Singularities for Waves And Turbulent Flows (SWAT)

The 4th Clay Millennium problem has a very simple formulation: may viscous incompressible fluids form a singularity in finite time? The answer is no for two dimensional flows as proved by Leray in 1934, but is out of reach in three dimensions where the problem becomes super critical. More generally, the description of energy concentration mechanisms for non linear waves possibly leading to the formation of singularities is a canonical mathematical problem with deep physical roots in electromagnetism, non linear optics, astrophysics and fluid mechanics. Spectacular progress have been made in the last twenty years on the description of blow up dynamics for non linear waves models with focusing non linearities, but it is only in 2020 that the applicability of this approach to fluid mechanics has been demonstrated with the first description of shock formation for the three dimensional compressible Navier Stokes equations (blow up by implosion), and the parallel discovery of highly oscillatory singularities for defocusing models. The aim of the SWAT project is to bring together a highly competitive team of leading experts to address, in the continuation of these recent breakthroughs, some of the key challenges on singularity formation. A fundamental aspect of the project will be the training of young scientists and the organization of frequent attractive workshops and schools within the European Mathematical community.