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

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