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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-08-04494-2
PII: S 0002-9947(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.