Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations

TL;DR

This paper establishes a weak Harnack inequality for non-negative solutions of kinetic Fokker-Planck equations with bounded coefficients by using a Log-transform approach and a new weak Poincaré inequality, advancing the understanding of regularity in kinetic equations.

Contribution

It introduces a novel weak Poincaré inequality for kinetic equations and applies a Log-transform technique to derive a weak Harnack inequality, extending previous methods to this class of equations.

Findings

Derived a weak Harnack inequality for kinetic Fokker-Planck equations.

Developed a new weak Poincaré inequality for ultraparabolic equations.

Applied a covering argument adapted for kinetic equations.

Abstract

This article deals with kinetic Fokker-Planck equations with essentially bounded coefficients. A weak Harnack inequality for non-negative super-solutions is derived by considering their Log-transform and following S. N. Kruzhkov (1963). Such a result rests on a new weak Poincar{\'e} inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.