New a priori estimate for stochastic 2D Navier-Stokes equation with applications to invariant measure

Abstract

The paper deals with the stochastic two-dimensional Navier-Stokes equation for incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $G\, dW$, 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 Sobolevski\u{\i}-Kato-Fujita approach to the stochastic N-S equations. This method provides new \textit{a priori} estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.