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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On some dyadic models of the Euler equations
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by Fabian Waleffe PDF
Proc. Amer. Math. Soc. 134 (2006), 2913-2922 Request permission

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.
References
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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
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2913-2922
  • MSC (2000): Primary 35Q30, 35Q35, 76B03
  • DOI: https://doi.org/10.1090/S0002-9939-06-08293-1
  • MathSciNet review: 2231615