---
abstract: 'We consider the (repulsive) Euler-Poisson system for the electrons in two dimensions and prove that small smooth perturbations of a constant background exist for all time and remain smooth (never develop shocks). This extends to $2D$ the work of Guo [@Guo].'
address:
- Princeton University
- Courant Institute
author:
- 'Alexandru D. Ionescu'
- Benoit Pausader
title: 'The Euler–Poisson system in $2D$: global stability of the constant equilibrium solution'
---

[^1]

Introduction {#intro}
============

In this paper we investigate the question of global existence for small perturbations of a constant background for the following [*Euler–Poisson*]{} system for the electrons in $2$ dimensions: $$\label{EP}
\begin{split}
\partial _{t}n_{-}+\nabla \cdot \left( n_{-}v\,_{-}\right) & =0 \\
n_{-}m_{-}(\partial _{t}v_{-}+v_{-}\cdot \nabla v_{-})+\nabla p(n_{-})&
=en_{-}\nabla \phi  \\
\Delta \phi & =4\pi e(n_{-}-n_{0}).
\end{split}$$ Here the unknowns are $n_-\ge 0$, the density of electrons, and $v_-\in\mathbb{R}^2$, the velocity field of the electrons. These are functions defined for $(t,x)\in\mathbb{R}\times\mathbb{R}^2$. The positive constants $m_-$, $e$ and $n_0$ correspond respectively to the mass of an electron, its charge and the average charge of an ion background. Finally, $p=p(n_-)$ is a pressure function, given by a constitutive relation which for simplicity we assume to be quadratic.

These equations model the 