# American Institute of Mathematical Sciences

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

 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 $\kappa\rightarrow0$.
Citation: Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete and 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. [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. [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. [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. [5] J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge, Commun. Math. Phys., 82 (1981/82), 1-28. doi: 10.1007/BF01206943. [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. [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. [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, arXiv:1209.3841. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [5] J. Ginibre and G. Velo, The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge, Commun. Math. Phys., 82 (1981/82), 1-28. doi: 10.1007/BF01206943. [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. [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. [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, arXiv:1209.3841. [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. [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. [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. [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. [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. [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.
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