Global well-posedness and scattering for the energy-critical nonlinear Schr\"odinger equation in R^3

TL;DR

This paper proves global existence, scattering, and spacetime bounds for energy-critical nonlinear Schrödinger equations in three dimensions, extending previous radial results to nonradial solutions using advanced analytical techniques.

Contribution

It introduces a novel approach combining frequency and physical space analysis with an interaction Morawetz estimate to handle nonradial solutions in energy-critical NLS.

Findings

Established global well-posedness and scattering for nonradial solutions.

Developed a new interaction Morawetz estimate that is not spatially localized.

Controlled energy concentration and mass movement in frequency space.

Abstract

We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$ spacetime bounds for energy-class solutions to the quintic defocusing Schr\"odinger equation in $\R^{1+3}$, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain and Grillakis, which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain, but we perform the induction analysis in both frequency space and physical space simultaneously, 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 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 $L^2$ mass in frequency space, rules out the possibility of energy concentration.