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

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 $\mu(\vartheta)$, $\xi(\vartheta) \sim (1+\vartheta)^\alpha$ for $0\leq \alpha \leq 1$. This extends the known theory for $\alpha=1$ from and improves significantly the results for $\alpha =0$. 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 $\gamma >\frac 32$ for the pressure law $p(\varrho,\vartheta) \sim \varrho^\gamma + \varrho\vartheta$. 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 $\gamma >1$ only is based on different type of pressure estimates and will be the content of the subsequent paper.