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Regularity and Blow up for Active Scalars

Published online by Cambridge University Press:  12 May 2010

A. Kiselev
A. Kiselev*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, USA
*
1 E-mail: kiselev@math.wisc.edu
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Abstract

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.

Keywords

active scalarsglobal regularityfinite time blow upnonlocal maximum principle
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 225 - 255
Copyright
© EDP Sciences, 2010

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