Analysis of a Candidate Flow For Hydrodynamic Blowup

  • Richard B. Pelz
Part of the NATO Science Series book series (NAII, volume 47)

Abstract

An octahedral, vortical (divergence-free) flow is defined, and reasons are given for why it is a candidate for self-similar, point-collapse and blowup under evolution of the Euler equations. Results from a vortex filament model of such flows are then reviewed and the data subsequently analyzed. The Gauss map, a mapping of the vorticity tangent field, is examined. A Leray-Beltrami flow, defined as a force-free flow in the Leray collapse frame, is shown to develop in the inner region. Finally, the vorticity is found to scale as the inverse square of the distance from the origin, the center of the point collapse. A discussion follows on the ramification of these findings to possible blowup in the Navier-Stokes equations.

Keywords

Clay Vortex Vorticity Elon 

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References

  1. 1.
    Boratav, O.N. & Pelz, R.B. (1994) Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. Phys. Fluids 6, 2757–2784.ADSMATHCrossRefGoogle Scholar
  2. 2.
    Caffarelli, L., Kohn, R.V. & Nirenberg, L. (1982) Partial regularity of suitable weak solution of the Navier-Stokes equations. Comm. Pure Appl. Math 35, 771–831.MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Constantin, P. (2000) The Euler equations and nonlocal conservative Riccati equations. Int. Math. Res. Notices 9, 455–465.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Constantin, P, Lax, P.D. & Majda, A. (1996) A simple one-dimensional model of the three-dimensional vorticity equation. Comm. Pure Appl. Math 38, 715–724.MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Constantin, P., Majda, A., & Tabac, E. (1994) Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533.MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Cordoba, D. (1998) Non-existence of simple hyperbolic blow-up for the quasigeostrophic equation. Annals of Mathematics 148, 1135–1152.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fefferman, C. (2000) Existence and smoothness of the Navier-Stokes equations. www.claymath.org/prizeproblems/navierstokes.htm.
  8. 8.
    Grauer, R., Marliani, C. & Germaschewski, K. (1998) Adaptive mesh refinement for singular solutions of the incompressible Euler equations. Phys. Rev. Letts. 80(19), 4177–4180.ADSCrossRefGoogle Scholar
  9. 9.
    Greene, J.M. & Pelz, R.B. (2000) Stability of postulated, self-similar, hydrodynamic blowup solutions. Phys. Rev. E 62, 7982.MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Kerr, R.M. (1993) Evidence for a singularity in the three-dimensional Euler equations. Phys. Fluids 6, 1725–1746.MathSciNetADSGoogle Scholar
  11. 11.
    Kettle, S.F.A. (1985) Symmetry and Structure. Wiley.Google Scholar
  12. 12.
    Leray, J. (1934) Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 193–248.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Malham, S.J.A. (2000) Collapse of a class of three-dimensional Euler vortices. Proc. R. Soc. Lond. A 456, 2823–2833.MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Moffatt, H.K. (2000) The interaction of skewed vortex pairs: a model for blow-up of the Navier-Stokes equations. J. Fluid Mech. 409, 51–68.MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Nečas, J., Ru̇žička, M. & Šverák, V. (1996) On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math. 176, 283–294.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ohkitani K. & Gibbon, J.D. (2000) Numerical study of singularity formation in a class of Euler and Navier-Stokes flows. Phys. Fluids 12, 3181–3194.MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Pelz, R.B. (1997) Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55, 1617–1626.ADSCrossRefGoogle Scholar
  18. 18.
    Pelz, R.B. (2001) Symmetry and the hydrodynamic blowup problem. J. Fluid Mech., in press.Google Scholar
  19. 19.
    Pelz, R.B. & Gulak, Y. (1997) Evidence for a real-time singularity in hydrodynamics from time series analysis. Phys. Rev. Lett. 79(25), 4998–5001.ADSCrossRefGoogle Scholar
  20. 20.
    Plechac, P & Šverák, V. (2001) On self-similar singular solutions of the complex Ginzburg-Landau equation. Comm. Pure Appl. Math., in press.Google Scholar
  21. 21.
    Scheffer, V. (1977) Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55, 97–112.MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Tsai, T.-P. (1998) On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rat. Mech. Anal. 143, 29–51.MATHCrossRefGoogle Scholar
  23. 23.
    Wherrett, B.S. (1986) Group Theory for Atoms, Molecules and Solids. Prentice Hall International.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Richard B. Pelz
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringRutgers UniversityPiscatawayUSA

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