向雅捷

(華北電力大學(xué) 數(shù)理學(xué)院,北京,102206)

一維二階非線性薛定諤方程的局部適定性

向雅捷

(華北電力大學(xué) 數(shù)理學(xué)院,北京,102206)

討論了一維二階非線性薛定諤方程在?？臻gM2,p中的局部適定性問題,通過對(duì)頻率進(jìn)行一致分解,將解在全空間中的整體估計(jì)轉(zhuǎn)化為單位區(qū)間中的局部估計(jì);通過討論不同頻率間的相互關(guān)系,運(yùn)用Strichartz估計(jì)和Bilinear Strichart估計(jì)得到方程的局部適定性。

非線性薛定諤方程;局部適定性;低正則性;?？臻g

1 預(yù)備知識(shí)

本文旨在研究如下一維二階非線性薛定諤方程的局部適定性,

?？臻g由Feichtinger引進(jìn),并被廣泛用來研究非線性dispersive(色散)方程,相關(guān)結(jié)果見文獻(xiàn)[6?7]。

定義 1 對(duì)于k∈Z,用表示區(qū)間上的特征函數(shù),設(shè)頻率投射算子則模范數(shù)定義為

2Up和Vp空間

3 Strichartz估計(jì)與Bilinear Strichartz估計(jì)

4 定理1的證明

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[9]Koch H,Tataru D.A priori bounds for the 1D cubic NLS in negative Sobolev spaces [J].Int Math Res Not IMRN,2007,16:36-53.

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[11]Strichartz R.Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations [J].Duke Math J,1977,44(3):705-714.

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(責(zé)任編校:劉曉霞)

Local well-posedness of 1-D nonlinear second ordered schr?dinger 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 schr?dinger 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