Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: 35A01, 35Q41, 35Q55.

 Citation:

•  [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.