 07130 I. Chueshov, S. Kuksin
 Stochastic 3D NavierStokes equations in a thin domain and its $\alpha$approximation
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May 25, 07

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Abstract. In the thin domain $O_\e=T^2\times (0,\e)$,
where $T^2$ is a twodimensional torus, we consider the
3D NavierStokes equations, perturbed by a white in time random force,
and the Leray $\alpha$approximation for this system.
We study ergodic properties of these models and their connection with the corresponding $2D$ models in the limit $\e\to0$. In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure $\mu$ comprises asymptotical in time statistical properties of solutions for the 3D~NavierStokes equations in $O_\e$, when $\e\ll 1$.
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