Kružkov-type uniqueness theorem for the chemical flood conservation law system. Generalizations and applications.

We consider the system of two hyperbolic conservation laws most commonly used to describe the flood of the oil reservoir with a chemical solution. This system is neither strictly hyperbolic nor genuinely non-linear, therefore known results for strictly hyperbolic genuinely non-linear systems of conservation laws are not directly applicable.
The solutions for some boundary-initial problems for this system were explored (for example, Riemann problem and slug injection) using the Lagrange coordinate transformation to split the equations and the characteristics method to construct solutions. However, the question of the uniqueness of the constructed solutions is not covered. Moreover, for problems with slug injection an heuristic method based on Jouguet principle is utilised with no rigorous derivation.
We used the proposed coordinate change to prove a Kruz̆kov-type uniqueness theorem for the Cauchy problem with several limitations on the initial data and the class of weak solutions under consideration. The local vanishing viscosity method is utilized to determine admissible shocks. Now we generalize this theorem and apply it to certain boundary-initial problems. For the Jouguet principle an applicability criterion is formulated.
This talk is based on the joint works with S. G. Matveenko and Yu.P.Petrova.