On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions

For a special class of discontinuous flux functions that can be associated to the limit case of a phase transition it has been introduced in [2] an appropriate notion of entropy weak solution to the Cauchy problem and some existence results were proved. In this paper, for the discontinuous scalar case, we give a counter-example to uniqueness and we prove an estimate based in Kruskov's method. Then, for a class of discontinuous p-systems, we prove, by applying a variant of the regularization method introduced by Dafermos in [1], an existence result for the Riemann problem.