Existence Theory for the Cauchy Problem

Abstract

This chapter is devoted to the general existence theory for scalar conservation laws in the setting of functions with bounded variation. We begin, in Section 1, with an existence result for the Cauchy problem when the flux-function is convex. We exhibit a solution given by an explicit formula (Theorem 1.1) and prove the uniqueness of this solution (Theorem 1.3). The approach developed in Section 1 is of particular interest as it reveals important features of classical entropy solutions. However, it does not extend to non-convex fluxes or nonclassical solutions, and an entirely different strategy based on Riemann solvers and wave front tracking is developed in the following sections. In Sections 2 and 3, we discuss the existence of classical and of nonclassical entropy solutions to the Cauchy problem, respectively; see Theorems 2.1 and 3.2 respectively. Finally in Section 4, we derive refined estimates for the total variation of solutions (Theorems 4.1 to 4.3) which represent a preliminary step toward the forthcoming discussion of the Cauchy problem for systems (in Chapters VII and VIII).