On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains

We consider the Navier-Stokes-Fourier system on an unbounded domain in the Euclidean space R³, supplemented by the far field conditions for the phase variables, specifically: ρ → 0,\ ϑ → ϑ_∞, \ u → 0 as \ |x| → ∞. We study the long time behaviour of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium ρₛ = 0,\ ϑₛ = ϑ_∞,\ uₛ = 0 in the sense of ergodic averages for time tending to infinity. As a consequence of the convergence result combined with the total mass conservation, we can show that the total momentum of global-in-time weak solutions is never globally conserved.