Geodesic
Homogenization of the Stokes equations with a~random potential
Matematičeskie zametki, Tome 59 (1996) no. 4, pp. 504-520.

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Homogenization of the Stokes equations in a random porous medium is considered. Instead of the homogeneous Dirichlet condition on the boundaries of numerous small pores, used in the existing work on the subject, we insert a term with a positive rapidly oscillating potential into the equations. Physically, this corresponds to porous media whose rigid matrix is slightly permeable to fluid. This relaxation of the boundary value problem permits one to study the asymptotics of the solutions and to justify the Darcy law for the limit functions under much fewer restrictions. Specifically, homogenization becomes possible without any connectedness conditions for the porous domain, whose verification would lead to problems of percolation theory that are insufficiently studied. 
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A. Yu. Belyaev; Ya. R. Èfendiev. Homogenization of the Stokes equations with a~random potential. Matematičeskie zametki, Tome 59 (1996) no. 4, pp. 504-520. http://geodesic.mathdoc.fr/item/MZM_1996_59_4_a3/

[1] Polubarinova-Kochina P. Ya., Teoriya dvizheniya gruntovykh vod, Nauka, M., 1977

[2] Sanches-Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984

[3] Polishevsky D., “On the homogenization of fluid flows through periodic media”, Rend. Sem. Mat. Univers. Politec. Torino, 45:2 (1987), 129–139 | MR

[4] Allaire G., “Homogenization of the Stokes flow in a connected porous medium”, Asymptotic Analysis, 2:3 (1989), 203–222 | MR | Zbl

[5] Allaire G., “Homogenization of the unsteady Stokes equations in porous media”, Progress in PDE: calculus of variations, applications, Research Notes in Math. Series, 267, New York, 1992, 109–122 | MR

[6] Zhikov V. V., “Ob usrednenii uravnenii Stoksa v perforirovannoi oblasti”, Dokl. RAN, 334:2 (1994), 144–147 | Zbl

[7] Beliaev A. Yu., Kozlov S. M., “Darcy equation for random porous”, Comm. Pure and Appl. Math., 49:1 (1996), 1–34 | 3.0.CO;2-J class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[8] Grimmet G., Perlocation, Springer-Verlag, New York, 1989 | Zbl

[9] Barenblatt G. I., Zheltov Yu. P., Kochina I. N., “Ob osnovnykh predstavleniyakh teorii filtratsii odnorodnykh zhidkostei v treschinovatykh porodakh”, PMM, 24:5 (1960), 852–864 | MR | Zbl

[10] Zhikov V. V., Kozlov S. M., Oleinik O. A., Kha Ten Ngoan, “Usrednenie i $G$-skhodimost differentsialnykh operatorov”, UMN, 34:5 (1979), 65–133 | MR | Zbl

[11] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | Zbl

[12] Levitan B. M., Zhikov V. V., Pochti periodicheskie funktsii i differentsialnye uravneniya, MGU, M., 1978

[13] Berdichevskii V. L., Variatsionnye printsipy mekhaniki sploshnoi sredy, Nauka, M., 1983

[14] Nečas J., Equations aux dérivées partielles, Presses de l'Université de Montréal, 1965

[15] Temam R., Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981 | Zbl