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, arXiv:1411.1207.

[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, arXiv:1411.1207.

[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.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Hartmut Pecher. Local well-posedness for the Maxwell-Dirac system in temporal gauge. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 3065-3076. doi: 10.3934/dcds.2022008

[7]

Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146

[8]

Hartmut Pecher. Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2965-2989. doi: 10.3934/cpaa.2021091

[9]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573

[10]

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

[11]

Jeongho Kim, Bora Moon. Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2541-2561. doi: 10.3934/dcds.2021202

[12]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135

[13]

Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097

[14]

Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271

[15]

Sitong Chen, Xianhua Tang. Improved results for Klein-Gordon-Maxwell systems with general nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2333-2348. doi: 10.3934/dcds.2018096

[16]

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

[17]

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

[18]

Zhi-You Chen, Chung-Yang Wang, Yu-Jen Huang. On the asymptotic behavior of solutions for the self-dual Maxwell-Chern-Simons $ O(3) $ Sigma model. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022077

[19]

Paulo Cesar Carrião, Patrícia L. Cunha, Olímpio Hiroshi Miyagaki. Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents. Communications on Pure and Applied Analysis, 2011, 10 (2) : 709-718. doi: 10.3934/cpaa.2011.10.709

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (141)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]