A note on asymptotics of linear dissipative kinetic equations in bounded domains

TL;DR

This paper proves exponential decay and diffusion asymptotics for linear dissipative kinetic equations in bounded domains, using energy estimates and hypocoercivity techniques, even with non-mass-conserving boundary conditions.

Contribution

It extends $L^2$ decay and diffusion results to a broader class of boundary conditions for kinetic equations, combining energy methods with hypocoercivity and entropy approaches.

Findings

Established $L^2$-exponential decay for kinetic equations in bounded domains.

Derived diffusion asymptotics under Maxwell boundary conditions.

Applied energy estimates and hypocoercivity techniques to non-conservative boundary conditions.

Abstract

We establish $L^2$-exponential decay properties for linear dissipative kinetic equations, including the time-relaxation and Fokker-Planck models, in bounded spatial domains with general boundary conditions that may not conserve mass. Their diffusion asymptotics in $L^2$ is also derived under general Maxwell boundary conditions. The proofs are simply based on energy estimates together with previous ideas from $L^2$-hypocoercivity and relative entropy methods.