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

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- by Alexey Cheskidov PDF
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**360**(2008), 5101-5120 Request permission

## 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 $\alpha \geq 1/2$.## References

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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
- MR Author ID: 680409
- ORCID: 0000-0002-2589-2047
- Email: acheskid@umich.edu, acheskid@uchicago.edu
- Received by editor(s): January 4, 2006
- Published electronically: May 19, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2415066