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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.17248 |
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Table of Contents:
- 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 Sobolevski\uı-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.