On linear hypocoercive BGK models

TL;DR

This paper develops entropy-based methods to prove exponential relaxation to equilibrium for linear and linearized BGK models in discrete and continuous phase spaces, also establishing local stability for a nonlinear variant.

Contribution

It introduces novel entropy and spectral techniques for hypocoercivity analysis in BGK models, including continuous phase space cases, and proves local stability of nonlinear models.

Findings

Exponential relaxation rates for linear BGK models established.

Entropy functionals constructed for hypocoercivity proofs.

Local asymptotic stability of nonlinear BGK model proven.

Abstract

We study hypocoercivity for a class of linear and linearized BGK models for discrete and continuous phase spaces. We develop methods for constructing entropy functionals that prove exponential rates of relaxation to equilibrium. Our strategies are based on the entropy and spectral methods, adapting Lyapunov's direct method (even for "infinite matrices" appearing for continuous phase spaces) to construct appropriate entropy functionals. Finally, we also prove local asymptotic stability of a nonlinear BGK model.