Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao

Research output: Contribution to journalArticlepeer-review

223 Scopus citations

Abstract

We obtain global well-posedness, scattering, and global L t,x10 spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation 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 frequency space, rules out the possibility of energy concentration.

Original languageEnglish (US)
Pages (from-to)767-865
Number of pages99
JournalAnnals of Mathematics
Volume167
Issue number3
DOIs
StatePublished - May 2008

Fingerprint Dive into the research topics of 'Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ<sup>3</sup>'. Together they form a unique fingerprint.

Cite this