Steady compressible Navier-Stokes-Fourier system with general temperature dependent viscosities and hard sphere pressure law

We study the existence theory for steady compressible Navier-Stokes-Fourier system in a three dimensional bounded domain for the case of viscosities depending on the temperature in the form $(1+\vartheta)^\alpha$ for $0\leq \alpha\leq 1$ and the hard sphere pressure $p(\varrho,\vartheta) = \vartheta\varrho h(\varrho)$, $h(\varrho)$ is increasing and singular at $\varrho = a > 0$. This paper considers both the heat-flux and Dirichlet boundary conditions for the temperature and Dirichlet boundary conditions for the velocity. The key point to controlling the pressure is based on some estimates of the Bogovskii operator.