Optimal $C^{\frac{1}{2}}$ regularity of the Boltzmann equation in non-convex domains

TL;DR

This paper proves the optimal $C^{1/2}$ regularity for the Boltzmann equation in non-convex domains with boundary conditions, overcoming previous difficulties caused by singular billiard trajectories.

Contribution

It introduces a new dynamical singular regime integration method to establish optimal regularity results in non-convex geometries.

Findings

Established $C^{1/2}$ regularity for the Boltzmann equation in non-convex domains.

Overcame previous challenges posed by grazing trajectories near the boundary.

Extended regularity results beyond convex obstacles.

Abstract

Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in [32], where $C^{\frac{1}{2}-}_{x,v}$ H\"{o}lder regularity was proven. In this paper, we introduce a novel dynamical singular regime integration methodology to establish the optimal $C^{\frac{1}{2}}_{x,v}$ regularity for the Boltzmann equation past a convex obstacle.