Vanishing viscosity solutions of Riemann problems for models of polymer flooding.

*(English)*Zbl 1402.35169
Gesztesy, Fritz (ed.) et al., Non-linear partial differential equations, mathematical physics, and stochastic analysis. The Helge Holden anniversary volume on the occasion of his 60th birthday. Based on the presentations at the conference ‘Non-linear PDEs, mathematical physics and stochastic analysis’, NTNU, Trondheim, Norway, July 4–7, 2016. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-186-6/hbk; 978-3-03719-686-1/ebook). EMS Series of Congress Reports, 261-285 (2018).

Summary: We consider the solutions of Riemann problems for polymer flooding models. In a suitable Lagrangian coordinate the systems take a triangular form, where the equation for thermodynamics is decoupled from the hydrodynamics, leading to the study of scalar conservation laws with discontinuous flux functions. We prove three equivalent admissibility conditions for shocks for scalar conservation laws with discontinuous flux. Furthermore, we show that a variation of minimum path of [T. Gimse and N. H. Risebro, in: Third international conference on hyperbolic problems. Theory, numerical methods and applications. Proceedings of the conference dedicated to Professor Heinz-Otto Kreiss on his 60th birthday, held in Uppsala, Sweden, June 11-15, 1990. Vol. I. Lund: Studentlitteratur; Bromley: Chartwell-Bratt Ltd.. 488–502 (1991; Zbl 0789.35102)] proposed in [W. Shen, J. Differ. Equations 261, No. 1, 627–653 (2016; Zbl 1382.35162)] is the vanishing viscosity limit of a partially viscous model with viscosity only in the hydro-dynamics.

For the entire collection see [Zbl 1390.35006].

For the entire collection see [Zbl 1390.35006].

##### MSC:

35L65 | Hyperbolic conservation laws |

35D40 | Viscosity solutions to PDEs |

35L45 | Initial value problems for first-order hyperbolic systems |

35L67 | Shocks and singularities for hyperbolic equations |

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\textit{G. Guerra} and \textit{W. Shen}, in: Non-linear partial differential equations, mathematical physics, and stochastic analysis. The Helge Holden anniversary volume on the occasion of his 60th birthday. Based on the presentations at the conference `Non-linear PDEs, mathematical physics and stochastic analysis', NTNU, Trondheim, Norway, July 4--7, 2016. Zürich: European Mathematical Society (EMS). 261--285 (2018; Zbl 1402.35169)

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