Kru\v{z}kov-type uniqueness theorem for a non-monotone flow function case with application to Riemann problem solutions

Abstract

We generalize the previously obtained Kru\v{z}kov-type uniqueness result for the initial-boundary value problem for the chemical flood conservation law system to the case of an almost arbitrary flow function, not restricted by the S-shaped condition or the monotonicity with respect to the chemical agent concentration. The result is applied to the analysis of the Riemann problem solutions for an S-shaped flow function changing monotonicity with respect to the chemical concentration exactly once. All possible Riemann problem solution structures are classified, including certain unique structures that have not been described in earlier studies. Keywords: Initial-boundary value problem; Riemann problem; first-order hyperbolic system; conservation laws; shock waves; uniqueness theorem; vanishing viscosity; chemical flood.