Vortex Dynamics and Singularities in the 3-D Euler Equations

  • Alain Pumir
  • Eric D. Siggia
Part of the NATO ASI Series book series (NSSB, volume 237)

Abstract

The serious study of singularities in the equations of three dimensional hydrodynamics goes back at least to 1934, and a proof still has not been given that solutions of the Navier-Stokes equations remain smooth.1 Far less is known rigorously about solutions to the Euler equations. Although on physical grounds one might expect a singularity here, the numerical evidence is unpersuasive.

Keywords

Vortex Nickel Vorticity Kato Incompressibility 

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References

  1. 1).
    J. Leray, Acta Math, 63, 193 (1934)MathSciNetMATHCrossRefGoogle Scholar
  2. L. Caffarelli, R. Kohn, L. Nirenberg, Commun. Math. Phys. 61, 41 (1978).CrossRefGoogle Scholar
  3. 2).
    G. L. Brown, A. Roshko, J. Fluid Mech. 64, 775 (1974).ADSCrossRefGoogle Scholar
  4. 3).
    H. Aref, E. D. Siggia, J. Fluid. Mech. 100, 705 (1980)ADSCrossRefGoogle Scholar
  5. H. Aref, E. D. Siggia, J. Fluid Mech. 109, 435 (1981).ADSCrossRefGoogle Scholar
  6. 4).
    M. Head and P. Bandyopadhyay, J. Fluid Mech. 107, 297 (1981).ADSCrossRefGoogle Scholar
  7. 5).
    W. Willmarth and T. Bogar, Phys. Fluids. 20, S9 (1977)ADSCrossRefGoogle Scholar
  8. S. Kline, W. Reynolds, F. Schraub, and P. Runstadler, J. Fluid Mech. 30, 741 (1967).ADSCrossRefGoogle Scholar
  9. 6).
    M. Brachet, D. Meiron, B. Nickel, S. Orszag, and U. Frisch, J. Fluid Mech. 130, 411 (1983).ADSMATHCrossRefGoogle Scholar
  10. 7).
    J. T. Beale, T. Kato and A. Majda, Commun. Math. Phys. 94, 61 (1984).MathSciNetADSMATHCrossRefGoogle Scholar
  11. 8).
    A. J. Chorin, Commun. Math Phys. 83, 517 (1982).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 9).
    A. Pumir and E. D. Siggia, Phys. Fluids, 30, 1606 (1987).ADSMATHCrossRefGoogle Scholar
  13. 10).
    A. Pumir and R. Kerr, Phys. Rev. Lett. 58, 1636 (1987).ADSCrossRefGoogle Scholar
  14. 11).
    W. Ashurst and D.Meiron, Phys. Rev. Lett. 58, 1632 (1987)ADSCrossRefGoogle Scholar
  15. 12).
    W. Ashurst, A. Kerstein, R. Kerr and C. Gibson, Phys. Fluids 30, 2343CrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Alain Pumir
    • 1
    • 2
  • Eric D. Siggia
    • 1
    • 3
  1. 1.Laboratoire de Physique StatistiqueEcole Normale SupérieureParis CedexFrance
  2. 2.SPTCEN SaclayGif sur YvetteFrance
  3. 3.LASSPCornell UniversityIthacaUSA

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