Global-in-time Well-posedness of the One-dimensional Hydrodynamic Gross-Pitaevskii Equations without Vacuum

Abstract: 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.