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

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Title
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations
Author
Cho, YonggeunHwang, GyeonghaKwon, SoonsikLee, Sanghyuk
Issue Date
2015-07
Publisher
AMER INST MATHEMATICAL SCIENCES
Citation
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, v.35, no.7, pp.2863 - 2880
Abstract
We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schr&ouml;dinger equations with L&eacute;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.
URI
https://scholarworks.unist.ac.kr/handle/201301/10622
URL
http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=10779
DOI
10.3934/dcds.2015.35.2863
ISSN
1531-3492
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