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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mass concentration phenomena for the $ L^2$-critical nonlinear Schrödinger equation


Authors: Pascal Bégout and Ana Vargas
Journal: Trans. Amer. Math. Soc. 359 (2007), 5257-5282
MSC (2000): Primary 35B05, 35B33, 35B40, 35Q55, 42B10
Published electronically: May 16, 2007
MathSciNet review: 2327030
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Abstract: In this paper, we show that any solution of the nonlinear Schrödinger equation $ iu_t+\Delta u\pm\vert u\vert^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on Bourgain's (1998), which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega (1999). We also generalize to higher dimensions the results in Keraani (2006) and Merle and Vega (1998).


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Additional Information

Pascal Bégout
Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 4, place Jussieu, 75252 Paris Cedex 05, France
Address at time of publication: Département de Mathématiques, Laboratoire d’Analyse et Probabilités, Université d’Évry Val d’Essone, Boulevard François Mitterrand, 91025 Évry Cedex, France
Email: begout@ann.jussieu.fr, Pascal.Begout@univ-evry.fr

Ana Vargas
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid 28049 Madrid, Spain
Email: ana.vargas@uam.es

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04250-X
PII: S 0002-9947(07)04250-X
Keywords: Schr\"{o}dinger equations, restriction theorems, Strichartz's estimate, blow-up
Received by editor(s): July 20, 2005
Published electronically: May 16, 2007
Additional Notes: This research was partially supported by the European network HPRN–CT–2001–00273–HARP (Harmonic analysis and related problems). The second author was also supported by Grant MTM2004–00678 of the MEC (Spain).
Article copyright: © Copyright 2007 American Mathematical Society