# American Institute of Mathematical Sciences

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

## 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

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.
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 [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

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
 [1] Yinxia Wang, Hengjun Zhao. Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating. Communications on Pure & Applied Analysis, 2018, 17 (2) : 347-374. doi: 10.3934/cpaa.2018020 [2] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [3] Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267 [4] Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 [5] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [6] Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723 [7] Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469 [8] Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088 [9] Mihai Bostan. On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 339-371. doi: 10.3934/dcdsb.2015.20.339 [10] Amer Rasheed, Aziz Belmiloudi, Fabrice Mahé. Dynamics of dendrite growth in a binary alloy with magnetic field effect. Conference Publications, 2011, 2011 (Special) : 1224-1233. doi: 10.3934/proc.2011.2011.1224 [11] Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188 [12] Shigui Ruan, Wendi Wang, Simon A. Levin. The effect of global travel on the spread of SARS. Mathematical Biosciences & Engineering, 2006, 3 (1) : 205-218. doi: 10.3934/mbe.2006.3.205 [13] Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235 [14] Boling Guo, Haiyang Huang. Smooth solution of the generalized system of ferro-magnetic chain. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 729-740. doi: 10.3934/dcds.1999.5.729 [15] Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations & Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713 [16] Cristina Brändle, Arturo De Pablo. Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1161-1178. doi: 10.3934/cpaa.2018056 [17] Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 [18] Bilal Al Taki. Global well posedness for the ghost effect system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 345-368. doi: 10.3934/cpaa.2017017 [19] Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030 [20] Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191

2018 Impact Factor: 0.545