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 form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(\hat{I}¾,X,Y,t):{\mathbb{R}}^m\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times \mathbb{R}\to {\mathbb{R}}^m$ is assumed to be continuous with respect to $\hat{I}¾$, and measurable with respect to $X,Y$ and $t$. $A=A(\hat{I}¾,X,Y,t)$ 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 H{ö}lder 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.

34 Pages