Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

TL;DR

This paper proves convergence to equilibrium for linear kinetic equations in bounded domains with general Maxwell boundary conditions by establishing hypocoercivity, unifying various boundary cases including specular reflection.

Contribution

It introduces a unified hypocoercivity framework for linear kinetic equations with general boundary conditions in bounded domains, covering cases like vanishing accommodation coefficient.

Findings

Proves convergence to equilibrium under broad boundary conditions

Develops a Hilbert norm for coercivity in the orthogonal of conservation laws

Includes cases like specular reflection boundary condition

Abstract

We establish the convergence to the equilibrium for various linear collisional kinetic equations (including linearized Boltzmann and Landau equations) with physical local conservation laws in bounded domains with general Maxwell boundary condition. Our proof consists in establishing an hypocoercivity result for the associated operator, in other words, we exhibit a convenient Hilbert norm for which the associated operator is coercive in the orthogonal of the global conservation laws. Our approach allows us to treat general domains with all type of boundary conditions in a unified framework. In particular, our result includes the case of vanishing accommodation coefficient and thus the specific case of the specular reflection boundary condition.