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

A note on the weak regularity theory for degenerate Kolmogorov equations

“…This framework is quite classical for the study of the weak regularity theory for the kinetic Kolmogorov-Fokker-Planck equation [4,11,13,14], that can be recovered from (1.1) by choosing N = 2d, κ = 1, m 0 = m 1 = d and c ≡ 0. Still, even if it is the most natural framework for (1.1), to the best of our knowledge it has never been considered in literature yet.…”

On the fundamental solution for degenerate Kolmogorov equations with rough coefficients

“…, x (κ) ). In particular, W is a Banach space, and it was firstly introduced in [4] as an extension of the natural functional setting that arises in the study of the weak regularity theory for the kinetic Kolmogorov-Fokker-Planck equation [5,17,18,19]. From now on, we consider the shorthand notation…”

Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

“…As an application, they also showed the conditional Hölder regularity for the Landau equation (1.6). More recently, weak Harnack inequality and some quantitative estimates for kinetic FPKEs were obtained in [17,18]. More historical backgrounds about Harnack inequalities and kinetic equations are referred to [17, Section 1.2].…”

“…Thus one can invoke the classical De-Giorgi method to show the apriori L ∞ -estimates. Compared with the recent works [16,17,18], the novelty of our boundedness estimates is that b can be in some L q1 t (L…”

“…In particular, we obtain the global bound estimate and stability of weak solutions, where the key point is to use the localization norm [39]. For the stability, we use the Hölder regularity estimate established in [16] (see also [34,17,18]).…”