MULTIPLE VISCOUS SOLUTIONS FOR SYSTEMS OF CONSERVATION-LAWS

We exhibit an example of mechanism responsible for multiple solutions in the Riemann problem for a mixed elliptic-hyperbolic type system of two quadratic polynomial conservation laws. In this example, multiple solutions result from folds in the set of Riemann solutions. The multi ple solutions occur despite the fact that they all satisfy the viscous profile entropy criterion. The failure of this criterion to provide u niqueness is evidence in support of a need for conceptual change in th e theory of shock waves for a system of conservation laws.