Three-dimensional time-periodic problem on the Boltzmann equation with external force

TL;DR

This paper proves the existence and stability of time-periodic solutions to the three-dimensional Boltzmann equation with small external forces, extending previous results to lower spatial dimensions.

Contribution

It provides the first affirmative solution to the 3D time-periodic Boltzmann problem with external force, using Serrin's method for stability analysis.

Findings

Existence of time-periodic solutions under small external forces.

Stability of stationary solutions in three dimensions.

Extension of previous high-dimensional results to 3D.

Abstract

The time-periodic problem on the Boltzmann equation with a given time-periodic external force in the three-dimensional whole space has remained open since it was first studied in [15] for only spatial dimensions not less than five. The goal of this paper is to give an affirmative answer to this problem provided that the external force is sufficiently small in the function space $\mathcal{C}(\mathbb{R};\dot{B}^{-3/2}_{2,\infty}\cap\dot{H}^N)$ with $N\geq 4$. The proof is based on Serrin's method through studying the global-in-time stability of the Cauchy problem with time-periodic external forces. As a direct consequence, the result also yields the existence and stability of stationary solutions to the physically realistic three-dimensional Boltzmann equation when the external force is time-independent.