# Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces

## TL;DR

This paper demonstrates the existence of stationary solutions for the stochastic compressible Navier-Stokes equations, revealing a noise-induced regularization effect that extends the class of admissible pressure laws.

## Contribution

It establishes the existence of stationary solutions for stochastic compressible Navier-Stokes equations under broad conditions, using new estimates and compactness methods.

## Key findings

- Existence of stationary solutions in Lebesgue--Sobolev spaces.
- Regularizing effect of stochastic forcing on solutions.
- Applicability to a full range of pressure constitutive relations.

## Abstract

We study the long-time behavior of solutions to a stochastically driven Navier-Stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation. The existence of stationary solutions is established in the framework of Lebesgue--Sobolev spaces pertinent to the class of weak martingale solutions. The methods are based on new global-in-time estimates and a combination of deterministic and stochastic compactness arguments. In contrast with the deterministic case, where related results were obtained only under rather restrictive constitutive assumptions for the pressure, the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory. This can be seen as a kind of regularizing effect of the noise on the global-in-time solutions.