Integral Equations Method and the Transmission Problem for the Stokes System

D. Medková

The transmission problem for the Stokes system is studied: $\Delta{\mathbf v}_\pm=\nabla p_\pm$, $\nabla\cdot{\mathbf v}_\pm=0$ in $G_\pm$, ${\mathbf v}_+-{\mathbf v}_-={\mathbf g}$, $a_+T({\mathbf v}_+,p_+)\mathbf n-a_-T({\mathbf v}_-,p_-)\mathbf n=\mathbf f$ on $\partial G_+$. Here $G_+\subset R^3$ is a bounded open set with Lipschitz boundary and $G_-$ is the corresponding complementary open set. Using the integral equation method we study the problem in homogeneous Sobolev spaces. Under assumption that $\partial G_+$ is of class ${\cal C}^1$ we study this problem also in Besov spaces and $L^q$-solutions of the problem. We show the unique solvability of the problem. Moreover, we solve the corresponding boundary integral equations by the successive approximation method.