Igor Chueshov and Sergei Kuksin
On the random kick-forced 3D Navier-Stokes equations in a thin 
domain
(495K, pdf)

ABSTRACT.  We consider the Navier-Stokes equations in the thin 3D domain $T^2\times(0,\epsilon)$, where $T^2$ is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick-force. We establish that, firstly, when $\epsilon\ll1$ the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as $\epsilon\to0$) 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.