Hyperbolic systems of conservation laws with Lipschitz continuous flux-functions: The Riemann problem

For strictly hyperbolic systems of conservation laws with Lipschitz continuous flux-functions we generalize Lax's genuine nonlinearity condition and shock admissibility inequalities and we solve the Riemann problem when the left- and right-hand initial data are sufficiently close. Our approach is based on the concept of multivalued representatives of L-infinity functions and a generalized calculus for Lipschitz continuous mappings. Several interesting features arising with Lipschitz continuous flux-functions come to light from our analysis.