Linear stability for a thermoelectromagnetic material with memory
Author:
Giovambattista Amendola
Journal:
Quart. Appl. Math. 59 (2001), 67-84
MSC:
Primary 35Q60; Secondary 35A05, 35B35, 78A25
DOI:
https://doi.org/10.1090/qam/1811095
MathSciNet review:
MR1811095
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Abstract: In this paper we study the behaviour of a three-dimensional linear thermoelectromagnetic material, which has constitutive equations with memory effects for both the heat flux and the electric current density. We develop a linearized theory of thermodynamics, in which context we are able to introduce a maximal free energy defined in the frequency domain. Using this free energy, a domain of dependence is obtained. Moreover, we prove a theorem of uniqueness, existence, and asymptotic stability.
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M. McCarthy, Constitutive equations for thermodynamical materials with memory, Internat. J. Engrg. Sci. 8, 467 474 (1970)
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- Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021
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E. B. Bykhousekis and N. V. Smirnov, On the orthogonal decomposition of the space of vector functions square summable in a given domain, Trudy Mat. Inst. Steklov 59, 6–36 (1960)
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L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960
C. Muller, Foundation of Mathematical Theory of Electromagnetic Waves, Springer, Berlin, 1969
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48, 1–50 (1972)
B. D. Coleman and D. R. Owen, A mathematical foundation of thermodynamics, Arch. Rational Mech. Anal. 54, 1–104 (1974)
M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113 126 (1968)
M. McCarthy, Constitutive equations for thermodynamical materials with memory, Internat. J. Engrg. Sci. 8, 467 474 (1970)
B. D. Coleman and E. H. Dill, Thermodynamic restrictions on the constitutive equations of electromagnetic theory, ZAMP 22, 691-702 (1971)
B. D. Coleman and E. H. Dill, On the thermodynamics of electromagnetic fields in materials with memory, Arch. Rational Mech. Anal. 41, 132-162 (1971)
M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, Philadelphia, 1992
M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism, Arch. Rational Mech. Anal. 136, 359 381 (1996)
E. B. Bykhousekis and N. V. Smirnov, On the orthogonal decomposition of the space of vector functions square summable in a given domain, Trudy Mat. Inst. Steklov 59, 6–36 (1960)
G. Amendola, On thermodynamic conditions for the stability of a thermoelectromagnetic system, Math. Methods Appl. Sci. 23, 17 39 (2000)
R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Monographs and Studies in Mathematics, Vol. 1, Pitman, London, 1977
F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York, 1975
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1960
C. Muller, Foundation of Mathematical Theory of Electromagnetic Waves, Springer, Berlin, 1969
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Article copyright:
© Copyright 2001
American Mathematical Society