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Blow-up in finite time for the dyadic model of the Navier-Stokes equations


Author: Alexey Cheskidov
Journal: Trans. Amer. Math. Soc. 360 (2008), 5101-5120
MSC (2000): Primary 35Q30, 76D03, 76D05
DOI: https://doi.org/10.1090/S0002-9947-08-04494-2
Published electronically: May 19, 2008
MathSciNet review: 2415066
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Abstract: We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlović. They showed a finite time blow-up in the case where the dissipation degree $ \alpha$ is less than $ 1/4$. In this paper we prove the existence of weak solutions for all $ \alpha$, energy inequality for every weak solution with nonnegative initial data starting from any time, local regularity for $ \alpha > 1/3$, and global regularity for $ \alpha \geq 1/2$. In addition, we prove a finite time blow-up in the case where $ \alpha<1/3$. It is remarkable that the model with $ \alpha=1/3$ enjoys the same estimates on the nonlinear term as the 4D Navier-Stokes equations. Finally, we discuss a weak global attractor, which coincides with a maximal bounded invariant set for all $ \alpha$ and becomes a strong global attractor for $ \alpha \geq 1/2$.


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  • 1. J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), 475-502. Erratum: J. Nonlinear Sci. 8 (1998), 233. MR 1462276 (98j:58071a); MR 1606601 (98j:58071b)
  • 2. L. Biferale, Shell models of energy cascade in turbulence, Annu. Rev. Fluid Mech. 35 (2003), 441-468. MR 1967019 (2004b:76074)
  • 3. A. Cheskidov, Global attractors of evolutionary systems, preprint, (2006).
  • 4. A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations 231 (2006), 714-754. MR 2287904
  • 5. A. Cheskidov, S. Friedlander, and N. Pavlović, An inviscid dyadic model of turbulence: the global attractor, preprint, (2006).
  • 6. P. Constantin and C. Foias, Navier-Stokes Equation, The University of Chicago Press, Chicago, 1989. MR 972259 (90b:35190)
  • 7. P. Constantin, B. Levant, and E. Titi, Analytic study of shell models of turbulence, Phys. D. 219 (2006), 120-141. MR 2251486 (2007h:76055)
  • 8. E. I. Dinaburg and Ya. G. Sinai, A quasi-linear approximation of three-dimensional Navier-Stokes system, Moscow Math. J. 1 (2001), 381-388. MR 1877599 (2002i:76035)
  • 9. S. Friedlander and N. Pavlović, Blow up in a three-dimensional vector model for the Euler equations, Comm. Pure Appl. Math. 57 (2004), 705-725. MR 2038114 (2005c:35241)
  • 10. S. Friedlander and N. Pavlović, Remarks concerning modified Navier-Stokes equations, Discrete and Continuous Dynamical Systems 10 (2004), 269-288. MR 2026195 (2005a:76043)
  • 11. U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995. MR 1428905 (98e:76002)
  • 12. E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Soviet Phys. Dokl. 18 (1973), 216-217.
  • 13. 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), 355-379. MR 1911664 (2003e:35243)
  • 14. N. H. Katz and N. Pavlović, Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc. 357 (2005), 695-708. MR 2095627 (2005h:35284)
  • 15. A. Kiselev and A. Zlatoš, On discrete models of the Euler equation, IMRN 38 (2005), 2315-2339. MR 2180809 (2007e:35229)
  • 16. V. S. L'vov, E. Podivilov, A. Pomyalov, I. Procaccia, and D. Vandembroucq, Improved shell model of turbulence, Phys. Rev. E (3) 58 (1998), 1811-1822. MR 1637121
  • 17. A. M. Obukhov, Some general properties of equations describing the dynamics of the atmosphere, Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer. i Okeana 7 (1971), 695-704.
  • 18. K. Ohkitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model of turbulence, Progr. Theoret. Phys. 81 (1989), 329-341. MR 997440 (90j:76065)
  • 19. R. M. S. Rosa, Asymptotic regularity condition for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations 229 (2006), 257-269. MR 2265627
  • 20. R. Temam, Navier-Stokes Equations: Theory and numerical analysis, North-Holland, Amsterdam, 1984. MR 769654 (86m:76003)
  • 21. F. Waleffe, On some dyadic models of the Euler equations, Proc. Amer. Math. Soc. 134 (2006), 2913-2922. MR 2231615 (2007g:35222)

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Additional Information

Alexey Cheskidov
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: acheskid@umich.edu, acheskid@uchicago.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04494-2
Received by editor(s): January 4, 2006
Published electronically: May 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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