Convergence Towards the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition

TL;DR

This paper proves the existence and convergence rate of a steady state for a collisionless gas with Cercignani-Lampis boundary conditions in a domain with variable wall temperature, using advanced probabilistic methods.

Contribution

It provides the first proof of steady state existence under variable wall temperature and characterizes the convergence rate, extending previous results to more general boundary conditions.

Findings

Existence of a steady state with variable wall temperature.

Optimal convergence rate in L1 norm towards the steady state.

Explicit form of the steady state for a mixed boundary condition model.

Abstract

We study the asymptotic behavior of the kinetic free-transport equation enclosed in a regular domain, on which no symmetry assumption is made, with Cercignani-Lampis boundary condition. We give the first proof of existence of a steady state in the case where the temperature at the wall varies, and derive the optimal rate of convergence towards it, in the L1 norm. The strategy is an application of a deterministic version of Harris subgeometric theorem, in the spirit of Ca\~nizo-Mischler (2021) and Bernou (2020). We also investigate rigorously the velocity flow of a model mixing pure diffuse and Cercignani-Lampis boundary conditions with variable temperature, for which we derive an explicit form for the steady state, providing new insights on the role of the Cercignani-Lampis boundary condition in this problem.