Skip to main content
SpringerLink
Log in

A Steady Weak Solution of the Equations of Motion of a Viscous Incompressible Fluid through Porous Media in a Domain with a Non-Compact Boundary

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We assume that Ω is a domain in ℝ2 or in ℝ3 with a non-compact boundary, representing a generally inhomogeneous and anisotropic porous medium. We prove the weak solvability of the boundary-value problem, describing the steady motion of a viscous incompressible fluid in Ω. We impose no restriction on sizes of the velocity fluxes through unbounded components of the boundary of Ω. The proof is based on the construction of appropriate Galerkin approximations and study of their convergence. In Sect. 4, we provide several examples of concrete forms of Ω and prescribed velocity profiles on ∂Ω, when our main theorem can be applied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Amick, C.J.: Existence of solutions to the nonhomogeneous steady Navier–Stokes equations. Indiana Univ. Math. J. 33, 817–830 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bresch, D., Sy, M.: Convection in rotating porous media: the planetary geostrophic equations, used in geophysical fluid dynamics. Contin. Mech. Thermodyn. 15(3), 247–263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Feistauer, M.: Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 67. Longman Scientific & Technical, Harlow (1993)

    MATH  Google Scholar 

  4. Flavin, J.N.: The evolution to a steady state for a porous medium model. J. Math. Anal. Appl. 322(1), 393–402 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Flavin, J.N., Rionero, S.: Stability properties for nonlinear diffusion in porous and other media. J. Math. Anal. Appl. 281(1), 221–232 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Franchi, F., Straughan, B.: Structural stability for the Brinkman equations of porous media. Math. Methods Appl. Sci. 19(16), 1335–1347 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I. Springer, New York (1994)

    Book  Google Scholar 

  8. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II. Springer, New York (1994)

    Book  Google Scholar 

  9. Galdi, G.P.: Further properties of steady-state solutions to the Navier–Stokes problem past a three-dimensional obstacle. J. Math. Phys. 48(6), 065207 (2007) (43 pp)

    Article  MathSciNet  Google Scholar 

  10. Kozono, H., Yanagisawa, T.: Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data. Math. Z. 262(1), 27–39 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kufner, A., John, O., Fučík, S.: Function Spaces. Academia, Prague (1977)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  13. Lombardo, S., Mulone, G.: Non-linear stability and convection for laminar flows in a porous medium with Brinkman law. Math. Methods Appl. Sci. 26(6), 453–462 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Morimoto, H.: Stationary Navier–Stokes equations with non-vanishing outflow condition. Hokkaido Math. J. 24(3), 641–648 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Mucha, P.B.: Flux problem for a certain class of two-dimensional domains. Nonlinearity 18(4), 1699–1704 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Neustupa, J.: On the steady Navier–Stokes boundary value problem in an unbounded 2D domain with arbitrary fluxes through the components of the boundary. Ann. Univ. Ferrara 55(2), 353–365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Neustupa, J.: A new approach to the existence of weak solutions of the steady Navier–Stokes system with inhomogeneous boundary data in domains with noncompact boundaries. Arch. Ration. Mech. Anal. 198(1), 331–348 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  19. Sohr, H.: The Navier–Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of the Petroleum Institute in Abu Dhabi, the Grant Agency of the Czech Academy of Sciences (grant No. IAA100190905) and the Academy of Sciences of the Czech Republic (Institutional Research Plan No. AV0Z10190503).

Author information

Authors and Affiliations

  1. Fac. of Education, Dept. of Elementary Math. Education, Zirve University, Gaziantep, Turkey

    Fahir Talay Akyildiz

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Jiří Neustupa

  3. Centro de Investigaciones en Creatividad y Educación Superior, Universidad de Santiago de Chile, Santiago, Chile

    Dennis Siginer

Authors
  1. Fahir Talay Akyildiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří Neustupa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dennis Siginer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří Neustupa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akyildiz, F.T., Neustupa, J. & Siginer, D. A Steady Weak Solution of the Equations of Motion of a Viscous Incompressible Fluid through Porous Media in a Domain with a Non-Compact Boundary. Acta Appl Math 119, 23–42 (2012). https://doi.org/10.1007/s10440-011-9659-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-011-9659-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation