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: 0, \ _, \ 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 ₛ = 0, \ ₛ = _, \ 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.