Communications in Mathematical Physics

, Volume 94, Issue 1, pp 61–66 | Cite as

Remarks on the breakdown of smooth solutions for the 3-D Euler equations

  • J. T. Beale
  • T. Kato
  • A. Majda


The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.


Neural Network Statistical Physic Vorticity Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brezis, H., Gallouet, T.: Nonlinear Schrödinger evolution equations. J. Nonlinear Anal.4, 677–681 (1980)Google Scholar
  2. 2.
    Chorin, A.: Estimates of intermittency, spectra, and blow-up in developed turbulence. Commun. Pure Appl. Math.34, 853–866 (1981)Google Scholar
  3. 3.
    Chorin, A.: The evolution of a turbulent vortex. Commun. Math. Phys.83, 517–535 (1982)Google Scholar
  4. 4.
    Frisch, U., Sulem, P. L., Nelkin, M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech.87, 719–736 (1978)Google Scholar
  5. 5.
    Kato, T.: Quasi-linear equations of evolution with applications to partial differential equations. In: Lecture Notes in Mathematics, Vol.448, Berlin, Heidelberg, New York: Springer 25–70 (1975)Google Scholar
  6. 6.
    Kato, T.: Nonstationary flows of viscous and ideal fluids in ℝ3. J. Funct. Anal.9, 296–305 (1972)Google Scholar
  7. 7.
    Kato, T.: Remarks on the Euler and Navier-Stokes equations in ℝ2. (to appear)Google Scholar
  8. 8.
    Klainerman, S., Majda, A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math.34, 481–524 (1981)Google Scholar
  9. 9.
    Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. In: Applied Math. Sciences Series, Vol. Berlin, Heidelberg, New York, Tokyo: Springer 1983Google Scholar
  10. 10.
    Marsden, J., Ebin, D., Fischer, A. E.: Diffeomorphism groups, hydrodynamics and relativity. Proceedings of the thirteenth biennial seminar of the Canadian Math. Congress, Vanstone, J. R. (ed.,) Montreal, 1972Google Scholar
  11. 11.
    Morf, R., Orszag, S., Frisch, U.: Spontaneous singularity in three-dimensional incompressible flow. Phys. Rev. Lett.44, 572–575 (1980)Google Scholar
  12. 12.
    Moser, J.: A rapidly convergent iteration method and nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa20, 265–315 (1966)Google Scholar
  13. 13.
    Temam, R.: Local existence ofC solutions of the Euler equations of incompressible perfect fluids. In: Lecture Notes in Mathematics, Vol.565, Berlin, Heidelberg, New York: Springer 184–194 (1976)Google Scholar
  14. 14.
    Klainerman, S., Ponce, G.: Global, small amplitude solutions to nonlinear wave equations. Commun. Pure Appl. Math.36, (1983)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • J. T. Beale
    • 1
  • T. Kato
    • 2
  • A. Majda
    • 2
  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations