$L^p$-estimates of the Botlzmann Equation around a traveling local Maxwellian

TL;DR

This paper develops $L^p$-estimates for the Boltzmann equation near a traveling local Maxwellian, using a regularization approach that extends the analysis to all Lebesgue exponents.

Contribution

It introduces a novel regularization method by dividing the distribution function, enabling $L^p$-estimates for the Boltzmann equation across all Lebesgue spaces.

Findings

Established $L^p$-estimates for all $0<p\, extless\,\infty$

Reformulated the Boltzmann equation into a regularized form

Applied Hölder inequalities to derive estimates

Abstract

In this paper, we are interested in the $L^p$-estimates of the Boltzmann equation in the case that the distribution function stays around a travelling local Maxwellian. For this, we divide both sides of the Boltzmann equation by the velocity distribution function with a fractional exponent and reformulate the Boltzmann equation into a regularized one. This amounts to endowing additional integrability on the collision kernel, which in turn enables us to apply simple H\"{o}lder type inequalities. Our results cover the whole range of Lebesgue exponents: $0<p\leq \infty$.