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

Nicholas Biglin, Joseph Crachiola, Jack Curtis, Thomas Kunz, Omkar Maralappanavar, Adrian Tudorascu

TL;DR

This work analyzes the one-dimensional repulsive pressureless Euler-Poisson system under sticky and generalized sticky frameworks, revealing that the attractive projection formula fails in the repulsive case and necessitates generalized sticky-particle solutions (GSPS). It develops a Hamiltonian energy framework, proves existence and uniqueness of perfect solutions for finite discrete masses, and derives a sharp quadratic envelope that governs finite-time collapse to equilibrium. The results establish both necessary and sufficient collapse conditions, connect discrete reductions via center-of-mass combining, and extend to general measures through strong convergence of discrete approximations; numerical experiments illustrate the delicate dependence on initial data. Collectively, the paper advances understanding of asymptotic behavior, energy dissipation, and collapse dynamics in repulsive kinetic systems with long-range interactions, and sets the stage for further GSPS analysis and unbounded-domain extensions.

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.

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

TL;DR

This work analyzes the one-dimensional repulsive pressureless Euler-Poisson system under sticky and generalized sticky frameworks, revealing that the attractive projection formula fails in the repulsive case and necessitates generalized sticky-particle solutions (GSPS). It develops a Hamiltonian energy framework, proves existence and uniqueness of perfect solutions for finite discrete masses, and derives a sharp quadratic envelope that governs finite-time collapse to equilibrium. The results establish both necessary and sufficient collapse conditions, connect discrete reductions via center-of-mass combining, and extend to general measures through strong convergence of discrete approximations; numerical experiments illustrate the delicate dependence on initial data. Collectively, the paper advances understanding of asymptotic behavior, energy dissipation, and collapse dynamics in repulsive kinetic systems with long-range interactions, and sets the stage for further GSPS analysis and unbounded-domain extensions.

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.
Paper Structure (19 sections, 23 theorems, 104 equations, 6 figures)

This paper contains 19 sections, 23 theorems, 104 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Sticky Particles and Generalized Sticky Particles Solutions
  5. The Projection Formula
  6. Describing Solutions
  7. Ill-Posedness and Discrete Approximations
  8. Instantaneous Failure of the Projection Formula
  9. Combining Particles in finite DSPS
  10. The Hamiltonian
  11. Strong Convergence of Discrete Approximations to GSPS
  12. Perfect Solutions
  13. Finite Time Collapse to Equilibrium
  14. A Quadratic Bound on Compactly Supported Solutions
  15. Equilibrium Conditions for Finite Particles
  16. ...and 4 more sections

Key Result

Proposition 2.2

Consider $(\rho_t,v_t)$, a distributional solution to the repulsive PEP equations with initial conditions $(\rho_0,v_0)$. Let $\rho_0\in\mathcal{P}_2(\mathbb{R})$, $v_0\in L^2(\rho_0)$. The following are properties of $(\rho_t,v_t)$.

Figures (6)

  • Figure 2.1: The trajectories in Example \ref{['attractive-repulsive-comparison']}. Left is the attractive case, right is the repulsive case.
  • Figure 2.2: Left: The discrete approximation $\rho_t^3$ from Example \ref{['bad-approximation']}. Right: $\rho_t$ as in Example \ref{['bad-approximation']}.
  • Figure 2.3: The distribution described in Example \ref{['breaks-instantaneous']}.
  • Figure 2.4: Glancing and non-glancing sticky collisions.
  • Figure 4.1: The quadratic bound described in Theorem \ref{['quad-envelope']} is shown in dashed lines. Left: the solution stays inside the bound and converges to equilibrium. Right: The solution exits the bound and does not converge to equilibrium.
  • ...and 1 more figures

Theorems & Definitions (54)

  • Definition 1.1
  • Example 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Lemma 2.8
  • ...and 44 more