American Institute of Mathematical Sciences

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March  2011, 10(2): 541-559. doi: 10.3934/cpaa.2011.10.541

A singular limit in a nonlinear problem arising in electromagnetism

Frank Jochmann 1,

1. 

Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften, Institut für Mathematik, Strasse des 17. Juni 136, 10623 Berlin

Received  February 2010 Revised  May 2010 Published  December 2010

This paper deals with a generally nonlinear mixed-type initial-boundary value problem for the description of the electromagnetic field in a conducting medium that is surrounded by an insulating medium with a high dielectric permittivity. The main goals are the existence, uniqueness and the asymptotic behavior of the solutions to this system.
Keywords: Maxwell's equations, asymptotic behavior., singular limit.
Mathematics Subject Classification: Primary: 35Q61, 35Q60; Secondary: 35L25, 78A4.
Citation: Frank Jochmann. A singular limit in a nonlinear problem arising in electromagnetism. Communications on Pure & Applied Analysis, 2011, 10 (2) : 541-559. doi: 10.3934/cpaa.2011.10.541
References:
[1]

J. M. Ball, Strongly continuous semi groups, weak solutions and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. Google Scholar

[2]

J. W. Barrett and L. Prigozhin, Bean's critical-state model as the $p\rightarrow\infty$ limit of an evolutionary $p$-Laplace equation, Nonlinear Anal., 42 (2000), 977-993. doi: doi:10.1016/S0362-546X(99)00147-9.  Google Scholar

[3]

C. P. Bean, Magnetization of high-field superconductors, Rev. Mod. Phys., 36 (1964), 31-39. doi: doi:10.1103/RevModPhys.36.31.  Google Scholar

[4]

F. Jochmann, Existence of weak solutions to the drift-diffusion model for semiconductors coupled with Maxwell's equations, J. Math. Anal. Appl., 204 (1996), 655-676. doi: doi:10.1006/jmaa.1996.0460.  Google Scholar

[5]

F. Jochmann, A semi-static limit for Maxwell's equations in an exterior domain, Comm. Part. Diff. Equations, 23 (1998), 2035-2076. doi: doi:10.1080/03605309808821410.  Google Scholar

[6]

F. Jochmann, Regularity of weak solutions to Maxwell's Equations with mixed boundary conditions, Math. Meth. Appl. Sci., 22 (1999), 1255-1274. doi: doi:10.1002/(SICI)1099-1476(19990925)22:14<1255::AID-MMA83>3.0.CO;2-N.  Google Scholar

[7]

F. Jochmann, Energy decay of solutions to Maxwells equations with conductivity and polarization, J. Diff. Equations, 203 (2004), 232-254. doi: doi:10.1016/j.jde.2004.05.005.  Google Scholar

[8]

F. Jochmann, On a first-order hyperbolic systems including Bean's model for superconductors with displacement current, J. Diff. Equations, 246 (2009), 2151-2191. doi: doi:10.1016/j.jde.2008.12.023.  Google Scholar

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", 2$^{nd}$ edition, ().   Google Scholar

[10]

R. Picard, An elementary proof for a compact embedding result in generalized electromagnetic theory, Math. Z., 187 (1984), 151-161. doi: doi:10.1007/BF01161700.  Google Scholar

[11]

C. Weber, A local compactness theorem for Maxwell's equations, Math. Methods Appl. Sci., 2 (1980), 12-25. doi: doi:10.1002/mma.1670020103.  Google Scholar

[12]

H. M. Yin, On a $p$-Laplacian type of evolution system and applications to Bean's model in the type-II superconductivity theory, Quarterly. Appl. Math., 59 (2001), 47-66.  Google Scholar

[13]

H. M. Yin, On a singular limit problem for nonlinear Maxwell equations, J. Diff. Equations, 156 (1999), 355-375. doi: doi:10.1006/jdeq.1998.3608.  Google Scholar

[14]

H. M. Yin, B. Q. Li and J. Zou, A degenerate evolution system modeling Bean's critical-state type-II superconductors, Discrete Continuous Dynam. Systems - B, 8 (2002), 781-794.  Google Scholar

show all references

References:
[1]

J. M. Ball, Strongly continuous semi groups, weak solutions and the variation of constants formula, Proc. Amer. Math. Soc., 63 (1977), 370-373. Google Scholar

[2]

J. W. Barrett and L. Prigozhin, Bean's critical-state model as the $p\rightarrow\infty$ limit of an evolutionary $p$-Laplace equation, Nonlinear Anal., 42 (2000), 977-993. doi: doi:10.1016/S0362-546X(99)00147-9.  Google Scholar

[3]

C. P. Bean, Magnetization of high-field superconductors, Rev. Mod. Phys., 36 (1964), 31-39. doi: doi:10.1103/RevModPhys.36.31.  Google Scholar

[4]

F. Jochmann, Existence of weak solutions to the drift-diffusion model for semiconductors coupled with Maxwell's equations, J. Math. Anal. Appl., 204 (1996), 655-676. doi: doi:10.1006/jmaa.1996.0460.  Google Scholar

[5]

F. Jochmann, A semi-static limit for Maxwell's equations in an exterior domain, Comm. Part. Diff. Equations, 23 (1998), 2035-2076. doi: doi:10.1080/03605309808821410.  Google Scholar

[6]

F. Jochmann, Regularity of weak solutions to Maxwell's Equations with mixed boundary conditions, Math. Meth. Appl. Sci., 22 (1999), 1255-1274. doi: doi:10.1002/(SICI)1099-1476(19990925)22:14<1255::AID-MMA83>3.0.CO;2-N.  Google Scholar

[7]

F. Jochmann, Energy decay of solutions to Maxwells equations with conductivity and polarization, J. Diff. Equations, 203 (2004), 232-254. doi: doi:10.1016/j.jde.2004.05.005.  Google Scholar

[8]

F. Jochmann, On a first-order hyperbolic systems including Bean's model for superconductors with displacement current, J. Diff. Equations, 246 (2009), 2151-2191. doi: doi:10.1016/j.jde.2008.12.023.  Google Scholar

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", 2$^{nd}$ edition, ().   Google Scholar

[10]

R. Picard, An elementary proof for a compact embedding result in generalized electromagnetic theory, Math. Z., 187 (1984), 151-161. doi: doi:10.1007/BF01161700.  Google Scholar

[11]

C. Weber, A local compactness theorem for Maxwell's equations, Math. Methods Appl. Sci., 2 (1980), 12-25. doi: doi:10.1002/mma.1670020103.  Google Scholar

[12]

H. M. Yin, On a $p$-Laplacian type of evolution system and applications to Bean's model in the type-II superconductivity theory, Quarterly. Appl. Math., 59 (2001), 47-66.  Google Scholar

[13]

H. M. Yin, On a singular limit problem for nonlinear Maxwell equations, J. Diff. Equations, 156 (1999), 355-375. doi: doi:10.1006/jdeq.1998.3608.  Google Scholar

[14]

H. M. Yin, B. Q. Li and J. Zou, A degenerate evolution system modeling Bean's critical-state type-II superconductors, Discrete Continuous Dynam. Systems - B, 8 (2002), 781-794.  Google Scholar

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