Existence in the Large for Riemann Problems for Systems of Conservation Laws

An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. Our principal assumptions are strict hyperbolicity, genuine nonlinearity in the strong sense, and the existence of a convex entropy function. The entropy inequality is used to obtain an a priori estimate of the strengths of the shocks and refraction waves forming a solution; existence of such a solution then follows by an application of finite-dimensional degree theory. The case of a single degenerate field is also included, with an additional assumption on the existence of Riemann invariants. 1. Main theorem. We consider initial-value problems for nonlinear systems of conservation laws of the form (1.1) ut + f(u)z =0, oo < x < oo, t > 0, u, f vectors of dimension n, / a smooth function of u, with given initial data of special form