Corrector results for a parabolic problem with a memory effect

Donato, Patrizia; Jose, Editha C.

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 3, pp. 421-454.

The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl. 54 (2009) 189-222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of order ε^γ. We suppose that -1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases -1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. As seen in [Jose, Rev. Roumaine Math. Pures Appl. 54 (2009) 189-222] and [Faella and Monsurrò, Topics on Mathematics for Smart Systems, World Sci. Publ., Hackensack, USA (2007) 107-121] (also in [Donato et al., J. Math. Pures Appl. 87 (2007) 119-143]), the case γ = 1 is more interesting because of the presence of a memory effect in the homogenized problem.

Classification : 35B27, 35K20, 82B24
Mots clés : periodic homogenization, correctors, heat equation, interface problems

[1] J.L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier. International J. Heat Mass Transfer 37 (1994) 2885-2892. | Zbl

[2] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). | Zbl

[3] S. Brahim-Otsman, G.A. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 8 (1992) 197-231. | Zbl

[4] M. Briane, A. Damlamian and P. Donato, H-convergence in Perforated Domains, in Nonlinear Partial Differential Equations and Their Applications - Collège de France Seminar XIII, D. Cioranescu and J.L. Lions Eds., Pitman Research Notes in Mathematics Series 391, Longman, New York, USA (1998) 62-100. | Zbl

[5] H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids. The Clarendon Press, Oxford, UK (1947). | Zbl

[6] D. Cioranescu and P. Donato, Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptot. Anal. 1 (1988) 115-138. | Zbl

[7] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains. J. Math. Pures Appl. 68 (1989) 185-213. | Zbl

[8] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications 17. Oxford Univ. Press, New York, USA (1999). | Zbl

[9] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. | Zbl

[10] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). | Zbl

[11] D. Cioranescu, P. Donato, F. Murat and E. Zuazua, Homogenization and corrector for the wave equation in domains with small holes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1999) 251-293. | Numdam | Zbl

[12] P. Donato, Some corrector results for composites with imperfect interface. Rend. Math. Ser. VII 26 (2006) 189-209. | Zbl

[13] P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl. 2 (2004) 1-27. | Zbl

[14] P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domains. ESAIM: COCV 6 (2001) 21-38. | Numdam | Zbl

[15] P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Chin. Ann. Math. B 25 (2004) 143-156. | Zbl

[16] P. Donato, A. Gaudiello and L. Sgambati, Homogenization of bounded solutions of elliptic equations with quadratic growth in periodically perforated domains. Asymptot. Anal. 16 (1998) 223-243. | Zbl

[17] P. Donato, L. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: a memory effect. J. Math. Pures Appl. 87 (2007) 119-143. | Zbl

[18] P. Donato, L. Faella and S. Monsurrò, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces. SIAM J. Math. Anal. 40 (2009) 1952-1978. | Zbl

[19] L. Faella and S. Monsurrò, Memory Effects Arising in the Homogenization of Composites with Inclusions, Topics on Mathematics for Smart Systems. World Sci. Publ., Hackensack, USA (2007) 107-121. | Zbl

[20] H.K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances. Appl. Anal. 75 (2000) 403-424. | Zbl

[21] E. Jose, Homogenization of a parabolic problem with an imperfect interface. Rev. Roumaine Math. Pures Appl. 54 (2009) 189-222. | Zbl

[22] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Volume 1. Dunod, Paris, France (1968). | Zbl

[23] R. Lipton, Heat conduction in fine scale mixtures with interfacial contact resistance . SIAM J. Appl. Math. 58 (1998) 55-72. | Zbl

[24] R. Lipton and B. Vernescu, Composite with imperfect interface. Proc. Soc. Lond. A 452 (1996) 329-358. | Zbl

[25] M.L. Mascarenhas, Linear homogenization problem with time dependent coefficient. Trans. Amer. Math. Soc. 281 (1984) 179-195. | Zbl

[26] S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 13 (2003) 43-63. | Zbl

[27] S. Monsurrò, Erratum for the paper “Homogenization of a two-component composite with interfacial thermal barrier” (in Vol. 13, pp. 43-63, 2003). Adv. Math. Sci. Appl. 14 (2004) 375-377. | Zbl

[28] S.E. Pastukhova, Homogenization of nonstationary problems in the theory of elasticity on thin periodic structures from the standpoint of the convergence of hyperbolic semigroups in a variable Hilbert space. Sovrem. Mat. Prilozh. 16, Differ. Uravn. Chast. Proizvod. (2004) 64-97 (Russian). Translation in J. Math. Sci. (N. Y.) 133 (2006) 949-998. | Zbl

[29] R.E. Showalter, Distributed microstructure models of porous media, in Flow in porous media (Oberwolfach (1992)), J. Douglas and U. Hornung Eds., Internat. Ser. Numer. Math. 114, Birkhäuser, Basel, Switzerland (1993) 155-163. | Zbl

[30] L. Tartar, Cours Peccot. Collège de France, France, unpublished (1977).

[31] L. Tartar, Quelques remarques sur l'homogénéisation, in Functional Analysis and Numerical Analysis, Proc. Japan-France Seminar 1976, Japanese Society for the Promotion of Science (1978) 468-482.

[32] L. Tartar, Memory effects and homogenization. Arch. Rational Mech. Anal. 3 (1990) 121-133. | Zbl