Advertisement

Science China Mathematics

, Volume 62, Issue 6, pp 1121–1142 | Cite as

A note on the regularity of the holes for permeability property through a perforated domain for the 2D Euler equations

  • Christophe LacaveEmail author
  • Chao Wang
Articles
  • 8 Downloads

Abstract

For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes dε is equal to or much larger than the size of the holes ε. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when (ε, dε) → (0, 0). Smaller distances were avoided for mathematical reasons and for these large distances, the geometry of the holes does not affect or rarely affect the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distances much smaller than the size of the holes. For this result, the boundary regularity of holes plays a crucial role, and the permeability criterion depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such that ε ln dε → 0.

Keywords

ideal fluids homogenization in perforated domains shrinking obstacles and the porous medium domains with corners 

MSC(2010)

35Q31 76B03 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The first author was supported by the CNRS (program Tellus), the Agence Nationale de la Recherche: Project IFSMACS (Grant No. ANR-15-CE40-0010) and Project SINGFLOWS (Grant No. ANR-18-CE40-0027-01). The second author was supported by National Natural Science Foundation of China (Grant No. 11701016). This work has been supported by the Sino-French Research Program in Mathematics (SFRPM), which made several visits between the authors possible. The first author would like to acknowledge the hospitality and financial support of Peking University to conduct part of this work.

References

  1. 1.
    Ahlfors L V. Lectures on Quasiconformal Mappings. Van Nostrand Mathematical Studies, 10. Toronto-New York-London: Van Nostrand, 1966zbMATHGoogle Scholar
  2. 2.
    Allaire G. 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, 1990, 113: 209–259MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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 Ration Mech Anal, 1990, 113: 261–298MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arsénio D, Dormy E, Lacave C. The vortex method for 2D ideal flows in exterior domains. ArXiv:1707.01458, 2017Google Scholar
  5. 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, 2015, 32: 159–182MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borsuk M, Kondratiev V. Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains. North-Holland Mathematical Library, vol. 69. Amsterdam: Elsevier, 2006zbMATHGoogle Scholar
  7. 7.
    DiPerna R J, Lions P-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math, 1989, 98: 511–547MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gérard-Varet D, Lacave C. The two dimensional Euler equations on singular exterior domains. Arch Ration Mech Anal, 2015, 218: 1609–1631MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grisvard P. Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Boston: Pitman, 1985zbMATHGoogle Scholar
  10. 10.
    Gustafsson B, Vasil’ev A. Conformal and Potential Analysis in Hele-Shaw Cells. Advances in Mathematical Fluid Mechanics. Basel: Birkhäuser, 2006zbMATHGoogle Scholar
  11. 11.
    Iftimie D, Lopes Filho M C, Nussenzveig Lopes H J. Two dimensional incompressible ideal flow around a small obstacle. Comm Partial Differential Equations, 2003, 28: 349–379MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jerison D, Kenig C E. The inhomogeneous Dirichlet problem in Lipschitz domains. J Funct Anal, 1995, 130: 161–219MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kozlov V A, Maz’ya V G, Rossmann J. Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85. Providence: Amer Math Soc, 2001zbMATHGoogle Scholar
  14. 14.
    Lacave C. Uniqueness for two-dimensional incompressible ideal flow on singular domains. SIAM J Math Anal, 2015, 47: 1615–1664MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lacave C, Masmoudi N. Impermeability through a perforated domain for incompressible two dimensional Euler equations. Arch Ration Mech Anal, 2016, 221: 1117–1160MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lacave C, Miot E, Wang C. Uniqueness for the two-dimensional Euler equations on domains with corners. Indiana Univ Math J, 2014, 63: 1725–1756MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lacave C, Zlatos A. The Euler equations in planar domains with corners. Arch Ration Mech Anal, 2019, in pressGoogle Scholar
  18. 18.
    Lions P-L. Mathematical Topics in Fluid Mechanics, Volume 1. Lecture Series in Mathematics and Its Applications, vol. 3. New York: Oxford University Press, 1996zbMATHGoogle Scholar
  19. 19.
    Lions P-L, Masmoudi N. Homogenization of the Euler system in a 2D porous medium. J Math Pures Appl (9), 2005, 84: 1–20MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Marchioro C, Pulvirenti M. Mathematical Theory of Incompressible Nonviscous Fluids. Applied Mathematical Sciences, vol. 96. New York: Springer-Verlag, 1994zbMATHGoogle Scholar
  21. 21.
    Mikelic A. Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary. Ann Mat Pura Appl (4), 1991, 158: 167–179MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mikelic A, Paoli L. Homogenization of the inviscid incompressible fluid flow through a 2D porous medium. Proc Amer Math Soc, 1999, 127: 2019–2028MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pommerenke C. Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften, vol. 299. Berlin: Springer-Verlag, 1992zbMATHGoogle Scholar
  24. 24.
    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. Research Notes in Mathematics, vol. 70. Boston: Pitman, 1982, 309–325Google Scholar
  25. 25.
    Tartar L. Incompressible fluid flow in a porous medium: Convergence of the homogenization process. In: Nonhomogeneous Media and Vibration Theory. Berlin: Springer, 1980, 368–377Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble Alpes, CNRSGrenobleFrance
  2. 2.School of Mathematical SciencesPeking UniversityBeijingChina

Personalised recommendations