Boundary Layer of Boltzmann Equation in 2D Convex Domains

TL;DR

This paper studies the boundary layer behavior of the 2D Boltzmann equation in convex domains, deriving the Navier-Stokes-Fourier system with new estimates for the Milne problem and boundary layers.

Contribution

It introduces novel weighted $W^{1,inity}$ estimates for the Milne problem with geometric correction and develops stronger remainder estimates in an $L^{2m}-L^{inity}$ framework.

Findings

Established hydrodynamic limits with boundary layers in 2D convex domains.

Derived the steady Navier-Stokes-Fourier system with non-slip boundary conditions.

Developed new weighted $W^{1,inity}$ estimates for the Milne problem.

Abstract

Consider the stationary Boltzmann equation in 2D convex domains with diffusive boundary condition. In this paper, we establish the hydrodynamic limits while the boundary layers are present, and derive the steady Navier-Stokes-Fourier system with non-slip boundary conditions. Our contribution focuses on novel weighted $W^{1,\infty}$ estimates for the Milne problem with geometric correction. Also, we develop stronger remainder estimates based on an $L^{2m}-L^{\infty}$ framework.