We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$ spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation in $\mathbb{R}^{1+3}$, 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 $L^2$ mass in frequency space, rules out the possibility of energy concentration.
James Colliander 1 ; Markus Keel 2 ; Gigiola Staffilani 3 ; Hideo Takaoka 4 ; Terence Tao 5
@article{10_4007_annals_2008_167_767,
author = {James Colliander and Markus Keel and Gigiola Staffilani and Hideo Takaoka and Terence Tao},
title = {Global well-posedness and scattering for the energy-critical {Schr\"odinger} equation in $\mathbb R^3$},
journal = {Annals of mathematics},
pages = {767--865},
year = {2008},
volume = {167},
number = {3},
doi = {10.4007/annals.2008.167.767},
mrnumber = {2415387},
zbl = {1178.35345},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.767/}
}
TY - JOUR AU - James Colliander AU - Markus Keel AU - Gigiola Staffilani AU - Hideo Takaoka AU - Terence Tao TI - Global well-posedness and scattering for the energy-critical Schrödinger equation in $\mathbb R^3$ JO - Annals of mathematics PY - 2008 SP - 767 EP - 865 VL - 167 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.767/ DO - 10.4007/annals.2008.167.767 LA - en ID - 10_4007_annals_2008_167_767 ER -
%0 Journal Article %A James Colliander %A Markus Keel %A Gigiola Staffilani %A Hideo Takaoka %A Terence Tao %T Global well-posedness and scattering for the energy-critical Schrödinger equation in $\mathbb R^3$ %J Annals of mathematics %D 2008 %P 767-865 %V 167 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.767/ %R 10.4007/annals.2008.167.767 %G en %F 10_4007_annals_2008_167_767
James Colliander; Markus Keel; Gigiola Staffilani; Hideo Takaoka; Terence Tao. Global well-posedness and scattering for the energy-critical Schrödinger equation in $\mathbb R^3$. Annals of mathematics, Tome 167 (2008) no. 3, pp. 767-865. doi: 10.4007/annals.2008.167.767
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