Random kick-forced 3D Navier-Stokes equations in a thin domain

We consider the Navier-Stokes equations in the thin 3D domain ² × (0, e), where ² is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when e ? 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as e ? 0) to a unique stationary measure for the Navier-Stokes equation on ². Thus, the 2D Navier-Stokes equations on surfaces describe asymptotic in time, and limiting in e, statistical properties of 3D solutions in thin 3D domains. © 2007 Springer-Verlag.