Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains

Benny Avelin; Mingyi Hou

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arXiv · 2405.04070

Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains

Benny Avelin·Mingyi Hou

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Abstract

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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Benny Avelin, Mingyi Hou. 2024-05-07. Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains. https://arxiv.org/abs/2405.04070

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