From bipolar Euler-Poisson system to unipolar Euler-Poisson system in the perspective of mass

TL;DR

This paper rigorously analyzes the transition from bipolar to unipolar Euler-Poisson systems by examining mass ratios, using asymptotic expansions to show decoupling as the key process rather than vanishing equations.

Contribution

It introduces a rigorous procedure to connect bipolar and unipolar Euler-Poisson systems via singular limits based on mass ratios, highlighting the decoupling process.

Findings

The limiting process is decoupling, not vanishing of equations.

Asymptotic expansion effectively handles the singular limits.

Mass ratio limits clarify the relationship between systems.

Abstract

The main purpose of this paper is to provide an effective procedure to study rigorously the relationship between unipolar and bipolar Euler-Poisson system in the perspective of mass. Based on the fact that the mass of an electron is far less than that of an ion, we amplify this property by letting $m_e/m_i\rightarrow 0$ and using two different singular limits to illustrate it, which are zero-electron mass limit and infinity-ion mass limit. We use the method of asymptotic expansion to handle the problem and find that the limiting process from bipolar to unipolar system is actually the process of decoupling, but not the vanishing of equations of the corresponding other particle.