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

TL;DR

This paper investigates the long-term behavior of solutions to the Navier-Stokes-Fourier system in unbounded 3D space, proving that solutions tend to equilibrium states over time in an ergodic sense.

Contribution

It establishes the convergence of global weak solutions to equilibrium states for the NSF system on unbounded domains, highlighting the non-conservation of total momentum.

Findings

Solutions approach equilibrium in ergodic averages as time goes to infinity.

Total momentum of solutions is not conserved over time.

Weak solutions tend to zero density and velocity, with temperature approaching a constant.

Abstract

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: $\rho \to 0,\ \vartheta \to \vartheta_\infty, \ u \to 0$ as $\ |x| \to \infty$. 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 $\rho_s = 0,\ \vartheta_s = \vartheta_\infty,\ u_s = 0$ 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.