On the long time behavior of compressible fluid flows excited by random forcing

TL;DR

This paper investigates the long-term behavior of stochastic compressible Navier-Stokes equations, demonstrating energy bounds, asymptotic compactness, and the existence of stationary and ergodic solutions in 2D and 3D.

Contribution

It establishes universal energy bounds and existence of stationary and ergodic solutions for stochastic compressible fluid flows, advancing understanding of their long-term dynamics.

Findings

Kinetic energy is universally and asymptotically bounded.

Time shifts of solutions are asymptotically compact.

Existence of stationary and ergodic solutions.

Abstract

We are concerned with the long time behavior of the stochastic Navier--Stokes system for compressible fluids in dimension two and three. In this setting, the part of the phase space occupied by the solution depends sensitively on the choice of the initial state. Our main results are threefold. (i) The kinetic energy of a solution is universally and asymptotically bounded, independent of the initial datum. (ii) Time shifts of a solution with initially controlled energy are asymptotically compact and generate an entire solution defined for all $t\in R$. (iii) Every solution with initially controlled energy generates a stationary solution and even an ergodic stationary solution on the closure of the convex hull of its $\omega$--limit set.