I. Chueshov, S. Kuksin
Stochastic 3D Navier-Stokes equations in a thin domain and its $\alpha$-approximation
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ABSTRACT. In the thin domain $O_\e=T^2\times (0,\e)$,
where $T^2$ is a two-dimensional torus, we consider the
3D Navier-Stokes 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~Navier-Stokes equations in $O_\e$, when $\e\ll 1$.