The goal of this project is to pursue new methods in the mathematical analysis of non-local and non-linearpartial differential equations. For this purpose we present several physical scenarios of interest in the context of incompressible fluids, from a mathematical point of view as well as for its applications: both from the standpoint of global well-posedness, existence and uniqueness of weak solutions and as candidates for blowup. The equations we consider are the incompressible Euler equations, incompressible porous media equation and the generalized Quasi-geostrophic equation. This research will lead to a deeper understanding of the nature of the set of initial data that develops finite time singularities as well as those solutions that exist for all time for incompressible flows.

ERC-2017-ADG