Dynamics of nearly parallel vortex filaments for the Gross–Pitaevskii equation

Abstract

Klein et al. (J Fluid Mech 288:201–248, 1995) have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross–Pitaevskii equation.

1 Introduction

The mathematical analysis of the evolution of vortex filaments within the framework of the classical equations for fluids is a challenging problem that dates back to the second half of the nineteenth century with the works of Kelvin and Helmholtz. Some “simplified” flows have long been considered as potential candidates for the description of the asymptotic regime of small vortex cores, the most well-known being the binormal curvature flow of Da Rios over a century ago, but the convergence proofs in all these cases are missing, and the validity of the convergence is sometimes questioned too in the literature.

Klein et al. [20] have proposed the system

$$\begin{aligned} \partial _t X_j = J \alpha _j \Gamma _j \partial _{zz} X_j + J\sum _{k \ne j} 2 \Gamma _k \frac{X_j - X_k}{|X_j - X_k|^2}, \qquad j=1,\ldots ,n \end{aligned}$$

(1)

as a simplified candidate model for the evolution of n nearly parallel vortex filaments in perfect incompressible fluids. This model extends a remark by Zakharov [26] for pairs of anti-parallel filaments, and is expected to be valid only when

(i):

the wavelength of the filaments perturbations are large with respect to the filaments mutual distances,

(ii):

the latter are large with respect to the size of the filaments cores, and

(iii):

the Reynolds number is sufficiently large.

In the above formulation, the filaments are assumed to be nearly parallel to the z-axis, and after rescaling¹ each of them is described by a function $z \mapsto (X_j(z,t),z)$, where $X_j(\cdot ,t)$ takes values in ${\mathbb {R}}^2$, which represents the horizontal displacement of the filament. The canonical two by two symplectic matrix is denoted by J, the constants $\Gamma _j \in {\mathbb {R}}$ are the circulations associated to each vortex filament, and the constants $\alpha _j \in {\mathbb {R}}$ are derived from assumptions on the vortex core profiles prior to passing in the limit.

From the fluid mechanics point of view, the case $n=1$ in (1) is already highly interesting and corresponds to a single weakly curved vortex filament. In that case, system (1) reduces to the free Schrödinger equation in one variable, and as a matter of fact this is also the linearized equation for the binormal curvature flow around a straight filament.

From a mathematical point of view, system (1) has been studied for his own (see e.g. [1, 2, 19, 21]) when $n>1$, in particular its well-posedness and the possibility of colliding filaments under (1). Nevertheless, as mentioned already, the justification of the model itself as a limit from a classical fluid mechanics model (such as the Euler equation or the Navier–Stokes equation in a vanishing viscosity limit) has so far only been obtained formally through matched asymptotic, even for $n=1$.

The goal the present work is to rigorously derive system (1), for arbitrary $n \ge 1$, as a limit from (yet another) PDE model whose relation to fluid mechanics is not new. In that framework, all the limiting circulations $\Gamma _j$ will end up being equal. Our object of study in this paper is indeed the Gross–Pitaevskii equation

$$\begin{aligned} i \partial _tu_{\varepsilon }- \Delta u_{\varepsilon }+ \frac{1}{\varepsilon ^2}(|u_{\varepsilon }|^2-1)u_{\varepsilon }=0 \qquad \text{ in } (0,T)\times \Omega , \end{aligned}$$

(2)

with initial data $u_{\varepsilon }(\cdot , 0) = u_{\varepsilon }^0(\cdot ).$ Here $0<\varepsilon \ll 1$ is a real parameter, $\Omega = \omega \times {\mathbb {T}}_{L}$ where $\omega \subset {\mathbb {R}}^2$ is a bounded open set with smooth boundary² and ${\mathbb {T}}_{L}= {\mathbb {R}}/\mathrm{L}{\mathbb {Z}}$ for some $L >0.$ Without loss of generality, we shall assume that $0 \in \omega .$ We also consider Neumann boundary conditions on $\partial \omega \times {\mathbb {T}}_{L}$:

$$\begin{aligned} \nu \cdot \nabla u_\varepsilon = 0 \text{ on } \partial \omega \times {\mathbb {T}}_{L}. \end{aligned}$$

Our main result will describe solutions of (2) associated to initial data $u_{\varepsilon }^0$ for vanishing families of $\varepsilon$, and corresponding in a sense to be described in detail below to n nearly parallel vortex filaments clustered around the vertical axis $\{0\}\times (0,L)$.

1.1 Statement of main result

We consider the system

$$\begin{aligned} i \partial _t f_j - \partial _{zz} f_j - 2\sum _{k\ne j} \frac{f_j-f_k}{|f_j-f_k|^2} = 0, \qquad j=1,\ldots , n \end{aligned}$$

(3)

for $f \equiv (f_1,\ldots , f_n)\, : {\mathbb {T}}_{L}\times {\mathbb {R}}\rightarrow {\mathbb {C}}^n$. This is the Klein Majda and Damodaran system (1) in the special case where all constants are equal and normalized to unity.

For $f\in H^1({\mathbb {T}}_{L},{\mathbb {C}}^n)$, we define

$$\begin{aligned} G_0(f) :=\pi \int _0^L \left( \frac{1}{2} \sum _{i=1}^n| f_i'|^2 - \sum _{i\ne j}\log |f_i-f_j| \right) dz, \end{aligned}$$

it is the Hamiltonian associated to the Eq. (3). We also set

$$\begin{aligned} \rho _f := \inf _{z\in (0,L), \ j\ne k}|f_j(z)-f_k(z)|. \end{aligned}$$

A sufficient condition for the Hamiltonian $G_0(f)$ to be finite is that $\rho _f > 0.$ For $f^0 \in H^1({\mathbb {T}}_{L},{\mathbb {C}}^n)$ such that $\rho _{f^0} > 0,$ system (3) possesses a unique solution $f \in \mathcal {C}((-T, T), H^1({\mathbb {T}}_{L},{\mathbb {C}}^n))$ for some $T>0$, and which satisfies $\rho _{f(\cdot ,t)} > 0$ for all $t \in (-T, T).$ Moreover, f can be approximated by (arbitrarily) smooth solutions of (3). If $\liminf _{t \rightarrow \pm T} \rho _{f(\cdot , t)} = 0,$ corresponding to a collision between filaments, the possibility to extend the solution past $\pm T$ is a delicate question, a situation which we won’t consider in this work.

Regarding the Ginzburg–Landau energy, we write points in $\Omega$ in the form $(x,z)\in \omega \times {\mathbb {T}}_{L}$, and define

$$\begin{aligned} e_\varepsilon (u) := \frac{1}{2}\left( |\nabla _xu|^2 + {|\partial _zu|^2} \right) + \frac{1}{4\varepsilon ^2}(|u|^2-1)^2 \ , \end{aligned}$$

and

$$\begin{aligned} G_\varepsilon (u):=\int _\Omega e_\varepsilon (u)\,dx\,dz - L \kappa (n, \varepsilon , \omega ) \end{aligned}$$

(4)

where $\kappa (n, \varepsilon ,\omega ) = n\pi {|\log {\varepsilon }|}+ n(n-1)\pi |\log h_\varepsilon | + O(1)$ is defined more precisely in (9) below. The Cauchy problem for the Gross–Pitaevskii equation is globally well posed for initial data with finite Ginzburg–Landau energy (i.e. in $H^1(\Omega )$ here), and solutions can be approximated by smooth ones too.

The quantity which will define and locate the vorticity of a solution $u_\varepsilon$ is the (horizontal³) Jacobian

$$\begin{aligned} Ju_\varepsilon := \nabla _x^\perp \cdot {\mathrm{Re} (u_\varepsilon \nabla _x \overline{u}_\varepsilon )}, \end{aligned}$$

it is therefore a real function of (x, z, t).

In order to measure the discrepancy between vorticity and an indefinitely thin filament, we will integrate in z some norms on the slices $\omega \times \{z\}.$ For $\mu \in W^{-1,1}(\omega )$ we let

$$\begin{aligned} \Vert \mu \Vert _{W^{-1,1}(\omega )} := \sup \left\{ \int \phi \,d\mu \ : \phi \in W^{1,\infty }_0(\omega ), \ \max \{ \Vert \phi \Vert _\infty , \Vert D\phi \Vert _\infty \} \le 1 \right\} . \end{aligned}$$

Among the various equivalent norms that induce the $W^{-1,1}(\omega )$ topology, this choice has the property that there exists $r(\omega )>0$ such that if $a_1,\ldots , a_n$ and $b_1,\ldots b_n$ are points in $B_r \subset \omega$, then

$$\begin{aligned} \Vert \sum _{i=1}^n \delta _{a_i} - \sum _{i=1}^n \delta _{b_i}\Vert _{W^{-1,1}(\omega )} = \min _{\sigma \in S_n} \sum _{i=1}^n |a_i - b_{\sigma (i)}| \end{aligned}$$

(5)

where $S_n$ denotes the group of permutations on n elements, see [4]. Indeed, this property holds whenever $r(\omega ) \le \min \{ \frac{1}{2} \text{ dist }(0,\partial \omega ),1 \}$, as then any 1-Lipschitz function on $B_r$ that equals zero at the origin can be extended to a function $\phi$ such that $\phi = 0$ on $\partial \omega$ and $\max \{ \Vert \phi \Vert _\infty , \Vert D\phi \Vert _\infty \} \le 1$.

Finally, we introduce the scale

$$\begin{aligned} h_\varepsilon := \frac{1}{\sqrt{{|\log {\varepsilon }|}}}. \end{aligned}$$

It will correspond to the amount of deformation of the filaments with respect to perfectly straight ones, and is also the typical separation distance between distinct filaments. At the same time, the scale $\varepsilon$ corresponds to the typical core size of the filaments, and therefore since $h_\varepsilon \gg \varepsilon$ as $\varepsilon \rightarrow 0,$ the displacements and mutual distances of filaments are much larger in this asymptotic regime than their core size.

Our main result is

Theorem 1

Let $f = (f_1,\ldots , f_n) \in \mathcal {C}((-T,T),H^1({\mathbb {T}}_{L},{\mathbb {C}}^n))$ be solution of the vortex filament system (3) with initial data $f^0$ and such that $\rho _{f(t)} \ge \rho _0>0$ for all $t\in (-T,T)$.

For $\varepsilon \in (0,1]$, let $u_\varepsilon$ solve the Gross–Pitaevskii equation (2) for initial data such that

$$\begin{aligned} \int _0^L \Big \Vert J_xu_\varepsilon ^0(\cdot ,z) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j^0(z)} \Big \Vert _{W^{-1,1}(\omega )} dz \ = \ o(h_\varepsilon ) \end{aligned}$$

(6)

and

$$\begin{aligned} G_\varepsilon (u_\varepsilon ^0)\rightarrow G_0(f^0) \end{aligned}$$

(7)

as $\varepsilon \rightarrow 0$. Then for every $t\in (-T,T)$,

$$\begin{aligned} \int _0^L \Big \Vert J_xu_\varepsilon (\cdot ,z, h_\varepsilon ^2 t) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j(z,t)} \Big \Vert _{W^{-1,1}(\omega )} dz \ = \ o(h_\varepsilon ), \end{aligned}$$

(8)

as $\varepsilon \rightarrow 0$.

Comments. The positivity of $\rho _0$ in Theorem 1 is essential, it implies that no collision between filaments occured over time, and the corresponding conclusion would very likely be incorrect without assuming it. Indeed, filaments collisions in superfluids experiments was observed to lead to highly complex reconnection dynamics, see for example [11], which exit the case of graph-like filaments considered here. Assumption (6) is responsible for the concentration of the initial vorticity of $u_\varepsilon$ around the filaments parametrized by (rescalings) of $f^0$. Assumption (7) can be understood as requiring that the former concentration holds in the most energy efficient way (at least asymptotically as $\varepsilon \rightarrow 0$); this follows from results in [7], building on earlier work of [10]. Below we will recall these results in detail and refine some of them. The conclusion (8) implies that the concentration of vorticity is preserved in time, and its location follows (after appropriate rescalings) the model of Klein Majda and Damodaran.

The periodicity assumption which we make on the vertical variable is probably only technical, but at the level of the Gross–Pitaevskii equation the framework needed to deal with local perturbations of straight filaments would involve some further renormalization process of the (otherwise infinite) energy. Periodic perturbations of the limit system (1) have been studied in particular in [8].

In the context of the 3D Gross–Pitaevskii equation, there are very few available mathematical results which rigorously derive a motion law for vortex filaments. Besides Theorem 1, the only one we are aware of which does not require a symmetry assumption reducing the actual problem to 2D is [15], where the case of a single vortex ring was treated (the limiting filament is symmetric but the field $u_\varepsilon$ is not assumed to be so). The situation is slightly better understood in the axisymmetric setting, in particular the case of a finite number of vortex rings was analized in [16], where the so-called leapfrogging phenomenon was established. In 2D the situation is of course brighter, and since vortex filaments are for the most part tensored versions of 2D vortex points, it is not surprising that the analysis of the latter is at the basis of all the 3D works we were referring to so far.

Vortex points and approximations of in 2D evolve according to the so-called point vortex system. That was established in [6] in the context of the Gross–Pitaevskii equation, but parallel results were also obtained (and actually earlier) in the framework of the incompressible 2D Euler equation [22, 23].

The analogy between Euler and Gross–Pitaevskii equations is expected to be valid not only in 2D, and as stated at the beginning of this introduction a common open challenge in both frameworks is to rigorously derive the binormal curvature flow equation for general vortex filament shapes. In this context, we emphasize the $n=1$ case of Theorem 1 establishes a linearized version of this so-called self-induction approximation for (2); the general case of the theorem describes evolution governed by a combination of the linearized self-induction of filaments and interaction with other filaments.

Contrary to the Euler equation, the Gross–Pitaevskii equation has a fixed “core length” $\varepsilon$ in its very definition: this simplifies some of the analysis and may explain why in particular the equivalent of the nonlinear 3D stability for one vortex ring or the leapfrogging phenomenon have not yet been proved in that context.⁴ On the other hand, there is no equivalent of the Biot–Savart law in the context of the Gross–Pitaveskii equation, the field is complex and the analysis often involves tricky controls of the phases. Partial results in the context of Euler in 3D include [12, 13] for the 3D spectral stability of a columnar vortex, [5] for the evolution of a finite number of axisymmetric vortex rings in a regime where they do not interact, and [9] for the existence of travelling helices.

Theorem 1 does not cover the case of anti-parallel vortex filaments, a situation which in (1) would correspond to constants $\Gamma _j \in \pm 1$ that do not all share the same sign. This is something that we wish to consider in the future.

In the remaining subsections of this introduction, after fixing a number of notations which we use throughout, we describe in details the strategy followed to prove Theorem 1 and we state the key intermediate lemmas and propositions. The proofs of the latter are presented latter in Sect. 2, for the key arguments related to the dynamics, in Sect. 3, for the results which do not depend on a time variable and which are for the most part extensions or variations of results in [7], and in Sect. 4, for those related to a priori compactness in time.

1.2 Further notations

In addition to the scale $h_\varepsilon := {|\log {\varepsilon }|}^{-1/2}$, we will always write $\omega _\varepsilon := h_\varepsilon ^{-1}\omega$ and $\Omega _\varepsilon := \omega _\varepsilon \times {\mathbb {T}}_{L}$ to denote the rescaled versions of $\omega$ and $\Omega$ respectively. Given $u_\varepsilon \in H^1(\Omega ,{\mathbb {C}})$ we will always let $v_\varepsilon$ denote the function in $H^1(\Omega _\varepsilon ,{\mathbb {C}})$ defined by

$$\begin{aligned} v_\varepsilon (x,z) = u_\varepsilon (h_\varepsilon x, z), \qquad (x,z)\in \Omega _\varepsilon . \end{aligned}$$

We will write

$$\begin{aligned} jv_\varepsilon := iv_\varepsilon \cdot \nabla _x v_\varepsilon , \end{aligned}$$

where here and throughout, a dot product of complex numbers denotes the real inner product:

$$\begin{aligned} \text{ for } v,w\in {\mathbb {C}}, \qquad v\cdot w = \text{ Re }( v\bar{w}) . \end{aligned}$$

Observe once more that $j v_\varepsilon$ contains only the horizontal components of the momentum vector $iv_\varepsilon \cdot D v_\varepsilon = (iv_\varepsilon \cdot \nabla _x v_\varepsilon ,iv_\varepsilon \cdot \partial _z v_\varepsilon )$.

In many places, we implicitly identify ${\mathbb {C}}^n$ with $({\mathbb {R}}^2)^n$ when no complex products are involded. We fix $\chi \in C^\infty ({\mathbb {R}})$ to be a nonnegative nonincreasing function such that

$$\begin{aligned} \chi (s) = 1 \text{ if } s<1, \qquad \chi (s)=0 \text{ if } s\ge 2, \end{aligned}$$

and for arbitrary $r>0$ we set $\chi _r(s):= \chi (s/r).$ For $f\in H^1((0,L), ({\mathbb {R}}^2)^n)$ such that $\rho _f>0$, and for $0<r<\rho _f/4,$ we also set

$$\begin{aligned} \begin{aligned} \chi _{r}^f (x,z)&:= \sum _{i=1}^n \chi _r( |x-f_i(z)| ) \, |x-f_i(z)|^2. \\ \chi _{r,\varepsilon }^f (x,z)&:= \frac{1}{h_\varepsilon ^2} \chi ^{h_\varepsilon f}_{h_\varepsilon r}(x,z) = \sum _{i=1}^n \chi _r(\frac{|x - h_\varepsilon f_i(z)|}{h_\varepsilon } ) \, \left| \frac{x -h_\varepsilon f_i(z)}{h_\varepsilon }\right| ^2. \end{aligned} \end{aligned}$$

Repeated indices $a,b,c,\ldots$ are implicitly summed from 1 to 2; these correspond to the horizontal x variables. We will also write $\varepsilon _{ab}$ to denote the usual antisymmetric symbol, with components

$$\begin{aligned} \varepsilon _{12}=-\varepsilon _{21} = 1, \qquad \varepsilon _{11} = \varepsilon _{22}=0. \end{aligned}$$

For $v = (v_1,v_2)\in {\mathbb {R}}^2$, we will write $v^\perp := (-v_2,v_1)$. Thus $(v^\perp )_b = \varepsilon _{ab}v_a$. We will similarly write $\nabla _x^\perp := (-\partial _y,\partial _x)$. In the same spirit,

$$\begin{aligned} v^\perp := (v_1^\perp ,\ldots ,v_n^\perp ) \quad \text{ for } v = (v_1,\ldots , v_n)\in ({\mathbb {R}}^2)^n, \end{aligned}$$

with a similar convention for $\nabla ^\perp W$, for $W:({\mathbb {R}}^2)^n\rightarrow {\mathbb {R}}$.

If $\mu _z$ is a family of signed measures on an open set $U\subset {\mathbb {R}}^2$, depending (measurably) on a parameter $z\in (0,L)$, then $\mu _z\otimes dz$ denotes the measure on $U\times (0,L)$ defined by

$$\begin{aligned} \int _{U\times (0,L)} f d\mu _z\otimes dz = \int _0^L (\int _U f(x,z) d\mu _z(x)) dz. \end{aligned}$$

For a smooth bounded $A\subset {\mathbb {R}}^2$ (typically $\omega$ or $\omega _\varepsilon$) and $a\in A^n$ we will write

$$\begin{aligned} j^*_A(x;a) := -\nabla ^\perp _x \psi _{A}^*, \end{aligned}$$

where $\psi _{A}^* = \psi _A^*(x;a)$ solves

Equivalently, $j^*_A(x;a): A\rightarrow {\mathbb {R}}^2$ is the unique solution of

$$\begin{aligned} \nabla _x \cdot j^*_A =0, \qquad \nabla _x^\perp \cdot j^*_A = 2\pi \sum _{i=1}^n \delta _{a_i} ,\qquad j^*_A(\cdot , a)\cdot \nu = 0 \text{ on } \partial A \end{aligned}$$

where $\nu$ denotes the outer unit normal to A. It is straightforward to check that

$$\begin{aligned} j^*_{\omega _\varepsilon }(x; a) = h_\varepsilon j^*_\omega (h_\varepsilon x; h_\varepsilon a) \end{aligned}$$

and that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} j^*_{\omega _\varepsilon }(x; a) = \sum _{i=1}^n \frac{(x-a_i)^\perp }{|x-a_i|^2} =: j^*_{{\mathbb {R}}^2}(x;a). \end{aligned}$$

Given $g: (0,L)\rightarrow A^n$, we will write $j^*_A(g)$ to denote the function $A\times (0,L)\rightarrow {\mathbb {R}}^2$ defined by

$$\begin{aligned} j^*_A(g)(x,z) = j^*_A(x; g(z)). \end{aligned}$$

We define a couple of other auxiliary functions related to $\psi _A$. First, note that

$$\begin{aligned} \psi _A(x;a) = - \sum _{i=1}^n\left( \log |x-a_i| + H_A(x, a_i)\right) \end{aligned}$$

where for $a_i\in \Omega$, we define $H_A(\cdot ,a_i)$ to be the solution of

$$\begin{aligned} -\Delta _x H_A(x, a_i) = 0 \text{ for } x\in A, \qquad H_A(x,a_i) = -\log |x-a_i| \text{ for } x\in \partial A. \end{aligned}$$

We define

$$\begin{aligned} W_A(a) = -\pi \Big (\sum _{i\ne j} \log |a_i-a_j| + \sum _{i,j} H_A(a_i,a_j)\Big ). \end{aligned}$$

The constant $\kappa (n,\varepsilon ,\omega )$ appearing in (4) is defined by

$$\begin{aligned} \kappa (n,\varepsilon ,\omega ) = n(\pi {|\log {\varepsilon }|}+\gamma ) + n(n-1)\pi |\log h_\varepsilon | - \pi n^2 H_\omega (0,0) \end{aligned}$$

(9)

where $\gamma$ is a universal constant⁵ introduced in the pioneering work of Béthuel, Brezis and Hélein [3], see Lemma IX.1.

1.3 Variational aspects of nearly parallel vortex filaments

In this section we first collect some information about the behaviour of nearly parallel vortex filaments under energy and localisation constraints, but without introducing any time dependence. Most of these results are contained in Contreras and Jerrard [7], or can be obtained by adapting and combining results in [7]. The necessary details are given in Sect. 3.

Our first result follows directly from arguments in [7], although it does not appear there in exactly this form.

Proposition 1

Assume that $(u_\varepsilon )\subset H^1(\Omega ,{\mathbb {C}})$ is a sequence satisfying

$$\begin{aligned} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z) - n\pi \delta _0\Vert _{W^{-1,1}(\omega )}dz&\le c_1 h_\varepsilon , \end{aligned}$$

(10)

$$\begin{aligned} G_\varepsilon (u_\varepsilon )&\le c_2. \end{aligned}$$

(11)

Then

$$\begin{aligned} \int _\Omega |\partial _z u_\varepsilon |^2 dx\;dz \le C(c_1,c_2) \end{aligned}$$

(12)

and there exists some $f = (f_1,\ldots , f_n)\in H^1({\mathbb {T}}_{L}, {\mathbb {C}}^n)$ such that after passing to a subsequence if necessary:

$$\begin{aligned} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j(z)}\Vert _{W^{-1,1}(\omega )} dz = o(h_\varepsilon ) \qquad \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(13)

Finally, f satisfies

$$\begin{aligned} G_0(f)\le \liminf _{\varepsilon \rightarrow 0}G_\varepsilon (u_\varepsilon ) , \qquad \qquad \Vert f\Vert _{H^1} \le C(c_1,c_2), \end{aligned}$$

(14)

where the lim inf refers to the subsequence for which (13) holds.

The arguments needed to extract Proposition 1 from facts established in [7] are presented in Sect. 3.2. Next we describe weak limits of products of derivatives of $v_\varepsilon$.

Proposition 2

Assume that $(u_\varepsilon )\subset H^1(\Omega ,{\mathbb {C}})$ satisfies (11) and (13) (and hence (10)), and let $v_\varepsilon (x,z) = u_\varepsilon (h_\varepsilon x, z)$. Then the following hold, in the weak sense of measures on $\Omega$

$$\begin{aligned} \frac{1}{{|\log {\varepsilon }|}}\partial _{x_k} v_\varepsilon \cdot \partial _{x_l} v_\varepsilon&\rightharpoonup \pi \delta ^{kl} \sum _{i=1}^n \delta _{f_i(z)} \otimes dz, \end{aligned}$$

(15)

$$\begin{aligned} \frac{1}{{|\log {\varepsilon }|}}\nabla _x v_\varepsilon \cdot \partial _z v_\varepsilon&\rightharpoonup -\pi \sum _{i=1}^n \partial _z f_i (z)\delta _{f_i(z)} \otimes dz , \end{aligned}$$

(16)

for all k, l in $\{1, 2\}.$ Moreover, for any nonnegative $\phi \in C_c({\mathbb {R}}^2\times {\mathbb {T}}_{L})$,

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \int _{\Omega _\varepsilon } \phi \frac{|\partial _z v_\varepsilon |^2}{{|\log {\varepsilon }|}}\, dx\,dz \ge \pi \sum _{i=1}^n\int _0^L |f_i'(z)|^2 \phi (f_i(z), z) \, dz. \end{aligned}$$

(17)

The proof of Proposition 2 is given in Sect. 3.3. Briefly, (15) and (17) are deduced by combining results from [7] with facts established in [14, 15, 24], and (16) is obtained via a short argument whose starting point is (15) and (17).

Finally we will need a refinement of a $\Gamma$-limit lower bound from [7]. The proof is given in Sect. 3.4.

Proposition 3

Let $r>0$ and $f\in H^1((0,L), {\mathbb {C}}^n)$ be given such that $r<\rho _f/4.$ Then given $\delta >0$, there exist $c_3, \varepsilon _3>0$, depending only on $\Vert f\Vert _{H^1}$ and r, such that for all $\Sigma \in (0,1]$ and any $\varepsilon \le \varepsilon _3$, if $u_\varepsilon \in H^1(\Omega ,{\mathbb {C}})$ and

$$\begin{aligned}&\int _0^L \Vert Ju_\varepsilon (\cdot , z) - \pi \sum _{i=1}^n \delta _{h_\varepsilon f_i(z)}\Vert _{W^{-1,1}(\omega )} dz \le c_3 h_\varepsilon , \end{aligned}$$

(18)

$$\begin{aligned}&G_\varepsilon (u_\varepsilon ) - G_0(f) \le \Sigma , \end{aligned}$$

(19)

then

$$\begin{aligned} \int _0^L \int _{\omega \setminus \cup _{i=1}^n B(h_\varepsilon f_i(x), h_\varepsilon r)} e_\varepsilon (|u_\varepsilon |) + \frac{1}{4} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f)\right| ^2 \le K_3\Sigma +\delta , \end{aligned}$$

(20)

where $K_3$ depends only on r, n, and $\Vert f\Vert _{H^1}$. Moreover, if

$$\begin{aligned} T^f_{r,\varepsilon }(u_\varepsilon ) := \int _\Omega J_x u_\varepsilon (x,z) \chi ^f_{r,\varepsilon } \,dx\,dz \le \frac{c_3^2}{4n\pi L} \end{aligned}$$

(21)

then

$$\begin{aligned} \frac{1}{h_\varepsilon }\int _0^L \Vert J_x u_\varepsilon (\cdot ,z) - \pi \sum _{i=1}^n \delta _{h_\varepsilon f_i(z)}\Vert _{W^{-1,1}(\omega )} dz \le \left( n\pi L T^f_{r,\varepsilon }(u_\varepsilon )\right) ^\frac{1}{2} + o(1) \le \frac{1}{2} c_3. \end{aligned}$$

(22)

1.4 Compactness in time

In this section we now assume that $u_\varepsilon$ is a solution of the Gross–Pitaevskii equation and we shall obtain sufficient compactness in time to pass to the limit as $\varepsilon \rightarrow 0$ on intervals of time of positive length.

Proposition 4

Let $r>0$ and $g\in W^{1,\infty }({\mathbb {T}}_{L}, {\mathbb {C}}^n)$ be given such that $r\le \rho _g/4.$ There exist $\varepsilon _4,\,c_4 >0$, depending only on $\Vert g\Vert _{H^1}$ and r, and there exist $C_4$, depending only on $\Vert g\Vert _{Lip}$ and r, with the following properties. If $u_\varepsilon$ solves the Gross–Pitaevskii equation (2) for some $0<\varepsilon \le \varepsilon _4$ for initial data $u_\varepsilon ^0$ satisfying

$$\begin{aligned}&G_\varepsilon (u_\varepsilon ^0) \le G_0(g)+1, \end{aligned}$$

(23)

$$\begin{aligned}&\int _0^L \Vert Ju_\varepsilon ^0(\cdot , z) - \pi \sum _{i=1}^n \delta _{h_\varepsilon g_i(z)}\Vert _{W^{-1,1}(\omega )} dz \le c_4 h_\varepsilon , \end{aligned}$$

(24)

and

$$\begin{aligned} T^g_{r,\varepsilon }(u_\varepsilon ^0) \le \frac{c_4^2}{4n\pi L}, \end{aligned}$$

(25)

then for every $0\le t \le t_4 := 3c_4^2 /(4 C_4n\pi L)$,

$$\begin{aligned}&T^g_{r,\varepsilon }(u_\varepsilon (\cdot ,\cdot ,h_\varepsilon ^2t)) \le T^g_{r,\varepsilon }(u_\varepsilon ^0) + C_4t, \end{aligned}$$

(26)

$$\begin{aligned}&\frac{1}{h_\varepsilon }\int _0^L \Vert J_x u_\varepsilon (\cdot ,z,h_\varepsilon ^2t) - \pi \sum _{i=1}^n \delta _{h_\varepsilon g_i(z)}\Vert _{W^{-1,1}(\omega )} dz \le \left( n\pi L (T^g_{r,\varepsilon }(u_\varepsilon ^0)+C_4 t)\right) ^\frac{1}{2} +o(1),\nonumber \\ \end{aligned}$$

(27)

and in particular

$$\begin{aligned} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z,h_\varepsilon ^2t) - \pi \sum _{i=1}^n \delta _{h_\varepsilon g_i(z)}\Vert _{W^{-1,1}(\omega )} dz \le (c_4 + o(1)) h_\varepsilon . \end{aligned}$$

(28)

The proof is given in Sect. 4, as is the proof of the following.

Corollary 1

Under the assumptions of Theorem 1, there exists $t_0>0$, depending only on $\rho _{f^0}$ and $\Vert f^0\Vert _{H^1}$, $f^*$ in $\mathcal {C}([0, t_0],L^1({\mathbb {T}}_{L},{\mathbb {C}}^n)) \cap L^\infty ([0, t_0],H^1({\mathbb {T}}_{L},{\mathbb {C}}^n))$, and a common sequence $\varepsilon \rightarrow 0$, such that for every $0\le t \le t_0$

$$\begin{aligned} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z,h_\varepsilon ^2t) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f^*_j(z,t)}\Vert _{W^{-1,1}(\omega )} dz = o(h_\varepsilon ) \qquad \text{ as } \varepsilon \rightarrow 0 \end{aligned}$$

and in addition the equivalent of (28) holds for all $t\in [0,t_0]$, for every $\varepsilon$ in the sequence.

Moreover, we have $f^*(0) = f(0)$ and

$$\begin{aligned} \sup _{s,t\in [0,t_0]}\max _{i,z} |f^*_i(z,t) - f_i(z,s)| \le \frac{\rho _0}{8},\qquad \text{ and } \text{ hence } \inf _{t \in [0,t_0]}\rho _{f^*(t)} \ge \frac{3}{4} \rho _0. \end{aligned}$$

(29)

Our main goal in the sequel is to show that f and $f^*$ coincide on $[0,t_0]$, from which Theorem 1 will follow by a straightforward continuation argument.

Proposition 5

In addition to the statements in Corollary 1, we have

$$\begin{aligned} \frac{j(v_\varepsilon )}{|v_\varepsilon |} \rightharpoonup j_{{\mathbb {R}}^2}^*(f^*) \quad \text {weakly in } L^2(O) \end{aligned}$$

for every open $O \subset \subset \{ (t,x,z)\in [0,t_0]\times {\mathbb {R}}^2 \times {\mathbb {T}}_{L}: x\ne f^*_k(z,t), \ k=1,\ldots , n\}$.

1.5 Proof of the main theorem

For points $a = (a_1,\ldots , a_n)\in ({\mathbb {R}}^2)^n$ such that $a_i\ne a_j$ for $i\ne j$, we will write

$$\begin{aligned} {\mathcal {W}}(a) = -\sum _{i\ne j} \log |a_i-a_j|. \end{aligned}$$

(30)

With this notation,

$$\begin{aligned} G_0(g) = \pi \int _0^L \frac{1}{2} |g'(z)|^2 + {\mathcal {W}}(g(z)) \, dz \quad \text{ for } g:{\mathbb {T}}_{L}\rightarrow ({\mathbb {R}}^2)^n. \end{aligned}$$

For $0\le t \le t_0$ (where $t_0$ appears in Corollary 1), we define

$$\begin{aligned} I_1(t)&:= \pi \int _0^L |f(z,t) - f^*(z, t)|^2 \, dz\\ I_2(t)&:= \pi \int _0^L \left( -\partial _{zz}f(z,t) + \nabla {\mathcal {W}}(f(z,t) \right) \cdot (f(z, t) - f^*(z, t)) dz\\ I_3(t)&:= G_0(f(\cdot , t)) - G_0(f^*(\cdot , t)). \end{aligned}$$

Note that, as a consequence of conservation of energy for both (2) and (3),

$$\begin{aligned} G_0(f(\cdot , t)) = G_0(f^0) = \lim _{\varepsilon \rightarrow 0}G_\varepsilon (u_\varepsilon ^0) = \lim _{\varepsilon \rightarrow 0}G_\varepsilon (u_\varepsilon (t)) \ge G_0(f^*(t)). \end{aligned}$$

The last inequality follows from (14), as discussed following the statement of Proposition 1. Thus $I_3(t)\ge 0$ for all $t\in [0,t_0]$. In addition, $I_3(0)=0$, due to Corollary 1.

We aim to apply Proposition 3 to control quantities such as $\frac{ju_\varepsilon }{|u_\varepsilon |}(t) - j_\omega ^*(h_\varepsilon f^*(t))$ for a range of t. To this end, we will need

$$\begin{aligned} \Sigma _\varepsilon (t) := G_\varepsilon (u_\varepsilon (t)) - G_0(f^*(t)) \le 1. \end{aligned}$$

(31)

Arguing as above, we see that $\lim _{\varepsilon \rightarrow 0}\Sigma _\varepsilon (t) = I_3(t)$. Thus $\Sigma _\varepsilon (t)\le 1$ if $\varepsilon$ is sufficiently small and $I_3(t)\le \frac{1}{2}$. We therefore define

$$\begin{aligned} t^* := \sup \{ t\in [0,t_0] : 0\le I_3(s) \le \frac{1}{2} \text{ for } \text{ all } s\in [0,t]\}. \end{aligned}$$

The positivity of $t^*$ is a consequence of the weak $H^1$ lowersemicontinuity of $f\mapsto G_0(f)$ and the continuity properties of $f^*$ as stated in Corollary 1. (The other hypothesis of Proposition 3 follows directly from Corollary 1.)

Theorem 1 will be an easy consequence of the following three lemmas.

Lemma 1

There exists a constant $C_2$ such that for every $t\in [0,t^*]$,

$$\begin{aligned} I_3(t) \le I_2(t) + C_2I_1(t). \end{aligned}$$

Proof

First, it follows from (29) that for every $z\in [0,L]$ and $t\in [0,t^*]$,

$$\begin{aligned} {\mathcal {W}}(f(z,t)) - {\mathcal {W}}(f^*(z,t)) \le \nabla {\mathcal {W}}(f(z,t))\cdot (f(z,t) - f^*(z,t)) + C|f(z,t) - f^*(z,t)|^2, \end{aligned}$$

for C depending only on $\rho _{f(0)}$. The conclusion of the lemma follows by integrating this inequality with respect to z and combining the result with the estimate

$$\begin{aligned} \frac{\pi }{2}\int _0^L |\partial _z f|^2 - |\partial _z f^*|^2 \, dz&= \frac{\pi }{2}\int _0^L 2\partial _z f \cdot \partial _z(f-f^*) - |\partial _z (f-f^*)|^2 \, dz\\&\le - \pi \int _0^L \partial _{zz}f \cdot (f-f^*)\, dz. \end{aligned}$$

$\square$

The proofs of the next two lemmas are presented in Sect. 2 below.

Lemma 2

For every $\tau \in [0,t^*]$,

$$\begin{aligned} I_1(\tau ) \le I_1(0) + C\int _0^\tau \left( I_1(t) + I_3(t)\right) dt. \end{aligned}$$

Lemma 3

For every $\tau \in [0,t^*]$,

$$\begin{aligned} I_2(\tau ) \le I_2(0) + C \int _0^\tau \left( I_1(t) + I_3(t)\right) dt. \end{aligned}$$

With these, we can complete the

Proof of Theorem 1

Let $I_4(t) = I_2(t)+(1 + C_2)I_1(t)$. It follows from Lemma 1 that $I_4(t)\ge I_3(t) + I_1(t) \ge 0$ for all $t\in [0,t^*]$, moreover $I_4(0) = 0$ by Corollary 1 and Lemmas 1–3 imply that

$$\begin{aligned} I_4(\tau ) \le (1 + C + C_2)\int _0^\tau I_4(t)\, dt \qquad \text{ for } \text{ all } \tau \in [0,t^*]. \end{aligned}$$

It follows by Grönwall’s inequality that $I_4(\tau ) = 0$ for all $\tau \in [0,t^*]$, and therefore also that $I_1(\tau ) = 0$ for all $\tau \in [0,t^*]$, in other words, that $f=f^*$ on $[0, t^*]$. A straightforward continuation argument now shows that this equality holds on (0, T), and then by reversibility on $(-T,T)$, thus completing the proof. $\square$

2 Dynamics

The object of this section is to present the proofs of Lemmas 2 and 3, from which (together with Lemma 1) our main Theorem was derived in the Introduction. We will find it useful to rescale the Gross–Pitaevskii equation (2), setting

$$\begin{aligned} v_{\varepsilon }(x,z,t) := u_{\varepsilon }(h_\varepsilon x, z, h_\varepsilon ^2 t), \end{aligned}$$

(32)

where

$$\begin{aligned} h_\varepsilon := {|\log {\varepsilon }|}^{-1/2}. \end{aligned}$$

Thus

$$\begin{aligned} i \partial _tv_{\varepsilon }- \Delta _x v_{\varepsilon }- \frac{\partial _{zz}v_{\varepsilon }}{{|\log {\varepsilon }|}} + \frac{1}{{|\log {\varepsilon }|}\varepsilon ^2}(|v_{\varepsilon }|^2-1)v_{\varepsilon }= 0. \end{aligned}$$

(33)

We will write

$$\begin{aligned} j_xv_{\varepsilon }&:= iv_{\varepsilon }\cdot \nabla _xv_{\varepsilon },\\ j_zv_{\varepsilon }&:= iv_{\varepsilon }\cdot \partial _zv_{\varepsilon }. \end{aligned}$$

For the rescaled equation (33), the equation for conservation of mass takes the form

$$\begin{aligned} \frac{1}{2} \partial _t |v_{\varepsilon }|^2&= \nabla _x \cdot j_x v_{\varepsilon }+ h_{\varepsilon }^2\, \partial _z j_zv_{\varepsilon }. \end{aligned}$$

(34)

We will rely mainly on the equation for vorticity, and in fact only for the z component of the vorticity vector, which is precisely $J_x v_\varepsilon$. By rescaling standard identities we have

$$\begin{aligned} \partial _t J_xv_{\varepsilon }= \varepsilon _{ab} \partial _{a c}( \partial _bv_{\varepsilon }\cdot \partial _c v_{\varepsilon }) + \varepsilon _{ab} \partial _{a z} (\frac{\partial _bv_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}). \end{aligned}$$

Thus,

$$\begin{aligned} \frac{d}{dt}\int {\varphi }J_xv_{\varepsilon }dx\,dz&= \int \partial _t {\varphi }\, J_x v_\varepsilon dx\, dz + \int \varepsilon _{ab}\partial _{ac} {\varphi }\ \partial _bv_{\varepsilon }\cdot \partial _cv_{\varepsilon }\,dx\,dz \nonumber \\&\quad + \int \varepsilon _{ab}\partial _{az}{\varphi }\frac{\partial _b v_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}\,dx\,dz, \end{aligned}$$

(35)

for smooth ${\varphi }:\Omega _\varepsilon \times (0,T)\rightarrow {\mathbb {R}}$ for some $T>0$, with compact support in $\Omega _\varepsilon = \omega _\varepsilon \times {\mathbb {T}}_{L}$. (That is, test functions are only required to have compact support with respect to the horizontal x variables, not the periodic z variable.)

Lemma 4

Assume that ${\varphi }\in C^2_c(\Omega _\varepsilon \times [0,t^*] )$ is a function such that for some $k\in \{1,\ldots , n\}$,

$$\begin{aligned} \text{ supp }({\varphi }) \subset \{ (x,z,t) : |x-f_k(z,t)|\le \frac{\rho _0}{2}\}, \end{aligned}$$

and

$$\begin{aligned} \partial _{ac}{\varphi }(x,z,t) = c(z,t)\delta ^{ac}\qquad \text{ in } \{ (z,t) : |x-f_k(z,t)|\le \frac{\rho _0}{4}\} \end{aligned}$$

(36)

for some continuous c(z, t). Then for any $\tau \in [0,t^*]$,

$$\begin{aligned} \begin{aligned}&\int _0^L{\varphi }(f^*_k(z,t), z, t)\,dz \Big |_{t=0}^{t=\tau }\\&\quad \le C \int _0^\tau I_3(t)\,dt \\&\qquad + \int _0^\tau \int _0^L \partial _t{\varphi }( f^*_k(z,t), z,t)\, dz\, dt \\&\qquad -\int _0^\tau \int _0^L \nabla ^\perp \partial _z{\varphi }(f_k^*(z),z,t) \cdot \partial _zf^*_k(z,t) \, dz\,dt\\&\qquad +\int _0^\tau \int _0^L\nabla {\varphi }(f_k^*(z,t),z,t) \cdot \nabla _k^\perp {\mathcal {W}}(f^*(z,t)) \, dz\,dt\ , \end{aligned} \end{aligned}$$

where C depends only $\rho _0$, $\Vert f\Vert _{L^\infty H^1}$ and $\Vert \nabla _x^2 {\varphi }\Vert _{L^\infty }$.

Proof

We apply (35) to ${\varphi }$, integrate both sides from 0 to $\tau$, and send $\varepsilon \rightarrow 0$. We consider the various terms that arise.

1.:

Corollary 1 and properties of the support of ${\varphi }$ imply that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{\Omega _\varepsilon } {\varphi }(x,z,t) \, J_x v_\varepsilon (x,z,t) dx \, dz = \pi \int _0^L{\varphi }(f^*_k(z,t), z, t)\,dz \end{aligned}$$

(37)

for every $t\in [0,t^*]$, and in particular for $t=0,\tau$.

2.:

Similarly, (37) holds with ${\varphi }$ replaced by $\partial _t{\varphi }$. In addition, it follows from (28) that $|\int _{\Omega _\varepsilon } \partial _t {\varphi }(x,z,t) \, J_x v_\varepsilon (x,z,t) dx \, dz|$ is bounded uniformly in t. Thus

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _0^\tau \int _{\Omega _\varepsilon }\partial _t {\varphi }\, J_x v_\varepsilon \,dx\,dz\,dt = \pi \int _0^\tau \int _0^L \partial _t {\varphi }(f_k^*(z,t), z,t) \, dz\,dt. \end{aligned}$$

3.:

The last term on the right-hand side of (35) is similar. First note that there exists some C such that

$$\begin{aligned} \int _{\Omega _\varepsilon } \varepsilon _{ab}\partial _{az}{\varphi }\frac{\partial _b v_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}\,dx\,dz\ \le C \end{aligned}$$

for every t. This is a consequence of (12) (which is available for all $t\in [0,t^*]$ by Corollary 1) and (7), since

$$\begin{aligned} \int _{\Omega _\varepsilon }|\partial _z v_\varepsilon (y,z,t)|^2 \, dy \,dz \ = \ \ \int _\Omega \frac{ |\partial _z u_\varepsilon (x,z,h_\varepsilon ^2 t)|^2 }{{|\log {\varepsilon }|}}\,dx \,dz \, \end{aligned}$$

and $\int _{\Omega _\varepsilon }\frac{1}{2} |\nabla _x v_\varepsilon (y,z,t)|^2 \, dy \le G_\varepsilon (u_\varepsilon (\cdot , \cdot , h_\varepsilon ^2 t)) = G_\varepsilon (u_\varepsilon ^0)$. Also,

$$\begin{aligned} \int _{\Omega _\varepsilon } \varepsilon _{ab}\partial _{az}{\varphi }\frac{\partial _b v_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}\,dx\,dz \rightarrow -\pi \int _0^L \nabla ^\perp \partial _z{\varphi }(f_k^*(z),z,t) \cdot \partial _zf^*_k(z,t) \, dz \end{aligned}$$

for every t, due to (16). It follows that

$$\begin{aligned} \int _0^\tau \int _{\Omega _\varepsilon } \varepsilon _{ab}\partial _{az}{\varphi }\frac{\partial _b v_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}\,dx\,dz\,dt \rightarrow -\pi \int _0^\tau \int _0^L \nabla ^\perp \partial _z{\varphi }(f_k^*(z),z,t) \cdot \partial _zf^*_k(z,t) \, dz\,dt. \end{aligned}$$

4.:

To describe the limit of the remaining term coming from (35), first note that (36), together with our assumptions on the support of ${\varphi }$, implies that

$$\begin{aligned} \text{ supp }(\varepsilon _{ab}\partial _{ac}{\varphi }\partial _b v_\varepsilon \cdot \partial _c v_\varepsilon )(\cdot ,t) \subset \Omega _{\varepsilon ,k}(t) := \{ (x,z)\in \Omega _\varepsilon : \ |x - f_k(z,t)| \in [\frac{\rho _0}{4} ,\frac{\rho _0}{2}] \} . \end{aligned}$$

Next, we follow standard arguments and write

$$\begin{aligned} \partial _b v_\varepsilon \cdot \partial _c v_\varepsilon = \partial _c|v_\varepsilon | \, \partial _c|v_\varepsilon | + \frac{j_b(v_\varepsilon ) j_c(v_\varepsilon )}{|v_\varepsilon |^2}. \end{aligned}$$

For the rest of this proof we will write $j^*_\varepsilon$ as an abbreviation for $j^*_{\omega _\varepsilon }(f^*)$, and $j^* := \lim _{\varepsilon \rightarrow 0} j^*_\varepsilon = j^*_{{\mathbb {R}}^2}(f^*)$. With this notation, we further decompose the last term above as

$$\begin{aligned}&\frac{j_b(v_\varepsilon ) j_c(v_\varepsilon )}{|v_\varepsilon |^2} = j^*_{\varepsilon ,b} \ j^*_{\varepsilon ,c}+ \left( \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right) _b \left( \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right) _c \\&\quad +j^*_{\varepsilon ,b}\left( \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right) _c +j^*_{\varepsilon ,c}\left( \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right) _b. \end{aligned}$$

Thus,

$$\begin{aligned}&\int _0^\tau \int _{\Omega _\varepsilon } \varepsilon _{ab}\partial _{ac}{\varphi }\, \partial _b v_\varepsilon \cdot \partial _c v_\varepsilon \,dz\,dz\,dt \le \int _0^\tau \int _{ \Omega _{\varepsilon ,k}(t) }\varepsilon _{ab} \partial _{ac}{\varphi }\, j^*_{\varepsilon ,b} j^*_{\varepsilon ,c} \\&\quad +\int _0^\tau \int _{ \Omega _{\varepsilon ,k}(t) }\varepsilon _{ab}\partial _{ac}{\varphi }\, \left[ j^*_{\varepsilon ,b} \left( \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right) _c + j^*_{\varepsilon ,c}\left( \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right) _b \right] \\&\quad + \int _0^\tau \int _{\Omega _{\varepsilon ,k}(t) } |\nabla _x^2{\varphi }|\, \left( |\nabla _x |v_\varepsilon ||^2 + \left| \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon }\right| ^2\right) . \end{aligned}$$

It follows from Proposition 5 that the second term on the right-hand side converges to 0 as $\varepsilon \rightarrow 0$.

Using (19) and (20) of Proposition 3 for a sequence $\delta _n\rightarrow 0$ and recalling that $\Sigma _\varepsilon (t)$, as defined in (31), satisfies $\Sigma _\varepsilon (t) \rightarrow I_3(t)$ as $\varepsilon \rightarrow 0$, we find that

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\int _0^\tau \int _{\Omega _\varepsilon } |\nabla _x^2{\varphi }|\left( |\nabla _x |v_\varepsilon ||^2 + \left| \frac{j(v_\varepsilon )}{|v_\varepsilon |} - j^*_{\varepsilon } \right| ^2\right) \le C \int _0^\tau I_3(t)\,dt. \end{aligned}$$

Since $j_\varepsilon ^* \rightarrow j^*$ locally uniformly on ${\mathbb {R}}^2$, it is clear that

$$\begin{aligned} \int _0^\tau \int _{ \Omega _{\varepsilon ,k}(t) }\varepsilon _{ab} \partial _{ac}{\varphi }\, j^*_{\varepsilon ,b} j^*_{\varepsilon ,c} \rightarrow \int _0^\tau \int _{ \Omega _{\varepsilon ,k}(t) }\varepsilon _{ab} \partial _{ac}{\varphi }\, j^*_{b} j^*_{c} \end{aligned}$$

as $\varepsilon \rightarrow 0$. Finally, we claim that

$$\begin{aligned} \int _0^\tau \int _{ \Omega _{\varepsilon ,k}(t) }\varepsilon _{ab} \partial _{ac}{\varphi }\, j^*_{b} j^*_{c} = \pi \int _0^\tau \int _0^L\nabla {\varphi }(f_k^*,z,t)) \cdot \nabla _k^\perp {\mathcal {W}}(f^*(z,t)) \, dz\,dt. \end{aligned}$$

This is a small variant of a classical fact. We recall the proof for the reader’s convenience. First note that for every t and every $z\in (0,L)$,

$$\begin{aligned} \int _{ \{ x\in \omega : |x-f_k(z,t)|\in [ \frac{\rho _0}{4}, \frac{\rho _0}{2}]\}} \varepsilon _{ab}\partial _{ac}{\varphi }\, j^*_{b} j^*_{c} \, dx = \lim _{s\rightarrow 0^+} \int _{\omega \setminus B_s(f_k(z,t))}\varepsilon _{ab} \partial _{ac}{\varphi }\, j^*_{b} j^*_{c} \,dx \end{aligned}$$

(where all integrands are evaluated at the fixed value of t). Indeed, the right-hand side is independent of s for $0<s<\rho _0/4$, since the integrand vanishes identically in $B_{\rho _0/4}(f_k(z,t))$. For every $s<\rho _0/4$,

$$\begin{aligned} \int _{\omega \setminus B_s(f_k(z,t))} \varepsilon _{ab}\partial _{ac}{\varphi }\, j^*_{b} j^*_{c} \,dx&= \int _{\omega \setminus B_s(f_k(z,t))} \varepsilon _{ab}\partial _{ac}{\varphi }\,( j^*_{b} j^*_{c} - \frac{1}{2} \delta ^{bc}|j^*|^2) \,dx \nonumber \\&= -\int _{\partial B_{s}(f_k(z,t))} \varepsilon _{ab}\partial _a{\varphi }\,( j^*_{b} j^*_{c} - \frac{1}{2} \delta ^{bc}|j^*|^2)\nu _c \nonumber \\&= -\int _{\partial B_{s}(f_k(z,t))}(\nabla ^\perp {\varphi }\cdot j^*)(\nu \cdot j^*) - \frac{1}{2} \nabla ^\perp {\varphi }\cdot \nu |j^*|^2. \end{aligned}$$

(38)

Note that

$$\begin{aligned} j^*(x,z,t) = \frac{(x- f_k(z,t))^\perp }{|x-f_k(z,t)|^2} +\widetilde{j}(x;k),\quad \text{ where } \widetilde{j}(x;k) = \sum _{\ell \ne k} \frac{(x- f_\ell (z,t))^\perp }{|x-f_\ell (z,t)|^2}. \end{aligned}$$

We decompose $j^*$ in this way on the right-hand side of (38), then expand and let s tend to zero. This leads to

$$\begin{aligned} \int _{ \{ x\in \omega : |x-f_k(z,t)|\in [ \frac{\rho _0}{4}, \frac{\rho _0}{2}]\}} \varepsilon _{ab}\partial _{ac}{\varphi }\, j^*_{b} j^*_{c} \, dx = -2\pi \nabla {\varphi }(f_k^*(z,t),z,t)\cdot \widetilde{j}(f_k^*(z,t);k). \end{aligned}$$

Since

$$\begin{aligned} \nabla _k^\perp {\mathcal {W}}(a) := -2\sum _{\ell \ne k} \frac{(a_k-a_\ell )^\perp }{|a_k-a_\ell |^2} = -2\widetilde{j}(f_k^*(z,t);k), \end{aligned}$$

this implies the claim, and the proof of Lemma 4 is completed. $\square$

Proof of Lemma 2

We first assume that f is of class $\mathcal {C}^2$ and we apply Lemma 4 with

$$\begin{aligned} {\varphi }(x,z,t) = \chi _{\rho _0/4}(|x-f_k(z,t)|)\; |x - f_k(z,t)|^2, \end{aligned}$$

and then sum the resulting inequalities over k. This leads to the estimate

$$\begin{aligned}&I_1(\tau ) \le I_1(0) + \int _0^\tau \int _0^L (f -f^*)\cdot \partial _t f + \partial _z f^\perp \cdot \partial _z f^* \ dz\, dt\\&\quad - \int _0^\tau \int _0^L (f-f^*)\cdot \nabla ^\perp {\mathcal {W}}(f^*) \ dz\, dt + C\int _0^\tau I_3(t)\,dt. \end{aligned}$$

The equation (3) satisfied by f may be written

$$\begin{aligned} \partial _t f^\perp = \partial _{zz} f - \nabla {\mathcal {W}}(f). \end{aligned}$$

(39)

Substituting this into the above inequality and integrating by parts, we obtain

$$\begin{aligned} I_1(\tau ) \le I_1(0) + \int _0^\tau \int _0^L (f-f^*)\cdot (\nabla ^\perp {\mathcal {W}}(f) - \nabla ^\perp {\mathcal {W}}(f^*)) \ dz\, dt + C\int _0^\tau I_3(t)\,dt. \end{aligned}$$

It follows from the definition of $t_0$ that

$$\begin{aligned} |\nabla ^\perp {\mathcal {W}}(f) - \nabla ^\perp {\mathcal {W}}(f^*)| \le C |f-f^*|, \end{aligned}$$

and the conclusion follows immediately.

It remains to remove the smoothness assumption on f. For that purpose, it suffices to replace f, in the definition of ${\varphi }$ above, by $\mathcal {C}^2$ solutions $f^\delta$ of (1) which converge towards f in $L^\infty H^1$ as $\delta \rightarrow 0$ and then to send $\delta$ to zero in the resulting inequality. The key point is that in the statement of Lemma 4, the constant C only depends on $\rho _0$, $\Vert f\Vert _{L^\infty H^1}$ and bounds on the second derivatives of ${\varphi }$ with respect to the variable x only. $\square$

Proof of Lemma 3

As for the proof of Lemma 2 we may assume without loss of generality that f is regular, the general case can then be obtained by approximation in $L^\infty H^1$. We apply Lemma 4 with

$$\begin{aligned} {\varphi }(x,z,t) = \chi _{\rho _0/4}(|x-f_k(z,t)|) \left( -\partial _{zz}f_k(z,t) + \nabla _k {\mathcal {W}}(f(z,t) \right) \cdot (f(z, t) - x)_k, \end{aligned}$$

and then (implicitly) sum the resulting inequalities over k. This leads to the estimate

$$\begin{aligned}&I_2(\tau ) \le I_2(0) + \int _0^\tau \int _0^L \partial _t\left( -\partial _{zz}f_k + \nabla _k {\mathcal {W}}(f )\right) \cdot (f - f^*)_k \ dz\, dt\\&\quad + \int _0^\tau \int _0^L\partial _z \left( -\partial _{zz}f_k + \nabla _k {\mathcal {W}}(f) \right) ^\perp \cdot \partial _z f_k^* \ dz\, dt\\&\quad + \int _0^\tau \int _0^L \left( \partial _{zz}f_k - \nabla _k {\mathcal {W}}(f) \right) \cdot \nabla _k^\perp {\mathcal {W}}(f^*) \ dz\, dt + C\int _0^\tau I_3(t)\,dt. \end{aligned}$$

The middle integral on the right-hand side can be rewritten

$$\begin{aligned} \int _0^\tau \int _0^L\partial _z \partial _t f_k \cdot \partial _z f_k^* \ dz\, dt = -\int _0^\tau \int _0^L\partial _{tzz}f_k \cdot f_k^* \ dz\, dt, \end{aligned}$$

and hence cancels out part of the first integral. We then integrate by parts and expand $\partial _t\nabla _k{\mathcal {W}}(f)$ to obtain

$$\begin{aligned}&I_2(\tau )\le I_2(0) - \int _0^\tau \int _0^L \partial _t f_j \cdot \partial _{zz}f_j \ dz\, dt\\&\quad + \int _0^\tau \int _0^L \partial _t f_j \cdot \nabla _j\nabla _k{\mathcal {W}}(f) \cdot ( f - f^*)_k \ dz\, dt\\&\quad + \int _0^\tau \int _0^L \left( \partial _{zz}f_k - \nabla _k {\mathcal {W}}(f) \right) \cdot \nabla _k^\perp {\mathcal {W}}(f^*) \ dz\, dt + C\int _0^\tau I_3(t)\,dt . \end{aligned}$$

Using the PDE (39) to eliminate $\partial _{zz}f$, we rewrite this as

$$\begin{aligned} I_2(\tau )&\le I_2(0) + C\int _0^\tau I_3(t)\,dt\\&\quad +\int _0^\tau \int _0^L \partial _t f_j\cdot \left[ \nabla _j{\mathcal {W}}(f^*) - \nabla _j{\mathcal {W}}(f) - \nabla _k\nabla _j{\mathcal {W}}(f) \cdot (f^*-f)_k \right] dz\,dt. \end{aligned}$$

Finally, it follows from the definition of $t_0$ that

$$\begin{aligned} \left| \nabla _j{\mathcal {W}}(f^*) - \nabla _j{\mathcal {W}}(f) - \nabla _k\nabla _j{\mathcal {W}}(f) \cdot (f^*-f)_k \right| \le C|f^*-f|^2. \end{aligned}$$

The conclusion of the lemma follows immediately. $\square$

3 Proofs of variational results

In this section we present the proofs of Propositions 1, 2, and 3.

3.1 Tools

We start by assembling some tools that give information about the vortex structure of a function satisfying (10), (11) for small but fixed $\varepsilon >0$, rather than in the limit $\varepsilon \rightarrow 0$. All of these are established in [7], but in some cases our presentation here differs a little. We therefore give short proofs that sketch the arguments needed to obtain the precise statements given here from those in [7].

Our first result of this sort states that under assumptions (10), (11), for every $z\in (0,L)$, if $\varepsilon$ is small enough then $u_\varepsilon (\cdot ,z)$ has either n distinct, well-localized vortices clustered near the vertical axis, or a certain amount of “extra energy”. We will write

$$\begin{aligned} e_\varepsilon ^{2d}(u) := \frac{1}{2} |\nabla _xu_{\varepsilon }|^2 + \frac{1}{4\varepsilon ^2}(|u_{\varepsilon }|^2-1)^2 \end{aligned}$$

the Ginzburg–Landau energy density with respect to horizontal variables.

Lemma 5

Assume that $u_\varepsilon \in H^1(\Omega ,{\mathbb {C}})$ satisfies (10) and (11).

There exist positive numbers $\theta , a,b, C$ and $\varepsilon _0$ depending on $n, c_1,c_2$ such that $b<a$, and if $0<\varepsilon <\varepsilon _0$, then for every $z\in (0,L)$ such that

$$\begin{aligned} \int _{\omega \times \{z\}} e^{2d}_\varepsilon (u_\varepsilon ) \,dx \le \pi (n+\theta ){|\log {\varepsilon }|}\ , \end{aligned}$$

(40)

there exist $g^\varepsilon _j(z)\in {\mathbb {R}}^2$ for $j = 1,\ldots , n$ such that

$$\begin{aligned}&\Vert J_x u_\varepsilon (\cdot , z) - \pi \sum _{j=1}^n \delta _{g^\varepsilon _j(z)}\Vert _{F(\omega )} \le \varepsilon ^a \ , \end{aligned}$$

(41)

$$\begin{aligned}&|g^\varepsilon _j (z)- g^\varepsilon _k(z)| \ge \varepsilon ^b \ \ \text{ for } \text{ all } j\ne k, \ \qquad \text{ dist }(g^\varepsilon _j(z), \partial \omega ) \ge C^{-1} \ \ \text{ for } \text{ all } j, \end{aligned}$$

(42)

$$\begin{aligned}&|g^\varepsilon _j(z)| \le C h_\varepsilon \ \ \text{ for } \text{ all } j, \end{aligned}$$

(43)

$$\begin{aligned}&\int _{\omega \times \{z\}} e^{2d}_\varepsilon (w) dx \ \ge \ n(\pi \vert \log \varepsilon \vert +\gamma ) + W_{\omega }(g^\varepsilon _1(z),\ldots , g^\varepsilon _n(z)) - C\varepsilon ^{(a-b)/2}, \end{aligned}$$

(44)

where $W_\omega$ is the renormalized energy defined in Sect. 1.2.

Proof of Lemma 5, excluding estimate (43)

Given a sequence of functions $u_\varepsilon \in H^1(\Omega ,{\mathbb {C}})$ satisfying (10) and (11), a set ${\mathcal {G}}^\varepsilon _1 = {\mathcal {G}}^\varepsilon _1(u_\varepsilon )\subset (0,L)$ is defined in equation (3.11) of [7] with the following properties. First, if $z\not \in {\mathcal {G}}^\varepsilon _1$ then

$$\begin{aligned} \int _\omega e^{2d}_\varepsilon (u_\varepsilon )(x,z) \,dx \ge \varepsilon ^{-1/2}, \end{aligned}$$

for all sufficiently small $\varepsilon$ (where “sufficiently small” may depend on the given sequence). And second, if $z\in {\mathcal {G}}^\varepsilon _1$ and (40) holds, then there exist $g^\varepsilon _j(z)\in \omega$, for $j=1,\dots , n$, satisfying (41), (44) and (42). These are proved in [7], Proposition 1 and Lemma 3 respectively, which actually assume a somewhat weaker condition in place of (10).

The conclusions of the lemma, apart from (43) (proved below), follow directly from these facts. $\square$

We will henceforth write

$$\begin{aligned} {\mathcal {G}}(u_\varepsilon ) := \{ z\in (0,L): (40) \text{ holds } \}, \qquad {\mathcal {B}}(u_\varepsilon ) := (0,L)\setminus {\mathcal {G}}(u_\varepsilon ) \ . \end{aligned}$$

(45)

Thus, for every $z\in {\mathcal {G}}(u_\varepsilon )$, Lemma 5 provides a detailed description of the vorticity of $u_\varepsilon (\cdot , z)$.

For $z\in {\mathcal {G}}(u_\varepsilon )$ we will write

$$\begin{aligned} f^\varepsilon _j(z) := g^\varepsilon _j(z)/h_\varepsilon . \end{aligned}$$

(46)

Rescaling (41), we find that $\Vert J_x v_\varepsilon (\cdot , z) - \pi \sum _{j=1}^n \delta _{f^\varepsilon _j(z)}\Vert _{W^{-1,1}(\omega _\varepsilon )} \le \varepsilon ^a/h_\varepsilon$, where $v_\varepsilon (x,z) = u_\varepsilon (h_\varepsilon x,z)$ as usual.

Remark 1

It is clear from the proof in [7] that $z\mapsto \chi _{{\mathcal {G}}(u_\varepsilon )} g_j^\varepsilon (z)$ may be taken to be measurable.

We next collect some conclusions that follow rather easily from Lemma 5.

Lemma 6

Assume that $0<\varepsilon < 1/2$ and that $u_\varepsilon \in H^1(\Omega ,{\mathbb {C}})$ satisfies (10) and (11). Then there exists a positive constant $C = C(c_1,c_2,n)$ such that

$$\begin{aligned}&\int _\Omega e^{2d}_\varepsilon (u_\varepsilon ) \ge n\pi L {|\log {\varepsilon }|}+\pi n (n-1)L |\log h_\varepsilon |- C, \end{aligned}$$

(47)

$$\begin{aligned}&|{\mathcal {B}}(u_\varepsilon )| \le C{|\log {\varepsilon }|}^{-1}, \end{aligned}$$

(48)

$$\begin{aligned}&\int _{z\in {\mathcal {B}}(u_\varepsilon )}\int _\omega e_\varepsilon ^{2d}(u_\varepsilon ) \, dx\,dz \le C, \end{aligned}$$

(49)

$$\begin{aligned}&\int _\Omega |\partial _z u_\varepsilon |^2 dx\;dz \le C. \end{aligned}$$

(50)

We will later improve on some of these estimates under the hypotheses of our main theorem.

Proof of Lemma 6

Conclusions (47) and (50) are proved in Lemma 9 of [7]. The proof relies on the parts of Lemma 5 proved above, together with properties of the renormalized energy $W_\omega$ (see Lemma 4 of [7]) and a short argument using Jensen’s inequality. The proof also easily yields the other conclusions (48), (49) stated here. Indeed, the proof of Lemma 9 in [7] actually shows⁶ that

$$\begin{aligned} \int _{z\in {\mathcal {G}}(u_\varepsilon )}\int _\omega e^{2d}_\varepsilon (u_\varepsilon )\, dx\,dz \ge \Big (n \pi {|\log {\varepsilon }|}+ n(n-1)\pi (|\log h_\varepsilon | - C\Big )|{\mathcal {G}}(u_\varepsilon )|. \end{aligned}$$

On the other hand it is clear from the definitions that

$$\begin{aligned} \int _{z\in {\mathcal {B}}(u_\varepsilon )}\int _\omega e^{2d}_\varepsilon (u_\varepsilon )\, dx\,dz \ge (n\pi +\theta ){|\log {\varepsilon }|}\, |{\mathcal {B}}(u_\varepsilon )|. \end{aligned}$$

Since $e_\varepsilon (u_\varepsilon ) = e_\varepsilon ^{2d}(u_\varepsilon ) + \frac{1}{2} |\partial _z u_\varepsilon |^2$ and $|{\mathcal {G}}(u_\varepsilon )| + |{\mathcal {B}}(u_\varepsilon )|=L$, by comparing these estimates with the hypothesis (11), we easily obtain (48) and (49). $\square$

We now state a result that establishes a sort of approximate equicontinuity of the map $z\in {\mathcal {G}}(u_\varepsilon )\mapsto \pi \sum \delta _{f^\varepsilon _j(z)}$ for finite $\varepsilon >0$.

Lemma 7

Assume that (10), (11) hold. Then for every $\delta >0$, there exists positive constants $\varepsilon _0, C$ such that if $0<\varepsilon <\varepsilon _0$, then the following holds:

Assume that $z_1,z_2$ are points in ${\mathcal {G}}(u_\varepsilon )$ such that $|z_1 - z_2| > \delta$, and let $g_j^\varepsilon (z_\ell )$ denote the points provided by Lemma 5 for $\ell = 1,2$. Then for $f^\varepsilon _j(z_\ell ) := g_j^\varepsilon (z_\ell )/h_\varepsilon$,

$$\begin{aligned} \pi \min _{\sigma \in S_n} \sum _{j=1}^n \frac{|f_{\sigma (j)}^\varepsilon (z_2) - f_j^\varepsilon (z_1)|^2}{|z_2-z_1|} \le C. \end{aligned}$$

(51)

Proof of conclusion (43) of Lemma 5 and of Lemma 7

Estimate (43) is shown to hold in Step 3 of the proof of Lemma 12 in [7], via a compactness argument based on Lemma 8, see below.

Lemma 7 then follows from Lemma 8 by almost exactly the same compactness argument. The constant C appearing in (51) may be chosen to be a multiple of the uniform bound for $\int _\Omega |\partial _z u_\varepsilon |^2$, established in Lemma 6 and depending only on $c_1,c_2$ from (10), (11). $\square$

The last result in this section is the lemma used in the compactness arguments described above. It will be used again in the proof of Proposition 3. In [7] it provides the basic estimate that eventually implies that $z\mapsto f(z) = (f_1(z),\ldots , f_n(z))$ belongs to $H^1((0,L), ({\mathbb {R}}^2)^n)$, see Proposition 1.

Lemma 8

Assume that $(u_\varepsilon )$ satisfies (10), (11). Let $v_\varepsilon (x,z) := u_\varepsilon (h_\varepsilon x,z)$.

Assume that $\{z^\varepsilon _1\}$ and $\{z^\varepsilon _2\}$ are sequences in [0, L] such that $z^\varepsilon _j\rightarrow z_j$ for $j=1,2$, with $0\le z_1 < z_2 \le L$, and that the following conditions hold for $j=1,2$ (perhaps after passing to a subsequence):

$$\begin{aligned}&J_x v_\varepsilon (\cdot , z^\varepsilon _j) \rightarrow \pi \sum _{i=1}^{n(z)} \delta _{p_i(z_j)} \qquad \text{ in } W^{-1,1}(B(R)), \ \ \text{ for } \text{ all } R>0, \end{aligned}$$

(for certain points $\{ p_i(z_j)\}_{i=1}^{n(z_j)}$, not necessarily distinct) and

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} {|\log {\varepsilon }|}^{-1} \int _{\omega } e^{2d}_\varepsilon (u_\varepsilon (x,z^\varepsilon _j))dx \le M\pi \end{aligned}$$

for some $M >0$. Then $n(z_1) = n(z_2) =: m$, and

$$\begin{aligned} \frac{\pi }{2}\min _{\sigma \in S_m}\sum _{i=1}^m\frac{|p_i(z_1)-p_{\sigma (i)}(z_2)|^2}{z_2-z_1} \ \le \ \liminf _{\varepsilon \rightarrow 0} \int _{z_1}^{z_2} \int _{\omega _\varepsilon } \frac{1}{2} |\partial _z u_\varepsilon |^2\, dx\,dz . \end{aligned}$$

Proof

This is essentially Lemma 10 of [7]. Apart from some notational changes, the main difference is that Lemma 10 of [7] is proved under an assumption that is somewhat weaker than (10). As a result, it is stated there for a rescaling $v_\varepsilon (x,z) := u_\varepsilon (\ell _\varepsilon x,z)$ using a scaling factor $\ell _\varepsilon$ that is shown only later to equal $h_\varepsilon$. With the stronger assumption (10), the proof can be simplified, and one can work directly with the $\ell _\varepsilon = h_\varepsilon$. $\square$

3.2 Proof of Proposition 1

Proof

With a couple of exceptions, everything in Proposition 1 is taken directly from the statement of Theorem 3 in [7].

The first exception is the compactness assertion (13); in [7], compactness is proved to hold only with respect to a weaker topology. To prove (13), we argue as follows. First note that

$$\begin{aligned}&\int _{z\in {\mathcal {B}}(u_\varepsilon )} \Vert J_x u_\varepsilon (\cdot ,z) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j(z)}\Vert _{W^{-1,1}(\omega )} dz \nonumber \\&\quad \le n\pi |{\mathcal {B}}(u_\varepsilon )|+ \int _{z\in {\mathcal {B}}(u_\varepsilon )} \Vert J_xu_\varepsilon (\cdot ,z)\Vert _{W^{-1,1}(\omega )} dz \nonumber \\&\quad \le n\pi |{\mathcal {B}}(u_\varepsilon )|+ C {|\log {\varepsilon }|}^{-1} \int _{z\in {\mathcal {B}}(u_\varepsilon )} e_\varepsilon ^{2d}(u_\varepsilon )(x,z)dz \nonumber \\&\quad \le C{|\log {\varepsilon }|}^{-1} = C h_\varepsilon ^2 \end{aligned}$$

(52)

by standard Jacobian estimates (see for example [17] or [25]) and Lemma 6, for $C = C(c_1,c_2, n)$. On the other hand, by (41) and (46),

$$\begin{aligned}&\int _{z\in {\mathcal {G}}(u_\varepsilon )} \Vert J_x u_\varepsilon (\cdot ,z) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j(z)}\Vert _{W^{-1,1}(\omega )} dz \nonumber \\&\quad \le \int _{z\in {\mathcal {G}}(u_\varepsilon )} \Vert \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j^\varepsilon (z)} - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j(z)}\Vert _{W^{-1,1}(\omega )} dz + C \varepsilon ^a. \end{aligned}$$

(53)

It is also shown in [7], Lemmas 13 and 14 that after passing to a suitable subsequence $\varepsilon _k\rightarrow 0$, there is a set $H_G\subset (0,L)$ of full measure, such that if $z\in H_G$, then there exists $\ell = \ell (z)$ such that $z\in {\mathcal {G}}(u_{\varepsilon _k})$ for all $k\ge \ell$, and

$$\begin{aligned} \Vert \pi \sum _{j=1}^n \delta _{ f_j^{\varepsilon _k}(z)} - \pi \sum _{j=1}^n \delta _{f_j(z)}\Vert _{W^{-1,1}(B(R))} \rightarrow 0 \qquad \text{ for } \text{ every } R>0 \end{aligned}$$

as $k\rightarrow \infty$. This implies that

$$\begin{aligned} \Vert \pi \sum _{j=1}^n \delta _{h_{\varepsilon _k} f_j^{\varepsilon _k}(z)} - \pi \sum _{j=1}^n \delta _{h_{\varepsilon _k} f_j(z)}\Vert _{W^{-1,1}(\omega )} = o(h_{\varepsilon _k}) \qquad \text{ for } \text{ every } z\in H_G \end{aligned}$$

as $k\rightarrow \infty$. It also follows from (43) that

$$\begin{aligned} \Vert \pi \sum _{j=1}^n \delta _{h_{\varepsilon _k} f_j^{\varepsilon _k}(z)} - \pi \sum _{j=1}^n \delta _{h_{\varepsilon _k} f_j(z)}\Vert _{W^{-1,1}(\omega )} \le C h_{\varepsilon _k} \quad \text{ for } z\in {\mathcal {G}}(u_{\varepsilon _k}) \setminus H_G, \end{aligned}$$

so the conclusion follows from the dominated convergence theorem, together with (52) and (53).

The other assertion that is not taken directly from the statement of Theorem 3 in [7] is the estimate $\Vert f\Vert _{H^1}\le C(c_1,c_2)$. To prove this, we use (5) to deduce that for $z\in H_G$,

$$\begin{aligned} \sum _i |f_i(z)| = \sum _i |f_i(z) - 0|&= \lim _{k\rightarrow \infty }\frac{1}{h_\varepsilon } \Vert \sum _i \delta _{h_{\varepsilon _k}f_i(z) } - n \pi \delta _0 \Vert _{W^{-1,1}(\omega )}\\&= \lim _{k\rightarrow \infty }\frac{1}{\pi h_\varepsilon } \Vert J_x u_\varepsilon (\cdot , z)- n\pi \delta _0 \Vert _{W^{-1,1}(\omega )} . \end{aligned}$$

Thus Fatou’s Lemma and (10) imply that

$$\begin{aligned} \Vert f \Vert _{L^1}\le C(c_1). \end{aligned}$$

We may then use Jensen’s inequality and the fact from [7] that $G_0(f)\le c_2$ to estimate

$$\begin{aligned} \frac{\pi }{2}\int _0^L \sum _j |f_j'|^2 dz&= G_0(f) + \pi \sum _{i\ne j}\int _0^L \log |f_i-f_j| dz \\&\le c_2 + L\pi \sum _{i\ne j} \log \left( \frac{1}{L} \int _0^L |f_i- f_j| dz\right) \\&\le C(c_1,c_2). \end{aligned}$$

Finally, $\Vert f\Vert _{L^2}$ is controlled by interpolating between $\Vert f\Vert _{L^1}$ and $\Vert f'\Vert _{L^2}$. $\square$

3.3 Proof of Proposition 2

Proof of (15)

It suffices to show, given any subsequence satisfying (11), (13) for which

$$\begin{aligned} {|\log {\varepsilon }|}^{-1}\partial _a v_\varepsilon \cdot \partial _b v_\varepsilon \rightharpoonup \text{ some } \text{ limit, } \text{ weakly } \text{ as } \text{ measures } \end{aligned}$$

that this limit can only equal $\pi \delta ^{ab} \sum _i \delta _{f_i(z)}\otimes dz$. For $z\in (0,L)$, let

$$\begin{aligned} E^{2d}_\varepsilon (z) :=\frac{1}{{|\log {\varepsilon }|}}\int _{\omega \times \{z\}} e_\varepsilon ^{2d}(u_\varepsilon ) \, dx = \frac{1}{{|\log {\varepsilon }|}}\int _{ \omega _\varepsilon \times \{z\}} e_{\varepsilon '}^{2d}(v_\varepsilon ) \, dx \end{aligned}$$

where $\varepsilon '= \varepsilon /h_\varepsilon$. It follows from the definition of ${\mathcal {B}}(u_\varepsilon )$ that $E_\varepsilon ^{2d}(z)\ge n\pi +\theta$ for $z\in {\mathcal {B}}(u_\varepsilon )$, and since (43) implies that $W_\omega (g_1^\varepsilon ,\ldots , g_n^\varepsilon )\ge n\pi |\log h_\varepsilon |-C$, we deduce from (44) that $E_\varepsilon ^{2d}(z)\ge n\pi - o(1)$ uniformly for $z\in {\mathcal {G}}(u_\varepsilon )$, as $\varepsilon \rightarrow 0$. On the other hand, the assumed energy scaling (11) implies that $\int _0^L E_\varepsilon ^{2d}(z)\, dz\rightarrow n\pi L$ as $\varepsilon \rightarrow 0$. In view of these facts, after passing to a further subsequence if necessary, we may assume that

$$\begin{aligned} \frac{1}{{|\log {\varepsilon }|}}\int _{ \omega _\varepsilon \times \{z\}} e_{\varepsilon '}^{2d}(v_\varepsilon ) \, dx \rightarrow n \pi \qquad \text{ for } \text{ a.e. } z\in (0,L). \end{aligned}$$

(54)

Next, upon rescaling (13) and passing to a further subsequence,

$$\begin{aligned} \Vert J v_\varepsilon - \pi \sum _{i=1}^n \delta _{f_i(z)}\Vert _{W^{-1,1}(\omega _\varepsilon )} \rightarrow 0 \qquad \text{ for } \text{ a.e. } z\in (0,L). \end{aligned}$$

(55)

It follows from Theorem 5 in [15] or Corollary 4 in [24] that whenever the above two conditions hold (i.e. a.e.),

$$\begin{aligned} \frac{1}{{|\log {\varepsilon }|}} \partial _a v_\varepsilon \cdot \partial _b v_\varepsilon (\cdot , z) \rightharpoonup \delta ^{ab} \pi \sum _{i=1}^n \delta _{f_i(z)}\qquad \text{ weakly } \text{ as } \text{ measures }. \end{aligned}$$

Now fix any $\phi \in C_c({\mathbb {R}}^2\times [0,L])$, and let

$$\begin{aligned} \Phi _\varepsilon (z) := \frac{1}{{|\log {\varepsilon }|}}\int _{\omega _\varepsilon \times \{z\}} \phi (x,z) \partial _a v_\varepsilon \cdot \partial _b v_\varepsilon (x, z) \, dx. \end{aligned}$$

We write $\Phi _\varepsilon = \Phi _{{\mathcal {G}}, \varepsilon } + \Phi _{{\mathcal {B}},\varepsilon }$, where $\Phi _{{\mathcal {G}}, \varepsilon } = \chi _{z\in {\mathcal {G}}(u_\varepsilon )}\Phi _\varepsilon (z)$. It follows immediately from (49) that $\Phi _{{\mathcal {B}},\varepsilon }\rightarrow 0$ in $L^1((0,L))$. We may assume after passing to a subsequence that $\chi _{{\mathcal {B}}(u_\varepsilon )}\rightarrow 0$ a.e.. It then follows that

$$\begin{aligned} \Phi _{{\mathcal {G}},\varepsilon }(z)\rightarrow \delta ^{ab}\pi \sum _{i=1}^n \phi ( f_i(z),z)\qquad \text{ for } \text{ a.e. } z. \end{aligned}$$

The definition of ${\mathcal {G}}(u_\varepsilon )$ implies that $\sup _z |\Phi _{{\mathcal {G}},\varepsilon }(z)|\le (n\pi +\theta )\sup _{(x,z)} |\phi (x,z)| \le C$. Thus the dominated convergence theorem implies that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0 } \int _0^L \Phi _\varepsilon (z) dz = \delta ^{ab}\pi \sum _{i=1}^n \phi ( f_i(z),z)\, dz. \end{aligned}$$

This is (15). $\square$

Proof of (17)

For $\delta >0$, let

$$\begin{aligned} {\mathcal {I}}_\delta := \{ z\in (0,L) : \min _{i\ne j} |f_i(z)-f_j(z)| > \delta \}. \end{aligned}$$

We know from (14) that $G_0(f)<\infty$, which implies that $|{\mathcal {I}}_\delta | \rightarrow L$ as $\delta \rightarrow 0$. It thus suffices to prove that for any nonnegative $\phi \in C_c({\mathbb {R}}^2\times [0,L])$ and for every $\delta >0$,

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\int _{\omega _\varepsilon {\mathcal {I}}_\delta } \phi \frac{|\partial _z v_\varepsilon |^2}{{|\log {\varepsilon }|}}\, dx\,dz \ge \pi \sum _{i=1}^n\int _{{\mathcal {I}}_\delta } |f_i'(z)|^2 \phi (f_i(z), z) \, dz . \end{aligned}$$

(56)

We may write ${\mathcal {I}}_\delta$ as a disjoint union of open intervals. Let I denote one such interval. In view of arguments in the proof of (15), it suffices to prove that if $f\in H^1(I, ({\mathbb {R}}^2)^n)$ is such that (54), (55) hold for a.e. $z\in I$ and $\min _{z\in I}\min _{i\ne h} |f_i(z)-f_j(z)| \ge \delta >0$, then (56) is satisfied (with ${\mathcal {I}}_\delta$ replaced by I).

There are a number of proofs of this fact⁷ when $\phi \equiv 1$; see for example [14] Proposition 3 or [24], Corollary 7. These proofs proceed by considering separately the energetic contributions associated to each trajectory $z\mapsto (f_i(z),z)$, and they show that for any $r>0$, and every $i\in \{1,\ldots , n\}$, and every interval $J\subset I$

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\int _{z\in J }\int _{B_r(f_i(z))} \frac{|\partial _z v_\varepsilon |^2}{{|\log {\varepsilon }|}}\, dx\,dz \ge \pi \int _{J} |f_i'(z)|^2 \, dz . \end{aligned}$$

This easily implies the desired estimate. $\square$

Proof of (16)

First, recalling that $v_\varepsilon (x,z) = u_\varepsilon (h_\varepsilon x, z)$ and using (12),

$$\begin{aligned} \int _\Omega |\partial _z u_\varepsilon |^2\, dx\,dz = \int _\Omega \frac{ |\partial _z v_\varepsilon |^2}{{|\log {\varepsilon }|}}dx\,dz \le C(c_1,c_2,n). \end{aligned}$$

We may thus assume that ${|\log {\varepsilon }|}^{-1} \partial _z v_\varepsilon \cdot \nabla _x v_\varepsilon$ converges weakly to a limiting ${\mathbb {R}}^2$-valued measure, say $\lambda$ on ${\mathbb {R}}^2\times [0,L]$.

Now fix some $g\in C^1(( 0,L), {\mathbb {R}}^2)$, and let

$$\begin{aligned} \tilde{u}_\varepsilon (x,z) := u_\varepsilon (x - h_\varepsilon g(z), z), \qquad \tilde{v}_\varepsilon (x,z) := \tilde{u}_\varepsilon (h_\varepsilon x, z) = v_\varepsilon (x - g(z), z). \end{aligned}$$

If we fix some $\tilde{\omega } \subset \subset \omega$ such that $0\in \tilde{\omega }$, we may then take the domain of $\tilde{u}_\varepsilon$ to be $\tilde{\Omega } := \tilde{\omega }\times (0,L)$, for all sufficiently small $\varepsilon$. (We remark that although we are ultimately interested in $u_\varepsilon$ that is periodic in the z variable, here we do not assume that g is periodic.)

It is straightforward to check from (13) and the definition of $\tilde{u}_\varepsilon$ that

$$\begin{aligned} \int _0^L \Vert J_x \tilde{u}_\varepsilon (\cdot ,z) - \pi \sum _{j=1}^n \delta _{h_\varepsilon (f_j(z) + g(z))} \Vert _{W^{-1,1}(\tilde{\omega })} dz = o(h_\varepsilon ) \qquad \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

Also, since $h_\varepsilon = {|\log {\varepsilon }|}^{-1/2}$ and extending the definition (4) of $G_\varepsilon$ to include a dependence in the domain, we have

$$\begin{aligned} G_\varepsilon (\tilde{u}_\varepsilon ;\tilde{\Omega })&\le G_\varepsilon (u_\varepsilon ;\Omega ) + \int _\Omega \frac{ |g'(z)\cdot \nabla _x u_\varepsilon |}{\sqrt{|\log {\varepsilon }|}} |\partial _z u_\varepsilon | + \frac{1}{2} \frac{ |g'(z)\cdot \nabla _x u_\varepsilon |^2}{{|\log {\varepsilon }|}}\, dx\, dz \\&\le c_2 + C\int _\Omega |\partial _z u_\varepsilon |^2 + \frac{ |\nabla _x u_\varepsilon |^2}{{|\log {\varepsilon }|}}dx\,dz \\&\le \tilde{K}_1 \end{aligned}$$

for some suitable $\tilde{K}_1$, whenever $\varepsilon$ is sufficiently small. Thus (17) implies that for any continuous $\tilde{\phi } \ge 0$,

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \int \tilde{\phi }(x,z) \frac{ |\partial _z \tilde{v}_\varepsilon (x,z)|^2}{{|\log {\varepsilon }|}}\, dx\,dz\, \ge \sum _i\pi \int _0^L |\partial _z(f_i +g)(z)|^2\tilde{\phi }(f_i(z)+g (z),z)\, dz. \end{aligned}$$

Taking $\tilde{\phi }$ of the form $\tilde{\phi }(x,z) = \phi (x-g(z),z)$, we get the more convenient expression

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \int \phi (x-g(z),z) \frac{ |\partial _z \tilde{v}_\varepsilon (x,z)|^2}{{|\log {\varepsilon }|}}\, dx\,dz\, \ge \sum _i \pi \int _0^L |\partial _z(f_i +g)(z)|^2\phi (f_i(z),z)\, dz. \end{aligned}$$

On the other hand, by using the definition of $\tilde{v}_\varepsilon$ and making the change of variables $(x-g(x), z) \mapsto (x,z)$, we obtain

$$\begin{aligned}&\int \phi (x-g(z),z) |\partial _z \tilde{v}_\varepsilon (x,z)|^2\, dx\,dz\, = \int \phi (x,z) |\partial _z v_\varepsilon (x,z)|^2\, dx\,dz\\&\quad +\int \phi (x,z)\left( -2 g'(z)\cdot \nabla _x v_\varepsilon (x,z) \cdot \partial _z v_\varepsilon (x,z) + |g'(z)\cdot \nabla _x v_\varepsilon (x,z)|^2\right) dx\,dz. \end{aligned}$$

Dividing by ${|\log {\varepsilon }|}$, letting $\varepsilon \rightarrow 0$, and invoking (12) and (15), we find that

$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0} \int \phi (x-g(z),z) \frac{ |\partial _z \tilde{v}_\varepsilon (x,z)|^2}{{|\log {\varepsilon }|}}\, dx\,dz\, \\&\quad \le C - 2\int _{{\mathbb {R}}^2\times (0,L)}\phi (x,z) g'(z)\cdot d\lambda + \sum _i \pi \int _0^L |\partial _z g(z)|^2\phi (f_i,z)\, dz. \end{aligned}$$

Combining this with the previous inequality and rewriting, we conclude that

$$\begin{aligned} \int _{{\mathbb {R}}^2\times (0,L)} \phi (x,z)g'(z)\cdot d\lambda + \pi \int _0^L \phi (x,z)g'(z) \cdot d\left( \sum _i f_i'(z)\delta _{f_i(z)}\otimes dz\right) \le C \end{aligned}$$

for $g, \phi$ as above, with C depending on $c_1,c_2,f,n, \phi$ but independent of g . Since we may multiply a given g by an arbitrary real constant, it follows that in fact

$$\begin{aligned} \int \phi (x,z)g'(z) \cdot d\lambda + \pi \int \phi (x,z)g'(z) \cdot d\left( \sum _i f_i'(z)\delta _{f_i(z)}\otimes dz\right) = 0 \end{aligned}$$

and hence that

$$\begin{aligned} \lambda = - \pi \sum _i f_i'(z)\delta _{f_i(z)}\otimes dz. \end{aligned}$$

This is (16). $\square$

3.4 Proof of Proposition 3

Define

$$\begin{aligned} \sigma ^{2d}_\varepsilon (z) = \sigma ^{2d}_\varepsilon (z; u_\varepsilon , h_\varepsilon f) = \int _{\omega } e^{2d}_\varepsilon (u_\varepsilon (x,z)) dx - W_\varepsilon (h_\varepsilon f(z) ; \omega ) \end{aligned}$$

where for $a\in \omega ^n$,

$$\begin{aligned} W_\varepsilon (a;\omega ) =n(\pi {|\log {\varepsilon }|}+\gamma ) - \pi \sum _{i\ne j} \log |a_i(z) - a_j(z)| + \pi \sum _{i,j} H_\omega (a_i, a_j)\ . \end{aligned}$$

Recall that $H_\omega$ is defined in Sect. 1.2. We interpret $\sigma ^{2d}_\varepsilon (z)$ as the surplus 2d (horizontal) energy of $u_\varepsilon$ at height z, with respect to the vortex positions $h_\varepsilon f(z)$. Further define

$$\begin{aligned} \Sigma ^{2d}_\varepsilon = \Sigma ^{2d}_\varepsilon (u_\varepsilon , h_\varepsilon f) = \int _0^L \sigma ^{2d}_\varepsilon (z) dz. \end{aligned}$$

Proof of estimate (20)

Assume toward a contradiction that there exists a sequence $(u_\varepsilon )_{\varepsilon \in (0,1]}$ in $H^1(\Omega ,{\mathbb {C}})$ such that

$$\begin{aligned} \int _0^L \Vert Ju_\varepsilon (\cdot , z) - \pi \sum _{i=1}^n \delta _{h_\varepsilon f_i(z)}\Vert _{W^{-1,1}(\omega )} dz = o(h_\varepsilon ) \end{aligned}$$

and $G_\varepsilon (u_\varepsilon ) - G_0(f) \le \Sigma _\varepsilon \le 1$, but

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} \int _0^L \int _{\omega \setminus \cup _{i=1}^n B(h_\varepsilon f_i(x), h_\varepsilon r)} e_\varepsilon (|u_\varepsilon |) + \frac{1}{4} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_{h_\varepsilon f}\right| ^2 - K_3\Sigma _\varepsilon >0 \end{aligned}$$

(57)

for $K_3$ to be chosen in a moment, and depending only on $\Vert f\Vert _{H^1}$ and $r< \frac{1}{4} \rho _f$.

This sequence satisfies the hypotheses (10), (11) of Lemma 5 with and $c_1=1+ n\pi L \Vert f\Vert _\infty$ and $c_2 = G_0(f)+1$, which are both controlled by $\Vert f\Vert _{H^1}$ and r. Let $\theta = \theta (n,c_1,c_2)$ be the constant found in Lemma 5. We will obtain a contradiction to (57) with $K_3 = \frac{4}{\theta }n\pi + 4$, thereby proving (20) for that value of $K_3$.

For this choice of $\theta$, we define sets ${\mathcal {G}}(u_\varepsilon )$ and ${\mathcal {B}}(u_\varepsilon )$ as in (45). For $z\in {\mathcal {G}}(u_\varepsilon )$, Lemma 5 provides points $g_j^\varepsilon (z)$ satisfying (41), (42) for $0<\varepsilon <\varepsilon _0(n, \Vert f\Vert _{H^1}, \rho _f, \Sigma )$, with constants such as a in (41) depending on the same quantities.

Setting $f^\varepsilon _j(z) = h_\varepsilon ^{-1}g^\varepsilon _j(z)$, it follows from (41) that

$$\begin{aligned} \int _{z\in {\mathcal {G}}(u_\varepsilon )} \Vert \sum _{i=1}^n \delta _{h_\varepsilon f^\varepsilon _j(z)} - \sum _{i=1}^n \delta _{h_\varepsilon f_i(z)}\Vert _{W^{-1,1}(\omega )} dz = o(h_\varepsilon ) \quad \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(58)

Our first goal is to strengthen this to read

$$\begin{aligned} \sup _{z\in {\mathcal {G}}(u_\varepsilon )} \Vert \sum _{i=1}^n \delta _{h_\varepsilon f^\varepsilon _j(z)} - \sum _{i=1}^n \delta _{h_\varepsilon f_i(z)}\Vert _{W^{-1,1}(\omega )} = o(h_\varepsilon ) \quad \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(59)

In brief, this follows from a compactness argument based on (58) and Lemma 8. Here are the details:

Assume toward a contradiction that (59) fails. Then there exists a (sub)sequence $\varepsilon \rightarrow 0$ and points $z_{\varepsilon }\in {\mathcal {G}}(u_{\varepsilon })$ such that

$$\begin{aligned} \Vert \sum _{i=1}^n \delta _{ h_\varepsilon f^\varepsilon _j(z_\varepsilon )} - \sum _{i=1}^n \delta _{ h_\varepsilon f_i(z_\varepsilon )}\Vert _{W^{-1,1}(\omega )} \ge c h_\varepsilon >0\quad \text{ for } \text{ all } \varepsilon . \end{aligned}$$

(60)

It follows from (48) and (58) that for all sufficiently small terms in the same subsequence, we may find points $\zeta _\varepsilon \in {\mathcal {G}}(u_\varepsilon )$ such that

$$\begin{aligned} \Vert \sum _{i=1}^n \delta _{ h_\varepsilon f^\varepsilon _j(\zeta _\varepsilon )} - \sum _{i=1}^n \delta _{ h_\varepsilon f_i(\zeta _\varepsilon )}\Vert _{W^{-1,1}(\omega )} = o(h_\varepsilon ), \quad \text{ and } \quad \alpha< | z_\varepsilon - \zeta _\varepsilon |< 2\alpha \end{aligned}$$

for some $\alpha$ to be fixed below. Extracting a further subsequence we may assume that $z_\varepsilon \rightarrow z$ and $\zeta _\varepsilon \rightarrow \zeta$, and that there exist $m\le n$ and $p_1,\ldots , p_m\in {\mathbb {R}}^2$ such that

$$\begin{aligned} \sum _{i=1}^n \delta _{ f_i^\varepsilon (\zeta _\varepsilon )} \rightarrow \sum _{i=1}^n \delta _{ f_i(\zeta )}, \quad \text{ and } \quad \sum _{i=1}^n \delta _{ f_i^\varepsilon (z_\varepsilon )} \rightarrow \sum _{i=1}^m \delta _{ p_i(z)} \end{aligned}$$

in $W^{-1,1}(B(R))$ for every $R>0$. (In fact both limits hold in stronger topologies as well.) These facts and (41) imply that for $v_\varepsilon (x,z) := u_\varepsilon (h_\varepsilon x,z)$,

$$\begin{aligned} J_x v(\cdot , \zeta _\varepsilon ) \rightarrow \pi \sum _{i=1}^n \delta _{ f_i(\zeta )} , \qquad J_x v(\cdot , z_\varepsilon ) \rightarrow \pi \sum _{i=1}^m \delta _{ p_i(z)} \end{aligned}$$

in the same topology. Then Lemma 8 and conclusion (12) from Proposition 1 imply that $m=n$ and that

$$\begin{aligned} \min _{\sigma \in S_n} \sum _{i=1}^n |f_i(\zeta ) - p_{\sigma (i)}(z)|^2 \le |z-\zeta | C \le 2\alpha C. \end{aligned}$$

(Here and below, the constant depends on f and $\Sigma$.) On the other hand, since f is Hölder continuous, it follows from (60) that

$$\begin{aligned} \min _{\sigma \in S_n} \sum _{i=1}^n |f_i(\zeta )- p_{\sigma (i)}(z))| \ge \min _{\sigma \in S_n} \sum _{i=1}^n |f_i(z)- p_{\sigma (i)}(z))| -nC|z-\zeta |^{1/2} \ge c - nC\alpha ^{1/2}. \end{aligned}$$

A contradiction is reached by choosing $\alpha$ sufficiently small, depending only on $f, \Sigma$, and c. This completes the proof of (59).

Next, we remark that in view of the fact that $\rho _f>0$, it follows from (59) and (5) that the labels on $f^\varepsilon _i$ may be chosen so that

$$\begin{aligned} \sup _{z\in {\mathcal {G}}(u_\varepsilon )} |f^\varepsilon _i(z) - f_i(z)| \rightarrow 0 \quad \text{ as } \varepsilon \rightarrow 0. \end{aligned}$$

(61)

We will write

$$\begin{aligned} \omega (z, \varepsilon , f) := \omega \setminus \cup _{i=1}^n B(h_\varepsilon f_i(z), h_\varepsilon r). \end{aligned}$$

For $z\in {\mathcal {G}}(u_\varepsilon )$, Theorem 2 of [18], for which the main hypothesis is a consequence of (41), provides certain integral estimates on $\omega \setminus \cup _{i=1}^n B( h_\varepsilon f^\varepsilon _i(z),C \varepsilon ^{a/2} )$, where $a>0$ comes from (41) and C depends on various ingredients that are fixed. It follows from (59) and (5) that if $\varepsilon$ is sufficiently small, then for every $z\in {\mathcal {G}}(u_\varepsilon )$, this set contains $\omega (z, \varepsilon , f)$. Theorem 2 of [18] thus implies that for every $z\in {\mathcal {G}}(u_\varepsilon )$,

$$\begin{aligned}&\int _{ \omega (z, \varepsilon , f) \times \{z\}} e_\varepsilon ^{2d}(|u_\varepsilon |) + \frac{1}{4} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f^\varepsilon (z))\right| ^2 \, dx\\&\qquad \le \int _{\omega \times \{z\}} e_\varepsilon ^{2d}(u)\, dx - \left[ n(\pi {|\log {\varepsilon }|}+ \gamma ) + W_\omega (h_\varepsilon f^\varepsilon (z)) \right] + C \varepsilon ^{a/2}. \end{aligned}$$

We recall that $W_\omega$ is defined in Sect. 1.2. It is easy to check from the definition there that

$$\begin{aligned} n(\pi {|\log {\varepsilon }|}+ \gamma ) + W_\omega (h_\varepsilon f^\varepsilon (z)) = \pi {\mathcal {W}}(f^\varepsilon (z))+ \kappa (n,\varepsilon ,\omega ) + O(h_\varepsilon ) \end{aligned}$$

where ${\mathcal {W}}$ is introduced in (30). Thus

$$\begin{aligned} \begin{aligned}&\int _{z\in {\mathcal {G}}(u_\varepsilon )} \int _{\omega (z, \varepsilon , f) \times \{z\}} e_\varepsilon ^{2d}(|u_\varepsilon |) + \frac{1}{8} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f(z))\right| ^2 \, dx\,dz\\&\qquad \le \int _{z\in {\mathcal {G}}(u_\varepsilon )} \int _{\omega (z, \varepsilon , f) \times \{z\}} \frac{1}{4} \left| j^*_\omega (h_\varepsilon f(z))- j^*_\omega (h_\varepsilon f^\varepsilon (z))\right| ^2 \, dx\,dz\\&\qquad + \int _{z\in {\mathcal {G}}(u_\varepsilon )} \left( \int _{\omega \times \{z\} } e^{2d}_\varepsilon (u)\, dx - \kappa (n,\varepsilon ,\omega ) - \pi {\mathcal {W}}(f^\varepsilon (z))\right) dz +O(h_\varepsilon ). \end{aligned} \end{aligned}$$

It follows from (61) and Lemma 9 below that the first term on the right-hand side vanishes as $\varepsilon \rightarrow 0$. Using this, we add and subtract various terms to rewrite the above inequality as

$$\begin{aligned} \begin{aligned}&\int _{z\in {\mathcal {G}}(u_\varepsilon )} \int _{\omega (z, \varepsilon , f) \times \{z\}} e_\varepsilon ^{2d}(|u_\varepsilon |) + \frac{1}{8} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f(z))\right| ^2 \, dx\,dz\\&\qquad \le G_\varepsilon (u_\varepsilon ) - G_0(f) - \left( \int _\Omega \frac{| \partial _z u_\varepsilon |^2}{2}\, dx\,dz - \frac{\pi }{2}\int _0^L |f'(z)|^2\,dz \right) \\&\qquad - \int _{z\in {\mathcal {B}}(u_\varepsilon )} \left( \int _{\omega \times \{z\} } e^{2d}_\varepsilon (u)\, dx - \kappa (n,\varepsilon ,\omega ) - \pi {\mathcal {W}}(f(z))\right) \, dz + o(1). \end{aligned} \end{aligned}$$

(62)

Clearly $|{\mathcal {W}}(f)|$ is bounded by a constant depending on $n, \rho _0$ and $\Vert f\Vert _{H^1}$, and it follows that $\kappa (n,\varepsilon ,\omega ) + \pi {\mathcal {W}}(f(z)) \le (\pi n + \frac{\theta }{2}){|\log {\varepsilon }|}$ for all sufficiently small $\varepsilon$. Then the definition of ${\mathcal {B}}(u_\varepsilon )$ implies that $\int _{\omega \times \{z\} } e^{2d}_\varepsilon (u)\, dx - \kappa (n,\varepsilon ,\omega ) -\pi {\mathcal {W}}(f(z)) \ge \frac{\theta }{2}{|\log {\varepsilon }|}$ when $z\in {\mathcal {B}}(u_\varepsilon )$. Taking $\varepsilon$ smaller, if necessary, we may assume by (17) that

$$\begin{aligned} \int _\Omega \frac{| \partial _z u_\varepsilon |^2}{2}\, dx\,dz - \frac{\pi }{2}\int _0^L |f'(z)|^2\,dz \ge - \varpi \delta \end{aligned}$$

for $\varpi >0$ to be chosen. Employing this in (62) and discarding the left-hand side, we deduce that

$$\begin{aligned} |{\mathcal {B}}(u_\varepsilon )| \le \frac{4}{\theta }(\Sigma _\varepsilon +\varpi \delta ) {|\log {\varepsilon }|}^{-1} \end{aligned}$$

for all sufficiently small $\varepsilon >0$. Returning to (62) with this new information, we deduce that

$$\begin{aligned} \int _{z\in {\mathcal {B}}(u_\varepsilon )} \int _{\omega \times \{z\} } e^{2d}_\varepsilon (u)\, dx\, dz&\le \Sigma _\varepsilon + \varpi \delta + \frac{4}{\theta }(\Sigma _\varepsilon +\varpi \delta )(n\pi +\frac{\theta }{2}) \\&\le (3+\frac{4n\pi }{\theta })\Sigma _\varepsilon +\frac{\delta }{4} + o(1) \end{aligned}$$

provided $\varpi \le \frac{1}{4}$ is chosen small enough, depending only on n and $\theta$, which itself is universal. Then, since

$$\begin{aligned} e_\varepsilon ^{2d}(|u_\varepsilon |)+\frac{1}{8} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f(z))\right| ^2 \le e^{2d}_\varepsilon (u) + \frac{1}{4}|j^*_\omega (h_\varepsilon f(z))|^2, \end{aligned}$$

we use (62) and the above estimate of $|{\mathcal {B}}(u_\varepsilon )|$ to find that

$$\begin{aligned} \begin{aligned}&\int _0^L \int _{\omega (z, \varepsilon , f) \times \{z\}} e_\varepsilon ^{2d}(|u_\varepsilon |) + \frac{1}{8} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f(z))\right| ^2 \, dx\,dz\\&\quad \le (4+\frac{4n\pi }{\theta })\Sigma _\varepsilon +\frac{\delta }{2} + \int _{z\in {\mathcal {B}}(u_\varepsilon )} \int _{\omega (z,\varepsilon , f) \times \{z\} } \frac{1}{4}|j^*_\omega (h_\varepsilon f(z))|^2 dx \, dz + o(1). \end{aligned} \end{aligned}$$

Finally,

$$\begin{aligned} \int _{\omega (z,\varepsilon , f) \times \{z\} } \frac{1}{4}|j^*_\omega (h_\varepsilon f(z))|^2 dx \, dz \le C|\log h_\varepsilon | = o({|\log {\varepsilon }|}) \end{aligned}$$

for a constant that depends only on n and $\Vert f\Vert _{H^1}$ and r; this can be verified by arguments similar to those in Lemma 9 below. Using this in the above inequality, we conclude that

$$\begin{aligned} \int _0^L \int _{\omega (z, \varepsilon , f) \times \{z\}} e_\varepsilon ^{2d}(|u_\varepsilon |) + \frac{1}{8} \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon f(z))\right| ^2 \, dx\,dz \le \left( \frac{4}{\theta }n\pi + 4\right) \Sigma _\varepsilon + \frac{3}{4} \delta \end{aligned}$$

for all sufficiently small $\varepsilon$. This contradicts (57) and completes the proof of (20). $\square$

Note that one can repeat the above proof with essentially no change, after replacing f in (57) and the two preceding assumptions by a sequence $\tilde{f}^\varepsilon$ with a uniform upper bound on $\Vert \tilde{f}^\varepsilon \Vert _{H^1}$ and the uniform lower bound on $\rho _{ \tilde{f}^\varepsilon }\ge 4r$, for r fixed. Then essentially⁸ the same argument as above leads to the same contradiction, establishing (20) with $\varepsilon _3,c_3$ that depend only on $\Vert f \Vert _{H^1}$ and r.

Next is the lemma that was used above.

Lemma 9

Assume that $a,a'\in \omega ^n$ and that there exist $r_0 \ge r_1>0$ such that

$$\begin{aligned} {\text {dist}}(a_i,\partial \omega )>r_0\qquad \text{ and } \qquad |a_i-\tilde{a}_i|\le \frac{1}{2} r_1\le \frac{1}{4}\rho _a \quad \text{ for } \text{ all } i. \end{aligned}$$

Then

$$\begin{aligned} \int _{\omega \setminus \cup B_{r_1}(a_i)} |j^*_\omega (a) - j^*_\omega (a')|^2\, dx \le C(n,r_0,\omega )|a-a'|^2 + C(n)(\frac{ |a-a'|}{r_1})^2. \end{aligned}$$

In particular, the above constants are independent of $r_1$.

Proof

Using notation from Sect. 1.2,

$$\begin{aligned} |j^*_\omega (x;a) - j^*_\omega (x;a') |^2&\le 2n \sum _i \left| \frac{x-a_i}{|x-a_i|^2} - \frac{x-a_i'}{|x-a_i'|^2}\right| ^2\\&\qquad + 2n\sum _i |\nabla H_\omega (x,a_i) - \nabla H_\omega (x, a_i')|^2. \end{aligned}$$

The definition of $H_\omega$ and the maximum principle imply that

$$\begin{aligned} |\nabla H_\omega (x,a_i) - \nabla H_\omega (x, a_i')| \le C(r_0)|a_i-a_i'|, \end{aligned}$$

and a short computation shows that if $|x-a|\ge 2 |a-a'|$, then

$$\begin{aligned} \left| \frac{x-a_i}{|x-a_i|^2} - \frac{x-a_i'}{|x-a_i'|^2}\right| ^2 \le 4 \frac{|a_i-a_i'|^2}{|x-a_i|^4}. \end{aligned}$$

Thus

$$\begin{aligned}&\int _{\omega \setminus \cup B_{r_1}(a_i)} |j^*_\omega (x;a) - j^*_\omega (x;a') |^2 \\&\qquad \le 2n|a-a'|^2 \int _{{\mathbb {R}}^2\setminus B_{r_1}(0)} |x|^{-4}dx + C(n,r_0,\omega )|a-a'|^2 \end{aligned}$$

from which the conclusion of the lemma follows.

Proof of (22)

Assume toward a contradiction that there is a subsequence along which (18), (19) and (21) hold for every $\varepsilon$, but there exists $\eta _1>0$ such that

$$\begin{aligned} \begin{aligned} \lim _{\varepsilon \rightarrow 0} h_\varepsilon ^{-1}\int _0^L \Vert J_x u_\varepsilon (\cdot ,z) - \pi \sum _{i=1}^h \delta _{h_\varepsilon f_i(z)}\Vert _{F(\omega )} dz&\ge \lim _{\varepsilon \rightarrow 0} \left( \pi n L (T^f_{r,\varepsilon }(u_\varepsilon )+\eta _1)\right) ^\frac{1}{2} \\&=: (\pi n L(T_{lim}+\eta _1))^{1/2}. \end{aligned} \end{aligned}$$

(63)

Clearly (18), (19) imply that the hypotheses of Proposition 1 are satisfied (with a larger constant in (10) than in (18)), so we may use the proposition to find a subsequence, still denoted $(u_\varepsilon )$, and a function $f^0\in H^1((0,L),({\mathbb {R}}^2)^n)$ such that

$$\begin{aligned} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f_j^0(z)}\Vert _{W^{-1,1}(\omega )} dz = o(h_\varepsilon ) \end{aligned}$$

(64)

as $\varepsilon \rightarrow 0$.

We will first show that, after choosing $c_3$ suitably small and possibly relabelling,

$$\begin{aligned} \Vert f_j - f^0_j\Vert _{L^\infty ((0,L))} \le r \quad \text{ for } j=1,\ldots , n. \end{aligned}$$

(65)

We start by noting from (18), (63), and (64) that

$$\begin{aligned} \left( \pi nL (T_{lim} + \eta _1)\right) ^\frac{1}{2} \le \lim _{\varepsilon \rightarrow 0}\frac{1}{h}_\varepsilon \int _0^L \Vert \pi \sum _j (\delta _{h_\varepsilon f_j(z)} - \delta _{h_\varepsilon f_j^0(z)}) \Vert _{W^{-1,1}(\omega )} \le c_3. \end{aligned}$$

It follows from (5) that for all sufficiently small $\varepsilon$ and all z,

$$\begin{aligned} \Vert \pi \sum _j (\delta _{h_\varepsilon f_j^0(z)} - \delta _{h_\varepsilon f_j(z)}) \Vert _{W^{-1,1}(\omega )} = \pi h_\varepsilon \min _{\sigma \in S_n} \sum _j |f_j(z) - f_{\sigma (j)}^0(z)|. \end{aligned}$$

Thus

$$\begin{aligned} \left( \pi nL T_{lim}\right) ^\frac{1}{2} +\eta _1 \le \pi \int _0^L \min _{\sigma \in S_n} \sum _j |f_j(z) - f_{\sigma (j)}^0(z)| \ dz \le c_3. \end{aligned}$$

(66)

In particular, this implies that

$$\begin{aligned} \Vert f\Vert _{L^1} \le C(f^0, c_3). \end{aligned}$$

It follows from a Sobolev embedding and (14) that there exists $C = C(f^0,c_2,c_3)$ such that

$$\begin{aligned} {[} f ]_{C^{0,1/2}} \le \Vert f'\Vert _{L^2} \le C,\qquad \text{ and } \text{ thus } [ f - f^0 ]_{C^{0,1/2}} \le C . \end{aligned}$$

(67)

Next, we deduce from (66) and Chebyshev’s inequality that

$$\begin{aligned} \left| \left\{ z\in (0,L) : \min _{\sigma \in S_n} \sum _j |f_j(z) - f_{\sigma (j)}^0(z)| > r/2 \right\} \right| \le \frac{2c_3}{r}. \end{aligned}$$

If $\min _{\sigma \in S_n} \sum _j |f_j(z_0) - f_{\sigma (j)}^0(z_0)|>r$ for any $z_0\in (0,L)$, then it follows from (67) that

$$\begin{aligned} \min _{\sigma \in S_n} \sum _j |f_j(z) - f_{\sigma (j)}^0(z)| >r/2 \quad \text{ for } \text{ all } z\in (0,L) \text{ such } \text{ that } |z-z_0| < r^2/C. \end{aligned}$$

Fixing $c_3$ small enough (which only decreases the constant $C(f_0,c_2, c_3)$ in (67)), we can arrange that the two above estimates are incompatible. (This adjustment to $c_3$ again depends only on $\rho _f\ge 4r$ and $\Vert f\Vert _{H^1}$.) It follows that for this choice of $c_3$,

$$\begin{aligned} \min _{\sigma \in S_n} \sum _j |f_j(z) - f_{\sigma (j)}^0(z)| \le r \qquad \text{ for } \text{ every } z\in (0,L). \end{aligned}$$

As a result, we can find a single permutation $\pi$, independent of z, such that $\sum _j |f_j(z) - f_{\pi (j)}^0(z)| = \min _\sigma \sum _j |f_j(z) - f_{\sigma (j)}^0(z)| \le r$ for all z. Using this permutation $\pi$ to relabel the indices, we obtain (65).

If we write $\varphi (x) := \chi _r(\frac{|x|}{h_\varepsilon })(\frac{|x|}{h_\varepsilon })^2$, then since $\Vert \nabla _x \varphi \Vert _\infty \le C/h_\varepsilon$, it follows from (18), (64) that

$$\begin{aligned} T_{lim}&= \pi \sum _{i,j} \int _0^L \chi _r(|f_j(z) - f^0_i (z)|) |f_j(z) - f^0_i(z)|^2\, dz. \end{aligned}$$

However, since $|f^0_i - f^0_j|\ge 4r$, we see from (65) that

$$\begin{aligned} \chi _r(|f_j(z) - f^0_i(z) |) = \delta ^{ij}\quad \text{ for } \text{ all } i,j \text{ and } \text{ all } z\in (0,L). \end{aligned}$$

So we obtain

$$\begin{aligned} \pi \Vert f - f^0\Vert _{L^2}^2 = T_{lim}. \end{aligned}$$

On the other hand, since we have by now arranged that

$$\begin{aligned} \min _\sigma \sum _j |f_j(z) - f_{\sigma (j)}^0(z)| = \sum _j |f_j(z) - f_j^0(z)| \le \sqrt{n} |f(z)-f^0(z)|\quad \text{ for } \text{ all } z, \end{aligned}$$

we pass to the limit in (63) to find that

$$\begin{aligned} \sqrt{n\pi L} (\sqrt{\pi }\Vert f - f^0\Vert _{L^2} +\eta _1) \le \sqrt{n} \pi \Vert f - f^0\Vert _{L^1}, \end{aligned}$$

in contradiction to the Cauchy-Schwarz inequality. Thus (22) holds. $\square$

4 Compactness in time

In this last section we present the proofs of Proposition 4, Corollary 1 and Proposition 5.

4.1 Proof of Proposition 4

Proof

We only need to prove (26), since all other conclusions follow from that and Proposition 3.

To prove (26), we define the stopping time

$$\begin{aligned} t^* := \sup \{ t>0 : u_\varepsilon (\cdot , \cdot , h_\varepsilon ^2 s) \text{ satisfies } (18), (21) \text{ for } \text{ all } s\in (0,t)\} \end{aligned}$$

where f should be replaced by g in (18), (21). By a change of variables,

$$\begin{aligned} T^g_{r,\varepsilon }(u_\varepsilon (\cdot , \cdot , h_\varepsilon ^2 t)) = T^g_{r}(v_\varepsilon (\cdot , \cdot , t)) , \end{aligned}$$

where $T^g_r := T^g_{r,1}$ and $u_\varepsilon , v_\varepsilon$ are related by (32). We use (35) with ${\varphi }(x,z,t) = \chi ^g_r(x,z)$ to find that

$$\begin{aligned} \frac{d}{dt} T^g_r(v_\varepsilon (\cdot , \cdot , t)) \le \left| \int \varepsilon _{ab}\partial _{ac} \chi ^g_r \ \partial _bv_{\varepsilon }\cdot \partial _cv_{\varepsilon }\,dx\,dz \right| + \left| \int \varepsilon _{ab}\partial _{az} \chi ^g_r \frac{\partial _b v_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}\,dx\,dz \right| . \end{aligned}$$

The definition of $\chi ^g_r$ and assumption $r \le \rho _r/4$ implies that $\partial _{ac}\chi ^g_r(x,z) = 2 \delta ^{ac}$ when $|x-g_i(z)|<r$ for some i, and hence that

$$\begin{aligned} \varepsilon _{ab}\partial _{ac} \chi ^g_r \ \partial _bv_{\varepsilon }\cdot \partial _cv_{\varepsilon }= 0 \text{ in } \cup _i B(g_i(z),r). \end{aligned}$$

In addition,

$$\begin{aligned} |\nabla _x v_\varepsilon |^2 \le 2 e_\varepsilon (|v_\varepsilon |) + \frac{|j(v_\varepsilon )|^2 }{|v_\varepsilon |^2} \le 2 e_\varepsilon (|v_\varepsilon |) + 2 \left| \frac{ju_\varepsilon }{|u_\varepsilon |} - j^*_\omega (h_\varepsilon g)\right| ^2 + 2\left| j^*_\omega (h_\varepsilon g)\right| ^2. \end{aligned}$$

The definition of $t^*$ allows us to apply estimates from Proposition 1 (with $c_1 = c_4 + n\pi L \Vert g\Vert _\infty$ and $c_2 = G_0(g)+1$) and Proposition 3 (with $\delta = \Sigma = 1$ for example) to $v_\varepsilon (\cdot ,\cdot , t)$, for any $t\in (0, t^*)$, as long as $c_4, \varepsilon _4$ are taken to be small enough, depending only on $\Vert g \Vert _{H^1}, n$ and r. We may therefore deduce from (20) that

$$\begin{aligned} \left| \int \varepsilon _{ab}\partial _{ac} \chi ^g_r \ \partial _bv_{\varepsilon }\cdot \partial _cv_{\varepsilon }\,dx\,dz \right| \le C(K_3 + 1 ) \Vert \nabla ^2_x \chi ^g_r\Vert _\infty = C(r, n,g). \end{aligned}$$

The remaining integral on the right-hand side is estimated by using (12) (which after rescaling to $v_\varepsilon$ acquires a factor of ${|\log {\varepsilon }|}^{-1}$) to find that

$$\begin{aligned} \left| \int \varepsilon _{ab}\partial _{az} \chi ^g_r \frac{\partial _b v_{\varepsilon }\cdot \partial _z v_{\varepsilon }}{{|\log {\varepsilon }|}}\,dx\,dz \right|&\le \frac{1}{{|\log {\varepsilon }|}}\Vert \nabla _x \partial _z \chi ^g_r\Vert _{L^\infty } \, \Vert \nabla _x v_\varepsilon \Vert _{L^2} \Vert \nabla _z v_\varepsilon \Vert _{L^2} \\&\le \Vert g\Vert _{Lip} C(c_1,c_2, n) . \end{aligned}$$

Thus

$$\begin{aligned} \frac{d}{dt} T^g_r(v_\varepsilon (\cdot , \cdot , t)) \le C(r, n,\Vert g\Vert _{H^1})+\Vert g\Vert _{Lip} C(c_1,c_2, n) =: C_4. \end{aligned}$$

It follows that (26) holds for all $t\in (0,t^*)$. Then, thanks to (27) and (28), we conclude that $t^*\ge t_4$, completing the proof of (26). $\square$

4.2 Proof of Corollary 1

Proof

Since f(0) may not be a Lipschitz function, we first mollify it to a function which we call g and which we require to satisfy $\sup _{i,z}|f_i(0,z) - g_i(z)| < \alpha \rho _{f(0)}$ for some $\alpha <1/8$ to be chosen, and thus $\rho _g > (1-2\alpha )\rho _{f(0)}.$ Since f(0) is already in $H^1$, we have that $|g|_{H^1} \le |f(0)|_{H^1}$. Proposition 4, applied to g, $r=\rho _g/4$, provides us with constants $\varepsilon _4,t_4,c_4,C_4$, the important point being that $\varepsilon _4$ and $c_4$ do not depend on the strength of the mollification. Without loss of generality, we may also assume that $c_4 \le \frac{1}{8} \rho _{f(0)}$. In view of the assumptions of Theorem 1, we may assume, decreasing the value of $\varepsilon _4$ if necessary, that (23) and (24) hold for every $\varepsilon \le \varepsilon _4.$ Finally, it is clear that $|| \chi ^g_{r,\varepsilon }(\cdot , z)\Vert _{W^{1,\infty }(\omega )} \le C(r)h_\varepsilon ^{-1}$ for every $z\in (0,L)$, so assumption (6) implies that $\limsup _{\varepsilon \rightarrow 0}T^g_{r,\varepsilon }(u_\varepsilon ^0) \le \pi \Vert f(0)-g\Vert _{L^2}^2$. We may therefore assume, decreasing $\varepsilon _4$ further if necessary, that $T^g_{r,\varepsilon }(u^0_\varepsilon )\le 2\pi \Vert f(0)-g\Vert _{L^2}^2 \le 2n\pi ^2\alpha ^2L \rho _{f(0)}^2$ for every $\varepsilon \le \varepsilon _4$, and in particular that (25) holds. In view of (23) and (28), we may then apply Proposition 1 for each fixed time $t \in [0,t_4]$ and derive some limiting $f^*(t)$ after passing to a possible subsequence.

The potential difficulty at this level is that the subsequence may depend on the value of t; to overcome this we will rely on the form of continuity in time provided by estimate (26). We first derive some estimates that apply to any limit $f^*(t)$ produced by the above argument. Note that (27) and (13) imply that

$$\begin{aligned} \frac{1}{h_\varepsilon }\int _0^L \Vert \pi \sum _{i=1}^n \delta _{h_\varepsilon f^*_i(z,t)} - \pi \sum _{i=1}^n \delta _{h_\varepsilon g_i(z)}\Vert _{W^{-1,1}(\omega )} dz \le \left( n\pi L (T^g_{r,\varepsilon }(u_\varepsilon ^0)+C_4 t)\right) ^\frac{1}{2}, \end{aligned}$$

and (14) implies that $\Vert f^*(t)\Vert _{H^1}\le C(G_0(g))$. Using (5),

$$\begin{aligned} \int _0^L \min _{\sigma \in S_n} | f^*_{\sigma (i)}(z,t) - g_i(z)| dz \ \le \left( n \pi L (T^g_{r,\varepsilon }(u_\varepsilon ^0)+C_4 t)\right) ^\frac{1}{2} . \end{aligned}$$

Since $f^*(t)-g$ is uniformly bounded in $H^1$, by choosing $t_0\le t_4$ and $\alpha$ sufficiently small, we conclude that

$$\begin{aligned} \max _z \min _{\sigma \in S_n} | f^*_{\sigma (i)}(z,t) - f^0_i(z)|&< \max _z\min _{\sigma \in S_n} | f^*_{\sigma (i)}(z,t) - g_i(z)| + |f^0_i(z)-g_i(z)|\\&\le \frac{1}{16}\rho _{f(0)} \end{aligned}$$

for all $t\in [0,t_0]$. It follows that there is a single permutation $\sigma$ that attains the min for all z. After relabelling $f^*$ if necessary, we deduce that (29) holds when $s=0$. Finally, using the $L^\infty$ continuity of $s\mapsto f(\cdot , s)$ and decreasing $t_0$ as needed, we deduce that (29) holds for all $s,t\in [0,t_0]$.

To prove continuity in time, we start by using a Cantor diagonal argument to fix a subsequence $\varepsilon \rightarrow 0$ such that

$$\begin{aligned} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z,h_\varepsilon ^2t) - \pi \sum _{j=1}^n \delta _{h_\varepsilon f^*_j(z,t)}\Vert _{W^{-1,1}(\omega )} dz = o(h_\varepsilon ) \qquad \text{ as } \varepsilon \rightarrow 0 \end{aligned}$$

for every time t in ${\mathbb {Q}}\cap [0,t_0]$. We claim that the mapping $t \mapsto f^*(t)$ is uniformly continuous from ${\mathbb {Q}}\cap [0,t_0]$ into $L^1([0,L]).$ Indeed, let $\eta >0$ be given, and let $s_0,s_1 \in {\mathbb {Q}}\cap [0,t_0]$ be arbitrary. We write

$$\begin{aligned} \sum _i \Vert f_i^*(s_0)-f_i^*(s_1)\Vert _{L^1} \le \sum _i \Vert f_i^*(s_0)-g_i^*(s_0)\Vert _{L^1} + \sum _i \Vert g_i^*(s_0)-f_i^*(s_1)\Vert _{L^1} \end{aligned}$$

(68)

where $g^*(s_0)$ is a mollification of $f^*(s_0)$. It follows from (14) that $t\mapsto f^*(t)$ is uniformly bounded with values into $H^1$ , so we may fix the mollification parameter sufficiently fine, but independently of $s_0$, such that

$$\begin{aligned} \sum _i \Vert f_i^*(s_0)-g_i^*(s_0)\Vert _{L^1} \le \eta /2. \end{aligned}$$

(69)

Next, we pass to the limit in the conclusions of Proposition 4 applied this time to $g=g^*(s_0)$ and conclude that

$$\begin{aligned} \begin{aligned}&\pi \sum _i \Vert g_i^*(s_0)-f_i^*(s_1)\Vert _{L^1}\\&\quad = \lim _{\varepsilon \rightarrow 0}h_\varepsilon ^{-1} \int _0^L \Vert J_x u_\varepsilon (\cdot ,z,h_\varepsilon ^2s_1) - \pi \sum _{i=1}^n \delta _{h_\varepsilon g^*_i(z,s_0)}\Vert _{W^{-1,1}(\omega )} dz\\&\quad \le \lim _{\varepsilon \rightarrow 0} \left( n \pi L (T^{g^*(s_0)}_{r,\varepsilon }(u_\varepsilon ^0) + C_4 |s_1-s_0|)\right) ^\frac{1}{2}\\&\quad \le \left( n \pi L ( \pi \Vert g^*(s_0)-f^*(s_0)\Vert _{L^2}^2 +C_4 |s_1-s_0|)\right) ^\frac{1}{2}, \end{aligned} \end{aligned}$$

(70)

where $C_4$ depends only on the mollification parameter. (We have implicitly used the fact that components of $f^*$ have been labelled correctly, as reflected in (29).) We therefore further decrease the mollification parameter if necessary, yet independently of $s_0$, so that $n\pi ^2 L \Vert g^*(s_0)-f^*(s_0)\Vert _{L^2}^2 \le \eta ^2/32.$ Once this, and hence $C_4$ are fixed, we require $|s_0-s_1|$ to be small enough so that $n\pi LC_4 |s_1-s_0| \le \eta ^2/32.$ Combining (69) and (70) in (68) yields the uniform continuity of $f^*.$ In the sequel we denote still by $f^*$ the unique continuous extension of $f^*$ to the whole interval $[0,t_0].$ We claim that the conclusion of Corollary 1 holds for any $t \in [0,t_0],$ with no need of further subsequences. Indeed, this follows from the fact that for each fixed t in $[0,t_0]$ there exist at least some further subsequence for which the convergence to some $f^{**}(t)$ holds (this is by Proposition 1 as we already saw it), and on the other hand by our previous argument (equally applied to the countable set $({\mathbb {Q}}\cap [0,t_0]) \cup \{t\}$) the only possible limit along any such subsequence is necessarily equal to $f^*(t).$ $\square$

4.3 Proof of Proposition 5

Proof

For $r,R>0$, define

$$\begin{aligned} {\mathcal {G}}_{r,R } := \{ (t,x,z)\in [0,t_0]\times B(R) \times [0,L] : |x - f_k^*(z,t)|\ge r, \ k=1,\ldots , n \}. \end{aligned}$$

Given ${\mathcal {O}}$ as in the statement of the Proposition, we may fix $r,R>0$ such that ${\mathcal {O}} \subset {\mathcal {G}}_{r,R}$.

We will only consider $\varepsilon$ small enough that $B(R)\subset \omega _\varepsilon$. It is then rather clear that

$$\begin{aligned} j_{\omega _\varepsilon }^*(f^*(t)) \rightarrow j^*_{{\mathbb {R}}^2}(f^*(t)) \ \ \text{ locally } \text{ uniformly } \text{ on } {\mathcal {G}}_{r,R} \text{ for } \text{ every } r>0. \end{aligned}$$

It thus follows from Proposition 3 (with $\Sigma =\delta =1$, rewritten in terms of $v_\varepsilon$) that

$$\begin{aligned} \left\| \frac{jv_\varepsilon }{|v_\varepsilon |} - j^*_{{\mathbb {R}}^2} \right\| _{{\mathcal {G}}_{r,R}} \le C \end{aligned}$$

for all sufficiently small $\varepsilon$, where C is independent of r and R. By extracting weak limits and employing a Cantor diagonal argument, we conclude that there exists a vector field $H\in L^2([0,t_0]\times {\mathbb {R}}^2\times {\mathbb {T}}_{L})$ such that

$$\begin{aligned} \frac{jv_\varepsilon }{|v_\varepsilon |} - j^*_{{\mathbb {R}}^2}\rightharpoonup H \text{ weakly } \text{ in } L^2({\mathcal {G}}_{r,R}) \text{ for } \text{ every } r,R>0. \end{aligned}$$

Now fix ${\varphi }\in {\mathcal {D}}((0,t_0)\times {\mathbb {R}}^2\times {\mathbb {T}}_{L})$ and compute, for $\varepsilon$ sufficiently small,

$$\begin{aligned} \left| \int \nabla _x^\perp {\varphi }\cdot \frac{jv_\varepsilon }{|v_\varepsilon |} - \int \nabla _x^\perp {\varphi }\cdot jv_\varepsilon \right|&\le \int |\nabla _x^\perp {\varphi }| \, \left| \frac{jv_\varepsilon }{|v_\varepsilon |}\right| \, \left| 1-|v_\varepsilon | \right| = o(1) \end{aligned}$$

(71)

as $\varepsilon \rightarrow 0$, in view of the pointwise inequality $\left| \frac{jv_\varepsilon }{|v_\varepsilon |}\right| \, \left| 1-|v_\varepsilon | \right| \le \varepsilon e_\varepsilon (v_\varepsilon )$ and the energy bound on $v_\varepsilon$. Next, integrating by parts and using Corollary 1 and the definition of $j^*_{{\mathbb {R}}^2}$,

$$\begin{aligned} \int \nabla _x^\perp {\varphi }\cdot jv_\varepsilon = 2 \int {\varphi }Jv_\varepsilon \rightarrow \int _0^{t_0} \int _0^L \sum _{i=1}^n {\varphi }(f_i(z),z)\, dz\, dt = \int \nabla _x^\perp {\varphi }\cdot j^*_{{\mathbb {R}}^2}(f^*) . \end{aligned}$$

By combining these and using the fact that $H\in L^2$, which implies that the singularities along $\{ (t, f_i(z),z) : t\in [0,t_0], z\in [0,L], i=1,\ldots n \}$ are removable, we infer that $\nabla ^\perp \cdot H = 0$ on ${\mathbb {R}}\times {\mathbb {R}}^2\times {\mathbb {T}}_{L}$. Similarly, by (34),

$$\begin{aligned} \int \nabla _x {\varphi }\cdot jv_\varepsilon = -\int \partial _t{\varphi }(|v_\varepsilon |^2-1) + h_\varepsilon ^2 \partial _z {\varphi }\, j_zv_\varepsilon \rightarrow 0, \end{aligned}$$

since $(v_\varepsilon |^2-1)^2 \le 4 \varepsilon ^2e_\varepsilon (v_\varepsilon )$ and $\left| h_\varepsilon ^2 \partial _z {\varphi }\, j_zv_\varepsilon \right| \le h_\varepsilon |\partial _z{\varphi }| (\frac{ |\partial _z v_\varepsilon |^2}{{|\log {\varepsilon }|}}+ |v_\varepsilon |^2 ),$ together with (12), rescaled to read $\Vert \nabla v_\varepsilon (t)\Vert _{L^2(dx\,dz)}^2 \le C{|\log {\varepsilon }|}$ for every $t\in [0,t_0]$. Arguing as in (71) to eliminate the factor of $|v_\varepsilon |$ in the denominator and recalling that $\nabla _x\cdot j^*_{{\mathbb {R}}^2}(f^*)=0$ by definition, we conclude that

$$\begin{aligned} \int \nabla _x \cdot (\frac{jv_\varepsilon }{|v_\varepsilon |} - j^*_{{\mathbb {R}}^2}) \, \rightarrow 0, \end{aligned}$$

and hence that $\nabla _x \cdot H = 0$ in ${\mathcal {D}}'$. We conclude by applying Lemma 10 below to the vector field $w(t,x,z) = \zeta (t)H(t,x,z)$, where $\zeta$ is an arbitrary function with compact support in $[0,t_0]$. $\square$

The proof of Proposition 5 used the following

Lemma 10

Assume that $w\in L^2({\mathbb {R}}\times {\mathbb {R}}^2\times {\mathbb {T}}_{L})$ satisfies

$$\begin{aligned} \nabla _x\cdot w = 0, \qquad \nabla _x^\perp \cdot w = 0 \qquad \text{ in } \mathcal {D}' . \end{aligned}$$

(72)

Then $w=0$.

Proof

If w is smooth, then since $\nabla _x^\perp \cdot w=0$, we may write $w = \nabla _x f$ for some scalar function f. Then the fact that $\nabla \cdot w = 0$ implies that f is harmonic, and hence that w is harmonic. For a.e. $t\in {\mathbb {R}}$ and $z\in {\mathbb {T}}_{L}$,

$$\begin{aligned} \int _{{\mathbb {R}}^2} |w(t,x,z)|^2\, dx = 0, \end{aligned}$$

so Liouville’s Theorem implies that $w(t,\cdot ,z)=0$ for such (t, z),and therefore everywhere in ${\mathbb {R}}\times {\mathbb {R}}^2\times {\mathbb {T}}_{L}$.

If w is not smooth, then we fix an approximate identity $(\eta _\varepsilon )$, and we write $w_\varepsilon := \eta _\varepsilon * w$. Then $w_\varepsilon$ satisfies conditions (72), with $\Vert w_\varepsilon \Vert _{L^2}\le \Vert w\Vert _{L^2} <\infty$ for every $\varepsilon >0$, and $w_\varepsilon \rightarrow w$ in $L^2$, so it follows that $w=0$ a.e. $\square$