New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

The paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma >1/4$$ , previous results show its uniqueness only for $$\gamma > 3/8$$ . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.