The Cauchy Problem for a non Strictly Hyperbolic $3\times3$ System of Conservation Laws Arising in Polymer Flooding

TL;DR

This paper investigates the existence of solutions for a complex 3x3 conservation law system modeling polymer flooding in porous media, addressing challenges due to loss of strict hyperbolicity and discontinuous coefficients.

Contribution

It extends front tracking and compensated compactness methods to prove solution existence under mild conditions, despite the system's non-strict hyperbolicity.

Findings

Existence of solutions established for the system.

Effective adaptation of front tracking method.

Handling of discontinuous permeability functions.

Abstract

We study the Cauchy problem of a $3\times 3$ system of conservation laws modeling two--phase flow of polymer flooding in rough porous media with possibly discontinuous permeability function. The system loses strict hyperbolicity in some regions of the domain where the eigenvalues of different families coincide, and BV estimates are not available in general. For a suitable $2\times 2$ system, a singular change of variable introduced by Temple could be effective to control the total variation. An extension of this technique can be applied to a $3\times 3$ system only under strict hypotheses on the flux functions. In this paper, through an adapted front tracking algorithm we prove the existence of solutions for the Cauchy problem under mild assumptions on the flux function, using a compensated compactness argument.