Critical thresholds in the Euler-Poisson-alignment system

TL;DR

This paper investigates the conditions for global existence and finite-time blow-up in the Euler-Poisson-alignment system, introducing invariant regions to control oscillations and extend results to weakly singular alignment forces.

Contribution

It develops a novel method to construct invariant regions for the EPA system, enabling analysis of both subcritical and supercritical initial data, including cases with weakly singular alignment forces.

Findings

Global wellposedness for subcritical initial data in invariant regions.

Finite-time singularity formation for supercritical initial data.

Extension of results to weakly singular alignment interactions.

Abstract

This paper is concerned with the global wellposedness of the Euler-Poisson-alignment (EPA) system. This system arises from collective dynamics, and features two types of nonlocal interactions: the repulsive electric force and the alignment force. It is known that the repulsive electric force generates oscillatory solutions, which is difficult to be controlled by the nonlocal alignment force using conventional comparison principles. We construct \emph{invariant regions} such that the solution trajectories cannot exit, and therefore obtain global wellposedness for subcritical initial data that lie in the invariant regions. Supercritical regions of initial data are also derived which leads to finite-time singularity formations. To handle the oscillation and the nonlocality, we introduce a new way to construct invariant regions piece by piece in the phase plane of a reformulation of the EPA system. Our result is extended to the case when the alignment force is weakly singular. The singularity leads to the loss of a priori bounds crucial in our analysis. With the help of improved estimates on the nonlocal quantities, we design non-trivial invariant regions that guarantee global wellposedness of the EPA system with weakly singular alignment interactions.