Boundary layer of the Boltzmann equation in 2-dimensional convex domains

Abstract

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