Diffusion limit for Vlasov-Fokker-Planck Equation in bounded domains

TL;DR

This paper establishes a diffusion approximation for the Vlasov-Fokker-Planck equation in bounded domains with specular reflection, using a novel test function construction and weak compactness methods.

Contribution

It introduces a new approach to derive diffusion limits in bounded domains by leveraging weak compactness and specialized test functions, avoiding complex L^1 analysis.

Findings

Diffusion approximation derived for Vlasov-Fokker-Planck in bounded domains.

Method relies on weak compactness in weighted Hilbert spaces.

Applicable to specular reflection boundary conditions.

Abstract

We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to be chosen in the weak formulation of the kinetic model. This involves the analysis of the underlying Hamiltonian dynamics of the kinetic equation coupled with the reflection laws at the boundary. This approach only demands the solution family to be weakly compact in some weighted Hilbert space rather than the much tricky $\mathrm L^1$ setting.