Global-in-time convergence in infinity-ion-mass limit for bipolar Euler-Poisson equations

TL;DR

This paper proves the global-in-time convergence of multi-dimensional bipolar Euler-Poisson equations to unipolar equations in the infinity-ion mass limit, using uniform estimates and approximation techniques.

Contribution

It provides the first rigorous proof of global-in-time convergence for this singular limit in multi-dimensional bipolar Euler-Poisson systems.

Findings

Established uniform tame estimates for solutions with respect to the ion mass ratio.

Proved error estimates between bipolar and unipolar systems in smooth function spaces.

Demonstrated the global existence of solutions via compactness arguments.

Abstract

In this paper, the Cauchy problem for the multi-dimensional (M-D) bipolar Euler-Poisson equations with far field vacuum is considered. Based on physical observations and some elaborate analysis of this system's intrinsic symmetric hyperbolic-elliptic coupled structures, for a class of smooth initial data that are of small scaled density but possibly large mean velocity, we give one rigorous global-in-time convergence proof for regular solutions from M-D bipolar Euler-Poisson equations to M-D unipolar Euler-Poisson equations through the infinity-ion mass limit. Here the initial scaled density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial mean velocity are all positive. In order to deal with such kind of singular limits, the global-in-time uniform tame estimates of regular solutions to M-D bipolar Euler-Poisson equations with respect to the ratio of electron mass over ion mass are established, based on which the corresponding error estimates in smooth function spaces between the two systems considered are also given. To achieve these, our main strategy is to regard the original problem for M-D bipolar Euler-Poisson equations as the limit of a series of carefully designed approximate problems which have truncated convection operators and compactly supported initial data. For such artificial problems, we can derive careful a-priori estimates that are independent of the mass ratio, the size of the initial data' supports and the truncation parameters. Then the global uniform existence of regular solutions of the original problem are attained via careful compactness.