Existence of ergodic invariant measures for 3D Navier-Stokes eqs with superlinear noise

Abstract: We consider the Navier-Stokes equations in a smooth bounded domain $D\subset \mathbb R<sup>3$. It has been proved in arXiv:2609.08284 that these equations have a unique solution - globally defined in time - when a superlinear noise is introduced and the initial velocity belongs to the space $V$ of divergence-free $H^{1_0(D;\mathbb} R<sup>3)$-vector fields. Here we show that the same result holds when there is also an additive noise and a deterministic forcing term. Moreover, when the deterministic forcing term is time-independent we prove the existence of at least one invariant measure for this stochastic equation. This is obtained by the Krylov-Bogoliubov method, based on tightness and Feller property in $V$. Finally, we show that the set of invariant measures contains one ergodic invariant measure.