# IJPAM: Volume 62, No. 1 (2010)

**THE DIRICHLET PROBLEM FOR THE STOKES SYSTEM**

AND THE INTEGRAL EQUATIONS' METHOD

AND THE INTEGRAL EQUATIONS' METHOD

Mathematical Institute

Academy of Sciences of the Czech Republic

25, Zitná, Praha 1, 115 67, CZECH REPUBLIC

e-mail: medkova@math.cas.cz

Faculty of Mathematics

University of Kassel

40, Heinrich Plett Str., Kassel, 34132, GERMANY

e-mail: varnhorn@mathematik.uni-kassel.de

**Abstract.**A boundary value problem for the Stokes
system is studied in a cracked domain in , , 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 the
unknown source densities via Fredholm integral equations' systems of
the second kind on the boundary of the domain. Here the boundary
value problem

for the Stokes system plays an important role. The solution of which is given explicitly in the form of a series. As a consequence, also a maximum modulus estimate for the Stokes system can be proved.

**Received: **May 6, 2010

**AMS Subject Classification: **76D10, 76D07, 65N38

**Key Words and Phrases: **Stokes system, layer potential, integral equation, crack

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 62

**Issue:** 1