We consider a linearized Euler–Maxwell model for the propagation and absorption of electromagnetic waves in a magnetized plasma. We present the derivation of the model, and we show its well-posedeness, its strong and polynomial stability under suitable and fairly general assumptions, its exponential stability in the same conditions as the Maxwell system, and finally its convergence to the time-harmonic regime. No homogeneity assumption is made, and the topological and geometrical assumptions on the domain are minimal. These results appear strongly linked to the spectral properties of various matrices describing the anisotropy and other plasma properties.
Accepté le :
Première publication :
Publié le :
Keywords: Maxwell equations, plasma, hydrodynamic models, stabilization, absorbing boundary condition, evolution semigroups, strong stability, exponential stability
@article{COCV_2021__27_1_A62_0,
author = {Labrunie, Simon and Zaafrani, Ibtissem},
title = {Linearized electrodynamics and stabilization of a cold magnetized plasma},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021056},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021056/}
}
TY - JOUR AU - Labrunie, Simon AU - Zaafrani, Ibtissem TI - Linearized electrodynamics and stabilization of a cold magnetized plasma JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021056/ DO - 10.1051/cocv/2021056 LA - en ID - COCV_2021__27_1_A62_0 ER -
%0 Journal Article %A Labrunie, Simon %A Zaafrani, Ibtissem %T Linearized electrodynamics and stabilization of a cold magnetized plasma %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021056/ %R 10.1051/cocv/2021056 %G en %F COCV_2021__27_1_A62_0
Labrunie, Simon; Zaafrani, Ibtissem. Linearized electrodynamics and stabilization of a cold magnetized plasma. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 60. doi: 10.1051/cocv/2021056
[1] , , and , Vector potentials in three-dimensional non-smooth domains. Math. Meth. Appl. Sci. 21 (1998) 823–864.
[2] and , Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 305 (1988) 837–852.
[3] , and , Mathematical foundations of computational electromagnetism . Appl. Math. Sci. 198 (2018).
[4] , , , and , Electromagnetic wave propagation and absorption in magnetised plasmas: variational formulations and domain decomposition. ESAIM: M2AN 49 (2015) 1239–1260.
[5] and , Asymptotic behavior of solutions to Maxwell’s system in bounded domains with absorbing boundary Silver–Mïler’s condition on the exterior boundary. Asympt. Anal. 15 (1997) 25–40.
[6] and , Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478.
[7] and , On traces for functional spaces related to Maxwell’s equations. Part I: An integration by parts formula in Lipschitz polyhedra. An integration by parts formula in Lipschitz polyhedra. Math. Meth. Appl. Sci. 24 (2001) 9–30.
[8] and , On traces for functional spaces related to Maxwell’s equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Meth. Appl. Sci. 24 (2001) 31–48.
[9] and , An Introduction to Semilinear Evolution Equations. Vol. 13 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (1998).
[10] , Introduction à l’analyse numérique matricielle et à l’optimisation. Masson Paris Milan Barcelone Mexico (1988).
[11] , and , Decay rates for solutions of a Maxwell system with nonlinear boundary damping. Comput. Appl. Math. 21 (2002) 135–165.
[12] and , Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Meth. App. Sci. 7 (1997) 957–991.
[13] , Décomposition de domaine pour la simulation Full-Wave dans un plasma froid. Thèse, Université de Lorraine (2014). http://docnum.univ-lorraine.fr/public/DDOC_T_2014_0380_HATTORI.pdf.
[14] , Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43–56.
[15] , Boundary stabilization, observation and control of Maxwell’s equations. PanAmer. Math. J. 4 (1994) 47–61.
[16] and , Dynamique d’un plasma magnétique froid. Prépublication HAL no. 01572067, version 2, 2017. Online: . | HAL
[17] and , Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 37–42.
[18] , Stabilization and asymptotic behavior of dispersive medium models. Syst. Control Lett. 61 (2012), 638–648.
[19] , Stabilization of a Drude / vacuum model. J. Anal. Appl. 37 (2018) 349–375.
[20] and , Boundary stabilization of Maxwell?s equations with space-time variable coefficients. ESAIM: COCV 9 (2003) 563–578.
[21] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Vol. 44 of Applied Mathematical Sciences. Springer-Verlag (1983).
[22] , Contrôle et stabilisation d’ondes électromagnétiques. ESAIM: COCV 5 (2000) 87–137.
[23] , On the spectrum of C0-semigroups. Trans. Am. Math. Soc. 284 (1984) 847–857.
[24] , Waves in plasmas. American Institute of Physics, New York (1992).
[25] , , , and , Theoretical and numerical analysis of local dispersion models coupled to a discontinuous Galerkin time-domain method for Maxwell’s equations. Technical Report 8298, INRIA (2013). | HAL
[26] , A local compactness theorem for Maxwell’s equations. Math. Meth. Appl. Sci. 2 (1980) 12–25.
Cité par Sources :
The second author thanks the Campus France Eiffel Excellence Programme for its financial support.
