The incompleteness of the Born-Infeld model for non-linear multi-d Maxwell’s equations
Authors:
Wladimir Neves and Denis Serre
Journal:
Quart. Appl. Math. 63 (2005), 343-367
MSC (2000):
Primary 35Q60
DOI:
https://doi.org/10.1090/S0033-569X-05-00964-6
Published electronically:
April 19, 2005
MathSciNet review:
2150780
Full-text PDF Free Access
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Abstract: We study the Born-Infeld system of conservation laws, which is the most famous model for non-linear Maxwell’s equations. This system is totally linear degenerated and there exists a conjecture, see Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rational Mech. Anal. 172 (2004), 65–91, that shocks are not allowed to form. In fact, we show that this conjecture is false and that the Born-Infeld model is not complete by itself. It means that a further theory is needed to complete the model.
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BI Born, M., Infeld, L., Foundations of a new field theory, Proc. Roy. London, A 144 (1934), 425–451.
B1 Brenier, Y., Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rational Mech. Anal. 172 (2004), 65–91.
BF Bressan, A., LeFloch, P. Uniqueness of weak solutions to systems of conservation laws, Arch. Rational Mech. Anal. 140 (1997), 301–317.
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BCP Bressan, A., Crasta, G., Piccoli, B., Well-posedness of the Cauchy problem for $n \times n$ systems of conservation laws, Mem. Am. Math. Soc. 146, no. 694 (2000).
CD Coleman, B., D., Dill, E., H. Thermodynamic restrictions on the constitutive equations of electromagnetic theory , Z. Angew. Math. Phys., 22 (1971), 691–702.
D1 Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics, Springer, 2000.
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EG Evans, L., C., Gariepy, R., F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
FS Fan, H., Slemrod M., Dynamic Flows with Liquid/Vapor Phase Transitions, Handbook of Mathematical Fluid Dynamics, North-Holland, vol. 1 (2002), 373–420.
KS Kulikovskii, A., Sveshnikova, E., Nonlinear Waves in Elastic Media, CRC Press, Boca Raton, FL, 1995.
LF1 LeFloch , P. G., Propagation Phase Boundaries: Formulation of the Problem and Existence via the Glimm Method, Arch. Rational Mech. Anal. 123 (1993), 153–197.
M Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, New York: Springer, 1984.
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S4 Serre, D., Hyperbolicity of the nonlinear models of Maxwell’s equations, Archive Ration. Mech. Anal. 172 (2004), 309–331.
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Additional Information
Wladimir Neves
Affiliation:
Instituto de Matematica, Universidade Federal do Rio de Janeiro, C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil
Email:
wladimir@im.ufrj.br
Denis Serre
Affiliation:
UMPA, Ecole Normale Superieure de Lyon, UMR 5669 CNRS, Lyon Cedex 07, France
MR Author ID:
158965
Email:
serre@umpa.ens-lyon.fr
Received by editor(s):
October 5, 2004
Published electronically:
April 19, 2005
Article copyright:
© Copyright 2005
Brown University
