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Global well-posedness and scattering for the defocusing, $ L^{2}$-critical nonlinear Schrödinger equation when $ d \geq3$


Author: Benjamin Dodson
Journal: J. Amer. Math. Soc. 25 (2012), 429-463
MSC (2010): Primary 35Q55
DOI: https://doi.org/10.1090/S0894-0347-2011-00727-3
Published electronically: December 21, 2011
MathSciNet review: 2869023
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Abstract: In this paper we prove that the defocusing, $ d$-dimensional mass critical nonlinear Schrödinger initial value problem is globally well-posed and solutions scatter for $ u_{0} \in L^{2}(\mathbf {R}^{d})$, $ d \geq 3$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made by Colliander, Keel, Staffilani, Takaoka, and Tao. Since we are considering an $ L^{2}$-critical initial value problem we will localize to low frequencies. The main new ingredient in this proof is a long time Strichartz estimate for the solution to the first equation given in the paper at high frequencies. The long term Strichartz estimates allow us to estimate the error in the interaction Morawetz estimate caused by localizing to low frequencies.


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Additional Information

Benjamin Dodson
Affiliation: Department of Mathematics, University of California, Berkeley, 970 Evans Hall 3840, Berkeley, California 94720-3840
Email: benjadod@math.berkeley.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00727-3
Received by editor(s): August 25, 2010
Received by editor(s) in revised form: October 25, 2010, May 20, 2011, September 30, 2011, and November 7, 2011
Published electronically: December 21, 2011
Additional Notes: The author was supported by the National Science Foundation postdoctoral fellowship DMS-1103914 during some of the writing of this paper.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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