For nonlinear kinetic Fokker-Planck equations, the velocity gradient is controlled pointwise by kinetic Riesz potentials of the data, yielding new Hölder, BMO, and Calderón-Zygmund regularity criteria.
Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains
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In this paper, we investigate weak solutions and Perron-Wiener-Brelot solutions to the linear stationary Kramers-Fokker-Planck equation in bounded domains. We establish the existence of weak solutions in product domains by applying the Lions-Lax-Milgram theorem and the vanishing viscosity method. Furthermore, we show that these solutions coincide in well-behaved domains. Building on the existence of weak solutions in product domains, we develop the foundational theory of Perron-Wiener-Brelot solutions in arbitrary bounded domains. Our results rely on recent advancements in the theory of kinetic Fokker-Planck equations with rough coefficients.
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Gradient estimates for nonlinear kinetic Fokker-Planck equations
For nonlinear kinetic Fokker-Planck equations, the velocity gradient is controlled pointwise by kinetic Riesz potentials of the data, yielding new Hölder, BMO, and Calderón-Zygmund regularity criteria.