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

Abstract

We consider the Navier–Stokes–Fourier (NSF) system on an unbounded domain in the Euclidean space $R^3$, supplemented by the far-field conditions for the phase variables, specifically: $\varrho \to 0,\vartheta \to {\vartheta }_{\infty },u\to 0$ as $\left|x\right|\to \infty$. We study the long-time behavior of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium ${\varrho }_s=0,{\vartheta }_s={\vartheta }_{\infty },u_s=0$ 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.