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 ${Bbb R^n(n = 2,3)$. We prove 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 $C^1$ sense. Using the method of separation of variables we also give a result about the asymptotic behaviour of the free boundary at the well. For 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 [AD2,3].