Global existence of small classical solutions to nonlinear Schrödinger equations

Tohru Ozawa; Jian Zhai

doi:10.1016/j.anihpc.2006.11.010

Global existence of small classical solutions to nonlinear Schrödinger equations

Tohru Ozawa^*, Jian Zhai

^*Corresponding author for this work

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

20 Citations (Scopus)

Abstract

We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n ≥ 3. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.

Original language	English
Pages (from-to)	303-311
Number of pages	9
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	25
Issue number	2
DOIs	https://doi.org/10.1016/j.anihpc.2006.11.010
Publication status	Published - 2008
Externally published	Yes

Keywords

Nonlinear Schrödinger equations
Schrödinger maps

ASJC Scopus subject areas

Analysis
Mathematical Physics
Applied Mathematics

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10.1016/j.anihpc.2006.11.010

Cite this

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title = "Global existence of small classical solutions to nonlinear Schr{\"o}dinger equations",

abstract = "We study the global Cauchy problem for nonlinear Schr{\"o}dinger equations with cubic interactions of derivative type in space dimension n ≥ 3. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.",

keywords = "Nonlinear Schr{\"o}dinger equations, Schr{\"o}dinger maps",

author = "Tohru Ozawa and Jian Zhai",

note = "Funding Information: This work started when T.O. visited Zhejiang University in 2004 and took a preliminary shape when J.Z. visited Hokkaido University in 2005. Part of this work was done while T.O. visited Courant Institute in 2005. T.O. would like to thank Professor Jalal Shatah for his hospitality and enlightning discussions. J.Z. is partly supported by NSFC 10571157.",

year = "2008",

doi = "10.1016/j.anihpc.2006.11.010",

language = "English",

volume = "25",

pages = "303--311",

journal = "Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis",

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publisher = "Elsevier Masson SAS",

number = "2",

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TY - JOUR

T1 - Global existence of small classical solutions to nonlinear Schrödinger equations

AU - Ozawa, Tohru

AU - Zhai, Jian

N1 - Funding Information: This work started when T.O. visited Zhejiang University in 2004 and took a preliminary shape when J.Z. visited Hokkaido University in 2005. Part of this work was done while T.O. visited Courant Institute in 2005. T.O. would like to thank Professor Jalal Shatah for his hospitality and enlightning discussions. J.Z. is partly supported by NSFC 10571157.

PY - 2008

Y1 - 2008

N2 - We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n ≥ 3. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.

AB - We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n ≥ 3. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.

KW - Nonlinear Schrödinger equations

KW - Schrödinger maps

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DO - 10.1016/j.anihpc.2006.11.010

M3 - Article

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SN - 0294-1449

VL - 25

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EP - 311

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

IS - 2

ER -

Global existence of small classical solutions to nonlinear Schrödinger equations

Abstract

Keywords

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