The path decomposition technique for systems of hyperbolic conservation laws

We are concerned with the problem of the global (in time) existence of weak solutions to hyperbolic systems of conservation laws, in one spatial 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 second 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.