Time-periodic solutions of the Boltzmann equation with soft potentials in $\mathbb{R}^3$

Yuanjie Lei; Shuangqian Liu; Jiaying Zhang; Huijiang Zhao

Time-periodic solutions of the Boltzmann equation with soft potentials in $\mathbb{R}^3$

Analysis of PDEs 2026-07-06 v1

Authors: Yuanjie Lei , Shuangqian Liu , Jiaying Zhang , Huijiang Zhao

Abstract

We establish the existence of time-periodic solutions to the Boltzmann equation for the full range of soft potentials and prove their global stability. The proof relies on refined energy estimates combining Besov and Sobolev regularity with velocity-weighted energy methods. The analysis is complicated by the lack of a spectral gap in the soft potential regime together with the absence of time integrability of the periodic forcing over $\mathbb{R}^+$ . This provides a framework for the study of time-periodic behavior for the Boltzmann equation.

Keywords

Boltzmann equation soft potentials time-periodic solutions global stability Besov regularity energy estimates

Cite

@article{arxiv.2607.04618,
  title  = {Time-periodic solutions of the Boltzmann equation with soft potentials in $\mathbb{R}^3$},
  author = {Yuanjie Lei and Shuangqian Liu and Jiaying Zhang and Huijiang Zhao},
  journal= {arXiv preprint arXiv:2607.04618},
  year   = {2026}
}