Existence of BV solution for the Euler–Poisson system in one dimension with large initial data

This paper deals with the existence of BV solution for the Euler–Poisson system en dowed with a [Formula: see text] pressure law. More precisely, we prove the existence of weak solution in the BV framework with arbitrary large initial data when [Formula: see text] satisfies a smallness condition. We use the Glimm scheme combined with a splitting method as introduced in [Poupaud, Rascle and Vila, J. Differ. Equ. (1995)]. Existence of BV solution of 1-D isentropic Euler equation for large data and [Formula: see text] is proved in [Nishida and Smoller, Commun. Pure Appl. Math. (1973)]. Due to the presence of electric field, the difficulty arises while controlling the Glimm functional for the Euler–Poisson system. It requires a subtle study of wave interaction. In the later part of this paper, we discuss the initial-boundary value problem for the Euler–Poisson system. We prove the existence of [Formula: see text] solution for the initial-boundary value problem with large initial and boundary data. By an explicit example, we also show ill-posedness of initial-boundary value problem for the isentropic Euler equation.