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

Article Contents

2015, Volume 35, Issue 7: 2863-2880. Doi: 10.3934/dcds.2015.35.2863

This issue Previous Article Morse decomposition of global attractors with infinite components Next Article Exponential attractors for abstract equations with memory and applications to viscoelasticity

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

1.
Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756
2.
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, 689-798
3.
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon, Korea 305-701
4.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747

Received: May 2014

Revised: December 2014

Published: May 2017

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Abstract

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy 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 $s \geq \frac {2-\alpha}4$. 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 $\frac {2 - 3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.

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Mathematics Subject Classification: Primary: 35Q55, 35Q40.

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