A free boundary problem involving a cusp: Breakthrough of salt water (Q1971106)

The paper deals with the study of a two-phase free boundary value problem describing the stationary flow of fresh and salt water in a porous medium when both fluids are drawn into a well. The well is located on the free separation interface into a 2d or 3d axial symmetric flow domain and the model is described by Darcy's law and the fluid balance equation with adequate boundary and interface conditions. Formulating the problem in terms of the stream function, one demonstrates the existence of the weak solution and investigates the shape of the free boundary. Using the method of separation of variables one gives a result concerning the asymptotic behaviour of the free boundary at the well. Considering the case of the vanishing salt water discharge one shows that a free boundary arises with a cusp having a positive distance from the well.