MATHEMATICA BOHEMICA, Vol. 127, No. 2, pp. 211-218 (2002)

Probabilistic analysis of singularities for
the 3D Navier-Stokes equations

Franco Flandoli, Marco Romito

Franco Flandoli, Dipartimento di Matematica Applicata, Universita di Pisa, Via Bonanno 25/b, 56126 Pisa, Italy, e-mail: flandoli@dma.unipi.it; Marco Romito, Dipartimento di Matematica, Universita di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy, e-mail: romito@math.unifi.it

Abstract: 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.

Keywords: singularities, Navier-Stokes equations, Brownian motion, stationary solutions

Classification (MSC2000): 76D06, 35Q30, 60H15

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