Existence of stationary weak solutions for the heat conducting flows

TL;DR

This paper proves the existence of stationary weak solutions for the steady compressible Navier--Stokes--Fourier system under various boundary conditions, parameters, and dimensions, including cases with temperature-dependent viscosity.

Contribution

It introduces new methods for constructing weak and variational entropy solutions for 3D flows with temperature-dependent viscosity without data restrictions.

Findings

Existence of weak solutions for 3D flows with temperature-dependent viscosity.

Results for 2D flows and bounded density solutions.

Review of solutions for more complex systems involving Navier--Stokes--Fourier equations.

Abstract

The steady compressible Navier--Stokes--Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e. the adiabatic constant $\gamma$ appearing in the pressure law $p(\vr,\vt) \sim \vr^\gamma + \vr \vt$ and the growth exponent in the heat conductivity, i.e. $\kappa(\vt) \sim (1+ \vt^m)$, and without any restriction on the size of the data, the main ideas of the construction of weak and variational entropy solutions for the three-dimensional flows with temperature dependent viscosity coefficients are explained. Further, the case when it is possible to prove existence of solutions with bounded density is reviewed. The main changes in the construction of solutions for the two-dimensional flows are mentioned and finally, results for more complex systems are reviewed, where the steady compressible Navier--Stokes--Fourier equations play an important role.