An existence theorem for the multi-fluid Stokes problem

Time-dependent flows of viscous incompressible immiscible fluids are studied in the limit of vanishing Reynolds numbers. The velocity fields associated to each fluid solve Stokes equations in a time-dependent domain. Classical immiscibility conditions on the varying fluids interfaces are taken into account by a new formulation of the problem: the viscosity solves a transport equation and the velocity field solves a Stokes problem with this nonconstant viscosity. This formulation, based on the use of a pseudoconcentration function, has already been used for numerical computations (see [9] and [4]). For this nonlinear system of equations, existence of solutions is proved, using the Schauder fixed point theorem and the concept of renormalized solutions introduced recently by R. J. DiPerna and P. L. Lions.