Global-in-time convergence from bipolar Euler-Poisson equations to unipolar ones

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 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 vanishing electron-to-ion mass-ratio limit. Here the initial scaled density is required to vanish in the far field, and the spectrum of the Jacobian matrix of the initial mean velocity stays uniformly away from the nonpositive real axis. In order to deal with singular limits of this kind, the global-in-time uniform tame estimates of regular solutions to M-D bipolar Euler-Poisson equations with respect to the mass ratio are established, based on which the corresponding error estimates in smooth function spaces between the two systems considered are given. To this end, we establish global a priori estimates for solutions with compactly supported initial densities, uniformly in the mass ratio and the initial support radii. Compactness then yields global regular solutions for general initial densities.