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

We study steady compressible Navier–Stokes–Fourier system in a bounded three– dimensional domain. Considering a general pressure law of the form p = (γ − 1)̺e, we show existence of a variational entropy solution (i.e. solution satisfying balance of mass, momentum, entropy inequality and global balance of total energy) for γ > 3+ √ 41 8 which is a weak solution (i.e. also the weak formulation of total energy balance is satisfied) provided γ > 4 3. These results cover at least two physically reasonable cases, namely γ = 5 3 (monoatomic gas) and γ = 4 3 (relativistic gas).