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Homogenization of the evolutionary Navier–Stokes system

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Abstract

We study the homogenization problem for the evolutionary Navier–Stokes system under the critical size of obstacles. Convergence towards the limit system of Brinkman’s type is shown under very mild assumptions concerning the shape of the obstacles and their mutual distance.

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References

  1. Acosta G., Durán R.G., Muschietti M.A.: Solutions of the divergence operator on John domains. Adv. Math. 206(2), 373–401 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allaire A.: Continuity of the Darcy’s law in the low-volume fraction limit. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18(4), 475–499 (1991)

    MathSciNet  MATH  Google Scholar 

  3. 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. 113(3), 209–259 (1990)

    Article  MathSciNet  Google Scholar 

  4. 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. 113(3), 261–298 (1990)

    Article  MathSciNet  Google Scholar 

  5. Bucur D., Feireisl E., Nečasová Š.: Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions. Arch. Ration. Mech. Anal. 197, 117–138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bucur D., Feireisl E., Nečasová Š.: On the asymptotic limit of flows past a ribbed boundary. J. Math. Fluid Mech. 10, 554–568 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dal Maso G., Skrypnik I.V.: Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains. J. Anal. Math. 79, 63–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dal Maso G., Skrypnik I.V.: A monotonicity approach to nonlinear Dirichlet problems in perforated domains. Adv. Math. Sci. Appl. 11(2), 721–751 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Desvillettes L., Golse F., Ricci V.: The mean-field limit for solid particles in a Navier–Stokes flow. J. Stat. Phys. 131(5), 941–967 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Diening L., Růžička M., Schumacher K.: A decomposition technique for John domains. Annal. Acad. Sci. Fenn. Math. 35 1, 87–114 (2010)

    Article  Google Scholar 

  11. Hopf E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)

    MATH  Google Scholar 

  13. Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  14. Marchenko, V.A., Hruslov, E.Ja.: Boundary Value Problems in Domains with a Fine-Grained Boundary (Russian). Naukova Dumka, Kiev (1974)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Skrypnik I.V.: Averaging of nonliner Dirichlet problems in punctured domains of general structure. Dokl. Akad. Nauk. 353(2), 163–166 (1997)

    MathSciNet  Google Scholar 

  17. Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Translated from the Russian by Dan D. Pascali. Translation edited by Simeon Ivanov, Translations of Mathematical Monographs, vol. 139. American Mathematical Society (AMS), Providence, RI (1994)

  18. Tartar, L.: Incompressible fluid flow in a porous medium: convergence of the homogenization process. In: Sánchez-Palencia, E. Nonhomegenous Media and Vibration Theory, pp. 368–377. Springer, Berlin (1980)

  19. Temam R.: Navier–Stokes Equations. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Prague 1, Czech Republic

    Eduard Feireisl, Yuliya Namlyeyeva & Šárka Nečasová

  2. Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg St. 74, Donetsk, 83114, Ukraine

    Yuliya Namlyeyeva

Authors
  1. Eduard Feireisl
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  2. Yuliya Namlyeyeva
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  3. Šárka Nečasová
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Corresponding author

Correspondence to Yuliya Namlyeyeva.

Additional information

The work of E. F. was supported by the GA ČR (Czech Science Foundation) Project 13-00522S in the framework of RVO: 67985840. The work of Š. N. and Yu. N. was supported by project between the Academy of Sciences of the Czech Republic and the National Academy of Sciences of Ukraine (2008–2013). The work of Š. N. was supported by the GA ČR (Czech Science Foundation) Project 13-00522S in the framework of RVO: 67985840.

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Feireisl, E., Namlyeyeva, Y. & Nečasová, Š. Homogenization of the evolutionary Navier–Stokes system. manuscripta math. 149, 251–274 (2016). https://doi.org/10.1007/s00229-015-0778-y

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  • DOI: https://doi.org/10.1007/s00229-015-0778-y

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