Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ 3

J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao

doi:10.1002/cpa.20029

Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ ³

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao

Mathematics

Research output: Contribution to journal › Article › peer-review

140 Scopus citations

Abstract

We prove global existence and scattering for the defocusing, cubic, nonlinear Schrödinger equation in H ^S(ℝ ³) for s > 4/5. The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds ||φ(x,t)|| _{L x,t}4(ℝ ³×ℝ), whereas the well-known Morawetz-fype estimate of Lin-Strauss controls ∫ ₀ ^∞ ∫ _ℝ3 (φ(x,t)) ⁴/|x| dx dt.

Original language	English (US)
Pages (from-to)	987-1014
Number of pages	28
Journal	Communications on Pure and Applied Mathematics
Volume	57
Issue number	8
DOIs	https://doi.org/10.1002/cpa.20029
State	Published - Aug 2004

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title = "Global existence and scattering for rough solutions of a nonlinear Schr{\"o}dinger equation on ℝ 3 ",

abstract = "We prove global existence and scattering for the defocusing, cubic, nonlinear Schr{\"o}dinger equation in H S(ℝ 3) for s > 4/5. The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds ||φ(x,t)|| L x,t4(ℝ 3×ℝ), whereas the well-known Morawetz-fype estimate of Lin-Strauss controls ∫ 0 ∞ ∫ ℝ3 (φ(x,t)) 4/|x| dx dt.",

author = "J. Colliander and M. Keel and G. Staffilani and H. Takaoka and T. Tao",

year = "2004",

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language = "English (US)",

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journal = "Communications on Pure and Applied Mathematics",

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T1 - Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ 3

AU - Colliander, J.

AU - Keel, M.

AU - Staffilani, G.

AU - Takaoka, H.

AU - Tao, T.

PY - 2004/8

Y1 - 2004/8

N2 - We prove global existence and scattering for the defocusing, cubic, nonlinear Schrödinger equation in H S(ℝ 3) for s > 4/5. The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds ||φ(x,t)|| L x,t4(ℝ 3×ℝ), whereas the well-known Morawetz-fype estimate of Lin-Strauss controls ∫ 0 ∞ ∫ ℝ3 (φ(x,t)) 4/|x| dx dt.

AB - We prove global existence and scattering for the defocusing, cubic, nonlinear Schrödinger equation in H S(ℝ 3) for s > 4/5. The main new estimate in the argument is a Morawetz-type inequality for the solution φ. This estimate bounds ||φ(x,t)|| L x,t4(ℝ 3×ℝ), whereas the well-known Morawetz-fype estimate of Lin-Strauss controls ∫ 0 ∞ ∫ ℝ3 (φ(x,t)) 4/|x| dx dt.

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EP - 1014

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 8

ER -

Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ ³

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