On the dam problem

Abstract

This paper is concerned with the qualitative behavior of solutions of the "dam problem'' for flow of an incompressible fluid through a two-dimensional porous medium. The cross section of the flow region is allowed to have a complicated boundary (the flow domain is assumed to have a Lipschitz boundary and minimal assumptions are made on the parts where various boundary conditions are to be satisfied). The main contribution of the paper is to determine the corresponding properties of the solution. After giving a concise proof of the known result that a solution exists, the authors introduce the so-called "S 3 -connected''solutions (S 3 is the impervious part of the boundary), and prove several interesting results about them. Solutions which do not have these properties are such that there are pools of fluid not in contact with the outside reservoir, and they prove that any solution can be written as a sum of an S 3 -connected solution and pools. Also, it is shown that S 3 -connected solutions are unique. For some boundaries it can be shown that only S 3 -connected solutions exist, so that uniqueness holds without qualification.