IJPAM: Volume 62, No. 1 (2010)

THE DIRICHLET PROBLEM FOR THE STOKES SYSTEM
AND THE INTEGRAL EQUATIONS' METHOD

Dagmar Medková$^1$, Werner Varnhorn$^2$
$^1$Mathematical Institute
Academy of Sciences of the Czech Republic
25, Zitná, Praha 1, 115 67, CZECH REPUBLIC
e-mail: medkova@math.cas.cz
$^2$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 $R^n$, $n>2$, 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 the unknown source densities via Fredholm integral equations' systems of the second kind on the boundary of the domain. Here the boundary value problem

\begin{displaymath}-\Delta \vec u+\nabla p\:=\:0\quad\mbox{in}\quad G, \qquad \... ...d G, \qquad \vec u\:=\:\vec b\quad\mbox{on}\quad \partial G, \end{displaymath}

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