  We prove global existence of finite energy weak solutions to the quantum
Navier-Stokes equations in the whole space with non trivial far-field condition
in dimensions d = 2,3. The vacuum regions are included in the weak formulation
of the equations. Our method consists in an invading domains approach. More
precisely, by using a suitable truncation argument we construct a sequence of
approximate solutions. The energy and the BD entropy bounds allow for the
passage to the limit in the truncated formulation leading to a finite energy
weak solution. Moreover, the result is also valid in the case of compressible
Navier-Stokes equations with degenerate viscosity.
