Skip to main content
SpringerLink
Log in

Behaviour of the Stokes operators under domain perturbation

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

    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.

Abstract

Depending on the geometry of the domain, one can define—at least—three different Stokes operators with Dirichlet boundary conditions. We describe how the resolvents of these Stokes operators converge with respect to a converging sequence of domains.

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. Arendt W. Approximation of degenerate semigroups. Taiwanese J Math, 2001, 5: 279–295

    Article  MathSciNet  MATH  Google Scholar 

  2. Arendt W, Daners D. Varying domains: Stability of the Dirichlet and the Poisson problem. Discrete Contin Dyn Syst, 2008, 21: 21–39

    Article  MathSciNet  MATH  Google Scholar 

  3. Daners D. Dirichlet problems on varying domains. J Differential Equations, 2003, 188: 591–624

    Article  MathSciNet  MATH  Google Scholar 

  4. de Rham G. Differentiable Manifolds. Grundlehren der Mathematischen Wissenschaften, vol. 266. Berlin: Springer-Verlag, 1984

    Google Scholar 

  5. Heywood J G. On uniqueness questions in the theory of viscous flow. Acta Math, 1976, 136: 61–102

    Article  MathSciNet  MATH  Google Scholar 

  6. Kato T. Perturbation Theory for Linear Operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995

    Google Scholar 

  7. Ladyzhenskaya O A, Solonnikov V A. Some problems of vector analysis, and generalized formulations of boundary-value problems for the Navier-Stokes equation. J Math Sci, 1978, 10: 257–286

    Article  MATH  Google Scholar 

  8. Monniaux S. Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme. Math Res Lett, 2006, 13: 455–461

    Article  MathSciNet  MATH  Google Scholar 

  9. Sohr H. The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts. Basel: Birkhauser Verlag, 2001

    Google Scholar 

  10. Stummel F. Perturbation theory for Sobolev spaces. Proc Roy Soc Edinburgh Sect A, 1975, 73: 5–49

    Article  MathSciNet  MATH  Google Scholar 

  11. Temam R. Navier-Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. Amsterdam-New York: North-Holland, 1979

    Google Scholar 

Download references

Acknowledgements

This work was supported by the ANR Project INFAMIE (Grant No. ANR-15-CE40-001). The understanding of this subject has benefited from discussions with Tom ter Elst. The author thanks the anonymous referees whose remarks greatly improved this manuscript.

Author information

Authors and Affiliations

  1. Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, 13453, France

    Sylvie Monniaux

Authors
  1. Sylvie Monniaux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvie Monniaux.

Additional information

Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Monniaux, S. Behaviour of the Stokes operators under domain perturbation. Sci. China Math. 62, 1167–1174 (2019). https://doi.org/10.1007/s11425-019-9517-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-9517-x

Keywords

MSC(2010)

Search

Navigation