New preprint about kinetic Fokker-Planck equations

I recently uploaded a new preprint on HAL and arxiv entitled « Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation ». It is written in collaboration with François Golse, Clément Mouhot and Alexis F. Vasseur.

Cylindres itérés dans la démonstration de l'inégalité de Harnack

Cylindres itérés dans la démonstration de l’inégalité de Harnack

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 Hölder 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 γ in [-d,1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the Hölder regularity of these solutions.


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