  We construct global weak solutions to isothermal quantum Navier-Stokes
equations, with or without Korteweg term, in the whole space of dimension at
most three. Instead of working on the initial set of unknown functions, we
consider an equivalent reformulation, based on a time-dependent rescaling, that
we introduced in a previous paper to study the large time behavior, and which
provides suitable a priori estimates, as opposed to the initial formulation
where the potential energy is not signed. We proceed by working on tori whose
size eventually becomes infinite. On each fixed torus, we consider the
equations in the presence of drag force terms. Such equations are solved by
regularization, and the limit where the drag force terms vanish is treated by
resuming the notion of renormalized solution developed by I. Lacroix-Violet and
A. Vasseur. We also establish global existence of weak solutions for the
isothermal Korteweg equation (no viscosity), when initial data are
well-prepared, in the sense that they stem from a Madelung transform.
