Hypocoercivity for kinetic equations with linear relaxation terms

TL;DR

This paper introduces a straightforward method to establish hypocoercivity for kinetic equations with linear relaxation, using an adapted Lyapunov functional that separates microscopic coercivity from macroscopic spectral gap properties.

Contribution

The paper presents a novel, simplified approach to proving hypocoercivity that enhances previous results by clearly distinguishing microscopic and macroscopic effects.

Findings

Method effectively proves hypocoercivity for kinetic equations with linear relaxation.

Illustrated with linear BGK model and fast diffusion relaxation operator.

Improves upon existing hypocoercivity results.

Abstract

This note is devoted to a simple method for proving hypocoercivity of the solutions of a kinetic equation involving a linear time relaxation operator, i.e. the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion.