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

Kazumasa Fujiwara; Tohru Ozawa

doi:10.3934/eect.2018013

Article Contents

2018, Volume 7, Issue 2: 275-280. Doi: 10.3934/eect.2018013

This issue Previous Article Robust Stackelberg controllability for linear and semilinear heat equations Next Article On the viscoelastic equation with Balakrishnan-Taylor damping and acoustic boundary conditions

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
* Corresponding author: Kazumasa Fujiwara

Received: May 2017

Revised: January 2018

Published: May 2018

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Abstract

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.

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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 H^1/2, 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 H^1/2, 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.