A class of hyperbolic systems of conservation laws satisfying weaker conditions than genuine nonlinearity

We discuss hyperbolic systems of nonlinear conservation laws in one space variable, for which a convex entropy function exists but which satisfy a structural assumption weaker than that of genuine nonlinearity in the large. Our principal result is an existence theorem in the large for the Riemann problem for such systems, extendng a previous result for the genuinely nonlinear case. In the present framework, our strongest assumption is equivalent to the statement that any (k - 1)-shock with a given state u on the right travels more slowly than any k-shock with the same state u on the left. If this fails, then entropy solutions of the Riemann problem can suddenly disappear as the data is perturbed. We also include some partial results on the viscous profile problem for such systems.