Solutions to a Moving Boundary Problem on the Boltzmann Equation

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.