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 $R^3$, supplemented by the far field conditions for the phase variables, specifically: $\rho \to 0,\vartheta \to {\vartheta }_{\mathrm{\infty }},u\to 0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. 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 ${\rho }_s=0,{\vartheta }_s={\vartheta }_{\mathrm{\infty }},u_s=0$ 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.