747 research outputs found
Regularity and blow up for active scalars
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.Comment: 33 page
Regularity of H\"older continuous solutions of the supercritical quasi-geostrophic equation
We present a regularity result for weak solutions of the 2D quasi-geostrophic
equation with supercritical () dissipation : If
a Leray-Hopf weak solution is H\"{o}lder continuous with on the time interval , then it is actually a classical solution on
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