Hölder spaces solvability of a model initial-boundary value problem generated by a problem on a motion of two fluids

The initial-boundary value problem for the Stokes system with discontinuous coefficients of viscosity and density on a plane $\{x_3=0\}$ is considered. This model problem is given rise by a problem on an unsteady motion of two fluids separated by a free surface. We take into account a surface tension which enters in the boundary conditions for a jump of normal stresses on the plane $\{x_3=0\}$ аs а non-coercetiv term containing the integral with respect to time.
The existence of unique solution of this problem is proved in Hölder spaces. The proof of the solvability and Hölder estimates of the solution is based on modifications of a theorem of the Fourier multipliers.