We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible viscous fluids. This enables us to apply Banach's fixed point theorem to show the existence of global solution. If time allows, we also comment on the decay estimates of the solution. This talk is based on a joint work with T. Kobayashi (Osaka University).