Global Solutions to the Isothermal Euler-Poisson System with Arbitrarily Large Data

We prove the global existence of a solution to the Euler-Poisson system, with arbitrarily large data, in a one-dimensional geometry. The pressure law we consider, is deduced from an isothermal assumption for the electrons gas. In this case, Nishida has already pointed out that the linear part of the Glimm functional is decreasing with respect to time. Using a Glimm scheme, he used this property to construct globally defined weak solutions for the Euler system with arbitrary large data. We follow his outline of proof. Here, a new difficulty arises with the source term, due to the electric field. A key point is that the Glimm scheme is almost conservative. This quasi-conservation of charge leads to a uniform estimate of the total variation of the electric field. This estimate allows to prove the convergence of the scheme.