Global-in-time well-posedness of the one-dimensional hydrodynamic Gross–Pitaevskii equations without vacuum

Abstract

Abstract We establish global-in-time well-posedness of the one-dimensional hydrodynamic Gross–Pitaevskii equations in the absence of vacuum in $$(1 + H^s) \times H^{s-1}$$ ( 1 + H s ) × H s - 1 with $$s \ge 1$$ s ≥ 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 Koch and Liao (Adv Math 377, 2021; Adv Math 420, 2023). Our core result is a local bilipschitz equivalence of the relevant function spaces, which enables the transfer of results between the two equations.