Regularity for the Boltzmann equation conditional to pressure and moment bounds

TL;DR

This paper establishes uniform $L^ Infty$ bounds and smoothness estimates for solutions to the Boltzmann and Landau equations under certain macroscopic bounds, advancing understanding of their regularity properties.

Contribution

It proves regularity and decay estimates for solutions to the Boltzmann and Landau equations based on pointwise bounds on observables, extending previous results to non-cutoff cases.

Findings

Uniform $L^ Infty$ bounds for Boltzmann solutions with pressure and moment bounds

Derivation of $C^{ Infty}$ regularity and decay estimates

Results extend to the Landau equation in the limit case

Abstract

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in $L^\infty$ in the case of hard potentials. As a consequence, we derive $C^{\infty}$ estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our $L^\infty$ estimates are uniform in the limit $s \nearrow 1$ and hence we recover the same results also for the Landau equation.