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Regularity and blow up for active scalars. (English) Zbl 1194.35490
Summary: 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.

MSC:
35R11 Fractional partial differential equations
35Q35 PDEs in connection with fluid mechanics
76U05 General theory of rotating fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B50 Maximum principles in context of PDEs
35B44 Blow-up in context of PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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