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

We establish global-in-time well-posedness of the one-dimensional hydrodynamic Gross–Pitaevskii equations in the absence of vacuum in (1+Hs)×Hs-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 + H^s) \times H^{s-1}$$\end{document} with s≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge 1$$\end{document}. 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.


Introduction
We consider in one dimension the Gross-Pitaevskii equation where q(t, x) : R × R −→ C represents an unknown wave function, subject to the boundary condition at infinity lim |x|→∞ |q(t, x)| = 1. The Gross-Pitaevskii equation has a hydrodynamic formulation (hGP) which we call the hydrodynamic Gross-Pitaevskii equations. Here ρ(t, x), v(t, x) : R × R −→ R may be understood as the unknown density and velocity of a quantum fluid. The relation between (GP) and (hGP) is given by the Madelung transform which formally transforms a solution q of (GP) into a solution (ρ, v) = M(q) of (hGP). Note that ρ and v are real-valued. One immediately sees that the Madelung transform M only makes sense when q = 0, which represents an absence of vacuum. We may recover q from its Madelung transform by the formula q = √ ρe iϕ , where ϕ is some spatial primitive of v, i.e.
One furthermore sees that the inverse Madelung transform (ρ, v) −→ q is only defined up to multiplication with S 1 , i.e. a constant rotation in phase (see (1.13) for more details). We refer the reader to [8] for a survey of the Madelung transform and the hydrodynamic Gross-Pitaevskii equations.
1.1. Overview of well-posedness results for the Gross-Pitaevskii equation. E.P. Gross [14] and L.P. Pitaevskii [20] 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 L. Erdős, B. Schlein, and H. Yau [10], 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 non-zero boundary condition, finite-energy solutions to (GP) can clearly not be in traditional function spaces that require global integrability, such as L p (R). For integers k ≥ 1 and in any dimension n ≥ 1, P.E. Zhidkov [22] established local well-posedness in the so-called Zhidkov space Z k (R n ), which is the closure of {u ∈ C k b (R n ) : ∂ x u ∈ H k−1 (R n )} under the norm This lead to a first global-in-time well-posedness result in Z 1 (R), as the Ginzburg-Landau energy is conserved. The Gross-Pitaevskii equation (GP) can be interpreted as the Hamiltonian evolutionary equation associated to this energy. The well-posedness result in Zhidkov spaces was expanded to the cases n = 2, 3 by C. Gallo [11]. Global-in-time well-posedness in the energy space {q ∈ H 1 loc (R n ) : E(q) < ∞} was obtained by P. Gérard [12,13] for n = 1, 2, 3, and for n = 4 under smallness assumptions. Later R. Killip, T. Oh, O. Pocovnicu, and M. Vis , an [15] established global-in-time well-posedness in the energy space for n = 4. We are concerned with the case n = 1. For s ∈ R, we associate with solutions of (GP) the energy functionals Note that indeed E 1 = E. Our results are consequences of a pair of papers [17,18] by H. Koch  .
Theorem 1.1 (Global-in-time well-posedness of (GP) [17,18]). Let s ≥ 0. The pair (X s , d s ) is a complete metric space, and the energy functional E s : X s −→ R is continuous. There exists a constant C 0 > 0 such that d s (1, q) ≤ C 0 E s (q) for all q ∈ 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 ∈ X s there exists a unique global-in-time solution q ∈ C(R; X s ) of (GP) (see Definition 3.2 below). For any t ≥ 0 the Gross-Pitaevskii flow map X s ∋ q 0 → q ∈ C([−t, t]; X s ) is continuous. There exists a constant C 1 (s, E s (q 0 )) such that and in the case s ≥ 1 the energy E(q(t)), defined in (1.3), is conserved.

1.2.
Functional analytic framework. Our goal is to show a novel global-in-time well-posedness result for (hGP) with (ρ, v) ∈ (1 + H s ) × H s−1 . We achieve this under the assumptions s ≥ 1 and E < 4 3 by passing the well-posedness result for (GP) in Theorem 1.1 through the Madelung transform (1.1). The first assumption s ≥ 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 ≥ 1 implies v ∈ L 2 (R), and so the problematic square of a distribution v 2 appearing in (hGP) 2 does indeed exist. The second assumption E < 4 3 can also be understood as a "regularity" assumption: Solutions below the critical energy of 4 3 can not have vacuum, that is points or intervals where |q| = √ ρ = 0. 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) = 4 3 .
The problem of dealing with the possibility of vacuum was previously overcome by P. Antonelli and P. Marcati [1], who constructed global-in-time weak solutions in n = 3 to (hGP) for initial data in L 2 . Their approach does not yield uniqueness though.
The well-posedness of the Euler-Korteweg system, a special case of the Euler equations which includes (hGP), was studied in higher dimensions by C. Audiard and B. Haspot [3,4]. Similar to the approach we take is a paper by C. Audiard [2], in which global-in-time well-posedness of (hGP) under smallness assumptions is shown in certain spaces for n ≥ 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 < 4 3 .
This is a strictly decreasing bijection whose inverse we denote bỹ This Lemma is a stronger version of [6, Lemma 1]. The proof of a slightly more general Lemma A.1 is given in the appendix. As a consequence of Lemma 1.2, the "energy gap" 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 ∈ C(R; X 1 ) of (GP) (see Definition 3.2), we have We thus consider solutions q of (GP) in X s , s ≥ 1 with energy recalling the definitions (1.6) and (1.3) of X s and E. We look for solutions (ρ, v) of (hGP) in the function space equipped with the metric We define the analogous energy Here the inverse Madelung transform is defined as where ϕ is any spatial primitive of v, i.e. ∂ x ϕ = v. Note that the energy E is indeed well-defined on equivalence classes under multiplication by S 1 , and furthermore that the space X s consists of such equivalence classes, and is hence a suitable domain for the Madelung transform 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 (Y s , θ s ). Specifically, we prove a local bilipschitz equivalence between the distance functions d s and θ s for all s > 1 2 . While our main result only holds for s ≥ 1, our approach has the potential to be extended to the case 1 2 < s < 1 if one finds a way to make sense of (hGP) 2 in such a low regularity setting. Here the absence of vacuum can still be ensured by a smallness assumption of the form where µ > 1 2 (see (1.16)). This smallness condition can also replace E < 4 3 in the case s ≥ 1, µ ≤ s. Specifically, we have the following Lemma 1.4 as a replacement for Lemma 1.2.
Then E µ 1 = 0, the function δ → E µ δ is decreasing, and there exists a constantC(µ) > 0 so that This Lemma is also a special case of Lemma A.1. By (1.7) there exists for any µ > 1 2 a constant c(µ) > 0 such that for all ε ∈ (0, 1) and any solution q ∈ C(R; X µ ) of (GP). Not attempting to obtain a sharp bound, we state the analogous of Corollary (1.3) Corollary 1.5. Let µ > 1 2 and define .
For any solution q ∈ C(R; X µ ) of (GP) (see Definition 3.2), we have Proof. We prove the contrapositive. Suppose inf (t,x)∈R 2 |q(t, x)| ≤ δ := 1 − √ ε c(µ)C(µ) and note that δ ∈ (0, 1). Using the definition of E μ δ and (1.14), this implies that for anyδ > δ there exists t ∈ R with In particular As the energies E µ still provide a lower bound for the distance of |q| to zero, we can use the smallness assumption E µ < ε 0 (µ) as a substitute for E < 4 3 . We define for µ > 1 2 the energies

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 §1.2 in the following diagram: Here the Madelung transform and its inverse are given in (1.1) and (1.13) respectively. Recall also the explicit forms of the energies E and E in the most important case s = 1: 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 θ s . We require s > 1 2 to use L ∞ embeddings and certain product estimates. Theorem 1.6 (Local bilipschitz equivalence of d s and θ s ). Let s > 1 2 and r, δ > 0. Consider measurable functions ρ, η, ϕ, ψ : R −→ R so that q, p ∈ S ′ (R) ∩ H s loc (R) and |q|, |p| > δ, where q = √ ρe iϕ and p = √ ηe iψ . There exists a constant C = C(s, δ, r) > 0 so that the following hold: (ii) If θ s ((1, 0), (ρ, ∂ x ϕ)), θ s ((1, 0), (η, ∂ x ψ)) < r, then d s (q, p) ≤ C θ s ((ρ, ∂ x ϕ), (η, ∂ x ψ)) .
Our second main result is the global-in-time well-posedness of the hydrodynamic Gross-Pitaevskii equations.
is continuous. For all 1 2 < µ < 1 there exist constants c(µ), ε 0 (µ) > 0, defined in (1.15) and (1. 16), so that if we replace the assumption E(ρ 0 , v 0 ) < 4 3 by E µ (ρ 0 , v 0 ) < ε < ε 0 (µ), then the above statement holds with (1.20) replaced by E µ (ρ(t), v(t)) < c(µ) ε. Remark 1.10. Previously, P.E. Zhidkov [21, Theorem III.3.1] studied the stability of solutions in the Zhidkov space Z 1 (R) (see (1.2)) near space-homogeneous solutions Φ, such as the constant solution Φ = 1, with respect to the distance θ 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 [21, Cor. III.3.5] he proved furthermore that for any ball B ⊂ R, if the initial θ 1 -distance between the perturbed and the space-homogeneous solution is small, then for all times also the distance inf is small. This can be interpreted as a weaker form of the estimate d 1 θ 1 we derive (see Lemma 2.6 and Remark 2.11 below). Remark 1.11. As both Theorem 1.1 and Theorem 1.6 work for all s > 1 2 , it may be possible to extend Theorem 1.9 to the case 1 2 < s < 1. The problem is that for v ∈ H s−1 ⊆ L 2 the product of distributions v 2 = v · v is not necessarily defined. Nevertheless, it may be possible to find global distributional solutions. For example, in the paper [16] by R. Killip

and M. Vis
, an global-in-time well-posedness of the KdV equation in H −1 is first shown in the sense that the solution map R ×S −→ S extends to a continuous mapping R ×H −1 −→ H −1 , and some other conditions are fulfilled. In our case, it is similarly true that the solution map has a unique continuous extension to a map This extension is given by the conjugation of the corresponding solution map for (GP) at regularity s with the Madelung transform. R. Killip

and M. Vis
, an then furthermore show a local smoothing result, which implies that the solution map produces functions in L 2 loc,t,x . As a result, (KdV) 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 §2 we prove Theorem 1.6, the local bilipschitz equivalence of (X s , d s ) and (Y s , θ s ). In §3 we prove Theorem 1.9, the global-in-time well-posedness of the hydrodynamic Gross-Pitaevskii equations.
Acknowledgements. I would like to thank Sarah Hofbauer for her help with reviewing the literature. I am especially thankful to my supervisor Xian Liao for proposing this problem and strategy, and for her patience during many hours of discussion.
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 258734477 -SFB 1173.
2. Local bilipschitz equivalence of (X s , d s ) and (Y s , θ s ) The goal of this section is to prove Theorem 1.6. In the §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 §2.2 and §2.3.

2.1.
Notations and preliminaries. We use the notation R ≥ = {r ∈ R : r ≥ 0}. We write C or C(...) for various constants with possible dependence on other quantities. These may change from one line to the next. We denote by D ′ = D ′ (R) = D ′ (R; C) the space of distributions and by S ′ = S ′ (R) = S ′ (R; 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 (R; C).
Let s ∈ R. We write f for the Fourier transform of a tempered distribution f ∈ S ′ . We define the Sobolev space Here ξ = 1 + |ξ| 2 . We also define the quasinorm of the homogeneous Sobolev space (see [19, p. 77 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 The third inequality in the sequence follows trivially. We show the second inequality for any

2.1.2.
Estimates in H s . The following Lemma is a crucial estimate. Such kinds of product estimates are well-known in the literature, see for example [9, Proposition 2.7].
Proof. See Appendix B.
In this section we often write f ′ for the spacial derivative ∂ x f . Recall the definitions (1.19). For notational convenience, we sometimes prefer to use the variables 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 [5].
Proof. We apply Lemma 2.3 with F (u) = u 2 and obtain 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, δ, R > 0. There exists C(s, δ, R) > 0 such that for any ball B 0 ⊂ R of radius R and all u ∈ W s,2 (B 0 ) with |u| > δ > 0 we have Proof. This follows by applying Lemma 2.3 with any function F ∈ C ∞ (R; R) so that F (0) = F ′ (0) = 0 and F (x) = x −1 for |x| > δ 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.

2.2.
Proof of Theorem 1.6 (i). Recall the definitions (1.19), in particular q = √ ρe iϕ and p = √ ηe iψ , as well as A = √ ρ and B = √ η. We assume s > 1 2 , d s (1, q), d s (1, p) < r and |q|, |p| > δ > 0. We have to prove that We do this by showing an estimate of the form Let us elaborate on the quantity in the middle before we start the proof. Given a ball B ⊂ R, we define for convenience the following notations: Lemma 2.6. Let s > 1 2 and let B 0 = {x ∈ R : |x| < R} be an open ball of radius R > 0 with center 0. There exists C(s, R) > 0 so that for all q, p ∈ S ′ ∩ H s loc . As a consequence, for families of balls B k = B 0 + kR with k ∈ Z we have Proof. As {y − x : x, y ∈ B} ⊆ {x ∈ R : |x| < 2R}, there exists a finite constant C(s, R) > 0 such that sup y∈B sech(y − ·) −1 2 W s,2 (B 0 ) ≤ C(s, R). The first estimate follows: Using this and Lemma 2.1 (iv), we obtain the second estimate: Proof of Theorem 1.6 (i). Let B 0 = {x ∈ R : |x| < 1} and observe that We can estimate 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 It remains to estimate ϕ ′ − ψ ′ H s−1 . Applying Lemma 2.2 yields It therefore suffices to derive the estimate for the quantity (e i(ϕ−ψ) ) ′ 2 H s−1 . Observe with Lemma 2.1 (iv) that We now carefully introduce the amplitudes: As A, B > δ > 0, we can apply Lemma 2.5. Together with (2.10) we obtain and similarly B ±1 W s,2 (B k ) , q ±1 W s,2 (B k ) , p ±1 W s,2 (B k ) ≤ C(s, δ, r). We conclude again by reducing the situation to an application of Lemma 2.6 and the previously shown estimate (2.10): ≤ C(s, δ, r) d s (q, p) 2 .
As mentioned above, due to Lemma 2.4 it suffices to prove this with ρ, η replaced by A = √ ρ and B = √ η. Recall the definitions (1.19). We definẽ where we have replaced the sech in the definition of d s with √ sech. Some of the hard work for this direction has already been done in the proof of the following Lemma 2.7. This was proven for d s in [17, Lemma 6.1], but the proof is identical ford s as √ sech is positive and still has sufficiently fast decay. . For all s ≥ 0 the energy E s : X s −→ R ≥ is continuous with respect to d s , and there exists C(s) > 0 so that for all q ∈ 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 [17] correspond to √ 2E s in our notation.
We first prove two Lemmas.
Lemma 2.9. Let s > 1 2 . There exists a constant C(s) > 0 so that for all ϕ ∈ S ′ ∩ H s loc we have Proof. We assume ϕ ′ H s−1 = 0. By Lemma 2.2 there exists a constant C(s) so that . This has the scaling estimates (2.12) Then we can rewrite the above as Combining this with (2.11) yields We conclude with the scaling estimates (2.12) that Proof. For the amplitudinal part of the energy, we know from Lemma 2.4 that For the remainder, we use Lemma 2.2: We now conclude by estimating both appearances of (e iϕ ) ′ H s−1 with Lemma 2.9.
Proof of Theorem 1.6 (ii). We split the distance d s (q, p) into two parts: It follows that Note that We can deal with the second term as before. For the first one, we use Lemma 2.1 (iv) and Young's convolution inequality: Remark 2.11. Recall the definition of d s * B (see (2.6)). We have shown in particular that there exist constants such that for all q, p ∈ X s with |q|, |p| > δ > 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 S 1 , as v = ϕ ′ ignores changes by a constant in the phase ϕ. Note also that s ≥ 1, and so for any (ρ, v) ∈ Y s we have v ∈ L 2 ⊂ L 1 loc . Therefore we can define Recall that b < 4 3 and ε < ε 0 (µ) (see (1.16)). Due to (1.17) and (1.8), there exists δ > 0 such that |q| > δ for all q ∈ X s with E(q) < b or E µ (q) < ε. With Lemma 2.7 we find some r = r(s, ε, b) > 0 such that d s (1, q) < r. Then Theorem 1.6 establishes the bilipschitz estimates.

Proof of Theorem 1.9
Given that Theorem 1.6 establishes an equivalence between the relevant function spaces (X s , d s ) and (Y s , θ 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. This result is necessary because Theorem 1.1, in the way it is stated in [17], only yields uniqueness for the following class of solutions, which a priori is smaller. (ii)q projects to q, which means thatqS 1 = q.
(iii) We have t →q(t) −q(0) ∈ C(I; L 2 (R)) . (iv) For all compact intervals [a, b] ⊂ I and for some (and hence for all) regularized initial datã . The uniqueness result in Theorem 1.1 for s ≥ 1 is therefore weaker than the one in Lemma 3.1. The proofs, however, are almost identical: In [17] uniqueness is shown by a classical argument with an energy estimate and Grönwall's inequality. We extend this argument for s ≥ 1 to gain Lemma 3.1.
Remark 3.3. Ifp ∈ C(I; L 2 loc ) ∩ L ∞ (I; L ∞ ∩Ḣ 1 ) is a distributional solution to (GP), as in Lemma 3.1, with initial datap(0)S 1 ∈ X 1 , thenpS 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 ∈ C(I; X 1 ) in the sense of Definition 3.2 with initial data q(0) =p(0)S 1 . One can see that this has a representativẽ q ∈ C(I; L 2 loc ) ∩ L ∞ (I; L ∞ ∩Ḣ 1 ) which solves (GP) in distribution, so Lemma 3.1 impliesq =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 < 4 3 or E µ < ε 0 (µ) (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 −→ R ≥ are continuous. Recall furthermore the definitions (1.19).
We show that (ρ, v) is a distributional solution of (hGP) in the sense of Definition 1.8. We fix a ball B 0 ⊂ R and a time interval J = (a, b) ⊂ R with 0 ∈ J. It suffices to verify that (hGP) holds in distribution, i.e. when tested against any test function f ∈ D(J × B 0 ).
As a consequence of duality and the algebra property of H 1 , one obtains the product estimate f g H −1 ≤ C f H 1 g H −1 . From this we obtain some regularity for some of the more difficult terms appearing in the subsequent calculations, for example ∂ tqq , ∂ xxqq ∈ L ∞ (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). Setq ε = η ε * q for a standard mollifier (η ε ) ε>0 , i.e. some 2 for sufficiently small ε > 0 as we have sufficient regularity. We define ρ ε = |q ε | 2 and v ε = Im ∂xqε qε . Note furthermore the identity ∂xqε qε = 1 2 ∂xρε ρε + iv ε , which we use below. Equation (hGP) 1 can be obtained by multiplying (GP) forq ε withq ε , taking the imaginary part, and then the limit: We have to justify the limits in distribution on both sides. Observe that With the same estimates, we can take the limit of the distribution ∂ xxqεqε . The convergence of the nonlinear term follows similarly. We have shown that and we can similarly Obtaining (hGP) 2 from (GP). We now repeat these arguments for the second equation (hGP) 2 . Here we divide (GP) forq ε byq ε , take a further derivative, the real part, and then the limit: Of course, the limits have to be justified again. For the left-hand side, we can proceed just as before sinceq −1 ∈ L ∞ (J; W 1,2 (B 0 )). On the right hand side the difficult terms are v 2 and 1 2 ∂xρ ρ 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 ≥ 1. In our case we indeed have v, ∂xρ ρ ∈ C(J; L 2 (B 0 )), which implies that the squares are trivially defined. Furthermore Uniqueness. Let 0 ∈ I ⊂ R be a bounded open interval and let (ρ 1 , v 1 ), (ρ 2 , v 2 ) ∈ C(I; Y s ) be two solutions to (hGP) in the sense of the theorem, both with initial data (ρ 0 , v 0 ) ∈ Y s . In particular, they satisfy one of the energy bounds E < b < 4 3 or E µ < c(µ)ε < c(µ)ε 0 (µ) (see (1.15) and (1.16)). As before, this implies that there exists a δ > 0 so that √ ρ k > δ, where k ∈ {0, 1, 2}.
Since v k ∈ L 2 ⊂ L 1 loc , we can define ϕ k (x) =´x 0 v k (y) dy andq k = √ ρ k e iϕ k . Note thatq k having uniformly bounded energy E 1 impliesq k ∈ L ∞ (I; L ∞ ∩Ḣ 1 ). We now fix j ∈ {1, 2}. Writing q j =q j S 1 for the equivalence class, we know from Corollary 1.7 that q j ∈ C(I; X s ).
Just as in the existence part of the proof, one can show that (ρ j , v j ) solving (hGP) implies that for the quantity in the sense of distributions. We sketch the argument that follows with a diagram.
Due to (3.2) we have in particular and hence for every t ∈ I there exists a g j (t) ∈ R so that We see that, in fact,q j does not necessarily solve (GP). The reason is that for each time t ∈ 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 ϕ(t) =´t 0 v(s) ds. In order to find solutions to (GP), we would now like to define Then This argument requires g j : I −→ R to be locally integrable. We show that g j ∈ C(I; R) by verifying that Q j ∈ C(I; W −1,2 (B 0 )) for any ball B 0 ⊂ R. With the same reasoning as in the existence part of the proof, ρ j ∈ C(I; W 1,2 (B 0 )) and v j ∈ C(I; L 2 (B 0 )) solving (hGP) in distribution implies ρ j ∈ C 1 (I; W −1,2 (B 0 )) and v j ∈ C 1 (I; W −1,1 (B 0 )). In particular we have ∂ t ϕ j ∈ C(I; L 1 (B 0 )). Observe that 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 ∋ 0, both p 1 and p 2 are distributional solutions to (GP) with initial data p 1 (0) = p 2 (0) =q 0 . At the same time p j ∈ C(I; L 2 loc ) ∩ L ∞ (I; L ∞ t,x ∩Ḣ 1 ). Therefore Lemma 3.1 implies p 1 = p 2 , from which q 1 = q 2 in C(I; X s ) and (ρ 1 , v 1 ) = (ρ 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.
We have There exists a strictly decreasing inverse functionδ : [0, 4 3 Proof. We see that E s 1 = 0 by choosing q = 1. Clearly the set over which the infimum is taken increases with δ, and hence the infimum is decreasing. Recall that Lemma 2.6 implies where B k = B 0 + k, k ∈ Z are balls of radius 1. Estimating with Lemma 2.7 on the right and the Sobolev embedding W 1,2 (B k ) ֒−→ L ∞ on the left, we obtain for every q ∈ H s loc with inf x∈R |q(x)| ≤ δ. This proves (A.1). Now we assume s = 1. We first rewrite the problem as E δ = inf ν∈[0,δ]Ẽν with Of course we expect thatẼ ν is decreasing in ν and hence E δ =Ẽ δ . This will be verified once we have calculatedẼ ν . Using invariance under translations, phase rotations, and mirror symmetry, we can equivalently consider the minimization problem We now follow the same arguments as in [6, Lemma 1] to find a minimizer. Consider a minimizing sequence (q n ) n∈N . As E s (q n ) is uniformly bounded, so is ∂ x q n L 2 (R ≥ ) . The Banach-Alaoglu theorem then implies, up to a subsequence, that ∂ x q n −→ p ′ ν for some p ′ ν ∈ L 2 (R ≥ ). Furthermore as q n (0) = ν is fixed, we have a Poincare inequality q n W 1,2 (B 0 ) ≤ C(s, B) ∂ x q n L 2 (B 0 ) on any finite interval B 0 ⊂ R ≥ . Then we can use compactness of the Sobolev embedding H 1 ֒−→ L ∞ to find, up to a subsequence, that q n −→ p ν in L ∞ loc (R ≥ ) for some p ν ∈ H 1 loc (R ≥ ), with p ′ ν indeed being its distributional derivative. Now we can conclude with Fatou's lemma that p ν is a minimizer forẼ ν : For the case ν = 0, we obtain the Euler-Lagrange equation Then as p 0 (0) = 0 and E(p 0 ) < ∞, [6, Theorem 1] implies that p 0 = tanh is the unique solution. Consequently, it must be the case that for a > 0 the function p ν (x) = tanh(x + a) is a minimizer for the problem with ν = tanh(a), as otherwise one could modify p 0 on [r, ∞) to find an admissible function with strictly smaller energy for the minimization problem ofẼ 0 . This implies that the q δ defined in (A.2) are minimizers forẼ ν . With a = tanh −1 (ν), and noting the identities we compute Evaluating the integral yields where we define In the Littlewood-Paley setting it is easy to define the Besov spaces B s p,q for 1 ≤ p, r ≤ ∞, Proof of Lemma 2.2. We only prove (2.2) as the proof of (2.1) is analogous and strictly simpler.
We estimate x . We have shown that there exists some C > 0, depending on q 1 , q 2 but independent of time, such that 1 2 In particular x . Now Grönwall's inequality implies for any fixed t > 0 that hence q 1 = q 2 for positive times. The argument for negative times is analogous.