On the local Maxwellians solving the Boltzmann equation with boundary condition

Abstract

We derive the expressions of the local Maxwellians that solve the Boltzmann equation in the interior of a regular open domain, 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.