  In this paper we consider the Navier-Stokes-Korteweg equations for a viscous
compressible fluid with capillarity effects in three space dimensions. We prove
global existence of finite energy weak solutions for large initial data.
Contrary to previous results regarding this system, vacuum regions are allowed
in the definition of weak solutions and no additional damping terms are
considered. The convergence of the approximating solutions is obtained by
introducing suitable truncations in the momentum equations of the velocity
field and the mass density at different scales and use only the a priori bounds
obtained by the energy and the BD entropy. Moreover, the approximating
solutions enjoy only a limited amount of regularity, and the derivation of the
truncations of the velocity and the density is performed by a suitable
regularization procedure.
