Degenerate Two-Phase Incompressible Flow: I. Existence, Uniqueness and Regularity of a Weak Solution

This is the first paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we first show that this system possesses a weak solution under physically reasonable hypotheses on the data. Then we prove that this weak solution is unique. Finally, we establish regularity on the weak solution which is needed in the uniqueness proof. In particular, the Hölder continuity of the saturation in space and time and the Lipschitz continuity of the pressure in space are obtained.