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
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