Permeability through a perforated domain for the incompressible 2D Euler equations
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 1, p. 159-182
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We investigate the influence of a perforated domain on the 2D Euler equations. Small inclusions of size ε are uniformly distributed on the unit segment or a rectangle, and the fluid fills the exterior. These inclusions are at least separated by a distance ${ϵ}^{\alpha }$ and we prove that for α small enough (namely, less than 2 in the case of the segment, and less than 1 in the case of the square), the limit behavior of the ideal fluid does not feel the effect of the perforated domain at leading order when $ϵ\to 0$.

@article{AIHPC_2015__32_1_159_0,
author = {Bonnaillie-No\"el, V. and Lacave, C. and Masmoudi, N.},
title = {Permeability through a perforated domain for the incompressible 2D Euler equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {1},
year = {2015},
pages = {159-182},
doi = {10.1016/j.anihpc.2013.11.002},
zbl = {1318.35070},
mrnumber = {3303945},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_1_159_0}
}

Bonnaillie-Noël, V.; Lacave, C.; Masmoudi, N. Permeability through a perforated domain for the incompressible 2D Euler equations. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 1, pp. 159-182. doi : 10.1016/j.anihpc.2013.11.002. http://www.numdam.org/item/AIHPC_2015__32_1_159_0/

[1] L.V. Ahlfors, Lectures on Quasiconformal Mappings, Jr. Van Nostrand Math. Stud. vol. 10 , D. Van Nostrand Co., Inc., Toronto, Ont., New York, London (1966) | MR 200442 | Zbl 0138.06002

[2] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal. 113 no. 3 (1990), 209 -259 | MR 1079189 | Zbl 0724.76020

[3] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Ration. Mech. Anal. 113 no. 3 (1990), 261 -298 | MR 1079190 | Zbl 0724.76021

[4] A. Bendali, M. Fares, E. Piot, S. Tordeux, Mathematical justification of the Rayleigh conductivity model for perforated plates in acoustics, SIAM J. Appl. Math. 73 no. 1 (2013), 438 -459 | MR 3033157 | Zbl 06171242

[5] V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial, Multiscale expansion and numerical approximation for surface defects, 40e Congrès National d'Analyse Numérique, CANUM 2010, ESAIM Proc. vol. 33 , EDP Sci., Les Ulis (2011), 22 -35 | MR 2863307 | Zbl 1302.35035

[6] V. Bonnaillie-Noël, M. Dambrine, Interactions between moderately close circular inclusions: the Dirichlet–Laplace equation in the plane, Asymptot. Anal. 84 (2013), 197 -227 , http://dx.doi.org/10.3233/ASY-131174 | MR 3136108 | Zbl 1279.35012

[7] V. Bonnaillie-Noël, M. Dambrine, S. Tordeux, G. Vial, Interactions between moderately close inclusions for the Laplace equation, Math. Models Methods Appl. Sci. 19 no. 10 (2009), 1853 -1882 | MR 2573145 | Zbl 1191.35112

[8] D. Cioranescu, F. Murat, Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. II, Paris, 1979/1980, Res. Notes Math. vol. 60 , Pitman, Boston, MA (1982), 98 -138 | MR 652509 | Zbl 0496.35030

[9] C. Conca, M. Sepúlveda, Numerical results in the Stokes sieve problem, Rev. Int. Métodos Numér. Cálc. Diseño Ing. 5 no. 4 (1989), 435 -452 | MR 1036224

[10] J.I. Díaz, Two problems in homogenization of porous media, Proceedings of the Second International Seminar on Geometry, Continua and Microstructure, vol. 14, Getafe, 1998 (1999), 141 -155 | MR 1758958 | Zbl 0942.35021

[11] J. Diaz-Alban, N. Masmoudi, Asymptotic analysis of acoustic waves in a porous medium: initial layers in time, Commun. Math. Sci. 10 no. 1 (2012), 239 -265 | MR 2901309 | Zbl 1284.35047

[12] D. Gérard-Varet, C. Lacave, The two-dimensional Euler equations on singular domains, Arch. Ration. Mech. Anal. 209 no. 1 (2013), 131 -170 | MR 3054600 | Zbl 1286.35200

[13] G. Geymonat, E. Sánchez-Palencia, Dégénérescence de la loi de Darcy pour un écoulement à travers des obstacles de petite concentration, C. R. Acad. Sci. Paris Sér. I Math. 293 no. 2 (1981), 179 -181 | MR 637121 | Zbl 0485.76072

[14] B. Gustafsson, A. Vasil'Ev, Conformal and potential analysis in Hele–Shaw cells, Adv. Math. Fluid Mech. , Birkhäuser Verlag, Basel (2006) | MR 2245542 | Zbl 1122.76002

[15] H. Hakobyan, D.A. Herron, Euclidean quasiconvexity, Ann. Acad. Sci. Fenn. Math. 33 no. 1 (2008), 205 -230 | MR 2386847 | Zbl 1155.30012

[16] D. Iftimie, M.C. Lopes Filho, H.J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Commun. Partial Differ. Equ. 28 no. 1–2 (2003), 349 -379 | MR 1974460 | Zbl 1094.76007

[17] V.I. Judovič, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. Mat. Fiz. 3 (1963), 1032 -1066 | MR 158189

[18] K. Kikuchi, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 30 no. 1 (1983), 63 -92 | MR 700596 | Zbl 0517.76024

[19] C. Lacave, Two dimensional incompressible ideal flow around a thin obstacle tending to a curve, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 no. 4 (2009), 1121 -1148 | Numdam | MR 2542717 | Zbl 1166.76300

[20] C. Lacave, M.C. Lopes Filho, H.J. Nussenzveig Lopes, Asymptotic behavior of 2d incompressible ideal flow around small disks, 2013, in preparation.

[21] P.-L. Lions, N. Masmoudi, Homogenization of the Euler system in a 2D porous medium, J. Math. Pures Appl. (9) 84 no. 1 (2005), 1 -20 | MR 2112870 | Zbl 1072.35032

[22] M.C. Lopes Filho, Vortex dynamics in a two-dimensional domain with holes and the small obstacle limit, SIAM J. Math. Anal. 39 no. 2 (2007), 422 -436 | MR 2338413 | Zbl 1286.76018

[23] A.J. Majda, A.L. Bertozzi, Vorticity and incompressible flow, Camb. Texts Appl. Math. vol. 27 , Cambridge University Press, Cambridge (2002) | MR 1867882 | Zbl 0983.76001

[24] N. Masmoudi, Homogenization of the compressible Navier–Stokes equations in a porous medium, ESAIM Control Optim. Calc. Var. vol. 8 (2002), 885 -906 | Numdam | MR 1932978 | Zbl 1071.76047

[25] F.J. Mcgrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Ration. Mech. Anal. 27 (1967), 329 -348 | MR 221818 | Zbl 0187.49508

[26] A. Mikelić, Homogenization of nonstationary Navier–Stokes equations in a domain with a grained boundary, Ann. Mat. Pura Appl. (4) 158 (1991), 167 -179 | MR 1131849 | Zbl 0758.35007

[27] A. Mikelić, L. Paoli, Homogenization of the inviscid incompressible fluid flow through a 2D porous medium, Proc. Am. Math. Soc. 127 no. 7 (1999), 2019 -2028 | MR 1626446 | Zbl 0922.35136

[28] C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren Math. Wiss. vol. 299 , Springer-Verlag, Berlin (1992) | MR 1217706 | Zbl 0762.30001

[29] E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, Springer-Verlag, Berlin (1980) | MR 578345

[30] E. Sánchez-Palencia, Boundary value problems in domains containing perforated walls, Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. III, Paris, 1980/1981, Res. Notes Math. vol. 70 , Pitman, Boston, MA (1982), 309 -325 | MR 670282

[31] L. Tartar, Incompressible fluid flow in a porous medium: convergence of the homogenization process, E. Sánchez-Palencia (ed.), Nonhomogeneous Media and Vibration Theory (1980), 368 -377