Narrow Results

For non-conservative hyperbolic systems several definitions of shock waves have been introduced in the literature. In this paper, we propose a new and simple definition in the case of genuinely nonlinear fields. Relying on a vanishing viscosity process we prove the existence of shock curves for viscosity matrix commuting with the matrix of the hyperbolic system. This setting generalizes a recent result by Bianchini and Bressan. Furthermore we prove that all definitions agree to third order near a given state.

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.