  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 $\alpha \geq 1/2$.
