Variational boundary integral equations for the Stokes system§

§Dedicated to Prof. Dr. Dr.h.c. Ioan A. Rus on the occasion of his 70th birthday.

Abstract

In this article, we analyze the variational formulations of the direct boundary integral equations for the Dirichlet, the Neumann, and the mixed boundary value problems of the Stokes system in on Lipschitz boundaries. Although the Stokes system does not belong to the class of formally positive elliptic second-order systems in the sense of Vishik and Schechter, its close relation to the Lamé system of elastostatics, together with Green's identities imply also here Gårding inequalities and coerciveness properties of the hydrodynamic boundary potentials. As a consequence, the well-developed fast multipole boundary element methods of 3D elasticity can be applied for solving the Stokes system as well.

§Dedicated to Prof. Dr. Dr.h.c. Ioan A. Rus on the occasion of his 70th birthday.

Keywords:

• Coerciveness and mapping properties of Stokes boundary layer potentials in Sobolev–Slobodetski spaces on Lipschitz boundaries
• Boundary integral equations
• Fast multipole boundary element methods

Keywords:

• AMS (MOS): 35Q30
• 47G10
• 65N38

Acknowledgements

This work was partially supported by the Collaborative Research Center “Sonderforschungsbereich” SFB 404, Multifieldproblems, of the German Reseach Foundation DFG at the University of Stuttgart (Project C10) and partially done while the second author was guest professor at the Babes–Bolyai University in Cluj–Napoca within the Gottfried Herder Program of the German Academic Exchange Service DAAD in 2005. The second author also gratefully acknowledges the helpful discussions with Marius Mitrea 2006 in Bedlewo on the space .

Notes

§Dedicated to Prof. Dr. Dr.h.c. Ioan A. Rus on the occasion of his 70th birthday.

¹Instead of (H ^ϱ(Γ))³ we shall write H ^ϱ(Γ).