Partial regularity for the stochastic Navier-Stokes equations

Abstract:

The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time $t$ the set of singular points is empty. The same result holds true for every martingale solution starting from $\mu _0$-a.e. initial condition $u_0$, where $\mu _0$ is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure $\mu _0$ is supported on the whole space $H$ of initial conditions.