Scalar conservation equations

Abstract

This chapter gives a brief description of the theory of scalar conservation equations. We introduce the notions of weak and entropy solutions and state existence and uniqueness results. Even if the initial data is smooth, the solution of a generic scalar conservation equation may lose smoothness in finite time, and weak solutions are in general nonunique. Uniqueness is recovered by enforcing constraints that are called entropy conditions. We finish this chapter by exploring the structure of a one-dimensional Cauchy problem called Riemann problem where the initial data is composed of two constant states. Understanding the structure of the solution to the Riemann problem is important to understand the approximation techniques discussed later on.