On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding.

*(English)*Zbl 1379.35003Summary: We consider the vanishing viscosity solutions of Riemann problems for polymer flooding models. The models reduce to triangular systems of conservation laws in a suitable Lagrangian coordinate, which connects to scalar conservation laws with discontinuous flux. These systems are parabolic degenerate along certain curves in the domain. A vanishing viscosity solution based on a partially viscous model is given in a parallell paper [G. Guerra and W. Shen, “Vanishing viscosity solutions of Riemann problems for models of polymer flooding”, Partial Differ. Equations, Math. Phys., Stochastic Anal. (to appear)]. In this paper the fully viscous model is treated. Through several counter examples we show that, as the ratio of the viscosity parameters varies, infinitely many vanishing viscosity limit solutions can be constructed. Under some further monotonicity assumptions, the uniqueness of vanishing viscosity solutions for Riemann problems can be proved.

##### MSC:

35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |

35L65 | Hyperbolic conservation laws |

35L80 | Degenerate hyperbolic equations |

35L60 | First-order nonlinear hyperbolic equations |

35D40 | Viscosity solutions to PDEs |

PDF
BibTeX
Cite

\textit{W. Shen}, NoDEA, Nonlinear Differ. Equ. Appl. 24, No. 4, Paper No. 37, 25 p. (2017; Zbl 1379.35003)

Full Text:
DOI

##### References:

[1] | Andreianov, B, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, ESAIM Proc. Surv., 50, 40-65, (2015) · Zbl 1342.35174 |

[2] | Buckley, SE; Leverett, M, Mechanism of fluid displacement in sands, Trans. AIME, 146, 107-116, (1942) |

[3] | Gimse, T., Risebro, N.H.: Riemann problems with a discontinuous flux function. In: Engquist, B., Gustafsson, B. (eds). Proceedings of Third International Conference on Hyperbolic Problems. Theory, Numerical Method and Applications. Studentlitteratur/Chartwell-Bratt, Lund-Bromley, pp. 488-502 (1991) · Zbl 0789.35102 |

[4] | Guerra, G., Shen, W.: Vanishing viscosity solutions of Riemann problems for models of polymer flooding. In: Gesztesy, F., Hanche-Olsen, H., Jakobsen, E., Lyubarskii, Y., Risebro, N., Seip, K. (eds). To appear in “Partial Differential Equations, Mathematical Physics, and Stochastic Analysis”. A Volume in Honor of Helge Holden’s 60th Birthday. EMS Congress Reports. (2017) |

[5] | Oleinik, O.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation. Uspehi Mat. Nauk. 14(2) (86), pp. 165-170 (1959) (Russian). English Translation in Amer. Math. Soc. Transl. Ser. 2, 33, pp. 285-290 (1964) |

[6] | Pires, AP; Bedrikovetsky, PG; Shapiro, AA, A splitting technique for analytical modelling of two-phase multicomponent flow in porous media, J. Pet. Sci. Eng., 51, 54-67, (2006) |

[7] | Shen, W, On the Cauchy problems for polymer flooding with gravitation, J. Differ. Equ., 261, 627-653, (2016) · Zbl 1382.35162 |

[8] | Shen, W.: Scilab codes for all the simulations presented in this paper can be found at: http://www.personal.psu.edu/wxs27/SIM/BressanSISSA · Zbl 0741.65071 |

[9] | Temple B.: Stability and decay in systems of conservation laws. In: Carasso, C., Raviart, P.A., Serre, D. (eds). Proceeding of Nonlinear Hyperbolic Problems. Springer-Verlag, New York, (1986) · Zbl 0647.76049 |

[10] | Tveito, A; Winther, R, Existence, uniqueness, and continuous dependence for a system of hyperbolic conservation laws modeling polymer flooding, SIAM J. Math. Anal., 22, 905-933, (1991) · Zbl 0741.65071 |

[11] | Wagner, D, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differ. Equ., 68, 118-136, (1987) · Zbl 0647.76049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.