Nonlinear double porosity models with non-standard growth

Catherine Choquet; Leonid Pankratov

doi:10.1016/j.crme.2009.09.004

Nonlinear double porosity models with non-standard growth
[Modèles non linéaires de type double-porosité à croissance non standart]

Présenté par : Évariste Sanchez-Palencia

Catherine Choquet ¹ ; Leonid Pankratov ²

¹ Université P. Cézanne, LATP UMR 6632, and Délégation CNRS LAMA UMR 5127, université de Savoie; FST, case cour A, 13397 Marseille cedex 20, France
² Mathematical Division, Institute for Low Temperature Physics, 47, Lenin ave., 310164 Kharkov, Ukraine

Comptes Rendus. Mécanique, Volume 337 (2009) no. 9-10, pp. 659-666.

Résumé
Abstract

Nous étudions les solutions d'équations quasilinéaires elliptiques à coefficients fortement contrastés. La formulation variationnelle associée conduit à travailler dans des espaces de Lebesgue à exposant variable $L^{p_{ε} (\cdot)}$ dans des domaines à la microstructure complexe caractérisée par un petit paramètre ε. Nous obtenons rigoureusement le problème homogénéisé correspondant. Il est déterminé par les caractéristiques variationnelles locales de la microstructure.

We study the solutions to quasilinear elliptic equations with high contrast coefficients. The energy formulation leads to work with variable exponent Lebesgue spaces $L^{p_{ε} (\cdot)}$ in domains with a complex microstructure scaled by a small parameter ε. We derive rigorously the corresponding homogenized problem. It is completely described in terms of local energy characteristics of the original domain.

Reçu le : 2009-06-24
Accepté le : 2009-08-31
Publié le : 2009-09-01

DOI : 10.1016/j.crme.2009.09.004

Keywords: Fluid mechanics, Homogenization, Double porosity, Non-standard growth
Mot clés : Mécanique des fluides, Homogénéisation, Double porosité, Croissance non standard

Affiliations des auteurs :

Catherine Choquet ¹ ; Leonid Pankratov ²

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     title = {Nonlinear double porosity models with non-standard growth},
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     language = {en},
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Catherine Choquet; Leonid Pankratov. Nonlinear double porosity models with non-standard growth. Comptes Rendus. Mécanique, Volume 337 (2009) no. 9-10, pp. 659-666. doi : 10.1016/j.crme.2009.09.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.09.004/

Bibliographie
Cité par

[1] M. Ruzicka Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000

[2] E. Acerbi; G. Mingione Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., Volume 164 (2002), pp. 213-259

[3] Y. Chen; S. Levine; M. Rao Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., Volume 66 (2006) no. 4, pp. 1383-1406

[4] G.I. Barenblatt; I.P. Zheltov; I.N. Kochina Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, Prikl. Mat. Mech., Volume 24 (1960) no. 5, pp. 1286-1303

[5] U. Hornung Homogenization and Porous Media, Interdisciplinary Applied Mathematics, vol. 6, Springer, 1997

[6] T. Arbogast; J. Douglas; U. Hornung Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Appl. Math. Anal., Volume 21 (1990), pp. 823-836

[7] B. Amaziane; L. Pankratov; A. Piatnitski Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media, Proc. Roy. Soc. Edinburgh Sect. A, Volume 136 (2006), pp. 1131-1155

[8] C. Choquet Derivation of the double porosity model of a compressible miscible displacement in naturally fractured reservoirs, Appl. Anal., Volume 83 (2004), pp. 477-500

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

[10] A. Bourgeat; A. Mikelic; A. Piatnitski On the double porosity model of a single phase flow in random media, Asympt. Anal., Volume 327 (2003), pp. 311-332

[11] V.A. Marchenko; E.Ya. Khruslov Homogenization of Partial Differential Equations, Birkhäuser, Boston, 2006

[12] S.N. Antontsev; S.I. Shmarev Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, Handbook of Differential Equations, vol. 3, Stationary Partial Differential Equations, Elsevier, 2006 (Chapter 1, pp. 1–100)

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

[14] P. Tolksdorf Regularity for a more general class of quasi-linear elliptic equations, J. Diff. Equations, Volume 51 (1984), pp. 126-150

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