Nonlinear “double porosity” type model

Pankratov, Leonid; Piatnitski, Andrey

Comptes Rendus. Mathématique, Tome 334 (2002) no. 5, pp. 435-440.

On étudie un problème variationnel ${\inf }_{\mathrm{u}\in {\mathrm{H}}^1\left(\Omega \right)}{\int }_{\Omega }\{{\mathrm{a}}^{ϵ}|\nabla {\mathrm{u}}^{ϵ}{|}^m+\mathrm{g}|{\mathrm{u}}^{ϵ}{|}^m-{\mathrm{mf}}^{ϵ}{u}^{ϵ}\}\mathrm{d}x$ dans un ouvert borné $\Omega ={ℱ}^{\left(ϵ\right)}\cup {\overline{ℳ}}^{\left(ϵ\right)}$ avec une microstructure ${ℱ}^{\left(ϵ\right)}$ non périodique ; a^ε=a^ε(x) vaut 1 dans ${ℱ}^{\left(ϵ\right)}$ et ${\sup }_{\mathrm{x}\in {ℳ}^{\left(ϵ\right)}}{a}^{ϵ}\left(\mathrm{x}\right)\to 0$ lorsque ε→0. Un modèle homogénéisé est construit.

We consider a variational problem ${\inf }_{\mathrm{u}\in {\mathrm{H}}^1\left(\Omega \right)}{\int }_{\Omega }\{{\mathrm{a}}^{ϵ}|\nabla {\mathrm{u}}^{ϵ}{|}^m+\mathrm{g}|{\mathrm{u}}^{ϵ}{|}^m-{\mathrm{mf}}^{ϵ}{u}^{ϵ}\}\mathrm{d}x$ in a bounded domain $\Omega ={ℱ}^{\left(ϵ\right)}\cup {\overline{ℳ}}^{\left(ϵ\right)}$ with a microstructure ${ℱ}^{\left(ϵ\right)}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in ${ℱ}^{\left(ϵ\right)}$ and ${\sup }_{\mathrm{x}\in {ℳ}^{\left(ϵ\right)}}{a}^{ϵ}\left(\mathrm{x}\right)\to 0$ as ε→0. A homogenized model is constructed.

[1] Acerbi, E.; Chiadò Piat, V.; Dal Maso, G.; Percivale, D. An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., Volume 18 (1992), pp. 481-496

[2] Arbogast, T.; Douglas, J.; Hornung, U. Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Appl. Math., Volume 21 (1990), pp. 823-826

[3] Bourgeat, A.; Goncharenko, M.; Panfilov, M.; Pankratov, L. A general double porosity model, C. R. Acad. Sci. Paris, Série IIb, Volume 327 (1999), pp. 1245-1250

[4] Bourgeat, A.; Mikelic, A.; Piatnitski, A. Modèle de double porosité aléatoire, C. R. Acad. Sci. Paris, Série I, Volume 327 (1998), pp. 99-104

[5] Braides, A.; Defranceschi, A. Homogenization of Multiple Integrals, Oxford Lecture Ser. Math. Appl., 12, Clarendon Press, Oxford, 1998

[6] Cioranescu, D.; Saint Jean Paulin, J. Homogenization of Reticulated Structures, Appl. Math. Sci., 136, Springer-Verlag, New York, 1999

[7] Homogenization and Porous Media (Hornung, U., ed.), Interdisciplinary Appl. Math., 6, Springer-Verlag, New York, 1997

[8] Khruslov, E. Asymptotic behavior of the solutions of the second boundary value problem in the case of the refinement of the boundary of the domain, Mat. Sb., Volume 106 (1978), pp. 604-621

[9] Khruslov, E. Averaged models of diffusion in fractured–porous media, Dokl. Akad. Nauk SSSR, Volume 309 (1989), pp. 332-335 English translation in Soviet Phys. Dokl. 34 (1989) 980–981

[10] Khruslov, E.; Pankratov, L. Homogenization of boundary value problems for the Ginzburg–Landau equation in weakly connected domains (Marchenko, V., ed.), Spectral Theory and Related Topics, 19, American Mathematical Society, Providence, RI, 1994, pp. 233-268

[11] Ladyzhenskaya, O.; Ural'tseva, N. Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968

[12] Pankratov, L. Homogenization of nonlinear Neumann elliptic and parabolic problems, Homogenization and Applications to Material Sciences, Math. Sci. Appl., 9, 1997, pp. 341-353

[13] Samarskii, A.; Galaktionov, V.; Kurdyumov, S.; Mikhailov, A. Blow-up in Quasilinear Parabolic Equations, De Gruyter, Berlin, 1995

[14] Zhikov, V.; Kozlov, S.; Oleinik, O. Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, New York, 1994