Global well-posedness and scattering for the energy-critical nonlinear

We obtain global well-posedness, scattering, and global L10t,x spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equa-tion in R1+3, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain [4], but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant (first used in [12], [13]). The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of L2 mass in fre-quency space, rules out the possibility of energy concentration