Global well-posedness and scattering in the energy space for critical nonlinear Schrödinger equation in 3D.

 

Abstract: In this talk I will present the main steps of the proof of global well-posedness, scattering and global L¹⁰ spacetime bounds for energy class solutions to the quintic defocusing Schrödinger equation in 3D. This proof was recently obtained in collaboration with J. Colliander, M. Keel, H. Takaoka and T. Tao and improves upon the results of Bourgain and Grillakis, which handled the radial case. The method is similar in spirit to the induction-on-energy stategy of Bourgain, but we perform the induction analysis in both frequency space and physical space simultaneousely, and replace the Morawetz inequality by an interaction variant. The principal advantage  of the interaction Morawetz estimate is that it is not localized to the spacial origin and so is better able to handle non-radial solutions. In particular, this interaction estimate together with an almost-conservation argument controlling the movement of the L² mass in frequncy space, rules out the possibility of energy concentration."