American Institute of Mathematical Sciences

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June  2018, 7(2): 275-280. doi: 10.3934/eect.2018013

On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation

Kazumasa Fujiwara 1,, and  Tohru Ozawa 2,

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

* Corresponding author: Kazumasa Fujiwara

The first author was partly supported by Grant-in-Aid for JSPS Fellows no 16J30008.

Received  May 2017 Revised  January 2018 Published  May 2018

An explicit lifespan estimate is presented for the derivative Schrödinger equations with periodic boundary condition.

Keywords: Periodic DNLS, power type nonlinearity without gauge invariance, lifespan estimate, finite time blowup, ODE approach.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B10.
Citation: Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013
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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