  We are concerned with a system of equations in $\mathbb{R}^{d}$ governing the
evolution of compressible fluids of Euler-Korteweg type, which can be used as a
phase transition model. In the case of zero sound speed $P'(\rho^{\ast})=0$, it
is found that the perturbation problem of irrotational fluids could be
reformulated into a quasi-linear Schr$\ddot{o}$dinger equation. Based on
techniques of dispersive estimates and methods of normal form, we shall
establish global-in-time well-posedness of Cauchy problems for $d=4$ and almost
global existence in $d=3$ without stability condition on the pressure.
Moreover, we would construct a class of global scattering solutions for 3D
case.
