The Riemann Problem

Abstract

In this chapter, we study the Riemann problem for scalar conservation laws. In Section 1 we discuss several formulations of the entropy condition. Then, in Section 2 we construct the classical entropy solution satisfying, by definition, all of the entropy inequalities; see Theorems 2.1 to 2.4. Next in Section 3, imposing only that solutions satisfy a single entropy inequality, we show that undercompressive shock waves are also admissible and we determine a one-parameter family of solutions to the Riemann problem; see Theorem 3.5. Finally in Sections 4 and 5, we construct nonclassical entropy solutions which, by definition, satisfy a single entropy inequality together with a kinetic relation; see Theorem 4.1 for concave-convex flux-functions and Theorem 5.4 for convex-concave flux-functions.