Kru\v{z}kov-type uniqueness theorem for the chemical flood conservation law system with local vanishing viscosity admissibility

TL;DR

This paper establishes a Kruzhkov-type uniqueness theorem for a chemical flood conservation law system, employing local vanishing viscosity and entropy inequalities to handle discontinuities and ensure solution uniqueness.

Contribution

It introduces a novel approach using local vanishing viscosity and Lagrange transformation to prove uniqueness for the chemical flood conservation law system.

Findings

Uniqueness of solutions is proven under certain restrictions.

The method handles discontinuities via local viscosity.

Entropy inequalities are used to establish the theorem.

Abstract

We study the uniqueness of solutions of the initial-boundary value problem in the quarter-plane for the chemical flood conservation law system in the class of piece-wise $\mathcal C^1$-smooth functions under certain restrictions. The vanishing viscosity method is used locally on the discontinuities of the solution to determine admissible and inadmissible shocks. The Lagrange coordinate transformation is utilized in order to split the equations. The proof of uniqueness is based on an entropy inequality similar to the one used in the classical Kru\v{z}kov's theorem.