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

Prashanta Garain; Kaj Nyström; Prashanta Garain; Kaj Nyström

doi:10.3934/mine.2023043

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-37. doi: 10.3934/mine.2023043

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

Prashanta Garain ,
Kaj Nyström ^,

Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

Received: 28 April 2022 Revised: 15 June 2022 Accepted: 15 June 2022 Published: 28 June 2022
MSC : 35H20, 35K65, 35K70, 35R03

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.
- Kolmogorov equation,
- parabolic,
- ultraparabolic,
- hypoelliptic,
- nonlinear Kolmogorov-Fokker-Planck equations,
- existence,
- uniqueness,
- regularity
Citation: Prashanta Garain, Kaj Nyström. On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients[J]. Mathematics in Engineering, 2023, 5(2): 1-37. doi: 10.3934/mine.2023043

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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.

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