# Stochastic tamed Navier-Stokes equations on $\mathbb{R}^3$:existence, uniqueness of solution and existence of an invariant measure

## TL;DR

This paper extends the analysis of stochastic tamed 3D Navier-Stokes equations to the whole space, proving existence, uniqueness, and invariant measures with a simplified, self-contained approach.

## Contribution

It improves previous results by generalizing estimates and establishing invariant measures for the stochastic tamed Navier-Stokes equations on ^3.

## Key findings

- Existence and uniqueness of solutions on ^3.
- Existence of an invariant measure for the system.
- Generalization of L^4-norm estimates from torus to ^3.

## Abstract

R\"ockner and Zhang in [27] proved the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space and for the periodic boundary case using a result from [31]. In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $L^4-$norm of the solution from the torus to $\mathbb{R}^3$, see Lemma 5.1 and thus establish the existence of an invariant measure on $\mathbb{R}^3$ for a time-homogeneous damped tamed 3D Navier-Stokes equation, given by (6.1).