We investigate the Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat conducting fluid on large class of unbounded domains with no slip and slip boundary conditions. We propose a definition of weak solutions, that is particularly convenient for the treatment of the Navier-Stokes-Fourier system on unbounded domains. We prove existence of weak solutions for arbitrary large initial data for potential forces with an arbitrary growth at large distances. We show, that any weak solution satisfies the so called relative entropy inequality. Finally we prove the weak-strong uniqueness principle, meaning that the weak solutions coincide with strong solutions emanating from the same initial data (as long as the latter exist), at least when the potential force vanishes at large distances.

We consider a problem modeling the steady flow of a compressible heat conducting Newtonian fluid subject to the slip boundary condition for the velocity. Assuming the pressure law of the form p(ϱ, ϑ) ~ ϱγ + ϱϑ, we show (under additional assumptions on the heat conductivity and the viscosity) that for any γ > 1 there exists a variational entropy solution to our problem (i.e. the weak formulation of the total energy balance is replaced by the entropy inequality and the global total energy balance). Moreover, if [Formula: see text] (together with further restrictions on the heat conductivity), the solution is in fact a weak one. The results are obtained without any restriction on the size of the data.

International audienceno abstrac