1 Introduction

We consider in one dimension the Gross–Pitaevskii equation

$$\begin{aligned} i \partial _t q + \partial _{xx} q - 2 (|q|^2 - 1) q = 0 , \end{aligned}$$
(GP)

where \(q(t,x): {{\,\mathrm{\mathbb {R}}\,}}\times {{\,\mathrm{\mathbb {R}}\,}}\longrightarrow {{\,\mathrm{\mathbb {C}}\,}}\) represents an unknown wave function, subject to the boundary condition at infinity \(\lim _{|x| \rightarrow \infty } |q(t,x)| = 1\). In the absence of vacuum, meaning that

$$\begin{aligned} |q| > 0 , \end{aligned}$$

the Gross–Pitaevskii equation has a hydrodynamic formulation

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \rho + 2 \partial _x (\rho v) = 0 , \\ \partial _t v + \partial _x (v^2) + 2 \partial _x \rho = \partial _x \left( \partial _x \left( \frac{1}{2} \frac{\partial _x \rho }{\rho } \right) + \left( \frac{1}{2} \frac{\partial _x \rho }{\rho } \right) ^2 \right) , \end{array}\right. \end{aligned}$$
(hGP)

which we call the hydrodynamic Gross–Pitaevskii equations. Here the functions \(\rho (t,x): {{\,\mathrm{\mathbb {R}}\,}}\times {{\,\mathrm{\mathbb {R}}\,}}\longrightarrow {{\,\mathrm{\mathbb {R}}\,}}_+\) and \(v(t,x): {{\,\mathrm{\mathbb {R}}\,}}\times {{\,\mathrm{\mathbb {R}}\,}}\longrightarrow {{\,\mathrm{\mathbb {R}}\,}}\) may be understood as the unknown density and velocity of a quantum fluid. The system (hGP) belongs to the class of quantum hydrodynamical models, which may be used to model various physical phenomena such as Bose–Einstein condensation [22, 27], superfluidity [21, 34, 35] and quantum semiconductors [24]. We refer to [4, 17, 29] for more information on quantum hydrodynamical models and their relation to nonlinear Schrödinger equations.

The relation between (GP) and (hGP) is given by the Madelung transform:

$$\begin{aligned} \mathcal {M}(q) = \left( |q|^2, {{\,\textrm{Im}\,}}\left[ \frac{\partial _x q}{q}\right] \right) , \end{aligned}$$
(1.1)

which formally transforms a solution q of (GP) into a solution \((\rho , v) = \mathcal {M}(q)\) of (hGP). Note that \(\rho \) and v are real-valued. One immediately sees that the Madelung transform \(\mathcal {M}\) only makes sense when \(|q| > 0\). In this case, we may recover q from its Madelung transform by the formula

$$\begin{aligned} q = \sqrt{\rho } e^{i \varphi } , \end{aligned}$$

where \(\varphi \) is some spatial primitive of v, i.e.

$$\begin{aligned} \partial _x \varphi = v. \end{aligned}$$

One furthermore sees that the inverse Madelung transform \((\rho , v) \longmapsto q\) is only defined up to multiplication with \(\mathbb {S}^1\), i.e., a constant rotation in phase [see (1.13) for more details]. We refer the reader to [18] for a survey of the Madelung transform and the hydrodynamic Gross–Pitaevskii equations.

1.1 Related results for the Gross–Pitaevskii equation

Gross [28] and Pitaevskii [41] introduced the Gross–Pitaevskii equation as a model for a Bose–Einstein condensate, a type of Boson gas at very low density and temperature. For rigorous justification of the model, we refer to the mean-field approximation established by Erdős et al. [20], as well as references therein. As the Gross–Pitaevskii equation is a kind of defocusing cubic nonlinear Schrödinger equation, its well-posedness has been extensively studied. Due to the nonzero boundary condition, finite-energy solutions to (GP) can clearly not be in traditional function spaces that require global integrability, such as \(L^p({{\,\mathrm{\mathbb {R}}\,}})\). For integers \(k \ge 1\) and in any dimension \(n \ge 1\), Zhidkov [47] established local-in-time well-posedness in the so-called Zhidkov space \(Z^k({{\,\mathrm{\mathbb {R}}\,}}^n)\), which is the closure of \(\{u \in C_b^k({{\,\mathrm{\mathbb {R}}\,}}^n): \partial _x u \in H^{k-1}({{\,\mathrm{\mathbb {R}}\,}}^n)\}\) under the norm

$$\begin{aligned} \Vert u\Vert _{Z^k({{\,\mathrm{\mathbb {R}}\,}}^n)} = \Vert u\Vert _{L^\infty ({{\,\mathrm{\mathbb {R}}\,}}^n)} + \sum _{1 \le |\alpha | \le k} \Vert \partial _x^\alpha u\Vert _{L^2({{\,\mathrm{\mathbb {R}}\,}}^n)}. \end{aligned}$$
(1.2)

This led to a first global-in-time well-posedness result in one dimension in \(Z^1({{\,\mathrm{\mathbb {R}}\,}})\), as the Ginzburg–Landau energy

$$\begin{aligned} E(q) = \frac{1}{2} \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}^n} |\partial _x q|^2 + (|q|^2 - 1)^2 \textrm{d}x \end{aligned}$$
(1.3)

is conserved. The Gross–Pitaevskii equation (GP) can be interpreted as the Hamiltonian evolutionary equation associated with this energy. The well-posedness result in Zhidkov spaces was expanded to the cases \(n = 2, 3\) by Gallo [23]. Global-in-time well-posedness in the energy space \(\{q \in H^1_{{{\,\textrm{loc}\,}}}({{\,\mathrm{\mathbb {R}}\,}}^n): E(q) < \infty \}\) equipped with the metric

$$\begin{aligned} d_E(q, p) = \Vert q - p\Vert _{Z^1({{\,\mathrm{\mathbb {R}}\,}}^n) + H^1({{\,\mathrm{\mathbb {R}}\,}}^n)} + \Vert |q|^2 - |p|^2\Vert _{L^2({{\,\mathrm{\mathbb {R}}\,}}^n)} \end{aligned}$$

was obtained by Gérard [25, 26] for \(n=1, 2, 3\) and for \(n = 4\) under smallness assumptions. Later, Killip et al. [31] established global-in-time well-posedness in the energy space for \(n = 4\). More recently, the problem has been studied by Antonelli et al. [7] in \(n = 2, 3\) for general nonlinearities f satisfying a Kato-type assumption. Local-in-time well-posedness is obtained in the energy space and extended globally in the defocusing case under some further assumptions. Regarding the case of non-finite energy, Pecher [40] established global-in-time well-posedness in three dimensions in \(1 + H^s({{\,\mathrm{\mathbb {R}}\,}}^3)\) for \(s \in \left( \frac{5}{6}, 1\right) \).

We are concerned with the case \(n = 1\). For \(s \in {{\,\mathrm{\mathbb {R}}\,}}\), we associate with solutions of (GP) the energy functionals

$$\begin{aligned} E^s(q) = \frac{1}{2} \big \Vert \partial _x q\big \Vert _{H^{s-1}({{\,\mathrm{\mathbb {R}}\,}})}^2 + \frac{1}{2} \big \Vert |q|^2 - 1\big \Vert _{H^{s-1}({{\,\mathrm{\mathbb {R}}\,}})}^2. \end{aligned}$$
(1.4)

Note that indeed \(E^1 = E\). Our results are consequences of a pair of papers [32, 33] by Koch and Liao, where for \(s \ge 0\) they proved the global-in-time well-posedness of (GP) in the complete metric space

$$\begin{aligned} X^s = \{q \in H^s_{{{\,\textrm{loc}\,}}}({{\,\mathrm{\mathbb {R}}\,}}): E^s(q) < \infty \} / \mathbb {S}^1 , \end{aligned}$$
(1.5)

equipped with the distance function

$$\begin{aligned} d^s(q, p) = \left( \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \inf _{\lambda \in \mathbb {S}^1} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) (\lambda q - p)\Vert _{H^s}^2 \textrm{d}y \right) ^{\frac{1}{2}} . \end{aligned}$$
(1.6)

We summarize several of their results, taken from [32, Theorem 1.2, 1.3, Lemma 6.1] and [33, Theorem 1.5], in the following theorem.

Theorem 1.1

(Global-in-time well-posedness of (GP) [32, 33]) Let \(s \ge 0\). The pair \((X^s, d^s)\) is a complete metric space, and the energy functional \(E^s: X^s \longrightarrow {{\,\mathrm{\mathbb {R}}\,}}\) is continuous. There exists a constant \(C_0 > 0\) such that \(d^s(1, q) \le C_0 \sqrt{E^s(q)}\) for all \(q \in X^s\).

The Gross–Pitaevskii equation (GP) is globally-in-time well-posed in the metric space \((X^s, d^s)\) in the following sense: For any initial data \(q_0 \in X^s\), there exists a unique global-in-time solution \(q \in C({{\,\mathrm{\mathbb {R}}\,}}; X^s)\) of (GP) (see Definition 3.2). For any \(t \ge 0\), the Gross–Pitaevskii flow map \(X^s \ni q_0 \mapsto q \in C([-t, t]; X^s)\) is continuous. There exists a constant \(\tilde{C}_0(s, E^s(q_0))\) such that

$$\begin{aligned} \sup _{t \in {{\,\mathrm{\mathbb {R}}\,}}} E^s(q(t)) \le \tilde{C}_0 (s, E^s(q_0)) \, E^s(q_0) , \end{aligned}$$
(1.7)

and in the case \(s \ge 1\) the energy E(q(t)), defined in (1.3), is conserved.

1.2 Related results for the hydrodynamic Gross–Pitaevskii equations

The question of equivalence between Schrödinger equations and quantum hydrodynamical equations is relevant for the validity of classical approaches to quantum mechanics such as de Broglie–Bohm theory [15] and stochastic mechanics [39]. It became a topic of controversy when Wallstrom raised some objections [44,45,46]. We recommend [42] for a review of these issues. The difficulty arises from possible vacuum regions, which complicate the definition of the inverse Madelung transform. This can be resolved via an additional “Takabayasi’s quantization condition” [14, 43], which requires the winding numbers of the velocity field on closed loops to be quantized. Note that this condition trivially holds in one dimension. Wallstrom also raised objections regarding uniqueness [46]. There are indeed non-uniqueness results for weak solutions to quantum hydrodynamical systems, which are related to a change in the number of non-vacuum connected components [36].

Nevertheless, Antonelli and Marcati [2, 3] constructed weak solutions vanishing at infinity with finite but arbitrarily large energy, meaning that vacuum may appear, in \(n = 2, 3\) for a general quantum hydrodynamical system. They use a polar decomposition technique in order to define the velocity field in the vacuum regions. In collaboration with Zheng, they extended this to \(n = 1\) via a purely hydrodynamical approach [6, 8]. An alternative approach to finite-energy weak solutions is explored in [1, 5].

The well-posedness of the Euler–Korteweg system, a generalization of the compressible Euler equations which includes capillarity effects and contains (hGP) as a special case, was studied in higher dimensions by Audiard and Haspot [10, 11]. Similar to the approach we take is a paper by Audiard [9], in which global-in-time well-posedness of (hGP) under smallness assumptions is shown in certain spaces for \(n \ge 2\) by applying the Madelung transform to solutions to (GP). While they used scattering results to bound the solution away from 0, we use a rather elementary argument that leads us to the aforementioned energy bound \(E < \frac{4}{3}\).

A closely related paper is a work by Mohamad [38], which in [38, Proposition 1.1] states similar relations to our Theorem 1.6, and then uses a well-posedness result for (GP) to obtain a well-posedness result for (hGP) in \((1 + H^{k+1}) \times H^k\) with \(k \in {{\,\mathrm{\mathbb {N}}\,}}_{\ge 0}\) up to the appearance of vacuum. We discuss the similarities and differences in Remark 1.10.

1.3 Functional analytic framework

Our goal is to show a novel global-in-time well-posedness result for (hGP) with \((\rho , v) \in (1 + H^s) \times H^{s-1}\) where \(s \ge 1\) and \(n = 1\). We achieve this under the assumptions \(s \ge 1\) and \(E < \frac{4}{3}\) by passing the well-posedness result for (GP) in Theorem 1.1 through the Madelung transform (1.1). The first assumption \(s \ge 1\) ensures sufficient regularity for the energy E to be defined and for (hGP) to be interpretable in the sense of distributions. As an example, consider that \(s \ge 1\) implies \(v \in L^2({{\,\mathrm{\mathbb {R}}\,}})\), and so the problematic square of a distribution \(v^2\) appearing in (hGP)2 does indeed exist. The second assumption \(E < \frac{4}{3}\) can also be understood as a “regularity” assumption: solutions below the critical energy of \(\frac{4}{3}\) cannot have vacuum, that is points or intervals where \(|q| = \sqrt{\rho } = 0\) (see Corollary 1.3). As a result, singularities are avoided in the hydrodynamic formulation. Due to conservation of energy, the absence of vacuum is guaranteed for all times. Note that this energy assumption is sharp in the sense that the black soliton solution \(q(t,x) = \tanh (x)\) to (GP) has a zero \(\tanh (0) = 0\), while also having energy \(E(\tanh ) = \frac{4}{3}\).

We collect now some results that ensure the absence of vacuum, given certain energy bounds. We start with the following lemma.

Lemma 1.2

Consider the function \(\tilde{b}: [0, 1] \longrightarrow [0, \frac{4}{3}]\) defined by

$$\begin{aligned} \tilde{b}(\delta ) = \frac{4}{3} - 2 \delta + \frac{2}{3} \delta ^3. \end{aligned}$$

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

$$\begin{aligned} \tilde{b}(\delta )&= \min \left\{ E(q): q \in H^1_{{{\,\textrm{loc}\,}}}({{\,\mathrm{\mathbb {R}}\,}}), \, \inf _{x \in {{\,\mathrm{\mathbb {R}}\,}}} |q(x)| \le \delta \right\} , \\ \tilde{\delta }(b)&= \min \left\{ \inf _{x \in {{\,\mathrm{\mathbb {R}}\,}}} |q(x)| : q \in H^1_{{{\,\textrm{loc}\,}}}({{\,\mathrm{\mathbb {R}}\,}}), E(q) \le b \right\} . \end{aligned}$$
Fig. 1
figure 1

Graph of \(\tilde{b}\)

Full size image

This lemma is a stronger version of [13, Lemma 1]. The proof of a slightly more general Lemma A.1 is given in appendix. As a consequence of Lemma 1.2, the “energy gap” \(\frac{4}{3} - E(q)\) yields an explicit lower bound for the distance of |q| to zero. Due to conservation of the energy E(q), we obtain the following corollary.

Corollary 1.3

For any solution \(q \in C({{\,\mathrm{\mathbb {R}}\,}}; X^1)\) of (GP) (see Definition 3.2), we have

$$\begin{aligned} E(q_0)< b < \frac{4}{3} \Longrightarrow \inf _{(t, x) \in {{\,\mathrm{\mathbb {R}}\,}}^2} |q(t, x)|> \tilde{\delta }(b) > 0. \end{aligned}$$
(1.8)

We thus consider solutions q of (GP) in \(X^s\), \(s \ge 1\) with energy

$$\begin{aligned} E(q) < \frac{4}{3} , \end{aligned}$$
(1.9)

recalling the definitions (1.3)–(1.6) of \(X^s\) and E. We look for solutions \((\rho , v)\) of (hGP) in the function space

$$\begin{aligned} \mathcal {Y}^s = (1 + H^s({{\,\mathrm{\mathbb {R}}\,}};{{\,\mathrm{\mathbb {R}}\,}})) \times H^{s-1}({{\,\mathrm{\mathbb {R}}\,}};{{\,\mathrm{\mathbb {R}}\,}}) , \end{aligned}$$
(1.10)

equipped with the metric

$$\begin{aligned} \theta ^s((\rho , v), (\eta , w)) = \Vert \rho - \eta \Vert _{H^s} + \Vert v - w\Vert _{H^{s-1}} . \end{aligned}$$
(1.11)

We define the analogous energy

$$\begin{aligned} \mathcal {E}(\rho , v) = E(\mathcal {M}^{-1}(\rho , v)) = \frac{1}{2} \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \frac{(\partial _x \rho )^2}{4 \rho } + \rho v^2 + (\rho - 1)^2 \textrm{d}x. \end{aligned}$$
(1.12)

Here, the inverse Madelung transform is defined as:

$$\begin{aligned} \mathcal {M}^{-1}(\rho , v) (x) = \left( \sqrt{\rho (x)} e^{i \varphi (x)}\right) \, \mathbb {S}^1 = \left\{ \lambda \sqrt{\rho (x)} e^{i \varphi (x)}: \lambda \in \mathbb {S}^1\right\} , \end{aligned}$$
(1.13)

where \(\varphi \) is any spatial primitive of v, i.e. \(\partial _x \varphi = v\). Note that the energy E is indeed well-defined on equivalence classes under multiplication by \(\mathbb {S}^1\), and furthermore that the space \(X^s\) consists of such equivalence classes, and is hence a suitable domain for the Madelung transform \(\mathcal {M}\), given in (1.1).

In order to transform solutions of (GP) into solutions of (hGP) via the Madelung transform, we establish an equivalence between the relevant function spaces \((X^s, d^s)\) and \((\mathcal {Y}^s, \theta ^s)\). Specifically, we prove a local bilipschitz equivalence between the distance functions \(d^s\) and \(\theta ^s\) for all \(s > \frac{1}{2}\). While our main result only holds for \(s \ge 1\), our approach has the potential to be extended to the case \(\frac{1}{2}< s < 1\) if one finds a way to make sense of (hGP)2 in such a low regularity setting. For example, this may be possible via a local smoothing result, as in [30] (see Remark 1.12). When \(\frac{1}{2}< s < 1\), the absence of vacuum can still be ensured by a smallness assumption of the form

$$\begin{aligned} E^\mu (q) < \varepsilon _0(\mu ) \ll 1 , \end{aligned}$$

where \(\mu > \frac{1}{2}\) [see (1.16)]. This smallness condition can also replace \(E < \frac{4}{3}\) in the case \(s \ge 1\), \(\mu \le s\). Specifically, we have the following Lemma 1.4 as a replacement for Lemma 1.2.

Lemma 1.4

For \(\delta \in [0, 1]\) and \(\mu > \frac{1}{2}\), define

$$\begin{aligned} E^\mu _\delta = \inf \left\{ E^\mu (q): q \in H^\mu _{{{\,\textrm{loc}\,}}}, \inf _{x \in {{\,\mathrm{\mathbb {R}}\,}}} |q(x)| \le \delta \right\} . \end{aligned}$$

Then \(E^\mu _1 = 0\), the function \(\delta \mapsto E^\mu _\delta \) is decreasing, and there exists a constant \(\tilde{C}(\mu ) > 0\) so that

$$\begin{aligned} E^\mu _\delta \ge \frac{(1 - \delta )^2}{\tilde{C}(\mu )}. \end{aligned}$$
(1.14)

This lemma is also a special case of Lemma A.1. By (1.7), there exists for any \(\mu > \frac{1}{2}\) a constant \(c(\mu ) > 0\) such that

$$\begin{aligned} E^\mu (q_0)< \varepsilon \Longrightarrow \sup _{t \in {{\,\mathrm{\mathbb {R}}\,}}} E^\mu (q(t)) < c(\mu ) \, \varepsilon \end{aligned}$$
(1.15)

for all \(\varepsilon \in (0, 1)\) and any solution \(q \in C({{\,\mathrm{\mathbb {R}}\,}}; X^\mu )\) of (GP). Not attempting to obtain a sharp bound, we state the analogous of Corollary 1.3.

Corollary 1.5

Let \(\mu > \frac{1}{2}\) and define

$$\begin{aligned} \varepsilon _0(\mu ) = \max \left\{ \frac{1}{2}, \frac{1}{4 c(\mu ) \tilde{C}(\mu )} \right\} . \end{aligned}$$
(1.16)

For any solution \(q \in C({{\,\mathrm{\mathbb {R}}\,}}; X^\mu )\) of (GP) (see Definition 3.2), we have

$$\begin{aligned} E^\mu (q_0)< \varepsilon < \varepsilon _0(\mu ) \Longrightarrow \inf _{(t, x) \in {{\,\mathrm{\mathbb {R}}\,}}^2} |q(t, x)|> 1 - \sqrt{\varepsilon } \sqrt{c(\mu ) \tilde{C}(\mu )} > \frac{1}{2}. \end{aligned}$$
(1.17)

Proof

We prove the contrapositive. Suppose \(\inf _{(t, x) \in {{\,\mathrm{\mathbb {R}}\,}}^2} |q(t, x)| \le \delta :=1 - \sqrt{\varepsilon } \sqrt{c(\mu ) \tilde{C}(\mu )}\) and note that \(\delta \in (0, 1)\). Using the definition of \(E^\mu _{\tilde{\delta }}\) and (1.14), this implies that for any \(\tilde{\delta } > \delta \) there exists \(t \in {{\,\mathrm{\mathbb {R}}\,}}\) with

$$\begin{aligned} E^\mu (q(t)) \ge E^\mu _{\tilde{\delta }} \ge \frac{(1 - \tilde{\delta })^2}{\tilde{C}(\mu )} . \end{aligned}$$

In particular,

$$\begin{aligned} \sup _{t \in {{\,\mathrm{\mathbb {R}}\,}}} E^\mu (q(t)) \ge \frac{(1 - \delta )^2}{\tilde{C}(\mu )} = c(\mu ) \varepsilon , \end{aligned}$$

so (1.15) implies \(E^\mu (q_0) \ge \varepsilon \). \(\square \)

As the energies \(E^\mu \) still provide a lower bound for the distance of |q| to zero, we can use the smallness assumption \(E^\mu < \varepsilon _0(\mu )\) as a substitute for \(E < \frac{4}{3}\). We define for \(\mu > \frac{1}{2}\) the energies

$$\begin{aligned} \mathcal {E}^\mu (\rho , v) = E^\mu (\mathcal {M}^{-1}(\rho , v)) . \end{aligned}$$
(1.18)

1.4 Main results

For both the Gross–Pitaevskii equation (GP) and its hydrodynamic formulation (hGP), there are three key objects in our function framework: the energy, the space and the metric. We summarize the definitions given in Sect. 1.2 in the following diagram:

(1.19)

Here, the Madelung transform and its inverse

$$\begin{aligned} \mathcal {M}(q) = \left( |q|^2, {{\,\textrm{Im}\,}}\left[ \frac{\partial _x q}{q}\right] \right) \qquad \mathcal {M}^{-1}(\rho , v) = \left( \sqrt{\rho } e^{i \varphi }\right) \, \mathbb {S}^1, \quad \partial _x \varphi = v \end{aligned}$$

are given in (1.1) and (1.13), respectively. Recall also the explicit forms of the energies E and \(\mathcal {E}\) in the most important case \(s = 1\):

$$\begin{aligned} \begin{aligned} E(q) = \frac{1}{2} \int \limits _{{{\,\mathrm {\mathbb {R}}\,}}} |\partial _x q|^2 + (|q|^2 - 1)^2 \text {d}x \qquad \mathcal {E}(\rho , v) = \frac{1}{2} \int \limits _{{{\,\mathrm {\mathbb {R}}\,}}} \frac{(\partial _x \rho )^2}{4 \rho } + \rho v^2 + (\rho - 1)^2 \text {d}x . \end{aligned} \end{aligned}$$

Our first main result is the following theorem, which is central to our strategy as it establishes a local bilipschitz equivalence between the metrics \(d^s\) and \(\theta ^s\). We require \(s > \frac{1}{2}\) to use \(L^\infty \) embeddings and certain product estimates.

Theorem 1.6

(Local bilipschitz equivalence of \(d^s\) and \(\theta ^s\)) Let \(s > \frac{1}{2}\) and \(r, \delta > 0\). Consider measurable functions \(\rho , \eta , \varphi , \psi : {{\,\mathrm{\mathbb {R}}\,}}\longrightarrow {{\,\mathrm{\mathbb {R}}\,}}\) so that \(q, p \in \mathcal {S}'({{\,\mathrm{\mathbb {R}}\,}}) \cap H_{{{\,\textrm{loc}\,}}}^s({{\,\mathrm{\mathbb {R}}\,}})\) and \(|q|, |p| > \delta \), where \(q = \sqrt{\rho } e^{i \varphi }\) and \(p = \sqrt{\eta } e^{i \psi }.\) There exist constants \(C_1(s, \delta , r), C_2(s, r) > 0\) so that the following hold:

  1. (i)

    If \(d^s(1, q), d^s(1, p) < r\), then

    $$\begin{aligned} \theta ^s((\rho , \partial _x \varphi ), (\eta , \partial _x \psi )) \le C_1(s, \delta , r) \, d^s(q, p) . \end{aligned}$$
  2. (ii)

    If \(\theta ^s((1,0), (\rho , \partial _x \varphi )), \theta ^s((1,0), (\eta , \partial _x \psi )) < r\), then

    $$\begin{aligned} d^s(q, p) \le C_2(s, r) \, \theta ^s((\rho , \partial _x \varphi ), (\eta , \partial _x \psi )) . \end{aligned}$$

Corollary 1.7

Let \(s \ge 1\), \(\frac{1}{2}< \mu < 1\). For all \(b < \frac{4}{3}\) and \(\varepsilon < \varepsilon _0(\mu )\), the maps

$$\begin{aligned} \big (\big \{q \in X^s: E(q)< b \big \}, d^s\big ) \xrightarrow {\quad \mathcal {M} \quad } \big (\big \{(\rho , v) \in \mathcal {Y}^s: \mathcal {E}(\rho ,v) < b \big \}, \theta ^s\big ) \end{aligned}$$

and

$$\begin{aligned} \big (\big \{q \in X^s: E^\mu (q)< \varepsilon \big \}, d^s\big ) \xrightarrow {\quad \mathcal {M} \quad } \big (\big \{(\rho , v) \in \mathcal {Y}^s: \mathcal {E}^\mu (\rho , v) < \varepsilon \big \}, \theta ^s\big ) \end{aligned}$$

are bilipschitz equivalences. Recall that a bilipschitz equivalence is a map which is bijective, Lipschitz continuous, and has a Lipschitz continuous inverse. Here \(\varepsilon _0(\mu )\) is a constant defined in (1.16).

Our second main result is the global-in-time well-posedness of the hydrodynamic Gross–Pitaevskii equations.

Definition 1.8

(Solution to (hGP)) Let \(s \ge 1\) and \(0 \in I \subset {{\,\mathrm{\mathbb {R}}\,}}\) be an open time interval or the real line, and let \((\rho _0, v_0) \in \mathcal {Y}^s\) with \(\rho _0 > 0\). A solution to (hGP) with initial data \((\rho _0, v_0)\) is a pair \((\rho , v) \in C(I; \mathcal {Y}^s)\) with \(\rho > 0\), which solves (hGP) in the sense of distributions and fulfills \((\rho , v)(0) = (\rho _0, v_0)\).

Theorem 1.9

(Global-in-time well-posedness of (hGP) for \(s \ge 1\)) Let \(s \ge 1\). The hydrodynamic Gross–Pitaevskii equations (hGP) are globally-in-time well-posed in the metric space \((\mathcal {Y}^s, \theta ^s)\) for initial data \((\rho _0, v_0) \in \mathcal {Y}^s\) with \(\mathcal {E}(\rho _0, v_0) < \frac{4}{3}\) in the following sense:

There exists a solution \((\rho , v) \in C_b({{\,\mathrm{\mathbb {R}}\,}}; \mathcal {Y}^s)\) to (hGP) (see Definition 1.8). It is the unique solution that fulfills

$$\begin{aligned} \mathcal {E}(\rho (t), v(t)) = \mathcal {E}(\rho _0, v_0) < \frac{4}{3} \end{aligned}$$
(1.20)

for all \(t \in {{\,\mathrm{\mathbb {R}}\,}}\). For any \(T \ge 0\) the solution map

$$\begin{aligned} \left\{ (\rho _0, v_0) \in \mathcal {Y}^s: \mathcal {E}(\rho _0, v_0) < \frac{4}{3} \right\}&\longrightarrow C_b([-T, T]; \mathcal {Y}^s) \\ (\rho _0, v_0)&\longmapsto (\rho , v) \end{aligned}$$

is continuous.

For all \(\frac{1}{2}< \mu < 1\) there exist constants \(c(\mu ), \varepsilon _0(\mu ) > 0\), defined in (1.15) and (1.16), so that if we replace the assumption \(\mathcal {E}(\rho _0, v_0) < \frac{4}{3}\) by \(\mathcal {E}^\mu (\rho _0, v_0)< \varepsilon < \varepsilon _0(\mu )\), then the above statement holds with (1.20) replaced by \(\mathcal {E}^\mu (\rho (t), v(t)) < c(\mu ) \, \varepsilon \).

Remark 1.10

In [38, Theorem 1.2], the well-posedness of (hGP) until the appearance of vacuum is shown in \((1 + H^k) \times H^{k-1}\) for \(k \in {{\,\mathrm{\mathbb {N}}\,}}_{\ge 1}\). Furthermore, continuity properties for the Madelung transform between \((\mathcal {Y}^k, \theta ^k)\) and the space

$$\begin{aligned} \tilde{E}^k = \{q \in L^\infty ({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {C}}\,}}): 1 - |q|^2 \in L^2({{\,\mathrm{\mathbb {R}}\,}}), \partial _x q \in H^{k-1}({{\,\mathrm{\mathbb {R}}\,}})\} \end{aligned}$$

are established. Specifically, a strong metric

$$\begin{aligned} \tilde{d}^k(q, p) = \Vert q - p\Vert _{L^\infty ({{\,\mathrm{\mathbb {R}}\,}})} + \Vert |q|^2 - |p|^2\Vert _{L^2({{\,\mathrm{\mathbb {R}}\,}})} + \Vert \partial _x q - \partial _x p\Vert _{H^{k-1}({{\,\mathrm{\mathbb {R}}\,}})} \end{aligned}$$

and a weak metric

$$\begin{aligned} \tilde{d}_{{{\,\textrm{loc}\,}}}^k(q, p) = \Vert q - p\Vert _{L^\infty ([-1, 1])} + \Vert |q|^2 - |p|^2\Vert _{L^2({{\,\mathrm{\mathbb {R}}\,}})} + \Vert \partial _x q - \partial _x p\Vert _{H^{k-1}({{\,\mathrm{\mathbb {R}}\,}})}\ \end{aligned}$$

are considered, and the following is shown [38, Proposition 1.1]:

figure a

There are several ways in which Theorem 1.6 improves upon [38, Theorem 1.2]. The first is that our result covers the fractional cases as well. The second is that we obtain a full local bilipschitz equivalence

figure b

and do so down to \(s > \frac{1}{2}\), while they have to work with two different metrics on the side of (GP) for the two directions of estimates. Furthermore, they state well-posedness up to the appearance of vacuum, while we use energy bounds to ensure this absence of vacuum for all times. Lastly, we find solutions which are uniformly bounded.

Remark 1.11

Previously, Zhidkov [48, Theorem III.3.1] studied the stability of solutions in the Zhidkov space \(Z^1({{\,\mathrm{\mathbb {R}}\,}})\) [see (1.2)] near-space homogeneous solutions \(\Phi \), such as the constant solution \( \Phi = 1\), with respect to the distance \(\theta ^1\). For the case \(s = 1\) he derived similar estimates as above under smallness assumptions, although he did not formulate a well-posedness result. Curiously, in [48, Cor. III.3.5] he proved furthermore that for any ball \(B \subset {{\,\mathrm{\mathbb {R}}\,}}\), if the initial \(\theta ^1\)-distance between the perturbed and the space-homogeneous solution is small, then for all times also the distance

$$\begin{aligned} \inf _{\lambda \in \mathbb {S}^1} \Vert \lambda q - \Phi \Vert _{W^{1,2}(B)} \end{aligned}$$

is small. This can be interpreted as a weaker form of the estimate \(d^1 \lesssim \theta ^1\) we derive (see Lemma 2.6 and Remark 2.11).

Remark 1.12

As both Theorems 1.1 and 1.6 work for all \(s > \frac{1}{2}\), it may be possible to extend Theorem 1.9 to the case \(\frac{1}{2}< s < 1\). The problem is that for \(v \in H^{s-1} \not \subseteq L^2\) the product of distributions \(v^2 = v \cdot v\) is not necessarily defined. Nevertheless, it may be possible to find global distributional solutions. For example, in the paper [30] by Killip and Vişan global-in-time well-posedness of the KdV equation in \(H^{-1}\) is first shown in the sense that the solution map \({{\,\mathrm{\mathbb {R}}\,}}\times \mathcal {S} \longrightarrow \mathcal {S}\) extends to a continuous mapping \({{\,\mathrm{\mathbb {R}}\,}}\times H^{-1} \longrightarrow H^{-1}\), and some other conditions are fulfilled. In our case, it is similarly true that for any \(T \ge 0\) the solution map

$$\begin{aligned} \Big \{(\rho _0, v_0) \in \mathcal {Y}^1: \mathcal {E}^s(\rho _0, v_0) < \varepsilon _0(s) \Big \} \longrightarrow C_b([-T,T]; \mathcal {Y}^1) \end{aligned}$$

has a unique continuous extension to a map

$$\begin{aligned} \Big \{(\rho _0, v_0) \in \mathcal {Y}^s: \mathcal {E}^s(\rho _0, v_0) < \varepsilon _0(s) \Big \} \longrightarrow C_b([-T,T]; \mathcal {Y}^s) . \end{aligned}$$

This extension is given by the conjugation of the corresponding solution map for (GP) at regularity s with the Madelung transform. Killip and Vişan then furthermore show a local smoothing result, which implies that the solution map produces functions in \(L^2_{{{\,\textrm{loc}\,}},t,x}\). As a result, the equation is indeed solved in the sense of distributions. We do not know if such a local smoothing result holds in our case.

Organization of the paper In Sect. 2, we prove Theorem 1.6, the local bilipschitz equivalence of \((X^s, d^s)\) and \((\mathcal {Y}^s, \theta ^s)\). In Sect. 3, we prove Theorem 1.9, the global-in-time well-posedness of the hydrodynamic Gross–Pitaevskii equations.

2 Local bilipschitz equivalence of \((X^s, d^s)\) and \((\mathcal {Y}^s, \theta ^s)\)

The goal of this section is to prove Theorem  1.6. In Sect. 2.1, we introduce the necessary notations, definitions, and basic results required for the rest of the paper. We split the proof of the two statements (i) and (ii) of Theorem 1.6 into Sects. 2.2 and 2.3.

2.1 Notations and preliminaries

We use the notations \({{\,\mathrm{\mathbb {R}}\,}}_\ge = \{r \in {{\,\mathrm{\mathbb {R}}\,}}: r \ge 0\}\) and \({{\,\mathrm{\mathbb {R}}\,}}_+ = \{r \in {{\,\mathrm{\mathbb {R}}\,}}: r > 0\}\). We write C or \(C(\ldots )\) for various constants with possible dependence on other quantities. These may change from one line to the next. We denote by \(\mathcal {D}' = \mathcal {D}'({{\,\mathrm{\mathbb {R}}\,}}) = \mathcal {D}'({{\,\mathrm{\mathbb {R}}\,}};{{\,\mathrm{\mathbb {C}}\,}})\) the space of distributions and by \(\mathcal {S}' = \mathcal {S}'({{\,\mathrm{\mathbb {R}}\,}}) = \mathcal {S}'({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {C}}\,}})\) the space of tempered distributions. In general, if for a family of function spaces, such as the \(L^p\)-spaces, we write just “\(L^p\)”, then we mean \(L^p({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {C}}\,}})\).

We write \(\widehat{f}\) for the Fourier transform

$$\begin{aligned} \hat{f}(\xi ) = \frac{1}{\sqrt{2 \pi }} \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} e^{- i x \xi } f(x) \textrm{d}x \end{aligned}$$

for a Schwartz function \(f \in \mathcal {S}\) and extend the definition as usual to the tempered distributions \(f \in \mathcal {S}'\). Let \(s \in {{\,\mathrm{\mathbb {R}}\,}}\). We define the Sobolev space

$$\begin{aligned} H^s = H^s({{\,\mathrm{\mathbb {R}}\,}}) = H^s({{\,\mathrm{\mathbb {R}}\,}};{{\,\mathrm{\mathbb {C}}\,}}) = \{f \in \mathcal {S}'({{\,\mathrm{\mathbb {R}}\,}};{{\,\mathrm{\mathbb {C}}\,}}): \Vert f\Vert _{H^s} < \infty \} \end{aligned}$$

with norm

$$\begin{aligned} \Vert f\Vert _{H^s} = \Vert f \Vert _{H^s({{\,\mathrm{\mathbb {R}}\,}})} = \big \Vert \langle \xi \rangle ^s \widehat{f}\, \big \Vert _{L^2({{\,\mathrm{\mathbb {R}}\,}})}. \end{aligned}$$

Here \(\langle \xi \rangle = \sqrt{1 + |\xi |^2}\). We also define the quasinorm of the homogeneous Sobolev space

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s} = \Vert f \Vert _{\dot{H}^s({{\,\mathrm{\mathbb {R}}\,}})} = \big \Vert |\xi |^s \widehat{f}\, \big \Vert _{L^2({{\,\mathrm{\mathbb {R}}\,}})}. \end{aligned}$$

Let \(p \in [1,\infty )\). For \(s \ge 0\), let \(\alpha \in [0, 1)\) and \(m \in {{\,\mathrm{\mathbb {Z}}\,}}\) so that \(s = m + \alpha \). Let \(B \subset {{\,\mathrm{\mathbb {R}}\,}}\) be a non-empty open interval. We define the Sobolev–Slobodeckij space

$$\begin{aligned} W^{s,p}(B) = \{f \in \mathcal {D}'(B): \Vert f\Vert _{W^{s,p}(B)} < \infty \} \end{aligned}$$

with norm

$$\begin{aligned} \Vert f\Vert _{W^{s,p}(B)} = \left( \sum _{k = 0}^m \Vert \partial ^k f\Vert _{L^p(B)}^p + \int \limits _B \int \limits _B \frac{|\partial ^m f(x) - \partial ^m f(y)|^p}{|x - y|^{1 + \alpha p}} \textrm{d}x \textrm{d}y \right) ^{\frac{1}{p}} . \end{aligned}$$

We define \(W_0^{s,p}(B) = \overline{\mathcal {D}(B)}^{W^{s,p}(B)}\). For \(s < 0\) we define \(W^{s,p'}(B) = (W_0^{-s,p}(B))^*\), where \(\frac{1}{p} + \frac{1}{p'} = 1\). We refer the reader to the book [37] by McLean for a comprehensive exposition. For the convenience of the reader, let us recall some well-known results on Sobolev spaces that may be used without mention.

2.1.1 Fractional Sobolev spaces on B and \({{\,\mathrm{\mathbb {R}}\,}}\)

The only bounded domains we use are balls B, and on those we use the Sobolev–Slobodeckij spaces \(W^{s,2}(B)\). On the whole real line \({{\,\mathrm{\mathbb {R}}\,}}\), we use \(H^s = H^s({{\,\mathrm{\mathbb {R}}\,}})\).

Lemma 2.1

(\(W^{s,2}(B)\) and \(H^s({{\,\mathrm{\mathbb {R}}\,}})\)) Let \(s \in {{\,\mathrm{\mathbb {R}}\,}}\) and \(R > 0\). Let \(\tilde{B} \subset B \subseteq {{\,\mathrm{\mathbb {R}}\,}}\) be concentric balls of radius \(\frac{R}{2}\) and R. Set \(B_k = B + k R\) and \(\tilde{B}_k = \tilde{B} + k R\) for \(k \in {{\,\mathrm{\mathbb {Z}}\,}}\).

  1. (i)

    There exists a natural isomorphism \(H^{-s} \cong (H^s)^*\) (see [37, p. 76]).

  2. (ii)

    \(H^s = W^{s,2}({{\,\mathrm{\mathbb {R}}\,}})\) and

    $$\begin{aligned} \Vert f\Vert _{W^{s,2}(B)} \le C_1 \min \{\Vert F\Vert _{H^s}: F \big \vert _B = f\} \le C_2 \Vert f\Vert _{W^{s,2}(B)}. \end{aligned}$$

    (see [37, p. 77, (3.23) + Theorem 3.18, 3.19]).

  3. (iii)

    For \(s \ge 0\), there exists a bounded linear extension operator \(E: W^{s,2}(B) \longrightarrow H^s\) with \(E f \big \vert _B = f\) (see [37, Theorem A.4]).

  4. (iv)

    For \(s \ge 0\), there exists a constant C(s, R) so that

    $$\begin{aligned} \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert f\Vert _{W^{s,2}(\tilde{B}_k)}^2 \le \Vert f\Vert _{H^s}^2 \le C(s, R) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert f\Vert _{W^{s,2}(B_k)}^2 \le 4 C(s, R) \Vert f\Vert _{H^s}^2 \end{aligned}$$

    and

    $$\begin{aligned} \Vert f\Vert _{H^{-s}}^2 \le C(s, R) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert f\Vert _{W^{-s,2}(B_k)}^2. \end{aligned}$$
  5. (v)

    If \(s > \frac{1}{2}\), then \(\Vert f g\Vert _{H^s} \le C(s) \Vert f\Vert _{H^s} \Vert g\Vert _{H^s}\) (see [12, Cor. 2.87]). By use of the extension operator, we also have \(\Vert f g\Vert _{W^{s,2}(B)} \le C(s) \Vert f\Vert _{W^{s,2}(B)} \Vert g\Vert _{W^{s,2}(B)}\).

Proof

We only have to prove (iv). We start with the first inequality in the sequence. The case \(s = 0\) is trivial, so we assume \(s > 0\). Here the statement is trivial for the terms with integer regularity, and for the fractional terms we estimate

$$\begin{aligned} \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \int \limits _{\tilde{B}_k} \int \limits _{\tilde{B}_k} \frac{|\partial ^m f(x) - \partial ^m f(y)|^2}{|x - y|^{1 + 2 \alpha }} \textrm{d}x \textrm{d}y&\le \sum _{j, k \in {{\,\mathrm{\mathbb {Z}}\,}}} \int \limits _{\tilde{B}_j} \int \limits _{\tilde{B}_k} \frac{|\partial ^m f(x) - \partial ^m f(y)|^2}{|x - y|^{1 + 2 \alpha }} \textrm{d}x \textrm{d}y \\&= \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \frac{|\partial ^m f(x) - \partial ^m f(y)|^2}{|x - y|^{1 + 2 \alpha }} \textrm{d}x \textrm{d}y . \end{aligned}$$

The third inequality in the sequence follows trivially. We show the second inequality for any \(s \in {{\,\mathrm{\mathbb {R}}\,}}\). We can decompose \(f = \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \eta _k f\), where \(\eta _k\) is a smooth partition of unity with \({{\,\textrm{supp}\,}}\eta _k \subset B_k\), \(\sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \eta _k = 1\) and \(\eta _k(x) = \eta _k(x + k R)\). Then,

$$\begin{aligned} \langle f, f \rangle _{H^s}&= \sum _{j,k \in {{\,\mathrm{\mathbb {Z}}\,}}} \langle \eta _j f, \eta _k f \rangle _{H^s} \overset{(ii)}{=}\ \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \langle (\eta _{k - 1} + \eta _k + \eta _{k+1}) f, \eta _k f \rangle _{W^{s,2}(B_k)} \\&\le C(s, R, \eta _0) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert f\Vert _{W^{s,2}(B_k)}^2 . \end{aligned}$$

\(\square \)

2.1.2 Estimates in \(H^s\)

The following lemma states two crucial estimates. Such kinds of product estimates are well-known in the literature, see for example [19, Proposition 2.7].

Lemma 2.2

Let \(s > \frac{1}{2}\) and \(f, g \in \mathcal {S}'\). There exists a constant C(s) so that

$$\begin{aligned} \Vert f g\Vert _{H^s} \le C(s) \Vert g\Vert _{H^s} \big (\Vert f\Vert _{L^\infty } + \Vert f'\Vert _{H^{s-1}}\big ) \end{aligned}$$
(2.1)

and

$$\begin{aligned} \Vert f g\Vert _{H^{s-1}} \le C(s) \Vert g\Vert _{H^{s-1}} \big (\Vert f\Vert _{L^\infty } + \Vert f'\Vert _{H^{s-1}}\big ) . \end{aligned}$$
(2.2)

Proof

See Appendix B. \(\square \)

In this section, we often write \(f'\) for the spatial derivative \(\partial _x f\). Recall the definitions (1.19). For notational convenience, we sometimes prefer to use the variables

$$\begin{aligned} A = \sqrt{\rho } \quad \text {and} \quad B = \sqrt{\eta }. \end{aligned}$$

These variables are equivalent for the sake of our estimates, by which we mean specifically Lemma 2.4. In order to prove this, we state two estimates regarding the action of a smooth function on Sobolev spaces. They are a direct consequence of some results in [12].

Lemma 2.3

[12, Theorem 2.87, Corollary 2.91] Let \(s > \frac{1}{2}\) and \(F \in C^\infty ({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}})\) with \(F'(0) = F(0) = 0\). Let \(u, v \in H^s({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}}) \cap L^\infty ({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}})\). We have the estimates

$$\begin{aligned} \Vert F \circ u\Vert _{H^s} \le C(s, F', \Vert u\Vert _{L^\infty }) \Vert u\Vert _{H^s} \end{aligned}$$
(2.3)

and

$$\begin{aligned} \Vert F \circ u - F \circ v\Vert _{H^s} \le C(s, F'', \Vert u\Vert _{H^s}, \Vert v\Vert _{H^s}) \Vert u - v\Vert _{H^s} . \end{aligned}$$
(2.4)

An analysis of the proof in [12] reveals that, more precisely, the constants depend on \(\Vert F'\Vert _{C^{\lceil s \rceil + 1}(B_{\Vert u\Vert _{L^\infty }})}\) and \(\Vert F''\Vert _{C^{\lceil s \rceil + 1}(B_{\Vert u\Vert _{L^\infty }})}\), respectively.

Lemma 2.4

Let \(s > \frac{1}{2}\) and \(\rho , \eta \in \mathcal {S}'({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}}) \cap H^s_{{{\,\textrm{loc}\,}}}({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}})\) with \(\rho , \eta > 0\). Define \(A = \sqrt{\rho }\) and \(B = \sqrt{\eta }\). We have the estimates

$$\begin{aligned} \Vert \rho - \eta \Vert _{H^s} \le C_1 \big (s, \Vert A - 1\Vert _{H^s}, \Vert B - 1\Vert _{H^s}\big )\, \Vert A - B\Vert _{H^s} \end{aligned}$$

and

$$\begin{aligned} \Vert A - B\Vert _{H^s} \le C_2 \big (s, \Vert \rho - 1\Vert _{H^s}, \Vert \eta - 1\Vert _{H^s}\big )\, \Vert \rho - \eta \Vert _{H^s} . \end{aligned}$$

Proof

We apply Lemma 2.3 with \(F(u) = u^2\) and obtain

$$\begin{aligned} \Vert A^2 - B^2\Vert _{H^s}&\le \Vert (A-1)^2 - (B-1)^2\Vert _{H^s} + 2 \Vert A - B\Vert _{H^s} \\&\le C\big (s, \Vert A - 1\Vert _{H^s}, \Vert B - 1\Vert _{H^s}\big ) \, \Vert A - B\Vert _{H^s}. \end{aligned}$$

Similarly with any function \(F \in C^\infty ({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}})\) that fulfills \(F(u) = \sqrt{u + 1} - \frac{1}{2} u - 1\) for \(x \ge 0\), we obtain

$$\begin{aligned} \Vert \sqrt{\rho } - \sqrt{\eta }\Vert _{H^s}&\le \left\| \left( \sqrt{(\rho -1) + 1} - \frac{1}{2} (\rho - 1)\right) - \left( \sqrt{(\eta -1) + 1} - \frac{1}{2} (\eta - 1)\right) \right\| _{H^s} \\&\quad + \frac{1}{2} \Vert \rho - \eta \Vert _{H^s} \\&\le C\big (s, \Vert \rho - 1\Vert _{H^s}, \Vert \eta - 1\Vert _{H^s}\big ) \, \Vert \rho - \eta \Vert _{H^s}. \end{aligned}$$

\(\square \)

The following lemma is also a consequence of Lemma 2.3 and will be used frequently in the subsequent section.

Lemma 2.5

Let \(s, \delta , R > 0\). There exists \(C(s, \delta , R) > 0\) such that for any ball \(B_0 \subset {{\,\mathrm{\mathbb {R}}\,}}\) of radius R and all \(u \in W^{s,2}(B_0)\) with \(|u|> \delta > 0\) we have

$$\begin{aligned} \Vert u^{-1}\Vert _{W^{s,2}(B_0)} \le C(s, \delta ) \Vert u\Vert _{W^{s,2}(B_0)}. \end{aligned}$$

Proof

This follows by applying Lemma 2.3 with any function \(F \in C^\infty ({{\,\mathrm{\mathbb {R}}\,}}; {{\,\mathrm{\mathbb {R}}\,}})\) so that \(F(0) = F'(0) = 0\) and \(F(x) = x^{-1}\) for \(|x| > \frac{\delta }{2}\), and using the existence of an extension operator from Lemma 2.1(iii). Note that Lemma 2.3 requires real-valued functions, so we apply it to the real and imaginary parts of \(u^{-1}\) separately. \(\square \)

2.2 Proof of Theorem 1.6(i)

Recall the definitions (1.19), in particular \(q = \sqrt{\rho } e^{i \varphi }\) and \(p = \sqrt{\eta } e^{i \psi }\), as well as \(A = \sqrt{\rho }\) and \(B = \sqrt{\eta }\). We assume \(s > \frac{1}{2}\), \(d^s(1, q), d^s(1, p) < r\) and \(|q|, |p|> \delta > 0\). We have to prove that

$$\begin{aligned} \theta ^s((\rho , \partial _x \varphi ), (\eta , \partial _x \psi )) \le C (s, \delta , r) \, d^s(q, p) . \end{aligned}$$

We do this by showing an estimate of the form

$$\begin{aligned} \theta ^s \lesssim \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d^s_*\big \vert _{B_k} \lesssim d^s. \end{aligned}$$
(2.5)

Let us elaborate on the quantity in the middle before we start the proof. Given a ball \(B \subset {{\,\mathrm{\mathbb {R}}\,}}\), we define for convenience the following notations:

$$\begin{aligned} d_*^s\big \vert _B(q, p)&= \inf _{\lambda \in \mathbb {S}^1} \Vert \lambda q - p\Vert _{W^{s,2}(B)} , \end{aligned}$$
(2.6)
$$\begin{aligned} d^s\big \vert _B(q, p)&= \left( \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \inf _{\lambda \in \mathbb {S}^1} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) (\lambda q - p)\Vert _{W^{s,2}(B)}^2 \textrm{d}y \right) ^{\frac{1}{2}} . \end{aligned}$$
(2.7)

Lemma 2.6

Let \(s > \frac{1}{2}\) and let \(B_0 = \{x \in {{\,\mathrm{\mathbb {R}}\,}}: |x| < R\}\) be an open ball of radius \(R > 0\) with center 0. There exists \(C(s, R) > 0\) so that

$$\begin{aligned} d_*^s\big \vert _{B_0}(q, p) \le C(s, R) \, d^s\big \vert _{B_0}(q, p) \end{aligned}$$
(2.8)

for all \(q, p \in \mathcal {S}' \cap H^s_{{{\,\textrm{loc}\,}}}\). As a consequence, for families of balls \(B_k = B_0 + k R\) with \(k \in {{\,\mathrm{\mathbb {Z}}\,}}\) we have

$$\begin{aligned} \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d_*^s\big \vert _{B_k}(q, p)^2 \le C(s, R) \, d^s(q, p)^2 . \end{aligned}$$
(2.9)

Proof

As \(\{y - x: x, y \in B_0\} \subseteq \{x \in {{\,\mathrm{\mathbb {R}}\,}}: |x| < 2R\}\), there exists a finite constant \(C(s, R) > 0\) such that \(\sup _{y \in B_0} \Vert {{\,\textrm{sech}\,}}(y - \cdot )^{-1}\Vert _{W^{s,2}(B_0)}^2 \le C(s, R)\). The first estimate follows:

$$\begin{aligned} \inf _{\lambda \in \mathbb {S}^1} \Vert \lambda q - p\Vert _{W^{s,2}(B_0)}^2&\le C(s, R) \inf _{y \in B_0} \inf _{\lambda \in \mathbb {S}^1} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) (\lambda q - p)\Vert _{W^{s,2}(B_0)}^2 \\&\le C(s, R) \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \inf _{\lambda \in \mathbb {S}^1} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) (\lambda q - p)\Vert _{W^{s,2}(B_0)}^2 \textrm{d}y \,. \end{aligned}$$

Using this and Lemma 2.1(iv), we obtain the second estimate:

$$\begin{aligned} \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d_*^s\big \vert _{B_k}(q, p)^2&\le C(s, R) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d^s\big \vert _{B_k}(q, p)^2 \\&\le C(s, R) \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \inf _{\lambda \in \mathbb {S}^1} \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) (\lambda q - p)\Vert _{W^{s,2}(B_k)}^2 \textrm{d}y \\&\le C(s, R) \, d^s(q, p)^2 . \end{aligned}$$

\(\square \)

Proof of Theorem 1.6(i)

Let \(B_0 = \{x \in {{\,\mathrm{\mathbb {R}}\,}}: |x| < 1\}\) and observe that

$$\begin{aligned}&\Vert |q|^2 - |p|^2\Vert _{W^{s,2}(B_0)} \\&\quad = \inf _{\lambda , \nu \in \mathbb {S}^1} \Vert |\lambda q|^2 - |\nu p|^2\Vert _{W^{s,2}(B_0)} \\&\quad = \inf _{\lambda , \mu , \nu \in \mathbb {S}^1} \Vert |\lambda q - \mu |^2 - |\nu p - \mu |^2 + 2 ({{\,\textrm{Re}\,}}(\lambda \overline{\mu } q) - {{\,\textrm{Re}\,}}(\overline{\mu } \nu p))\Vert _{W^{s,2}(B_0)} . \end{aligned}$$

We can estimate

$$\begin{aligned}&\Vert |q|^2 - |p|^2\Vert _{W^{s,2}(B_0)} \nonumber \\&\quad \le \inf _{\lambda , \mu , \nu \in \mathbb {S}^1} \Big \Vert {{\,\textrm{Re}\,}}\Big ( \big ((\lambda q - \mu ) - (\nu p - \mu )\big ) \big (\overline{(\lambda q - \mu ) + (\nu p - \mu )}\big ) \Big ) \Big \Vert _{W^{s,2}(B_0)} \nonumber \\&\qquad + 2 \Vert {{\,\textrm{Re}\,}}(\lambda \overline{\mu } q - \overline{\mu } \nu p)\Vert _{W^{s,2}(B_0)} \nonumber \\&\quad \le C(s) \inf _{\lambda , \nu \in \mathbb {S}^1} \Vert \lambda q - \nu p\Vert _{W^{s,2}(B_0)} \inf _{\mu \in \mathbb {S}^1} \big ( \Vert \lambda q - \mu \Vert _{W^{s,2}(B_0)} + \Vert \nu p - \mu \Vert _{W^{s,2}(B_0)} + 2\big ) \nonumber \\&\quad = C(s) \inf _{\lambda \in \mathbb {S}^1} \Vert \lambda q - p\Vert _{W^{s,2}(B_0)} \inf _{\mu \in \mathbb {S}^1} \inf _{\nu \in \mathbb {S}^1} \big ( \Vert \lambda q - \mu \Vert _{W^{s,2}(B_0)} + \Vert \nu p - \mu \Vert _{W^{s,2}(B_0)} + 2\big ) \nonumber \\&\quad \le C(s) \, d^s_{*} \big \vert _{B_0}(q, p) \big ( 2 + d^s_{*}\big \vert _{B_0}(1, q) + d^s_{*}\big \vert _{B_0}(1, p) \big ) \nonumber \\&\quad \le C(s, r) \, d^s_{*} \big \vert _{B_0} (q, p), \end{aligned}$$
(2.10)

where in the last line we used (2.8). Now we set \(B_k = B_0 + k\) and see with Lemma 2.1(iv) and (2.9) that

$$\begin{aligned} \Vert \rho - \eta \Vert _{H^s}^2&\le C(s) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert |q|^2 - |p|^2\Vert _{W^{s,2}(B_k)}^2 \\&\le C(s, r) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d_{*}^s\big \vert _{B_k}(q, p)^2 \\&\le C(s, r) \, d^s(q, p)^2. \end{aligned}$$

It remains to estimate \(\Vert \varphi ' - \psi '\Vert _{H^{s-1}}\). Applying Lemma 2.2 yields

$$\begin{aligned} \Vert (\varphi - \psi )'\Vert _{H^{s-1}}^2&= \Vert (e^{i (\varphi - \psi )})' e^{-i (\varphi - \psi )}\Vert _{H^{s-1}}^2 \\&\le C(s) \Vert (e^{i (\varphi - \psi )})'\Vert _{H^{s-1}}^2 \big ( \Vert e^{- i (\varphi - \psi )}\Vert _{L^\infty }^2 + \Vert (e^{-i (\varphi - \psi )})'\Vert _{H^{s-1}}^2 \big ) \\&\le C(s) \Vert (e^{i (\varphi - \psi )})'\Vert _{H^{s-1}}^2 \big (1 + \Vert (e^{i (\varphi - \psi )})'\Vert _{H^{s-1}}^2 \big ) . \end{aligned}$$

It therefore suffices to derive the estimate for the quantity \(\Vert (e^{i (\varphi - \psi )})'\Vert _{H^{s-1}}^2\). Observe with Lemma 2.1 (iv) that

$$\begin{aligned} \Vert (e^{i (\varphi - \psi )})'\Vert _{H^{s-1}}^2&\le C(s) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \inf _{\lambda \in \mathbb {S}^1} \Vert (e^{i (\varphi - \psi )} - \lambda )'\Vert _{W^{s-1,2}(B_k)}^2 \\&= C(s) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \inf _{\theta \in {{\,\mathrm{\mathbb {R}}\,}}} \Vert e^{i (\varphi - \psi - \theta )} - 1\Vert _{W^{s,2}(B_k)}^2 \\&= C(s) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \inf _{\theta \in {{\,\mathrm{\mathbb {R}}\,}}} \Vert e^{-i \psi } (e^{i (\varphi + \theta )} - e^{i \psi })\Vert _{W^{s,2}(B_k)}^2 . \end{aligned}$$

We now carefully introduce the amplitudes:

$$\begin{aligned}&\inf _{\theta \in {{\,\mathrm{\mathbb {R}}\,}}} \Vert e^{-i \psi } (e^{i (\varphi + \theta )} - e^{i \psi })\Vert _{W^{s,2}(B_k)}^2 \\&\quad = \inf _{\theta \in {{\,\mathrm{\mathbb {R}}\,}}} \left\| B e^{- i \psi (x)} B^{-1} \left( \frac{ A e^{i (\varphi (x) + \theta )} - B e^{i \psi (x)}}{A} + B e^{i \psi (x)} \left( \frac{1}{A} - \frac{1}{B}\right) \right) \right\| _{W^{2,s}(B_k)}^2 \\&\quad \le C(s)\Vert B e^{-i \psi }\Vert _{W^{s,2}(B_k)}^2 \Vert B^{-1}\Vert _{W^{s,2}(B_k)}^2 \\&\qquad \times \left( \inf _{\theta \in {{\,\mathrm{\mathbb {R}}\,}}} \left\| \frac{A e^{i (\varphi (x) + \theta )} - B e^{i \psi (x)}}{A}\right\| _{W^{s,2}(B_k)}^2 + \left\| B e^{i \psi (x)} \left( \frac{1}{A} - \frac{1}{B}\right) \right\| _{W^{s,2}(B_k)}^2\right) . \end{aligned}$$

As \(A, B> \delta > 0\), we can apply Lemma 2.5. Together with (2.10), we obtain

$$\begin{aligned} \Vert A^{-1}\Vert _{W^{s,2}(B_k)}^2 \le C(s, \delta ) \Vert A\Vert _{W^{s,2}(B_k)}^2 \le C(s, \delta , r) \left( \inf _{\lambda \in \mathbb {S}^1} \Vert A - \lambda \Vert _{W^{s,2}(B_k)}^2 + |B_k| \right) \le C(s, \delta , r), \end{aligned}$$

and similarly \(\Vert B^{\pm 1}\Vert _{W^{s,2}(B_k)}, \Vert q^{\pm 1}\Vert _{W^{s,2}(B_k)}, \Vert p^{\pm 1}\Vert _{W^{s,2}(B_k)} \le C(s, \delta , r)\). We conclude again by reducing the situation to an application of Lemma 2.6 and the previously shown estimate (2.10):

$$\begin{aligned} \Vert (e^{i (\varphi - \psi )})'\Vert _{H^{s-1}}^2&\le C(s) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert p\Vert _{W^{s,2}(B_k)}^2 \Vert B^{-1}\Vert _{W^{s,2}(B_k)}^2 \\&\quad \times \Big (\inf _{\lambda \in \mathbb {S}^1} \Vert \lambda q - p\Vert _{W^{s,2}(B_k)}^2 \Vert A^{-1}\Vert _{W^{s,2}(B_k)}^2 \\&\quad + \Vert A - B\Vert _{W^{s,2}(B_k)}^2 \Vert p\Vert _{W^{s,2}(B_k)}^2 \Vert A^{-1}\Vert _{W^{s,2}(B_k)}^2 \Vert B^{-1}\Vert _{W^{s,2}(B_k)}^2 \Big ) \\&\le C(s, \delta , r) \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d^s_*\big \vert _{B_k}(q, p)^2 \\&\le C(s, \delta , r)\, d^s(q, p)^2. \end{aligned}$$

\(\square \)

2.3 Proof of Theorem 1.6(ii)

We assume

$$\begin{aligned} \theta ^s((1,0), (\rho , \partial _x \varphi )), \theta ^s((1,0), (\eta , \partial _x \psi )) < r, \end{aligned}$$

and \(\sqrt{\rho }, \sqrt{\eta }> \delta > 0\). We have to prove that

$$\begin{aligned} d^s(q, p) \le C(s, r) \, \theta ^s((\rho , \partial _x \varphi ), (\eta , \partial _x \psi )) . \end{aligned}$$

As mentioned above, due to Lemma 2.4 it suffices to prove this with \(\rho , \eta \) replaced by \(A = \sqrt{\rho }\) and \(B = \sqrt{\eta }\). Recall the definitions (1.19). We define

$$\begin{aligned} \tilde{d}^s(q, p) = \left( \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \inf _{\lambda \in \mathbb {S}^1} \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} (\lambda q - p)\Vert _{H^s}^2 \textrm{d}y \right) ^{\frac{1}{2}}, \end{aligned}$$

where we have replaced the \({{\,\textrm{sech}\,}}\) in the definition of \(d^s\) with \(\sqrt{{{\,\textrm{sech}\,}}}\). Some of the hard work for this direction has already been done in the proof of Lemma 2.7. This was proven for \(d^s\) in [32, Lemma 6.1], but the proof is identical for \(\tilde{d}^s\) as \(\sqrt{{{\,\textrm{sech}\,}}}\) is positive and still has sufficiently fast decay.

Lemma 2.7

[32, Lemma 6.1] For all \(s \ge 0\), the energy \(E^s: X^s \longrightarrow {{\,\mathrm{\mathbb {R}}\,}}_\ge \) is continuous with respect to \(d^s\), and there exists \(C(s) > 0\) so that

$$\begin{aligned} d^s(1, q) \le C(s) \sqrt{E^s(q)} \quad \text {and} \quad \tilde{d}^s(1, q) \le C(s) \sqrt{E^s(q)} \end{aligned}$$

for all \(q \in X^s\).

Remark 2.8

The appearance of the square root is explained by a clash of notation: the energies \(E^s\) as defined in [32] correspond to \(\sqrt{2 E^s}\) in our notation.

We first prove two Lemmas.

Lemma 2.9

Let \(s > \frac{1}{2}\). There exists a constant \(C(s) > 0\) so that for all \(\varphi \in \mathcal {S}' \cap H^s_{{{\,\textrm{loc}\,}}}\) we have

$$\begin{aligned} \Vert (e^{i \varphi })'\Vert _{H^{s-1}} \le C(s) (1 + \Vert \varphi '\Vert _{H^{s-1}})^{\gamma } \Vert \varphi '\Vert _{H^{s-1}} , \end{aligned}$$

where \(\gamma = 2s - 2\) if \(s \ge 1\) and \(\gamma = \frac{1 - s}{s - \frac{1}{2}}\) if \(s < 1\).

Proof

We assume \(\Vert \varphi '\Vert _{H^{s-1}} \ne 0\). By Lemma 2.2, there exists a constant C(s) so that

$$\begin{aligned} \Vert \varphi ' e^{i \varphi }\Vert _{H^{s-1}} \le C(s) \Vert \varphi '\Vert _{H^{s-1}} \big ( \Vert e^{i \varphi }\Vert _{L^\infty } + \Vert (e^{i \varphi })'\Vert _{H^{s-1}} \big ) . \end{aligned}$$
(2.11)

For \(\varepsilon \in (0, 1)\) and \(f \in H^s_{{{\,\textrm{loc}\,}}}\) define \(f_\varepsilon (x) = f(\varepsilon x)\). This has the scaling estimates

$$\begin{aligned} \min \{\varepsilon ^{s - \frac{1}{2}}, \varepsilon ^{\frac{1}{2}}\} \Vert f'\Vert _{H^{s-1}} \le \Vert (f_\varepsilon )'\Vert _{H^{s-1}} \le \max \{\varepsilon ^{s - \frac{1}{2}}, \varepsilon ^{\frac{1}{2}}\} \Vert f'\Vert _{H^{s-1}} . \end{aligned}$$
(2.12)

Define \(s_{\min } \le s_{\max }\) so that \(\{s_{\min }, s_{\max }\} = \{s - \frac{1}{2}, \frac{1}{2}\}\). Then we can rewrite the above as

$$\begin{aligned} \varepsilon ^{s_{\max }} \Vert f'\Vert _{H^{s-1}} \le \Vert (f_\varepsilon )'\Vert _{H^{s-1}} \le \varepsilon ^{s_{\min }} \Vert f'\Vert _{H^{s-1}} . \end{aligned}$$
(2.13)

We choose \(\varepsilon = (1 + 2 C(s) \Vert \varphi '\Vert _{H^{s-1}})^{- \frac{1}{s_{\min }}}\) so that

$$\begin{aligned} \Vert (\varphi _\varepsilon )'\Vert _{H^{s-1}} \le \varepsilon ^{s_{\min }} \Vert \varphi '\Vert _{H^{s-1}} = \frac{\Vert \varphi '\Vert _{H^{s-1}}}{1 + 2 C(s) \Vert \varphi '\Vert _{H^{s-1}}} \le \frac{1}{2 C(s)} . \end{aligned}$$

Combining this with (2.11) yields

$$\begin{aligned} \Vert (e^{i \varphi _\varepsilon })'\Vert _{H^{s-1}} \le C(s) \Vert (\varphi _\varepsilon )'\Vert _{H^{s-1}} + \frac{1}{2} \Vert (e^{i \varphi _\varepsilon })'\Vert _{H^{s-1}} , \end{aligned}$$

so we obtain

$$\begin{aligned} \Vert (e^{i \varphi _\varepsilon })'\Vert _{H^{s-1}} \le 2 C(s) \Vert (\varphi _\varepsilon )'\Vert _{H^{s-1}} . \end{aligned}$$

We conclude with the scaling estimates (2.12) that

$$\begin{aligned} \Vert (e^{i \varphi })'\Vert _{H^{s-1}} \le \varepsilon ^{- s_{\max }} \Vert (e^{i \varphi _\varepsilon })'\Vert _{H^{s-1}} \le 2 C(s) \varepsilon ^{- s_{\max }} \Vert (\varphi _\varepsilon )'\Vert _{H^{s-1}} \le 2 C(s) \varepsilon ^{s_{\min } - s_{\max }} \Vert \varphi '\Vert _{H^{s-1}} . \end{aligned}$$

Lastly, note that

$$\begin{aligned} \varepsilon ^{s_{\min } - s_{\max }} = \big (1 + 2 C(s) \Vert \varphi '\Vert _{H^{s-1}}\big )^{\frac{|s - 1|}{s_{\min }}} = \big (1 + 2 C(s) \Vert \varphi '\Vert _{H^{s-1}}\big )^{\gamma } . \end{aligned}$$

\(\square \)

Lemma 2.10

Let \(s > \frac{1}{2}\) and \(r > 0\). There exists \(C(s, r) > 0\) so that for all \(q = A e^{i \varphi } \in \mathcal {S}' \cap H^s_{{{\,\textrm{loc}\,}}}\) with \(\theta ^s((1,0), (A,\varphi ')) < r\) we have

$$\begin{aligned} E^s(q) \le C(s, r) \, \theta ^s((1, 0), (A, \varphi '))^2 . \end{aligned}$$

Proof

For the amplitudinal part of the energy, we know from Lemma 2.4 that

$$\begin{aligned} \Vert |q|^2 - 1\Vert _{H^{s-1}} \le \Vert A^2 - 1\Vert _{H^s} \le C(s, r) \Vert A - 1\Vert _{H^s} . \end{aligned}$$

For the remainder, we use Lemma 2.2:

$$\begin{aligned} \Vert q'\Vert _{H^{s-1}}&\le \Vert A' e^{i \varphi }\Vert _{H^{s-1}} + \Vert A (e^{i \varphi })'\Vert _{H^{s-1}} \\&\le C(s) \Vert A'\Vert _{H^{s-1}} \big ( \Vert e^{i\varphi }\Vert _{L^\infty } + \Vert (e^{i \varphi })'\Vert _{H^{s-1}} \big ) \\&\quad + \Vert (e^{i \varphi })'\Vert _{H^{s-1}} \big ( \Vert A - 1\Vert _{L^\infty } + 1 + \Vert A'\Vert _{H^{s-1}} \big ). \end{aligned}$$

We now conclude by estimating both appearances of \(\Vert (e^{i \varphi })'\Vert _{H^{s-1}}\) with Lemma 2.9. \(\square \)

Proof of Theorem 1.6(ii)

We split the distance \(d^s(q, p)\) into two parts:

$$\begin{aligned} d^s(q, p)^2&\le 2 \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \inf _{\theta \in {{\,\mathrm{\mathbb {R}}\,}}} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) A (e^{i (\varphi + \theta )} - e^{i \psi })\Vert _{H^s}^2 \textrm{d}y \\&\quad + 2 \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \Vert {{\,\textrm{sech}\,}}(y - \cdot ) (B - A) e^{i \psi }\Vert _{H^s}^2 \textrm{d}y \\&= (I) + (II). \end{aligned}$$

We use the algebra property of \(H^s\) and Lemma 2.7 to estimate

$$\begin{aligned} (I)&\le C(s) \sup _{y \in {{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} A e^{i \psi } \Big \Vert _{H^s}^2 \tilde{d}^s(1, e^{i (\varphi - \psi )})^2 \\&\le C(s, r) \big (\Vert A - 1\Vert _{H^s}^2 + 1 \big ) \sup _{y \in {{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} e^{i \psi } \Big \Vert _{H^s} E^s(e^{i (\varphi - \psi )}) . \end{aligned}$$

With Lemma 2.10, we can estimate \(E^s(e^{i (\varphi - \psi )})\) by \(\Vert \varphi ' - \psi '\Vert _{H^{s-1}}^2\), and Lemma 2.2 yields

$$\begin{aligned} \sup _{y \in {{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} e^{i \psi } \Big \Vert _{H^s} \le C(s) \sup _{y \in {{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} \Big \Vert _{H^s} (\Vert e^{i \psi }\Vert _{L^\infty } + \Vert (e^{i \psi })'\Vert _{H^{s-1}}) \le C(s, r) . \end{aligned}$$

It follows that

$$\begin{aligned} (I) \le C(s, r) \, \theta ^s((A, \varphi '), (B, \psi '))^2 . \end{aligned}$$

Note that

$$\begin{aligned} (II)&\le C(s) \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} (A - B) \Big \Vert _{H^s}^2 \left( 1 + \inf _{\lambda \in \mathbb {S}^1} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} (e^{i \psi } - \lambda ) \Big \Vert _{H^s}^2 \right) \textrm{d}y \\&\le C(s) \left( \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} (A - B) \Big \Vert _{H^s}^2 \textrm{d}y + \big \Vert \sqrt{{{\,\textrm{sech}\,}}}\big \Vert _{H^s}^2 \Vert A - B\Vert _{H^s}^2 \, \tilde{d}^s(1, e^{i \psi })^2 \right) . \end{aligned}$$

We can deal with the second term as before. For the first one, we use Lemma 2.1(iv) and Young’s convolution inequality:

$$\begin{aligned} \int \limits _{{{\,\mathrm{\mathbb {R}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} (A - B) \Big \Vert _{H^s}^2 \textrm{d}y&\le \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \sup _{y \in [k, k+1]} \sum _{j \in {{\,\mathrm{\mathbb {Z}}\,}}} \Big \Vert \sqrt{{{\,\textrm{sech}\,}}(y - \cdot )} (A - B) \Big \Vert _{W^{s,2}([j, j+3])}^2 \\&\le \sum _{j, k \in {{\,\mathrm{\mathbb {Z}}\,}}} \big \Vert \sqrt{{{\,\textrm{sech}\,}}} \big \Vert _{C^{\lceil s \rceil +1}([k-j-3, k-j+1]))}^2 \Vert A - B\Vert _{W^{s,2}([j,j+3])}^2 \\&\le \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} \big \Vert \sqrt{{{\,\textrm{sech}\,}}} \big \Vert _{C^{\lceil s \rceil +1}([k-3, k+1]))}^2 \sum _{j \in {{\,\mathrm{\mathbb {Z}}\,}}} \Vert A - B\Vert _{W^{s,2}([j, j+3])}^2 \\&\le C(s) \Vert A - B\Vert _{H^s}^2 . \end{aligned}$$

Therefore,

$$\begin{aligned} (II) \le C(s, r) \, \theta ^s((A, \varphi ), (B, \psi ))^2 . \end{aligned}$$

To conclude, we have shown that

$$\begin{aligned} d^s(q, p)^2 \le (I) + (II) \le C(s, r) \, \theta ^s((A, \varphi ), (B, \psi ))^2 . \end{aligned}$$

\(\square \)

Remark 2.11

Recall the definition of \(d^s_*\big \vert _B\) [see (2.6)]. We have shown in particular that there exist constants such that

$$\begin{aligned} \left( \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d^s_*\big \vert _{B_k}(q, p)^2 \right) ^{\frac{1}{2}} \le C(s, r)\, d^s(q, p) \le C(s, \delta , r) \left( \sum _{k \in {{\,\mathrm{\mathbb {Z}}\,}}} d^s_*\big \vert _{B_k}(q, p)^2 \right) ^{\frac{1}{2}} \end{aligned}$$

for all \(q, p \in X^s\) with \(|q|, |p|> \delta > 0\) and \(d^s(1, q), d^s(1, p) < r\). Here the first estimate is Lemma 2.6, while the second estimate actually follows from (ii) together with the fact that we showed (i) by proving (2.5).

Let us say a few words on how Corollary 1.7 follows from Theorem 1.6.

Proof of Corollary 1.7

The Madelung transform is well-defined on equivalence classes under multiplication by \(\mathbb {S}^1\), as \(v = \varphi '\) ignores changes by a constant in the phase \(\varphi \). Note also that \(s \ge 1\), and so for any \((\rho , v) \in \mathcal {Y}^s\) we have \(v \in L^2 \subset L^1_{{{\,\textrm{loc}\,}}}\). Therefore we can define

$$\begin{aligned} \varphi (x) = \int \limits _0^x v(y) \textrm{d}y . \end{aligned}$$

Recall that \(b < \frac{4}{3}\) and \(\varepsilon < \varepsilon _0(\mu )\) [see (1.16)]. Due to (1.17) and (1.8), there exists \(\delta > 0\) such that \(|q| > \delta \) for all \(q \in X^s\) with \(E(q) < b\) or \(E^\mu (q) < \varepsilon \). With Lemma 2.7, we find some \(r = r(s, \varepsilon , b) > 0\) such that \(d^s(1, q) < r\). Then Theorem 1.6 establishes the bilipschitz estimates. \(\square \)

3 Proof of Theorem 1.9

Given that Theorem 1.6 establishes an equivalence between the relevant function spaces \((X^s, d^s)\) and \((\mathcal {Y}^s, \theta ^s)\), the proof of Theorem 1.9 is now primarily a matter of carefully carrying over the results of Theorem 1.1. This is straightforward for the existence and continuity results. Uniqueness requires a further Lemma.

Lemma 3.1

Let \(I \ni 0\) be an open time interval and \(q_0 \in L^\infty \cap \dot{H}^1\). Suppose

$$\begin{aligned} q_1, q_2 \in C(I; L^2_{{{\,\textrm{loc}\,}}}) \cap L^\infty (I; L^\infty \cap \dot{H}^1) \end{aligned}$$

are two distributional solutions to (GP) with \(q_1(0) = q_2(0) = q_0\). Then \(q_1 = q_2\).

Proof

See Appendix C. \(\square \)

This result is necessary because Theorem 1.1, in the way it is stated in [32], only yields uniqueness for the following class of solutions, which for the case \(s \ge 1\) is a priori smaller.

Definition 3.2

(Solutions to (GP) [32]) Let \(s \ge 0\). We say that \(q \in C(I; X^s)\) is a solution of the Gross–Pitaevskii equation (GP) with initial data \(q_0 \in X^s\) on the open time interval \(I \ni 0\) if there exists \(\tilde{q}: I \longrightarrow H^s_{{{\,\textrm{loc}\,}}}\) such that the following hold:

  1. (i)

    \(\tilde{q}\) solves (GP) in the sense of distributions on \(I \times {{\,\mathrm{\mathbb {R}}\,}}\).

  2. (ii)

    \(\tilde{q}\) projects to q, which means that \(\tilde{q} \mathbb {S}^1 = q\).

  3. (iii)

    We have

    $$\begin{aligned} \big [t \mapsto \tilde{q}(t) - \tilde{q}(0)\big ] \in C(I; L^2({{\,\mathrm{\mathbb {R}}\,}})) . \end{aligned}$$
  4. (iv)

    For all compact intervals \([a, b] \subset I\) and for some (and hence for all) regularized initial data \(\tilde{q}_0^*\) of \(\tilde{q}(0)\) we have

    $$\begin{aligned} \big [t \mapsto \tilde{q}(t) - \tilde{q}^*_0 \big ] \in L^4([a, b] \times {{\,\mathrm{\mathbb {R}}\,}}) . \end{aligned}$$

The uniqueness result in Theorem 1.1 for \(s \ge 1\) is therefore weaker than the one in Lemma 3.1. The proofs, however, are almost identical: in [32] uniqueness is shown by a classical argument with an energy estimate and Grönwall’s inequality. We extend this argument for \(s \ge 1\) to gain Lemma 3.1.

Remark 3.3

If \(\tilde{p} \in C(I; L^2_{{{\,\textrm{loc}\,}}}) \cap L^\infty (I; L^\infty \cap \dot{H}^1)\) is a distributional solution to (GP), as in Lemma 3.1, with initial data \(\tilde{p}(0) \mathbb {S}^1 \in X^1\), then \(\tilde{p} \mathbb {S}^1\) is also a solution in the sense of Definition 3.2. The reason is that by Theorem 1.1 there exists a solution \(q \in C(I; X^1)\) in the sense of Definition 3.2 with initial data \(q(0) = \tilde{p}(0) \mathbb {S}^1\). One can see that this has a representative \(\tilde{q} \in C(I; L^2_{{{\,\textrm{loc}\,}}}) \cap L^\infty (I; L^\infty \cap \dot{H}^1)\) which solves (GP) in distribution, so Lemma 3.1 implies \(\tilde{q} = \tilde{p}\).

Theorem 1.9 states that (hGP) is globally-in-time well-posed, meaning that there exist solutions, they are unique, and the flow map is continuous. The structure of the proof is to transfer the existence and continuity result for (GP) from Theorem 1.1 via the Madelung transform over to (hGP). This requires the absence of vacuum, which we obtain by the energy assumptions \(E < \frac{4}{3}\) or \(E^\mu < \varepsilon _0(\mu )\) [see (1.17) and (1.8)]. Uniqueness for (hGP) is similarly inferred from the uniqueness result for (GP) in Lemma 3.1.

Recall that by Lemma 2.7 the energy functionals \(E^s: X^s \longrightarrow {{\,\mathrm{\mathbb {R}}\,}}_\ge \) are continuous. Recall furthermore the definitions (1.19).

Proof of Theorem 1.9

Existence. We are given an initial data \((\rho _0, v_0) \in \mathcal {Y}^s\) which fulfills one of the bounds \(\mathcal {E}(\rho _0, v_0) < \frac{4}{3}\) or \(\mathcal {E}^\mu (\rho _0, v_0) < \varepsilon _0(\mu )\). We define \(q_0 = \mathcal {M}^{-1}(\rho _0, v_0)\) and obtain via Theorem 1.1 a solution \(q \in C_b({{\,\mathrm{\mathbb {R}}\,}}; X^s)\) of (GP) in the sense of Definition 3.2. Our solution q has a special representative \(\tilde{q} \in \mathcal {S}'({{\,\mathrm{\mathbb {R}}\,}}\times {{\,\mathrm{\mathbb {R}}\,}})\). In both cases \(E(q_0) < \frac{4}{3}\) and \(E^\mu (q_0) < \varepsilon _0(\mu )\), we obtain either (1.8) or (1.17), so there exists some \(\delta > 0\) depending on the initial data such that \(|\tilde{q}|> \delta > 0\). Now Corollary 1.7 implies \((\rho , v) = \mathcal {M}(\tilde{q}) \in C_b({{\,\mathrm{\mathbb {R}}\,}}; \mathcal {Y}^s)\).

figure c

We show that \((\rho , v)\) is a distributional solution of (hGP) in the sense of Definition 1.8. We fix a ball \(B_0 \subset {{\,\mathrm{\mathbb {R}}\,}}\) and a time interval \(J = (a, b) \subset {{\,\mathrm{\mathbb {R}}\,}}\) with \(0 \in J\). It suffices to verify that (hGP) holds in distribution, i.e. when tested against any test function \(f \in \mathcal {D}(J \times B_0)\).

On regularity. Due to Lemmas 2.5 and 2.6 for \(s \ge 1\), we know that \(\tilde{q}, \tilde{q}^{-1} \in L^\infty (J; W^{1,2}(B_0))\). From these considerations \(\partial _{xx} \tilde{q} \in L^\infty (J; W^{-1,2}(B_0))\) and \((|\tilde{q}|^2 - 1) \tilde{q} \in L^\infty (J; W^{1,2}(B_0))\) directly follow. Then \(\partial _t \tilde{q} \in L^\infty (J; W^{-1,2}(B_0))\) holds because \(\tilde{q}\) solves (GP) in the sense of distributions.

As a consequence of duality and the algebra property of \(H^1\), one obtains the product estimate \(\Vert f g\Vert _{H^{-1}} \le C \Vert f\Vert _{H^1} \Vert g\Vert _{H^{-1}} \). From this, we obtain some regularity for some of the more difficult terms appearing in the subsequent calculations, for example \(\partial _t \tilde{q} \overline{\tilde{q}},\, \partial _{xx} \tilde{q} \overline{\tilde{q}} \in L^\infty (J; W^{-1,2}(B_0))\). We now present approximation arguments that derive (hGP)1 and (hGP)2 from (GP).

Obtaining (hGP)1 from (GP). Set \(\tilde{q}_\varepsilon = \eta _\varepsilon *\tilde{q}\) for a standard mollifier \((\eta _\varepsilon )_{\varepsilon > 0}\), i.e. some \(\eta _\varepsilon (x) = \eta (\varepsilon ^{-1}(\varepsilon ^{-1} x))\) where \(\eta \in C_c^\infty ({{\,\mathrm{\mathbb {R}}\,}};{{\,\mathrm{\mathbb {R}}\,}}_\ge )\) with \(\int \eta \textrm{d}x = 1\). Note that \(|\tilde{q}| > \delta \) implies \(|\tilde{q}_\varepsilon | > \frac{\delta }{2}\) for sufficiently small \(\varepsilon > 0\) as we have sufficient regularity. We define \(\rho _\varepsilon = |\tilde{q}_\varepsilon |^2\) and \(v_\varepsilon = {{\,\textrm{Im}\,}}\big [\frac{\partial _x \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon }\big ]\). Note furthermore the identity \(\frac{\partial _x \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } = \frac{1}{2} \frac{\partial _x \rho _\varepsilon }{\rho _\varepsilon } + i v_\varepsilon \), which we use below. Equation (hGP)1 can be obtained by multiplying (GP) for \(\tilde{q}_\varepsilon \) with \(\overline{\tilde{q}}_\varepsilon \), taking the imaginary part, and then the limit:

$$\begin{aligned} 0 = {{\,\textrm{Im}\,}}\left( \overline{\tilde{q}} \, \text {(GP)} \right)&\xleftarrow {\varepsilon \rightarrow 0} {{\,\textrm{Im}\,}}\left[ i \partial _t \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } + \partial _{xx} \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } - 2 \tilde{q}_\varepsilon (|\tilde{q}_\varepsilon |^2 - 1) \overline{\tilde{q}_\varepsilon } \right] \\&={{\,\textrm{Re}\,}}\left[ \partial _t \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } \right] + \partial _x {{\,\textrm{Im}\,}}\left[ \frac{\partial _x \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } \right] - {{\,\textrm{Im}\,}}\left[ \partial _x \tilde{q}_\varepsilon \partial _x \overline{\tilde{q}_\varepsilon } \right] - {{\,\textrm{Im}\,}}\left[ 2 |\tilde{q}_\varepsilon |^2 (|\tilde{q}_\varepsilon |^2 - 1) \right] \\&= \frac{1}{2} \partial _t (|\tilde{q}_\varepsilon |^2) + \partial _x \left( |\tilde{q}_\varepsilon |^2 {{\,\textrm{Im}\,}}\left[ \frac{\partial _x \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } \right] \right) \\&=\frac{1}{2} \partial _t \rho _\varepsilon + \partial _x (\rho _\varepsilon v_\varepsilon ) \\&\xrightarrow {\varepsilon \rightarrow 0} \frac{1}{2} \partial _t \rho + \partial _x(\rho v) . \end{aligned}$$

We have to justify the limits in distribution on both sides. Observe that

$$\begin{aligned} \left| \int \limits _J \int \limits _{B_0} (\partial _t \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } - \partial _t \tilde{q} \overline{\tilde{q}}) \overline{f} \right|&\lesssim \Vert \partial _t \tilde{q}_\varepsilon - \partial _t \tilde{q}\Vert _{L^\infty (J; W^{-1,2}(B_0))} \Vert \tilde{q}_\varepsilon \Vert _{L^\infty (J; W^{1,2}(B_0))} \Vert f\Vert _{L^\infty (J; W^{1,2}(B_0))} \\&\quad +\Vert \partial _t \tilde{q}\Vert _{L^\infty (J; W^{-1,2}(B_0))} \Vert \tilde{q}_\varepsilon - \tilde{q}\Vert _{L^\infty (J; W^{1,2}(B_0))} \Vert f\Vert _{L^\infty (J; W^{1,2}(B_0))} \\&\xrightarrow {\varepsilon \rightarrow 0} 0 . \end{aligned}$$

With the same estimates, we can take the limit of the distribution \(\partial _{xx} \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon }\). The convergence of the nonlinear term follows similarly. We have shown that

$$\begin{aligned} i \partial _t \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } + \partial _{xx} \tilde{q}_\varepsilon \overline{\tilde{q}_\varepsilon } - 2 \tilde{q}_\varepsilon (|\tilde{q}_\varepsilon |^2 - 1) \overline{\tilde{q}_\varepsilon } \xrightarrow {\varepsilon \rightarrow 0} \overline{\tilde{q}} \big ( i \partial _t \tilde{q} + \partial _{xx} \tilde{q} - 2 \tilde{q} (|\tilde{q}|^2 - 1) \big ) = 0 \end{aligned}$$

in distribution on \(J \times B_0\). As

$$\begin{aligned}&\left| \int \limits _J \int \limits _{B_0} (\rho _\varepsilon v_\varepsilon - \rho v) \overline{\partial _x f} \textrm{d}x \textrm{d}t \right| \\&\quad \lesssim \big (\Vert \tilde{q}\Vert _{L^\infty (J \times B_0)} + \Vert \tilde{q}_\varepsilon \Vert _{L^\infty (J \times B_0)}\big ) \Vert \tilde{q}_\varepsilon - \tilde{q}\Vert _{L^\infty (J \times B_0)} \Vert v\Vert _{L^\infty (J; L^2(B_0))}\Vert \partial _x f\Vert _{L^2(J \times B_0)} \\&\qquad + \big (\Vert \tilde{q}\Vert _{L^\infty (J \times B_0)} + \Vert \tilde{q}_\varepsilon \Vert _{L^\infty (J \times B_0)}\big ) \Vert \tilde{q}_\varepsilon \Vert _{L^\infty (J \times B_0)} \Vert v_\varepsilon - v\Vert _{L^\infty (J; L^2(B_0))} \Vert \partial _x f\Vert _{L^2(J \times B_0)} \xrightarrow {\varepsilon \rightarrow 0} 0 , \end{aligned}$$

and we can similarly show \(\partial _t \rho _\varepsilon \xrightarrow {\varepsilon \rightarrow 0} \partial _t \rho \), we have

$$\begin{aligned} \frac{1}{2} \partial _t \rho _\varepsilon + \partial _x (\rho _\varepsilon v_\varepsilon ) \xrightarrow {\varepsilon \rightarrow 0} \frac{1}{2} \partial _t \rho + \partial _x (\rho v) \end{aligned}$$

in distribution on \(J \times B_0\).

Obtaining (hGP)\(_{2}\) from (GP). We now repeat these arguments for the second equation (hGP)2. Here we divide (GP) for \(\tilde{q}_\varepsilon \) by \(\tilde{q}_\varepsilon \), take a further derivative, the real part, and then the limit:

$$\begin{aligned} 0&= {{\,\textrm{Re}\,}}\partial _x \left( \frac{\text {(GP)}}{\tilde{q}}\right) \xleftarrow {\varepsilon \rightarrow 0} {{\,\textrm{Re}\,}}\left[ \partial _x \left( \frac{i \partial _t \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } + \frac{ \partial _{xx} \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } - 2 (|\tilde{q}_\varepsilon |^2 - 1) \right) \right] \\&= - {{\,\textrm{Im}\,}}\left[ \partial _x \partial _t (\ln \tilde{q}_\varepsilon )\right] + {{\,\textrm{Re}\,}}\left[ \partial _x \left( \partial _x \left( \frac{\partial _x \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } \right) + \left( \frac{\partial _x \tilde{q}_\varepsilon }{\tilde{q}_\varepsilon } \right) ^2 \right) \right] - 2 \partial _x (|\tilde{q}_\varepsilon |^2) \\&= - \partial _t v_\varepsilon - \partial _x(v_\varepsilon ^2) - 2 \partial _x \rho _\varepsilon + \partial _x \left( \partial _x \left( \frac{1}{2} \frac{\partial _x \rho _\varepsilon }{\rho _\varepsilon } \right) + \left( \frac{1}{2} \frac{\partial _x \rho _\varepsilon }{\rho _\varepsilon } \right) ^2 \right) \\&\xrightarrow {\varepsilon \rightarrow 0} - \partial _t v - \partial _x(v^2) - 2 \partial _x \rho + \partial _x \left( \partial _x \left( \frac{1}{2} \frac{\partial _x \rho }{\rho } \right) + \left( \frac{1}{2} \frac{\partial _x \rho }{\rho } \right) ^2 \right) . \end{aligned}$$

Of course, the limits have to be justified again. For the left-hand side, we can proceed just as before since \(\tilde{q}^{-1} \in L^\infty (J; W^{1,2}(B_0))\). On the right-hand side, the difficult terms are \(v^2\) and \(\left( \frac{1}{2} \frac{\partial _x \rho }{\rho }\right) ^2\), as here the square of a distribution in \(H^{s-1}\) is taken. The situation would be much more difficult if we did not assume \(s \ge 1\). In our case, we indeed have \(v, \frac{\partial _x \rho }{\rho } \in C(J; L^2(B_0))\), which implies that the squares are trivially defined. Furthermore,

$$\begin{aligned}&\left| \int \limits _J \int \limits _{B_0} (v_\varepsilon ^2 - v^2) \partial _x f \textrm{d}x \right| \\&\quad \le \Vert v_\varepsilon - v\Vert _{L^\infty (J; L^2(B_0))} \big (\Vert v_\varepsilon \Vert _{L^\infty (J; L^2(B_0))} + \Vert v\Vert _{L^\infty (J; L^2(B_0))}\big ) \Vert \partial _x f\Vert _{L^\infty (J \times B_0)} \xrightarrow {\varepsilon \rightarrow 0} 0 , \end{aligned}$$

and the same estimate works for \(\left( \frac{1}{2} \frac{\partial _x \rho }{\rho } \right) ^2\). The remaining terms are strictly easier to deal with.

Uniqueness. Let \(0 \in I \subset {{\,\mathrm{\mathbb {R}}\,}}\) be a bounded open interval and let \((\rho _1, v_1), (\rho _2, v_2) \in C(I; \mathcal {Y}^s)\) be two solutions to (hGP) in the sense of the theorem, both with initial data \((\rho _0, v_0) \in \mathcal {Y}^s\). In particular, they satisfy one of the energy bounds \(\mathcal {E}< b < \frac{4}{3}\) or \(\mathcal {E}^\mu< c(\mu ) \varepsilon < c(\mu ) \varepsilon _0(\mu )\) (see (1.15) and (1.16)). As before, this implies that there exists a \(\delta > 0\) so that \(\sqrt{\rho _k} > \delta \), where \(k \in \{0, 1, 2\}\). Since \(v_k \in L^2 \subset L^1_{{{\,\textrm{loc}\,}}}\), we can define \(\varphi _k(x) = \int \limits _0^x v_k(y) \textrm{d}y\) and \(\tilde{q}_k = \sqrt{\rho _k} e^{i \varphi _k}\). Note that \(\tilde{q}_k\) having uniformly bounded energy \(E^1\) implies \(\tilde{q}_k \in L^\infty (I; L^\infty \cap \dot{H}^1)\). We now fix \(j \in \{1, 2\}\). Writing \(q_j = \tilde{q}_j \mathbb {S}^1\) for the equivalence class, we know from Corollary 1.7 that \(q_j \in C(I; X^s)\).

Just as in the existence part of the proof, one can show that \((\rho _j, v_j)\) solving (hGP) implies that for the quantity

$$\begin{aligned} Q_j = i \partial _t \tilde{q}_j + \partial _{xx} \tilde{q}_j - 2 \tilde{q}_j (|\tilde{q}_j|^2 - 1) \end{aligned}$$
(3.1)

we have

$$\begin{aligned} {{\,\textrm{Im}\,}}\left[ Q_j \overline{\tilde{q}_j} \,\right] = 0 \quad \text {and} \quad \partial _x {{\,\textrm{Re}\,}}\left[ \frac{Q_j}{\tilde{q}_j}\right] = 0 \end{aligned}$$
(3.2)

in the sense of distributions. We sketch the argument that follows with a diagram.

figure d

Due to (3.2), we have in particular

$$\begin{aligned} {{\,\textrm{Im}\,}}\left[ \frac{Q_j}{\tilde{q}_j}\right] = {{\,\textrm{Im}\,}}\left[ \frac{Q_j \overline{\tilde{q}_j}}{|\tilde{q}_j|^2}\right] = \frac{{{\,\textrm{Im}\,}}[Q_j \overline{\tilde{q}_j}]}{|\tilde{q}_j|^2} = 0 , \end{aligned}$$

and hence for every \(t \in I\) there exists a \(g_j(t) \in {{\,\mathrm{\mathbb {R}}\,}}\) so that

$$\begin{aligned} g_j(t) = \frac{Q_j}{\tilde{q}_j} . \end{aligned}$$

We see that, in fact, \(\tilde{q}_j\) does not necessarily solve (GP). The reason is that for each time \(t \in I\) we had to make an arbitrary choice of a constant-in-space phase rotation, as this information is lost in the Madelung transform. This choice was the arbitrary lower limit 0 in the integral \(\varphi (t) = \int \limits _0^t v(s) \textrm{d}s\). In order to find solutions to (GP), we would now like to define

$$\begin{aligned} p_j(t) = e^{i G_j(t)} \tilde{q}_j(t) \quad \text {where} \quad G_j(t) = \int \limits _0^t g_j(s) \textrm{d}s . \end{aligned}$$

Then,

$$\begin{aligned} i \partial _t p_j + \partial _{xx} p_j - 2 p_j (|p_j|^2 - 1) = Q_j - G'(t) \tilde{q}_j = 0 . \end{aligned}$$

This argument requires \(g_j: I \longrightarrow {{\,\mathrm{\mathbb {R}}\,}}\) to be locally integrable. We show that \(g_j \in C(I; {{\,\mathrm{\mathbb {R}}\,}})\) by verifying that \(Q_j \in C(I; W^{-1,2}(B_0))\) for any ball \(B_0 \subset {{\,\mathrm{\mathbb {R}}\,}}\). With the same reasoning as in the existence part of the proof, \(\rho _j \in C(I; W^{1,2}(B_0))\) and \(v_j \in C(I; L^2(B_0))\) solving (hGP) in distribution implies \(\rho _j \in C^1(I; W^{-1,2}(B_0))\) and \(v_j \in C^1(I; W^{-1,1}(B_0))\). In particular, we have \(\partial _t \varphi _j \in C(I; L^1(B_0))\). Observe that

$$\begin{aligned} Q_j = i \frac{\partial _t \rho _j}{\rho _j} \tilde{q}_j + i (\partial _t \varphi _j) \tilde{q}_j + \partial _{xx} \tilde{q}_j - 2 \tilde{q}_j (|\tilde{q}_j|^2 - 1) . \end{aligned}$$

Verifying the products of distributions, each term can now be seen to be in \(C(I; W^{-1, 2}(B_0))\). We have shown that for any bounded interval \(I \ni 0\), both \(p_1\) and \(p_2\) are distributional solutions to (GP) with initial data \(p_1(0) = p_2(0) = \tilde{q}_0\). At the same time, \(p_j \in C(I; L^2_{{{\,\textrm{loc}\,}}}) \cap L^\infty (I; L^\infty _{t,x} \cap \dot{H}^1)\). Therefore, Lemma 3.1 implies \(p_1 = p_2\), from which \(q_1 = q_2\) in \(C(I; X^s)\) and \((\rho _1, v_1) = (\rho _2, v_2)\) follow.

Continuity. This is a direct consequence of the continuity result for (GP) from Theorem 1.1, the continuity of the energy functionals from Lemma 2.7, and the local bilipschitz equivalence from Theorem 1.6. \(\square \)