Global well-posedness of critical nonlinear Schrödinger equations below L2

Yonggeun Cho; Gyeongha Hwang; Tohru Ozawa

doi:10.3934/dcds.2013.33.1389

Global well-posedness of critical nonlinear Schrödinger equations below L²

Yonggeun Cho^*, Gyeongha Hwang, Tohru Ozawa

^*Corresponding author for this work

School of Advanced Science and Engineering

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

9 Citations (Scopus)

Abstract

The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below L² in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index s _c is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

Original language	English
Pages (from-to)	1389-1405
Number of pages	17
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	33
Issue number	4
DOIs	https://doi.org/10.3934/dcds.2013.33.1389
Publication status	Published - 2013 Apr

Keywords

Angular regularity
Critical nonlinearity below L
Global well-posedness
Hartree equations
Weighted Strichartz estimate

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.3934/dcds.2013.33.1389

Cite this

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title = "Global well-posedness of critical nonlinear Schr{\"o}dinger equations below L2",

abstract = "The global well-posedness on the Cauchy problem of nonlinear Schr{\"o}dinger equations (NLS) is studied for a class of critical nonlinearity below L2 in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index s c is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.",

keywords = "Angular regularity, Critical nonlinearity below L, Global well-posedness, Hartree equations, Weighted Strichartz estimate",

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T1 - Global well-posedness of critical nonlinear Schrödinger equations below L2

AU - Cho, Yonggeun

AU - Hwang, Gyeongha

AU - Ozawa, Tohru

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N2 - The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below L2 in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index s c is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

AB - The global well-posedness on the Cauchy problem of nonlinear Schrödinger equations (NLS) is studied for a class of critical nonlinearity below L2 in small data setting. We consider Hartree type (HNLS) and inhomogeneous power type NLS (PNLS). Since the critical Sobolev index s c is negative, it is rather difficult to analyze the nonlinear term. To overcome the difficulty we combine weighted Strichartz estimates in polar coordinates with new Duhamel estimates involving angular regularity.

KW - Angular regularity

KW - Critical nonlinearity below L

KW - Global well-posedness

KW - Hartree equations

KW - Weighted Strichartz estimate

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