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Stochastic 3D Navier-Stokes equations in a thin domain and its \(\alpha \)-approximation. (English) Zbl 1143.76422
Summary: In the thin domain \(\mathcal O _\varepsilon = \mathbb T^2 \times (0,\varepsilon)\), where \(\mathbb T^2\) 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 \(\epsilon \rightarrow 0\). 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 \(\mathcal O _\varepsilon\), when \(\varepsilon \ll 1\).

MSC:
76D06 Statistical solutions of Navier-Stokes and related equations
35Q30 Navier-Stokes equations
76M35 Stochastic analysis applied to problems in fluid mechanics
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