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active scalars

Mathematical Modelling of Natural Phenomena (1)

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Regularity and Blow up for Active Scalars
A. Kiselev
Journal:

Mathematical Modelling of Natural Phenomena / Volume 5 / Issue 4 / 2010

Published online by Cambridge University Press:

12 May 2010, pp. 225-255

Print publication:

2010
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We review some recent results for a class of fluid mechanics equations called activescalars, with fractional dissipation. Our main examples are the surface quasi-geostrophicequation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discussnonlocal maximum principle methods which allow to prove existence of global regularsolutions for the critical dissipation. We also recall what is known about the possibilityof finite time blow up in the supercritical regime.