Finite time blow-up for a dyadic model of the Euler equations

@article{Katz2004FiniteTB,
  title={Finite time blow-up for a dyadic model of the Euler equations},
  author={Nets Hawk Katz and Nata{\vs}a Pavlovi{\'c}},
  journal={Transactions of the American Mathematical Society},
  year={2004},
  volume={357},
  pages={695-708}
}
  • N. KatzN. Pavlović
  • Published 12 March 2004
  • Mathematics
  • Transactions of the American Mathematical Society
We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small. 
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References

SHOWING 1-10 OF 18 REFERENCES

Blowup in a three‐dimensional vector model for the Euler equations

We present a three‐dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations. This model has structural similarities with the Euler…

Remarks on the breakdown of smooth solutions for the 3-D Euler equations

The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially…

Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations

A cheap Caffarelli—Kohn—Nirenberg inequality for the Navier—Stokes equation with hyper-dissipation

Abstract. We prove that for the Navier—Stokes equation with dissipation $ (-\Delta)^\alpha $ where 1 < α < 5 /4, and smooth initial data, the Hausdorff dimension of the singular set at time of first…

System of hydrodynamic type admitting two quadratic integrals of motion

A pneumatic braking system includes two first pneumatic braking circuits of a motor vehicle and a second pneumatic braking circuit of a trailer towed by the vehicle and which second braking circuit…

Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations

Con ten t s Introduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 1. Prel iminaries . . . . . . . . . . . . . . . . . . . . . . .…

Temporal Intermittency in the Energy Cascade Process and Local Lyapunov Analysis in Fully-Developed Model Turbulence

Le processus de cascade d'energie est etudie numeriquement sur un modele scalaire de la turbulence tridimensionnelle entierement developpee. L'energie se propage a travers l'intervalle inerte par…

Partial regularity of suitable weak solutions of the navier‐stokes equations

Quasi-linear equations of evolution, with applications to partial differential equations

Local existence of ℓ∞ solutions of the euler equations of incompressible perfect fluids