Gradient estimates for nonlinear kinetic Fokker-Planck equations

TL;DR

This paper develops a detailed gradient regularity theory for nonlinear kinetic Fokker-Planck equations, providing novel pointwise estimates and fine regularity results even for linear cases without forcing.

Contribution

It introduces new gradient estimates for a broad class of nonlinear kinetic Fokker-Planck equations, advancing the understanding of their regularity properties.

Findings

Established pointwise gradient estimates in terms of data

Achieved fine gradient regularity under borderline assumptions

Provided novel results even for linear equations without forcing

Abstract

In this work, we provide a comprehensive gradient regularity theory for a broad class of nonlinear kinetic Fokker-Planck equations. We achieve this by establishing precise pointwise estimates in terms of the data in the spirit of nonlinear potential theory, leading to fine gradient regularity results under borderline assumptions on the data. Notably, our gradient estimates are novel already in the absence of forcing terms and even for linear kinetic Fokker-Planck equations in divergence form.