THE SHOCK CURVE APPROACH TO THE RIEMANN PROBLEM FOR 2 × 2 HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

Abstract

We consider the Riemann problem for 2 × 2 hyperbolic systems of conservation laws in one space variable. Our main assumptions are that the product of non-diagonal elements within the Fréchet derivative (Jacobian) of the flux is positive, and that the system is genuinely nonlinear. The first assumption implies that the system is strictly hyperbolic, but we do not require a convexity-like condition such as the Smoller–Johnson condition. By using the shock curve approach, we show that those two assumptions are sufficient to establish the uniqueness of self-similar solutions satisfying the Lax entropy conditions at discontinuities.