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singularities; Navier-Stokes equations; Brownian motion; stationary solutions
The classical result on singularities for the 3D Navier-Stokes equations says that the $1$-dimensional Hausdorff measure of the set of singular points is zero. For a stochastic version of the equation, new results are proved. For statistically stationary solutions, at any given time $t$, with probability one the set of singular points is empty. The same result is true for a.e. initial condition with respect to a measure related to the stationary solution, and if the noise is sufficiently non degenerate the support of such measure is the full energy space.
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