  In this paper a time dependent Stokes problem that is motivated by a standard
sharp interface model for the fluid dynamics of two-phase flows is studied.
This Stokes interface problem has discontinuous density and viscosity
coefficients and a pressure solution that is discontinuous across an evolving
interface. This strongly simplified two-phase Stokes equation is considered to
be a good model problem for the development and analysis of finite element
discretization methods for two-phase flow problems. In view of the unfitted
finite element methods that are often used for two-phase flow simulations, we
are particularly interested in a well-posed variational formulation of this
Stokes interface problem in a Euclidean setting. Such well-posed weak
formulations, which are not known in the literature, are the main results of
this paper. Different variants are considered, namely one with suitable spaces
of divergence free functions, a discrete-in-time version of it, and variants in
which the divergence free constraint in the solution space is treated by a
pressure Lagrange multiplier. The discrete-in-time variational formulation
involving the pressure variable for the divergence free constraint is a natural
starting point for a space-time finite element discretization. Such a method is
introduced and results of numerical experiments with this method are presented.
