Regularity and Blow up for Active Scalars

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 dissi- pation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.