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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Finite time blow-up for a dyadic model of the Euler equations
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by Nets Hawk Katz and Nataša Pavlović PDF
Trans. Amer. Math. Soc. 357 (2005), 695-708 Request permission

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
  • 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, DOI 10.1007/BF01212349
  • 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, DOI 10.1002/cpa.3160350604
  • M. Cannone: Harmonic analysis tools for solving the incompressible Navier-Stokes equations, To appear in Handbook of Mathematical Fluid Dynamics 3 (2004).
  • C. Fefferman: Existence and smoothness of the Navier-Stokes equation, http://www. claymath.org, (2000).
  • S. Friedlander, and N. Pavlović: Blow up in a three-dimensional vector model for the Euler equations, Preprint, (2003).
  • E. B. Gledzer: System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl. 18, No. 4 (1973), 216–217.
  • 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), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
  • 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, DOI 10.1007/s00039-002-8250-z
  • F. Nazarov: Personal communication, (2001).
  • 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, DOI 10.1143/PTP.81.329
  • R. Temam, Local existence of $C^{\infty }$ solutions of the Euler equations of incompressible perfect fluids, Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975) Lecture Notes in Math., Vol. 565, Springer, Berlin, 1976, pp. 184–194. MR 0467033
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Additional Information
  • Nets Hawk Katz
  • Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130
  • MR Author ID: 610432
  • Email: nets@math.wustl.edu
  • Nataša Pavlović
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • MR Author ID: 697878
  • Email: natasa@math.princeton.edu
  • Received by editor(s): July 25, 2003
  • Published electronically: March 12, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 695-708
  • MSC (2000): Primary 35Q30, 35Q35, 76B03
  • DOI: https://doi.org/10.1090/S0002-9947-04-03532-9
  • MathSciNet review: 2095627