A Galerkin type method for kinetic Fokker Planck equations based on Hermite expansions

Abstract

In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $(0,T)\times D\times {\mathbb{R}}^d$, where $D$ is either ${\mathbb{T}}^d$ or ${\mathbb{R}}^d$. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\text{href}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.

24 pages, corrected the references, version submitted to Kinet. Relat. Models, to appear