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 |