Large-time behavior of pressureless Euler--Poisson equations with background states

TL;DR

This paper investigates the long-term behavior of solutions to the damped pressureless Euler-Poisson equations with background states, proving convergence to equilibrium states and exponential stability in plasma models.

Contribution

It establishes the asymptotic convergence and stability of solutions with variable background states, combining phase plane analysis and hypocoercivity techniques.

Findings

Solutions converge to equilibrium when background density approaches a positive constant.

Exponential convergence to steady states in plasma ion dynamics.

Rigorous characterization of stability for Euler-Poisson systems with background structures.

Abstract

We study the large-time asymptotic behavior of solutions to the one-dimensional damped pressureless Euler-Poisson system with variable background states, subject to a neutrality condition. In the case where the background density converges asymptotically to a positive constant, we establish the convergence of global classical solutions toward the corresponding equilibrium state. The proof combines phase plane analysis with hypocoercivity-type estimates. As an application, we analyze the damped pressureless Euler--Poisson system arising in cold plasma ion dynamics, where the electron density is modeled by a Maxwell-Boltzmann relation. We show that solutions converge exponentially to the steady state under suitable a priori bounds on the density and velocity fields. Our results provide a rigorous characterization of asymptotic stability for damped Euler-Poisson systems with nontrivial background structures.