On singularity formation for energy super critical problems

The 4th Clay Millenium problem has a very simple formulation: may viscous incompressible fluids form a singularity in finite time? The answer is no in dimension two as proved by Leray in 1932, but the three dimensional problem is out of reach. 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. These possibly highly unstable dynamics are difficult to detect numerically, and there is still little theoretical understanding of the underlying mechanisms. I will in this first lecture introduce model canonical problems and illustrate the concept of energy critical versus super critical models through the exposition of classical open problems in the field.