On regularity of a Kinetic Boundary layer

TL;DR

This paper investigates the regularity of solutions to the nonlinear steady Boltzmann equation in a half-space with phase transition, introducing new weighted estimates and addressing unbounded, non-convex domains.

Contribution

It develops a novel kinetic weight and establishes weighted $C^1$ and unweighted $W^{1,p}$ estimates for the Boltzmann equation in complex half-space geometries.

Findings

Established a weighted $C^1$ estimate for the solution.

Proved a $W^{1,p}$ estimate without weight for $p<2$.

Addressed regularity in unbounded, non-convex domains.

Abstract

We study the nonlinear steady Boltzmann equation in the half space, with phase transition and Dirichlet boundary condition. In particular, we study the regularity of the solution to the half-space problem in the situation that the gas is in contact with its condensed phase. We propose a novel kinetic weight and establish a weighted $C^1$ estimate under the spatial domain $x\in [0,\infty)$, which is unbounded and not strictly convex. Additionally, we prove the $W^{1,p}$ estimate without any weight for $p<2$.