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

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 $α$ is less than 1/4. In this paper we prove the existence of weak solutions for all $α$, energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for $α> 1/3$, and global regularity for $α\geq 1/2$. In addition, we prove a finite time blow-up in the case where $α<1/3$. It is remarkable that the model with $α=1/3$ 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 $α$ and becomes a strong global attractor for $α\geq 1/2$.