Long-term regularity of the periodic Euler--Poisson system for electrons in 2D

TL;DR

This paper proves that solutions to the 2D Euler--Poisson system for electrons remain regular over long timescales when starting close to equilibrium, with the duration depending on the domain size and initial data.

Contribution

It establishes long-term regularity of periodic solutions for the 2D Euler--Poisson system near equilibrium, extending understanding of plasma dynamics stability.

Findings

Solutions stay regular for at least R/ε²(log R)^{O(1)} time when initial data is close to constant.

The result applies to the system on a square torus of side length R.

The analysis provides bounds depending on initial data size and domain size.

Abstract

We study a basic plasma physics model--the one-fluid Euler--Poisson system on the square torus, in which a compressible electron fluid flows under its own electrostatic field. In this paper we prove long-term regularity of periodic solutions of this system in 2 spatial dimensions. Our main conclusion is that on a square torus of side length $R$, if the initial data is sufficiently close to a constant solution, then the solution is wellposed for a time at least $R/\epsilon^2(\log R)^{O(1)}$, where $\epsilon$ is the size of the initial data.