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*} (\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): \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.

Keywords

• kolmogorov equation
• parabolic
• ultraparabolic
• hypoelliptic
• nonlinear kolmogorov-fokker-planck equations
• existence
• uniqueness
• regularity