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

TL;DR

This paper establishes optimal regularity and detailed boundary behavior of solutions to linear kinetic Fokker-Planck equations, showing solutions are smoother near the grazing set than previously known, with precise expansions beyond the critical regularity.

Contribution

It proves the sharp $C^{1/2}$ regularity for solutions and characterizes their behavior near the grazing set with higher order expansions, advancing understanding of boundary regularity.

Findings

Solutions are $C^{1/2}$ regular near the boundary.

Higher order expansions describe solution behavior near the grazing set.

Solutions exhibit increased smoothness up to the grazing set.

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.