×

Corrector results for a parabolic problem with a memory effect. (English) Zbl 1195.35038

The authors here present corrector results for a parabolic problem studied by the second author in [Rev. Roum. Math. Pures Appl. 54, No. 3, 189–222 (2009; Zbl 1199.35015)]. The problem consists of two parabolic equations posed in the domains \(\Omega _{1\varepsilon }\) and \(\Omega _{2\varepsilon}\), respectively defined as the connected and the disconnected union of \( \varepsilon \)-periodic translated sets of \(\varepsilon Y_{1}\) and \( \varepsilon Y_{2}\) when the unit cell \(Y=(0,1)^{N}\) is decomposed as \( Y=Y_{1}\cup Y_{2}\). These two parabolic equations are completed with two boundary conditions on the interface \(\Gamma ^{\varepsilon }\times (0,T)=\partial \Omega _{2\varepsilon }\times (0,T)\), with an homogeneous Dirichlet boundary condition on \(\partial \Omega \times (0,T)\) and with initial data. The main results of the paper present the structure of the first-order correctors for this problem, the authors distinguishing between the cases \(-1<\gamma <1\) and \(\gamma =1\), where \(\gamma \) is an order parameter which appears in one boundary condition on \(\Gamma ^{\varepsilon }\times (0,T)\). These correctors involve the solution of the homogenized problem, which has been obtained in the above-indicated paper, and a matrix \( C^{\varepsilon }\) which is based on the solution of cell problems. The proof of these results is mainly based on the derivation of estimates on the solution of the original problem, on the corresponding energy and on the difference between the solution of the original problem and these correctors.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics

Citations:

Zbl 1199.35015
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] J.L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier. International J. Heat Mass Transfer37 (1994) 2885-2892. Zbl0900.73453 · Zbl 0900.73453
[2] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). · Zbl 0404.35001
[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 0837.35016
[4] M. Briane, A. Damlamian and P. Donato, H-convergence in Perforated Domains, in Nonlinear Partial Differential Equations and Their Applications - Collège de France SeminarXIII, D. Cioranescu and J.L. Lions Eds., Pitman Research Notes in Mathematics Series391, Longman, New York, USA (1998) 62-100. Zbl0927.00031 · Zbl 0927.00031
[5] H.S. Carslaw and J.C. Jaeger, Conduction of heat in solids. The Clarendon Press, Oxford, UK (1947). · Zbl 0029.37801
[6] D. Cioranescu and P. Donato, Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptot. Anal.1 (1988) 115-138. · Zbl 0683.35026
[7] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains. J. Math. Pures Appl.68 (1989) 185-213. · Zbl 0627.35057
[8] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications17. Oxford Univ. Press, New York, USA (1999). · Zbl 0939.35001
[9] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl.71 (1979) 590-607. Zbl0427.35073 · Zbl 0427.35073
[10] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). · Zbl 0929.35002
[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. · Zbl 0807.35077
[12] P. Donato, Some corrector results for composites with imperfect interface. Rend. Math. Ser. VII26 (2006) 189-209. · Zbl 1129.35008
[13] P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl.2 (2004) 1-27. Zbl1083.35014 · Zbl 1083.35014
[14] P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domains. ESAIM: COCV6 (2001) 21-38. Zbl0964.35015 · Zbl 0964.35015
[15] P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Chin. Ann. Math. B25 (2004) 143-156. · Zbl 1085.35022
[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 0944.35009
[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 1112.35017
[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. Zbl1197.35029 · Zbl 1197.35029
[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 1114.74048
[20] H.K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances. Appl. Anal.75 (2000) 403-424. · Zbl 1024.80005
[21] E. Jose, Homogenization of a parabolic problem with an imperfect interface. Rev. Roumaine Math. Pures Appl.54 (2009) 189-222. Zbl1199.35015 · Zbl 1199.35015
[22] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Volume 1. Dunod, Paris, France (1968). · Zbl 0165.10801
[23] R. Lipton, Heat conduction in fine scale mixtures with interfacial contact resistance . SIAM J. Appl. Math.58 (1998) 55-72. Zbl0913.35010 · Zbl 0913.35010
[24] R. Lipton and B. Vernescu, Composite with imperfect interface. Proc. Soc. Lond. A452 (1996) 329-358. · Zbl 0872.73033
[25] M.L. Mascarenhas, Linear homogenization problem with time dependent coefficient. Trans. Amer. Math. Soc.281 (1984) 179-195. · Zbl 0536.45003
[26] S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl.13 (2003) 43-63. · Zbl 1052.35022
[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 1069.35500
[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 1089.74041
[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 0805.76082
[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 0725.45012
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.