Existence of radial global smooth solutions to the pressureless Euler-Poisson equations with quadratic confinement

TL;DR

This paper establishes explicit criteria for the existence of radial global smooth solutions to the pressureless Euler-Poisson equations with quadratic confinement in dimensions other than four, revealing more restrictive conditions than typical critical thresholds.

Contribution

It provides a necessary and sufficient condition for global smooth solutions based on initial data, using the periodicity of characteristic ODE systems.

Findings

Global smooth solutions exist under explicit initial data conditions

Characteristic periods must be identical for solutions to be global

Conditions are more restrictive than standard critical-threshold criteria

Abstract

We consider the pressureless Euler-Poisson equations with quadratic confinement. For spatial dimension $d\ge 2,\,d\ne 4$, we give a necessary and sufficient condition for the existence of radial global smooth solutions, which is formulated explicitly in terms of the initial data. This condition appears to be much more restrictive than the critical-threshold conditions commonly seen in the study of Euler-type equations. To obtain our results, the key observation is that every characteristic satisfies a periodic ODE system, and the existence of global smooth solution requires the period of every characteristic to be identical.