The Euler-Poisson system as a second order differential inclusion

We consider the one-dimensional Euler-Poisson system in the repulsive regime and in Lagrangian coordinates. Assuming a sticky particle dynamics, we show that the system can be interpreted as a second-order differential inclusion on the space of optimal transport maps. We discuss a stability result for solutions of this system. Global existence then follows by approximating the initial data by finite linear combinations of Dirac measures, for which the solution can be constructed explicitly.