On the local Maxwellians solving the Boltzmann equation with boundary condition

TL;DR

This paper characterizes local Maxwellians solving the Boltzmann equation with boundary conditions, proving that only global Maxwellians do so under bounce-back conditions and classifying solutions for specular reflection in symmetric domains.

Contribution

It provides a complete classification of local Maxwellians solving the Boltzmann equation with boundary conditions, including conditions for domains with symmetries and the uniqueness of solutions.

Findings

Only global Maxwellians solve the Boltzmann equation with bounce-back boundary conditions.

Complete classification of domains where only global Maxwellians solve the equation with specular reflection.

Description of all local Maxwellians in symmetric domains.

Abstract

We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of an open domain. We determine which of these local Maxwellians satisfy the Boltzmann equation in a regular domain with boundary, without assuming the boundedness of the domain. We investigate separately, on the one hand, the case of the bounce-back boundary condition in any dimension, and on the other hand the case of the specular reflection boundary condition, in dimension $d = 2$ and $d = 3$. In the case of the bounce-back boundary condition, we prove that the only local Maxwellians solving the Boltzmann equation with boundary condition are the global Maxwellians. In the case of the specular reflection, we provide a complete classification of the domains for which only the global Maxwellians solve the Boltzmann equation with boundary condition, and we describe all the local Maxwellians that solve the equation for the domains presenting symmetries.