Optimal regularity for kinetic Fokker-Planck equations in domains

We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with specular reflection condition, including Kolmogorov’s equation ${\mathrm{\partial }}_{t}f+v\cdot {\mathrm{\nabla }}_{x}f-{\mathrm{\Delta }}_{v}f=h$. Our main results establish the following: - Solutions are always $C^{\mathrm{\infty }}$ in $t,v,x$ away from the grazing set $\{x\in \mathrm{\partial }\mathrm{\Omega },v\cdot n_x=0\}$. - They are $C_{\text{kin}}^{4,1}$ up to the grazing set. - This regularity is optimal, i.e. we show that that they are in general not $C_{\text{kin}}^5$. These results show for the first time that solutions are classical up to boundary, i.e. $C_{t,x}^1$ and $C_v^2$.