Measure-valued solutions of scalar hyperbolic conservation laws, Part 2: Uniqueness

We prove well-posedness of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, where the singular part of the initial data is a finite superposition of Dirac masses and the flux is a continuous function with possible linear growth at infinity. The uniqueness class consists of signed Radon measure-valued entropy solutions, called admissible, whose regular and singular parts satisfy so-called compatibility conditions and suitable continuity requirements with respect to time.