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