Existence and uniqueness results for the pressureless Euler–Poisson system in one spatial variable

Abstract

We study the Euler-Poisson system describing the evolution of a fluid without pressure e¤ect and, more generally, also treat a class of nonlinear hyperbolic systems with an analogous structure. We investigate the initial value problem by generalizing a method first introduced by LeFloch in 1990 and based on Volpert’s product and Lax’s explicit formula for scalar conservation laws. We establish several existence and uniqueness results when one component of the system (the density) is measure-valued and the second one (the velocity) has bounded variation. Existence is proven for general initial data, while uniqueness is guaranteed only when the initial data does not generate rarefaction centers. Our proof proceeds by solving first a nonconservative version of the problem and constructing solutions with bounded variation, while the solutions of the Euler-Poisson system is then deduced by differentiation.