On the asymptotic behavior of the Repulsive Pressureless Euler-Poisson System

TL;DR

This paper analyzes the long-term behavior of solutions to the one-dimensional repulsive pressureless Euler-Poisson system, focusing on energy conservation, equilibrium states, collapse conditions, and illustrative examples.

Contribution

It establishes existence and uniqueness of perfect states, characterizes collapse conditions, and provides analytical and simulation-based insights into the system's dynamics.

Findings

Existence and uniqueness of perfect equilibrium states.

Necessary and sufficient conditions for finite-time collapse.

Counterexamples illustrating the system's behavior.

Abstract

The main objective of this paper is a study of the asymptotic behavior of distributional solutions to the one-dimensional repulsive pressureless Euler-Poisson system. The system is a model for the dynamics of a mass distribution evolving on \mathbb{R} whose masses exert outward forces on one another. A discrete (describing the evolution of finitely many particles) solution is called sticky if, upon collision, particles stick together and move as one for all subsequent time, according to the conservation of mass and momentum principles. We prove results on the total energy (Hamiltonian) of the system and demonstrate the existence and uniqueness of so-called "perfect" states, where the Hamiltonian is constant over all time and the solution converges to equilibrium, a single stationary particle. We provide a necessary and a sufficient condition for finite-time collapse, and present a quadratic envelope within which a solution must remain in order to collapse. We demonstrate various (counter)examples that illustrate the unique behavior of the repulsive scheme with the sticky condition, analytically and with a computer simulation.