  We establish global-in-time well-posedness of the one-dimensional
hydrodynamic Gross-Pitaevskii equations in the absence of vacuum in $(1 + H^s)
\times H^{s-1}$ with $s \geq 1$. We achieve this by a reduction via the
Madelung transform to the previous global-in-time well-posedness result for the
Gross-Pitaevskii equation in arXiv:1801.08386v2 [math.AP] and
arXiv:2204.06293v1 [math.AP]. Our core result is a local bilipschitz
equivalence between the relevant function spaces.
