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September  2020, 40(9): 5541-5570. doi: 10.3934/dcds.2020237

## Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system

 School of Artificial Intelligence and Information Technology, Nanjing University of Chinese Medicine, Nanjing 210046, China

Received  December 2019 Revised  March 2020 Published  June 2020

In this paper, we consider the following equivariant defocusing Chern-Simons-Schrödinger system,
 $\begin{eqnarray}i\partial_{t}\phi+\Delta\phi = \frac{2m}{r^2}A_{\theta}\phi+A_{0}\phi+\frac{1}{r^2}A_{\theta}^2\phi-\lambda|\phi|^{p-2}\phi,\\ \partial_rA_{0} = \frac{1}{r}(m+A_{\theta})|\phi|^2,\\ \partial_tA_{\theta} = rIm(\bar{\phi}\partial_{r}\phi),\\ \partial_rA_{\theta} = -\frac{1}{2}|\phi|^2r,\\ A_r = 0.\end{eqnarray}$
where
 $\phi(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R}$
is a complex scalar field,
 $A_\mu(t, x_1, x_2): \mathbb{R}^{1+2}\rightarrow \mathbb{R}$
is the gauge field for
 $\mu = 0, 1, 2$
,
 $A_r = \frac{x_1}{|x|}A_1+\frac{x_2}{|x|}A_2$
,
 $A_{\theta} = -x_2A_1+x_1A_2$
,
 $\lambda<0$
and
 $p>4$
.
When
 $p>4$
, the system is in the mass supercritical and energy subcrtical range. By using the conservation law of the system and the concentration compactness method introduced in [17], we show that the solution of the system exists globally and scatters.
Citation: Jianjun Yuan. Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5541-5570. doi: 10.3934/dcds.2020237
##### References:
 [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  doi: 10.1353/ajm.1999.0001. [2] L. Bergé, A. De Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007. [3] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024. [4] J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order $N$ for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004. [5] X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934.  doi: 10.1016/j.jde.2016.04.031. [6] G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Physics Monographs, 36, Springer, Berlin, Heidelberg, 1995. doi: 10.1007/978-3-540-44777-1. [7] T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13. [8] D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9. [9] J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pur. Appl. (9), 64 (1985), 363-401. [10] H. Huh, Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.  doi: 10.1088/0951-7715/22/5/003. [11] H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53 (2012), 8pp. doi: 10.1063/1.4726192. [12] H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), 7pp. doi: 10.1155/2013/590653. [13] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y. [14] R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D (3), 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500. [15] R. Jackiw and S.-Y. Pi, Self-dual Chern-Simons solitons, Prog. Theor. Phys. Suppl., 107 (1992), 1-40.  doi: 10.1143/PTPS.107.1. [16] Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrödinger equation with a vortex point, Commun. Contemp. Math., 18 (2016), 20pp. doi: 10.1142/S0219199715500741. [17] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4. [18] G. Li and X. Luo, Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^2$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428.  doi: 10.5186/aasfm.2017.4223. [19] Z. M. Lim, Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system, J. Differential Equations, 264 (2018), 2553-2597.  doi: 10.1016/j.jde.2017.10.026. [20] B. Liu and P. Smith, Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.  doi: 10.4171/RMI/898. [21] B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not. IMRN, 2014 (2013), 6341-6398.  doi: 10.1093/imrn/rnt161. [22] F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, 1998 (1998), 399-425.  doi: 10.1155/S1073792898000270. [23] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225.  doi: 10.1006/jfan.1999.3503. [24] S.-J. Oh and F. Pusateri, Decay and scattering for the Chern-Simons-Schrödinger equations, Int. Math. Res. Not. IMRN, 2015 (2015), 13122-13147.  doi: 10.1093/imrn/rnv093. [25] A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535. [26] K. Sahbi, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.  doi: 10.1006/jdeq.2000.3951. [27] J. Yuan, Multiple normalized solutions of Chern-Simons-Schrödinger system, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1801-1816.  doi: 10.1007/s00030-015-0344-z. [28] J. Yuan, Global existence and scattering of radial Chern-Simons-Schrödinger system, submitted.

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##### References:
 [1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  doi: 10.1353/ajm.1999.0001. [2] L. Bergé, A. De Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007. [3] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024. [4] J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order $N$ for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004. [5] X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934.  doi: 10.1016/j.jde.2016.04.031. [6] G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Physics Monographs, 36, Springer, Berlin, Heidelberg, 1995. doi: 10.1007/978-3-540-44777-1. [7] T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13. [8] D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9. [9] J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pur. Appl. (9), 64 (1985), 363-401. [10] H. Huh, Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.  doi: 10.1088/0951-7715/22/5/003. [11] H. Huh, Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53 (2012), 8pp. doi: 10.1063/1.4726192. [12] H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), 7pp. doi: 10.1155/2013/590653. [13] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y. [14] R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D (3), 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500. [15] R. Jackiw and S.-Y. Pi, Self-dual Chern-Simons solitons, Prog. Theor. Phys. Suppl., 107 (1992), 1-40.  doi: 10.1143/PTPS.107.1. [16] Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrödinger equation with a vortex point, Commun. Contemp. Math., 18 (2016), 20pp. doi: 10.1142/S0219199715500741. [17] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4. [18] G. Li and X. Luo, Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^2$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428.  doi: 10.5186/aasfm.2017.4223. [19] Z. M. Lim, Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system, J. Differential Equations, 264 (2018), 2553-2597.  doi: 10.1016/j.jde.2017.10.026. [20] B. Liu and P. Smith, Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.  doi: 10.4171/RMI/898. [21] B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not. IMRN, 2014 (2013), 6341-6398.  doi: 10.1093/imrn/rnt161. [22] F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, 1998 (1998), 399-425.  doi: 10.1155/S1073792898000270. [23] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225.  doi: 10.1006/jfan.1999.3503. [24] S.-J. Oh and F. Pusateri, Decay and scattering for the Chern-Simons-Schrödinger equations, Int. Math. Res. Not. IMRN, 2015 (2015), 13122-13147.  doi: 10.1093/imrn/rnv093. [25] A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535. [26] K. Sahbi, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392.  doi: 10.1006/jdeq.2000.3951. [27] J. Yuan, Multiple normalized solutions of Chern-Simons-Schrödinger system, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1801-1816.  doi: 10.1007/s00030-015-0344-z. [28] J. Yuan, Global existence and scattering of radial Chern-Simons-Schrödinger system, submitted.
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