A complete characterization of sharp thresholds to spherically symmetric multidimensional pressureless Euler-Poisson systems

TL;DR

The paper provides a complete characterization of sharp critical thresholds distinguishing global regularity from finite-time blow-up for radially symmetric, multidimensional pressureless Euler–Poisson systems. It reduces the dynamics to a 4×4 ODE system along characteristics, introduces a novel nonlinear quantity and compatible transformations, and derives explicit subcritical regions Θ_4 for c>0 (under a periodicity assumption) and Σ_N/Σ_2 for c=0, establishing precise blow-up vs global existence criteria. A key insight is that Poisson forcing prevents concentration at the origin in multiple dimensions, enabling global smooth solutions in regimes previously thought fragile, with N=4 exhibiting particularly rich threshold structure. The work opens paths to applying the framework to related models (e.g., EP–alignment) and provides a detailed, implementable threshold theory for multidimensional EP dynamics.

Abstract

The Euler-Poisson (EP) system models the dynamics of a variety of physical processes, including charge transport, collisional plasmas, and certain cosmological wave phenomena. In this work, we establish sharp critical threshold conditions that distinguish global-in-time regularity from finite-time breakdown for solutions of the radially symmetric, multidimensional pressureless EP system. Overall, there are two cases: with and without background ($c>0, c=0$ respectively). For $c>0$, we obtain precise thresholds assuming a periodicity condition. A key feature of our approach is that it extends seamlessly to the zero background case, where we obtain sharp thresholds without imposing any additional assumptions. In particular, the framework accommodates initial velocities that may be negative, allowing the flow to be directed toward the origin. The main analytical challenge of deriving threshold conditions for EP systems stems from the intricate coupling of various local/nonlocal forces. To overcome this, we identify a novel nonlinear quantity that plays a decisive role in the analysis and enables a unified treatment of all relevant scenarios. Our results provide a comprehensive characterization of critical thresholds for the pressureless EP system in multiple dimensions.

Figures (9)

• Figure 1: Illustration of Theorem \ref{['ctcn']}.
• Figure 2: Illustration of Theorem \ref{['ctcn']} with $N = 5, k=1, c = 1, q_0 = 0.2, s_0 = -0.1$.
• Figure 3: $A$ vs $\Gamma$ with $N=5, k=1, c=1, q_0 = 0.2, s_0 = -0.1$.
• Figure 4: Visualization of Lemma \ref{['lemequivalence']} with $k=1, c=1, q_0=0.1, s_0=-0.15,A_0=0.2$. The top figure has $N=5$ (generic solutions not periodic) and the bottom with $N=4$ (periodic solutions).
• Figure 5: Left figure: when $\eta_1\equiv\eta_2$. Right figure: when $\eta_1,\eta_2$ are distinct.

Theorems & Definitions (62)

• Theorem 1.1: Local wellposedness
• Remark 1.2
• Theorem 2.1: Sharp threshold condition
• Definition 2.2
• Theorem 2.3: Sufficient condition for blow-up for $N\geq 3$
• Remark 2.4
• Theorem 2.5: Global solution for $N\geq 3$ with zero background
• Definition 2.6
• Theorem 2.7: Global solution for $N=2$ with zero background
• Definition 2.8