On multi-dimensional hypocoercive BGK models

TL;DR

This paper develops new methods to analyze hypocoercivity in linearized BGK models, providing explicit exponential relaxation rates, understanding the waiting time before decay, and establishing local stability for nonlinear variants.

Contribution

It introduces a novel hypocoercivity index, a decomposition technique for entropy functionals, and extends hypocoercivity analysis to continuous phase space models with jump processes.

Findings

Proven exponential relaxation to equilibrium with explicit rates.

Identified a long waiting time before exponential decay begins.

Established local asymptotic stability of nonlinear BGK models.

Abstract

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.