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Stability of the anisotropic Maxwell equations with a conductivity term
Department of Mathematics, Georgetown University, Washington, DC 20057, USA |
The dynamic Maxwell equations with a conductivity term are considered. Conditions for the exponential and strong stability of an initial-boundary value problem are given. The permeability and the permittivity are assumed to be $ 3\times 3 $ symmetric, positive definite tensors. A result concerning solutions of higher regularity is obtained along the way.
References:
| [1] |
M. Belishev and A. Glasman, Boundary control of the Maxwell dynamical system: Lack of controllability by topological reasons, in Mathematical and Numerical Aspects of Wave Propagation—WAVES 2003, Springer, Berlin, 2003,177-182. |
| [2] |
J. Cagnol and M. Eller,
Boundary regularity for Maxwell's equations with applications to shape optimization, J. Differential Equations, 250 (2011), 1114-1136.
doi: 10.1016/j.jde.2010.08.004. |
| [3] |
M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co. Inc., River Edge, NJ, 1996, Linear theory and applications.
doi: 10.1142/2938. |
| [4] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990, Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson. |
| [5] |
W. Desch, R. Grimmer and W. Schappacher,
Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234.
doi: 10.1016/0022-247X(84)90044-1. |
| [6] |
M. Eller and D. Toundykov,
A global holmgren theorem for multidimensional hyperbolic partial differential equations, Applicable Analysis, 91 (2012), 69-90.
doi: 10.1080/00036811.2010.538685. |
| [7] |
M. M. Eller,
Continuous observability for the anisotropic Maxwell system, Appl. Math. Optim., 55 (2007), 185-201.
doi: 10.1007/s00245-006-0886-x. |
| [8] |
M. M. Eller and M. Yamamoto,
A Carleman inequality for the stationary anisotropic Maxwell system, J. Math. Pures Appl. (9), 86 (2006), 449-462.
doi: 10.1016/j.matpur.2006.10.004. |
| [9] |
N. Iwasaki,
Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains, Publ. Res. Inst. Math. Sci., 5 (1969), 193-218.
doi: 10.2977/prims/1195194630. |
| [10] |
P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967.
|
| [11] |
R. Leis,
über die eindeutige Fortsetzbarkeit der Lösungen der Maxwellschen Gleichungen in anisotropen inhomogenen Medien, Bul. Inst. Politehn. Iașsi (N.S.), 14 (1968), 119-124.
|
| [12] |
A. Majda and S. Osher,
Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675.
doi: 10.1002/cpa.3160280504. |
| [13] |
C. S. Morawetz,
The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561-568.
doi: 10.1002/cpa.3160140327. |
| [14] |
S. Nicaise and C. Pignotti, Internal stabilization of Maxwell's equations in heterogeneous media, Abstr. Appl. Anal., 2005,791-811.
doi: 10.1155/AAA.2005.791. |
| [15] |
T. Ohkubo,
Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary, Hokkaido Math. J., 10 (1981), 93-123.
doi: 10.14492/hokmj/1381758116. |
| [16] |
A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [17] |
K. D. Phung,
Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137.
doi: 10.1051/cocv:2000103. |
| [18] |
R. Picard and W. Zajaczkowski,
Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Mathematical Methods in the Applied Sciences, 18 (1995), 169-199.
doi: 10.1002/mma.1670180302. |
| [19] |
J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, vol. 133 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2012.
doi: 10.1090/gsm/133. |
| [20] |
J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86, URL https://doi.org/10.1512/iumj.1974.24.24004.
doi: 10.1512/iumj.1975.24.24004. |
| [21] |
J. B. Rauch and F. J. Massey Ⅲ,
Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318.
doi: 10.2307/1996861. |
| [22] |
V. A. Solonnikov,
Overdetermined elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 199 (1971), 279-281.
|
| [23] |
M. Spitz, Local Wellposedness of Nonlinear Maxwell Equations, Karlsruhe Institute of Technology, https://doi.org/10.5445/IR/1000078030, 2017, Ph.D Thesis. |
show all references
References:
| [1] |
M. Belishev and A. Glasman, Boundary control of the Maxwell dynamical system: Lack of controllability by topological reasons, in Mathematical and Numerical Aspects of Wave Propagation—WAVES 2003, Springer, Berlin, 2003,177-182. |
| [2] |
J. Cagnol and M. Eller,
Boundary regularity for Maxwell's equations with applications to shape optimization, J. Differential Equations, 250 (2011), 1114-1136.
doi: 10.1016/j.jde.2010.08.004. |
| [3] |
M. Cessenat, Mathematical Methods in Electromagnetism, vol. 41 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co. Inc., River Edge, NJ, 1996, Linear theory and applications.
doi: 10.1142/2938. |
| [4] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3, Springer-Verlag, Berlin, 1990, Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson. |
| [5] |
W. Desch, R. Grimmer and W. Schappacher,
Some considerations for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234.
doi: 10.1016/0022-247X(84)90044-1. |
| [6] |
M. Eller and D. Toundykov,
A global holmgren theorem for multidimensional hyperbolic partial differential equations, Applicable Analysis, 91 (2012), 69-90.
doi: 10.1080/00036811.2010.538685. |
| [7] |
M. M. Eller,
Continuous observability for the anisotropic Maxwell system, Appl. Math. Optim., 55 (2007), 185-201.
doi: 10.1007/s00245-006-0886-x. |
| [8] |
M. M. Eller and M. Yamamoto,
A Carleman inequality for the stationary anisotropic Maxwell system, J. Math. Pures Appl. (9), 86 (2006), 449-462.
doi: 10.1016/j.matpur.2006.10.004. |
| [9] |
N. Iwasaki,
Local decay of solutions for symmetric hyperbolic systems with dissipative and coercive boundary conditions in exterior domains, Publ. Res. Inst. Math. Sci., 5 (1969), 193-218.
doi: 10.2977/prims/1195194630. |
| [10] |
P. D. Lax and R. S. Phillips, Scattering Theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967.
|
| [11] |
R. Leis,
über die eindeutige Fortsetzbarkeit der Lösungen der Maxwellschen Gleichungen in anisotropen inhomogenen Medien, Bul. Inst. Politehn. Iașsi (N.S.), 14 (1968), 119-124.
|
| [12] |
A. Majda and S. Osher,
Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675.
doi: 10.1002/cpa.3160280504. |
| [13] |
C. S. Morawetz,
The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561-568.
doi: 10.1002/cpa.3160140327. |
| [14] |
S. Nicaise and C. Pignotti, Internal stabilization of Maxwell's equations in heterogeneous media, Abstr. Appl. Anal., 2005,791-811.
doi: 10.1155/AAA.2005.791. |
| [15] |
T. Ohkubo,
Regularity of solutions to hyperbolic mixed problems with uniformly characteristic boundary, Hokkaido Math. J., 10 (1981), 93-123.
doi: 10.14492/hokmj/1381758116. |
| [16] |
A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [17] |
K. D. Phung,
Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137.
doi: 10.1051/cocv:2000103. |
| [18] |
R. Picard and W. Zajaczkowski,
Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Mathematical Methods in the Applied Sciences, 18 (1995), 169-199.
doi: 10.1002/mma.1670180302. |
| [19] |
J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, vol. 133 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2012.
doi: 10.1090/gsm/133. |
| [20] |
J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86, URL https://doi.org/10.1512/iumj.1974.24.24004.
doi: 10.1512/iumj.1975.24.24004. |
| [21] |
J. B. Rauch and F. J. Massey Ⅲ,
Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318.
doi: 10.2307/1996861. |
| [22] |
V. A. Solonnikov,
Overdetermined elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 199 (1971), 279-281.
|
| [23] |
M. Spitz, Local Wellposedness of Nonlinear Maxwell Equations, Karlsruhe Institute of Technology, https://doi.org/10.5445/IR/1000078030, 2017, Ph.D Thesis. |
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