Optimal regularity for kinetic Fokker-Planck equations in domains
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| Format: | Preprint |
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2025
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| _version_ | 1866916742025969664 |
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| author | Ros-Oton, Xavier Weidner, Marvin |
| author_facet | Ros-Oton, Xavier Weidner, Marvin |
| contents | We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $Ω\subset \mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\partial_t f +v\cdot\nabla_x f-Δ_v f=h$. Our main results establish the following:
- Solutions are always $C^\infty$ in $t,v,x$ away from the grazing set $\{x\in\partialΩ,\ v\cdot n_x=0\}$.
- They are $C^{4,1}_{\text{kin}}$ up to the grazing set.
- This regularity is optimal, i.e. we show that that they are in general not $C^5_{\text{kin}}$.
These results show for the first time that solutions are classical up to boundary, i.e. $C^1_{t,x}$ and $C^2_v$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_11943 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal regularity for kinetic Fokker-Planck equations in domains Ros-Oton, Xavier Weidner, Marvin Analysis of PDEs 35Q84, 35B65, 82C40 We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $Ω\subset \mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\partial_t f +v\cdot\nabla_x f-Δ_v f=h$. Our main results establish the following: - Solutions are always $C^\infty$ in $t,v,x$ away from the grazing set $\{x\in\partialΩ,\ v\cdot n_x=0\}$. - They are $C^{4,1}_{\text{kin}}$ up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not $C^5_{\text{kin}}$. These results show for the first time that solutions are classical up to boundary, i.e. $C^1_{t,x}$ and $C^2_v$. |
| title | Optimal regularity for kinetic Fokker-Planck equations in domains |
| topic | Analysis of PDEs 35Q84, 35B65, 82C40 |
| url | https://arxiv.org/abs/2505.11943 |