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.

Partial regularity for the stochastic Navier-Stokes equations / F. FLANDOLI; M. ROMITO. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 354, no. 6:(2002), pp. 2207-2241. [10.1090/S0002-9947-02-02975-6]

Partial regularity for the stochastic Navier-Stokes equations

ROMITO, MARCO
2002

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.
2002
354, no. 6
2207
2241
F. FLANDOLI; M. ROMITO
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/3043
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