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 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 .

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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.