Sharp regularity near the grazing set for kinetic Fokker-Planck equations

Kyeongbae Kim; Marvin Weidner

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arXiv · 2603.04121

Sharp regularity near the grazing set for kinetic Fokker-Planck equations

Kyeongbae Kim·Marvin Weidner

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Abstract

We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp $C^{1/2}$ regularity for either diffuse reflection or prescribed in-flow boundary conditions. Previously, in this setting, it was only known that solutions are $C^{\alpha}$ for some small $\alpha > 0$. Second, we provide a complete characterization of the solution behavior near the grazing set by proving higher order expansions beyond the critical regularity threshold of $\frac{1}{2}$. These results demonstrate for the first time that solutions maintain higher smoothness up to the grazing set near the incoming boundary.

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Kyeongbae Kim, Marvin Weidner. 2026-03-04. Sharp regularity near the grazing set for kinetic Fokker-Planck equations. https://arxiv.org/abs/2603.04121

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