Existence and uniqueness of two dimensional Euler-Poisson system and WKB approximation to the nonlinear Schrödinger-Poisson system

In this paper, we study a dispersive Euler-Poisson system in two dimensional Euclidean space. Our aim is to show unique existence and the zero-dispersion limit of the time-local weak solution. Since one may not use dispersive structure in the zero-dispersion limit, when reducing the regularity, lack of critical embedding H¹ ⊆ L^∞ becomes a bottleneck. We hence employ an estimate on the best constant of the Gagliardo-Nirenberg inequality. By this argument, a reasonable convergence rate for the zero-dispersion limit is deduced with a slight loss. We also consider the semiclassical limit problem of the Schrödinger-Poisson system in two dimensions.