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

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).