Archive for Rational Mechanics and Analysis

, Volume 114, Issue 4, pp 313–333

A well-posed problem for the exterior Stokes equations in two and three dimensions

  • Vivette Girault
  • Adelia Sequeira
Article

Abstract

This paper treats the Stokes problem in exterior Lipschitz-continuous domains of ℝ2 and ℝ3. Using the weighted Sobolev spaces of Hanouzet (in ℝ3) and Giroire (in ℝ2), we establish the inf-sup condition between the velocity and pressure spaces. This fundamental result shows that the variational Stokes problem is well-posed in those spaces. In the last paragraph, we obtain additional regularity of the solution when the data are smoother.

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References

  1. [Am]
    C. J. Amick, On Leray's problem of steady Navier-Stokes flow past a body in the plane, Acta Math. 161 (1988), 71–130.Google Scholar
  2. [Ba]
    K. I. Babenko, On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Mat. Sbornik 91 (133) (1973), 3–26.Google Scholar
  3. [Bo]
    W. Borchers & H. Sohr, On the semigroup of the Stokes operator for exterior domains in L q-spaces, Math. Z. 196 (1987), 415–425.Google Scholar
  4. [Br]
    F, Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, R.A.I.R.O., Anal. Numer. R2 (1974), 129–151.Google Scholar
  5. [Bu]
    I. Babuska, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1973), 179–192.Google Scholar
  6. [Ca]
    L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308–340.Google Scholar
  7. [Fi]1
    R. Finn, On the steady-state solutions of the Navier-Stokes equations III, Acta Math. 105 (1961), 197–244.Google Scholar
  8. [Fi]2
    R. Finn, On the exterior stationary problem for the Navier-Stokes equations and associated perturbation problems, Arch. Rational Mech. Anal. 19 (1965), 363–406.Google Scholar
  9. [Fu]
    H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 9 (1961), 59–102.Google Scholar
  10. [Gi]
    V. Girault & P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Comp. Math. 5, Springer-Verlag, Berlin (1986).Google Scholar
  11. [Gl]
    D. Gilbarg & H. F. Weinberger, Asymptotic properties of Leray's solution of the stationary two-dimensional Navier-Stokes equations, Russian Math. Surveys 29 (1974), 109–123.Google Scholar
  12. [Gu]
    G. Guirguis, On the existence, uniqueness and regularity of the exterior Stokes problem in ℝ 3, Comm. in Partial Diff. Eq. 11 (1986), 567–594.Google Scholar
  13. [Gr]
    J. Giroire, Etude de Quelques Problèmes aux Limites Extérieurs et Résolution par Equations Intégrales, Thèse, Université Paris VI (1987).Google Scholar
  14. [Ha]
    B. Hanouzet, Espaces de Sobolev avec poids — application au problème de Dirichlet dans un demi-espace, Rend. Sem. Mat. Univ. Padova 46 (1971), 227–272.Google Scholar
  15. [He]1
    J. G. Heywood, On stationary solutions of the Navier-Stokes equations and limits of nonstationary solutions, Arch. Rational Mech. Anal. 37 (1970), 48–60Google Scholar
  16. [He]2
    J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), 639–681.Google Scholar
  17. [Hr]
    G. G. Hardy, D. E. Littlewood & G. Polya, Inequalities, Cambridge Univ. Press (1959).Google Scholar
  18. [Jo]
    C. Johnson & J. C. Nedelec, On the coupling of boundary integrals and finite element methods, Math. of Comp. 35 (1980), 1063–1079.Google Scholar
  19. [Ko]
    H. Kozono & H. Sohr, L q-regularity theory of the Stokes equations in exterior domains, (preprint).Google Scholar
  20. [La]
    O. A. Ladyzhenskaya & V. A. Solonnikov, On the solvability of boundary and initial-boundary value problems in regions with non-compact boundaries, Vestnik Leningrad Univ. 13 (1977), 39–47.Google Scholar
  21. [Le]1
    J. Leray, Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1–82.Google Scholar
  22. [Le]2
    J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193–248.Google Scholar
  23. [Ma]
    C. M. Ma, On square-summability and uniqueness questions concerning nonstationary Stokes flow in an exterior domain, Arch. Rational Mech. Anal. 6 (1979), 99–112.Google Scholar
  24. [Ms]
    K. Masuda, On the stability of incompressible viscous fluid motions past objects, J. Math. Soc. Japan 27 (1975), 294–327.Google Scholar
  25. [Ne]
    J. C. Nedelec, Equations intégrales in Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques (R. Dautray & J. L. Lions, eds.) 2, Ch. XI, Collection C.E.A., Masson, Paris (1985).Google Scholar
  26. [Se]1
    A. Sequeira, Couplage entre la Méthode des Eléments Finis et la Méthode des Equations Intégrales. Application au Problème Extérieur de Stokes Stationnaire dans le Plan, Thèse, Université Paris VI (1981).Google Scholar
  27. [Se]2
    A. Sequeira, The coupling of boundary integral and finite element methods for the bidimensional exterior steady Stokes problem, Math. Meth. in Appl. Sci. 5 (1983), 356–375.Google Scholar
  28. [Se]3
    A. Sequeira, On the computer implementation of a coupled boundary and finite element method for the bidimensional exterior steady Stokes problem, Math. Meth. in Appl. Sci. 8 (1986), 117–133.Google Scholar
  29. [Sm]
    D. R. Smith, Estimates at infinity for stationary solutions of the Navier-Stokes equations in two dimensions, Arch. Rational Mech. Anal. 20 (1965), 341–372.Google Scholar
  30. [So]
    H. Sohr & W. Varnhorn, On decay properties of the Stokes equations in exterior domains (preprint).Google Scholar
  31. [Sp]
    M. Specovius-Neugebauer, Exterior Stokes problem and decay at infinity, Math. Meth. in Appl. Sci. 8 (1986), 351–367.Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Vivette Girault
    • 1
    • 2
  • Adelia Sequeira
    • 1
    • 2
  1. 1.Laboratoire d'Analyse Numérique Tour 55-655ème étage Université Pierre et Marie CurieParis Cedex 05France
  2. 2.INIC/CMAF Universidade de LisboaLisboa CodexPortugal

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