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On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation
1. | Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri, 3, 56126 Pisa, Italy |
2. | Department of Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan |
An explicit lifespan estimate is presented for the derivative Schrödinger equations with periodic boundary condition.
References:
[1] |
D. M. Ambrose and G. Simpson,
Local existence theory for derivative nonlinear Schrödinger equations with noninteger power nonlinearities, SIAM J. Math. Anal., 47 (2015), 2241-2264.
doi: 10.1137/140955227. |
[2] |
H. A. Biagioni and F. Linares,
Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.
doi: 10.1090/S0002-9947-01-02754-4. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[4] |
K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys., 57 (2016), 082103, 8pp.
doi: 10.1063/1.4960725. |
[5] |
K. Fujiwara and T. Ozawa,
Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance, J. Evol. Equ., 17 (2017), 1023-1030.
doi: 10.1007/s00028-016-0364-0. |
[6] |
A. Grünrock and S. Herr,
Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.
doi: 10.1137/070689139. |
[7] |
M. Hayashi and T. Ozawa,
Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differential Equations, 261 (2016), 5424-5445.
doi: 10.1016/j.jde.2016.08.018. |
[8] |
N. Hayashi,
The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.
doi: 10.1016/0362-546X(93)90071-Y. |
[9] |
N. Hayashi and T. Ozawa,
On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.
doi: 10.1016/0167-2789(92)90185-P. |
[10] |
N. Hayashi and T. Ozawa,
Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.
doi: 10.1137/S0036141093246129. |
[11] |
S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., 2006 (2006), Art. ID 96763, 33pp.
doi: 10.1155/IMRN/2006/96763. |
[12] |
X. Liu, G. Simpson and C. Sulem,
Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation, J. Nonlinear Sci., 23 (2013), 557-583.
doi: 10.1007/s00332-012-9161-2. |
[13] |
K. Mio, T. Ogino, K. Minami and S. Takeda,
Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan, 41 (1976), 265-271.
doi: 10.1143/JPSJ.41.265. |
[14] |
R. Mosincat,
Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in H1/2, J. Differential Equations, 263 (2017), 4658-4722.
doi: 10.1016/j.jde.2017.05.026. |
[15] |
A. R. Nahmod, T. Oh, L. Rey-Bellet and G. Staffilani,
Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS), 14 (2012), 1275-1330.
doi: 10.4171/JEMS/333. |
[16] |
T. Ozawa and Y. Yamazaki,
Life-span of smooth solutions to the complex Ginzburg-Landau type equation on a torus, Nonlinearity, 16 (2003), 2029-2034.
doi: 10.1088/0951-7715/16/6/309. |
[17] |
G. d. N. Santos,
Existence and uniqueness of solution for a generalized nonlinear derivative Schrödinger equation, J. Differential Equations, 259 (2015), 2030-2060.
doi: 10.1016/j.jde.2015.03.023. |
[18] |
C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, Springer New York, 1999. |
[19] |
H. Sunagawa,
The lifespan of solutions to nonlinear Schrödinger and Klein-Gordon equations, Hokkaido Math. J., 37 (2008), 825-838.
doi: 10.14492/hokmj/1249046371. |
[20] |
H. Takaoka,
Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580.
|
[21] |
H. Takaoka,
A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below H1/2, J. Differential Equations, 260 (2016), 818-859.
doi: 10.1016/j.jde.2015.09.011. |
[22] |
S. B. Tan,
Blow-up solutions for mixed nonlinear Schrödinger equations, Acta Math. Sin. (Engl. Ser.), 20 (2004), 115-124.
doi: 10.1007/s10114-003-0295-x. |
[23] |
L. Thomann and N. Tzvetkov,
Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 2771-2791.
doi: 10.1088/0951-7715/23/11/003. |
[24] |
M. Tsutsumi and I. Fukuda,
On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259-277.
|
[25] |
Y. Y. S. Win,
Global well-posedness of the derivative nonlinear Schrödinger equations on T, Funkcial. Ekvac., 53 (2010), 51-88.
doi: 10.1619/fesi.53.51. |
show all references
The first author was partly supported by Grant-in-Aid for JSPS Fellows no 16J30008.
References:
[1] |
D. M. Ambrose and G. Simpson,
Local existence theory for derivative nonlinear Schrödinger equations with noninteger power nonlinearities, SIAM J. Math. Anal., 47 (2015), 2241-2264.
doi: 10.1137/140955227. |
[2] |
H. A. Biagioni and F. Linares,
Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.
doi: 10.1090/S0002-9947-01-02754-4. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.
doi: 10.1137/S0036141001394541. |
[4] |
K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys., 57 (2016), 082103, 8pp.
doi: 10.1063/1.4960725. |
[5] |
K. Fujiwara and T. Ozawa,
Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance, J. Evol. Equ., 17 (2017), 1023-1030.
doi: 10.1007/s00028-016-0364-0. |
[6] |
A. Grünrock and S. Herr,
Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.
doi: 10.1137/070689139. |
[7] |
M. Hayashi and T. Ozawa,
Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differential Equations, 261 (2016), 5424-5445.
doi: 10.1016/j.jde.2016.08.018. |
[8] |
N. Hayashi,
The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.
doi: 10.1016/0362-546X(93)90071-Y. |
[9] |
N. Hayashi and T. Ozawa,
On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.
doi: 10.1016/0167-2789(92)90185-P. |
[10] |
N. Hayashi and T. Ozawa,
Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.
doi: 10.1137/S0036141093246129. |
[11] |
S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., 2006 (2006), Art. ID 96763, 33pp.
doi: 10.1155/IMRN/2006/96763. |
[12] |
X. Liu, G. Simpson and C. Sulem,
Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation, J. Nonlinear Sci., 23 (2013), 557-583.
doi: 10.1007/s00332-012-9161-2. |
[13] |
K. Mio, T. Ogino, K. Minami and S. Takeda,
Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan, 41 (1976), 265-271.
doi: 10.1143/JPSJ.41.265. |
[14] |
R. Mosincat,
Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in H1/2, J. Differential Equations, 263 (2017), 4658-4722.
doi: 10.1016/j.jde.2017.05.026. |
[15] |
A. R. Nahmod, T. Oh, L. Rey-Bellet and G. Staffilani,
Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS), 14 (2012), 1275-1330.
doi: 10.4171/JEMS/333. |
[16] |
T. Ozawa and Y. Yamazaki,
Life-span of smooth solutions to the complex Ginzburg-Landau type equation on a torus, Nonlinearity, 16 (2003), 2029-2034.
doi: 10.1088/0951-7715/16/6/309. |
[17] |
G. d. N. Santos,
Existence and uniqueness of solution for a generalized nonlinear derivative Schrödinger equation, J. Differential Equations, 259 (2015), 2030-2060.
doi: 10.1016/j.jde.2015.03.023. |
[18] |
C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, Springer New York, 1999. |
[19] |
H. Sunagawa,
The lifespan of solutions to nonlinear Schrödinger and Klein-Gordon equations, Hokkaido Math. J., 37 (2008), 825-838.
doi: 10.14492/hokmj/1249046371. |
[20] |
H. Takaoka,
Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580.
|
[21] |
H. Takaoka,
A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below H1/2, J. Differential Equations, 260 (2016), 818-859.
doi: 10.1016/j.jde.2015.09.011. |
[22] |
S. B. Tan,
Blow-up solutions for mixed nonlinear Schrödinger equations, Acta Math. Sin. (Engl. Ser.), 20 (2004), 115-124.
doi: 10.1007/s10114-003-0295-x. |
[23] |
L. Thomann and N. Tzvetkov,
Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 2771-2791.
doi: 10.1088/0951-7715/23/11/003. |
[24] |
M. Tsutsumi and I. Fukuda,
On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259-277.
|
[25] |
Y. Y. S. Win,
Global well-posedness of the derivative nonlinear Schrödinger equations on T, Funkcial. Ekvac., 53 (2010), 51-88.
doi: 10.1619/fesi.53.51. |
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