Seminar

This course will be an introduction to the problem of singularity formation in nonlinear evolution equations, both dispersive and parabolic, which has seen spectacular developments for the past ten years. We will start with reviewing some of the basis concepts in the field (local well posedness, global existence, scattering and blow up) and in particular the role of solitary waves in the qualitative description of the flow. We will then introduce some of the basic tools for the construction of blow up solutions. We will in particular detail the construction of minimal blow up elements and explain their role in describing the flow near solitary waves. This will give us the first keys to understand the differences between type I and type II blow up which we will illustrate on various problems, We will for example give partial answers to a very simple question: how fast do ice balls melt? Time permitting, we will conclude the class towards energy super critical models.

The course will be completely self contained with the prior knowledge of basis functional analysis (Hilbert and Banach spaces), PDE’s (well posedness, energy estimates) and dynamical systems (perturbation theory), and will start with a crash course on the Sobolev H^s(\R^d) space and its embedding properties.