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 the 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.
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Dong Li, Yifei Wu, The Cauchy problem for the two dimensional Euler–Poisson system. J. Eur. Math. Soc. 16 (2014), no. 10, pp. 2211–2266DOI 10.4171/JEMS/486