GLOBAL WELL-POSEDNESS, SCATTERING AND BLOW UP FOR THE ENERGY-CRITICAL, FOCUSING, NON-LINEAR SCHRÖDINGER AND WAVE EQUATIONS by

In these lectures I will discuss recent joint works with F. Merle. In them we have developed an approach to the study of non-linear critical problems of dispersive type. The issues studied are global well-posedness and scattering. The approach works for both focusing and defocusing problems, but in these lectures I will concentrate on two focusing problems. The approach proceeds in steps, some of which are general and hence apply to “all problems” and some which are specific to each particular problem. The concrete problems to be discussed here are the energy critical, focusing non-linear Schrödinger equation and wave equation. I will try to separate both kinds of arguments in the exposition. I will start out by discussing (NLS). Consider thus the Cauchy problem for the Ḣ1 critical non-linear Schrödinger equation