Steady Compressible Navier-Stokes-Fourier System with Slip Boundary Conditions Arising from Kinetic Theory

Abstract

This paper studies the boundary value problem on the steady compressible Navier-Stokes-Fourier system in a channel domain \((0,1)\times \mathbb {T}^2\) with a class of generalized slip boundary conditions that were systematically derived from the Boltzmann equation by Coron [9] and later by Aoki et al [1]. We establish the existence and uniqueness of strong solutions in \((L_{0}^{2}\cap H^{2}(\Omega ))\times V^{3}(\Omega )\times H^{3}(\Omega )\) provided that the wall temperature is near a positive constant. The proof relies on the construction of a new variational formulation for the corresponding linearized problem and employs a fixed point argument. The main difficulty arises from the interplay of velocity and temperature derivatives together with the effect of density dependence on the boundary.