The path decomposition technique for systems of hyperbolic conservation laws

We are concerned with the problem of the global (in time) exis- tence of weak solutions to hyperbolic systems of conservation laws, in one spa- tial dimension. First, we provide a survey of the different facets of a technique that has been used in several papers in the last years: the path decomposition. Then, we report on two very recent results that have been achieved by means of suitable applications of this technique. The first one concerns a system of three equations arising in the dynamic modeling of phase transitions, the sec- ond one is the famous Euler system for nonisentropic fluid flow. In both cases, the results concern classes of initial data with possibly large total variation.