Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

Yonggeun Cho; Gyeongha Hwang; Soonsik Kwon; Sanghyuk Lee

doi:10.3934/dcds.2015.35.2863

Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

Yonggeun Cho
, Gyeongha Hwang^*
, Soonsik Kwon
, Sanghyuk Lee

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

57 Scopus citations

Abstract

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices 1 < α < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in H^s for s ≥ 2-α/4. This is shown via a trilinear estimate in Bourgain's X^s,b space. We also show that non-periodic equations are ill-posed in H^s for 2-3α/4(α+1) < s < 2-α/ 4 in the sense that the flow map is not locally uniformly continuous.

Original language	English
Pages (from-to)	2863-2880
Number of pages	18
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	35
Issue number	7
DOIs	https://doi.org/10.3934/dcds.2015.35.2863
State	Published - 2015.07.1

Keywords

Cubic nonlinearity
Fractional Schrödinger equation
Ill-posedness
Well-posedness

Quacquarelli Symonds(QS) Subject Topics

Mathematics

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10.3934/dcds.2015.35.2863

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@article{769fdce3d84d403d9c22e80892dd1b2c,

title = "Well-posedness and ill-posedness for the cubic fractional Schr{\"o}dinger equations",

abstract = "We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schr{\"o}dinger equations with L{\'e}vy indices 1 < α < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in Hs for s ≥ 2-α/4. This is shown via a trilinear estimate in Bourgain's Xs,b space. We also show that non-periodic equations are ill-posed in Hs for 2-3α/4(α+1) < s < 2-α/ 4 in the sense that the flow map is not locally uniformly continuous.",

keywords = "Cubic nonlinearity, Fractional Schr{\"o}dinger equation, Ill-posedness, Well-posedness",

author = "Yonggeun Cho and Gyeongha Hwang and Soonsik Kwon and Sanghyuk Lee",

year = "2015",

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doi = "10.3934/dcds.2015.35.2863",

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journal = "Discrete and Continuous Dynamical Systems- Series A",

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TY - JOUR

T1 - Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

AU - Cho, Yonggeun

AU - Hwang, Gyeongha

AU - Kwon, Soonsik

AU - Lee, Sanghyuk

PY - 2015/7/1

Y1 - 2015/7/1

N2 - We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices 1 < α < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in Hs for s ≥ 2-α/4. This is shown via a trilinear estimate in Bourgain's Xs,b space. We also show that non-periodic equations are ill-posed in Hs for 2-3α/4(α+1) < s < 2-α/ 4 in the sense that the flow map is not locally uniformly continuous.

AB - We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices 1 < α < 2. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in Hs for s ≥ 2-α/4. This is shown via a trilinear estimate in Bourgain's Xs,b space. We also show that non-periodic equations are ill-posed in Hs for 2-3α/4(α+1) < s < 2-α/ 4 in the sense that the flow map is not locally uniformly continuous.

KW - Cubic nonlinearity

KW - Fractional Schrödinger equation

KW - Ill-posedness

KW - Well-posedness

UR - https://www.scopus.com/pages/publications/84921872388

U2 - 10.3934/dcds.2015.35.2863

DO - 10.3934/dcds.2015.35.2863

M3 - Journal article

AN - SCOPUS:84921872388

SN - 1078-0947

VL - 35

SP - 2863

EP - 2880

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 7

ER -

Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

Abstract

Keywords

Quacquarelli Symonds(QS) Subject Topics

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