Vol. 71

Latest Volume
All Volumes
All Issues

On Uniqueness and Continuity for the Quasi-Linear, Bianisotropic Maxwell Equations, Using an Entropy Condition

By Daniel Sjoberg
Progress In Electromagnetics Research, Vol. 71, 317-339, 2007


The quasi-linear Maxwell equations describing electromagnetic wave propagation in nonlinear media permit several weak solutions, which may be discontinuous (shock waves). It is often conjectured that the solutions are unique if they satisfy an additional entropy condition. The entropy condition states that the energy contained in the electromagnetic fields is irreversibly dissipated to other energy forms, which are not described by the Maxwell equations. We use the method employed by Krûzkov to scalar conservation laws to analyze the implications of this additional condition in the electromagnetic case, i.e., systems of equations in three dimensions. It is shown that if a cubic term can be ignored, the solutions are unique and depend continuously on given data.


 (See works that cites this article)
Daniel Sjoberg, "On Uniqueness and Continuity for the Quasi-Linear, Bianisotropic Maxwell Equations, Using an Entropy Condition," Progress In Electromagnetics Research, Vol. 71, 317-339, 2007.


    1. Bloom, F., Mathematical Problems of Classical Nonlinear Electromagnetic Theory, Longman Scientific & Technical, Burnt Mill, Harlow, England, 1993.

    2. Coleman, B. D., "B. D. and E. H. Dill. Thermodynamic restrictions on the constitutive equations of electromagnetic theory," Z. Angew. Math. Phys., Vol. 22, 691-702, 1971.

    3. Courant, R. and K. O. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1948.

    4. Dafermos, C. M., "The entropy rate admissibility criteria for solutions of hyperbolic conservation laws," Journal of Differential Equations, Vol. 14, 202-212, 1973.

    5. Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics, 325, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2000.

    6. Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998.

    7. Godlewski, E. and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, Berlin, 1996.

    8. Gustafsson, M., "Wave splitting in direct and inverse scattering problems," Ph.D. thesis, 2000.

    9. Hadamard, J., Lectures on the Cauchy Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923.

    10. Hopf, E., "The partial differential equation ut + uux = μuxx," Comm. Pure Appl. Math., Vol. 3, 201-230, 1950.

    11. Hörmander, L., The Analysis of Linear Partial Differential Operators I, Grundlehren der mathematischen Wissenschaften 256, Springer-Verlag, Berlin Heidelberg, 1983.

    12. Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations, Number 26 in Mathemathiques & Applications, Springer-Verlag, Berlin, 1997.

    13. Jackson, J. D., Classical Electrodynamics, 3rd Ed., John Wiley & Sons, New York, 1999.

    14. Jouguet, E., "Sur la propagation des discontinuites dans les fluides," C. R. Acad. Sci., Vol. 132, 673-676, 1901.

    15. Kreiss, H.-O. and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press, San Diego, 1989.

    16. Kristensson, G. and D. J. N. Wall, "Direct and inverse scattering for transient electromagnetic waves in nonlinear media," Inverse Problems, Vol. 14, 113-137, 1998.

    17. Kržkov, S., "First order quasilinear equations with several space variables," Math. USSR Sbornik, Vol. 10, 217-273, 1970.

    18. Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskiǐ, Electrodynamics of Continuous Media, 2nd Ed., Pergamon, Oxford, 1984.

    19. Lax, P. D., "Shock waves and entropy," Contributions to Nonlinear Functional Analysis, 603-634, 1971.

    20. Lax, P. D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conf. Board. Math. Sci. Regional Conference Series in Applied Mathematics 11, 1973.

    21. Lindell, I. V., A. H. Sihvola, and K. Suchy, "Six-vector formalism in electromagnetics of bi-anisotropic media," J. Electro. Waves Applic., Vol. 9, No. 7/8, 887-903, 1995.

    22. Liu, T.-P., "The entropy condition and the admissibility of shocks," J. Math. Anal. Appl., Vol. 53, 78-88, 1976.

    23. Maugin, G. A., "On shock waves and phase-transition fronts in continua," ARI, Vol. 50, 141-150, 1998.

    24. Maugin, G. A., "On the universality of the thermomechanics of forces driving singular sets," Archive of Applied Mechanics, Vol. 70, 31-45, 2000.

    25. Oleǐnik, O. A., "Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation," Amer. Math. Soc. Transl., Vol. 14, No. 2(86), 87-158, 1959.

    26. Serre, D., "Systems of conservation laws, A challenge for the XXIst century," Mathematics Unlimited - 2001 and Beyond, 1061-1080, 2001.

    27. Sjöberg, D., "Reconstruction of nonlinear material properties for homogeneous, isotropic slabs using electromagnetic waves," Inverse Problems, Vol. 15, No. 2, 431-444, 1999.

    28. Sjöberg, D., "Simple wave solutions for the Maxwell equations in bianisotropic, nonlinear media, with application to oblique incidence," Wave Motion, Vol. 32, No. 3, 217-232, 2000.

    29. Styer, D. F., "Insight into entropy," Am. J. Phys, Vol. 68, No. 12, 1090-1096, 2000.

    30. Taylor, M., Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, 1996.

    31. Åberg, I., "High-frequency switching and Kerr effect — Nonlinear problems solved with nonstationary time domain techniques," Progress In Electromagnetics Research, Vol. 17, 185-235, 1997.

    32. Kung, F. and H. T. Chuah, "Stability of classical finite-difference time-domain (FDTD) formulation with nonlinear elements — A new perspective," Progress In Electromagnetics Research, Vol. 42, 49-89, 2003.

    33. Makeeva, G. S., O. A. Golovanov, and M. Pardavi-Horvath, "Mathematical modeling of nonlinear waves and oscillations in gyromagnetic structures by bifurcation theory methods," J. of Electromagn. Waves and Appl., Vol. 20, No. 11, 1503-1510, 2006.

    34. Norgren, M. and S. He, "Effective boundary conditions for a 2D inhomogeneous nonlinear thin layer coated on a metallic surface," Progress In Electromagnetics Research, Vol. 23, 301-314, 1999.