Scattering for bounded solutions to the cubic, defocusing NLS in 3 dimensions
Authors:
Carlos E. Kenig and Frank Merle
Journal:
Trans. Amer. Math. Soc. 362 (2010), 19371962
MSC (2010):
Primary 35Q55
Published electronically:
November 18, 2009
MathSciNet review:
2574882
Fulltext PDF Free Access
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Abstract: We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.
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 J. Holmer and S Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), no. 2, 435467. MR 2421484
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 M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955980. MR 1646048 (2000d:35018)
 [Ke]
 C. Kenig, Global wellposedness, scattering and blowup for the energycritical, focusing, nonlinear Schrödinger and wave equations, Lecture notes for a minicourse given at ``Analyse des equations aux derivées partielles'', Evian, France, June 2007.
 [KM1]
 C. Kenig and F. Merle, Global wellposedness, scattering and blowup for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math. 166 (2006), 645675. MR 2257393 (2007g:35232)
 [KM2]
 C. Kenig and F. Merle, Global wellposedness, scattering and blowup for the energy critical, focusing, nonlinear wave equation, Acta Math. 201 (2008), 147212.
 [KPV]
 C. Kenig, G. Ponce, and L. Vega, Wellposedness and scattering results for the generalized KortewegdeVries equation via the contraction principle, Comm. Pure. Appl. Math 46 (1993), 527620. MR 1211741 (94h:35229)
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 J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, Jour. Funct. Anal. 30 (1978), 245263. MR 515228 (80k:35056)
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 F. Merle, Existence of blowup solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), 555578. MR 1824989 (2002f:35193)
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 F. Merle and L. Vega, Compactness at blowup time for solutions of the critical nonlinear Schrödinger equation in 2D, Intern. Math. Res. Notices 8 (1998), 399425. MR 1628235 (99d:35156)
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 E. M. Stein and G. Weiss, Fractional integrals on dimensional Euclidean space, J. Math. Mech. 7 (1958), 503514. MR 0098285 (20:4746)
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 R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705714. MR 0512086 (58:23577)
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 M. Vilela, Regularity of solutions to the free Schrödinger equation with radial initial data, Illinois J. Math. 45 (2001), 361370. MR 1878609 (2002k:35061)
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Additional Information
Carlos E. Kenig
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email:
cek@math.uchicago.edu
Frank Merle
Affiliation:
Departement de Mathematiques, Universite de Cergy–Pontoise, Pontoise 95302 Cergy–Pontoise, France
Email:
Frank.Merle@math.ucergy.fr
DOI:
http://dx.doi.org/10.1090/S0002994709047229
Received by editor(s):
September 20, 2007
Published electronically:
November 18, 2009
Additional Notes:
The first author was supported in part by NSF
The second author was supported in part by CNRS. Part of this research was carried out during visits of the second author to the University of Chicago and IHES. This research was also supported in part by ANR ONDE NONLIN
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
