On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

Abstract

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the formThe functionassumed to be continuous with respect to ξ, and measurable with respect to X, Y and t. A = A(ξ, X, Y, t) is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded X, Y and t dependent domains.

Regularity for rough hypoelliptic equations

“…hyp (Ω ∩{ > in } + 0 H −1 hyp (Ω ∩{ > in } , which implies the thought bound (10) by Sobolev embedding for some 1 > 2 and 0 = 1 = 2.…”

“…The combination of these ideas saw a lot of recent interest [2,3,10,12,13,14,29,30,31] as it is a path for regularity results for nonlinear kinetic equations, where the solution satisfies schematically…”