On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding. (English) Zbl 1379.35003

Summary: 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.


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
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