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Partial regularity for the stochastic Navier-Stokes equations

Authors: Franco Flandoli and Marco Romito
Journal: Trans. Amer. Math. Soc. 354 (2002), 2207-2241
MSC (2000): Primary 76D05; Secondary 35A20, 35R60
Published electronically: February 14, 2002
MathSciNet review: 1885650
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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.

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Additional Information

Franco Flandoli
Affiliation: Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno 25/b, 56126 Pisa, Italia

Marco Romito
Affiliation: Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italia

Keywords: Navier-Stokes equations, singularities, partial regularity, suitable weak solutions, martingale solutions, stationary solutions
Received by editor(s): January 11, 2001
Received by editor(s) in revised form: July 21, 2001
Published electronically: February 14, 2002
Article copyright: © Copyright 2002 American Mathematical Society