Weak and variational solutions to steady equations for compressible heat conducting fluids

Abstract

We study steady compressible Navier--Stokes--Fourier system in a bounded three--dimensional domain. We consider a general pressure law of the form $p = (\gamma-1) \vr e$ which includes in particular the case $p=a_1 \vr \vt + a_2 \vr^\gamma$.

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 $\gamma >\frac{3+\sqrt{41}}{8}$ which is a weak solution (i.e. also the weak formulation of total energy balance is satisfied) provided $\gamma>\frac 43$. These results cover at least two physically reasonable cases, namely $\gamma = \frac 53$ (monoatomic gas) and $\gamma = \frac 43$ (relativistic gas).

Keywords

• Steady Navier--Stokes--Fourier system
• variational entropy solution
• weak solution
• large data
• renormalized solution