Recent development in the theory of linear partial differential equations.
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...
Résonances en limite semi-classique
Revisiting the method of sub- and supersolutions for nonlinear elliptic problems.