Stochastic 3D Navier–Stokes equations in a thin domain and its α -approximation

In the thin domain Oε=T2×(0,ε)Oε=T2×(0,ε), where T2T2 is a two-dimensional torus, we consider the 3D Navier–Stokes equations, perturbed by a white in time random force, and the Leray αα-approximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D models in the limit ε→0ε→0. In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure μμ comprises asymptotical in time statistical properties of solutions for the 3D Navier–Stokes equations in OεOε, when ε≪1ε≪1.