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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finite time blow-up for a dyadic model of the Euler equations


Authors: Nets Hawk Katz and Natasa Pavlovic
Journal: Trans. Amer. Math. Soc. 357 (2005), 695-708
MSC (2000): Primary 35Q30, 35Q35, 76B03
Published electronically: March 12, 2004
MathSciNet review: 2095627
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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.


References [Enhancements On Off] (What's this?)

  • 1. J.T. Beale, T. Kato, and A. Majda: Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. Math. Phys. 94 , No. 1 (1984), 61-66. MR 85j:35154
  • 2. L. Caffarelli, R. Kohn, and L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35, No. 6 (1982), 771-831. MR 84m:35097
  • 3. M. Cannone: Harmonic analysis tools for solving the incompressible Navier-Stokes equations, To appear in Handbook of Mathematical Fluid Dynamics 3 (2004).
  • 4. C. Fefferman: Existence and smoothness of the Navier-Stokes equation, http://www. claymath.org, (2000).
  • 5. S. Friedlander, and N. Pavlovic: Blow up in a three-dimensional vector model for the Euler equations, Preprint, (2003).
  • 6. E. B. Gledzer: System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl. 18, No. 4 (1973), 216-217.
  • 7. Kato, T.: Quasi-linear equations of evolution with applications to partial differential equations, Lecture Notes in Mathematics 448, Berlin, Heidelberg, New York, Springer, ( 1975), 25-70. MR 53:11252
  • 8. N.H. Katz and N. Pavlovic: A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, GAFA 12 (2002), 355-379. MR 2003e:35243
  • 9. F. Nazarov: Personal communication, (2001).
  • 10. K. Ohkitani and M. Yamada: Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model turbulence, Prog. Theor. Phys. 81, No. 2 (1989), 329-341. MR 90j:76065
  • 11. R. Temam: Local existence of $C^{\infty}$ solutions of the Euler equations of incompressible perfect fluids,Lecture Notes in Mathematics 565, Berlin, Heidelberg, New York,Sp ringer, (1976), 184-195. MR 57:6902

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Additional Information

Nets Hawk Katz
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
Email: nets@math.wustl.edu

Natasa Pavlovic
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: natasa@math.princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03532-9
PII: S 0002-9947(04)03532-9
Received by editor(s): July 25, 2003
Published electronically: March 12, 2004
Article copyright: © Copyright 2004 American Mathematical Society