Weak and Variational Solutions to Steady Equations for Compressible Heat Conducting Fluids

We study a steady compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain. We consider a general pressure law of the form $p=(\gamma-1)\varrho e$ which includes in particular the case $p=a_1\varrho\vartheta+a_2\varrho^\gamma$. We show the existence of a variational entropy solution (i.e., a solution satisfying balance of mass, momentum, entropy inequality, and global balance of total energy) for $\gamma>\frac{3+\sqrt{41}}{8}$ which is a weak solution (i.e., also the weak formulation of total energy balance is satisfied), provided $\gamma>\frac{4}{3}$. These results cover at least two physically reasonable cases, namely, $\gamma=\frac{5}{3}$ (monoatomic gas) and $\gamma=\frac{4}{3}$ (relativistic gas).

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