Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

Abstract

We consider the Navier–Stokes equations in the thin 3D domain \({\mathbb{T}}_2 \times (0, \epsilon)\) , where \({\mathbb{T}}_2\) 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 \({\mathbb{T}}_2\) . 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.