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On some dyadic models of the Euler equations

Fabian Waleffe

Abstract

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the $H^{3/2+ε}$ Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in $H^α$, for $α\ge 1/2$, but its dyadic restriction is even more singular, exhibiting blow-up for any $α> 0$. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in $H^{3/2+ε}$. Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the $H^α$ norm, with $α> 0$, is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.

On some dyadic models of the Euler equations

Abstract

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in , for , but its dyadic restriction is even more singular, exhibiting blow-up for any . Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in . Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the norm, with , is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.

Paper Structure

This paper contains 6 sections, 2 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Dyadic Inviscid Burgers equation
  3. Reduction of the Katz-Pavlovic model
  4. Friedlander-Pavlovic Model
  5. Finite-time blow-up in $H^{\alpha}$
  6. Concluding remarks

Key Result

Lemma 1

For some $q$ with $\lambda^{-2} < q < 1$, and for some $j$ sufficiently large, if then there is $\rho$ with $\left(\lambda^2 q \right)^{-1/2} < \rho <1$, and $t$ in $[t_0, t_0+\rho^j]$, such that

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof