Sur quelques problèmes d’homogénéisation non locale et de fluides en milieu poreux : une contribution de Abdelhamid Ziani
Annales Mathématiques Blaise Pascal, Tome 14 (2007) no. 2, pp. 149-186.

Dans cet article nous présentons quelques problèmes et résultats d’homogénéisation non locale pour certaines équations de type dégénéré. Nous considérons des équations de transport, une équation des ondes dégénérée et une équation différentielle de Riccati, et nous décrivons dans chacun des cas les effets non locaux induits par homogénéisation. Nous donnons aussi quelques résultats sur l’analyse mathématique des équations des fluides miscibles en milieu poreux.

@article{AMBP_2007__14_2_149_0,
     author = {Amirat, Youcef and Hamdache, Kamel},
     title = {Sur quelques probl\`emes d'homog\'en\'eisation non locale et de fluides en milieu poreux~: une contribution de Abdelhamid Ziani},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {149--186},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     number = {2},
     year = {2007},
     doi = {10.5802/ambp.231},
     mrnumber = {2369870},
     zbl = {1158.35011},
     language = {fr},
     url = {www.numdam.org/item/AMBP_2007__14_2_149_0/}
}
Amirat, Youcef; Hamdache, Kamel. Sur quelques problèmes d’homogénéisation non locale et de fluides en milieu poreux : une contribution de Abdelhamid Ziani. Annales Mathématiques Blaise Pascal, Tome 14 (2007) no. 2, pp. 149-186. doi : 10.5802/ambp.231. http://www.numdam.org/item/AMBP_2007__14_2_149_0/

[1] Aheizer, N. I.; Krein, M. Some questions in the theory of moments, translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2, American Mathematical Society, Providence, R.I., 1962 | MR 167806 | Zbl 0117.32702

[2] Alexandre, R. Some results in homogenization tacking memory effects, Asymp. Anal., Volume Vol. 15 (1997), pp. 229-259 | MR 1487712 | Zbl 0894.35007

[3] Alexandre, R. Asymptotic behavior of transport equations, Appl. Anal., Volume Vol. 70, 3-4 (1999), pp. 405-430 | MR 1688867 | Zbl 1026.35099

[4] Allaire, G. Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23(6) (1992), pp. 1482-1518 | Article | MR 1185639 | Zbl 0770.35005

[5] Amirat, Y.; Hamdache, K.; Ziani, A. Homogénéisation d’équations hyperboliques du premier ordre – Application aux milieux poreux, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 6, no. 5 (1989), pp. 397-417 | Numdam | Zbl 0699.35170

[6] Amirat, Y.; Hamdache, K.; Ziani, A. Comportement limite de modèles d’équations de convection-diffusion dégénérées, C. R. Acad. Sci. Paris Sér. I Math., Volume 310, no. 11 (1990), pp. 765-768 | Zbl 0713.76097

[7] Amirat, Y.; Hamdache, K.; Ziani, A. Étude d’une équation de transport à mémoire, C. R. Acad. Sci. Paris Sér. I Math., Volume 311, no. 11 (1990), pp. 685-688 | Zbl 0709.76004

[8] Amirat, Y.; Hamdache, K.; Ziani, A. Homogénéisation d’un modèle d’écoulements miscibles en milieu poreux, Asymptotic Analysis, Volume 3 ̲ (1990), pp. 77-89 | Zbl 0702.35213

[9] Amirat, Y.; Hamdache, K.; Ziani, A. Homogénéisation non locale pour des équations dégénérées à coefficients périodiques, C. R. Acad. Sci. Paris Sér. I Math., Volume 312, no. 13 (1991), pp. 963-966 | MR 1113085 | Zbl 0762.35007

[10] Amirat, Y.; Hamdache, K.; Ziani, A. Homogénéisation par décomposition en fréquences d’une équation de transport dans n , C. R. Acad. Sci. Paris Sér. I Math., Volume 312, no. 1 (1991), pp. 37-40 | Zbl 0722.35010

[11] Amirat, Y.; Hamdache, K.; Ziani, A. Homogenization of a model of compressible miscible flow in porous media, Boll. U.M.I., Volume 7 (5-B) (1991), pp. 463-487 | MR 1111133 | Zbl 0727.76093

[12] Amirat, Y.; Hamdache, K.; Ziani, A. Kinetic formulation for a transport equation with memory, Comm. in P.D.E., Volume 16 (8 & 9) (1991), pp. 1287-13311 | Article | MR 1132786 | Zbl 0749.35021

[13] Amirat, Y.; Hamdache, K.; Ziani, A. Some results on homogenization of convection-diffusion equations, Arch. Rational Mech. Anal., Volume 114, no. 2 (1991), pp. 155-178 | Article | MR 1094434 | Zbl 0742.35007

[14] Amirat, Y.; Hamdache, K.; Ziani, A. Homogenization of parametrized families of hyperbolic problems, Proceedings of the Royal Society of Edinburgh, Volume 120 A (1992), pp. 199-221 | Article | Zbl 0758.35006

[15] Amirat, Y.; Hamdache, K.; Ziani, A. Homogenization of degenerate wave equations with periodic coefficients, SIAM Journal of Mathematical Analysis, Volume 24, no. 5 (1993), pp. 1226-1253 | Article | MR 1234013 | Zbl 0803.35008

[16] Amirat, Y.; Hamdache, K.; Ziani, A. Remarques sur l’interaction d’oscillations dans une équation de transport (1993) (Technical report)

[17] Amirat, Y.; Hamdache, K.; Ziani, A. Existence globale de solutions faibles pour un système parabolique-hyperbolique intervenant en dynamique des milieux poreux, C. R. Acad. Sci. Paris Sér. I Math., Volume 321, no. 2 (1995), pp. 253-258 | MR 1345458 | Zbl 0838.76086

[18] Amirat, Y.; Hamdache, K.; Ziani, A. On homogenization of ordinary differential equations and linear transport equations, Homogénéisation et Méthodes de Convergence en Calcul des Variations, Ed. G. Bouchitte, G. Buttazzo, and P. Suquet (Advances in Math. for Applied Sciences) Volume 18s, World Scientific-Singapore, 1995, pp. 29-50 | MR 1428690 | Zbl 0894.34056

[19] Amirat, Y.; Hamdache, K.; Ziani, A. Mathematical Analysis for compressible miscible displacement models in porous media, M³AS, Volume 6, no. 6 (1996), pp. 729-747 | MR 1404826 | Zbl 0859.35087

[20] Amirat, Y.; Hamdache, K.; Ziani, A. On Homogenisation of a Riccati Equation (1999) (Technical report)

[21] Amirat, Y.; Ziani, A. Global weak solutions to a parabolic system modeling a one-dimensional compressible miscible flow in porous media, Journal of Mathematical Analysis and Applications, Volume 220 (1998), pp. 697-718 | Article | MR 1614944 | Zbl 0911.35063

[22] Amirat, Y.; Ziani, A. Classical solutions of a parabolic-hyperbolic system modelling a three-dimensional compressible miscible flow in porous media, Applicable Analysis, Volume 72 (1–2) (1999), pp. 155-168 | Article | MR 1775438 | Zbl 1020.76044

[23] Amirat, Y.; Ziani, A. Asymptotic behavior of the solutions of an elliptic-parabolic system arising in flow in porous media, Z. Anal. Anwend., Volume 23, no. 2 (2004), pp. 335-351 | Article | MR 2085294 | Zbl 1072.35039

[24] Amirat, Y.; Ziani, A. Classical solutions for a multicomponent flow model in porous media, Differential and Integral Equations, Volume 17, no. 7-8 (2004), pp. 893-920 | MR 2075412 | Zbl 1150.35511

[25] Amirat, Y.; Ziani, A. Global weak solutions for a nonlinear degenerate parabolic system modelling a one-dimensional compressible miscible flow in porous media, Boll. U.M.I., Volume 8 (7-B) (2004), pp. 109-128 | MR 2044263 | Zbl 05147129

[26] Bear, J. Dynamics of fluids in porous media, Elsevier, 1972

[27] Bensoussan, A.; Lions, J. L.; Papanicolaou, G. C. Asymptotic analysis for periodic structures, North-Holland, 1978 | MR 503330 | Zbl 0404.35001

[28] Bonnetier, E.; Conca, C. Approximation of Young measures by functions and an application to an optimal design problem for plates with variable thickness, Proceedings of the Royal Society of Edinburgh, Volume 124 A, no. 3 (1994), pp. 399-422 | Article | MR 1286912 | Zbl 0824.49006

[29] Briane, M. Nonlocal effects in two-dimensional conductivity, Arch. Rat. Mech. Anal., Volume 182(2) (2006), pp. 255-267 | Article | MR 2259333 | Zbl 05059701

[30] Chavent, G.; Jaffré, J. Mathematical models and finite elements for reservoir simulation, North-Holland, 1986 | Zbl 0603.76101

[31] DiPerna, R. J. Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., Volume 88(3) (1985), pp. 223-270 | Article | MR 775191 | Zbl 0616.35055

[32] DiPerna, R. J.; Majda, A. J. Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., Volume 108 (1987), pp. 667-689 | Article | MR 877643 | Zbl 0626.35059

[33] Donoghue, W. F. Monotone matrix functions and analytic continuation, Springer-Verlag: New York, Heidelberg, Berlin, 1974 | MR 486556 | Zbl 0278.30004

[34] Douglas, J.; Roberts, J. E. Numerical methods for a model of compressible miscible displacement in porous media, Math. Comput., Volume 41 (1983), pp. 441-459 | Article | MR 717695 | Zbl 0537.76062

[35] E, Weinan Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math., Volume 45 (1992) no. 3, pp. 301-326 | Article | MR 1151269 | Zbl 0794.35014

[36] Ene, H. I.; Mascarenhas, M. L.; Saint Jean Paulin, J. Fading memory effects in elastic-viscoelastic composites, M²AN, Volume 31(7) (1997), pp. 927-952 | Numdam | MR 1489178 | Zbl 0895.73045

[37] Engquist, B.; Hou, T. Y. Particle method approximation of oscillatory solutions to hyperbolic differential equations, SIAM J. Numer. Anal., Volume 26(2) (1989), pp. 289-319 | Article | MR 987391 | Zbl 0675.65093

[38] Fabrie, P.; Langlais, M. Mathematical analysis of miscible displacement in porous media, SIAM Journal of Mathematical Analysis, Volume 23 (1992), pp. 1375-1392 | Article | MR 1185634 | Zbl 0772.76070

[39] Feng, X. On existence and uniqueness results for a coupled system modeling miscible displacement in porous media, J. Math. Anal. Appl., Volume 194 (1995), pp. 883-910 | Article | MR 1350201 | Zbl 0856.35030

[40] Gel’fand, I. M.; Graev, M. I.; Vilenkin, N. Ya. Generalized Functions 5, Academic Press, 1966 | Zbl 0144.17202

[41] Helgason, Sigurdur The Radon transform, Progress in Mathematics, Volume 5, Birkhäuser Boston, Mass., 1980 | MR 573446 | Zbl 0453.43011

[42] Hruslov, Ē. Ya. Homogenized models of composite media, Composite media and homogenization theory (Trieste, 1990) (Progr. Nonlinear Differential Equations Appl.) Volume 5, Birkhäuser Boston, Boston, MA, 1991, pp. 159-182 | MR 1145750 | Zbl 0737.73009

[43] Krommes, J. A. Statistical desciptions and plasmas physics, Handbook of Plasmas Physics 2, eds A.A. Galeev and R.N. Sudan, North Holland, 1984

[44] Lions, J. L. Homogénéisation non locale, Proceeding of the international meeting on recent methods in nonlinear analysis , eds E. De Giorgi , E. Magenes and U. Mosco, Bologna : Pitagora Editrice, 1979, pp. 189-203 | MR 533167 | Zbl 0408.35026

[45] Marčenko, V. A.; Ja.Hruslov, E. Boundary value-problems in domains with a fine-grained boundary, Naukova Dumka, Kiev, 1974 | MR 601059 | Zbl 0289.35002

[46] Mascarenhas, L. A linear homogenization problem with time dependent coefficient, Trans. Am. Math. Soc., Volume 281, 1 (1984), pp. 179-195 | Article | MR 719664 | Zbl 0536.45003

[47] Mikelic, A. Mathematical theory of stationary miscible filtration, J. Differential Equations, Volume 90 (1991), pp. 186-202 | Article | MR 1094455 | Zbl 0735.35100

[48] Murat, François; Tartar, Luc H-convergence, Topics in the mathematical modelling of composite materials (Progr. Nonlinear Differential Equations Appl.) Volume 31, Birkhäuser Boston, Boston, MA, 1997, pp. 21-43 | MR 1493039 | Zbl 0920.35019

[49] Nguetseng, G. A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., Volume 20(3) (1989), pp. 608-623 | Article | MR 990867 | Zbl 0688.35007

[50] Pazy, A. Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983 | MR 710486 | Zbl 0516.47023

[51] Peaceman, D. W. Fundamentals of numerical reservoir simulation, Elsevier, 1977

[52] Renardy, M.; Hrusa, J. A.; Nohel, W. J. Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics 35, Longman, 1987 | Zbl 0719.73013

[53] Sanchez-Palencia, E. Méthodes d’homogénéisation pour l’étude de matériaux hétérogènes  : phénomène de mémoire, Rend. Sem. Mat. Torino, Volume 36 (1978), pp. 15-25 | Zbl 0394.35001

[54] Sanchez-Palencia, E. Non-homogeneous media and vibration theory, Lecture Notes in Physics, 127, Springer-Verlag, 1980 | Zbl 0432.70002

[55] Shvidler, M. I. Dispersion of a filtration stream in a medium with random inhomogeneities, Sov. Phys. Dokl., Volume 20(3) (1975), pp. 171-173 | Zbl 0341.76053

[56] Shvidler, M. I. Dispersion of a filtration flow, Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, Volume 4 (1976), pp. 65-69

[57] Shvidler, M. I. Conditional Conditional averaging of the equations of flow in random composite porous media, Transl. Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, Volume 1 (1987), pp. 75-81 | Zbl 0636.76103

[58] Smith, R. A delay-diffusion description for contaminant dispersion, J. Fluid Mech., Volume 105, 9 (1981), pp. 469-486 | Article | Zbl 0463.76086

[59] Smith, R. Longitudinal dispersion coefficients for varying channels, J. Fluid Mech., Volume 130 (1983), pp. 299-314 | Article | Zbl 0524.76081

[60] Tartar, L. Remarks on Homogenization, Homogenization and Effective Moduli of Materials and Media (The IMA Volumes in Mathematics and its Applications) Volume 1, Springer, New York, 1986, pp. 228-246 | MR 859418 | Zbl 0652.35012

[61] Tartar, L. Memory effects and Homogenization, Arch. Rat. Mech. Anal., Volume 111, No. 2 (1990), pp. 121-133 | Article | MR 1057651 | Zbl 0725.45012

[62] Tartar, Luc Nonlocal effects induced by homogenization, Partial differential equations and the calculus of variations, Vol. II (Progr. Nonlinear Differential Equations Appl.) Volume 2, Birkhäuser Boston, 1989, pp. 925-938 | MR 1034036 | Zbl 0682.35028

[63] Taylor, G. I. Dispersion of soluble matter in solvent flowing slowly through a tube, Proceedings of the Royal Society of London, Volume A 219 (1953), pp. 186-203

[64] Taylor, G. I. The dispersion of matter in turbulent flow through a pipe, Proceedings of the Royal Society of London, Volume A 223 (1954), pp. 446-468