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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • Marco Romito
  • Affiliation: Dipartimento di Matematica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italia
  • Email:
  • 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:
  • MathSciNet review: 1885650