ABSTRACT:

I will talk about the 2D Navier-Stokes equation (NSE) on a torus, perturbed by a nondegenerate random force, treating it as a random dynamical system in a function space of divergence-free vector fields. This equation has a unique stationary measure $\mu_\nu$ ($\nu$ is the fluid's viscosity) and all solutions converge to $\mu_\nu$ in distribution as time goes to infinity. Now assume that $\nu$ is small and that the force is scaled in such a way that equation's solutions remain of order one when $\nu$ goes to zero. Then, when $\nu\to0$ along a subsequence, the stationary measure $\mu_\nu$ converges to a limiting measure $\mu$ which is an invariant measure for the free 2D Euler equation. This measure and the corresponding stationary in time $t$ process $U(t,x)$, formed by solutions of the Euler equation, describe the stationary space-periodic 2D turbulence. The pair $\mu, U$ is called the {\it Eulerian limit}. The goal of my lectures is to discuss its properties.