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

Abstract

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 (Qf for fresh water and Qs 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:=Qf + Qs we consider the vanishing Qs 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].