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 (partial derivative(t) + X . del(Y))(u) = del(X) . (A(del(X)u, X, Y, t)). The function A = A(xi, X, Y, t) : R-m x R-m x R-m x R -> 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 Holder 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.