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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Partial regularity for the stochastic Navier-Stokes equations
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by Franco Flandoli and Marco Romito PDF
Trans. Amer. Math. Soc. 354 (2002), 2207-2241 Request permission

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
  • Email: flandoli@dma.unipi.it
  • Marco Romito
  • Affiliation: Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italia
  • Email: romito@math.unifi.it
  • Received by editor(s): January 11, 2001
  • Received by editor(s) in revised form: July 21, 2001
  • Published electronically: February 14, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2207-2241
  • MSC (2000): Primary 76D05; Secondary 35A20, 35R60
  • DOI: https://doi.org/10.1090/S0002-9947-02-02975-6
  • MathSciNet review: 1885650