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Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

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Abstract

We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size \({\varepsilon}\) separated by distances \({d_{\varepsilon}}\) and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of \({\frac{d_{\varepsilon}}\varepsilon}\) when \({\varepsilon}\) goes to zero. If \({\frac{d_{\varepsilon}}\varepsilon \to \infty}\), then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, \({\frac{d_{\varepsilon}}\varepsilon \to 0}\), then the fluid cannot penetrate the porous region, namely, the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of \({\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}}\) where \({\gamma \in (0,\infty]}\) is related to the geometry of the lateral boundaries of the obstacles. If \({\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty}\), then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions; in particular, it is equal to \({\varepsilon^{3}}\) for balls.

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References

  1. Allaire G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113(3), 209–259 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Allaire G.: 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. Rational Mech. Anal. 113(3), 261–298 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Allaire G.: Homogenization of the Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math. 44(6), 605–641 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bendali A., Fares M., Piot E., Tordeux S.: Mathematical justification of the Rayleigh conductivity model for perforated plates in acoustics. SIAM J. Appl. Math. 73(1), 438–459 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bonnaillie-Noël V., Lacave C., Masmoudi N.: Permeability through a perforated domain for the incompressible 2D Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 159–182 (2015)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Borsuk, M., Kondratiev, V.: Elliptic boundary value problems of second order in piecewise smooth domains, volume 69 of North-Holland Mathematical Library. Elsevier Science B.V., Amsterdam, 2006.

  7. Cardone G., Nazarov S.A., Sokolowski J.: Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary. Asymptot. Anal. 62(1-2), 41–88 (2009)

    MATH  MathSciNet  Google Scholar 

  8. Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), volume 60 of Res. Notes in Math., pp. 98–138, 389–390. Pitman, Boston, Mass., 1982.

  9. Conca C.: Étude d’un fluide traversant une paroi perforée. I. Comportement limite près de la paroi. J. Math. Pures Appl. (9) 66(1), 1–43 (1987)

    MATH  MathSciNet  Google Scholar 

  10. Conca C.: Étude d’un fluide traversant une paroi perforée. II. Comportement limite loin de la paroi. J. Math. Pures Appl. (9) 66(1), 45–70 (1987)

    MATH  MathSciNet  Google Scholar 

  11. Díaz, J.I.: Two problems in homogenization of porous media. Proceedings of the Second International Seminar on Geometry, Continua and Microstructure (Getafe, 1998), volume 14, pp. 141–155, 1999.

  12. Diaz-Alban J., Masmoudi N.: Asymptotic analysis of acoustic waves in a porous medium: initial layers in time. Commun. Math. Sci. 10(1), 239–265 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. Springer Monographs in Mathematics. Springer, New York, second edition, 2011. Steady-state problems.

  14. Gérard-Varet D., Lacave C.: The Two-Dimensional Euler Equations on Singular Domains. Arch. Ration. Mech. Anal. 209(1), 131–170 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gérard-Varet D., Lacave C.: The Two Dimensional Euler Equations on Singular Exterior Domains. Arch. Ration. Mech. Anal. 218(3), 1609–1631 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  16. Iftimie D., Lopes Filho M.C., Nussenzveig Lopes H.J.: Two dimensional incompressible ideal flow around a small obstacle. Comm. Partial Differential Equations, 28(1–2), 349–379 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kikuchi K.: Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30(1), 63–92 (1983)

    MATH  MathSciNet  Google Scholar 

  18. Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations, volume 85 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.

  19. Lacave C.: Two dimensional incompressible ideal flow around a thin obstacle tending to a curve. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1121–1148 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Lacave C.: Uniqueness for two-dimensional incompressible ideal flow on singular domains. SIAM J. Math. Anal. 47(2), 1615–1664 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lacave, C., Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Asymptotic behavior of 2D incompressible ideal flow around small disks. arXiv preprint arXiv:1510.05864, 2015.

  22. Lacave, C., Mazzucato, A.: The vanishing viscosity limit in the presence of a porous medium. To appear in Math. Ann., 2016.

  23. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications.

  24. Lions P.-L., Masmoudi N.: Une approche locale de la limite incompressible. (french) [a local approach to the incompressible limit]. C. R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Lions P.-L., Masmoudi N.: Homogenization of the Euler system in a 2D porous medium. J. Math. Pures Appl. (9) 84(1), 1–20 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lopes Filho, M.C.: Vortex dynamics in a two-dimensional domain with holes and the small obstacle limit. SIAM J. Math. Anal., 39(2):422–436, 2007 (electronic)

  27. Masmoudi, N.: Homogenization of the compressible Navier-Stokes equations in a porous medium. ESAIM Control Optim. Calc. Var., 8:885–906 , 2002 ((electronic) A tribute to J. L. Lions).

  28. Masmoudi N.: Some uniform elliptic estimates in a porous medium. C. R. Math. Acad. Sci. Paris, 339(12), 849–854 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mikelić A.: Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary. Ann. Mat. Pura Appl. (4) 158, 167–179 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mikelić A., Paoli L.: Homogenization of the inviscid incompressible fluid flow through a 2D porous medium. Proc. Am. Math. Soc., 127(7), 2019–2028 (1999)

    Article  MATH  Google Scholar 

  31. Munnier, A., Ramdani, K.: Asymptotic analysis of a neumann problem in a domain with cusp. application to the collision problem of rigid bodies in a perfect fluid. arXiv preprint arXiv:1405.5446, 2014

  32. Sánchez-Palencia E.: Nonhomogeneous media and vibration theory. Springer-Verlag, Berlin (1980)

    MATH  Google Scholar 

  33. Sánchez-Palencia, E.: Boundary value problems in domains containing perforated walls. In Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), volume 70 of Res. Notes in Math., pp. 309–325. Pitman, Boston, Mass., 1982

  34. Tartar, L.: Incompressible fluid flow in a porous medium: convergence of the homogenization process. in Nonhomogeneous media and vibration theory (E. Sánchez-Palencia), pp. 368–377, 1980

  35. Wolibner W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z., 37(1), 698–726 (1933)

    Article  MATH  MathSciNet  Google Scholar 

  36. Yudovic V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)

    MathSciNet  Google Scholar 

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Correspondence to Christophe Lacave.

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Communicated by L. Saint-Raymond

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Lacave, C., Masmoudi, N. Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations. Arch Rational Mech Anal 221, 1117–1160 (2016). https://doi.org/10.1007/s00205-016-0980-4

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