Local Existence and Stability for a Hyperbolic-Elliptic System Modeling Two-Phase Reservoir Flow

A system arising in the modeling of oil--recovery processes is analyzed. It consists of a hyperbolic conservation law governing the saturation and a elliptic equation for the pressure. By an operator splitting approach, an approximate solution is constructed. For this approximation appropriate a--priori bounds are derived. Applying the Arzela--Ascoli theorem, local existence and uniqueness of a classical solution for the original hyperbolic--elliptic system is proved. Furthermore, convergence of the approximation generated by operator splitting towards the unique solution follows. It is also proved that the unique solution is stable with respect to perturbations of the initial data. 1 Introduction The purpose of this paper is to study the following system of partial differential equations, s t +r \Delta [f(s)v] = g; \Gammar \Delta [(s)rp] = q; (1) v = \Gamma(s)rp: * Numerische Mathematik, RWTH--Aachen, Templergraben 55, D-52056 Aachen, Germany, (schroll@igpm.rwth-aachen.de). This ..