The weak Harnack inequality for the Boltzmann equation without cut-off

TL;DR

This paper establishes the weak Harnack inequality and H"older continuity for a broad class of kinetic integro-differential equations, including the non-cutoff Boltzmann equation, under certain boundedness conditions.

Contribution

It introduces a novel approach to derive regularity estimates for the Boltzmann equation without cut-off, extending the applicability of Harnack inequalities in kinetic theory.

Findings

Proves the weak Harnack inequality for the non-cutoff Boltzmann equation.

Establishes local H"older continuity of solutions.

Provides a quantitative lower bound for solutions.

Abstract

In this paper, we obtain the weak Harnack inequality and H\"older estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cut-off can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local H\"older estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cut-off.