Semi-local behaviour of non-local hypoelliptic equations: Boltzmann

TL;DR

This paper proves a semi-local Strong Harnack inequality for the Boltzmann equation with soft potentials without cutoff, filling a gap in previous proofs and deriving bounds on the fundamental solution.

Contribution

It establishes the semi-local Strong Harnack inequality for the Boltzmann equation in a new setting, using divergence form properties to handle non-divergent operators.

Findings

Proved semi-local Strong Harnack inequality for Boltzmann equation.

Derived upper and lower bounds on the fundamental solution.

Extended the inequality to non-cutoff, soft potential cases.

Abstract

The purpose of this note is to demonstrate the announced result in [Loher, The Strong Harnack inequality for the Boltzmann equation, S\'eminaire Laurent Schwartz proceeding] by filling the gap in the proof sketch. We prove the semi-local Strong Harnack inequality for the Boltzmann equation for moderately soft potentials without cutoff assumption. The non-local operator in the Boltzmann equation is in non-divergence form, and thus the method developed in [arXiv:2404.05612] does not apply. However, we exploit that the Boltzmann equation is on average in divergence form, and we show that the non-divergent part of the collision operator is of lower order in a suitable sense, which proves to be sufficient to deduce the Strong Harnack inequality. Consequentially, we derive upper and lower bounds on the fundamental solution of the linearised Boltzmann equation.