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

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

2006-01-04

arxiv.org/abs/math/0601074v2

Mathematics

Analysis of PDEs

19 pages, corrected typos

Scientific paper

We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovi\'c. 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 datum 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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