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.