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January  2017, 37(1): 257-264. doi: 10.3934/dcds.2017010

Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$

Zihua Guo 1,, and  Yifei Wu 2,

1. 

School of Mathematical Sciences, Monash University, VIC 3800, Australia
2. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

* Corresponding author: zihua.guo@monash.edu

Received  April 2016 Revised  June 2016 Published  November 2016

Fund Project: The second author is supported by NSFC grant 11571118.

We prove that the derivative nonlinear Schrödinger equation is globally well-posed in $H^{\frac 12} (\mathbb{R} )$ when the mass of initial data is strictly less than 4π.

Keywords: Nonlinear Schrödinger equation with derivative, global well-posedness, low regularity.
Mathematics Subject Classification: Primary:35Q55;Secondary:35A0.
Citation: Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010
References:
[1]

M. Agueh, Sharp Gagliardo-Nirenberg Inequalities and Mass Transport Theory, J.Dyn.Differ.Equ., 18 (2006), 1069-1093.  doi: 10.1007/s10884-006-9039-9.  Google Scholar

[2]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[3]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness result for Schrödinger equations with derivatives, SIAM J. Math. Anal., 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.  Google Scholar

[4]

Z. Guo, N. Hayashi, Y. Lin and P. I. Naumkin, Modified scattering operator for the derivative nonlinear Schrodinger equation, SIAM J. Math. Anal., 45 (2013), 3854-3871.  doi: 10.1137/12089956X.  Google Scholar

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N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonl. Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar

[6]

N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Physica D., 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar

[7]

N. Hayashi and T. Ozawa, Finite energy solution of nonlinear Schr¨odinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129.  Google Scholar

[8]

D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 17 (1978), 789-801.   Google Scholar

[9]

K. Kondo, K. Kajiwara and K. Matsui, Solution and integrability of a generalized derivative nonlinear Schrödinger equation, J. Phys. Soc. Japan, 66 (1997), 60-66.  doi: 10.1143/JPSJ.66.60.  Google Scholar

[10]

X. Liu, G. Simpson and C. Sulem, Focusing singularity in a derivative nonlinear Schr¨odinger equation, Physica D: Nonlinear Phenomena, 262 (2013), 48-58.  doi: 10.1016/j.physd.2013.07.011.  Google Scholar

[11]

C. Miao, Y. Wu and G. Xu, Global well-posedness for Schr¨odinger equation with derivative in $H^{\frac 12} (\mathbb{R} )$, J. Diff. Equ., 251 (2011), 2164-2195.  doi: 10.1016/j.jde.2011.07.004.  Google Scholar

[12]

W. Mio, T. Ogino, K. Minami and S. Takeda, Modified nonlinear Schrödinger equation for Alfven waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan, 41 (1976), 265-271.  doi: 10.1143/JPSJ.41.265.  Google Scholar

[13]

E. Mj∅lhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys., 16 (1976), 321-334.   Google Scholar

[14]

T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137-163.  doi: 10.1512/iumj.1996.45.1962.  Google Scholar

[15]

H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Equ., 4 (1990), 561-680.   Google Scholar

[16]

H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Equ., 42 (2001), 1-23.   Google Scholar

[17]

T. Tsuchida and M. Wadati, Complete integrability of derivative nonlinear Schrödinger-type equations, Inverse Problems, 15 (1999), 1363-1373.  doi: 10.1088/0266-5611/15/5/317.  Google Scholar

[18]

Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Analysis & PDE, 6 (2013), 1989-2002.  doi: 10.2140/apde.2013.6.1989.  Google Scholar

[19]

Y. Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Analysis & PDE, 8 (2015), 1101-1112.  doi: 10.2140/apde.2015.8.1101.  Google Scholar

show all references

References:
[1]

M. Agueh, Sharp Gagliardo-Nirenberg Inequalities and Mass Transport Theory, J.Dyn.Differ.Equ., 18 (2006), 1069-1093.  doi: 10.1007/s10884-006-9039-9.  Google Scholar

[2]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[3]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness result for Schrödinger equations with derivatives, SIAM J. Math. Anal., 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.  Google Scholar

[4]

Z. Guo, N. Hayashi, Y. Lin and P. I. Naumkin, Modified scattering operator for the derivative nonlinear Schrodinger equation, SIAM J. Math. Anal., 45 (2013), 3854-3871.  doi: 10.1137/12089956X.  Google Scholar

[5]

N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonl. Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar

[6]

N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Physica D., 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar

[7]

N. Hayashi and T. Ozawa, Finite energy solution of nonlinear Schr¨odinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129.  Google Scholar

[8]

D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 17 (1978), 789-801.   Google Scholar

[9]

K. Kondo, K. Kajiwara and K. Matsui, Solution and integrability of a generalized derivative nonlinear Schrödinger equation, J. Phys. Soc. Japan, 66 (1997), 60-66.  doi: 10.1143/JPSJ.66.60.  Google Scholar

[10]

X. Liu, G. Simpson and C. Sulem, Focusing singularity in a derivative nonlinear Schr¨odinger equation, Physica D: Nonlinear Phenomena, 262 (2013), 48-58.  doi: 10.1016/j.physd.2013.07.011.  Google Scholar

[11]

C. Miao, Y. Wu and G. Xu, Global well-posedness for Schr¨odinger equation with derivative in $H^{\frac 12} (\mathbb{R} )$, J. Diff. Equ., 251 (2011), 2164-2195.  doi: 10.1016/j.jde.2011.07.004.  Google Scholar

[12]

W. Mio, T. Ogino, K. Minami and S. Takeda, Modified nonlinear Schrödinger equation for Alfven waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan, 41 (1976), 265-271.  doi: 10.1143/JPSJ.41.265.  Google Scholar

[13]

E. Mj∅lhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys., 16 (1976), 321-334.   Google Scholar

[14]

T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137-163.  doi: 10.1512/iumj.1996.45.1962.  Google Scholar

[15]

H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Equ., 4 (1990), 561-680.   Google Scholar

[16]

H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Diff. Equ., 42 (2001), 1-23.   Google Scholar

[17]

T. Tsuchida and M. Wadati, Complete integrability of derivative nonlinear Schrödinger-type equations, Inverse Problems, 15 (1999), 1363-1373.  doi: 10.1088/0266-5611/15/5/317.  Google Scholar

[18]

Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Analysis & PDE, 6 (2013), 1989-2002.  doi: 10.2140/apde.2013.6.1989.  Google Scholar

[19]

Y. Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Analysis & PDE, 8 (2015), 1101-1112.  doi: 10.2140/apde.2015.8.1101.  Google Scholar

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