On the random kick-forced 3D Navier-Stokes equations in a thin domain

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