# American Institute of Mathematical Sciences

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

## Global solutions of two coupled Maxwell systems in the temporal gauge

 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.
Citation: Jianjun Yuan. Global solutions of two coupled Maxwell systems in the temporal gauge. Discrete and Continuous Dynamical Systems, 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), 5470-5482. doi: 10.1063/1.1507609. [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, S. T. Kuroda and H. Okamoto, Distributed by Yurinsha), Sūrikaisekikenkyūsho Kōkyūroku, 1234 (2001), 206-212. [3] 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. [4] 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. [5] H. Pecher, Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge,, preprint, (). [6] S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein- Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100. [7] S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Cont. Dyn. Syst., 33 (2013), 2531-2546. doi: 10.3934/dcds.2013.33.2531. [8] T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, JDE, 189 (2003), 366-382. doi: 10.1016/S0022-0396(02)00177-8. [9] J. Yuan, Well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 2389-2403. doi: 10.3934/dcds.2014.34.2389. [10] J. Yuan, Local well-posedness of the Maxwell-Chern-Simons-Higgs system in the temporal gauge, Nonlinear Analysis: Theory, Methods & Applications, 99 (2014), 128-135. doi: 10.1016/j.na.2013.12.018.

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), 5470-5482. doi: 10.1063/1.1507609. [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, S. T. Kuroda and H. Okamoto, Distributed by Yurinsha), Sūrikaisekikenkyūsho Kōkyūroku, 1234 (2001), 206-212. [3] 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. [4] 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. [5] H. Pecher, Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge,, preprint, (). [6] S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein- Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100. [7] S. Selberg and A. Tesfahun, Global well-posedness of the Chern-Simons-Higgs equations with finite energy, Discrete Cont. Dyn. Syst., 33 (2013), 2531-2546. doi: 10.3934/dcds.2013.33.2531. [8] T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, JDE, 189 (2003), 366-382. doi: 10.1016/S0022-0396(02)00177-8. [9] J. Yuan, Well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge, Discrete and Continuous Dynamical Systems-Series A, 34 (2014), 2389-2403. doi: 10.3934/dcds.2014.34.2389. [10] J. Yuan, Local well-posedness of the Maxwell-Chern-Simons-Higgs system in the temporal gauge, Nonlinear Analysis: Theory, Methods & Applications, 99 (2014), 128-135. doi: 10.1016/j.na.2013.12.018.
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