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

In the thin domain O ε = T 2 × ( 0 , ε ) , 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 α -approximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D models in the limit ε → 0 . In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure μ comprises asymptotical in time statistical properties of solutions for the 3D Navier–Stokes equations in O ε , when ε ≪ 1 .