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arXiv · 2409.02903

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

Amélie Loher
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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.

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Amélie Loher. 2024-09-04. Semi-local behaviour of non-local hypoelliptic equations: Boltzmann. https://arxiv.org/abs/2409.02903

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