On the behavior of multidimensional axisymmetric solutions of the repulsive Euler-Poisson equations

TL;DR

This paper proves that radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background generally blow up in multiple dimensions, except in four dimensions, and characterizes the long-term behavior of smooth solutions.

Contribution

It establishes blow-up results for multidimensional solutions of the repulsive Euler-Poisson equations and describes the asymptotic behavior of globally smooth solutions.

Findings

Solutions blow up in many dimensions except for four.

Simple wave initial data may avoid blow-up.

Globally smooth solutions tend to affine functions over time.

Abstract

It is proved that the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations blow up in many spatial dimensions except for $\bd=4$ for almost all initial data. The initial data, for which the solution may not blow up, correspond to simple waves. Moreover, if a solution is globally smooth in time, then it is either affine or tends to affine as $t\to\infty$.