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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

Jianjun Yuan

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.
Keywords: Chern-Simons-Schrödinger, concentration compactness, profile decomposition, scatter, global existence.
Mathematics Subject Classification: 35A01, 35Q41, 35Q55.
Citation: Jianjun Yuan. Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), 7pp. doi: 10.1155/2013/590653.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Yuan, Global existence and scattering of radial Chern-Simons-Schrödinger system, submitted. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., 2013 (2013), 7pp. doi: 10.1155/2013/590653.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Yuan, Global existence and scattering of radial Chern-Simons-Schrödinger system, submitted. Google Scholar

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