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

TL;DR

The authors address the stochastic 2D Navier–Stokes equation in a bounded domain with Dirichlet boundary conditions driven by additive noise $G\,dW_t$, under a degenerate noise regime ${\rm Ran}(G)\subset D(A^{1/4+\varepsilon})$. They adapt the Sobolevskii–Kato–Fujita framework to obtain new a priori estimates and prove a path regularity result $u\in C([0,T];D(A^{\gamma}))$ for $\gamma<1/4+\varepsilon$, which then yields irreducibility and strong Feller properties of the associated Markov semigroup. Using these, they establish the existence, uniqueness, ergodicity, and strong mixing of the invariant measure for $\gamma\in(1/4,3/8]$, extending prior results that required $\gamma>3/8$. The work also shows concentration of the invariant measure on $D(A^{\gamma})$ and provides a robust variational approach combining finite-dimensional approximations and stochastic convolution analysis. Overall, the paper advances the understanding of stochastic NS dynamics under degenerate noise and strengthens the statistical description via invariant measures and mixing properties.

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 $A^γ$, $A$ being the Stokes operator. While it is known that existence of invariant measure holds for $γ>1/4$, previous results show its uniqueness only for $γ> 3/8$. We fill this gap and prove uniqueness and strong mixing property in the range $γ\in (1/4, 3/8]$ by adapting the so-called Sobolevskiĭ-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.

Theorems & Definitions (54)

• Remark 1.1
• Lemma 2.1: Interpolation inequality
• Lemma 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Example 2.5
• Definition 2.6
• Theorem 2.7
• proof