×

Homogenization of the wave equation in composites with imperfect interface: a memory effect. (English) Zbl 1112.35017

The authors describe the asymptotic behaviour of the solution \( u_{\varepsilon }\) of the linear wave equation \(u_{\varepsilon }^{\prime \prime }- \text{div}(A^{\varepsilon }\nabla u_{\varepsilon })=f_{\varepsilon } \), posed in \(\Omega \times (0,T)\), where \(\Omega =\Omega _{1\varepsilon }\cup \Omega _{2\varepsilon }\) is a domain of \(\mathbb R^{n}\), \(\Omega _{1\varepsilon }\) being connected and \(\Omega _{2\varepsilon }\) being the union of disjoint inclusions of size \(\varepsilon \). Here \(A^{\varepsilon }(x)\) is equal to \(A(x/\varepsilon )\), where \(A\) is a periodic, bounded, symmetric and positive definite matrix. Homogeneous Dirichlet boundary conditions are imposed on the boundary \(\partial \Omega \times (0,T)\). The following boundary conditions are imposed on \(\Gamma ^{\varepsilon }\times (0,T)\), with \(\Gamma ^{\varepsilon }=\partial \Omega _{2\varepsilon }\): \( [A^{\varepsilon }\nabla u_{\varepsilon }]\cdot n_{1\varepsilon }=0\), \( A^{\varepsilon }\nabla u_{1\varepsilon }\cdot n_{1\varepsilon }=-\varepsilon ^{\gamma }h^{\varepsilon }[u_{\varepsilon }]\). Here \([\;]\) means the jump of the function across \(\Gamma ^{\varepsilon }\), \(n_{1\varepsilon }\) is the unit outward normal to \(\Omega _{1\varepsilon }\) and \(h^{\varepsilon }(x)\) is equal to \(h(x/\varepsilon )\) for some positive, bounded and periodic function \(h\). The initial data \(U_{\varepsilon }^{0}=u_{\varepsilon }(0)\) and \(U_{\varepsilon }^{1}=u_{\varepsilon }^{\prime }(0)\) are supposed to converge in some weak-\(L^{2}\) sense, together with the sources \( f_{\varepsilon }\).
The authors prove the convergence of the solution \( u_{\varepsilon }\) to the solution \(u_{1}\) of the wave equation \( u_{1}^{\prime \prime }- \text{div}(A_{\gamma }^{0}\nabla u_{1})=f_{1}+f_{2}\), plus the homogeneous Dirichlet boundary conditions and some associated initial data. The matrix \(A_{\gamma }^{0}\) is proved to take different values according to the position of \(\gamma >-1\) with respect to 1. When \( \gamma =1\) a term associated to some memory effect appears in this limit matrix. The authors first prove estimates on the solution using some Galerkin method. They then prove the convergence results building appropriate test-functions.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35L05 Wave equation
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
35L20 Initial-boundary value problems for second-order hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arbogast, T.; Douglas, J.; Hornung, U., Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., 21, 4, 823-836 (1990) · Zbl 0698.76106
[2] Auriault, J. L.; Ene, H., Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Internat. J. Heat Mass Transfer, 37, 2885-2892 (1994) · Zbl 0900.73453
[3] Bensoussan, A.; Lions, J. L.; Papanicolaou, G., Asymptotic Analysis for Periodic Structures (1978), North-Holland: North-Holland Amsterdam · Zbl 0411.60078
[4] Canon, E.; Pernin, J. N., Homogenization of diffusion in composite media with interfacial barrier, Rev. Roumaine Math. Pures Appl., 44, 1, 23-36 (1999) · Zbl 0994.35023
[5] Cioranescu, D.; Saint-Jean Paulin, J., Homogenization in open sets with holes, J. Math. Anal. Appl., 71, 590-607 (1979) · Zbl 0427.35073
[6] Cioranescu, D.; Donato, P., Exact internal controllability in perforated domains, J. Math. Pures Appl., 68, 185-213 (1989) · Zbl 0627.35057
[7] Cioranescu, D.; Donato, P., An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, vol. 17 (1999), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0939.35001
[8] Colombini, F.; Spagnolo, S., On the Convergence of Solutions of Hyperbolic Equations, Comm. Partial Differential Equations, Ser. 3, 1, 77-103 (1978) · Zbl 0375.35034
[9] Donato, P., Some corrector results for composites with imperfect interface, Rend. Mat. Ser. VII, 26, 189-209 (2006) · Zbl 1129.35008
[10] Donato, P.; Monsurrò, S., Homogenization of two heat conductors with interfacial contact resistance, Anal. Appl., 2, 3, 247-273 (2004) · Zbl 1083.35014
[11] Ene, H.; Polisevski, D., Model of diffusion in partially fissured media, Z. Angew. Math. Phys., 53, 1052-1059 (2002) · Zbl 1017.35016
[12] Hummel, H. C., Homogenization for heat transfer in polycrystals with interfacial resistances, Appl. Anal., 75, 3-4, 403-424 (2000) · Zbl 1024.80005
[13] Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes et applications (1968), Dunod: Dunod Paris · Zbl 0165.10801
[14] Lipton, R., Heat conduction in fine scale mixtures with interfacial contact resistance, SIAM J. Appl. Math., 58, 1, 55-72 (1998) · Zbl 0913.35010
[15] Lipton, R.; Vernescu, B., Composite with imperfect interface, Proc. Soc. Lond. A, 452, 329-358 (1996) · Zbl 0872.73033
[16] Mascarenhas, M. L., Linear homogenization problem with time dependent coefficient, Trans. Amer. Math. Soc., 281, 179-195 (1984) · Zbl 0536.45003
[17] Monsurrò, S., Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13, 1, 43-63 (2003) · Zbl 1052.35022
[18] Pernin, J. N., Homogénéisation d’un problème de diffusion en milieu composite à deux composantes, C. R. Acad. Sci. Paris Sér. I, 321, 949-952 (1995) · Zbl 0842.73047
[19] Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 127 (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0432.70002
[20] Tartar, L., Quelques remarques sur l’homogénéisation, (Fujita, M., Functional Analysis and Numerical Analysis, Proc. Japan-France Seminar 1976 (1978), Japanese Society for the Promotion of Science), 468-482
[21] Tartar, L., Memory effects an homogenization, Arch. Rational Mech. Anal., 2, 121-133 (1990) · Zbl 0725.45012
[22] Zeidler, E., Nonlinear Functional Analysis and Its Applications, II/B. Nonlinear Monotone Operators (1990), Springer-Verlag: Springer-Verlag New York
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.