We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form (∂t + X · ∇Y)u = ∇X · (A(∇Xu, X, Y, t)). The function A = A(ξ, X, Y, t): ℝm × ℝm × ℝm × ℝ → ℝm is assumed to be continuous with respect to ξ, and measurable with respect to X, Y and t. A = A(ξ, 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.
|Journal||Mathematics In Engineering|
|State||Published - 1 Jan 2022|
- Kolmogorov equation
- nonlinear Kolmogorov-Fokker-Planck equations
ASJC Scopus subject areas
- Mathematical Physics
- Applied Mathematics