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

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 [2] 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.