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一维二阶非线性薛定谔方程的局部适定性

2017-06-01向雅捷

关键词:薛定谔数理二阶

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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[2]Ginibre J,Velo G.On a class of nonlinear Schrödinger equations [J].J Funct Anal,1979,32:61-71.

[3]Tsutsumi Y.L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups [J].Funk Ekva,1987,30:115-125.

[4]Kenig Carlos E,Ponce G,Vega L.Quadratic forms for the 1-D semilinear Schrödinger equation [J].Transactions of the American Mathematical Society,1996,348:3 323-3 353.

[5]Guo S M.On the 1-D cubic nonlinear schr-ödinger equation in an almost critical space [J].Journal of Fourier Analysis &Applications,2016,22:1-34.

[6]Wang B,Huang C.Frequency uniform decomposition method forthe generalized BO,KdV and NLS equations [J].J Differential Equations,2007,239:213-250.

[7]Wang B,Hudzik H.The global Cauchy problem for the NLS and NLKG with small rough data [J].J Differential Equations,2007,232:36-73.

[8]Koch H,Tataru D.Dispersive estimates for principlally normal pseudo differential operators [J].Comm Pure Appl Math,2005,58(2):217-284.

[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.

[10]Hadac M,Herr S,Koch H.Well-posedness and scattering for the KP-II equation in a critical space [J].Ann Inst H Poincar´ e Anal Non Lin´ eaire,2009,26(3):917-941.

[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.

[12]Grünrock A.Bi- and trilinear Schrödin -ger estimates in one space dimension with application to cubic NLS and DNLS [J].Int Math Res Not,2005,41:2 525-2 558

[13]Koch H,Tataru D.Energy and local energy bounds for the 1-D cubic NLS equation in H −1/4 [J].Ann Inst H Poincare Anal Non Lineaire,2012,29(6):955-988.

(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

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