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Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation
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 |
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), 823-864. |
| [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), 1024-1065.
doi: 10.1137/0330055. |
| [3] |
P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire," Ph.D thesis, Université de Rennes 1, 2005. |
| [4] |
N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett., 14 (2007), 35-47. |
| [5] |
M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications," World Scientific, Singapore, 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, Paris, 1988. |
| [7] |
G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique," Dunod, Paris, 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), 1375-1396.
doi: 10.1016/j.jmaa.2006.06.101. |
| [9] |
J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I," Masson, Paris, 1988. |
| [10] |
H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains, Math. Res. Lett., 16 (2009), 881-894. |
| [11] |
K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137.
doi: 10.1051/cocv:2000103. |
| [12] |
K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Diff. Eq., 240 (2007), 92-124.
doi: 10.1016/j.jde.2007.05.016. |
| [13] |
J. Ralston, Gaussian beams and propagation of singularities, in "Studies in Partial Differential Equations" (eds. W. Littman), MAA studies in Mathematics, 23 (1982), 206-248. |
| [14] |
W. Wei, H-M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Analysis, 75 (2012), 2024-2036.
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), 861-863.
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), 823-864. |
| [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), 1024-1065.
doi: 10.1137/0330055. |
| [3] |
P. Boissoles, "Problèmes mathématiques et numériques issus de l'imagerie par résonance magnétique nucléaire," Ph.D thesis, Université de Rennes 1, 2005. |
| [4] |
N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett., 14 (2007), 35-47. |
| [5] |
M. Cessenat, "Mathematical Method in Electromagnetism, Linear Theory and Applications," World Scientific, Singapore, 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, Paris, 1988. |
| [7] |
G. Duvaut and J.-L. Lions, "Les inéquations en mécanique et en physique," Dunod, Paris, 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), 1375-1396.
doi: 10.1016/j.jmaa.2006.06.101. |
| [9] |
J.-L. Lions, "Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués I," Masson, Paris, 1988. |
| [10] |
H. Nishiyama, Polynomial decay for damped wave equations on partially rectangular domains, Math. Res. Lett., 16 (2009), 881-894. |
| [11] |
K. D. Phung, Contrôle et stabilisation d'ondes électromagnétiques, ESAIM Control Optim. Calc. Var., 5 (2000), 87-137.
doi: 10.1051/cocv:2000103. |
| [12] |
K. D. Phung, Polynomial decay rate for the dissipative wave equation, J. Diff. Eq., 240 (2007), 92-124.
doi: 10.1016/j.jde.2007.05.016. |
| [13] |
J. Ralston, Gaussian beams and propagation of singularities, in "Studies in Partial Differential Equations" (eds. W. Littman), MAA studies in Mathematics, 23 (1982), 206-248. |
| [14] |
W. Wei, H-M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Analysis, 75 (2012), 2024-2036.
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), 861-863.
doi: 10.1063/1.526579. |
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