Homogenization of the compressible Navier-Stokes equations in a porous medium
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 885-906.

We study the homogenization of the compressible Navier-Stokes system in a periodic porous medium (of period ε) with Dirichlet boundary conditions. At the limit, we recover different systems depending on the scaling we take. In particular, we rigorously derive the so-called “porous medium equation”.

DOI : 10.1051/cocv:2002053
Classification : 76M50
Mots clés : compressible Navier-Stokes, homogenization, porous medium equation
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Masmoudi, Nader. Homogenization of the compressible Navier-Stokes equations in a porous medium. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 885-906. doi : 10.1051/cocv:2002053. http://archive.numdam.org/articles/10.1051/cocv:2002053/

[1] G. Allaire, Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2 (1989) 203-222. | MR | Zbl

[2] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR | Zbl

[3] G. Allaire, Homogenization of the unsteady Stokes equations in porous media, in Progress in partial differential equations: Calculus of variations, applications, Pont-à-Mousson, 1991. Longman Sci. Tech., Harlow (1992) 109-123. | MR | Zbl

[4] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl

[5] M.E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators div and grad , in Theory of cubature formulas and the application of functional analysis to problems of mathematical physics. Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk (1980) 5-40, 149. | MR

[6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31 (1961) 308-340. | Numdam | MR | Zbl

[7] H. Darcy, Les fontaines publiques de la ville de Dijon. Dalmont Paris (1856).

[8] J.I. Díaz, Two problems in homogenization of porous media, in Proc. of the Second International Seminar on Geometry, Continua and Microstructure, Getafe, 1998, Vol. 14 (1999) 141-155. | MR | Zbl

[9] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable. Comment. Math. Univ. Carolin. 42 (2001) 83-98. | Zbl

[10] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I. Springer-Verlag, New York (1994). Linearized steady problems. | Zbl

[11] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). | MR | Zbl

[12] J.-L. Lions, Some methods in the mathematical analysis of systems and their control. Kexue Chubanshe (Science Press), Beijing (1981). | MR | Zbl

[13] P.-L. Lions, Mathematical topics in fluid mechanics, Vol. 1. The Clarendon Press Oxford University Press, New York (1996). Incompressible models, Oxford Science Publications. | MR | Zbl

[14] P.-L. Lions, Mathematical topics in fluid mechanics, Vol. 2. The Clarendon Press Oxford University Press, New York (1998). Compressible models, Oxford Science Publications. | MR | Zbl

[15] R. Lipton and M. Avellaneda, Darcy's law for slow viscous flow past a stationary array of bubbles. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 71-79. | Zbl

[16] N. Masmoudi (in preparation).

[17] A. Mikelić, Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary. Ann. Mat. Pura Appl. (4) 158 (1991) 167-179. | Zbl

[18] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR | Zbl

[19] G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal. 21 (1990) 1394-1414. | MR | Zbl

[20] E. Sánchez-Palencia, Nonhomogeneous media and vibration theory. Springer-Verlag, Berlin (1980). | Zbl

[21] L. Tartar, Incompressible fluid flow in a porous medium: convergence of the homogenization process, in Nonhomogeneous media and vibration theory, edited by E. Sánchez-Palencia (1980) 368-377.

[22] R. Temam, Navier-Stokes equations and nonlinear functional analysis. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Second Edition (1995). | Zbl

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