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

TL;DR

This paper demonstrates that at small scales, the stochastic Navier-Stokes equations behave like an Ornstein-Uhlenbeck process, and establishes conditions for ergodicity and transition density equivalence with hyperviscosity.

Contribution

It proves the convergence of small-scale Fourier modes to an Ornstein-Uhlenbeck process and shows how hyperviscosity affects ergodicity and transition densities.

Findings

Small scales of stochastic Navier-Stokes approach Ornstein-Uhlenbeck process.

Transition densities become equivalent with sufficient hyperviscosity.

Hyperviscosity influences ergodic properties of the system.

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.