Channel of energy for outgoing waves and universality of blow up for wave maps
We will introduce a recently found channel of energy inequality for outgoing waves, which has been useful for semi-linear wave equations at energy criticality. Then we will explain an application of this channel of energy argument to the energy critical wave maps into the sphere. The main issue is to eliminate the so-called ``null concentration of energy". We will explain why this is an important issue in the wave maps. Combining the absence of null concentration with suitable coercive property of energy near traveling waves, we show a universality property for the blow up of wave maps with energy that are just above the co-rotational wave maps. Difficulties with extending to arbitrarily large wave maps will also be discussed. This is joint work with Duyckaerts, Kenig and Merle.