A powerful strategy for establishing regularity in a non-perturbative setting is to combine bubbling analysis with a rigidity argument. Assuming that a blow up occurs, one first magnifies near the blow up and identifies a nontrivial profile (bubbling analysis). Then one starts proving special properties of this profile, to the point such an object cannot exist and one arrives at a contradiction (rigidity). 

 

The goal of this talk is to describe a bubbling analysis scheme for energy critical wave equations introduced by J. Sterbenz and D. Tataru (in the context of the critical wave map equation), which is robust yet highly effective. I will begin by briefly describing my recent work with D. Tataru on the energy critical Maxwell-Klein-Gordon (MKG) equation, where we followed this scheme to establish global well-posedness and scattering for arbitrary finite energy data. I will then illustrate the scheme by considering a simple model equation, namely the cubic NLW on the 4+1-dimensional Minkowski space, which resembles MKG but is largely free of technical difficulties.