Global existence of Euler-Korteweg equations with the non-monotone pressure

We are concerned with the global solution of the compressible Euler-Korteweg equations in ${\mathbb{R}}^3$. In the case of zero sound speed $P^{\mathrm{\prime }}({\rho }^{\ast })=0$, 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 construct a class of global scattering solutions for 3D case.