Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
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
Full-text PDF Free Access

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 3-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR 763762 (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 (1982), no. 6, 771–831. MR 673830 (84m:35097), http://dx.doi.org/10.1002/cpa.3160350604
  • 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. Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Springer, Berlin, 1975, pp. 25–70. Lecture Notes in Math., Vol. 448. MR 0407477 (53 #11252)
  • 8. N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal. 12 (2002), no. 2, 355–379. MR 1911664 (2003e:35243), http://dx.doi.org/10.1007/s00039-002-8250-z
  • 9. F. Nazarov: Personal communication, (2001).
  • 10. Koji Ohkitani and Michio Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model turbulence, Progr. Theoret. Phys. 81 (1989), no. 2, 329–341. MR 997440 (90j:76065), http://dx.doi.org/10.1143/PTP.81.329
  • 11. R. Temam, Local existence of 𝐶^{∞} solutions of the Euler equations of incompressible perfect fluids, Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975) Springer, Berlin, 1976, pp. 184–194. Lecture Notes in Math., Vol. 565. MR 0467033 (57 #6902)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35Q30, 35Q35, 76B03

Retrieve articles in all journals with MSC (2000): 35Q30, 35Q35, 76B03


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