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

摘要

We consider the Navier-Stokes equations in the thin 3D domain T-2 x (0,epsilon), where T-2 is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when epsilon 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as epsilon -> 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 epsilon, statistical properties of 3D solutions in thin 3D domains.