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March  2016, 36(3): 1709-1719. doi: 10.3934/dcds.2016.36.1709

Global solutions of two coupled Maxwell systems in the temporal gauge

Jianjun Yuan 1,

1. 

The College of Information and Technology, Nanjing University of Chinese Medicine, Nanjing 210046

Received  December 2014 Revised  March 2015 Published  August 2015

In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.
Keywords: Maxwell-Chern-Simons-Higgs, temporal gauge, Coulomb gauge., Maxwell-Klein-Gordon.
Mathematics Subject Classification: 35Q61, 35A01, 35B3.
Citation: Jianjun Yuan. Global solutions of two coupled Maxwell systems in the temporal gauge. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1709-1719. doi: 10.3934/dcds.2016.36.1709
References:
[1]

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.  doi: 10.1063/1.1507609.  Google Scholar

[2]

D. Chae and M. Chae, On the Cauchy problem in the Maxwell-Chern-Simons-Higgs system,, Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics (eds. H. Fujita, 1234 (2001), 206.   Google Scholar

[3]

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

[4]

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

[5]

H. Pecher, Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge,, preprint, ().   Google Scholar

[6]

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

[7]

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

[8]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm,, JDE, 189 (2003), 366.  doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[9]

J. Yuan, Well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge,, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 2389.  doi: 10.3934/dcds.2014.34.2389.  Google Scholar

[10]

J. Yuan, Local well-posedness of the Maxwell-Chern-Simons-Higgs system in the temporal gauge,, Nonlinear Analysis: Theory, 99 (2014), 128.  doi: 10.1016/j.na.2013.12.018.  Google Scholar

show all references

References:
[1]

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.  doi: 10.1063/1.1507609.  Google Scholar

[2]

D. Chae and M. Chae, On the Cauchy problem in the Maxwell-Chern-Simons-Higgs system,, Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics (eds. H. Fujita, 1234 (2001), 206.   Google Scholar

[3]

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

[4]

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

[5]

H. Pecher, Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge,, preprint, ().   Google Scholar

[6]

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

[7]

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

[8]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm,, JDE, 189 (2003), 366.  doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[9]

J. Yuan, Well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge,, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 2389.  doi: 10.3934/dcds.2014.34.2389.  Google Scholar

[10]

J. Yuan, Local well-posedness of the Maxwell-Chern-Simons-Higgs system in the temporal gauge,, Nonlinear Analysis: Theory, 99 (2014), 128.  doi: 10.1016/j.na.2013.12.018.  Google Scholar

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