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

TL;DR

This paper studies nonlinear Kolmogorov-Fokker-Planck equations with rough coefficients, proving regularity, inequalities, and existence and uniqueness of weak solutions under minimal assumptions.

Contribution

It establishes regularity results, inequalities, and existence and uniqueness of weak solutions for nonlinear equations with rough coefficients, extending previous theories.

Findings

Higher integrability and local boundedness of weak sub-solutions

Harnack inequalities and Hölder continuity with quantitative estimates

Existence and uniqueness of weak solutions in bounded domains

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 $A=A(\xi,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ 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 H{\"o}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.