On some dyadic models of the Euler equations

Author:
Fabian Waleffe

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2913-2922

MSC (2000):
Primary 35Q30, 35Q35, 76B03

Published electronically:
April 11, 2006

MathSciNet review:
2231615

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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 a dyadic model of the inviscid Burgers equation. 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, for any , is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.

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Additional Information

**Fabian Waleffe**

Affiliation:
Departments of Mathematics and Engineering Physics, University of Wisconsin, Madison, Wisconsin 53706

Email:
waleffe@math.wisc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08293-1

Keywords:
Euler equations,
Burgers equation,
Navier-Stokes equations,
finite time blow-up

Received by editor(s):
October 8, 2004

Received by editor(s) in revised form:
April 21, 2005

Published electronically:
April 11, 2006

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.