向雅捷

(華北電力大學 數(shù)理學院,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 數(shù)理學院,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,通過對頻率進行一致分解,將解在全空間中的整體估計轉化為單位區(qū)間中的局部估計;通過討論不同頻率間的相互關系,運用Strichartz估計和Bilinear Strichart估計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 預備知識

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,并被廣泛用來研究非線性dispersive(色散)方程,相關結果見文獻[6?7]。

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

2Up和Vp空間

3 Strichartz估計與Bilinear Strichartz估計

4 定理1的證明

[1]Cazenave T,Weissler F.The Cauchy problem for the critical nonlinear Schr?dinger equation in H s [J].Nonlinear Anal TMA,1990,14:807-836.

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

[3]Tsutsumi Y.L 2 -solutions for nonlinear Schr?dinger 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 Schr?dinger 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-?dinger 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 Schr?din -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.

(責任編校: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