The Small Scales of the Stochastic Navier Stokes Equations under Rough Forcing

Abstract

We prove that the small scale structures of the stochastically forced Navier-Stokes equations approach those of the naturally associated Ornstein-Uhlenbeck process as the scales get smaller. Precisely, we prove that the rescaled k-th spatial Fourier mode converges weakly on path space to an associated Ornstein-Uhlenbeck process as |k| --> infty . In addition, we prove that the Navier-Stokes equations and the naturally associated Ornstein-Uhlenbeck process induce equivalent transition densities if the viscosity is replaced with sufficient hyperviscosity. This gives a simple proof of unique ergodicity for the hyperviscous Navier-Stokes system. We show how different strengthened hyperviscosity produce varying levels of equivalence.

Accepted for publication by the Journal of Statistical Physics