Navier-Stokes-Fourier System on Unbounded Domains: Weak Solutions, Relative Entropies, Weak-Strong Uniqueness

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.