American Institute of Mathematical Sciences

Advanced
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

Kim Dang Phung 1,

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

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.   Google Scholar

[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.  Google Scholar

[3]

P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire,", Ph.D thesis, (2005).   Google Scholar

[4]

N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.   Google Scholar

[5]

M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications,", World Scientific, (1996).   Google Scholar

[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).   Google Scholar

[7]

G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique,", Dunod, (1972).   Google Scholar

[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.  Google Scholar

[9]

J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I,", Masson, (1988).   Google Scholar

[10]

H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881.   Google Scholar

[11]

K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques,, ESAIM Control Optim. Calc. Var., 5 (2000), 87.  doi: 10.1051/cocv:2000103.  Google Scholar

[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.  Google Scholar

[13]

J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.   Google Scholar

[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.  Google Scholar

[15]

R. Ziolkowski, Exact solutions of the wave equation with complex source locations,, J. Math. Phys., 26 (1985), 861.  doi: 10.1063/1.526579.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire,", Ph.D thesis, (2005).   Google Scholar

[4]

N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 14 (2007), 35.   Google Scholar

[5]

M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications,", World Scientific, (1996).   Google Scholar

[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).   Google Scholar

[7]

G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique,", Dunod, (1972).   Google Scholar

[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.  Google Scholar

[9]

J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I,", Masson, (1988).   Google Scholar

[10]

H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains,, Math. Res. Lett., 16 (2009), 881.   Google Scholar

[11]

K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques,, ESAIM Control Optim. Calc. Var., 5 (2000), 87.  doi: 10.1051/cocv:2000103.  Google Scholar

[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.  Google Scholar

[13]

J. Ralston, Gaussian beams and propagation of singularities,, in, 23 (1982), 206.   Google Scholar

[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.  Google Scholar

[15]

R. Ziolkowski, Exact solutions of the wave equation with complex source locations,, J. Math. Phys., 26 (1985), 861.  doi: 10.1063/1.526579.  Google Scholar

[1]

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
[2]

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
[3]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[4]

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
[5]

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
[6]

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
[7]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[8]

Cleverson R. da Luz, Gustavo Alberto Perla Menzala. Uniform stabilization of anisotropic Maxwell's equations with boundary dissipation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 547-558. doi: 10.3934/dcdss.2009.2.547
[9]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
[10]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[11]

Marco Cappiello, Fabio Nicola. Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1869-1880. doi: 10.3934/dcds.2016.36.1869
[12]

Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69
[13]

Cristina Brändle, Arturo De Pablo. Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1161-1178. doi: 10.3934/cpaa.2018056
[14]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
[15]

S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590
[16]

Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407
[17]

Frank Jochmann. Decay of the polarization field in a Maxwell Bloch system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 663-676. doi: 10.3934/dcds.2003.9.663
[18]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[19]

Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032
[20]

Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences