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

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form

$$({\mathrm{\partial }}_t+X\cdot {\mathrm{\nabla }}_Y)u={\mathrm{\nabla }}_X\cdot (A({\mathrm{\nabla }}_{X}u,X,Y,t)).$$

The function $A=A(\xi ,X,Y,t):{\mathbb{R}}^m\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times \mathbb{R}\to {\mathbb{R}}^m$ is assumed to be continuous with respect to $\xi$, and measurable with respect to $X,Y$ and $t$. $A=A(\xi ,X,Y,t)$ 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.