Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas

TL;DR

This paper investigates how Maxwell-type boundary conditions influence the approach to equilibrium in highly rarefied gases, analyzing the interplay between boundary effects and molecular collisions in spherical domains.

Contribution

It provides new insights into the equilibrating effects of Maxwell-type boundary conditions and constructs steady states for the Boltzmann equation in large Knudsen number regimes.

Findings

Algebraic convergence rates to steady state without collisions

Dependence of convergence on accommodation coefficient

Existence and stability of Boltzmann steady states

Abstract

We study the equilibrating effects of the boundary and intermolecular collision in the kinetic theory for rarefied gases. We consider the Maxwell-type boundary condition, which has weaker equilibrating effect than the commonly studied diffuse reflection boundary condition. The gas region is the spherical domain in $\mathbb{R}^d$, $d=1,2$. First, without the equilibrating effect of collision, we obtain the algebraic convergence rates to the steady state of free molecular flow with variable boundary temperature. The convergence behavior has intricate dependence on the accommodation coefficient of the Maxwell-type boundary condition. Then we couple the boundary effect with the intermolecular collision and study their interaction. We are able to construct the steady state solutions of the full Boltzmann equation for large Knudsen numbers and small boundary temperature variation. We also establish the nonlinear stability with exponential rate of the stationary Boltzmann solutions. Our analysis is based on the explicit formulations of the boundary condition for symmetric domains.