Low-regularity global well-posedness for the Boltzmann equation near vacuum

TL;DR

This paper proves the global existence and uniqueness of solutions to the Boltzmann equation near vacuum in low-regularity Besov spaces, using new bilinear estimates and a div-curl lemma.

Contribution

It introduces a novel bilinear estimate for the collision operator and a div-curl lemma to establish global well-posedness in anisotropic low-regularity spaces.

Findings

Established global existence and uniqueness of solutions near vacuum.

Developed a new bilinear estimate for the collision operator.

Utilized a div-curl lemma to close a priori estimates.

Abstract

We study the Boltzmann equation near vacuum in anisotropic low-regularity Besov spaces. We establish the global existence and uniqueness of strong solutions with the critical regularity index $2/p$ for $p\in[1,\infty)$ in $\mathbb{R}^3$. The proof relies on a new bilinear estimate for the nonlinear collision operator. Combined with a div-curl type lemma we develop, this allows us to close the a priori estimates and thereby obtain global well-posedness.