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

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

* 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.

Citation: Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & 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. Google Scholar

[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. Google Scholar

[3]

J. CollianderM. KeelG. StaffilaniH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

X. LiuG. 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. Google Scholar

[13]

K. MioT. OginoK. 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. Google Scholar

[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. Google Scholar

[15]

A. R. NahmodT. OhL. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, Springer New York, 1999. Google Scholar

[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. Google Scholar

[20]

H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259-277. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

J. CollianderM. KeelG. StaffilaniH. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

X. LiuG. 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. Google Scholar

[13]

K. MioT. OginoK. 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. Google Scholar

[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. Google Scholar

[15]

A. R. NahmodT. OhL. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Applied Mathematical Sciences, Springer New York, 1999. Google Scholar

[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. Google Scholar

[20]

H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259-277. Google Scholar

[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. Google Scholar

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