Electron. J. Diff. Eqns., Vol. 2000(2000), No. 04, pp. 1-28.

Local existence and stability for a hyperbolic-elliptic system modeling two-phase reservoir flow

H. J. Schroll & A. Tveito

A system arising in the modeling of oil-recovery processes is analyzed. It consists of a hyperbolic conservation law governing the saturation and an 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.

Submitted March 10, 1999. Published January 5, 2000.
Math Subject Classifications: 35M10, 35L45, 35J25.
Key Words: Hyperbolic-elliptic system, two-phase flow, existence, stability, operator splitting, convergence.

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Hans Joachim Schroll
Numerische Mathematik, RWTH--Aachen,
Templergraben 55, D-52056 Aachen, Germany.
Mathematical Sciences,
The Norwegian University of Science and Technology,
N-7491 Trondheim, Norway.
e-mail: schroll@math.ntnu.no
Aslak Tveito
Department of Informatics, University of Oslo,
P.O. Box 1080 Blindern, N-0316 Oslo, Norway.
e-mail: aslak@ifi.uio.no

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