Maxwell equations with localized internal damping: strong and polynomial stability

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

We study the Maxwell system with localized conductivity σ and the boundary conditions of a perfect conductor on a simply connected domain Ω, assuming that there are no electric charges off the support of σ. For matrix-valued permittivity ε and permeability µ, we show strong stability of the underlying semigroup by checking the spectral criteria of the Arendt–Batty–Lyubich–V˜u Theorem. If ε = µ = 1, Ω is the cube (0, π)³ and supp σ contains a strip, the semigroup is polynomially stable of rate 1 2. To derive this result, we establish the resolvent estimate of the Borichev–Tomilov Theorem using an orthonormal basis of eigenfunctions of the Maxwell operator for σ = 0.

Keywords

stability / Maxwell system / localized conductivity / ABLV theorem / Borichev–Tomilov theorem

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] M. Eller, Stability of the anisotropic Maxwell equations with a conductivity term, Evol. Equ. Control Theory, 8 (2019), 343-357. https://doi.org/10.3934/eect.2019018

[2] S. Nicaise, C. Pignotti, Internal stabilization of Maxwell’s equations in heterogeneous media, Abstr. Appl. Anal., 2005 (2005), 791-811. https://doi.org/10.1155/AAA.2005.791

[3] S. Nicaise, C. Pignotti, Internal and boundary observability estimates for the heterogeneous Maxwell’s system, Appl. Math. Optim., 54 (2006), 47-70. https://doi.org/10.1007/s00245-005-0851-0

[4] K. D. Phung, Contrôle et stabilisation d’ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137. https://doi.org/10.1051/cocv:2000103

[5] I. Lasiecka, M. Pokojovy, R. Schnaubelt, Exponential decay of quasilinear Maxwell equations with interior conductivity, Nonlinear Differ. Equ. Appl., 26 (2019), 51. https://doi.org/10.1007/s00030-019-0595-1

[6] M. Eller, Continuous observability for the anisotropic Maxwell system, Appl. Math. Optim., 55 (2007), 185-201. https://doi.org/10.1007/s00245-006-0886-x

[7] M. Eller, J. E. Lagnese, S. Nicaise, Decay rates for solutions of a Maxwell system with nonlinear boundary damping, Comput. Appl. Math., 21 (2002), 135-165.

[8] B. V. Kapitonov, Stabilization and exact boundary controllability for Maxwell’s equations, SIAM J. Control Optim., 32 (1994), 408-420. https://doi.org/10.1137/S0363012991218487

[9] V. Komornik, Boundary stabilization, observation and control of Maxwell’s equations, PanAmer. Math. J., 4 (1994), 47-61.

[10] J. E. Lagnese, Exact boundary controllability of Maxwell’s equations in a general region, SIAM J. Control Optim., 27 (1989), 374-388. https://doi.org/10.1137/0327019

[11] R. Nutt, R. Schnaubelt, Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary, J. Math. Anal. Appl., 532 (2024), 127915. https://doi.org/10.1016/j.jmaa.2023.127915

[12] M. Cessenat, Mathematical Methods in Electromagnetism , World Scientific Publishing Co., Inc., River Edge, NJ, 1996. https://doi.org/10.1142/2938

[13] N. Skrepek, M. Waurick, Semi-uniform stabilization of anisotropic Maxwell’s equations via boundary feedback on split boundary, J. Differential Equations, 394 (2024), 345-374. https://doi.org/10.1016/j.jde.2024.03.021

[14] K. D. Phung, Energy decay for Maxwell’s equations with Ohm’s law in partially cubic domains, Commun. Pure Appl. Anal., 12 (2013), 2229-2266. https://doi.org/10.3934/cpaa.2013.12.2229

[15] Z. Liu, B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. https://doi.org/10.1007/s00033-004-3073-4

[16] M. Costabel, M. Dauge, Maxwell eigenmodes in product domains, In Maxwell’s equations-analysis and numerics, volume 24 of Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 2019, 171-198. https://doi.org/10.1515/9783110543612-006

[17] F. Assous, P. Ciarlet, S. Labrunie, Mathematical Foundations of Computational Electromagnetism , Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-70842-3

[18] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. https://doi.org/10.1090/S0002-9947-1988-0933321-3

[19] Y. I. Lyubich, Q. P. V˜u, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. https://doi.org/10.4064/sm-88-1-37-42

[20] T. Nguyen, J.-N. Wang, Quantitative uniqueness estimate for the Maxwell system with Lipschitz anisotropic media, Proc. Amer. Math. Soc., 140 (2012), 595-605. https://doi.org/10.1090/S0002-9939-2011-11137-7

[21] M. Eller, M. Yamamoto, A Carleman inequality for the stationary anisotropic Maxwell system, J. Math. Pures Appl., 86 (2006), 449-462. https://doi.org/10.1016/j.matpur.2006.10.004

[22] C. Amrouche, C. Bernardi, M. Dauge, V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Appl. Sci., 21 (1998), 823-864. https://doi.org/10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B

[23] M. Costabel, M. Dauge, Singularities of electromagnetic fields in polyhedral domains, Arch Rational Mech. Anal., 151 (2000), 221-276. https://doi.org/10.1007/s002050050197

[24] M. Costabel, M. Dauge, S. Nicaise, Singularities of Maxwell interface problems, RAIRO Modél. Math. Anal. Numér., 33 (1999), 627-649. https://doi.org/10.1051/m2an:1999155

[25] R. Picard, On the boundary value problems of electro- and magnetostatics, Proc. Roy. Soc. Edinburgh Sect. A, 92 (1982), 165-174. https://doi.org/10.1017/S0308210500020023

[26] D. Pauly, R. Picard, S. Trostorff, M. Waurick, On a class of degenerate abstract parabolic problems and applications to some eddy current models, J. Funct. Anal., 280 (2021), 108847. https://doi.org/10.1016/j.jfa.2020.108847

[27] M. Costabel, M. Dauge, S. Nicaise, Singularities of eddy current problems, M2AN Math. Model. Numer. Anal., 37 (2003), 807-831. https://doi.org/10.1051/m2an:2003056

[28] S. Goldberg, Unbounded Linear Operators , Dover Publications, Inc., Mineola, NY, 2006.

[29] A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. https://doi.org/10.1007/s00208-009-0439-0

[30] A. Bátkai, K.-J. Engel, J. Prüss, R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440. https://doi.org/10.1002/mana.200410429

[31] C. J. K. Batty, T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. https://doi.org/10.1007/s00028-008-0424-1