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

被引：54

作者：

Cho, Yonggeun ^{[1
,2
]}

Hwang, Gyeongha ^{[3
]}

Kwon, Soonsik ^{[4
]}

Lee, Sanghyuk ^{[5
]}

机构：

[1] Chonbuk Natl Univ, Dept Math, Jeonju 561756, South Korea

[2] Chonbuk Natl Univ, Inst Pure & Appl Math, Jeonju 561756, South Korea

[3] Ulsan Natl Inst Sci & Technol, Dept Math Sci, Ulsan 689798, South Korea

[4] Korea Adv Inst Sci & Technol, Dept Math Sci, Taejon 305701, South Korea

[5] Seoul Natl Univ, Dept Math Sci, Seoul 151747, South Korea

来源：

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS | 2015年 / 35卷 / 07期

关键词：

Fractional Schrodinger equation; cubic nonlinearity; well-posedness; ill-posedness;

D O I：

10.3934/dcds.2015.35.2863

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：2863 / 2880

页数：18

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