| We study the Navier-Stokes equation forced by white in time stochastic noise constructed as the weighted, infinite sum of standard Brownian motions. Using martingale and function-analytic techniques, we prove results characterizing the almost sure and expected time evolution of the energy, enstrophy, and higher Sobolev norms. In particular, we show how various norms regularize in time and obtain control over the time needed for their time averages to approach their mean values. Using these results, we make precise the idea that the longterm behavior can be described by a system with only a finite number of Fourier modes. This result derives from estimates on the contractive nature of the dynamics. At high enough viscosity these estimates are used to prove the existence of a distinguished, stationaxy solution, depending only on the noise realization, to which other solutions, staring from arbitrary initial data, are attracted. We discuss how this relates to the idea of a random attractor for a random dynamical system. We also prove that if the forcing is sufficiently smooth then the spatial Fourier modes are exponentially decaying as a function of wave number. |
