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

被引：53

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

共 14 条

[1] Burq N, 2002, MATH RES LETT, V9, P323
[2] GLOBAL WELL-POSEDNESS AND I METHOD FOR THE FIFTH ORDER KORTEWEG-DE VRIES EQUATION
Chen, Wengu
Guo, Zihua
[J]. JOURNAL D ANALYSE MATHEMATIQUE, 2011, 114 : 121 - 156
[3] Strichartz Estimates in Spherical Coordinates
Cho, Yonggeun
Lee, Sanghyuk
[J]. INDIANA UNIVERSITY MATHEMATICS JOURNAL, 2013, 62 (03) : 991 - 1020
[4] On the Cauchy Problem of Fractional Schrodinger Equation with Hartree Type Nonlinearity
Cho, Yonggeun
Hajaiej, Hichem
Hwang, Gyeongha
Ozawa, Tohru
[J]. FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 2013, 56 (02): : 193 - 224
[5] Profile decompositions and blowup phenomena of mass critical fractional Schrodinger equations
Cho, Yonggeun
Hwang, Gyeongha
Kwon, Soonsik
Lee, Sanghyuk
[J]. NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2013, 86 : 12 - 29
[6] REMARKS ON SOME DISPERSIVE ESTIMATES
Cho, Yonggeun
Ozawa, Tohru
Xia, Suxia
[J]. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 2011, 10 (04) : 1121 - 1128
[7] Christ M, 2003, AM J MATH, V125, P1235
[8] Demirbas S., ARXIV13125249
[9] Global Well-Posedness for the Fractional Nonlinear Schrodinger Equation
Guo, Boling
Huo, Zhaohui
[J]. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2011, 36 (02) : 247 - 255
[10] Nonlinear fractional Schrodinger equations in one dimension
Ionescu, Alexandru D.
Pusateri, Fabio
[J]. JOURNAL OF FUNCTIONAL ANALYSIS, 2014, 266 (01) : 139 - 176

← 1 2 →