Solutions to a moving boundary problem on the Boltzmann equation

TL;DR

This paper proves the existence and positivity of time-periodic solutions to the Boltzmann equation modeling rarefied gas flow between oscillating plates, extending understanding of boundary-driven kinetic problems.

Contribution

It establishes the existence of time-periodic solutions with small boundary oscillations and proves their positivity using uniform estimates and bootstrap methods.

Findings

Existence of time-periodic solutions under small oscillation amplitude

Positivity of solutions confirmed through stability analysis

Development of uniform estimates in the periodic setting

Abstract

Motivated by the numerical investigation by Aoki et al. [1], we study a rarefied gas flow between two parallel infinite plates of the same temperature governed by the Boltzmann equation with diffuse reflection boundaries, where one plate is at rest and the other one oscillates in its normal direction periodically in time. For such boundary-value problem, we establish the existence of a time-periodic solution with the same period, provided that the amplitude of the oscillating boundary is suitably small. The positivity of the solution is also proved basing on the study of its large-time asymptotic stability for the corresponding initial-boundary value problem. For the proof of existence, we develop uniform estimates on the approximate solutions in the time-periodic setting and make a bootstrap argument by reducing the coefficient of the extra penalty term from a large enough constant to zero.