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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...
2000 Mathematics Subject Classification: 35K55, 35K60.We investigate the blow-up of the solutions to a nonlinear parabolic system with Robin boundary conditions and time dependent coefficients. We derive sufficient conditions on the nonlinearities and the initial data in order to obtain explicit lower and upper bounds for the blow up time t*.
We study the Cauchy problem for the MHD system, and provide two regularity conditions involving horizontal components (or their gradients) in Besov spaces. This improves previous results.
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