Stabilities for Euler-Poisson Equations with Repulsive Forces in R^N

TL;DR

This paper investigates the stability properties of Euler-Poisson equations with repulsive forces in multi-dimensional space, extending previous work on attractive forces and showing the impossibility of density collapsing solutions with certain conditions.

Contribution

It extends stability analysis of Euler-Poisson equations to repulsive forces in higher dimensions and proves the non-existence of collapsing solutions with compact support for specific parameters.

Findings

Stability properties are similar to the attractive case.

Density collapsing solutions with compact support do not exist for b3 > 1.

The results generalize previous stability analyses to repulsive forces.

Abstract

This article extends the previous paper in "M.W. Yuen, \textit{Stabilities for Euler-Poisson Equations in Some Special Dimensions}, J. Math. Anal. Appl. \textbf{344} (2008), no. 1, 145--156.", from the Euler-Poisson equations for attractive forces to the repulsive ones in $R^{N}$ $(N\geq2)$. The similar stabilities of the system are studied. Additionally, we explain that it is impossible to have the density collapsing solutions with compact support to the system with repulsive forces for $\gamma>1$.