Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation
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- by Pascal Bégout and Ana Vargas PDF
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Abstract:
In this paper, we show that any solution of the nonlinear Schrödinger equation $iu_t+\Delta u\pm |u|^\frac {4}{N}u=0,$ 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).References
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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
- 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).
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 5257-5282
- MSC (2000): Primary 35B05, 35B33, 35B40, 35Q55, 42B10
- DOI: https://doi.org/10.1090/S0002-9947-07-04250-X
- MathSciNet review: 2327030