Regularity and blow up for active scalars

Details Regularity and blow up for active scalars Regularity and blow up for active scalars

2010-09-02

arxiv.org/abs/1009.0540v1

Mathematics

Analysis of PDEs

33 pages

Scientific paper

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.

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