Previous work on the Euler-Poisson system in $2D$ also includes the work of Jang [@Juhi] and Wei-Tadmore-Bae [@WTB] for radial data, and Jang-Li-Zhang [@JangLiZhang] for the existence of wave operators.

For simplicity, we assume that the pressure law is quadratic $p(n_-)=T_-(n_-)^2/2$. The purpose of this is only to minimize the number of terms in the nonlinearity, but other powers could be treated similarly. After rescaling, we can then reduce to the following system[^3] $$\label{EP1}
\begin{split}
\partial _{t}n+\nabla \cdot \left( nv\right)&=0,\\
\partial _{t}v+v\cdot \nabla v+a\nabla n&=b\nabla \phi,\\
\Delta \phi&=n-1,
\end{split}$$ where $a,b\in(0,\infty)$, $a=T_-n_0/m_-$ is the square of the speed of sound and $b=4\pi e^2n_0/m_-$ is the electron plasma frequency. Our main Theorem asserts that small, neutral irrotational perturbations of a constant equilibrium $(n,v)=(1,0)$ are global.

\[MainThm\] There exists $\varepsilon_0>0$ and a norm $\Vert \cdot\Vert_Y$ such that any initial data $(n(x,0),v(x,0))$ satisfying $$\Vert (n(\cdot,0)-1,v(\cdot,0))\Vert_Y\le\varepsilon_0,\quad \hbox{curl}_x(v(\cdot,0))=0$$ leads to a global solution $(n,v)$ of which returns to equilibrium in the sense that $$\Vert n(\cdot,t)-1\Vert_{L^\infty}+\Vert\nabla v(\cdot,t)\Vert_{L^\infty}\lesssim(1+t)^{-1}.$$

The precise nature of the norm is given in below. It controls a finite number of derivatives and requires localization of the initial perturbations. Its most notable feature is that finiteness of the norm implies that the perturbation is 