Boundary value problems for the Stokes equations with jumps in open sets

Abstract

A boundary value problem for the Stokes system is studied in a cracked domain in ℝ ⁿ , n > 2, where the Dirichlet condition is specified on the boundary of the domain. The jump of the velocity and the jump of the stress tensor in the normal direction are prescribed on the crack. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence, a maximum modulus estimate for the Stokes system is proved.

Keywords:

• Stokes system
• single-layer potential
• double-layer potential

AMS Classifications: :

• 76D10
• 76D07
• 65N38

Acknowledgements

The work of D.M. was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AVOZ10190503 and grant No. I AA 100190804 financed by the GA AVČR. The work of W.V. was supported by the Nečas Center for mathematical modelling LC06052 financed by MSMT. W.V. gratefully acknowledges the warm hospitality and the support of the Academy of Sciences of the Czech Republic where this research was performed.