Critical thresholds in pressureless Euler--Poisson equations with background states

TL;DR

This paper analyzes the critical threshold phenomena in one-dimensional pressureless Euler--Poisson equations with background states, establishing well-posedness, exploring neutrality conditions, and characterizing thresholds for attractive and repulsive forces, with applications to plasma physics.

Contribution

It provides the first detailed analysis of critical thresholds in pressureless Euler--Poisson systems with background states, including well-posedness and the impact of neutrality conditions.

Findings

Well-posedness established under neutrality condition

Neutrality condition is necessary for solution existence

Critical thresholds are characterized for attractive and repulsive cases

Abstract

We investigate the critical threshold phenomena in a large class of one dimensional pressureless Euler--Poisson (EP) equations, with non-vanishing background states. First, we establish local-in-time well-posedness in proper regularity spaces, which are adapted for a certain \textit{neutrality condition} to hold. The neutrality condition is shown to be necessary: we construct smooth solutions that exhibit instantaneous failure of the neutrality condition, which in turn yields non-existence of solutions, even locally in time, in the classical Sobolev spaces $H^s({\mathbb R})$, $s \geq 2$. Next, we study the critical threshold phenomena in the neutrality-condition-satisfying pressureless EP systems, where we distinguish between two cases. We prove that in the case of attractive forcing, the neutrality condition can further restrict the sub-critical region into its borderline, namely -- the sub-critical region is reduced to a single line in the phase plane. We then turn to provide a rather definitive answer for the critical thresholds in the case of repulsive EP systems with variable backgrounds. As an application, we analyze the critical thresholds for the damped EP system for cold plasma ion dynamics, where the density of electrons is given by the \textit{Maxwell--Boltzmann relation}.