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

• Friedrich K. Hebeker
• George C. Hsiao
Chapter
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).

## Keywords

Boundary Element Method Boundary Integral Equation Pseudodifferential Operator Jump Relation Boundary Integral Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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.
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.
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.
8. [8]
Hebeker, F. K., Efiicient boimdary element methods for 3D exterior viscous flows. Num. Meth. PDE, 2 (1986), 273–297.
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.
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.
16. [16]
Hsiao, G. C., Wendland, W. L., The Aubin-Nitsche lemma for integral equations, J. Integral Eqn., 3 (1981), 299–315.
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.
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.
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.
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.
26. [26]
Temam, R., Navier Stokes Equations, 1977, Amsterdam.
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