Enstropy Generation and Regularity of Solutions to the 3D Navier-Stokes Equations

  • Charles R. Doering
  • Lu Lu
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 4)


The question of existence of smooth solutions to the 3D Navier-Stokes equations is an outstanding open problem in applied mathematics and theoretical physics. It is known that solutions remain smooth as long as the enstrophy remains finite, but it is not known whether or not the enstrophy may diverge to infinity at some finite time. In this paper we report that the state-of-the-art mathematical estimates on the growth rate of enstrophy—estimates that do not rule out the existence of a finite-time singularity—are sharp and cannot be improved.


vorticity enstrophy Navier-Stokes regularity 


  1. 1.
    Doering CR, Gibbon JD (1995) Applied Analysis of the Navier‐Stokes Equations, Cambridge University Press, CambridgeMATHGoogle Scholar
  2. 2.
  3. 3.
    Lu L (2006) Bounds on the enstrophy growth rate for solutions of the 3D Navier‐ Stokes equations, PhD Dissertation, University of MichiganGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Charles R. Doering
    • 1
    • 2
  • Lu Lu
    • 1
    • 3
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor
  2. 2.Department of PhysicsUniversity of MichiganAnn Arbor
  3. 3.Wachovia InvestmentsNew York

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