Global existence for the Boltzmann equation in $L^r_v L^\infty_t L^\infty_x$ spaces

TL;DR

This paper proves the global existence and uniqueness of solutions to the Boltzmann equation near a Maxwellian in specific mixed Lebesgue spaces, utilizing conservation laws and entropy inequalities.

Contribution

It establishes the global well-posedness of the Boltzmann equation in $L^r_v L^ fty_t L^ fty_x$ spaces for a range of r, extending previous results.

Findings

Global existence of solutions in specified function spaces

Uniqueness of mild solutions near Maxwellian

Use of conservation laws and entropy inequalities

Abstract

We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $L^r_v L^\infty_t L^\infty_x $ spaces where $r \in (1,\infty]$ by using the excess conservation laws and entropy inequality introduced in [5].