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

In this paper we study a two-phase free boundary problem describing the stationary flow of fresh and salt water in a porous medium, when both fluids are drawn into a well. For given discharges at the well (Q_f for fresh water and Q_s for salt water) we formulate the problem in terms of the stream function in an axial symmetric flow domain in ℝⁿ(n = 2, 3).We prove the existence of a continuous free boundary which ends up in the well, located on the central axis. Moreover, we show that the free boundary has a tangent at the well and approaches it in a C¹ sense. Using the method of separation of variables we also give a result concerning the asymptotic behaviour of the free boundary at the well. For a given total discharge (Q:= Q_f + Q_s) we consider the vanishing Q_s limit.We show that a free boundary arises with a cusp at the central axis, having a positive distance from the well. This work is a continuation of [5, 6].