  Active scalars appear in many problems of fluid dynamics. The most common
examples of active scalar equations are 2D Euler, Burgers, and 2D surface
quasi-geostrophic equations. Many questions about regularity and properties of
solutions of these equations remain open. We develop the idea of nonlocal
maximum principle, formulating a more general criterion and providing new
applications. The most interesting application is finite time regularization of
weak solutions in the supercritical regime.
