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

33 pages

References in corpus (5)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
• Infinite superlinear growth of the gradient for the two-dimensional Euler equation
• Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations
• Local and Nonlocal Dispersive Turbulence