Dynamics of stochastic 2 D Navier–Stokes equations

Abstract

In this paper, we study the dynamics of a two-dimensional stochastic Navier–Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2 D Navier–Stokes equation generate a perfect and locally compacting C 1 , 1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier–Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.