Steady compressible Navier-Stokes-Fourier system with general temperature dependent viscosities I: density estimates based on Bogovskii operator

Abstract

The aim of this paper is to reconsider the existence theory for steady compressible Navier--Stokes--Fourier system assuming more general condition of the dependence of the viscosities on the temperature in the form $\hat{I}¼(\vartheta )$, $\hat{I}¾(\vartheta )\sim (1+\vartheta {)}^{\hat{I}}\pm$ for $0\le \hat{I}\pm \le 1$. This extends the known theory for $\hat{I}\pm =1$ from and improves significantly the results for $\hat{I}\pm =0$. This paper is the first of a series of two papers dealing with this problem and is connected with the Bogovskii-type estimates of the sequence of densities. This leads, among others, to the limitation ${\hat{I}}^3>\frac{3}{2}$ for the pressure law $p(\varrho ,\vartheta )\sim {\varrho }^3+\varrho \vartheta$. The paper considers both the heat-flux (Robin) and Dirichlet boundary conditions for the temperature as well as both the homogeneous Dirichlet and zero inflow/outflow Navier boundary conditions for the velocity. Further extension for ${\hat{I}}^3>1$ only is based on different type of pressure estimates and will be the content of the subsequent paper.