American Institute of Mathematical Sciences

Advanced
February  2009, 23(1&2): 435-454. doi: 10.3934/dcds.2009.23.435

Non-linear electromagnetism and special relativity

Denis Serre 1,

1. 

École Normale Supérieure de Lyon, UMPA (UMR 5669 CNRS), ENS de Lyon, 46, allée d’Italie, 69364 Lyon, cedex 07, France

Received  September 2007 Revised  July 2008 Published  September 2008

We continue the study of nonlinear Maxwell equations for electromagnetism in the formalism of B. D. Coleman & E. H. Dill. We exploit here the assumption of Lorentz invariance, following I. Białinicki-Barula. In particular, we show that nonlinearity forbids the convexity of the electromagnetic energy density. This justifies the study of rank-one convex and of polyconvex densities, begun in [8, 16]. We also show the alternative that either electrodynamics is linear, or dispersion is lost as the electromagnetic field becomes intense.
Keywords: Born--Infeld model, quasiconvexity, Maxwell equations.
Mathematics Subject Classification: Primary: 78A25, 83A05; Secondary: 35L45.
Citation: Denis Serre. Non-linear electromagnetism and special relativity. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[1]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[2]

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
[3]

Dinh-Liem Nguyen. The factorization method for the Drude-Born-Fedorov model for periodic chiral structures. Inverse Problems and Imaging, 2016, 10 (2) : 519-547. doi: 10.3934/ipi.2016010
[4]

Kun Li, Ting-Zhu Huang, Liang Li, Ying Zhao, Stéphane Lanteri. A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022084
[5]

Jonathan Ben-Artzi, Stephen Pankavich, Junyong Zhang. A toy model for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2022, 15 (3) : 341-354. doi: 10.3934/krm.2021053
[6]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[7]

Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations and Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018
[8]

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
[9]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631
[10]

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
[11]

Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305
[12]

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
[13]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic and Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865
[14]

Johannes Eilinghoff, Roland Schnaubelt. Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5685-5709. doi: 10.3934/dcds.2018248
[15]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867
[16]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
[17]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[18]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations and Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
[19]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503
[20]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems and Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (136)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy