Communications in Mathematical Physics

, Volume 303, Issue 2, pp 289–300 | Cite as

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

  • Yoshikazu Giga
  • Hideyuki MiuraEmail author


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.


Vorticity Mild Solution Slip Boundary Condition Regularity Criterion Liouville Theorem 
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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan
  2. 2.Department of MathematicsOsaka UniversityOsakaJapan

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