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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 is less than . In this paper we prove the existence of weak solutions for all , energy inequality for every weak solution with nonnegative initial data starting from any time, local regularity for , and global regularity for . In addition, we prove a finite time blow-up in the case where . It is remarkable that the model with 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 and becomes a strong global attractor for .

**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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35Q30,
76D03,
76D05

Retrieve articles in all journals with MSC (2000): 35Q30, 76D03, 76D05

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.