Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909541165170688 |
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| author | Avelin, Benny Hou, Mingyi |
| author_facet | Avelin, Benny Hou, Mingyi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04070 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains Avelin, Benny Hou, Mingyi Analysis of PDEs 35Q84, 35D30, 35D99, 35J25 (Primary) 35H10, 35J70, 31C45 (Secondary) 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. |
| title | Weak and Perron Solutions for Stationary Kramers-Fokker-Planck Equations in Bounded Domains |
| topic | Analysis of PDEs 35Q84, 35D30, 35D99, 35J25 (Primary) 35H10, 35J70, 31C45 (Secondary) |
| url | https://arxiv.org/abs/2405.04070 |