Homogenization of the compressible Navier–Stokes equations in a porous medium

Nader Masmoudi

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 885-906
  • ISSN: 1292-8119

Abstract

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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”.

How to cite

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Masmoudi, Nader. "Homogenization of the compressible Navier–Stokes equations in a porous medium." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 885-906. <http://eudml.org/doc/90676>.

@article{Masmoudi2010,
abstract = { 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”. },
author = {Masmoudi, Nader},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Compressible Navier–Stokes; homogenization; porous medium equation.; convergence; Dirichlet boundary conditions; porous medium equation},
language = {eng},
month = {3},
pages = {885-906},
publisher = {EDP Sciences},
title = {Homogenization of the compressible Navier–Stokes equations in a porous medium},
url = {http://eudml.org/doc/90676},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Masmoudi, Nader
TI - Homogenization of the compressible Navier–Stokes equations in a porous medium
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 885
EP - 906
AB - 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”.
LA - eng
KW - Compressible Navier–Stokes; homogenization; porous medium equation.; convergence; Dirichlet boundary conditions; porous medium equation
UR - http://eudml.org/doc/90676
ER -

References

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  1. G. Allaire, Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal.2 (1989) 203-222.  
  2. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482-1518.  
  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.  
  4. A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978).  
  5. M.E. Bogovski, Solutions of some problems of vector analysis, associated with the operators and , 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.  
  6. L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova31 (1961) 308-340.  
  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.  
  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.  
  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.  
  11. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969).  
  12. J.-L. Lions, Some methods in the mathematical analysis of systems and their control. Kexue Chubanshe (Science Press), Beijing (1981).  
  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.  
  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.  
  15. R. Lipton and M. Avellaneda, Darcy's law for slow viscous flow past a stationary array of bubbles. Proc. Roy. Soc. Edinburgh Sect. A114 (1990) 71-79.  
  16. N. Masmoudi (in preparation).  
  17. A. Mikelic, Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary. Ann. Mat. Pura Appl. (4)158 (1991) 167-179.  
  18. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608-623.  
  19. G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal.21 (1990) 1394-1414.  
  20. E. Sánchez-Palencia, Nonhomogeneous media and vibration theory. Springer-Verlag, Berlin (1980).  
  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).  

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