Quantitative De Giorgi methods in kinetic theory for non-local operators

Abstract

We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies Hölder continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate. This permits to quantitatively prove the intermediate value lemma for the full range of non-locality parameter $s\in (0,1)$. Our results recover the results from Imbert and Silvestre [22] for the inhomogeneous Boltzmann equation in the non-cutoff case. The paper is self-contained.

arXiv comment53 pages, 2 figures. This version corresponds to the version that is to appear in the Journal of Functional Analysis