Compressible Navier-Stokes-Fourier flows at steady-state

TL;DR

This paper proves the existence of weak solutions for steady-state compressible Navier-Stokes-Fourier flows with heat transfer and mixed boundary conditions, using a novel fixed point approach applicable in Lipschitz domains.

Contribution

It introduces a new fixed point method for establishing weak solutions of the NSF system with heat transfer and boundary conditions in Lipschitz domains.

Findings

Existence of weak solutions proved for the NSF system with mixed boundary conditions.

Applicable fixed point argument in Lipschitz domains using L^q-Neumann problems.

Quantitative estimates for solutions are established.

Abstract

The heat conducting compressible viscous flows are governed by the Navier-Stokes-Fourier (NSF) system. In this paper, we study the NSF system accomplished by the Newton law of cooling for the heat transfer at the boundary. On one part of the boundary, we consider the Navier slip boundary condition, while in the remaining part the inlet and outlet occur. The existence of a weak solution is proved via a new fixed point argument. With this new approach, the weak solvability is possible in Lipschitz domains, by making recourse to \(L^q\)-Neumann problems with \(q>n\).Thus, standard existence results can be applied to auxiliary problems and the claim follows by compactness techniques. Quantitative estimates are established.