Hypocoercivity of Linear Degenerately Dissipative Kinetic Equations

TL;DR

This paper investigates the hypocoercivity properties of linear kinetic equations with degenerately dissipative terms, establishing decay rates to equilibrium using energy methods and macro-micro decomposition across various settings.

Contribution

It introduces a unified approach to analyze hypocoercivity for several kinetic operators, including relaxation, Fokker-Planck, and Boltzmann, with new energy functionals and inequalities.

Findings

Derived explicit decay rates for solutions approaching equilibrium.

Established a Korn-type inequality with probability measure.

Unified treatment of hypocoercivity for multiple kinetic models.

Abstract

In this paper, we study the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann operator are considered. By constructing equivalent temporal energy functionals, time rates of the solution approaching equilibrium in some Hilbert spaces are obtained when the spatial domain takes the whole space or torus and when there is a confining force or not. The main tool of the proof is the macro-micro decomposition combined with Kawashima's argument on dissipation of the hyperbolic-parabolic system. Finally, a Korn-type inequality with probability measure is provided to deal with dissipation of momentum components.