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

Matteo Ferrari
Jan 2024
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摘要原文
The paper deals with the two-dimensional stochastic incompressible Navier-Stokes equation set in a bounded domain with Dirichlet boundary conditions. We consider an additive noise in the form of a cylindrical Wiener process regularized by a term $A^{-\gamma}$, where $A$ is the Stokes operator, and $\gamma\in(1/4,1/2)$. We prove uniqueness, ergodicity, and a strong mixing property for the invariant measure of the Markov semigroup. While previous results require $\gamma > 3/8$, we uncover the range $\gamma \in (1/4, 3/8]$ by adapting the so called Sobolevskii-Kato-Fujita approach to stochastic Navier-Stokes equations. By means of the mild formulation, this method gives a new \textit{a priori} estimate for the trajectories of the solution, which entails H\"older continuity in time and regularity $D\big(A^{\gamma'}\big)$ in space, where $\gamma'<\gamma$.
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