Partial regularity for the stochastic Navier-Stokes equations |
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Authors: | Franco Flandoli Marco Romito |
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Affiliation: | Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno 25/b, 56126 Pisa, Italia ; Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italia |
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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 the set of singular points is empty. The same result holds true for every martingale solution starting from -a.e. initial condition , where 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 is supported on the whole space of initial conditions. |
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Keywords: | Navier-Stokes equations singularities partial regularity suitable weak solutions martingale solutions stationary solutions |
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