Energy decay for Maxwell's equations with Ohm's law in partially cubic domains

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September 2013, 12(5): 2229-2266. doi: 10.3934/cpaa.2013.12.2229

Energy decay for Maxwell's equations with Ohm's law in partially cubic domains

1.	Université d'Orléans, Laboratoire MAPMO, CNRS UMR 7349, Fédération Denis Poisson, FR CNRS 2964, Bâtiment de Mathématiques, B.P. 6759, 45067 Orléans Cedex 2, France

Received January 2012 Revised June 2012 Published January 2013

Abstract
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We prove a polynomial energy decay for the Maxwell's equations with Ohm's law in partially cubic domains with trapped rays. We extend the results of polynomial decay for the scalar damped wave equation in partially rectangular or cubic domain. Our approach have some similitude with the construction of reflected gaussian beams.

Keywords: Maxwell's equations, trapped ray., decay estimates.

Mathematics Subject Classification: Primary: 35B40, 35Q61; Secondary: 93D1.

Citation: Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229

References:

[1]	C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains,, Math. Methods Appl. Sci., 21 (1998), 823.
[2]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.
[3]	P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire,", Ph.D thesis, (2005).
[4]	N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.
[5]	M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications,", World Scientific, (1996).
[6]	R. Dautray and J.-L. Lions, "Analyse mathématique et calcul numérique pour les sciences et les techniques, Volume 5, Spectre des opérateurs,", Masson, (1988).
[7]	G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique,", Dunod, (1972).
[8]	S. S. Krigman and C. E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability,, J. Math. Anal. Appl., 329 (2007), 1375. doi: 10.1016/j.jmaa.2006.06.101.
[9]	J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I,", Masson, (1988).
[10]	H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881.
[11]	K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques,, ESAIM Control Optim. Calc. Var., 5 (2000), 87. doi: 10.1051/cocv:2000103.
[12]	K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Diff. Eq., 240 (2007), 92. doi: 10.1016/j.jde.2007.05.016.
[13]	J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.
[14]	W. Wei, H-M. Yin and J. Tang, An optimal control problem for microwave heating,, Nonlinear Analysis, 75 (2012), 2024. doi: 10.1016/j.na.2011.10.003.
[15]	R. Ziolkowski, Exact solutions of the wave equation with complex source locations,, J. Math. Phys., 26 (1985), 861. doi: 10.1063/1.526579.

show all references

References:

[1]	C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains,, Math. Methods Appl. Sci., 21 (1998), 823.
[2]	C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.
[3]	P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire,", Ph.D thesis, (2005).
[4]	N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.
[5]	M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications,", World Scientific, (1996).
[6]	R. Dautray and J.-L. Lions, "Analyse mathématique et calcul numérique pour les sciences et les techniques, Volume 5, Spectre des opérateurs,", Masson, (1988).
[7]	G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique,", Dunod, (1972).
[8]	S. S. Krigman and C. E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability,, J. Math. Anal. Appl., 329 (2007), 1375. doi: 10.1016/j.jmaa.2006.06.101.
[9]	J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I,", Masson, (1988).
[10]	H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881.
[11]	K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques,, ESAIM Control Optim. Calc. Var., 5 (2000), 87. doi: 10.1051/cocv:2000103.
[12]	K. D. Phung, Polynomial decay rate for the dissipative wave equation,, J. Diff. Eq., 240 (2007), 92. doi: 10.1016/j.jde.2007.05.016.
[13]	J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.
[14]	W. Wei, H-M. Yin and J. Tang, An optimal control problem for microwave heating,, Nonlinear Analysis, 75 (2012), 2024. doi: 10.1016/j.na.2011.10.003.
[15]	R. Ziolkowski, Exact solutions of the wave equation with complex source locations,, J. Math. Phys., 26 (1985), 861. doi: 10.1063/1.526579.

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