Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

TL;DR

This paper extends the De Giorgi--Nash--Moser theory to kinetic Fokker-Planck equations with rough coefficients, establishing a Harnack inequality and applying it to prove Hölder regularity for solutions of the Landau equation with Coulomb interactions.

Contribution

The paper introduces new regularity results for kinetic Fokker-Planck equations with minimal coefficient regularity and applies these to the Landau equation, advancing understanding of solutions with rough coefficients.

Findings

Hölder regularity for solutions of kinetic Fokker-Planck equations with rough coefficients

Harnack inequality established for a broad class of hypoelliptic equations

Regularity results applied to the Landau-Coulomb equation with inverse-power law

Abstract

We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{\"o}lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law $\gamma$ $\in$ [--d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the H{\"o}lder regularity of these solutions.