On regularity of a Kinetic Boundary layer

TL;DR

This paper studies the regularity of the nonlinear steady Boltzmann equation in the half-space under phase transition and Dirichlet boundary data. It introduces a novel kinetic weight and proves a weighted $C^1$ estimate in the unbounded, non-strictly convex domain $x\in[0,\infty)$, as well as a $W^{1,p}$ estimate without weight for $p<2$, in the regime $0<|u|\ll 1$. The analysis is carried out in a perturbative framework around the global Maxwellian $M$ with the shifted velocity $\xi=v-(u,0,0)$, yielding the equation $(\xi_1+u)\partial_x f + \mathcal{L}f = \Gamma(f,f)$ and boundary data $f(0,\xi)=f_b(\xi)$. This work extends boundary-layer regularity theory from convex bounded domains to half-space with phase transition, providing new insights for hydrodynamic limits and kinetic-fluid boundary interactions.

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 $p<2$.