The Cauchy problem for the two dimensional Euler-Poisson system
  The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo first constructed a global smooth irrotational solution by using dispersive Klein-Gordon effect. It has been conjectured that same results should hold in the two-dimensional case. In our recent work, we proved the existence of a family of smooth solutions by constructing the wave operators for the 2D system. In this work we completely settle the 2D Cauchy problem. 
Dong Li and Yifei Wu
math-ph math.AP math.MP