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 & 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).  doi: 10.1090/cln/010.  Google Scholar

[6]

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

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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.  doi: 10.1090/S0002-9947-08-04295-5.  Google Scholar

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

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

[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.  doi: 10.3934/dcds.2012.32.827.  Google Scholar

[11]

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

[12]

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

[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.  doi: 10.1016/j.aim.2013.02.008.  Google Scholar

[14]

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

[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.   Google Scholar

[16]

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

[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.  doi: 10.1016/j.jmaa.2009.10.045.  Google Scholar

[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.  doi: 10.1353/ajm.1998.0011.  Google Scholar

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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.   Google Scholar

[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.  doi: 10.4310/CMS.2013.v11.n2.a4.  Google Scholar

[21]

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

[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.   Google Scholar

[23]

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

[24]

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

[25]

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

[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.  doi: 10.1017/S0022377800010588.  Google Scholar

[27]

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

[28]

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

[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.  doi: 10.1002/cpa.21461.  Google Scholar

[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.   Google Scholar

[31]

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

[32]

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

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).  doi: 10.1090/cln/010.  Google Scholar

[6]

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

[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.  doi: 10.1090/S0002-9947-08-04295-5.  Google Scholar

[8]

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

[9]

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

[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.  doi: 10.3934/dcds.2012.32.827.  Google Scholar

[11]

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

[12]

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

[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.  doi: 10.1016/j.aim.2013.02.008.  Google Scholar

[14]

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

[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.   Google Scholar

[16]

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

[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.  doi: 10.1016/j.jmaa.2009.10.045.  Google Scholar

[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.  doi: 10.1353/ajm.1998.0011.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.4310/CMS.2013.v11.n2.a4.  Google Scholar

[21]

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

[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.   Google Scholar

[23]

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

[24]

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

[25]

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

[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.  doi: 10.1017/S0022377800010588.  Google Scholar

[27]

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

[28]

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

[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.  doi: 10.1002/cpa.21461.  Google Scholar

[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.   Google Scholar

[31]

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

[32]

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

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