A nonlinear parabolic free boundary problem

Abstract

When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems.