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

TL;DR

The paper investigates the 1D pressureless Euler-Poisson-alignment system with variable background $c(x,t)$ and nonlocal alignment kernel $\psi$, addressing the critical threshold problem—whether subcritical data yield global smooth solutions or supercritical data lead to finite-time blowup. It introduces a reformulation using $G=\partial_x u+\psi*\rho$ and the variables $w=G/\rho$, $s=1/\rho$, transforming the dynamics into a two-dimensional system with nonlocal input bounded by $c_-,c_+$ and $\nu_-,\nu_+$. A four-region phase-plane framework, together with Lyapunov function families $\mathcal{L}$, is developed to construct explicit subcritical $\widehat{\Sigma}_\flat$ and supercritical $\widehat{\Sigma}_\sharp$ threshold regions, unifying prior results for constant backgrounds and damping cases as limiting instances. The analysis yields invariant-region arguments and comparison principles that prove global regularity for subcritical data and finite-time blowup for supercritical data, with explicit threshold expressions and phase-plane trajectories, including detailed auxiliary phase-plane results. The framework accommodates extensions such as variable backgrounds, different spatial domains, and various alignment protocols, providing a unified approach to critical thresholds in EPA-type models and offering insight into the influence of oscillations and nonlocal interactions on global dynamics.

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.

Figures (5)

• Figure 1: Illustration of the critical thresholds $\widehat{\Sigma}_\flat$ and $\widehat{\Sigma}_\sharp$. Left: weak alignment; middle: median alignment; right: strong alignment.
• Figure 2: Illustration of the subcritical region $\Sigma_\flat$ (shaded area)
• Figure 3: Illustration of the supercritical region $\Sigma_\sharp$ (shaded area)
• Figure 4: Illustration of the critical thresholds $\Sigma_\flat$ and $\Sigma_\sharp$. Left: weak alignment; middle: median alignment; right: strong alignment.
• Figure 5: Illustration of the flow in the supercritical region $\Sigma_\sharp$

Theorems & Definitions (45)

• Theorem 2.1: Local well-posedness
• Theorem 2.2: Critical thresholds
• Theorem 3.1
• proof
• Theorem 3.2: Global regularity
• Remark 3.1: Admissible conditions
• Theorem 3.3: Finite-time blowup
• Proposition 4.1: Weak comparison principles
• proof
• Proposition 4.2: Strong comparison principles