Holder estimates for kinetic Fokker-Planck equations up to the boundary

TL;DR

This paper establishes boundary regularity and asymptotic behavior for solutions to kinetic Fokker-Planck equations with rough coefficients, extending De Giorgi methods to boundary estimates and analyzing solution vanishing at the boundary.

Contribution

It extends boundary regularity results for kinetic Fokker-Planck equations using De Giorgi techniques and provides higher order asymptotic estimates near the boundary.

Findings

Established local Hölder continuity up to the boundary.

Derived higher order asymptotic estimates near the incoming boundary.

Proved solutions vanish at infinite order on the boundary with zero boundary conditions.

Abstract

We obtain local Holder continuity estimates up to the boundary for a kinetic Fokker-Planck equation with rough coefficients, with the prescribed influx boundary condition. Our result extends some recent developments that incorporate De Giorgi methods to kinetic Fokker-Planck equations. We also obtain higher order asymptotic estimates near the incoming part of the boundary. In particular, when the equation has a zero boundary conditions and no source term, we prove that the solution vanishes at infinite order on the incoming part of the boundary.