Optimal regularity for kinetic Fokker-Planck equations in domains

TL;DR

This paper investigates the regularity of solutions to kinetic Fokker-Planck equations with boundary conditions, establishing optimal smoothness results and demonstrating classical boundary regularity for the first time.

Contribution

It provides the first comprehensive analysis of boundary regularity for solutions to kinetic Fokker-Planck equations, showing optimal smoothness and classical regularity up to the boundary.

Findings

Solutions are $C^ abla$ in $t,v,x$ away from the grazing set.

Solutions are $C^{4,1}_{ ext{kin}}$ up to the grazing set.

Regularity is optimal; solutions are generally not $C^5_{ ext{kin}}$.

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 $C^2_v$.