Global Convergence and Error Estimates in Infinity-ion-mass Limits for Bipolar Euler-Poisson System

TL;DR

This paper proves the global convergence and provides error estimates for the bipolar Euler-Poisson system approaching the unipolar system as the ion mass ratio tends to infinity, using advanced mathematical techniques.

Contribution

It introduces a novel approach to analyze the infinity-ion-mass limit for bipolar Euler-Poisson systems, including separate stream functions for ions and electrons.

Findings

Global convergence from bipolar to unipolar system established.

Error estimates between solutions derived and quantified.

Method handles strong coupling via Poisson equation effectively.

Abstract

This paper is concerned with the global-in-time convergence from bipolar Euler-Poisson system (BEP) to unipolar one (UEP) through the infinity-ion-mass limit by letting the ratio of the mass of ion $m_i$ over that of electron $m_e$ goes to infinity. The global convergence of the limit is obtained for smooth solutions sufficiently close to constant equilibrium states. Furthermore, by applying the stream function method and taking advantage of the anti-symmetric structure of the error system, one obtains the corresponding global-in-time error estimates between smooth solutions of (BEP) and (UEP). It is worth mentioning that due to the strong coupling through the Poisson equation in bipolar system, stream functions for ions and electrons equations should be constructed separately based on asymptotic expansions of solutions, which is very different from the case of unipolar system.