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

Robert Wegner

TL;DR

This work proves global-in-time well-posedness for the one-dimensional hydrodynamic Gross-Pitaevskii equations without vacuum by reducing to the classical GP equation via the Madelung transform. The authors establish a local bilipschitz equivalence between the GP function space (X^s,d^s) and the hydrodynamic space (Y^s,θ^s) for s>1/2, enabling transfer of GP well-posedness results to the hydrodynamic formulation. Global well-posedness for the hydrodynamic system is obtained in (1+H^s)×H^{s-1} with s≥1 under the energy bound E<4/3 (with a small-energy μ-variant), guaranteeing absence of vacuum for all times. The approach provides a robust framework connecting quantum hydrodynamics and nonlinear Schrödinger dynamics in 1D, with potential extensions to lower regularity regimes and broader energy regimes.

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.

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

TL;DR

This work proves global-in-time well-posedness for the one-dimensional hydrodynamic Gross-Pitaevskii equations without vacuum by reducing to the classical GP equation via the Madelung transform. The authors establish a local bilipschitz equivalence between the GP function space (X^s,d^s) and the hydrodynamic space (Y^s,θ^s) for s>1/2, enabling transfer of GP well-posedness results to the hydrodynamic formulation. Global well-posedness for the hydrodynamic system is obtained in (1+H^s)×H^{s-1} with s≥1 under the energy bound E<4/3 (with a small-energy μ-variant), guaranteeing absence of vacuum for all times. The approach provides a robust framework connecting quantum hydrodynamics and nonlinear Schrödinger dynamics in 1D, with potential extensions to lower regularity regimes and broader energy regimes.

Abstract

We establish global-in-time well-posedness of the one-dimensional hydrodynamic Gross-Pitaevskii equations in the absence of vacuum in with . 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.
Paper Structure (15 sections, 18 theorems, 165 equations, 1 figure)

This paper contains 15 sections, 18 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related results for the Gross-Pitaevskii equation
  3. Related results for the hydrodynamic Gross-Pitaevskii equations
  4. Functional analytic framework
  5. Main results
  6. Local bilipschitz equivalence of $(X^s, d^s)$ and $(\mathcal{Y}^s, \theta^s)$
  7. Notations and preliminaries
  8. Fractional Sobolev spaces on $B$ and $\mathop{\mathrm{\mathbb{R}}}\nolimits$
  9. Estimates in $H^s$
  10. Proof of Theorem \ref{['thm:3']} (i)
  11. Proof of Theorem \ref{['thm:3']} (ii)
  12. Proof of Theorem \ref{['thm:2']}
  13. Absence of vacuum for small energies
  14. Littlewood-Paley theory and proof of Lemma \ref{['lem:2']}
  15. Uniqueness for the Gross-Pitaevskii equation

Key Result

Lemma 1.1

Consider the function $\tilde{b}: [0, 1] \longrightarrow [0, \frac{4}{3}]$ defined by This is a strictly decreasing bijection (see Fig. 1) whose inverse we denote by $\tilde{\delta}(b): [0,\frac{4}{3}] \longrightarrow [0, 1]$. We have

Figures (1)

  • Figure 1: Graph of $\tilde{b}$

Theorems & Definitions (42)

  • Lemma 1.1
  • Corollary 1.2
  • Lemma 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.5: Local bilipschitz equivalence of $d^s$ and $\theta^s$
  • Corollary 1.6
  • Definition 1.7: Solution to (hGP)
  • Theorem 1.8: Global-in-time well-posedness of (hGP) for $s \geq 1$
  • Remark 1.9
  • ...and 32 more