American Institute of Mathematical Sciences

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December  2016, 9(6): 1753-1773. doi: 10.3934/dcdss.2016073

Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect

Daiwen Huang 1, and  Jingjun Zhang 2,

1. 

Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China
2. 

College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China

Received  July 2015 Revised  September 2016 Published  November 2016

We consider the Cauchy problem of the nonlinear Schrödinger equation with magnetic effect, and prove global existence of smooth solutions and decay estimates for suitably small initial data. The key step in our analysis is to exploit the null structures for the phases, which allow us to close our argument in the framework of space-time resonance method.
Keywords: decay estimate, magnetic effect., Global solution.
Mathematics Subject Classification: Primary: 35Q55, 35A01; Secondary: 35B4.
Citation: Daiwen Huang, Jingjun Zhang. Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1753-1773. doi: 10.3934/dcdss.2016073
References:
[1]

H. Added and S. Added, Existence globle de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris, 299 (1984), 551-554.

[2]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506. doi: 10.1016/j.jfa.2011.03.015.

[3]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2d Zakharov system with $L^{2}$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.

[4]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Int. Math. Res. Not., 1996 (1996), 515-546. doi: 10.1155/S1073792896000359.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[7]

J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[8]

Z. Gan, B. Guo and D. Huang, Blow-up and nonlinear instability for the magnetic Zakharov system, J. Funct. Anal., 265 (2013), 953-982. doi: 10.1016/j.jfa.2013.05.017.

[9]

Z. Gan and J. Zhang, Nonlocal nonlinear Schrödinger equation in $\mathbbR^3$, Arch. Rational Mech. Anal., 209 (2013), 1-39. doi: 10.1007/s00205-013-0612-1.

[10]

Z. Gan and J. Zhang, Blow-up, global existence and standing waves for the magnetic nonlinear schrödinger equations, Discrete Contin. Dyn. Syst.-A, 32 (2012), 827-846. doi: 10.3934/dcds.2012.32.827.

[11]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not., 2009 (2009), 414-432. doi: 10.1093/imrn/rnn135.

[12]

Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry, Int. Math. Res. Not., 2014 (2014), 2327-2342.

[13]

Z. Guo, K. Nakanishi and S. Wang, Global dynamics below the ground state energy for the Zakharov system in the 3D radial case, Advances in Math., 238 (2013), 412-441. doi: 10.1016/j.aim.2013.02.008.

[14]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[15]

B. Guo and L. Shen, The existence and uniqueness of the classical solution on the periodic initial value problem for Zakharov equation (in Chinese), Acta Math. Appl. Sinica, 5 (1982), 310-324.

[16]

B. Guo and J. Zhang, Well-posedness of the Cauchy problem for the magnetic Zakharov type system, Nonlinearity, 24 (2011), 2191-2210. doi: 10.1088/0951-7715/24/8/004.

[17]

B. Guo, J. Zhang and X. Pu, On the existence and uniqueness of smooth solution for a generalized Zakharov equation, J. Math. Anal. Appl., 365 (2010), 238-253. doi: 10.1016/j.jmaa.2009.10.045.

[18]

N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369-389. doi: 10.1353/ajm.1998.0011.

[19]

N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified wave operators for nonlinear Schrödinger equations in one and two dimensions, Electronic J. Diff. Equa., 2004 (2004), 1-16.

[20]

L. Han, J. Zhang, Z. Gan and B. Guo, Cauchy problem for the Zakharov system arising from hot plasma with low regularity data, Commun. Math. Sci., 11 (2013), 403-420. doi: 10.4310/CMS.2013.v11.n2.a4.

[21]

Z. Hani, F. Pusateri and J. Shatah, Scattering for the Zakharov system in 3 dimensions, Commun. Math. Phys., 322 (2013), 731-753. doi: 10.1007/s00220-013-1738-6.

[22]

X. He, The pondermotive force and magnetic field generation effects resulting from the non-linear interaction between plasma-wave and particles (in Chinese), Acta Phys. Sinica, 32 (1983), 325-337.

[23]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027.

[24]

J. Kato and F. Pusateri, A new proof of long range scattering for critical nonlinear Schrödinger equations, Diff. Integral Equa., 24 (2011), 923-940.

[25]

C. Kenig and W. Wang, Existence of local smooth solution for a generalized Zakharov system, J. Fourier Anal. Appl., 4 (1998), 469-490. doi: 10.1007/BF02498221.

[26]

M. Kono, M. M. Skoric and D. Ter Haar, Spontaneous excitation of magnetic fields and collapse dynamics in a Langmuir plasma, J. Plasma Phys., 26 (1981), 123-146. doi: 10.1017/S0022377800010588.

[27]

C. Laurey, The Cauchy problem for a generalized Zakharov system, Diff. Integral Equ., 8 (1995), 105-130.

[28]

T. Ozawa and Y. Tsutsumi, Existence and smooth effect of solutions for the Zakharov equations, Pub. RIMS. Kyoto Univ., 28 (1992), 329-361. doi: 10.2977/prims/1195168430.

[29]

F. Pusateri and J. Shatah, Space-time resonances and the null condition for first order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540. doi: 10.1002/cpa.21461.

[30]

C. Sulem and P. L. Sulem, Quelques résulatats de régularité pour les équation de la turbulence de Langmuir, C. R. Acad. Sci. Paris, 289 (1979), 173-176.

[31]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP., 35 (1972), 908-914.

[32]

J. Zhang, C. Guo and B. Guo, On the Cauchy problem for the magnetic Zakharov system, Monatsh. Math., 170 (2013), 89-111. doi: 10.1007/s00605-012-0402-0.

show all references

References:
[1]

H. Added and S. Added, Existence globle de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris, 299 (1984), 551-554.

[2]

I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., 261 (2011), 478-506. doi: 10.1016/j.jfa.2011.03.015.

[3]

I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2d Zakharov system with $L^{2}$ Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. doi: 10.1088/0951-7715/22/5/007.

[4]

J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Int. Math. Res. Not., 1996 (1996), 515-546. doi: 10.1155/S1073792896000359.

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[7]

J. Colliander, J. Holmer and N. Tzirakis, Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems, Trans. Amer. Math. Soc., 360 (2008), 4619-4638. doi: 10.1090/S0002-9947-08-04295-5.

[8]

Z. Gan, B. Guo and D. Huang, Blow-up and nonlinear instability for the magnetic Zakharov system, J. Funct. Anal., 265 (2013), 953-982. doi: 10.1016/j.jfa.2013.05.017.

[9]

Z. Gan and J. Zhang, Nonlocal nonlinear Schrödinger equation in $\mathbbR^3$, Arch. Rational Mech. Anal., 209 (2013), 1-39. doi: 10.1007/s00205-013-0612-1.

[10]

Z. Gan and J. Zhang, Blow-up, global existence and standing waves for the magnetic nonlinear schrödinger equations, Discrete Contin. Dyn. Syst.-A, 32 (2012), 827-846. doi: 10.3934/dcds.2012.32.827.

[11]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not., 2009 (2009), 414-432. doi: 10.1093/imrn/rnn135.

[12]

Z. Guo and K. Nakanishi, Small energy scattering for the Zakharov system with radial symmetry, Int. Math. Res. Not., 2014 (2014), 2327-2342.

[13]

Z. Guo, K. Nakanishi and S. Wang, Global dynamics below the ground state energy for the Zakharov system in the 3D radial case, Advances in Math., 238 (2013), 412-441. doi: 10.1016/j.aim.2013.02.008.

[14]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.

[15]

B. Guo and L. Shen, The existence and uniqueness of the classical solution on the periodic initial value problem for Zakharov equation (in Chinese), Acta Math. Appl. Sinica, 5 (1982), 310-324.

[16]

B. Guo and J. Zhang, Well-posedness of the Cauchy problem for the magnetic Zakharov type system, Nonlinearity, 24 (2011), 2191-2210. doi: 10.1088/0951-7715/24/8/004.

[17]

B. Guo, J. Zhang and X. Pu, On the existence and uniqueness of smooth solution for a generalized Zakharov equation, J. Math. Anal. Appl., 365 (2010), 238-253. doi: 10.1016/j.jmaa.2009.10.045.

[18]

N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369-389. doi: 10.1353/ajm.1998.0011.

[19]

N. Hayashi, P. I. Naumkin, A. Shimomura and S. Tonegawa, Modified wave operators for nonlinear Schrödinger equations in one and two dimensions, Electronic J. Diff. Equa., 2004 (2004), 1-16.

[20]

L. Han, J. Zhang, Z. Gan and B. Guo, Cauchy problem for the Zakharov system arising from hot plasma with low regularity data, Commun. Math. Sci., 11 (2013), 403-420. doi: 10.4310/CMS.2013.v11.n2.a4.

[21]

Z. Hani, F. Pusateri and J. Shatah, Scattering for the Zakharov system in 3 dimensions, Commun. Math. Phys., 322 (2013), 731-753. doi: 10.1007/s00220-013-1738-6.

[22]

X. He, The pondermotive force and magnetic field generation effects resulting from the non-linear interaction between plasma-wave and particles (in Chinese), Acta Phys. Sinica, 32 (1983), 325-337.

[23]

A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027.

[24]

J. Kato and F. Pusateri, A new proof of long range scattering for critical nonlinear Schrödinger equations, Diff. Integral Equa., 24 (2011), 923-940.

[25]

C. Kenig and W. Wang, Existence of local smooth solution for a generalized Zakharov system, J. Fourier Anal. Appl., 4 (1998), 469-490. doi: 10.1007/BF02498221.

[26]

M. Kono, M. M. Skoric and D. Ter Haar, Spontaneous excitation of magnetic fields and collapse dynamics in a Langmuir plasma, J. Plasma Phys., 26 (1981), 123-146. doi: 10.1017/S0022377800010588.

[27]

C. Laurey, The Cauchy problem for a generalized Zakharov system, Diff. Integral Equ., 8 (1995), 105-130.

[28]

T. Ozawa and Y. Tsutsumi, Existence and smooth effect of solutions for the Zakharov equations, Pub. RIMS. Kyoto Univ., 28 (1992), 329-361. doi: 10.2977/prims/1195168430.

[29]

F. Pusateri and J. Shatah, Space-time resonances and the null condition for first order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540. doi: 10.1002/cpa.21461.

[30]

C. Sulem and P. L. Sulem, Quelques résulatats de régularité pour les équation de la turbulence de Langmuir, C. R. Acad. Sci. Paris, 289 (1979), 173-176.

[31]

V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP., 35 (1972), 908-914.

[32]

J. Zhang, C. Guo and B. Guo, On the Cauchy problem for the magnetic Zakharov system, Monatsh. Math., 170 (2013), 89-111. doi: 10.1007/s00605-012-0402-0.

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