New a priori estimate for stochastic 2D Navier-Stokes equation with applications to invariant measure
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| Format: | Preprint |
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2024
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| _version_ | 1866912368898867200 |
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| author | Ferrari, Matteo |
| author_facet | Ferrari, Matteo |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New a priori estimate for stochastic 2D Navier-Stokes equation with applications to invariant measure Ferrari, Matteo Probability Analysis of PDEs 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. |
| title | New a priori estimate for stochastic 2D Navier-Stokes equation with applications to invariant measure |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2401.17248 |