Exponential mixing of the 2D stochastic Navier-Stokes dynamics
The Navier-Stokes equation on two-dimensional torus with a random force is under consideration. It is supposed that the force is white noise in time and excites only a finite number of modes. The number of excited modes depends on the viscosity \(\nu\) and grows like \(\nu^{-3}\) as \(\nu\to 0\). The authors use a Lyapunov-Schmidt type reduction to transform the original infinite-dimensional Markov process to a certain non-Markovian finite-dimensional one, and employ standard ideas of statistical mechanics to deduce the mixing properties of the dynamics. It is proved that the Markov process (describing the solution) has a unique invariant measure and is exponentially mixing in time.
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