WELL-POSEDNESS AND ILL-POSEDNESS FOR THE CUBIC FRACTIONAL SCHRODINGER EQUATIONS

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We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrodinger equations with Levy indices 1 < alpha < 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-alpha/4. This is shown via a trilinear estimate in Bourgain's X-s,X-b space. We also show that non-periodic equations are ill-posed in H-S for 2-3 alpha/4(alpha+1) < 2-alpha/4 in the sense that the flow map is not locally uniformly continuous.
Publisher
AMER INST MATHEMATICAL SCIENCES
Issue Date
2015-07
Language
English
Article Type
Article
Citation

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v.35, no.7, pp.2863 - 2880

ISSN
1078-0947
DOI
10.3934/dcds.2015.35.2863
URI
http://hdl.handle.net/10203/195772
Appears in Collection
MA-Journal Papers(저널논문)
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