Boundary-domain integral equation systems to the Dirichlet bvp for an incompressible Stokes system with variable viscosity in 2D

Abstract

The Dirichlet problem for the steady-state Stokes system of partial differential equations for an incompressible viscous fluid with variable viscosity coefficient is considered in a two-dimensional bounded domain. Using an appropriate parametrix, this problem is reduced to two systems of direct segregated boundary-domain integral equations (bdies). The bdies in 2D have special properties in comparison with the three dimensional case, because of the logarithmic term in the parametrix for the associated partial differential equation. Consequently, we need to set conditions on the function spaces or on the domain to ensure the invertibility of the corresponding parametrix-based hydrodaynamic single layer potential and hence, guarantying the unique solvability of bdies. Equivalence of the obtained bdie systems to the original Dirichlet bvp and unique solvability of bdie systems is shown. Invertibility of the corresponding boundary-domain integral operators is proved in appropriate Sobolev-Slobodetski (Bessel potential) spaces.