Variational views of stokeslets and stresslets

Abstract

In this paper we present a self-contained variational theory of the layer potentials for the Stokes problem on Lipschitz boundaries. We use these weak definitions to show how to prove the main theorems about the associated Calderón projector. Finally, we relate these variational definitions to the integral forms. Instead of working these relations from scratch, we show some formulas parametrizing the Stokes layer potentials in terms of those for the Lamé and Laplace operators. While all the results in this paper are well known for smooth domains, and most might be known for non-smooth domains, the approach is novel and gives a solid structure to the theory of Stokes layer potentials.

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

References

  1. 1.

    Amrouche, C., Girault, V., Giroire, J.: Espaces de Sobolev avec poids et équation de Laplace dans R \(^n\). I. C. R. Acad. Sci. Paris Sér. I Math. 315(3), 269–274 (1992)

  2. 2.

    Amrouche, C., Girault, V., Giroire, J.: Weighted Sobolev spaces for Laplace’s equation in R \(^{n}\). J. Math. Pures Appl. (9) 73(6), 579–606 (1994)

    MATH  MathSciNet  Google Scholar 

  3. 3.

    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 129–151 (1974)

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, volume 15 of Springer Series in Computational Mathematics. Springer, New York (1991)

  5. 5.

    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19(3), 613–626 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Dijkstra, W., Kakuba, G., Mattheij, R.M.M.: Condition numbers and local errors in the boundary element method. In: Boundary element methods in engineering and sciences, volume 4 of Computer Experiment Methods Structure. pp. 365–402. Imperial College Press, London (2011)

  7. 7.

    Domínguez, V., Sayas, F.-J.: A BEM-FEM overlapping algorithm for the Stokes equation. Appl. Math. Comput. 182(1), 691–710 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Fabes, E.B., Kenig, C.E., Verchota, G.C.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57(3), 769–793 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Girault, V., Sequeira, A.: A well-posed problem for the exterior Stokes equations in two and three dimensions. Arch. Ration. Mech. Anal. 114(4), 313–333 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    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  Google Scholar 

  11. 11.

    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, volume 164 of Applied Mathematical Sciences. Springer, Berlin (2008)

  12. 12.

    Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow. Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, vol. 2. Gordon and Breach Science Publishers, New York (1969)

  13. 13.

    Le Roux, M.-N.: Équations intégrales pour le problème du potentiel électrique dans le plan. C. R. Acad. Sci. Paris Sér. A 278, 541–544 (1974)

    MATH  MathSciNet  Google Scholar 

  14. 14.

    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  15. 15.

    J.-C. Nédélec. Approximation des équations intégrales en mécanique et en physique, Cours de DEA (1977)

  16. 16.

    Pozrikidis, C.: Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Francisco-Javier Sayas.

Additional information

Partially supported by the NSF (DMS 1216356). Partially supported by MICINN (Project MTM2010-21135-C021-01) and the Universidad de Oviedo ‘Ayudas de Movilidad de Excelencia’ Program.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sayas, F., Selgas, V. Variational views of stokeslets and stresslets. SeMA 63, 65–90 (2014). https://doi.org/10.1007/s40324-014-0013-x

Download citation

Keywords

  • Stokes flow
  • Potential theory
  • Boundary integral equations

Mathematics Subject Classification

  • 31B10
  • 76D07