Homogenization of the compressible Navier-Stokes equations in a porous medium. (English) Zbl 1071.76047
Summary: We study the homogenization of the compressible Navier-Stokes system in a periodic porous medium (of period \(\varepsilon \)) 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”.
MSC:
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76M50 | Homogenization applied to problems in fluid mechanics |
| 35Q30 | Navier-Stokes equations |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 76S05 | Flows in porous media; filtration; seepage |
References:
| [1] | G. Allaire , Homogenization of the Stokes flow in a connected porous medium . Asymptot. Anal. 2 ( 1989 ) 203 - 222 . MR 1020348 | Zbl 0682.76077 · Zbl 0682.76077 |
| [2] | G. Allaire , Homogenization and two-scale convergence . SIAM J. Math. Anal. 23 ( 1992 ) 1482 - 1518 . MR 1185639 | Zbl 0770.35005 · Zbl 0770.35005 · doi:10.1137/0523084 |
| [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 1194192 | Zbl 0801.35103 · Zbl 0801.35103 |
| [4] | A. Bensoussan , J.-L. Lions and G. Papanicolaou , Asymptotic analysis for periodic structures . North-Holland Publishing Co., Amsterdam ( 1978 ). MR 503330 | Zbl 0404.35001 · Zbl 0404.35001 |
| [5] | M.E. Bogovskiĭ , Solutions of some problems of vector analysis, associated with the operators \({\mathrm div}\) and \({\mathrm grad}\) , 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 631691 |
| [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 138894 | Zbl 0116.18002 · Zbl 0116.18002 |
| [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 1758958 | Zbl 0942.35021 · Zbl 0942.35021 |
| [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 . Article | Zbl 1115.35096 · Zbl 1115.35096 |
| [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 0949.35004 · Zbl 0949.35004 |
| [11] | J.-L. Lions , Quelques méthodes de résolution des problèmes aux limites non linéaires . Dunod ( 1969 ). MR 259693 | Zbl 0189.40603 · Zbl 0189.40603 |
| [12] | J.-L. Lions , Some methods in the mathematical analysis of systems and their control . Kexue Chubanshe (Science Press), Beijing ( 1981 ). MR 664760 | Zbl 0542.93034 · Zbl 0542.93034 |
| [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 1422251 | Zbl 0866.76002 · Zbl 0866.76002 |
| [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 1637634 | Zbl 0908.76004 · Zbl 0908.76004 |
| [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 0850.76778 · Zbl 0850.76778 · doi:10.1017/S0308210500024276 |
| [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 0758.35007 · Zbl 0758.35007 · doi:10.1007/BF01759303 |
| [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 990867 | Zbl 0688.35007 · Zbl 0688.35007 · doi:10.1137/0520043 |
| [19] | G. Nguetseng , Asymptotic analysis for a stiff variational problem arising in mechanics . SIAM J. Math. Anal. 21 ( 1990 ) 1394 - 1414 . MR 1075584 | Zbl 0723.73011 · Zbl 0723.73011 · doi:10.1137/0521078 |
| [20] | E. Sánchez-Palencia , Nonhomogeneous media and vibration theory . Springer-Verlag, Berlin ( 1980 ). Zbl 0432.70002 · Zbl 0432.70002 |
| [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 0833.35110 · Zbl 0833.35110 |
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