一维二阶非线性薛定谔方程的局部适定性

向雅捷

(华北电力大学 数理学院,北京,102206)

一维二阶非线性薛定谔方程的局部适定性

向雅捷

(华北电力大学 数理学院,北京,102206)

讨论了一维二阶非线性薛定谔方程在模空间M2,p中的局部适定性问题,通过对频率进行一致分解,将解在全空间中的整体估计转化为单位区间中的局部估计;通过讨论不同频率间的相互关系,运用Strichartz估计和Bilinear Strichart估计得到方程的局部适定性。

非线性薛定谔方程;局部适定性;低正则性;模空间

1 预备知识

本文旨在研究如下一维二阶非线性薛定谔方程的局部适定性,

模空间由Feichtinger引进,并被广泛用来研究非线性dispersive(色散)方程,相关结果见文献[6−7]。

定义 1 对于k∈Z,用表示区间上的特征函数,设频率投射算子则模范数定义为

2Up和Vp空间

3 Strichartz估计与Bilinear Strichartz估计

4 定理1的证明

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(责任编校:刘晓霞)

Local well-posedness of 1-D nonlinear second ordered schrödinger equation

Xiang Yajie
(School of Mathematics and Physics,North China Electric Power University,Beijing 102206,China)

Local well-posedness problem is discussed.Through the frequency uniform decomposion of a solution in the whole space,the global well-posedness estimate of the solution is converted into the unit local well-posedness estimate.By discussing the relationship between different frequency and using the Strichartz estimates and the Bilinear Strichartz estimates,the local well-posedness of equation is obtained.

nonlinear schrödinger equation;local well-posedness;low regularity;modulation space

O 241.8

A

1672-6146(2017)02-0012-05

向雅捷,zhuangxiaomath@163.com。

2016−09−03

10.3969/j.issn.1672-6146.2017.02.004