Global well-posedness for Euler-Nernst-Planck-Possion system in   dimension two
  In this paper, we study the Cauchy problem of the Euler-Nernst-Planck-Possion system. We obtain global well-posedness for the system in dimension $d=2$ for any initial data in $H^{s_1}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)$ under certain conditions of $s_1$ and $s_2$. 
Zeng Zhang and Zhaoyang Yin
math.AP