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

Abstract

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 introduce suitable weak solutions as weak solutions that satisfy the relative entropy inequality. We prove existence of weak solutions and of suitable weak solutions for arbitrary large initial data for potential forces with an arbitrary growth at large distances. Finally we prove the weak-strong uniqueness principle, meaning that the suitable 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.