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Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$
1. | School of Mathematical Sciences, Monash University, VIC 3800, Australia |
2. | Center for Applied Mathematics, Tianjin University, Tianjin 300072, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
D. J. Kaup and A. C. Newell,
An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 17 (1978), 789-801.
|
[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. |
[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. |
[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. |
[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. |
[13] |
E. Mj∅lhus,
On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys., 16 (1976), 321-334.
|
[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. |
[15] |
H. Takaoka,
Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Equ., 4 (1990), 561-680.
|
[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.
|
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
D. J. Kaup and A. C. Newell,
An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 17 (1978), 789-801.
|
[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. |
[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. |
[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. |
[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. |
[13] |
E. Mj∅lhus,
On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys., 16 (1976), 321-334.
|
[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. |
[15] |
H. Takaoka,
Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Equ., 4 (1990), 561-680.
|
[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.
|
[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. |
[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. |
[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. |
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