Steady compressible Navier-Stokes-Fourier system with slip boundary conditions arising from kinetic theory

TL;DR

This paper proves the existence and uniqueness of strong solutions for a steady compressible Navier-Stokes-Fourier system with slip boundary conditions derived from kinetic theory, using a novel variational approach and fixed point method.

Contribution

It introduces a new variational formulation for the linearized problem and addresses the complex interplay of velocity, temperature, and density effects at the boundary.

Findings

Existence and uniqueness of strong solutions under near-constant wall temperature.

Development of a new variational formulation for the linearized system.

Successful application of fixed point argument to the nonlinear problem.

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 \cite{Coron-JSP-1989} and later by Aoki et al \cite{Aoki-Baranger-Hattori-Kosuge-Martalo-Mathiaud-Mieussens-JSP-2017}. 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.