# Stationary solutions for stochastic damped Navier-Stokes equations in $\mathbb R^d$

## TL;DR

This paper investigates the existence of stationary solutions and invariant measures for stochastic damped Navier-Stokes equations in two and three dimensions, addressing challenges posed by irregular noise covariance.

## Contribution

It establishes the existence of invariant measures in 2D and stationary solutions in 3D for these equations under less regular noise conditions.

## Key findings

- Existence of invariant measure in 2D
- Existence of stationary solution in 3D
- Addresses irregular noise covariance challenges

## Abstract

We consider the stochastic damped Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$), assuming as in our previous work [4] that the covariance of the noise is not too regular, so It\^o calculus cannot be applied in the space of finite energy vector fields. We prove the existence of an invariant measure when $d=2$ and of a stationary solution when $d=3$.