Critical Thresholds in One Dimensional Damped Euler-Poisson Systems

TL;DR

This paper investigates the critical initial conditions that determine whether solutions to one-dimensional damped Euler-Poisson equations remain smooth or break down, using phase plane analysis to explicitly characterize these thresholds.

Contribution

It provides a new phase plane analysis and explicit formulas for critical threshold curves in damped Euler-Poisson systems with three damping regimes.

Findings

Explicit critical threshold curves derived for all damping cases.

Solutions remain smooth if initial data is within the threshold region.

Finite time breakdown occurs if initial data exceeds the threshold.

Abstract

This paper is concerned with the critical threshold phenomenon for one dimensional damped, pressureless Euler-Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J., 50:109--157, 2001]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.