The coupled heat Maxwell equations with temperature-dependent permittivity. (English) Zbl 07412959

Summary: We consider the heat equation coupled with the Maxwell system, the Ampère-Maxwell equation being coupled to the heat equation by the permittivity, which depends on the temperature due to thermal agitation, and the heat equation being coupled to the Maxwell system by the volumic heat source term. Our purpose is to establish the existence of a local-in-time solution to this coupled problem. Firstly, fixing the temperature distribution, we study the resulting Maxwell system, a nonautonomous system due to the dependence of the permittivity on the temperature and consequently on time, by using the theory of evolution systems. Next, we return to our coupled problem, introducing a fixed-point problem in the closed convex set \(K(0; R) : = \{z \in \overline{B}(0; R); z(0) = 0 \}\) of the Banach space \(C^1([0, T]; C^1( \overline{\Omega}))\) and proving that the hypotheses of Schauder’s theorem are verified for \(R\) sufficiently large. The construction of the fixed-point problem is nontrivial as we need \(K(0; R)\) to be stable.


35Qxx Partial differential equations of mathematical physics and other areas of application
78Axx General topics in optics and electromagnetic theory
65Zxx Applications to the sciences
Full Text: DOI


[1] Agazzi, A.; Van Belle, L., Inverse thermal analysis of melting pool in selective laser melting process, (Material Forming ESAFORM. Material Forming ESAFORM, Key Engineering Materials (2015)), 1519-1524
[2] Ahmed, E.; Barrera, F.; Early, E.; Denton, M.; Clark, C.; Sardar, D., Maxwell’s equations-based dynamic laser-tissue interaction model, Comput. Biol. Med., 43, 2278-2286 (2013)
[3] Alam, T.-M.; Nicaise, S.; Paquet, L., An optimal control problem governed by the heat equation with nonconvex constraints applied to selective laser melting, Minimax Theory and Its Applications, 6, 2, 191-204 (2021) · Zbl 1469.49006
[4] Bossavit, A., Electromagnétisme en Vue de la Modélisation, Mathématiques Applications, SMAI, vol. 14 (1993), Springer-Verlag France: Springer-Verlag France Paris, (in French) · Zbl 0787.65090
[5] Clain, S.; Rappaz, J.; Swierkosz, M.; Touzani, R., Numerical modelling of induction heating for two-dimensional geometries, Math. Models Methods Appl. Sci., 3, 805-822 (1993) · Zbl 0801.65120
[6] Dieudonné, J., Foundations of Modern Analysis, Pure and Applied Mathematics, vol. X (1960), Academic Press: Academic Press New York · Zbl 0100.04201
[7] Duvaut, G.; Lions, J.-L., Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, vol. 219 (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0331.35002
[8] Friedman, A., Partial Differential Equations (1976), Robert E. Krieger: Robert E. Krieger New York
[9] Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5 (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0585.65077
[10] Kittel, C., Physique de l’État Solide (1983), Dunod: Dunod Paris, (in French)
[11] Ladyženskaja, O. A.; Solonnikov, V. A.; Uralceva, N. N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23 (1968), American Mathematical Society: American Mathematical Society Providence, Translated from the Russian by S. Smith
[12] Nicaise, S., Exact boundary controllability of Maxwell’s equations in heterogeneous media and an application to an inverse source problem, SIAM J. Control Optim., 38, 4, 1145-1170 (2000) · Zbl 0963.93041
[13] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44 (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[14] Pérez, J.-P., Optique Géométrique et Ondulatoire (1994), Masson: Masson Paris, (in French)
[15] Picard, R. H.; Zajaczkowski, W. M., Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations, Math. Methods Appl. Sci., 18, 169-199 (1995) · Zbl 0879.35151
[16] Showalter, R., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, vol. 49 (1997), American Mathematical Society: American Mathematical Society Providence · Zbl 0870.35004
[17] Spivak, M., Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Mathematics Monograph Series (1965), W.A. Benjamin · Zbl 0141.05403
[18] Taine, J.; Petit, J.-P., Heat Transfer (1993), Prentice Hall: Prentice Hall New York · Zbl 0818.73001
[19] Tröltzsch, F.; Yousept, I., PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages, ESAIM: Math. Model. Numer. Anal., 46, 4, 709-729 (2012) · Zbl 1288.78040
[20] Van Belle, L., Analysis, modeling and simulation of residual stresses during the SLM process of metallic powders (2013), INSA de Lyon: INSA de Lyon Lyon, Thesis
[21] Yin, H.-M., Regularity of weak solution to Maxwell’s equations and applications to microwave heating, J. Differ. Equ., 200, 137-161 (2004) · Zbl 1068.35169
[22] Ziman, J., Principles of the Theory of Solids (1964), Cambridge University Press: Cambridge University Press London · Zbl 0121.44801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.