Communications in Mathematical Sciences

Volume 19 (2021)

Number 6

The Cauchy problem for a non strictly hyperbolic $3 \times 3$ system of conservation laws arising in polymer flooding

Pages: 1491 – 1507



Graziano Guerra (Department of Mathematics and its Applications, University of Milano-Bicocca, Milan, Italy)

Wen Shen (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)


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 [B. Temple, Adv. Appl. Math., 3(3):335–375, 1982], [E.L. Isaacson and J.B. Temple, J. Diff. Eqs., 65(2):250–268, 1986] could be effective to control the total variation [W. Shen, J. Diff. Eqs., 261(1):627–653, 2016]. An extension of this technique can be applied to a $3 \times 3$ system only under strict hypotheses on the flux functions [G.M. Coclite and N.H. Risebro, SIAM J. Math. Anal., 36(4):1293–1309, 2005]. 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.


conservation laws, discontinuous flux, compensated compactness, polymer flooding, wave front tracking, degenerate systems

2010 Mathematics Subject Classification

35L40, 35L45, 35L60, 35L65, 35L80

The present work was supported by the PRIN 2015 project Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications, and by GNAMPA 2019 project Equazioni alle derivate parziali di tipo iperbolico o non locale ed applicazioni.

Received 7 July 2020

Accepted 18 January 2021

Published 2 August 2021