The Boltzmann equation with time-periodic boundary temperature

TL;DR

This paper proves the existence and stability of time-periodic solutions to the Boltzmann equation in bounded domains with time-periodic boundary temperature, extending understanding of kinetic equations with time-dependent boundary conditions.

Contribution

It establishes the existence and stability of time-periodic solutions for the Boltzmann equation with time-periodic boundary temperature, a novel extension to previous stationary boundary studies.

Findings

Existence of time-periodic solutions for both hard and soft potentials.

Stability of these solutions under small perturbations.

Non-negativity of the time-periodic profiles.

Abstract

This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.