Geodesic
Global well-posedness and scattering for the energy-critical Schrödinger equation in $\mathbb R^3$
James Colliander1 ; Markus Keel2 ; Gigiola Staffilani3 ; Hideo Takaoka4 ; Terence Tao5
1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
2 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
3 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States
4 Department of Mathematics, Kobe University, Kobe 657-8501, Japan
5 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States and Mathematical Sciences Institute, The Australian National University, Canberra 0200, Australia
Annals of mathematics, Tome 167 (2008) no. 3, pp. 767-865.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2008.167.767

James Colliander 1 ; Markus Keel 2 ; Gigiola Staffilani 3 ; Hideo Takaoka 4 ; Terence Tao 5

1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
2 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
3 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States
4 Department of Mathematics, Kobe University, Kobe 657-8501, Japan
5 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States and Mathematical Sciences Institute, The Australian National University, Canberra 0200, Australia 
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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. http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.767/

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