Skip to main content
SpringerLink
Log in

Linearized Navier–Stokes Equations with Boundary Conditions Involving the Pressure in an Exterior Domain of \({\mathbb {R}}^{3}\)

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the stationary Oseen equations in an exterior domain of \({\mathbb {R}}^ 3\) with boundary conditions involving the pressure. Our purpose is to prove the existence and the uniqueness of a weak solution in a Hilbertian framework. To prescribe the growth or decay of functions at infinity, we set the problem in weighted Sobolev spaces.

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

Similar content being viewed by others

References

  1. Alliot, F., Amrouche, C.: Weak solutions for the exterior Stokes problem in weighted Sobolev spaces. Math. Methods Appl. Sci. 23(6), 575–600 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Amrouche, C., Girault, V., Giroire, J.: Dirichlet and Neumann exterior problems for the \(n\)-dimensional Laplace operator: an approach in weighted Sobolev spaces. J. Math. Pures Appl. (9) 76(1), 55–81 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Amrouche, C., Meslameni, M.: Stokes problem with several types of boundary conditions in an exterior domain. Electron. J. Differ. Equ. 28, 196 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Amrouche, C., Meslameni, M.: Very weak solutions for the Stokes problem in an exterior domain. Annali Dell’Universita’ di Ferrara 59(1), 3–29 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Amrouche, C., Meslameni, M., Nečasová, Š.: The stationary Oseen equations in an exterior domain: an approach in weighted Sobolev spaces. J. Differ. Equ. 256(6), 1955–1986 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Amrouche, C., Rejaiba, A.: \({L}^p\)-theory for Stokes and Navier–Stokes equations with Navier boundary condition. J. Differ. Equ. 256, 1515–1547 (2014)

    MATH  Google Scholar 

  7. Amrouche, C., Seloula, N.E.H.: Stokes equations and elliptic systems with nonstandard boundary conditions. C. R. Math. Acad. Sci. Paris 349(11–12), 703–708 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Amrouche, C., Seloula, N.E.H.: \({L}^p\)-theory for the Navier–Stokes equations with pressure boundary conditions. Disc. Cont. Dyn. Syst. Ser. S 6(5), 1113–1137 (2013)

    MATH  Google Scholar 

  9. Amrouche, C., Seloula, N.E.H.: \(L^p\)-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23(1), 37–92 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Bègue, C., Conca, C., Murat, F., Pironneau, O.: À nouveau sur les équations de Stokes et de Navier–Stokes avec des conditions aux limites sur la pression. C. R. Acad. Sci. Paris Sér. I Math. 304(1), 23–28 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Bègue, C., Conca, C., Murat, F., Pironneau, O.: Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression. In: Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. IX (Paris, 1985–1986), volume 181 of Pitman Res. Notes Math. Ser., pp. 179–264. Longman Sci. Tech., Harlow (1988)

  12. Benjemaa, M., Louati, H., Meslameni, M.: On the stationary Navier–Stokes problem in \({\mathbb{R}}^3\): an approach in weighted Sobolev spaces. Mediterr. J. Math. 14(3), 1–24 (2017). Art. 138

    MATH  Google Scholar 

  13. Conca, C., Murat, F., Pironneau, O.: The Stokes and Navier–Stokes equations with boundary conditions involving the pressure. Japan J. Math. 20(2), 279–318 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Conca, C., Parés, C., Pironneau, O., Thiriet, M.: Navier–Stokes equations with imposed pressure and velocity fluxes. Int. J. Numer. Methods Fluids 20(4), 267–287 (1995)

    MathSciNet  MATH  Google Scholar 

  15. de Rham, G.: Variétés différentiables. Formes, courants, formes harmoniques. Actualités Sci. Ind., no. 1222 = Publ. Inst. Math. Univ. Nancago III. Hermann et Cie, Paris, (1955)

  16. Dhifaoui, A., Meslameni, M., Razafison, U.: Weighted Hilbert spaces for the stationary exterior Stokes problem with Navier slip boundary conditions. J. Math. Anal. Appl. 472(2), 1846–1871 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Farwig, R.: The stationary exterior \(3\)D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211(3), 409–447 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Farwig, R., Kozono, H., Sohr, H.: Very weak solutions of the Navier–Stokes equations in exterior domains with nonhomogeneous data. J. Math. Soc. Japan 59(1), 127–150 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. Vol. I, volume 38 of Springer Tracts in Natural Philosophy. Springer, New York (1994)

  20. Galdi, G.P., Simader, C.G.: Existence, uniqueness and \(L^q\)-estimates for the Stokes problem in an exterior domain. Arch. Rational Mech. Anal. 112(4), 291–318 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Galdi, G.P., Simader, C.G.: New estimates for the steady-state Stokes problem in exterior domains with applications to the Navier–Stokes problem. Differ. Integr. Equ. 7(3–4), 847–861 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Girault, V.: The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of \({\mathbb{R}}^3\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39(2), 279–307 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Girault, V.: The Stokes problem and vector potential operator in three-dimensional exterior domains: an approach in weighted Sobolev spaces. Differ, Integr. Equ. 7(2), 535–570 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Giroire, J.: Etude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Thèse de Doctorat d’Etat. Université Pierre et Marie Curie, Paris (1987)

  25. Hanouzet, B.: Espaces de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46, 227–272 (1971)

    MathSciNet  MATH  Google Scholar 

  26. Horst, H., Hyunseok, K., Hideo, K.: Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle. Math. Ann. 356, 653–681 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Kim, H., Kozono, H.: On the stationary Navier–Stokes equations in exterior domains. J. Math. Anal. Appl. 395(2), 486–495 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Louati, H., Meslameni, M., Razafison, U.: Weighted \(L^p\)-theory for vector potential operators in three-dimensional exterior domains. Math. Methods Appl. Sci. 39(8), 1990–2010 (2016)

    MATH  Google Scholar 

  29. Serrin, J.: Mathematical principles of classical fluid mechanics. In: Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), pages 125–263. Springer-Verlag, Berlin-Göttingen-Heidelberg (1959)

Download references

Author information

Authors and Affiliations

  1. Preparatory Institute for Engineering Studies of Sfax, University of Sfax, Sfax, Tunisia

    Mohamed Meslameni

  2. College of Arts and Science in Al-Mandaq, AL Baha University, Al Bahah, Kingdom of Saudi Arabia

    Mohamed Meslameni

Authors
  1. Mohamed Meslameni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Meslameni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meslameni, M. Linearized Navier–Stokes Equations with Boundary Conditions Involving the Pressure in an Exterior Domain of \({\mathbb {R}}^{3}\). Mediterr. J. Math. 17, 85 (2020). https://doi.org/10.1007/s00009-020-01524-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-01524-4

Mathematics Subject Classification

Keywords

Search

Navigation