American Institute of Mathematical Sciences

Advanced
January & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[2]

Dinh-Liem Nguyen. The factorization method for the Drude-Born-Fedorov model for periodic chiral structures. Inverse Problems & Imaging, 2016, 10 (2) : 519-547. doi: 10.3934/ipi.2016010
[3]

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
[4]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[5]

Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097
[6]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631
[7]

Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271
[8]

Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations & Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018
[9]

Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305
[10]

Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053
[11]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865
[12]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867
[13]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473
[14]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503
[15]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[16]

Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087
[17]

Laurent Boudin, Bérénice Grec, Francesco Salvarani. A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1427-1440. doi: 10.3934/dcdsb.2012.17.1427
[18]

Serge Nicaise, Fredi Tröltzsch. Optimal control of some quasilinear Maxwell equations of parabolic type. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1375-1391. doi: 10.3934/dcdss.2017073
[19]

Joel Spruck, Yisong Yang. Charged cosmological dust solutions of the coupled Einstein and Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 567-589. doi: 10.3934/dcds.2010.28.567
[20]

B. L. G. Jonsson. Wave splitting of Maxwell's equations with anisotropic heterogeneous constitutive relations. Inverse Problems & Imaging, 2009, 3 (3) : 405-452. doi: 10.3934/ipi.2009.3.405

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences