Math. Model. Nat. Phenom.
Volume 5, Number 4, 2010Spectral problems. Issue dedicated to the memory of M. Birman
|Page(s)||225 - 255|
|Published online||12 May 2010|
Regularity and Blow up for Active Scalars
Department of Mathematics, University of Wisconsin-Madison,
1 E-mail: email@example.com
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.
Mathematics Subject Classification: 35Q35 / 76U05
Key words: active scalars / global regularity / finite time blow up / nonlocal maximum principle
© EDP Sciences, 2010
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