Global wellposedness of defocusing critical nonlinear Schrödinger equation

in the radial case

Author:
J. Bourgain

Journal:
J. Amer. Math. Soc. **12** (1999), 145-171

MSC (1991):
Primary 35Q55, 35L15.

DOI:
https://doi.org/10.1090/S0894-0347-99-00283-0

MathSciNet review:
1626257

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Abstract | References | Similar Articles | Additional Information

Abstract: We establish global wellposedness and scattering for the -critical defocusing NLS in 3D

assuming radial data , . In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation

**[L-S]**J. Lin, W. Strauss,*Decay and scattering of solutions of a nonlinear Schrödinger equation*, JFA Vol. 30, no. 2 (1978), 245-263. MR**80k:35056****[Str]**M. Struwe,*Globally regular solutions to the -Klein-Gordon equations*, Ann. Scuola Norm Sup. Pisa, Ser. 4, 15 (1988), 495-513. MR**90j:35142****[Gr]**M. Grillakis,*Regularity and asymptotic behavior of the wave equation with a critical nonlinearity*, Annals Math. 132 (1990), 485-509. MR**92c:35080****[S-S]**J. Shatah, M. Struwe,*Regularity results for nonlinear wave equations*, Annals of Math. 138 (1993), 503-518. MR**95f:35164****[Caz]**T. Cazenave,*An introduction to nonlinear Schrödinger equations*, Textos de Metodes Matematicos 26 (Rio de Janeiro).**[C-W]**T. Cazenave, F. Weissler,*The Cauchy problem for the critical nonlinear Schrödinger equation in*, Nonlinear Anal., TMA 14 (1990), 807-836. MR**91j:35252****[G-V1]**J. Ginibre, G. Velo,*Scattering theory in the energy space for a class of nonlinear Schrödinger equations*, J. Math Pure Appl. 64 (1985), 363-401. MR**87i:35171****[G-V2]**J. Ginibre, G. Velo,*On a class of nonlinear Schrödinger equations with nonlocal interaction*, Math. Z. 170 (1980), 109-136. MR**82c:35018****[S]**R. Strichartz,*Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations*, Duke Math. J. 44 (1977), 705-714. MR**58:23577**

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Additional Information

**J. Bourgain**

Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Email:
bourgain@math.ias.edu

DOI:
https://doi.org/10.1090/S0894-0347-99-00283-0

Keywords:
Nonlinear Schr\"{o}dinger equation,
global wellposedness.

Received by editor(s):
April 20, 1998

Article copyright:
© Copyright 1999
American Mathematical Society