On Volterra Boundary Integral Equations of the First Kind for Nonstationary Stokes Equations

  • Friedrich K. Hebeker
  • George C. Hsiao
Part of the Springer Series in Computational Mechanics book series (SSCMECH)

Abstract

We are concerned here with slow viscous incompressible flow in a container or around a rigid body. The physical situation is described mathematically by the initial- boundary value problem for the nonstationary Stokes equations:
$$\upsilon _t - \nu \Delta \upsilon + \nabla _p = f\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$
(1.1)
$${\rm div}\,\upsilon = 0\,\,{\rm in}\,\Omega _T \,{\rm or}\,\Omega _T^c ,$$
(1.2)
$$\upsilon = - \upsilon _\infty \,\,{\rm on}\,\partial \Omega _T ,$$
(1.3)
$$\upsilon ,p \to 0\,\,{\rm as}\left| x \right| \to \infty ,t > 0,$$
(1.4)
$$\upsilon = \upsilon _0 \,\,{\rm as}\,t = 0,x \in \Omega \,\,{\rm or}\,x \in \Omega ^c $$
(1.5)
Here denotes υ(t) the vector of uniform onset flow at infinity, w = υ + υ the velocity field, and p the scalar pressure field of the flow, which evolves from an initial state υ0. Further, \(\nu \sim {1 \over { R}e}\) is the dynamic viscosity of the medium. In a body-fixed frame, the flow region is Ω ⊂ 3 or \(\Omega ^C = \mathbb{R}^3 /\bar \Omega\), the boundary ∂Ω of which is assumed as sufficiently smooth (∂Ω ∈ C , e.g.). Further, Ω T = Ω × (0, T) (and so Ω T c , ∂ΩT) where (0, T) denotes a fixed finite time interval. The vector field ƒ includes the exterior forces (and virtually some small convective terms).

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References

  1. [1]
    Brebbia, C. A., The solution of time-dependent problems using boundary elements, Ed. Whiteman J. R., The Mathematics of Finite Elements and Applications, 5 (1985), 229–255.Google Scholar
  2. [2]
    Costabel, M., Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal, (1988), 613–626.Google Scholar
  3. [3]
    Costabel, M., Onishi, K., Wendland, W. L., A boundary element collocation method for the Neumann problem of the heat equation. Inverse and Ill-posed Problems, (1987), 369–384.Google Scholar
  4. [4]
    Costabel, M., Boundary integral operators for the heat equation. Integral Equ. Oper. Theory (1990).Google Scholar
  5. [5]
    Costabel, M., Wendland, W. L., Strong eUipticity of boundary integral operators, J. Reine Angew. Math., 372(1986), 39–63.MathSciNetGoogle Scholar
  6. [6]
    Fabes, E. B., Lewis, J. E., Riviere, N. M., Boundary value problems for the Navier-Stokes equations, Amer. J. Math., 99 (1977), 626–668.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Fischer, T. M., An integral equtions procedure for the exterior 3D slow viscous flow. Integral Eqn. Oper. Th., 5 (1982), 490–505. 273–297.MATHCrossRefGoogle Scholar
  8. [8]
    Hebeker, F. K., Efiicient boimdary element methods for 3D exterior viscous flows. Num. Meth. PDE, 2 (1986), 273–297.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Hebeker, F. K., Characteristics and boundary elements for 3D Navier Stokes flows, The Mathematics of Finite Elements and Applications, Ed. J. R. Whiteman, 6, (1988), 305–312.Google Scholar
  10. [10]
    Hebeker, F. K., On Lagrangean and unsteady boundary element methods for in compressible Navier Stokes problems, The Navier Stokes Equations - Theory and Numerical Methods, Ed. R. Rautmann, (Oberwolfach, September 19–23, 1988), to appear.Google Scholar
  11. [11]
    Hebeker, F. K., Hsiao, G. C., On a boundary integral equation approach to a nonstationary problem of isothermal viscous compressible flows, Preprint 1134, Fb Mathematik, (May, 1988), Technische Hochschule Darmstadt.Google Scholar
  12. [12]
    Hsiao, G. C., Kress R., On an integral equation for the 2D exterior Stokes problem, Applied Numer. Math., 1(1985), 77–93.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Hsiao, G. C., Saranen, J., Integral equation solution of some heat conduction problems, interal Equations and Inverse Problems, Eds. V. Petkov and R. Lazarov (1991), 107–113.Google Scholar
  14. [14]
    Hsiao, G. C., Saranen, J., Coercivity of single layer heat operator, SI AM Math. Anal, Submitted.Google Scholar
  15. [15]
    Hsiao, G. C., Wendland, W. L., A finite element method for some integral equations of the first kind, J. Math. Anal. Appl, 58 (1977), 449–481.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Hsiao, G. C., Wendland, W. L., The Aubin-Nitsche lemma for integral equations, J. Integral Eqn., 3 (1981), 299–315.MathSciNetMATHGoogle Scholar
  17. [17]
    Ladyzhenskaja, O. A., The Mathematical Theory of Viscous Incompressible Flows, 1969, New York.Google Scholar
  18. [18]
    Ladyzhenskaja, O. A., Solonnikov, V. A., Uralzewa, N. N., Linear and Quasilinear Equations of Parabolic Type, 1968, Providence.Google Scholar
  19. [19]
    Leis, R., Initial Boundary Value Problems in Mathematical Physics, 1986, John Wily & Sons, and B. G. Teubner, Stuttgart, New York.MATHGoogle Scholar
  20. [20]
    Lions, J. L., Magenes, E., Non-homogeneous Boundary Value Problems and Applications, Vol. 2, 1972, Berlin.Google Scholar
  21. [21]
    Nedelec, J. C., Planchard, J., Une methode variationelle d’elements finis pour la resolution numerique d’un probleihe exterior dans R, RARI0, R-8, 7 (1973), 105–129.MathSciNetGoogle Scholar
  22. [22]
    Noon, P. J., The single layer heat potential and Galerkin boundary element methods for the heat equation, Ph.D. thesis, 1988, 108 pp. University of Maryland.Google Scholar
  23. [23]
    Piskorek, A., Zabrodski, E., Uber die instationaären hydrodynamischen Potentiale, ZAMM, 60 (1980), T267–269.MATHGoogle Scholar
  24. [24]
    Solonnikov, V. A., Estimates of the solutions of a nonstationary linearized system of Navier Stokes equations, AMS Transl. Ser. 2, 75, (1968).Google Scholar
  25. [25]
    Solonnikov, V. A., Estimates for solutions of nonstationaxy Navier Stokes equations, J, Soc, Math, 8 (1977), 467–529.MATHCrossRefGoogle Scholar
  26. [26]
    Temam, R., Navier Stokes Equations, 1977, Amsterdam.MATHGoogle Scholar
  27. [27]
    Zhu, J., A boundary integral equation method for the stationary Stokes problem in 3D, Boundary Elements, Ed. C. A. Brebbia, 5 (1983), 283–292, Berlin.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Friedrich K. Hebeker
    • 1
    • 2
  • George C. Hsiao
    • 1
    • 2
  1. 1.Institute of Supercomputing and Applied MathematicsIBM Scientific CenterHeidelbergGermany
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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