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

Alexey Cheskidov

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 $α\geq 1/2$.

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 1/4. In this paper we prove the existence of weak solutions for all , energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for , and global regularity for . In addition, we prove a finite time blow-up in the case where . It is remarkable that the model with 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 .

Paper Structure

This paper contains 6 sections, 11 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Dyadic model
  3. Functional setting
  4. A priori estimates and existence of weak and strong solutions
  5. Blow-up in finite time
  6. Global attractor

Key Result

Theorem 4.1

For every $u^0 \in H$ and $g \in H$, there exists a solution of model with $u(0)=u^0$. Moreover, the energy inequality holds for all $0 \leq t_0 \leq t$, $t_0$ a.e. in $[0,\infty)$.

Theorems & Definitions (21)

  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • Theorem 4.4
  • Lemma 5.1
  • proof
  • ...and 11 more