of [5], we prove regularity in the kinetic Sobolev spaces Sp and anisotropic Hölder spaces for such weak solutions. Such Sp regularity leads to the uniqueness of weak solutions.