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arXiv · 2505.11943

Optimal regularity for kinetic Fokker-Planck equations in domains

Xavier Ros-Oton·Marvin Weidner
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Abstract

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $\Omega\subset \mathbb{R}^n$ with specular reflection condition, including Kolmogorov's equation $\partial_t f +v\cdot\nabla_x f-\Delta_v f=h$. Our main results establish the following: - Solutions are always $C^\infty$ in $t,v,x$ away from the grazing set $\{x\in\partial\Omega,\ v\cdot n_x=0\}$. - 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 $C^2_v$.

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Xavier Ros-Oton, Marvin Weidner. 2025-05-17. Optimal regularity for kinetic Fokker-Planck equations in domains. https://arxiv.org/abs/2505.11943

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