Exponential mixing of the 2D stochastic Navier-Stokes dynamics

We consider the Navier-Stokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity v, and grows like v(-3) when v goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time