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Existence, uniqueness, and continuous dependence for a system of hyperbolic conservation laws modeling polymer flooding. (English) Zbl 0741.65071

The purpose of this paper is to study the Cauchy problem for the hyperbolic system of conservation laws (1) \(s_ t+f(s,c)_ x=0\), \((s\cdot c)_ t+(c\cdot f(s,c))_ x=0\). This model arises in enhanced oil recovery when oil is displaced in a porous rock by water containing dissolved polymer. The variable \(s\) denotes the saturation of the aqueous phase, consisting of water and polymer, while \(c\) denotes the concentration of polymer in the aqueous phase. The function \(f=f(s,c)\) is usually referred to as the fractional flow function.
The authors generalize the results based on a simple form of the fractional flow function to physically relevant fractional flow functions, for which one cannot formulate the finite difference scheme in Riemann invariants. Since the system is nonstrictly hyperbolic, the Riemann invariants do not constitute a global coordinate system in the state-space.
A finite difference scheme is used to prove the existence of an entropy solution with bounded variation. It is proved that the entropy solution of the system is unique, and that the solution depends continuously on its initial data in a proper topology.
The main contribution of the paper is the proof of uniqueness and continuous dependence results for the system (1). The analysis is based on a smoothness property of one of the Riemann invariants of the system.
As a consequence of the uniqueness result, the authors obtained the convergence of the entire family of approximate solutions, not only a subsequence of it. They hope to be able to investigate this convergence, with an eye to error estimates, in the future.
Reviewer: G.Dimitriu (Iaşi)

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76S05 Flows in porous media; filtration; seepage
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