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

TL;DR

This paper proves the existence of BV solutions for the one-dimensional Euler-Poisson system with large initial data under a smallness condition on gamma, using a Glimm scheme and wave interaction analysis.

Contribution

It extends the existence of BV solutions to the Euler-Poisson system with large data and electric field effects, incorporating boundary conditions and illustrating ill-posedness in certain cases.

Findings

Existence of BV solutions for large initial data when gamma satisfies a smallness condition.

Use of Glimm scheme combined with splitting method to handle wave interactions.

Ill-posedness demonstrated for the initial-boundary value problem of the Euler equation.

Abstract

This paper deals with the existence of BV solution for the Euler-Poisson system endowed with a $\gamma$ pressure law. More precisely, we prove the existence of weak solution in the BV framework with arbitrary large initial data when $\gamma=1+2\epsilon$ satisfies a smallness condition. We use the Glimm scheme combined with a splitting method as introduced in [Poupaud, Rascle and Vila, J. Differential Equations, 1995]. Existence of BV solution of 1-D isentropic Euler equation for large data and $\gamma=1+2\epsilon$ is proved in [Nishida and Smoller, Comm. 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 article, we discuss the initial-boundary value problem for the Euler-Poisson system. We prove the existence of $BV$ 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.