Exponential Mixing of the 2D Stochastic Navier-Stokes Dynamics

Abstract

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 $ν$, and grows like $ν^{-3}$ when $ν$ goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time.