On the Euler-Poisson equations with variable background states and nonlocal velocity alignment

TL;DR

This paper analyzes the critical threshold phenomena in 1D pressureless Euler-Poisson equations with variable backgrounds and nonlocal velocity alignment, unifying previous special cases and revealing the oscillatory solution structure.

Contribution

It provides a comprehensive critical threshold analysis for the general EPA system, integrating techniques from prior studies and addressing both nonlocal effects and variable backgrounds.

Findings

Unified framework for critical thresholds in EPA systems

Recovery of previous results under special assumptions

Identification of oscillatory solution behavior

Abstract

We study the 1D pressureless Euler-Poisson equations with variable background states and nonlocal velocity alignment. Our main focus is the phenomenon of critical thresholds, where subcritical initial data lead to global regularity, while supercritical data result in finite-time singularity formation. The critical threshold behavior of the Euler-Poisson-alignment (EPA) system has previously been investigated under two specific setups: (1) when the background state is constant, phase plane analysis was used in the work of Bhatnagar, Liu and Tan [J. Differ. Equ. 375 (2023) 82-119] to establish critical thresholds; and (2) when the nonlocal alignment is replaced by linear damping, comparison principles based on Lyapunov functions were employed in the work of Choi, Kim, Koo and Tadmor [arXiv:2402.12839]. In this work, we present a comprehensive critical threshold analysis of the general EPA system, incorporating both nonlocal effects. Our framework unifies the techniques developed in the aforementioned studies and recovers their results under the respective limiting assumptions. A key feature of our approach is the oscillatory nature of the solution, which motivates a decomposition of the phase plane into four distinct regions. In each region, we implement tailored comparison principles to construct the critical thresholds piece by piece.