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Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect
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 |
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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