ABSTRACT
This chapter deals with scalar conservation laws, that is, the case where s = 1. The vanishing viscosity method is not the only one used to study the solutions of the Cauchy problem. Another interesting approach was presented in the recent preprint by Cockburn, Gripenberg and Londen. The chapter proves the existence, uniqueness and also some qualitative properties of the weak solution of the parabolic equation. Employing the vanishing viscosity method we prove existence and uniqueness of the so-called weak entropy solution of the Cauchy problem. The approach is very close to Godlewski and Raviart. The chapter also presents an overview of analogous results for bounded domains proved recently by Otto. It presents similar results concerning the existence and uniqueness of solutions to the hyperbolic conservation law for bounded smooth domains.
