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On Vorticity Directions near Singularities for the Navier-Stokes Flows with Infinite Energy

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Abstract

We give a geometric nonblow-up criterion on the direction of the vorticity for the three dimensional Navier-Stokes flow whose initial data is just bounded and may have infinite energy. We prove that under a restriction on behavior in time (type I condition) the solution does not blow up if the vorticity direction is uniformly continuous at the place where the vorticity magnitude is large. This improves the regularity condition for the vorticity direction first introduced by P. Constantin and C. Fefferman (1993) for finite energy weak solution. Our method is based on a simple blow-up argument which says that the situation looks like two-dimensional under continuity of the vorticity direction. We also discuss boundary value problems.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro, Tokyo, 153-8914, Japan

    Yoshikazu Giga

  2. Department of Mathematics, Osaka University, Machikaneyamacho 1-1, Toyonaka, Osaka, 560-0043, Japan

    Hideyuki Miura

Authors
  1. Yoshikazu Giga
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  2. Hideyuki Miura
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Correspondence to Hideyuki Miura.

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Communicated by P. Constantin

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Giga, Y., Miura, H. On Vorticity Directions near Singularities for the Navier-Stokes Flows with Infinite Energy. Commun. Math. Phys. 303, 289–300 (2011). https://doi.org/10.1007/s00220-011-1197-x

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  • DOI: https://doi.org/10.1007/s00220-011-1197-x

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