NONUNIQUENESS OF SOLUTIONS OF RIEMANN PROBLEMS

We investigate a general mechanism, utilizing nonclassical shock waves , for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., tw o states U-1 and U-2 such that a traveling wave leads from U-1 to U-2 and another leads from U-2 to U-1. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solut ions of the Riemann problem, and me determine all solutions within a c ertain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of s olutions of the conservation laws, as augmented by viscosity terms.