## Abstract

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form (∂_{t} + X · ∇_{Y})u = ∇_{X} · (A(∇_{X}u, X, Y, t)). The function A = A(ξ, X, Y, t): ℝ^{m} × ℝ^{m} × ℝ^{m} × ℝ → ℝ^{m} 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.

Original language | English |
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Journal | Mathematics In Engineering |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2022 |

Externally published | Yes |

## Keywords

- Kolmogorov equation
- existence
- hypoelliptic
- nonlinear Kolmogorov-Fokker-Planck equations
- parabolic
- regularity
- ultraparabolic
- uniqueness

## ASJC Scopus subject areas

- Analysis
- Mathematical Physics
- Applied Mathematics