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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 $ H^{3/2+\epsilon}$ 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 $ H^{\alpha}$, for $ \alpha \ge 1/2$, but its dyadic restriction is even more singular, exhibiting blow-up for any $ \alpha >0$. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in $ H^{3/2+\epsilon}$. Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the $ H^{\alpha}$ norm, for any $ \alpha >0$, 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

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.

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