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

TL;DR

This paper establishes precise conditions that determine whether solutions to the multidimensional pressureless Euler-Poisson system remain smooth or develop singularities, covering both cases with and without background charge.

Contribution

It provides the first comprehensive characterization of sharp threshold conditions for the multidimensional pressureless Euler-Poisson system, including cases with negative initial velocities and no additional assumptions.

Findings

Derived sharp threshold conditions for global regularity versus finite-time breakdown.

Extended the analysis to include zero background charge without extra assumptions.

Identified a novel nonlinear quantity critical for the analysis.

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.