American Institute of Mathematical Sciences

Advanced
May  2014, 34(5): 2389-2403. doi: 10.3934/dcds.2014.34.2389

On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge

Jianjun Yuan 1,

1. 

Department of Basic Science, Yancheng Institute of Technology, Yancheng 224051, China

Received  March 2013 Revised  July 2013 Published  October 2013

In this paper, we investigate the well-posedness of the Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. In particular, we prove that the system is globally wellposed in the energy space. As an application, we prove that the solution of the Maxwell-Chern-Simons-Higgs system converges to that of Maxwell-Higgs system in $H^s\times H^{s-1}$($s\geq1$) as the Chern-Simons coupling constant $\kappa\rightarrow0$.
Keywords: Lorenz gauge., Maxwell-Higgs, Maxwell-Chern-Simons-Higgs.
Mathematics Subject Classification: Primary: 35L15, 35L45; Secondary: 35Q4.
Citation: Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389
References:
[1]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, Theory, Methods and Applications, 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[2]

N. Bournaveas, Low regularity solutions of the relativistic Chern-Simons-Higgs theory in the Lorentz gauge, Electronic Journal of Differential Equations (2009), 1-10.  Google Scholar

[3]

D. Chae and M. Chae, The global existence in the Cauchy problem of the Maxwell-Chern- Simons-Higgs system, Journal of Mathematical physics, 43 (2002),5470-5482. doi: 10.1063/1.1507609.  Google Scholar

[4]

P. D'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, In Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, Contemporary Mathematics, vol. 526, Amer. Math. Soc., Providence, RI, 2010, pp. 125-150. doi: 10.1090/conm/526/10379.  Google Scholar

[5]

J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge,, Commun. Math. Phys., 82 (): 1.  doi: 10.1007/BF01206943.  Google Scholar

[6]

H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations, Nonlinearity, 18 (2005), 2581-2589. doi: 10.1088/0951-7715/18/6/009.  Google Scholar

[7]

H. Huh, Local and global solutions of the Chern-Simons-Higgs system, Journal of Functional Analysis, 242 (2007), 526-549. doi: 10.1016/j.jfa.2006.09.009.  Google Scholar

[8]

H. Huh and S.-J. Oh, Low Regularity Solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs Equations in the Lorenz Gauge,, preprint, ().   Google Scholar

[9]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[10]

C. Lee, K. Lee and H. Min, Self-dual Maxwell-Chern-Simons solitons, phys. Lett. B, 252 (1990), 79-83. doi: 10.1016/0370-2693(90)91084-O.  Google Scholar

[11]

V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in (2+1)-dimensional space-time, Journal of mathematical physics, 21 (1980), 2291-2296. doi: 10.1063/1.524669.  Google Scholar

[12]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Communications in Partial Differential Equations, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[13]

S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Contin. Dyn. Syst., 33 (2013), 2531-2546. doi: 10.3934/dcds.2013.33.2531.  Google Scholar

[14]

J. Yuan, Local well-posedness of Chern-Simons-Higgs system in the Lorentz gauge, Journal of Mathematical Physics, 52 (2011), 103706, 14 pp. doi: 10.1063/1.3645365.  Google Scholar

show all references

References:
[1]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, Theory, Methods and Applications, 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[2]

N. Bournaveas, Low regularity solutions of the relativistic Chern-Simons-Higgs theory in the Lorentz gauge, Electronic Journal of Differential Equations (2009), 1-10.  Google Scholar

[3]

D. Chae and M. Chae, The global existence in the Cauchy problem of the Maxwell-Chern- Simons-Higgs system, Journal of Mathematical physics, 43 (2002),5470-5482. doi: 10.1063/1.1507609.  Google Scholar

[4]

P. D'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, In Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, Contemporary Mathematics, vol. 526, Amer. Math. Soc., Providence, RI, 2010, pp. 125-150. doi: 10.1090/conm/526/10379.  Google Scholar

[5]

J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge,, Commun. Math. Phys., 82 (): 1.  doi: 10.1007/BF01206943.  Google Scholar

[6]

H. Huh, Low regularity solutions of the Chern-Simons-Higgs equations, Nonlinearity, 18 (2005), 2581-2589. doi: 10.1088/0951-7715/18/6/009.  Google Scholar

[7]

H. Huh, Local and global solutions of the Chern-Simons-Higgs system, Journal of Functional Analysis, 242 (2007), 526-549. doi: 10.1016/j.jfa.2006.09.009.  Google Scholar

[8]

H. Huh and S.-J. Oh, Low Regularity Solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs Equations in the Lorenz Gauge,, preprint, ().   Google Scholar

[9]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[10]

C. Lee, K. Lee and H. Min, Self-dual Maxwell-Chern-Simons solitons, phys. Lett. B, 252 (1990), 79-83. doi: 10.1016/0370-2693(90)91084-O.  Google Scholar

[11]

V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in (2+1)-dimensional space-time, Journal of mathematical physics, 21 (1980), 2291-2296. doi: 10.1063/1.524669.  Google Scholar

[12]

S. Selberg and A. Tesfahun, Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Communications in Partial Differential Equations, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[13]

S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Contin. Dyn. Syst., 33 (2013), 2531-2546. doi: 10.3934/dcds.2013.33.2531.  Google Scholar

[14]

J. Yuan, Local well-posedness of Chern-Simons-Higgs system in the Lorentz gauge, Journal of Mathematical Physics, 52 (2011), 103706, 14 pp. doi: 10.1063/1.3645365.  Google Scholar

[1]

Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193
[2]

Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145
[3]

Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199
[4]

Jongmin Han, Juhee Sohn. On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 819-839. doi: 10.3934/dcds.2019034
[5]

Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531
[6]

Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053
[7]

Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693
[8]

Jianjun Yuan. Global solutions of two coupled Maxwell systems in the temporal gauge. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1709-1719. doi: 10.3934/dcds.2016.36.1709
[9]

Magdalena Czubak, Robert L. Jerrard. Topological defects in the abelian Higgs model. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1933-1968. doi: 10.3934/dcds.2015.35.1933
[10]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
[11]

Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034
[12]

Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317
[13]

Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064
[14]

Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2765-2786. doi: 10.3934/dcds.2017119
[15]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1967-1981. doi: 10.3934/dcdss.2021008
[16]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[17]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
[18]

Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2703-2728. doi: 10.3934/dcds.2014.34.2703
[19]

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
[20]

Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1931-1944. doi: 10.3934/dcdss.2021016

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences