Bubble decomposition for the harmonic map heat flow in the equivariant case

Abstract

We consider the harmonic map heat flow for maps $\mathbb {R}^{2} \rightarrow \mathbb {S}^2$, under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times—the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval from Jendrej and Lawrie (J Amer Math Soc).

1 Introduction

1.1 Setting of the problem

Consider the harmonic map heat flow (HMHF) for maps $\Psi : \mathbb {R}^2 \rightarrow \mathbb {S}^2 \subset \mathbb {R}^3$, that is, the heat flow associated to the Dirichlet energy

$$\begin{aligned} E(\Psi ):= \frac{1}{2} \int _{\mathbb {R}^2} \left|{\nabla \Psi (x)}\right|^2 \, \textrm{d}x. \end{aligned}$$

The initial value problem for the HMHF is given by

$$\begin{aligned} \begin{aligned} \partial _t \Psi - \Delta \Psi&= \Psi \left|{\nabla \Psi }\right|^2 \\ \Psi (0, x)&= \Psi _0(x). \end{aligned} \end{aligned}$$

(1.1)

We say a solution to (1.1) is k-equivariant if it takes the form

$$\begin{aligned} \Psi (t, re^{i\theta }) = (\sin u(t, r) \cos k \theta , \sin u(t, r) \sin k \theta , \cos u(t, r)) \in \mathbb {S}^2 \subset \mathbb {R}^3, \end{aligned}$$

where $k \in \mathbb {N}$ and $(r, \theta )$ are polar coordinates on $\mathbb {R}^2$. In this case the HMHF reduces to a scalar equation for the polar angle $u= u(t,r)$,

$$\begin{aligned} \begin{aligned} \partial _t u&= \partial _{r}^2 u + \frac{1}{r} \partial _r u - \frac{k^2}{r^2} \frac{\sin 2u}{2 }, \\ u(0)&= u_0, \end{aligned} \end{aligned}$$

(1.2)

and the energy $E = E(u)$ reduces to

$$\begin{aligned}&E( u(t) ) = 2 \pi \int _0^\infty \frac{1}{2}\left( (\partial _r u(t, r))^2 + k^2 \frac{\sin ^2 (u(t, r))}{r^2} \right) \, r \, \textrm{d}r, \end{aligned}$$

and formally satisfies

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}t} E( u(t)) = - 2 \pi \int _0^\infty (\partial _t u(t, r))^2 \, r \, \textrm{d}r = - 2 \pi \Vert \mathcal T(u(t)) \Vert _{L^2}^2, \end{aligned}$$

where in the k-equivariant setting $\mathcal T(u):= \partial _{r}^2 u + \frac{1}{r} \partial _r u - \frac{k^2}{2r^2} \sin (2u)$ is called the tension of u. Integrating in time from $t_0$ to t gives,

$$\begin{aligned} \begin{aligned}{ E( u(t)) + 2 \pi \int _{t_0}^t \Vert \mathcal T(u(s)) \Vert _{L^2}^2 \, \textrm{d}s = E( u(t_0)). } \end{aligned} \end{aligned}$$

(1.3)

The natural setting in which to consider the initial value problem for (1.2) is the space of initial data $u_0$ with finite energy, $E(u) < \infty$. This set is split into disjoint sectors, $\mathcal {E}_{\ell , m}$, which for $\ell , m\in \mathbb {Z}$, are defined by

$$\begin{aligned} \mathcal E_{\ell , m}:= \big \{ u \mid E(u) < \infty , \quad \lim _{r \rightarrow 0} u(r) = \ell \pi , \quad \lim _{r \rightarrow \infty } u(r) = m \pi \big \}. \end{aligned}$$

These sectors, which are preserved by the flow, are related to the topological degree of the full map $\Psi : \mathbb {R}^2 \rightarrow \mathbb {S}^2$: if $m - \ell$ is even and $u \in \mathcal {E}_{\ell , m}$, then the corresponding map $\Psi$ with polar angle u is topologically trivial, whereas for odd $m - \ell$ the map has degree k.

The sets $\mathcal E_{\ell , m}$ are affine spaces, parallel to the linear space $\mathcal {E}:= \mathcal {E}_{0, 0}$, which we endow with the norm,

$$\begin{aligned} \Vert u_0 \Vert _{\mathcal {E}}^2:= \int _0^\infty \Big ( ( \partial _r u_0(r))^2 + k^2 \frac{ (u_0(r))^2}{r^2} \Big ) \, r \textrm{d}r. \end{aligned}$$

We make note of the embedding $\Vert u_0 \Vert _{L^\infty } \le C \Vert u_0 \Vert _{\mathcal {E}}$.

The unique k-equivariant harmonic map is given explicitly by

$$\begin{aligned} Q(r):= 2 \arctan (r^k). \end{aligned}$$

Here uniqueness means up to scaling, sign change, and adding a multiple of $\pi$, i.e., every finite energy stationary solution to (1.2) takes the form $Q_{\mu , \sigma , m}(r) = m \pi + \sigma Q(r/ \mu )$ for some $\mu \in (0, \infty ), \sigma \in \{0, -1, 1\}$ and $m \in \mathbb {Z}$. The map Q and its rescaled versions $Q_\lambda (r):= Q(\lambda ^{-1}r)$ for $\lambda > 0$, are minimizers of the energy E within the class $\mathcal {E}_{0, 1}$; in fact, $E( Q_\lambda ) = 4 \pi k$.

1.2 Statement of the results

We prove the following theorem.

Theorem 1

(Bubble decomposition) Let $k \in \mathbb {N}$, let $\ell , m \in \mathbb {Z}$, and let u(t) be the solution to (1.2) with initial data $u(0) = u_0 \in \mathcal {E}_{\ell , m}$, defined on its maximal interval of existence $[0,T_+)$.

(Global solution) If $T_+ = \infty$, there exist a time $T_0>0$, an integer $N \ge 0$, continuous functions $\lambda _1(t), \dots , \lambda _N(t) \in C^0([T_0, \infty ))$, signs $\iota _1, \dots , \iota _N \in \{-1, 1\}$, and $g(t) \in \mathcal {E}$ defined by

$$\begin{aligned} \begin{aligned}{ u(t) = m \pi + \sum _{j =1}^N \iota _j ( Q_{\lambda _j(t)} - \pi ) + g(t), } \end{aligned} \end{aligned}$$

(1.4)

such that

$$\begin{aligned} \Vert g(t)\Vert _{\mathcal {E}} + \sum _{j =1}^{N} \frac{\lambda _{j}(t)}{\lambda _{j+1}(t)} \rightarrow 0 {\ \ \text {as} \ \ }t \rightarrow \infty , \end{aligned}$$

where above we use the convention that $\lambda _{N+1}(t) = \sqrt{t}$.

(Blow-up solution) If $T_+ < \infty$, there exist a time $T_0< T_+$, integers $m_{\infty }, m_\Delta$, a mapping $u^*\in \mathcal {E}_{0, m_{\infty }}$, an integer $N \ge 1$, continuous functions $\lambda _1(t), \dots , \lambda _N(t) \in C^0([T_0, T_+))$, signs $\iota _1, \dots , \iota _N \in \{-1, 1\}$, and $g(t) \in \mathcal {E}$ defined by

$$\begin{aligned} \begin{aligned}{ u(t) = m_\Delta \pi + \sum _{j =1}^N \iota _j( Q_{\lambda _j(t)} - \pi ) + u^* + g(t), } \end{aligned} \end{aligned}$$

such that

$$\begin{aligned} \Vert g(t)\Vert _{\mathcal {E}} + \sum _{j =1}^{N} \frac{\lambda _{j}(t)}{\lambda _{j+1}(t)} \rightarrow 0 {\ \ \text {as} \ \ }t \rightarrow T_+, \end{aligned}$$

where above we use the convention that $\lambda _{N+1}(t) = \sqrt{T_+-t}$.

Remark 1.1

Asymptotic decompositions of solutions to (1.2) (in fact for solutions to the Eq. (1.1) without symmetry assumptions) were proved along a sequence of times $t_n \rightarrow T_+$, in a series of works by Struwe [28], Qing [24], Ding-Tian [9], Wang [33], Qing-Tian [25], and Topping [31]. The main contribution of this paper is to show that the decomposition can be taken continuously in time for k-equivariant solutions.

Remark 1.2

In the non-equivariant setting, i.e., for (1.1), Topping [29, 30] made important progress on a related question in the global case, showing the uniqueness of the locations of the bubbling points under restrictions on the configurations of bubbles appearing in the sequential decomposition. His assumption, roughly, is that all of the bubbles concentrating at a certain point have to have the same orientation. We can contrast this assumption with the equivariant setting, where in the decomposition (1.4) subsequent bubbles have opposite orientations.

Remark 1.3

Given Theorem 1, it is natural to ask which configurations of bubbles are possible in the decomposition. Van der Hout [32] showed that there can only be one bubble in the decomposition in the case of equivariant finite time blow-up; see also [2]. In contrast, in the infinite time case, it is expected that there can be equivariant bubble trees of arbitrary size (see recent work of Del Pino, Musso, and Wei [8] for a construction in the case of the critical semi-linear heat equation).

Remark 1.4

There are solutions to the HMHF that develop a bubbling singularity in finite time, the first being the examples of Coron and Ghidaglia [5] (in dimension $d \ge 3$) and Chang, Ding, Ye [4] in the 2d case considered here. Guan, Gustafson, and Tsai [12] and Gustafson, Nakanishi, and Tsai [14] showed that the harmonic maps Q are asymptotically stable in equivariance classes $k \ge 3$, and thus there is no finite time blow up for energies close to Q in that setting. This asymptotic stability result was improved to energies up to 3E(Q) by Gustafson and Roxanas in [13] in equivariance classes $k \ge 4$. For $k=2$, [14] gave examples of solutions exhibiting infnite time blow up and eternal oscillations. Raphaël and Schweyer constructed a stable blow-up regime for $k=1$ in [26] and then blow up solutions with different rates in [27]. Recently, Davila, Del Pino, and Wei [7] constructed examples of solutions simultaneously concentrating a single copy of the ground state harmonic map at distinct points in space.

1.3 Summary of the proof

We give an informal description of the proof of Theorem 1 starting with a summary of the sequential bubbling results as in, e.g., [24, 31], adapted to our setting. A crucial ingredient is a sequential compactness lemma, which says that a sequence of maps with vanishing tension must converge (at least locally in space) to a multi-bubble, which we define as follows.

Definition 1.5

(Multi-bubble configuration) Given $M \in \{0, 1, \ldots \}$, $m \in \mathbb {Z}$, $\mathbf {\iota }= (\iota _1, \ldots , \iota _M) \in \{-1, 1\}^M$ and an increasing sequence $\mathbf {\lambda }= (\lambda _1, \ldots , \lambda _M) \in (0, \infty )^M$, a multi-bubble configuration is defined by the formula

$$\begin{aligned} \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }; r):= m\pi + \sum _{j=1}^M\iota _j\big ( Q_{\lambda _j}(r) - \pi \big ). \end{aligned}$$

Remark 1.6

If $M = 0$, it should be understood that $\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }; r) = m\pi$ for all $r \in (0, \infty )$, where $\mathbf {\iota }$ and $\mathbf {\lambda }$ are 0-element sequences, that is the unique functions $\emptyset \rightarrow \{-1, 1\}$ and $\emptyset \rightarrow (0, \infty )$, respectively.

With this definition, we define a localized distance function to multi-bubble configurations by

$$\begin{aligned} \begin{aligned}{ \varvec{\delta }_{R}(u):= \inf _{m, M, \mathbf {\iota }, \mathbf {\lambda }} \Big ( \Vert u - \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }) \Vert _{\mathcal {E}(r \le R)}^2 + \sum _{j =1}^M \Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^k \Big )^{\frac{1}{2}} } \end{aligned} \end{aligned}$$

(1.5)

where the infimum is taken over all $m \in \mathbb {Z}$, $M \in \{0, 1, 2, \dots \}$, all vectors $\iota \in \{-1, 1\}^M$, $\mathbf {\lambda }\in (0, \infty )^M$, and we use the convention that the last scale $\lambda _{M+1} = R$.

The localized sequential compactness lemma (see Lemma 3.1) says the following: given a sequence of maps $u_n$ with bounded energy, a sequence $\rho _n \in (0, \infty )$ of scales, and tension vanishing in $L^2$ relative to the scale $\rho _n$, i.e., $\lim _{n \rightarrow \infty } \rho _n \Vert \mathcal T(u_n) \Vert _{L^2} = 0$, there exists a subsequence of the $u_n$ that converges to a multi-bubble configuration up to large scales relative to $\rho _n$, i.e., $\lim _{n \rightarrow \infty } \varvec{\delta }_{R_n \rho _n} ( u_n) = 0$ for some sequence $R_n \rightarrow \infty$. An analogous result with no symmetry assumptions was proved by Qing [24] using the local bubbling theory of Struwe [28] together with a delicate elliptic analysis showing that no energy can accumulate on the “neck” regions between the bubbles. Here we give a mostly self-contained proof of this compactness result in the simpler equivariant setting using the theory of profile decompositions of Gérard [11] and an approach in the spirit of Duyckaerts, Kenig, and Merle’s work on nonlinear waves [10]. To control the energy on the neck regions we use a virial-type functional adapted from Jia and Kenig’s proof of sequential soliton resolution for equivariant wave maps [19].

With the compactness lemma in place, we now consider the heat flow. To fix ideas, let u(t) be a solution to (1.2) defined globally in time, i.e., $T_+ = \infty$. By the energy identity (1.3),

$$\begin{aligned} \begin{aligned}{ \int _0^\infty \Vert \mathcal T(u(t)) \Vert _{L^2}^2 \, \textrm{d}t < \infty , } \end{aligned} \end{aligned}$$

(1.6)

and thus we can find a sequence of times $t_n \rightarrow \infty$ so that $\lim _{n \rightarrow \infty }\sqrt{ t_n} \Vert \mathcal T(u(t_n)) \Vert _{L^2} = 0$. From the compactness lemma we deduce that after passing to a subsequence of the $t_n$, $u(t_n)$ converges to an N-bubble configuration up to the self-similar scale $r = \sqrt{t_n}$. In the exterior region $r \gtrsim \sqrt{t}$, we prove that u(t) has vanishing energy (continuously in time) using a localized energy inequality due to Struwe [28]; see Proposition 4.3.

Let $\textbf{d}(t)$ denote the distance to the particular N-bubble configuration obtained via the compactness lemma (which is defined analogously to (1.5), except without the spatial localization; see Definition 5.1). We have so far proved that

$$\begin{aligned} \lim _{n \rightarrow \infty } \textbf{d}(t_n) = 0. \end{aligned}$$

Theorem 1 follows from showing that in fact $\lim _{t \rightarrow \infty } \textbf{d}(t) = 0$. We assume that continuous-in-time convergence of $\textbf{d}(t)$ fails. To reach a contradiction we study time intervals on which bubbles come into collision (i.e., where $\textbf{d}(t)$ grows), adapting the notion of a collision interval from our paper [17].

We say that an interval [a, b] is a collision interval with parameters $0<\epsilon < \eta$ and $N-K$ exterior bubbles for some $1 \le K \le N$, if $\textbf{d}(a) \le \epsilon$, $\textbf{d}(b) \ge \eta$, and there exists a curve $r = \rho _K(t)$ outside of which u(t) is within $\epsilon$ of an $N-K$-bubble (in the sense of a localized version of $\textbf{d}(t)$); see Defintion 5.4. We now define K to be the smallest non-negative integer for which there exists $\eta >0$, a sequence $\epsilon _n \rightarrow 0$, and sequences $a_n, b_n \rightarrow \infty$, so that $[a_n, b_n]$ are collision intervals with parameters $\epsilon _n, \eta$ and $N-K$ exterior bubbles, and we write $[a_n, b_n] \in \mathcal C_K( \epsilon _n, \eta )$; see Sect. 5.1 for the proof that K is well-defined and $\ge 1$, under the contradiction hypothesis.

Consider a sequence of collision intervals $[a_n, b_n] \in \mathcal C_K( \epsilon _n, \eta )$. Near the endpoint $a_n$, u(t) is close to an N-bubble configuration and we denote the interior scales, which will come into collision, by $\mathbf {\lambda }= ( \lambda _1, \dots , \lambda _K)$ and the exterior scales, which stay coherent, by $\mathbf {\mu }= ( \mathbf {\mu }_{K+1}, \dots , \mathbf {\mu }_N)$. The crucial point is that the minimality of K allows us to relate the scale of the Kth bubble $\lambda _K$ to the lengths of the collision intervals $b_n - a_n$. We prove, roughly, that for sufficiently large n the collision intervals $[a_n, b_n]$ contain subintervals $[c_n, d_n]$ on which (1) $\inf _{t \in [c_n, d_n]}\textbf{d}(t) \ge \alpha$ for some $\alpha >0$, (2) the scale $\lambda _K(t)$ stays roughly constant on $[c_n, d_n]$, and (3) the lower bound $d_n - c_n \gtrsim n^{-1} \lambda _K(c_n)^2$ holds. The compactness lemma and the lower bound $\textbf{d}(t) \ge \alpha$ together yield a lower bound on the tension $\inf _{t \in [c_n, d_n]}\lambda _K(c_n)^2\Vert \mathcal T( u(t) ) \Vert _{L^2}^2 \gtrsim 1$ where the scale $\lambda _K$ appears again due to the definition of K. The last two sentences lead to an immediate contradiction from the boundedness of the integral (1.6), i.e.,

$$\begin{aligned} C \ge \int _0^\infty \Vert \mathcal T(u(t) ) \Vert _{L^2}^2 \, \textrm{d}t \ge \sum _n \int _{c_n}^{d_n} \Vert \mathcal T(u(t) ) \Vert _{L^2}^2 \, \textrm{d}t \gtrsim \sum _n n^{-1}, \end{aligned}$$

which proves that $\lim _{t \rightarrow \infty } \textbf{d}(t)= 0$.

1.4 Notational conventions

The energy is denoted E, $\mathcal {E}$ is the energy space, $\mathcal {E}_{\ell , m}$ are the finite energy sectors. We use the notation $\mathcal {E}(r_1, r_2)$ to denote the local energy norm

$$\begin{aligned} \Vert g\Vert _{\mathcal {E}(r_1, r_2)}^2:= \int _{r_1}^{r_2} \Big ( (\partial _r g)^2 + \frac{k^2}{r^2}g^2\Big )\,r\textrm{d}r, \end{aligned}$$

By convention, $\mathcal {E}(r_0):= \mathcal {E}(r_0, \infty )$ for $r_0 > 0$. The local nonlinear energy is denoted $E(\varvec{u}_0; r_1, r_2)$. We adopt similar conventions as for $\mathcal {E}$ regarding the omission of $r_2$, or both $r_1$ and $r_2$.

Given a function $\phi (r)$ and $\lambda >0$, we denote by $\phi _{\lambda }(r) = \phi (r/ \lambda )$, the $\mathcal {E}$-invariant re-scaling, and by $\phi _{\underline{\lambda }}(r) = \lambda ^{-1} \phi (r/ \lambda )$ the $L^2$-invariant re-scaling. We denote by $\Lambda :=r \partial _r$ and $\underline{\Lambda }:= r \partial r +1$ the infinitesimal generators of these scalings. We denote $\left\langle {\cdot \mid \cdot }\right\rangle$ the radial $L^2(\mathbb {R}^2)$ inner product given by,

$$\begin{aligned} \big \langle \phi \mid \psi \big \rangle := \int _0^\infty \phi (r) \psi (r) \,r \, \textrm{d}r. \end{aligned}$$

We denote k the equivariance degree and $f(u):= \frac{1}{2} \sin 2u$ the nonlinearity in (1.2). We let $\chi$ be a smooth cut-off function, supported in $r \le 2$ and equal 1 for $r \le 1$.

We call a “constant” a number which depends only on the equivariance degree k and the number of bubbles N. Constants are denoted $C, C_0, C_1, c, c_0, c_1$. We write $A \lesssim B$ if $A \le CB$ and $A \gtrsim B$ if $A \ge cB$. We write $A \ll B$ if $\lim _{n\rightarrow \infty } A / B = 0$.

For any sets X, Y, Z we identify $Z^{X\times Y}$ with $(Z^Y)^X$, which means that if $\phi : X\times Y \rightarrow Z$ is a function, then for any $x \in X$ we can view $\phi (x)$ as a function $Y \rightarrow Z$ given by $(\phi (x))(y):= \phi (x, y)$.

2 Preliminaries

2.1 Well-posedness

The starting point for our analysis is the following result of Struwe [28], which says that the initial value problem for the harmonic map flow is well-posed for data in the energy space.

Lemma 2.1

(Local well-posedness)[28, Theorem 4.1] For each $\ell , m \in \mathbb {Z}$ and $u_0\in \mathcal {E}_{\ell , m}$ there exists a maximal time of existence $T_+= T_+(u_0)$ and a unique solution $u(t) \in \mathcal {E}_{\ell , m}$ to (1.2) on the time interval $t \in [0, T_+)$ with $u(0)= u_0$. The maximal time is characterized by the following condition: if $T_+<\infty$, there exists $\epsilon _0>0$ such that

$$\begin{aligned} \begin{aligned}{ \limsup _{ t \rightarrow T_+} E(u(t); 0, r_0) \ge \epsilon _0, } \end{aligned} \end{aligned}$$

(2.1)

for all $r_0>0$. If there is no such $T_+<\infty$, we say $T_+ = \infty$ and the flow is globally defined.

The energy E(u(t)) is absolutely continuous and non-increasing as a function of $t \in [0, T]$ for any $T < T_+$, and for any $t_1 \le t_2 \in [0, T_+)$, there holds,

$$\begin{aligned}\begin{aligned}{ E(u(t_2)) + 2\pi \int _{t_1}^{t_2} \int _0^\infty ( \partial _t u(t, r))^2 \,r \, \textrm{d}r \textrm{d}t = E(u(t_1)). } \end{aligned} \end{aligned}$$

In particular,

$$\begin{aligned} \begin{aligned}{ \int _0^{T_+} \int _0^\infty ( \partial _t u(t, r))^2 \,r \, \textrm{d}r \textrm{d}t \le E(u_0). } \end{aligned} \end{aligned}$$

(2.2)

Remark 2.2

Local well-posedness is proved by Struwe for the HMHF without symmetry assumptions in the case of maps from a closed Riemann surface $\mathcal M\rightarrow \mathbb {S}^2$. For the case of maps from $\mathbb {R}^2$ we refer the reader to Lin and Wang [20, Theorem 5.2.1] for the short time existence of regular solutions. As equivariant symmetry is preserved by the flow, we obtain regular equivariant solutions to (1.2) by taking equivariant initial data. Solutions with finite energy initial data are then obtained as limits of smooth solutions, and in [28] Struwe proved these solutions are regular, e.g., $C^2$, on any compact time interval $[\tau , T] \subset (0, T_+)$. We note that in the equivariant case the energy can only concentrate at the origin $r=0$, giving the form of the blow-up criterion in (2.1).

2.2 Basic estimates

Lemma 2.3

Fix integers $\ell , m$. For every $\epsilon >0$ and $R_0 >1$, there exists a $\delta >0$ with the following property. Let $0 \le R_1 < R_2\le \infty$ with $R_2/ R_1 \ge R_0$, and $u \in \mathcal {E}_{\ell , m}$ be such that $E( u; R_1, R_2) < \delta$. Then, there exists $\ell _0 \in \mathbb {Z}$ such that $| u(r) - \ell _0 \pi |<\epsilon$ for almost all $r \in (R_1, R_2)$.

Moreover, there exist constants $C=C(R_0), \alpha = \alpha (R_0)>0$ such that if $E(u; R_1, R_2)< \alpha$, then

$$\begin{aligned} \begin{aligned}{ \Vert u - \ell _0 \pi \Vert _{\mathcal {E}(R_1, R_2)} \le C E( u; R_1, R_2). } \end{aligned} \end{aligned}$$

(2.3)

Proof

By an approximation argument we can assume $u \in \mathcal {E}_{\ell , m}$ is smooth. First, we show that for any $\epsilon _0>0$, there exists $r_0 \in [R_1, R_2]$ such that $| u(r_0) - \ell _0 \pi |< \epsilon _0$ for some $\ell _0 \in \mathbb {Z}$ as long as $E( u; R_1, R_2)$ is sufficiently small. If not, one could find $\epsilon _1>0$, $0< R_1< R_2$, and a sequence $u_n \in \mathcal {E}_{\ell , m}$ so that $E( u_n; R_1; R_2) \rightarrow 0$ as $n \rightarrow \infty$ but such that $\inf _{r \in [R_1, R_2], \ell \in \mathbb {Z}} | u_n(r) - \ell \pi | \ge \epsilon _1$. The latter condition gives a constant $c( \epsilon _1)>0$ such that $\inf _{r \in [R_1, R_2]} |\sin ( u_n(r))| \ge c(\epsilon _1)$. But then

$$\begin{aligned} E( u_n; R_1; R_2) \ge \frac{k^2}{2} \int _{R_1}^{R_2} \sin ^2( u_n(r)) \, \frac{\textrm{d}r}{r} \ge \frac{k^2}{2} c(\epsilon _1)^2 \log (R_2/R_1), \end{aligned}$$

which is a contradiction. Next define the function, $G(u) = \int _0^u \left|{ \sin \rho }\right| \, \textrm{d}\rho ,$ and for $r_1 \in (R_1, R_2)$ note the inequality,

$$\begin{aligned} \left|{G(u(r_0)) - G( u(r_1))}\right| = \Big |\int _{u(r_1)}^{u(r_0)} \left|{ \sin \rho }\right| \, \textrm{d}\rho \Big | \le \int _{r_1}^{r_0} \left|{ \sin u(r) }\right| \left|{ \partial _r u(r)}\right| \, \textrm{d}r \lesssim E(u; R_1, R_2). \end{aligned}$$

We conclude using that G is continuous and increasing that $| u(r) - \ell _0 \pi |<\epsilon$ for all $r \in (R_1, R_2)$. As long as $\epsilon >0$ is small enough we see that in fact, $\sin ^2(u(r)) \ge \frac{1}{2}| u(r) - \ell _0 \pi |^2$ for all $r \in (R_1, R_2)$ and (2.3) follows.$\square$

Given a mapping $u: (0, \infty ) \rightarrow \mathbb {R}$ we define its energy density,

$$\begin{aligned} \textbf{e}(u(r), r):= \frac{1}{2} \left( (\partial _r u(r))^2 + \frac{k^2}{r^2} \sin ^2(u(r)) \right) . \end{aligned}$$

Lemma 2.4

(Localized energy inequality) Let $I \subset [0, \infty )$ be a time interval, and let $\phi : I \times (0, \infty ) \rightarrow [0, \infty )$ be a smooth function. Let $u(t)\in \mathcal {E}_{\ell , m}$ be a solution to (1.2) on I. Then, for any $t_1<t_2 \in I$,

$$\begin{aligned} \begin{aligned}&\int _{t_1}^{t_2} \int _0^\infty (\partial _t u (t, r))^2 \phi (t, r)^2 \, r \, \textrm{d}r \textrm{d}t + \int _0^\infty \textbf{e}(u(t_2, r), r) \phi (t_2, r)^2 \, r \, \textrm{d}r \quad \quad \\&\quad =\int _0^\infty \textbf{e}(u(t_1, r), r) \phi (t_1, r)^2 \, r\, \textrm{d}r - 2\int _{t_1}^{t_2} \int _0^\infty \partial _t u(t, r) \partial _r u(t, r) \phi (t, r) \partial _r \phi (t, r)\,r \, \textrm{d}r \textrm{d}t \quad \quad \\&\quad \quad + 2\int _{t_1}^{t_2} \int _0^\infty \textbf{e}(u(t, r), r)\phi (t, r) \partial _t \phi (t, r) \, r \, \textrm{d}r \, \textrm{d}t\quad \quad \end{aligned} \end{aligned}$$

(2.4)

If $\phi (t, r)$ satisfies, $\partial _t \phi (t, r) \le 0$ for all $t \in [t_1, t_2]$ then,

$$\begin{aligned} \begin{aligned} \int _0^\infty \textbf{e}(u(t_2), r)&\phi (t_2, r)^2 \, r \, \textrm{d}r + \frac{1}{2} \int _{t_1}^{t_2} \int _0^\infty (\partial _t u (t, r))^2 \phi (t, r)^2 \, r \, \textrm{d}r \textrm{d}t \\&\le \int _0^\infty \textbf{e}(u(t_1), r) \phi (t_1, r)^2 \, r\, \textrm{d}r + 2\int _{t_1}^{t_2}\int _0^\infty (\partial _r u(t, r))^2 (\partial _r \phi (t, r))^2 \, r \,\textrm{d}r \textrm{d}t, \end{aligned} \end{aligned}$$

(2.5)

and,

$$\begin{aligned} \begin{aligned}&\int _0^\infty \textbf{e}(u(t_2), r) \phi (t_2, r)^2 \, r \, \textrm{d}r + \int _{t_1}^{t_2} \int _0^\infty (\partial _t u (t, r))^2 \phi (t, r)^2 \, r \, \textrm{d}r \textrm{d}t \\&\quad \le \int _0^\infty \textbf{e}(u(t_1), r) \phi (t_1, r)^2 \, r\, \textrm{d}r + 2\sqrt{E(u(t_1))}(t_2 - t_1)^{\frac{1}{2}}\\&\qquad \Big (\int _{t_1}^{t_2}\int _0^\infty (\partial _t u(t, r))^2 (\partial _r \phi (t, r))^2(\phi (t, r))^2 \, r \,\textrm{d}r \textrm{d}t \Big )^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

(2.6)

Proof

By an approximation argument we may assume that u is smooth. Then (2.4) is obtained for smooth solutions to (1.2) by multiplying the equation by $\partial _t u \phi ^2$ and integrating by parts. The subsequent inequalities follow from Cauchy-Schwarz.$\square$

2.3 Profile decomposition

We state a profile decomposition in the sense of Gérard [11], adapted to sequences of functions in the affine spaces $\mathcal {E}_{\ell , m}$; see also [1, 3, 21,22,23]. We use the analysis of sequences in $\mathcal {E}_{\ell , m}$ by Jia and Kenig in [19], which synthesized Côte’s analysis in [6].

Lemma 2.5

(Linear profile decomposition) Let $\ell , m \in \mathbb {Z}$ and let $u_n$ be a sequence in $\mathcal E_{\ell , m}$ with $\limsup _{n \rightarrow \infty } E( u_n) <\infty$. Then, there exists $K_0 \in \{0, 1, 2, \dots \}$, sequences $\lambda _{n, j} \in (0, \infty )$ for $j \in \{1, \dots , K_0\}$, $\sigma _{n, i} \in (0, \infty )$ for $i \in \mathbb {N}$, as well as mappings $\psi ^j \in \mathcal {E}_{\ell _j, m_j}$ with $E( \psi ^j) < \infty$, and mappings $v^{i} \in \mathcal {E}_{0, 0}$ such that for each $J \ge 1$,

$$\begin{aligned} u_n&= m \pi + \mathop \sum \limits _{{j = 1}}^{{K_{0} }} ( \psi ^j \big ( \frac{ \cdot }{\lambda _{n, j}} \big ) - m_j \pi ) + \mathop \sum \limits _{{i = 1}}^{J} v^i \big ( \frac{ \cdot }{\sigma _{n, i}} \big ) + w_{n}^J( \cdot ) \end{aligned}$$

so that,

• the parameters $\lambda _{n, j}$ satisfy

  $$\begin{aligned} \lambda _{n, 1} \ll \lambda _{n, 2} \ll \dots \ll \lambda _{n, K_0} {\ \ \text {as} \ \ }n \rightarrow \infty ; \end{aligned}$$

  and for each j one of $\lambda _{n, j} \rightarrow 0$, $\lambda _{n, j} = 1$ for all n, or $\lambda _{n, j} \rightarrow \infty$ as $n \rightarrow \infty$, holds;

• for each i either $\sigma _{n, i} \rightarrow 0$, $\sigma _{n, i} = 1$ for all n, or $\sigma _{n, i} \rightarrow \infty$ as $n \rightarrow \infty$;

• for each $i \in \mathbb {N}$,

  $$\begin{aligned} \frac{\lambda _{n, j}}{\sigma _{n, i}} + \frac{\sigma _{n, i}}{\lambda _{n, j}} \rightarrow \infty {\ \ \text {as} \ \ }n \rightarrow \infty \quad \forall j = 1, \dots , K_0; \end{aligned}$$

• the scales $\sigma _{n, i}$ satisfy,

  $$\begin{aligned} \frac{\sigma _{n, i}}{\sigma _{n, i'}} + \frac{ \sigma _{n, i'}}{\sigma _{n, i}} \rightarrow \infty {\ \ \text {as} \ \ }n \rightarrow \infty ; \end{aligned}$$

• the integers $\ell _j$ and $m_j$ satisfy, $\left|{\ell _j - m_j}\right| \ge 1$, and,

  $$\begin{aligned} \ell = m + \sum _{j=1}^{K_0} ( \ell _j - m_j); \end{aligned}$$

• the error term $w_{n}^J$ satisfies,

  $$\begin{aligned}&w_{n}^J( \lambda _{n, j} \cdot ) \rightharpoonup 0 \in \mathcal {E}{\ \ \text {as} \ \ }n \rightarrow \infty \\&w_{n }^J( \sigma _{n, i} \cdot ) \rightharpoonup 0\in \mathcal {E}{\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

  for each $J \ge 1$, each $j = 1, \dots , K_0$, and $i \in \mathbb {N}$, and vanishes strongly in the sense that

  $$\begin{aligned} \begin{aligned}{ \lim _{J \rightarrow \infty } \limsup _{n \rightarrow \infty } \Vert w_{n}^J \Vert _{L^\infty } = 0; } \end{aligned} \end{aligned}$$

  (2.7)

• the following pythagorean decomposition of the nonlinear energy holds: for each $J \ge 1$,

  $$\begin{aligned} E( u_n)&= \sum _{j =1}^{K_0} E( \psi ^j) + \sum _{i =1}^J E( v^j) + E( w_{n}^J) + o_n(1) \end{aligned}$$

  as $n \rightarrow \infty$.

Proof

(Sketch of Proof) We follow Jia and Kenig’s argument [19, Proof of Lemma 5.5] to first extract the profiles $\psi ^j \in \mathcal {E}_{\ell _j, m_j}$ at the scales $\lambda _{n, j}$, see [19, Pages 1594–1600]. Since these all have energy $\ge E( Q)$, there can only be finitely many of them, which defines the non-negative integer $K_0$. The conclusion of their argument yields a sequence,

$$\begin{aligned} h_n:= u_n - m \pi - \sum _{j= 1}^{K_0} (\psi ^j_{\lambda _{n, j}} - m_j \pi ) \in \mathcal {E}_{0, 0} \end{aligned}$$

with $\limsup _{n \rightarrow \infty } \Vert h_n \Vert _{\mathcal {E}} < \infty$. Setting $H_n:= r^{-k} h_n$ we see that $\limsup _{n \rightarrow \infty } \Vert H_n \Vert _{\dot{H}^{1}(\mathbb {R}^d)} < \infty$ for $d = 2k + 2$ (here we view $H_n$ as a sequence of radially symmetric functions on $\mathbb {R}^d$). Thus we may apply Gérard’s profile decomposition [11, Theorem 1.1] for sequences in $\dot{H}^1(\mathbb {R}^d)$ to the sequence $H_n$ obtaining sequences of scales $\sigma _{n, i}$ and profiles $V^i$ so that for $W_n^J$ defined by

$$\begin{aligned} H_n = \sum _{i=1}^J \sigma _{n, i}^{-\frac{d}{p^*}} V\Big ( \frac{\cdot }{\sigma _{n, i}} \Big ) + W_{n}^J \end{aligned}$$

we have

$$\begin{aligned} \lim _{J \rightarrow \infty } \limsup _{n \rightarrow \infty } \Vert W_{n}^J \Vert _{L^{p^*}} = 0, \end{aligned}$$

along with the usual orthogonality of the scales and the pythagorean expansion of the $\dot{H}^1$ norm. Note that here $p^*:= \frac{ 2d}{d-2}$ is the critical Sobolev exponent. We set $v^i(r):= r^k V^i(r)$ and $w_n^J(r):= r^{k} W_{n}^J(r)$ for each i, n, J. Note that $w_n^J \in \mathcal {E}$ and

$$\begin{aligned} \begin{aligned}{ \lim _{J \rightarrow \infty } \limsup _{n \rightarrow \infty } \int _0^\infty (w_n^J(r))^{p^*} \, \frac{\textrm{d}r}{r} = 0. } \end{aligned} \end{aligned}$$

(2.8)

We conclude by observing the inequality

$$\begin{aligned} \sup _{r>0} \left|{ w(r)}\right|^{\frac{p^*}{2} +1} \le C(p^*) \left( \int _0^\infty (w(r))^{p^*} \, \frac{\textrm{d}r}{r} \right) ^{\frac{1}{2}} \left( \int _0^\infty (\partial _r w(r))^{2} \, r \, \textrm{d}r\right) ^{\frac{1}{2}}, \end{aligned}$$

which holds for all $w \in \mathcal {E}$. Thus (2.8) combined with the above gives the vanishing of the error as in (2.7).$\square$

2.4 Multi-bubble configurations

We study properties of finite energy maps near a multi-bubble configuration as in Definition 1.5. We record here several lemmas proved in [17].

The operator $\mathcal {L}_{\mathcal Q}$ obtained by linearization of (1.2) about an M-bubble configuration $\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })$ is given by,

$$\begin{aligned}\begin{aligned}{ \mathcal {L}_{\mathcal Q} \, g:= {\text {D}}^2 E(\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })) g = - \partial _r^2 g - \frac{1}{r} \partial _r g + \frac{k^2}{r^2} f'(\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }) )g, } \end{aligned} \end{aligned}$$

where $f'( z) = \cos 2 z$. Given $g \in \mathcal {E}$,

$$\begin{aligned} \big \langle {\text {D}}^2 E(\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })) g \mid g \big \rangle = \int _0^\infty \left( (\partial _r g(r))^2 + \frac{k^2}{r^2} f'(\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })) g(r)^2 \, \right) r \textrm{d}r. \end{aligned}$$

An important instance of the operator $\mathcal {L}_{\mathcal Q}$ is given by linearizing (1.2) about a single harmonic map $\mathcal Q(m, M, \mathbf {\iota }, \mathbf {\lambda }) = Q_{\lambda }$. In this case we use the short-hand notation,

$$\begin{aligned}\begin{aligned}{ \mathcal {L}_{\lambda }:= \left( -\Delta + \frac{k^2}{r^2}\right) + \frac{k^2}{r^2} \left( f'(Q_{\lambda }) - 1\right) } \end{aligned} \end{aligned}$$

We write $\mathcal L:= \mathcal L_1$. For each $k \ge 1$,

$$\begin{aligned} \Lambda Q(r):= r \partial _r Q(r) = k \sin Q = 2 k \frac{ r^k}{1 + r^{2k}} \end{aligned}$$

When $k \ge 2$, $\Lambda Q$ is a zero energy eigenfunction for $\mathcal L$, i.e.,

$$\begin{aligned} \mathcal L\Lambda Q = 0, {\ \ \text {and} \ \ }\Lambda Q \in L^2_{\text {rad}}(\mathbb {R}^2). \end{aligned}$$

When $k=1$, $\mathcal L\Lambda Q = 0$ holds but $\Lambda Q \not \in L^2$ due to slow decay as $r \rightarrow \infty$ and 0 is called a threshold resonance.

We define a smooth non-negative function $\mathcal Z\in C^{\infty }(0, \infty ) \cap L^1((0, \infty ), r\, \textrm{d}r)$ by

$$\begin{aligned} \begin{aligned}{ \mathcal Z(r):= {\left\{ \begin{array}{ll} \chi (r) \Lambda Q(r) {\ \ \text {if} \ \ }k =1, 2 \\ \Lambda Q(r) {\ \ \text {if} \ \ }k \ge 3 \end{array}\right. } } \end{aligned} \end{aligned}$$

(2.9)

and note that

$$\begin{aligned} \left\langle { \mathcal Z\mid \Lambda Q}\right\rangle >0 . \end{aligned}$$

The precise form of $\mathcal Z$ is not so important, rather only that it is not perpendicular to $\Lambda Q$ and has sufficient decay and regularity. We fix it as above because of the convenience of setting $\mathcal Z= \Lambda Q$ if $k\ge 3$. We record the following localized coercivity lemma proved in [18].

Lemma 2.6

(Localized coercivity for $\mathcal {L}$)[18, Lemma 5.4] Fix $k \ge 1$. There exist uniform constants $c< 1/2, C>0$ with the following properties. Let $g \in \mathcal {E}$. Then,

$$\begin{aligned}\begin{aligned} \left\langle { \mathcal {L}g \mid g}\right\rangle \ge c \Vert g \Vert _{H}^2 - C\left\langle { \mathcal Z\mid g}\right\rangle ^2 \end{aligned} \end{aligned}$$

If $R>0$ is large enough then,

$$\begin{aligned}\begin{aligned} (1-2c)&\int _0^{R} \Big ((\partial _r g)^2 + k^2 \frac{g^2}{r^2} \Big ) \, r \textrm{d}r + c \int _{R}^\infty \Big ((\partial _r g)^2 + k^2 \frac{g^2}{r^2} \Big ) \, r \textrm{d}r + \left\langle \frac{ k^2}{r^2}(f'(Q) - 1) g\mid g \right\rangle \\&\ge - C\left\langle { \mathcal Z\mid g}\right\rangle ^2. \end{aligned} \end{aligned}$$

If $r>0$ is small enough, then

$$\begin{aligned}\begin{aligned} (1-2c)&\int _r^{\infty } \Big ((\partial _r g)^2 + k^2 \frac{g^2}{r^2} \Big ) \, r \textrm{d}r + c \int _{0}^r \Big ((\partial _r g)^2 + k^2 \frac{g^2}{r^2} \Big ) \, r \textrm{d}r + \left\langle \frac{ k^2}{r^2}(f'(Q) - 1) g\mid g\right\rangle \\&\ge - C\left\langle { \mathcal Z\mid g}\right\rangle ^2. \end{aligned} \end{aligned}$$

As a consequence, (see for example [16, Proof of Lemma 2.4] for an analogous argument) one obtains the following coercivity property of the operator $\mathcal {L}_{\mathcal Q}$.

Lemma 2.7

[17, Lemma 2.19] Fix $k \ge 1$, $M \in \mathbb {N}$. There exist $\eta , c_0>0$ with the following properties. Consider the subset of M-bubble configurations $\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })$ for $\mathbf {\iota }\in \{-1, 1\}^M$, $\mathbf {\lambda }\in (0, \infty )^M$ such that,

$$\begin{aligned} \begin{aligned}{ \sum _{j =1}^{M-1} \Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^k \le \eta ^2. } \end{aligned} \end{aligned}$$

(2.10)

Let $g \in H$ be such that

$$\begin{aligned} 0 = \left\langle { \mathcal Z_{\underline{\lambda _j}} \mid g}\right\rangle {\ \ \text {for} \ \ }j = 1, \dots M. \end{aligned}$$

for some $\mathbf {\lambda }$ as in (2.10). Then,

$$\begin{aligned} \left\langle { {\text {D}}^2 E( \mathcal Q( m, \mathbf {\iota }, \mathbf {\lambda })) g \mid g}\right\rangle \ge c_0 \Vert g \Vert _{\mathcal {E}}^2. \end{aligned}$$

The following technical lemma is useful when computing interactions between bubbles at different scales.

Lemma 2.8

For any $\lambda \le \mu$ and $\alpha , \beta > 0$ with $\alpha \ne \beta$ the following bound holds:

$$\begin{aligned} \int _0^{\infty } \max \Big (1, \frac{r}{\lambda }\Big )^{-\alpha }\max \Big (1, \frac{\mu }{r}\Big )^{-\beta } \frac{\textrm{d}r}{r} \lesssim _{\alpha , \beta } \Big (\frac{\lambda }{\mu }\Big )^{\min (\alpha , \beta )}. \end{aligned}$$

For any $\alpha > 0$ the following bound holds:

$$\begin{aligned} \int _0^\infty \max \Big (1, \frac{r}{\lambda }\Big )^{-\alpha }\max \Big (1, \frac{\mu }{r}\Big )^{-\alpha } \frac{\textrm{d}r}{r} \lesssim _{\alpha } \Big (\frac{\lambda }{\mu }\Big )^{\alpha }\Big ( 1+ \log \Big (\frac{\mu }{\lambda }\Big )\Big ). \end{aligned}$$

Proof

This is a straightforward computation, considering separately the regions $0 < r \le \lambda$, $\lambda \le r \le \mu$, and $r \ge \mu$.$\square$

Using the above, along with the formula for $\mathcal Z$ in (2.9) we obtain the following.

Corollary 2.9

Let $\mathcal Z$ be as in (2.9) and suppose that $\lambda , \mu >0$ satisfy $\lambda / \mu \le 1$. Then,

$$\begin{aligned} \left\langle { \mathcal Z_{\underline{\lambda }} \mid \Lambda Q_{\underline{\mu }}}\right\rangle \lesssim {\left\{ \begin{array}{ll} (\lambda /\mu )^{k+1} {\ \ \text {if} \ \ }k=1, 2 \\ (\lambda / \mu )^{k-1} {\ \ \text {if} \ \ }k \ge 3 \end{array}\right. }, \quad \left\langle { \mathcal Z_{\underline{\mu }} \mid \Lambda Q_{\underline{\lambda }}}\right\rangle \lesssim {\left\{ \begin{array}{ll} 1 {\ \ \text {if} \ \ }k =1 \\ (\lambda /\mu )^{k-1} {\ \ \text {if} \ \ }k \ge 2\end{array}\right. } \end{aligned}$$

Lemma 2.8 is also used to prove the following lemma from [17] giving leading order terms in an expansion of the nonlinear energy functional about an M-bubble configuration. We refer the reader to [17] for the proof.

Lemma 2.10

[17, Lemma 2.22] Fix $k\ge 1, M \in \mathbb {N}$. For any $\theta >0$, there exists $\eta >0$ with the following property. Consider the subset of M-bubble $\mathcal Q(m,\iota , \mathbf {\lambda })$ configurations such that

$$\begin{aligned} \sum _{j =1}^{M-1} \Big ( \frac{ \lambda _{j}}{\lambda _{j+1}} \Big )^k \le \eta . \end{aligned}$$

Then,

$$\begin{aligned}\begin{aligned}{ \Big | E( \mathcal Q( m, \mathbf {\iota }, \mathbf {\lambda })) - M E( Q) - 16 k \pi \sum _{j =1}^{M-1} \iota _j \iota _{j+1} \Big ( \frac{ \lambda _{j}}{\lambda _{j+1}} \Big )^k \Big | \le \theta \sum _{j =1}^{M-1} \Big ( \frac{ \lambda _{j}}{\lambda _{j+1}} \Big )^k. } \end{aligned} \end{aligned}$$

Moreover, there exists a uniform constant $C>0$ such that for any $g \in H$,

$$\begin{aligned}\begin{aligned}{ \left|{\left\langle { {\text {D}}E( \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })) \mid g}\right\rangle }\right| \le C \Vert g \Vert _{\mathcal {E}} \sum _{j =1}^M \Big ( \frac{\lambda _{j}}{\lambda _{j+1}} \Big )^k. } \end{aligned} \end{aligned}$$

The following (standard) modulation lemma plays an important role and we refer the reader to [17, Lemma 2.25] for its proof. Before stating it, we define a proximity function to M-bubble configurations. Fixing m, M we observe that $\mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }; r)$ is an element of $\mathcal {E}_{\ell , m}$, where

$$\begin{aligned} \begin{aligned}{ \ell = \ell (m, M, \mathbf {\iota }): = m - \sum _{j=1}^M \iota _j } \end{aligned} \end{aligned}$$

(2.11)

Definition 2.11

Fix m, M as in Definition 1.5 and let $v \in \mathcal {E}_{\ell , m}$ for some $\ell \in \mathbb {Z}$. Define,

$$\begin{aligned} \begin{aligned}{ \textbf{d}( v) = \textbf{d}_{ m, M}( v):= \inf _{\mathbf {\iota }, \mathbf {\lambda }} \left( \Vert v - \mathcal Q( m, \mathbf {\iota }, \mathbf {\lambda }) \Vert _{\mathcal {E}}^2 + \sum _{j =1}^{M-1} \Big ( \frac{\lambda _{j}}{\lambda _{j+1}} \right) ^k \Big )^{\frac{1}{2}}. } \end{aligned} \end{aligned}$$

(2.12)

where the infimum is taken over all vectors $\mathbf {\lambda }= (\lambda _1, \dots , \lambda _M) \in (0, \infty )^M$ and all $\mathbf {\iota }= \{ \iota _1, \dots , \iota _M\} \in \{-1, 1\}^M$ satisfying (2.11).

Lemma 2.12

(Static modulation lemma) [17, Lemma 2.25] Fix $k \ge 1$ and $M \in \mathbb {N}$. There exists $\eta \in (0, 1)$, $C>0$ with the following properties. Let m be as in Definition 1.5 and $\textbf{d}_{m, M}$ as in Definition 2.11. Let $\theta >0$, $\ell \in \mathbb {Z}$, and let $v \in \mathcal E_{\ell , m}$ be such that

$$\begin{aligned}\begin{aligned}{ \textbf{d}_{ m, M}( v) \le \eta , {\ \ \text {and} \ \ }E( v) \le ME( Q) + \theta ^2, } \end{aligned} \end{aligned}$$

Then, there exists a unique choice of $\mathbf {\lambda }= ( \lambda _1, \dots , \lambda _M) \in (0, \infty )^M$, $\mathbf {\iota }\in \{-1, 1\}^M$, and $g \in H$, such that

$$\begin{aligned}\begin{aligned} v&= \mathcal Q( m, \mathbf {\iota }, \mathbf {\lambda }) + g, \\ 0&= \big \langle \mathcal Z_{\underline{\lambda _j}} \mid g\big \rangle , \quad \forall j = 1, \dots , M, \end{aligned} \end{aligned}$$

along with the estimates,

$$\begin{aligned}\begin{aligned} \textbf{d}_{ m, M}( v)^2&\le \Vert g \Vert _{\mathcal {E}}^2 + \sum _{j =1}^{M-1} \Big ( \frac{\lambda _{j}}{\lambda _{j+1}} \Big )^k \le C \textbf{d}_{ m, M}( v)^2, \end{aligned} \end{aligned}$$

and,

$$\begin{aligned} \begin{aligned} \Vert g \Vert _{\mathcal {E}}^2 + \sum _{j \not \in \mathcal A} \Big ( \frac{ \lambda _j}{ \lambda _{j+1} }\Big )^k&\le C \max _{ j \in \mathcal A} \Big ( \frac{ \lambda _j}{ \lambda _{j+1} }\Big )^k + \theta ^2, \end{aligned} \end{aligned}$$

(2.13)

where $\mathcal A:= \{ j \in \{ 1, \dots , M-1\} \,: \, \iota _j \ne \iota _{j+1} \}$.

We also make use of the following lemma proved from [17] which says that a finite energy map cannot be close to two distinct multi-bubble configurations.

Lemma 2.13

[17, Lemma 2.27] Let $k \ge 1$. There exists $\eta >0$ sufficiently small with the following property. Let $m, \ell \in \mathbb {Z}$, $M, L \in \mathbb {N}$, $\mathbf {\iota }\in \{-1, 1\}^M, \mathbf {\sigma }\in \{-1, 1\}^L$, $\mathbf {\lambda }\in (0, \infty )^M, \mathbf {\mu }\in (0, \infty )^L$, and w be such that $E_{\textbf{p}}( w) < \infty$ and,

$$\begin{aligned} \Vert w - \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda })\Vert _{\mathcal {E}}^2 + \sum _{j =1}^{M-1} \Big (\frac{\lambda _j}{\lambda _{j+1}} \Big )^{k}&\le \eta , \\ \Vert w - \mathcal Q(\ell , \mathbf {\sigma }, \mathbf {\mu })\Vert _{\mathcal {E}}^2 + \sum _{j =1}^{L-1} \Big (\frac{\mu _j}{\mu _{j+1}} \Big )^{k}&\le \eta . \end{aligned}$$

Then, $m = \ell$, $M = L$, $\mathbf {\iota }= \mathbf {\sigma }$. Moreover, for every $\theta >0$ the number $\eta >0$ above can be chosen small enough so that

$$\begin{aligned}\begin{aligned}{ \max _{j = 1, \dots M} \left| \frac{\lambda _j}{\mu _j} - 1 \right| \le \theta . } \end{aligned} \end{aligned}$$

3 Localized sequential bubbling

We define a localized distance function

$$\begin{aligned} \begin{aligned}{ \varvec{\delta }_{R}(u):= \inf _{m, M, \mathbf {\iota }, \mathbf {\lambda }} \left( \Vert u - \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }) \Vert _{\mathcal {E}(r \le R)}^2 + \sum _{j =1}^M \Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^k \right) ^{\frac{1}{2}} } \end{aligned} \end{aligned}$$

(3.1)

where the infimum is taken over all $m \in \mathbb {Z}$, $M \in \{0, 1, 2, \dots \}$, all vectors $\iota \in \{-1, 1\}^M$, $\mathbf {\lambda }\in (0, \infty )^M$, and we use the convention that the last scale $\lambda _{M+1} = R$.

Lemma 3.1

Let $\ell , m \in \mathbb {Z}$ and let $u_n \in \mathcal {E}_{\ell , m}$ be a sequence of maps with $\limsup _{n \rightarrow \infty } E( u_n) < \infty$. Let $\rho _n \in (0, \infty )$ be a sequence and suppose that

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } (\rho _n \Vert \mathcal T(u_n) \Vert _{L^2}) = 0. } \end{aligned} \end{aligned}$$

(3.2)

Then, there exists a sequence $R_n \rightarrow \infty$ so that, up to passing to a subsequence of the $u_n$, we have,

$$\begin{aligned} \lim _{n \rightarrow \infty } \varvec{\delta }_{R_n\rho _n}( u_n) = 0. \end{aligned}$$

The subsequence of the $u_n$ can be chosen so that there are fixed $(M, m, \mathbf {\iota }) \in \mathbb {N}\cup \{0\} \times \mathbb {Z}\times \{-1, 1\}^M$, a sequence $\mathbf {\lambda }_n \in (0, \infty )^M$, and $C_0 >0$ with

$$\begin{aligned} \lim _{n \rightarrow \infty } \left( \Vert u_n - \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }_n) \Vert _{\mathcal {E}(r \le R_n \rho _n)}^2 + \sum _{j =1}^{M-1} \Big ( \frac{\lambda _{n,j}}{\lambda _{n, j+1}} \Big )^k\right) = 0, \end{aligned}$$

and,

$$\begin{aligned} \lambda _{n,M} \le C_0\rho _n, \quad \forall \, n. \end{aligned}$$

Remark 3.2

Lemma 3.1 is proved in the general (non-equivariant) setting by Qing [24]. Here we give a different (but related) treatment adapted to the equivariant setting using explicitly the notion of a profile decomposition of Gérard [11]. The proof that no energy can accumulate in the “neck” regions between the bubbles can be simplified in the equivariant setting and here we use an argument due to Jia and Kenig [19] from their proof of an analogous result for equivariant wave maps; see Lemma 3.4 below.

Lemma 3.3

If $a_{k, n}$ are positive numbers such that $\lim _{n\rightarrow \infty }a_{k, n} = \infty$ for all $k \in \mathbb {N}$, then there exists a sequence of positive numbers $b_n$ such that $\lim _{n\rightarrow \infty } b_n = \infty$ and $\lim _{n\rightarrow \infty } a_{k, n} / b_n = \infty$ for all $k \in \mathbb {N}$.

Proof

For each k and each n define $\widetilde{a}_{k, n} = \min \{ a_{1, n}, \dots , a_{k, n}\}$. Then the sequences $\widetilde{a}_{k, n} \rightarrow \infty$ as $n \rightarrow \infty$ for each k, but also satisfy $\widetilde{a}_{k, n} \le a_{k, n}$ for each k, n, as well as $\widetilde{a}_{j, n} \le \widetilde{a}_{k, n}$ if $j>k$. Next, choose a strictly increasing sequence $\{n_k \}_k \subset \mathbb {N}$ such that $\widetilde{a}_{k, n} \ge k^2$ as long as $n \ge n_k$. For n large enough, let $b_n \in \mathbb {N}$ be determined by the condition $n_{b_n} \le n < n_{b_n + 1}$. Observe that $b_n \rightarrow \infty$ as $n \rightarrow \infty$. Now fix any $\ell \in \mathbb {N}$ and let n be such that $b_n > \ell$. We then have

$$\begin{aligned} a_{\ell , n} \ge \widetilde{a}_{\ell , n} \ge \widetilde{a}_{b_n, n} \ge b_n^2 \gg b_n. \end{aligned}$$

Thus the sequence $b_n$ has the desired properties.$\square$

The proof of the Lemma 3.1 consists of several steps, which are designed to reduce the proof to a scenario already considered by Côte in [6, Proof of Lemma 3.5] and then by Jia-Kenig in [19, Proof of Theorem 3.2], albeit in a different context. In particular, we will seek to apply the following result from [19].

Lemma 3.4

[19, Theorem 3.2] Let $v_n$ be a sequence of maps such that $\limsup _{n \rightarrow \infty } E( v_n) < \infty$. Suppose that there exists a sequence an integer $M \ge 0$ and scales $\lambda _{n, 1} \ll \dots \ll \lambda _{n, M} \lesssim 1$ such that

$$\begin{aligned} {v}_n&= m_1 \pi + \sum _{j =1}^{M} \iota _j( Q \big ( \frac{ \cdot }{\lambda _{n, j}} \big ) - \pi ) + {w}_{n, 0}, \end{aligned}$$

where $\Vert {w}_{n} \Vert _{L^\infty } \rightarrow 0$ and $\Vert { w}_{n} \Vert _{\mathcal {E}(r \ge r_n^{-1})} \rightarrow 0$ as $n \rightarrow \infty$ for some sequence $r_{n} \rightarrow \infty$. Suppose in addition that, $\Vert {w}_{n} \Vert _{\mathcal {E}(A^{-1} \lambda _n \le r \le A \lambda _n)} \rightarrow 0$ as $n \rightarrow \infty$ for any sequence $\lambda _n \lesssim 1$ and any $A>1$, and finally, that

$$\begin{aligned} \begin{aligned}{ \limsup _{n\rightarrow \infty }\int _0^\infty \bigg ( k^2 \frac{\sin ^2(2 v_n)}{2r^2} + (\partial _r v_n)^2 2 \cos (2 v_n) \bigg ) \,r \, \textrm{d}r \le 0. } \end{aligned} \end{aligned}$$

(3.3)

Then,

$$\begin{aligned} \Vert {w}_{n} \Vert _{\mathcal {E}} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty . \end{aligned}$$

Remark 3.5

Lemma 3.4 is not stated in [19] exactly as given above. However, an examination of [19, Proof of Theorem 3.2] shows that this is precisely what is established. The heart of the matter lies in the fact that the Jia-Kenig virial functional (3.3) vanishes at Q, i.e.,

$$\begin{aligned} \int _0^\infty \bigg ( k^2 \frac{\sin ^2(2Q)}{2r^2} + (\partial _r Q)^2 2 \cos (2 Q) \bigg ) \,r \, \textrm{d}r = 0, \end{aligned}$$

but gives coercive control of the energy in regions where $v_n( r)$ is near integer multiples of $\pi$.

Proof of Lemma 3.1

By rescaling we may assume that $\rho _n =1$ for each n.

First, we observe that after passing to a subsequence, $u_n$ admits a profile decomposition,

$$\begin{aligned} u_n&= m \pi + \sum _{j = 1}^{K_0} \left( \psi ^j \big ( \frac{ \cdot }{\lambda _{n, j}} \big ) - m_j \pi \right) + \sum _{i =1}^J v^i \big ( \frac{ \cdot }{\sigma _{n, i}} \big ) + w_{n}^J( \cdot ). \end{aligned}$$

where the profiles $(\psi ^j, \lambda _{n,j}), (v^j, \sigma _{n, j})$ and the error satisfy the conclusions of Lemma 2.5.

Step 1 We make an initial restriction on the sequence $R_n \rightarrow \infty$, refining our choice of this sequence later in the proof. Consider the sets of indices

$$\begin{aligned} \mathcal J_{\infty }:= \left\{ j \in \{1, \dots , K_0\} \mid \lim _{n \rightarrow \infty } \lambda _{n, j} = \infty \right\} , \quad \mathcal {I}_{\infty }:= \left\{ i \in \mathbb {N}\mid \lim _{n \rightarrow \infty } \sigma _{n, i} = \infty \right\} \end{aligned}$$

By Lemma 3.3 we choose a sequence $R_{n,1} \rightarrow \infty$ so that $R_{n, 1} \ll \lambda _{n, j}, \sigma _{n, i}$ for each $\lambda _{n, j}$ with $j \in \mathcal J_{\infty }$ and each $\sigma _{n, i}$ with $i \in \mathcal I_{\infty }$. If follows that

$$\begin{aligned} \lim _{n \rightarrow \infty } E( \psi ^j( \cdot / \lambda _{n, j}); 0, R_{n, 1} ) = 0, \quad \lim _{n \rightarrow \infty } E( v^j( \cdot / \sigma _{n, i}); 0, R_{n, 1} ) = 0 \end{aligned}$$

for any of the indices $j \in \mathcal J_{\infty }$ or $i \in \mathcal I_{\infty }$, and thus these profiles do not factor into the distance $\varvec{\delta }_{R_{n}} (u_n)$ for any sequence $R_n \le R_{n, 1}$.

Step 2 Next we perform a bubbling analysis on the profiles with bounded scale. Define

$$\begin{aligned} \mathcal J_{0}:= \left\{ j \in \{1, \dots , K_0\} \mid \lim _{n \rightarrow \infty } \lambda _{n, j}< \infty \right\} , \quad \mathcal {I}_{0}:= \left\{ i \in \mathbb {N}\mid \lim _{n \rightarrow \infty } \sigma _{n, i} < \infty \right\} \end{aligned}$$

First, for $j \in \mathcal J_0$ and $i \in \mathcal I_0$, denote

$$\begin{aligned} u_n^j(r):= u_n( \lambda _{n, j} r), \quad u_n^i(r):= u_n( \sigma _{n, i} r) \end{aligned}$$

Then we have $u_n^j \rightarrow \psi ^j$ as $n \rightarrow \infty$ locally uniformly in $(0, \infty )$ and weakly in $\dot{H}^1(\mathbb {R}^2)$ (that is, if we view each $u_n^j$ as a radially symmetric function on $\mathbb {R}^2$). These convergence properties are by construction, see [19, pg. 1594]). Moreover, since $\lim _{n \rightarrow \infty } \lambda _{n, j} < \infty$ we have,

$$\begin{aligned} \Vert \mathcal T( u_n^j) \Vert _{L^2} = \lambda _{n, j} \Vert \mathcal T(u_n) \Vert _{L^2} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

It follows that,

$$\begin{aligned} \big \langle \mathcal T( \psi ^j) \mid \phi \big \rangle _{L^2} = 0 \end{aligned}$$

for all $\phi \in C^\infty _0(0, \infty )$, i.e., $\psi ^j$ is a weak harmonic map, and hence a smooth harmonic map by Hélein [15]. Since $\left|{m_j - \ell _j}\right| \ge 1$ we see that $E(\psi ^j) \ge E( Q)$, and thus $\psi ^j = \ell _j \pi + \iota _j Q_{\lambda _{j, 0}}$ for some $\iota _j \in \{-1, 1\}$ and some fixed scale $\lambda _{j, 0}$ and $m_j = \ell _j + \iota \pi$. We will abuse notation and replace $\lambda _{n, j}$ with $\lambda _{n, j} \lambda _{j, 0}$ while still calling this sequence $\lambda _{n, j}$.

We perform the same analysis with the $u_n^i$ and $v^i$, concluding that each $v^i$ is a smooth harmonic map. But since $v^i \in \mathcal {E}_{0, 0}$ we find that $v^i \equiv 0$ for every $i \in \mathcal I_0$.

Step 3: Next, by (3.2) and recalling that we have rescaled so that $\rho _n = 1$, we let $R_{2, n} \rightarrow \infty$ be a sequence such that

$$\begin{aligned} 1 \ll R_{2, n} \ll \Vert \mathcal T( u_n ) \Vert _{L^2}^{-1}. \end{aligned}$$

Then, by Cauchy-Schwarz

$$\begin{aligned} \begin{aligned}{ \Big |\big \langle \mathcal T(u_n) \mid \sin (2 u_n) \chi _{\widetilde{R}_{n}} \big \rangle \Big | \le \Vert \mathcal T( u_n ) \Vert _{L^2} \widetilde{R}_{ n} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty } \end{aligned} \end{aligned}$$

(3.4)

for any sequence $\widetilde{R}_{n} \le R_{2, n}$. We define $R_{3, n}:= \min ( R_{1, n}, R_{2, n})$.

Step 4: We close in on the final selection of the sequence $R_n$, choosing first $\sqrt{R_{3, n}} \le R_{4, n} \le (1/2) R_{3, n}$ so that

$$\begin{aligned} E \left( u_n; \frac{1}{4} R_n, 4R_n \right) \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

The existence of such a sequence is proved by pigeonholing; see for example [17, Eq. (3.12)]. Using Lemma 2.3 we can, after passing to a subsequence, find an integer $m_1 \in \mathbb {Z}$ so that $|u_n(r) - m_1\pi | \rightarrow 0$ for a.e., $r \in [\frac{1}{4} R_n, 4 R_n]$, and we define a truncated sequence

$$\begin{aligned} \widetilde{u}_n:= \chi _{R_{4, n}} u_n + (1- \chi _{R_{4, n}}) m_1 \pi \end{aligned}$$

By construction we have the following decomposition for $\widetilde{u}_n$,

$$\begin{aligned} \widetilde{u}_n = m_1 \pi + \sum _{j \in \mathcal J_0} ( \iota _j Q_{\lambda _j} - \pi ) + \widetilde{w}_n \end{aligned}$$

where the error $\widetilde{w}_n:= \chi _{R_{4,n}} w_{n}^J + o_n(1)$ (note we can drop the index J since any nontrivial profiles from the index sets $\mathcal J_\infty$ or $\mathcal I_\infty$ contribute a vanishing error in the region $r \le R_{4, n}$ by Step 1 and there are no nontrivial profiles from the index set $\mathcal I_{0}$ from Step 2). We define $M:= \# \mathcal J_0$ and we reorder/relabel the profiles so that $\lambda _{n, 1} \ll \lambda _{n, 2} \ll \dots \lambda _{n, M}$ for the indices $j \in \mathcal J_0$. Note that we have proved that

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \Vert \widetilde{w}_n \Vert _{L^\infty } = 0 } \end{aligned} \end{aligned}$$

(3.5)

After passing to a subsequence of the $u_n$, we claim there is a sequence $R_n \rightarrow \infty$ with the properties,

$$\begin{aligned} \begin{aligned}{ 1 \ll R_n \le R_{4, n}, \quad \Vert {\widetilde{w}}_{n}\Vert _{\mathcal {E}( \frac{1}{4} R_n^{-1} \le r \le 4 R_n)} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty . } \end{aligned} \end{aligned}$$

(3.6)

The existence of such a sequence is a consequence of the following property about $\widetilde{w}_{n}$: for any sequence $\lambda _n \lesssim 1$ and any $A>1$ we have,

$$\begin{aligned} \begin{aligned}{ \Vert w_{n}\Vert _{\mathcal {E}( \lambda _n A^{-1} \le r \le \lambda _n A)} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty . } \end{aligned} \end{aligned}$$

(3.7)

The property (3.7) was proved in [6, Step 2., p.1973–1975, Proof of Theorem 3.5] and  [19, Proof of (5.29) in Theorem 5.1] and we refer the reader to those works for details of the argument, which also applies in the current setting. The intuition is that at any scale $\lambda _{n} \lesssim 1$ at which $\widetilde{u}_n$ carries energy we have already extracted a profile $Q_{\lambda _{n, j}}$ with $\lambda _{n, j} \simeq \lambda _n$. To prove (3.6) we consider the case $\lambda _n =1$ in (3.7), and passing to a subsequence of the ${\widetilde{u}}_n$, we obtain a sequence as in (3.6).

We truncate to the region $r \le R_n$, following the same procedure used to define ${\widetilde{u}}_n$, using now $R_n$ in place of $R_{4, n}$. Indeed, set

$$\begin{aligned} {\breve{u}}_n(t_n, r):= \chi _{R_n}(r) {\widetilde{u}}_n(t, r) + (1- \chi _{R_n}(r) ) m_1 \pi . \end{aligned}$$

Defining ${\breve{w}}_{n, 0}:= \chi _{R_n}(r) {\widetilde{w}}_{n} + ( \chi _{R_n}(r) -1) \sum _{j =1}^{M} \iota _j( Q \big ( \frac{ \cdot }{\lambda _{n, j}} \big ) - \pi )$ and using that $\lambda _{n, 1} \ll \dots \ll \lambda _{n, M} \lesssim 1$ along with (3.5) and (3.6) we see that,

$$\begin{aligned} \begin{aligned}&{\breve{ u}}_n(t_n) = m_1 \pi + \sum _{j =1}^{M} \iota _j \left( Q \left( \frac{ \cdot }{\lambda _{n, j}} \right) - \pi \right) + { \breve{ w}}_{n, 0}, {\ \ \text {and} \ \ }\\&\lim _{n \rightarrow \infty }\Big ( \Vert {\breve{w}}_{n} \Vert _{\mathcal {E}(R_n^{-1} \le r < \infty )} + \Vert \breve{w}_{n} \Vert _{L^\infty } \Big ) = 0. \end{aligned} \end{aligned}$$

(3.8)

Moreover, by (3.7) we see that for any sequence $\lambda _n \lesssim 1$ and any $A>1$ that,

$$\begin{aligned}\begin{aligned}{ \lim _{n \rightarrow \infty } \Vert {\breve{w}}_{n}\Vert _{\mathcal {E}( \lambda _n A^{-1} \le r \le \lambda _n A)} = 0. } \end{aligned} \end{aligned}$$

Note that since $\breve{u}_n(r) = u_n(r)$ for $r \le R_n$, we deduce from (3.4) that,

$$\begin{aligned} \Big | \big \langle \mathcal T( \breve{u}_n) \mid \sin (2 \breve{u}_n) \chi _{\frac{1}{4} R_n} \big \rangle \Big | \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

We claim that

$$\begin{aligned} \Big | \big \langle \mathcal T( \breve{u}_n) \mid \sin (2 \breve{u}_n)(1- \chi _{\frac{1}{4} R_n}) \big \rangle \Big | \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

as well. To see this, note that by (3.8)

$$\begin{aligned} \lim _{n \rightarrow \infty } E( \breve{u}_n; r_n, \infty ) = 0 \end{aligned}$$

for any sequence $r_n \rightarrow \infty$. And after integration by parts we deduce the bound,

$$\begin{aligned} \Big | \big \langle \mathcal T( \breve{u}_n) \mid \sin (2 \breve{u}_n)(1- \chi _{\frac{1}{4} R_n}) \big \rangle \Big | \lesssim E(\breve{u}_n; 1/8 R_n, \infty ) \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

Hence,

$$\begin{aligned} \Big | \big \langle \mathcal T( \breve{u}_n) \mid \sin (2 \breve{u}_n) \big \rangle \Big | \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

Integrating by parts on the left hand side, we see that

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _0^\infty \bigg ( k^2 \frac{\sin ^2(2 \breve{u}_n)}{2r^2} + (\partial _r \breve{u}_n)^2 2 \cos (2 \breve{u}_n) \bigg ) \,r \, \textrm{d}r = 0. \end{aligned}$$

The sequence $\breve{u}_n$ then satisfies all the conditions of Lemma 3.4 and we conclude that $\lim _{ n \rightarrow \infty } \Vert \breve{w}_n \Vert _{\mathcal {E}} = 0$. Since $\breve{u}_n(r) = u_n(r)$ for $r \le R_n$ we conclude that $\lim _{n \rightarrow \infty } \varvec{\delta }_{R_n} ( u_n) = 0$. An examination of the decomposition (3.8) yields the remaining claims in from Lemma 3.1.$\square$

4 Sequential bubbling

4.1 Sequential bubbling for finite time blow-up solutions

Proposition 4.1

(Sequential bubbling for solutions that blow up in finite time) Let $\ell , m \in \mathbb {Z}$, $u_0 \in \mathcal {E}_{\ell , m}$, and let u(t) denote the solution to (1.2) with initial data $u_0$. Suppose that $T_+(u_0) < \infty$. There exist integers $m_{\infty }, m_{\Delta }$, a mapping $u^* \in \mathcal {E}_{0, m_\infty }$, an integer $N \ge 1$, a sequence of times $t_n \rightarrow T_+$, signs $\mathbf {\iota }\in \{-1, 1\}^N$, a sequence of scales $\mathbf {\lambda }_n \in (0, \infty )^N$, and an error $g_n$ defined by

$$\begin{aligned} u(t_n) = m_{\Delta } \pi + \sum _{j =1}^N \iota _j( Q_{\lambda _n} - \pi ) + u^* + g_n, \end{aligned}$$

with the following properties:

• (i) The integer $N \ge 1$ and the body map $u^*$ satisfy,

  $$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow T_+} E( u(t)) = N E(Q) + E( u^*); } \end{aligned} \end{aligned}$$

  (4.1)

• (i) for any $\alpha >0$,

  $$\begin{aligned}{} & {} \lim _{t \rightarrow T_+} E \left( u(t); 0, \alpha (T_+ - t)^{\frac{1}{2}} \right) = NE(Q), \end{aligned}$$

  (4.2)
  $$\begin{aligned}{} & {} \lim _{t \rightarrow T_+} E \left( u(t) - u^*; \alpha (T_+ - t)^{\frac{1}{2}}, \infty \right) = 0, \end{aligned}$$

  (4.3)

  and there exists $0< T_0 < T_+$ and function $\rho : [T_0, T_+) \rightarrow (0,\infty )$ satisfying,

  $$\begin{aligned} \lim _{t \rightarrow T_*} \big ((\rho (t) / \sqrt{T_+-t}) + \Vert u(t) - u^* - m_\Delta \pi \Vert _{\mathcal {E}(\rho (t))}\big ) = 0; \end{aligned}$$

  (4.4)

• (ii) the error $g_n$ and the scales $\mathbf {\lambda }_n$ satisfy,

  $$\begin{aligned} \begin{aligned}{ \lim _{ n \rightarrow \infty } \left( \Vert g_n \Vert _{\mathcal {E}}^2 + \sum _{j =1}^N \Big ( \frac{\lambda _{n, j}}{\lambda _{n, j+1}} \Big )^k \right) ^{\frac{1}{2}} = 0, } \end{aligned} \end{aligned}$$

  (4.5)

  where here we adopt the convention that $\lambda _{n, N+1}:= (T_+ - t_n)^{\frac{1}{2}}$.

Lemma 4.2

(Identification of the body map) Let $u_0 \in \mathcal {E}_{\ell , m}$ and let u(t) be the solution to (1.2). Suppose that $T_+(u_0)< \infty$ and let $I_* = [0, T_+)$. There exist $m_{\infty }, m_{\Delta } \in \mathbb {Z}$ and a mapping $u^* \in \mathcal {E}_{0, m_{\infty }}$ such that for any $r_0>0$,

$$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow T_*}\Vert u(t) - u^* - m_\Delta \pi \Vert _{\mathcal {E}(r \ge r_0)} = 0. } \end{aligned} \end{aligned}$$

(4.6)

Moreover, there exists $L>0$ such that for each $r_0 \in (0, \infty ]$,

$$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow T_+} E( u(t); 0, r_0) = L + E(u^*; 0, r_0), } \end{aligned} \end{aligned}$$

(4.7)

and in particular, $\lim _{r_0 \rightarrow 0} \lim _{t \rightarrow T_+} E( u(t); 0, r_0) = L$.

Proof of Lemma 4.2

In the general (non-equivariant) setting Struwe [28] proves the existence of the body map as the weak limit of the flow in $H^1$ as $t \rightarrow T_+$ and moreover that one has strong $C^2$ convergence on compact sets not containing the bubbling points (the origin in our case); see for example [20, Step 3, Proof of Theorem 6.16]. The existence of the limit L is proved by Qing in [24, Proposition 2.1], and an identical argument can be used in the equivariant setting.$\square$

Proof of Proposition 4.1

We follow, roughly, the arguments by Qing in [24, Proof of Theorem 1.1] and Topping in [31, Proof of Theorem 1.4]. The main ingredient is the compactness result, Lemma 3.1. Let $u(t) \in \mathcal {E}_{\ell , m}$ be a heat flow blowing up at time $T_+ >0$. By (2.2) we can find a sequence $t_n \rightarrow T_+$ so that,

$$\begin{aligned} (T_+ - t_n)^{\frac{1}{2}} \Vert \mathcal T( u(t_n)) \Vert _{L^2} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty . \end{aligned}$$

We can now apply Lemma 3.1 with $\rho _n:= (T_+ - t_n)^{\frac{1}{2}}$, which yields $N \ge 0$, $m_0\in \mathbb {Z}$, $\mathbf {\iota }\in \{-1, 1\}^N, \mathbf {\lambda }_n \in (0, \infty )^N$ such that after passing to a subsequence, we have

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \left( \Vert u(t_n) - \mathcal Q(m_0, \mathbf {\iota }, \mathbf {\lambda }_n) \Vert _{\mathcal {E}( r \le A (T_+- t_n)^{\frac{1}{2}})}^2 + \sum _{j =1}^{N-1} \Big ( \frac{ \lambda _{n, j}}{\lambda _{n, j+1}} \Big )^{k} \right) = 0 } \end{aligned} \end{aligned}$$

(4.8)

for each $A>0$, and moreover that $\lambda _{n, N} \lesssim (T_+- t_n)^{\frac{1}{2}}$. Next, for each $R>0$ define the localized energy,

$$\begin{aligned} \Theta _R(t):= \int _0^\infty \chi _R(r)^2 \textbf{e}(u(t, r)) \, r \, \textrm{d}r. \end{aligned}$$

along with the localized energy of the body map,

$$\begin{aligned} \Theta _R^*:= \int _0^\infty \chi _R(r)^2 \textbf{e}(u^*(r)) \, r \, \textrm{d}r. \end{aligned}$$

From (2.4) we see that for each $0<s<\tau < T_+$ we have,

$$\begin{aligned} \begin{aligned} \Big | \Theta _R(\tau ) - \Theta _R(s) \Big |&\lesssim \int _s^{\tau } \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t + \frac{(\tau - s)^{\frac{1}{2}}}{R} \left( \int _s^\tau \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t \right) ^{\frac{1}{2}}\\&\lesssim \int _s^{T_+} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t + \frac{(T_+ - s)^{\frac{1}{2}}}{R} \left( \int _s^{T_+} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t \right) ^{\frac{1}{2}} \end{aligned} \end{aligned}$$

(4.9)

Since the right-hand side tends to zero as $s \rightarrow T_+$, it follows that $\lim _{t \rightarrow T_+}\Theta _R(t):= \ell _R$ exists. Define,

$$\begin{aligned} \frac{1}{2\pi } L_R:= \ell _R - \Theta _R^* \end{aligned}$$

and we claim that in fact, $L_R = L:= \lim _{r_0 \rightarrow 0} \lim _{t \rightarrow T_+}E( u(t); 0, r_0)$, which is independent of $R>0$. To see this we write, for any $0<r_0 <R$,

$$\begin{aligned} \Theta _R(t) - \Theta _R^*&= \int _{r_0}^{4R} \chi _R(r)^2 (\textbf{e}(u(t, r)) - \textbf{e}(u^*(r))) \, r \, \textrm{d}r +\frac{1}{2\pi }E( u(t); 0, r_0) - \frac{1}{2\pi }E(u^*; 0, r_0) \end{aligned}$$

Letting $t \rightarrow T_+$, the right hand side tends to $\frac{1}{2\pi }L_R$. By (4.6) the first term on the left vanishes as $t \rightarrow T_+$. Sending $r_0 \rightarrow 0$ after letting $t \rightarrow T_+$ on the right, we see from (4.7) that $L_R = L = \lim _{r_0 \rightarrow 0} \lim _{t \rightarrow T_+}E( u(t); 0, r_0)$.

Next, let $\gamma >0$ and set $R = \gamma (T_+- s)^{\frac{1}{2}}$ in (4.9) we obtain, after letting $\tau \rightarrow T_+$,

$$\begin{aligned} \Big | \frac{1}{2 \pi } L + \Theta _{\gamma (T_+ -s)^{\frac{1}{2}}}^* - \Theta _{\gamma (T_+ -s)^{\frac{1}{2}}}(s) \Big |&\lesssim \int _s^{T_+} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t + \frac{1}{\gamma } \Big ( \int _s^{T_+} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t \Big )^{\frac{1}{2}} \end{aligned}$$

Letting $s \rightarrow T_+$ above we see that $\lim _{s \rightarrow T_+} \Theta _{\gamma (T_+ -s)^{\frac{1}{2}}}(s) = \frac{1}{2\pi } L$ for all $\gamma >0$.

Let $\alpha >0$ and note the inequality,

$$\begin{aligned} 2 \pi \Theta _{\frac{\alpha }{2}(T_+ -s)^{\frac{1}{2}}}(s) \le E( u(s); 0, \alpha (T_+ -s)^{\frac{1}{2}}) \le 2 \pi \Theta _{\alpha (T_+ -s)^{\frac{1}{2}}}(s) \end{aligned}$$

which implies that $\lim _{s \rightarrow T_+}E( u(s); 0, \alpha (T_+ -s)^{\frac{1}{2}}) = L$ for any $\alpha >0$. Hence, for any $0< \alpha< A < \infty$, $\lim _{s \rightarrow T_+} E(u(s); \alpha (T_+ -s)^{\frac{1}{2}}, A(T_+ -s)^{\frac{1}{2}}) = 0$. Returning to the decomposition (4.8) we find that

$$\begin{aligned} \begin{aligned}{ \frac{\lambda _{n, N} }{(T_+ - t_n)^{\frac{1}{2}}} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty , } \end{aligned} \end{aligned}$$

(4.10)

and as a consequence, $L = NE(Q)$ and (4.2) is proved. Further, we see from (4.7) that for every $r_0>0$,

$$\begin{aligned} \lim _{t \rightarrow T_+} E( u(t); 0, r_0) = N E(Q) + E(u^*; 0, r_0). \end{aligned}$$

and we see from (2.1) that $N \ge 1$. Combining the above with (4.2) we see that for every $\alpha >0$, $r_0 \in (0, \infty ]$,

$$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow T_+} E( u(t); \alpha (T_+ - t)^{\frac{1}{2}}, r_0) = E(u^*; 0, r_0) } \end{aligned} \end{aligned}$$

(4.11)

and (4.1) now follows. Next, if (4.3) were to fail, we could find $\alpha _1, \epsilon _1>0$ and a sequence $s_n \rightarrow T_+$ such that

$$\begin{aligned} E( u(s_n) - u^*; \alpha _1 (T_+- s_n)^{\frac{1}{2}}, \infty ) \ge \epsilon _1, \quad \forall \, n. \end{aligned}$$

To reach a contradiction, we choose $r_0>0$ sufficiently small so that $E( u^*; 0, r_0) \le \epsilon _1/8$, and then, using (4.6) and (4.11), n sufficiently large so that $E(u(s_n) - u^*; r_0, \infty ) \le \epsilon _1/8$ and $E(u(s_n); \alpha _1 (T_+- s_n)^{\frac{1}{2}}, r_0) \le \epsilon _1/4$. We then estimate,

$$\begin{aligned} E( u(s_n) - u^*;&\alpha _1 (T_+- s_n)^{\frac{1}{2}}, \infty ) \le E( u(s_n) - u^*; \alpha _1 (T_+- s_n)^{\frac{1}{2}}, r_0) + E( u(s_n) - u^*; r_0, \infty )\\&\le 2E( u(s_n); \alpha _1 (T_+- s_n)^{\frac{1}{2}}, r_0) + 2 E(u^*; \alpha _1 (T_+- s_n)^{\frac{1}{2}}, r_0) + \epsilon _1/8 \le 7\epsilon _1/8, \end{aligned}$$

a contradiction, proving (4.3). We see from (4.6) and (4.8) that $m_0 = m_{\Delta }$ and from Lemma 2.3 we have,

$$\begin{aligned} \lim _{t \rightarrow T_+} \Vert u(t) - u^* - m_{\Delta } \pi \Vert _{\mathcal {E}( r \ge \alpha (T_+ - t) )} = 0, \end{aligned}$$

which implies (4.4). Finally, the above together with (4.8) and (4.10) yield (4.5).$\square$

4.2 Sequential bubbling for global solutions

Proposition 4.3

(Sequential bubbling for global-in-time solutions) Let $\ell , m \in \mathbb {Z}$. Let $u_0 \in \mathcal {E}_{\ell , m}$ and let u(t) denote the solution to (1.2) with initial data $u_0$. Suppose that $T_+(u_0) = \infty$. Then there exist $T_0>0$, an integer $N \ge 0$, a sequence of times $t_n \rightarrow \infty$, signs $\mathbf {\iota }\in \{-1, 1\}^N$, a sequence of scales $\mathbf {\lambda }_n \in (0, \infty )^N$, and an error $g_n$ defined by

$$\begin{aligned} u(t_n) = m \pi + \sum _{j =1}^N \iota _j( Q_{\lambda _n} - \pi ) + g_n \end{aligned}$$

with the following properties:

• (i) the integer $N \ge 0$ satisfies,

  $$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow \infty } E( u(t)) = NE(Q); } \end{aligned} \end{aligned}$$

  (4.12)

• (ii) for every $\alpha >0$,

  $$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow \infty } E( u(t); \alpha \sqrt{t}, \infty ) = 0, } \end{aligned} \end{aligned}$$

  (4.13)

  and there exists $T_0>0$ and a function $\rho : [T_0, \infty ) \rightarrow (0, \infty )$ such that

  $$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow \infty } \Big ( \frac{\rho (t)}{ \sqrt{t}} + \Vert u(t) - m\pi \Vert _{\mathcal {E}( r \ge \rho (t))} \Big ) = 0; } \end{aligned} \end{aligned}$$

  (4.14)

• (iii) the scales $\mathbf {\lambda }_n$ and the sequence $g_n$ satisfy,

  $$\begin{aligned} \begin{aligned}{ \lim _{ n \rightarrow \infty } \left( \Vert g_n \Vert _{\mathcal {E}}^2 + \sum _{j =1}^N \Big ( \frac{\lambda _{n, j}}{\lambda _{n, j+1}} \Big )^k \right) ^{\frac{1}{2}} = 0 } \end{aligned} \end{aligned}$$

  (4.15)

  where here we adopt the convention that $\lambda _{n, j+1}:= t_n^{\frac{1}{2}}$.

Proof

Let $u(t) \in \mathcal {E}_{\ell , m}$ be a heat flow defined globally in time. By (2.2) we can find a sequence $t_n \rightarrow \infty$ so that,

$$\begin{aligned} t_n^{\frac{1}{2}} \Vert \mathcal T( u(t_n)) \Vert _{L^2} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty . \end{aligned}$$

We can now apply Lemma 3.1 with $\rho _n:= t_n^{\frac{1}{2}}$, which yields $N \ge 0$, $m_0\in \mathbb {Z}$, $\mathbf {\iota }\in \{-1, 1\}^N, \mathbf {\lambda }_n \in (0, \infty )^N$ such that after passing to a subsequence, we have

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \left( \Vert u(t_n) - \mathcal Q(m_0, \mathbf {\iota }, \mathbf {\lambda }_n) \Vert _{\mathcal {E}( r \le A t_n^{\frac{1}{2}})}^2 + \sum _{j =1}^{N-1} \Big ( \frac{ \lambda _{n, j}}{\lambda _{n, j+1}} \Big )^{k} \right) = 0 } \end{aligned} \end{aligned}$$

(4.16)

for each $A>0$, and moreover that $\lambda _{n, N} \lesssim t_n^{\frac{1}{2}}$.

Fix $\alpha >0$ and let $\epsilon >0$. By (2.2) and the fact that $E( u(0))< \infty$ we can find $T_0 = T_0(\epsilon ) >0$ such that,

$$\begin{aligned} \begin{aligned}{ \frac{4\sqrt{E(u(0))}}{\alpha }\Big (\int _{T_0}^\infty \int _0^\infty (\partial _t u (t, r))^2 \, r \, \textrm{d}r \, \textrm{d}t\Big )^{\frac{1}{2}} \le \epsilon } \end{aligned} \end{aligned}$$

(4.17)

Next, choose $T_1 \ge T_0$ so that

$$\begin{aligned} \begin{aligned}{ E( u(T_0); \alpha \sqrt{T}/4, \infty ) \le \epsilon } \end{aligned} \end{aligned}$$

(4.18)

for all $T \ge T_1$. Fixing any such T, we set

$$\begin{aligned} \phi (t, r) = \phi _T(r) = 1 - \chi ( 4r/ \alpha \sqrt{T}) {\ \ \text {for} \ \ }t \in [T_0, T] \end{aligned}$$

where $\chi (r)$ is a smooth function on $(0, \infty )$ such that $\chi (r) = 1$ for $r \le 1$, $\chi (r) = 0$ if $r \ge 4$, and $\left|{\chi '(r)}\right| \le 1$ for all $r \in (0, \infty )$. Since $\frac{\textrm{d}}{\textrm{d}t} \phi (t, r) = 0$ for $t \in [T_0, T]$ it follows from (2.6) that,

$$\begin{aligned} \int _0^\infty \textbf{e}(u(T, r)) \, \phi _T(r)^2 \, r \, \textrm{d}r\le & {} \int _0^\infty \textbf{e}(u(T_0, r)) \, \phi _T(r)^2 \, r \, \textrm{d}r \\{} & {} + \frac{4\sqrt{E(u(0))}}{\alpha }\left( \int _{T_0}^T \int _0^\infty (\partial _t u (t, r))^2 \, r \, \textrm{d}r \, \textrm{d}t\right) ^{\frac{1}{2}} \end{aligned}$$

Using the above together with (4.17) and (4.18) we find that

$$\begin{aligned} E( u(T); \alpha \sqrt{T}, \infty ) \le \epsilon . \end{aligned}$$

for all $T \ge T_1$, completing the proof of (4.13). It follows from (4.13) that there exists $T_0>0$ and a function $\rho : [T_0, \infty ) \rightarrow (0, \infty )$ with $\rho (t) \ll \sqrt{t}$ and $\lim _{t \rightarrow \infty } E( u(t); \rho (t), \infty ) = 0$. Thus, (4.14) is a consequence of Lemma 2.3.

Returning to the sequential decomposition we see from (4.16), the fact that $\lambda _{n, N} \lesssim t_n^{\frac{1}{2}},$ and from (4.13) that we must have

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\lambda _{n, N}}{t_n^{\frac{1}{2}}} = 0. \end{aligned}$$

Then, (4.15) follows from the above, (4.14) and (4.16). Moreover we see that $\lim _{n \rightarrow \infty } E( u(t_n)) = N E(Q)$ and the continuous limit (4.12) then follows from the fact that E(u(t)) is non-increasing.$\square$

5 Decomposition of the solution and collision intervals

For the remainder of the paper we fix a solution $u(t) \in \mathcal {E}_{\ell , m}$ of (1.2), defined on the time interval $I_*=[0, T_*)$ where $T_*:= T_+<\infty$ in the finite time blow-up case and $T_*= \infty$ in the global case. Let $u^* \in \mathcal {E}_{0, m_{\infty }}$ be the body map as defined in Proposition 4.1 and in the case of a global solution we adopt the convention that $u^* = 0$. Note that $m_\infty = 0$ if $T_* = \infty$. We let $m_\Delta$ be as in Proposition 4.1 so that $u(t) \sim m_{\Delta } \pi + u^*$ in the region $r \gtrsim (T_+ - t)^{\frac{1}{2}}$. To unify notation, we adopt the convention that $m_{\Delta } = m$ in the case of a global solution, so that we may again view $u(t) \sim m_{\Delta } \pi + u^*$ in the region $r \gtrsim \sqrt{t}$. By Propositions 4.1 and 4.3 there exists an integer $N \ge 0$ and a sequence of times $t_n \rightarrow T_*$ so that $u(t_n)- u^*$ approaches an N-bubble as $n \rightarrow \infty$.

We define a localized distance to an N-bubble.

Definition 5.1

(Proximity to a multi-bubble) For all $t \in I$, $\rho \in (0, \infty )$, and $K \in \{0, 1, \ldots , N\}$, we define the localized multi-bubble proximity function as

$$\begin{aligned} \textbf{d}_K(t; \rho ):= \inf _{\mathbf {\iota }, \mathbf {\lambda }}\left( \Vert u(t) - u^* - \mathcal Q(m_{\Delta }, \mathbf {\iota }, \mathbf {\lambda }) \Vert _{\mathcal {E}(\rho , \infty )}^2 + \sum _{j=K}^{N}\Big (\frac{ \lambda _{j}}{\lambda _{j+1}}\Big )^{k} \right) ^{\frac{1}{2}}, \end{aligned}$$

where $\mathbf {\iota }:= (\iota _{K+1}, \ldots , \iota _N) \in \{-1, 1\}^{N-K}$, $\mathbf {\lambda }:= (\lambda _{K+1}, \ldots , \lambda _N) \in (0, \infty )^{N-K}$, $\lambda _K:= \rho$ and $\lambda _{N+1}:= \sqrt{T_+ -t}$ in the finite time blow-up case and $\lambda _{N+1}:= \sqrt{t}$ in the case of a global solution.

The multi-bubble proximity function is defined by $\textbf{d}(t):= \textbf{d}_0(t; 0)$.

Remark 5.2

We emphasize that if $\textbf{d}_K(t; \rho )$ is small, this means that $u(t) - u^*$ is close to $N-K$ bubbles in the exterior region $r \in (\rho , \infty )$.

We can now rephrase a consequence of Propositions 4.1 and 4.3 in this notation: there exists a monotone sequence $t_n \rightarrow T_*$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \textbf{d}(t_n) = 0. \end{aligned}$$

(5.1)

We state and prove some simple consequences of the set-up above. We always assume $N \ge 1$, since Theorem 1 in the case $N = 0$ is immediate from (4.12).

A direct consequence of (4.14) is that u(t) always approaches a 0-bubble in some exterior region. With $\rho _N(t) = \rho (t)$ given by the function in Proposition 4.1 or 4.3 the following lemma is immediate from the conventions of Definition 5.1.

Lemma 5.3

There exists $T_0>0$ and function $\rho _N: [T_0, T_*) \rightarrow (0, \infty )$ such that

$$\begin{aligned} \lim _{t\rightarrow T_*}\textbf{d}_N(t; \rho _N(t)) = 0. \end{aligned}$$

(5.2)

5.1 Collision intervals

Theorem 1 will follow from showing that,

$$\begin{aligned} \lim _{t \rightarrow T_*} \textbf{d}(t) = 0. \end{aligned}$$

(5.3)

The approach which we adopt in order to prove (5.3) is to study colliding bubbles. A collision is defined as follows.

Definition 5.4

(Collision interval) Let $K \in \{0, 1, \ldots , N\}$. A compact time interval $[a, b] \subset I_*$ is a collision interval with parameters $0< \epsilon < \eta$ and $N - K$ exterior bubbles if

• $\textbf{d}(a) \le \epsilon$ and $\textbf{d}(b) \ge \eta$,

• there exists a function $\rho _K: [a, b] \rightarrow (0, \infty )$ such that $\textbf{d}_K(t; \rho _K(t)) \le \epsilon$ for all $t \in [a, b]$.

In this case, we write $[a, b] \in \mathcal C_K(\epsilon , \eta )$.

Definition 5.5

(Choice of K) We define K as the smallest nonnegative integer having the following property. There exist $\eta > 0$, a decreasing sequence $\epsilon _n \rightarrow 0$, and sequences $(a_n), (b_n)$ such that $[a_n, b_n] \in \mathcal C_K(\epsilon _n, \eta )$ for all $n \in \{1, 2, \ldots \}$.

Lemma 5.6

(Existence of $K \ge 1$) If (5.3) is false, then K is well defined and $K \in \{1, \ldots , N\}$.

Remark 5.7

The fact that $K \ge 1$ means that at least one bubble must lose its shape if (5.3) is false.

Proof of Lemma 5.6

Assume (5.3) does not hold, so that there exist $\eta > 0$ and a monotone sequence $b_n \rightarrow T_*$ such that

$$\begin{aligned} \textbf{d}(b_n) \ge \eta , \qquad \text {for all }n. \end{aligned}$$

We claim that there exist sequences $(\epsilon _n), (a_n)$ such that $[a_n, b_n] \in \mathcal C_N(\epsilon _n, \eta )$. Indeed, (5.1) implies that there exist $\epsilon _n \rightarrow 0$ and $a_n \le b_n$ such that $\textbf{d}(a_n) \le \epsilon _n$. Note that $a_n \rightarrow T_*$ and $b_n \rightarrow T_*$. Let $\rho _N: [a_n, b_n] \rightarrow (0, \infty )$ be the function given by Lemma 5.3, restricted to the time interval $[a_n, b_n]$. Then (5.2) yields

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{t\in [a_n, b_n]}\textbf{d}_N(t; \rho _N(t)) = 0. \end{aligned}$$

Upon adjusting the sequence $\epsilon _n$, we obtain that all the requirements of Definition 5.4 are satisfied for $K = N$.

We now prove that $K \ge 1$. Suppose $K = 0$. By Definition 5.4 of a collision interval, there exist $\eta > 0$, and sequences $a_n, b_n \rightarrow T_*$ and $\rho _0(b_n) \ge 0$ such that $\textbf{d}_0(b_n; \rho _0(b_n)) \le \epsilon _n$ and at the same time $\textbf{d}(b_n) \ge \eta$. We show that this is impossible.

Define $v_n:= u(b_n) - u^*$. Since $\textbf{d}_0(b_n; \rho _0(b_n)) \le \epsilon _n$ we can find parameters, $\rho _0(b_n) \ll \lambda _{n, 1} \ll \dots \ll \lambda _{n, N}$ and signs $\mathbf {\iota }_n$ such that defining ${ g}_n = v_n - \mathcal Q( m_\Delta , \mathbf { \iota }_n, \mathbf { \lambda }_n)$ we have

$$\begin{aligned} \begin{aligned}{ \textbf{d}_0(c_n; \rho _0(b_n)) \simeq \Vert {g}_n \Vert _{\mathcal {E}( \rho _0(b_n), \infty )}^2 + \sum _{j =0}^N\Big ( \frac{ \lambda _{n, j}}{\lambda _{n, j+1}}\Big )^k \lesssim \epsilon _n^2. } \end{aligned} \end{aligned}$$

(5.4)

If $T_*< \infty$, with $\rho (t)$ as in (4.4) we see that we must have $\lambda _{n, N} \ll \rho (b_n) \ll (T_*-b_n)^{\frac{1}{2}}$, and thus using (4.4) along with (5.4) and Lemma 2.10 we have

$$\begin{aligned}\begin{aligned} E( u(b_n); \rho _0(b_n), \infty )&= E( {g}_n + u^* + \mathcal Q( m_\Delta , \mathbf { \iota }_n, \mathbf { \lambda }_n); \rho _0(b_n), \rho (b_n)) \\&\quad + E( { g}_n + u^* + \mathcal Q( m_\Delta , \mathbf {\iota }_n, \mathbf {\lambda }_n); \rho (b_n), \infty ) \\&= N E( Q) + E( u^*) + o_n(1). \end{aligned} \end{aligned}$$

A similar argument in the case $T_*= \infty$ shows that

$$\begin{aligned}\begin{aligned} E( u(b_n); \rho _0(b_n), \infty )&= N E( Q) + o_n(1). \end{aligned} \end{aligned}$$

Since by (4.1) and (4.12) we know that $\lim _{ n \rightarrow \infty } E( u(b_n)) = N E( Q) + E( u^*)$, we conclude from the previous line that,

$$\begin{aligned} E( u(b_n); 0, \rho _0(b_n)) = o_n(1) {\ \ \text {as} \ \ }n \rightarrow \infty . \end{aligned}$$

Using the fact that $\rho _0(b_n) \ll \rho (b_n)$ it follows that $E( v_n; 0, \rho _0(b_n)) = o_n(1)$, and hence by (2.3) we conclude that

$$\begin{aligned} \Vert v_n - \ell \pi \Vert _{\mathcal {E}( 0, \rho _0(b_n))} \lesssim E( v_n; 0, \rho _0(b_n)) = o_n(1) {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

Thus, combining the above with (5.4) we have $\textbf{d}(b_n) = o_n(1)$ as $n \rightarrow \infty$, a contradiction.$\square$

Remark 5.8

For each collision interval we may assume without loss of generality that $\textbf{d}(a_n) = \epsilon _n$, $\textbf{d}(b_n) = \eta$, and $\textbf{d}(t) \in [\epsilon _n, \eta ]$ for each $t \in [a_n, b_n]$. Indeed, given some initial choice of $[a_n, b_n] \in \mathcal C_K( \epsilon _n, \eta )$, just set $a_n \le \widetilde{a}_n:= \sup \{ t \in [a_n, b_n] \mid \textbf{d}(t) \le \epsilon _n \}$ and $\widetilde{b}_n:= \inf \{t \in [\widetilde{a}_n, b_n] \mid \textbf{d}(t) \ge \eta \}$.

Similarly, given some initial choice $\epsilon _n \rightarrow 0, \eta >0$ and intervals $[a_n, b_n] \in \mathcal C_K( \eta , \epsilon _n)$ we are free to “enlarge” $\epsilon _n$ or “shrink” $\eta >0$, by choosing some other sequence $\epsilon _n \le \widetilde{\epsilon }_n \rightarrow 0$, and $0< \widetilde{\eta }\le \eta$, and new collision subintervals $[\widetilde{a}_n, \widetilde{b}_n] \subset [a_n, b_n] \cap \mathcal C_{K}(\widetilde{\eta }, \widetilde{\epsilon }_n)$ as in the previous paragraph. We will enlarge our initial choice of $\epsilon _n$ and shrink $\eta$ in this fashion over the course of the proof.

5.2 Decomposition of the solution

Lemma 5.9

(Basic modulation) Let $K \ge 1$ be the number given by Lemma 5.6. There exist $\eta >0$, a sequence $\epsilon _n \rightarrow 0$, and sequences $a_n, b_n \rightarrow \infty$ satisfying the requirements of Definition 5.5, and such that $\textbf{d}(a_n) = \epsilon _n$, $\textbf{d}(b_n) = \eta$ and $\textbf{d}(t) \in [\epsilon _n, \eta ]$ for all $t \in [a_n, b_n]$ and so that the following properties hold. There exist signs $\mathbf {\iota }\in \{-1, 1\}^N$, a function $\mathbf {\lambda }= ( \lambda _1, \dots , \lambda _N) \in C^1(\cup _{n \in \mathbb {N}} [a_n, b_n]; (0, \infty )^{N})$, sequences $\alpha _n \rightarrow 0$ and $\nu _n \rightarrow 0$, such that defining the functions,

$$\begin{aligned}{} & {} \nu :\cup _{n \in \mathbb {N}} [a_n, b_n] \rightarrow (0, \infty ), \quad \nu (t):= \nu _n \lambda _{K+1}(t), {\ \ \text {for} \ \ }\, \, t\in [a_n,b_n], \\{} & {} \alpha \cup _{n \in \mathbb {N}} [a_n, b_n] \rightarrow (0, \infty ), \quad \alpha (t):= {\left\{ \begin{array}{ll} \alpha _n\sqrt{T_+ - t_n}{\ \ \text {if} \ \ }T_+<\infty \\ \alpha _n \sqrt{t} {\ \ \text {if} \ \ }T_+ = \infty \end{array}\right. }, {\ \ \text {for} \ \ }\, \, t\in [a_n,b_n], \\{} & {} u^*(t):= {\left\{ \begin{array}{ll} (1 - \chi _{\alpha (t)}) \big ( u(t) - m_{\Delta } \pi \big ){\ \ \text {if} \ \ }T_+<\infty \\ 0 {\ \ \text {if} \ \ }T_+= \infty \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} g: \cup _{n \in \mathbb {N}} [a_n, b_n] \rightarrow \mathcal {E}; \quad g(t):= u(t) - u^*(t) - \mathcal Q(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }(t)), \end{aligned}$$

there hold,

• (i) the orthogonality conditions,

  $$\begin{aligned} \begin{aligned}{ 0 = \big \langle \mathcal Z_{\underline{\lambda (t)}} \mid g(t) \big \rangle , \quad \forall \, t \in [a_n, b_n], \quad \forall n; } \end{aligned} \end{aligned}$$

  (5.5)

• (ii) and the estimates,

  $$\begin{aligned}{} & {} \lim _{n \rightarrow \infty } \sup _{t \in [a_n, b_n]} \left( \frac{\nu (t)}{ \lambda _{K+1}(t)} + \sum _{j=K+1}^{N-1} \frac{\lambda _{j}(t)}{\lambda _{j+1}(t)} + \frac{\lambda _{N}(t)}{\alpha (t)}+ E( u(t); \frac{1}{4}{\nu (t)}, 4 \nu (t)) \right) = 0, \nonumber \\ \end{aligned}$$

  (5.6)
  $$\begin{aligned}{} & {} C_0^{-1}\textbf{d}(t) \le \Vert g(t) \Vert _{\mathcal {E}} + \sum _{j=1}^{N-1} \Big ( \frac{ \lambda _{j}(t)}{\lambda _{j+1}(t)} \Big )^{\frac{k}{2}} \le C_0\textbf{d}(t), \end{aligned}$$

  (5.7)
  $$\begin{aligned}{} & {} \Vert g(t) \Vert _{\mathcal {E}} + \sum _{j \not \in \mathcal A} \Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^{\frac{k}{2}} \le C_0 \sum _{j \in \mathcal A} \Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^{\frac{k}{2}} \end{aligned}$$

  (5.8)
  $$\begin{aligned}{} & {} \left|{\lambda _j'(t)}\right| \le C_0 \frac{1}{\lambda _j(t)} \textbf{d}(t), \end{aligned}$$

  (5.9)

  for all $t \in [a_n, b_n]$ and all $n \in \mathbb {N}$;

• (iii) for any sequence $s_n \in [a_n, b_n]$ and any sequence $R_n$ such that $\nu (s_n) \le R_n \ll \lambda _{K+1}(s_n)$ if $K < N$ and $\nu (s_n) \le R_n \le \alpha (s_n)$ if $K=N$, then,

  $$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } E( u(s_n); R_n, \infty ) = (N-K) E(Q) + E( u^*).} \end{aligned} \end{aligned}$$

  (5.10)

  and,

  $$\begin{aligned} \begin{aligned} \lim _{n \rightarrow \infty } ( \Vert u(s_n) - u^*(s_n) - \mathcal Q( m_{\Delta }, \iota _{K+1}, \dots , \iota _N,&\lambda _{K+1}(s_n), \dots , \lambda _{N}(s_n)) \Vert _{\mathcal {E}( r \ge R_n)} \\&\left. + \sum _{j=K+1}^N \Big ( \frac{\lambda _{j}(s_n)}{\lambda _{j+1}(s_n)} \Big )^{\frac{k}{2}} \right) = 0. \end{aligned} \end{aligned}$$

  (5.11)

Remark 5.10

One should think of $\nu (t)$ as the scale that separates the $N-K$ “exterior” bubbles, which stay coherent on the union of the collision intervals $[a_n, b_n]$ from the K “interior” bubbles that are coherent at the left endpoint $[a_n, b_n]$, but come into collision inside the interval and lose their shape. In the case $K =N$, there are no exterior bubbles, we set $\lambda _{K+1}(t):= \sqrt{T_+ - t}$ and $\nu _n \rightarrow 0$ is chosen using (4.4) in the blow up case, and $\lambda _{K+1}(t):= \sqrt{t}$ and $\nu _n \rightarrow 0$ is chosen using (4.14) in the global case.

Proof of Lemma 5.9

We carry out the argument in the case $T_+< \infty$, and note that the global case is similar, and in fact, slightly less involved since $u^*= 0$ in that case. Let $a_n, b_n, \epsilon _n, \eta$, and $K\in \{1, \dots , N\}$ be some initial choice of parameters given by Definition 5.5 and Lemma 5.6. Over the course of the proof we will shrink $\eta$ and enlarge $\epsilon _n$ as in Remark 5.8, but abuse notation by still denoting the resulting subintervals by $[a_n, b_n]$ after these modifications.

We first define the function $\alpha (t)$ and choose the sequence $\nu _n \rightarrow 0$. By Defintion 5.1, for each n we can find scales $\rho _K(t) \ll \mu _{K+1}(t) \ll \dots \ll \mu _{N}(t) \ll (T_+-t)^{\frac{1}{2}}$ and signs $\mathbf {\sigma }(t) \in \{-1, 1\}^{N-K}$ for $t \in [ a_n, b_n]$, such that defining $h_{\rho _K}(t)$ for $r \in ( \rho _K(t), \infty )$ by

$$\begin{aligned} u(t) - u^* = \mathcal Q(m_{\Delta }, \mathbf {\sigma }(t), \mathbf { \mu }(t)) + h_{\rho _K}(t) \end{aligned}$$

we have,

$$\begin{aligned} \begin{aligned}{ \textbf{d}(t; \rho _K(t)) \simeq \Vert h_{\rho _K}(t) \Vert _{\mathcal {E}( \rho _K(t), \infty )}^2 + \sum _{j =K}^N \Big ( \frac{ \mu _{j}(t)}{ \mu _{j+1}(t)} \Big )^k \lesssim \epsilon _n^2, } \end{aligned} \end{aligned}$$

(5.12)

keeping the convention $\mu _K(t):= \rho _K(t), \mu _{N+1}(t):=(T_+-t)^{\frac{1}{2}}$. Using $\lim _{n \rightarrow \infty } \sup _{t \in [a_n, b_n]}\textbf{d}_K( t; \rho _K(t)) = 0$ and the fact that

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \sup _{t \in [a_n, b_n]}E( \mathcal Q(m_{\Delta }, \mathbf {\sigma }(t), \mathbf { \mu }(t)); \nu _{n, 1} \widetilde{\mu }_{K+1}(t), \nu _{n, 2} \widetilde{\mu }_{K+1}(t)) = 0, } \end{aligned} \end{aligned}$$

(5.13)

for any two sequence $\nu _{n, 1} \ll \nu _{n, 2} \ll 1$, we can choose a sequence $\nu _n \rightarrow 0$ such that for any $A>1$,

$$\begin{aligned} \begin{aligned}{ \rho _{K}(t) \le \nu _n \mu _{K+1}(t), {\ \ \text {and} \ \ }\lim _{n \rightarrow \infty }\sup _{t \in [a_n, b_n]} E( u(t) - u^*; \frac{1}{A}\nu _n \mu _{K+1}(t), A \nu _n \mu _{K+1}(t)) = 0. } \end{aligned}\nonumber \\ \end{aligned}$$

(5.14)

Next, letting $\rho (t)$ be as in (4.4), we can use (5.12) to choose $\alpha _n \rightarrow 0$ to be a sequence such that,

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \sup _{t \in [a_n, b_n]} \left( \frac{\mu _{N}(t)}{ \alpha _n (T_+ - t)^{\frac{1}{2}}} + \frac{\rho (t)}{\alpha _n (T_+ - t)^{\frac{1}{2}}} \right) = 0, } \end{aligned} \end{aligned}$$

(5.15)

and we define $\alpha (t):= \alpha _n (T_+ - t)^{\frac{1}{2}}$ for $t \in [a_n, b_n]$. If $K=N$ we may assume that $\alpha _n \ge \nu _n$. Setting,

$$\begin{aligned} \begin{aligned}{ u^*(t):= (1 - \chi _{\alpha (t)}) \big ( u(t) - m_{\Delta } \pi \big ) } \end{aligned} \end{aligned}$$

(5.16)

we see from (4.4) and the fact that $\lim _{t \rightarrow T_+} E( u^*; \gamma (t)) = 0$ for any $\gamma (t) \rightarrow 0$ as $t \rightarrow T_+$, that

$$\begin{aligned} \begin{aligned}{ \lim _{t \rightarrow T_+}\Vert u^*(t) - u^*\Vert _{\mathcal {E}} = 0, } \end{aligned} \end{aligned}$$

(5.17)

and by definition,

$$\begin{aligned} u(t) - u^*(t) = \chi _{\alpha (t)} u(t) + (1- \chi _{\alpha (t)}) m_{\Delta } \pi \end{aligned}$$

and by (4.3) and (4.2) we have,

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \sup _{t \in [a_n, b_n]} \big | E( u(t) - u^*(t)) - NE(Q)\big | = 0 } \end{aligned} \end{aligned}$$

(5.18)

Now that $u^*(t)$ is defined, we find the parameters $\mathbf {\iota }\in \{-1, 1\}^N$ and $\mathbf {\lambda }(t) \in (0, \infty )^N$. By the definition of $\textbf{d}(t)$ we make an initial choice of signs $\mathbf {\widetilde{\iota }}(t) \in \{-1, 1\}$ and scales $\mathbf {\widetilde{\lambda }}(t) \in (0, \infty )^N$ such that defining

$$\begin{aligned} \begin{aligned}{ \widetilde{g}(t):= u(t) - u^* - \mathcal Q( m_{\Delta }, \mathbf {\widetilde{\iota }}(t), \mathbf {\widetilde{\lambda }}(t)) } \end{aligned} \end{aligned}$$

(5.19)

we have,

$$\begin{aligned} \begin{aligned}{ \textbf{d}(t) \le \Vert \widetilde{g}(t) \Vert _{\mathcal {E}} + \sum _{j = 1}^{N} \Big ( \frac{\widetilde{\lambda }_{j}(t)}{ \widetilde{\lambda }_{j+1}(t)} \Big )^{\frac{k}{2}} \le 2 \textbf{d}(t) \le 2 \eta } \end{aligned} \end{aligned}$$

(5.20)

keeping the convention that $\lambda _{N+1}(t) = (T_+-t)^{\frac{1}{2}}$.

By (5.17) (5.19), and (5.20) we see that $\textbf{d}(t) \le \eta$ implies that

$$\begin{aligned} \begin{aligned}{ \textbf{d}_{m_{\Delta }, N}(u(t) - u^*(t)) \le C_0 \textbf{d}(t) + o_n(1) \le 2 C_0 \eta , } \end{aligned} \end{aligned}$$

(5.21)

where $\textbf{d}_{m_{\Delta }, N}$ is as in (2.12) and $o_n(1)$ denotes a term that tends to zero as $n \rightarrow \infty$. We may then shrink $\eta >0$ as in Remark 5.8 small enough so that we can apply Lemma 2.12 to $u(t) - u^*(t)$, obtaining $\mathbf {\lambda }(t) \in (0, \infty )^N$ defined on $\cup _n [a_n, b_n]$, and signs $\mathbf {\iota }\in \{-1, 1\}^N$ (which can be taken independent of $t \in [a_n, b_n]$ using continuity of the flow and independently of n after passing to a subsequence of the $[a_n, b_n]$), and g(t) so that

$$\begin{aligned}\begin{aligned}{ u(t) - u^*(t) = m_\Delta \pi + \sum _{j=1}^N \iota _{ j} (Q_{\lambda _{j}(t)} - \pi ) + g(t), \quad \big \langle \mathcal Z_{\underline{\lambda (t)}} \mid g(t)\big \rangle = 0, \quad \forall t \in [a_n, b_n], } \end{aligned} \end{aligned}$$

and,

$$\begin{aligned}\begin{aligned} \textbf{d}_{m_{\Delta }, N}(u(t) - u^*(t))) \le \Vert g(t) \Vert _{\mathcal {E}} + \sum _{j=1}^{N-1} \Big ( \frac{\lambda _j(t)}{\lambda _{j+1}(t)} \Big )^{\frac{k}{2}} \le C_0 \textbf{d}_{m_{\Delta }, N}(u(t) - u^*(t)) \end{aligned} \end{aligned}$$

Using again (5.17) along with (5.21) we see that in fact,

$$\begin{aligned} \textbf{d}(t) - \zeta _{1, n} \le \Vert g(t) \Vert _{\mathcal {E}} + \sum _{j=1}^N \Big ( \frac{\lambda _j(t)}{\lambda _{j+1}(t)} \Big )^{\frac{k}{2}} \le C_0 \textbf{d}(t) + \zeta _{1, n} \end{aligned}$$

where $\zeta _{1, n}$ is a sequence tending to zero as $n \rightarrow \infty$. By enlarging $\epsilon _n$ so that $\epsilon _n \ge 2 \zeta _{1, n}$ for all n as in Remark 5.8 we prove (5.7) (note here that because of Remark 5.8, the act of “enlarging” $\epsilon _n$ does not affect the sequence $\zeta _{1, n}$).

Next, we compare the scales $\lambda _{K+1}, \dots , \lambda _{N}$ to $\mu _{K+1}, \dots , \mu _N$. Denoting by $\widetilde{\nu }(t):= \nu _n \mu _{K+1}(t)$ we claim that for each $j = 1, \dots , N$,

$$\begin{aligned} \begin{aligned}{ \lim _{ n \rightarrow \infty } \sup _{t \in [a_n, b_n]} \Big ( \frac{\widetilde{\nu }(t)}{ \lambda _j(t)} + \frac{\lambda _j(t)}{\widetilde{\nu }(t)} \Big ) = 0. } \end{aligned} \end{aligned}$$

(5.22)

If not, we could find $C>0$, $j \in \{1, \dots , N\}$, a subsequence of the $[a_n, b_n]$ and a sequence $s_n \in [a_n, b_n]$ such that

$$\begin{aligned} C^{-1} \widetilde{\nu }(s_n) \le \lambda _j(s_n) \le C \widetilde{\nu }(s_n) \end{aligned}$$

By (5.7) for all $\eta >0$ sufficiently small we can find $\delta = \delta (\eta ), R= R(\eta )>0$ so that for all n,

$$\begin{aligned} \delta \le E( u(s_n) - u^*(s_n); R^{-1} \lambda _j(s_n), R \lambda _j(s_n)) \le E( u(s_n) - u^*(s_n); C^{-1}R^{-1} \widetilde{\nu }(s_n), RC \widetilde{\nu }(s_n)) \end{aligned}$$

which contradicts (5.14).

By (5.14) and Lemma 2.3 we can find integers $m_n$ so that denoting

$$\begin{aligned} w(t) = m_n \pi \chi _{\widetilde{\nu }(t)} + ( 1- \chi _{\widetilde{\nu }(t)})( u(t) - u^*(t)) \end{aligned}$$

we have,

$$\begin{aligned} \begin{aligned}{ \Vert w(t) - \mathcal Q(m_\Delta , \mathbf {\sigma }(t), \mathbf {\mu }(t)) \Vert _{\mathcal {E}}^2 + \sum _{j=K+1}^{N-1} \Big (\frac{ \mu _{j}(t)}{\mu _{j+1}(t)} \Big )^k = o_n(1) } \end{aligned} \end{aligned}$$

(5.23)

On the other hand, by (5.22) we can find $j_0 \in \{1, \dots , N-1\}$ so that

$$\begin{aligned} \Vert w(t) - \mathcal Q(m_\Delta , \iota _{j_0}, \dots , \iota _{N}, \lambda _{j_0}(t), \dots , \lambda _{N}(t)) \Vert _{\mathcal {E}}^2 + \sum _{j=j_0}^{N-1} \Big (\frac{ \lambda _{j}(t)}{\lambda _{j+1}(t)} \Big )^k \le C_0 \eta \end{aligned}$$

An application of Lemma 2.13 yields $j_0 = K+1$, $\mathbf {\sigma }(t) = \{\iota _{K+1}, \dots , \iota _K\}$ and moreover, by shrinking $\eta >0$, we can ensure that

$$\begin{aligned} \sup _{t \in [a_n, b_n]} \Big | \frac{\lambda _j(t)}{\mu _j(t)} - 1 \Big | \le \frac{1}{4} \end{aligned}$$

and thus, defining $\nu (t):= \nu _n \lambda _{K+1}(t)$ we see that (5.6) follows from (5.12) (5.14), and (5.15). Let $s_n\in [a_n, b_n]$ and $R_n$ so that $\nu (s_n) \le R_n \ll \lambda _{K+1}(s_n)$. If $K < N$ then $R_n \ll \alpha (s_n)$, thus, using (5.23) and (5.15), we see that

$$\begin{aligned} E( u(s_n); R_n, \alpha (s_n)) \rightarrow (N-K) E(Q) {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

Since by (5.15), (5.16) and (5.17),

$$\begin{aligned} E( u(s_n); \alpha (s_n), \infty ) \rightarrow E( u^*) {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

we see that (5.10) follows. If $K =N$ then $E( u(s_n); R_n, \infty ) \rightarrow E( u^*)$. Similarly $N-K$ converge now follows from (5.23).

Next we prove (5.8). An application of (2.13) together with (5.18) gives,

$$\begin{aligned} \Vert g(t) \Vert _{\mathcal {E}} + \sum _{j \not \in \mathcal A} \Big ( \frac{\lambda _j(t)}{\lambda _{j+1}(t)} \Big )^{\frac{k}{2}} \le C_0 \sum _{j \in \mathcal A} \Big ( \frac{\lambda _j(t)}{\lambda _{j+1}(t)} \Big )^{\frac{k}{2}} + \zeta _{2, n} \end{aligned}$$

for some sequence $\zeta _{2, n} \rightarrow 0$, which is independent of $t \in [a_n, b_n]$. But then by enlarging $\epsilon _n \rightarrow 0$ as in Remark 5.8 so that $\epsilon _n \gg \zeta _{2, n}$ we obtain (5.8) via the above and (5.7) (note again here that because of Remark 5.8, the act of “enlarging” $\epsilon _n$ does not affect the sequence $\zeta _{2, n}$).

Lastly, we prove the modulation estimate (5.9). Differentiating in time the orthogonality conditions (5.5) yields, for each $j = 1, \dots , N$, the identity,

$$\begin{aligned} \begin{aligned}{ \big \langle \partial _t g \mid \mathcal Z_{\underline{\lambda _j}} \big \rangle = \frac{\lambda _j'}{\lambda _j} \big \langle \underline{\mathcal Z}_{\underline{\lambda }} \mid g \big \rangle } \end{aligned} \end{aligned}$$

(5.24)

Next, differentiating in time the expression for g(t) in (5.2) and recalling the definition of $u^*(t)$ gives,

$$\begin{aligned}\begin{aligned} \partial _t g&= \partial _t \chi _{\alpha } - \frac{\alpha '}{\alpha } \Lambda \chi _{\alpha } ( u(t) - m_{\Delta } \pi ) + \sum _{j=1}^N \iota _j \lambda _j' \Lambda Q_{\underline{\lambda _j}}\\&= (\Delta u) \chi _\alpha - \frac{k^2}{r^2} f(u) \chi _\alpha - \frac{\alpha '}{\alpha } \Lambda \chi _{\alpha } ( u(t) - m_{\Delta } \pi ) + \sum _{j=1}^N \iota _j \lambda _j' \Lambda Q_{\underline{\lambda _j}}\\&= \Delta (\chi _\alpha u + (1- \chi _\alpha ) m_{\Delta } \pi ) - \frac{k^2}{r^2}f\big ( \chi _\alpha u + (1- \chi _\alpha ) m_{\Delta } \pi \big )+ \sum _{j=1}^N \iota _j \lambda _j' \Lambda Q_{\underline{\lambda _j}}\\&\quad - ( u - m_\Delta \pi ) \Delta \chi _\alpha - 2 \partial _r u \partial _r \chi _{\alpha } - \frac{\alpha '}{\alpha } \Lambda \chi _{\alpha } ( u(t) - m_{\Delta } \pi ) \\&\quad - \frac{k^2}{r^2}\Big ( f(u) \chi _\alpha - f( \chi _\alpha u + (1- \chi _\alpha ) m_{\Delta } \pi )\Big ), \end{aligned} \end{aligned}$$

and we see that

$$\begin{aligned} \begin{aligned} \partial _t g = - \mathcal {L}_{\mathcal Q} g + \sum _{j=1}^N \iota _j \lambda _j' \Lambda Q_{\underline{\lambda _j}} + f_{\textbf{i}}( m_\Delta , \mathbf {\iota }, \mathbf {\lambda }) + f_{\textbf{q}}(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }, g) + \phi (u, \alpha ) \end{aligned} \end{aligned}$$

(5.25)

where

$$\begin{aligned}\begin{aligned} \phi (u, \alpha )&:= - ( u - m_\Delta \pi ) \Delta \chi _\alpha - 2 \partial _r u \partial _r \chi _{\alpha } - \frac{\alpha '}{\alpha } \Lambda \chi _{\alpha } ( u(t) - m_{\Delta } \pi ) \\&\quad - \frac{k^2}{r^2}\Big ( f(u) \chi _\alpha - f( \chi _\alpha u + (1- \chi _\alpha ) m_{\Delta } \pi )\Big ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned}\begin{aligned} f_{\textbf{i}}( m_\Delta , \mathbf {\iota }, \mathbf {\lambda })&:=- {\text {D}}E( \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }))= - \frac{k^2}{r^2} \left( f\big ( \mathcal Q(m_\Delta , \mathbf {\iota }, \mathbf {\lambda })\big ) - \sum _{j =1}^{N} \iota _j f(Q_{\lambda _{j}}) \right) \\ f_{\textbf{q}}(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }, g)&:= - \frac{k^2}{r^2} \left( f\big ( \mathcal Q(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }) + g \big ) - f\big (\mathcal Q(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }) \big ) - f'\big ( \mathcal Q(m_\Delta , \mathbf {\iota }, \mathbf {\lambda })\big ) g \right) . \end{aligned} \end{aligned}$$

The subscript $\textbf{i}$ above stands for “interaction” and $\textbf{q}$ stands for “quadratic.”

We make use of the estimates,

$$\begin{aligned} \begin{aligned}{ \Vert f_{\textbf{i}}( m_\Delta , \mathbf {\iota }, \mathbf {\lambda }) \Vert _{L^1} \lesssim \sum _{j=1}^{N-1}\Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^k, \quad \Vert f_{\textbf{q}}(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }, g) \Vert _{L^1} \lesssim \Vert g \Vert _{\mathcal {E}}^2 } \end{aligned} \end{aligned}$$

(5.26)

For the $f_{\textbf{i}}$ estimate we expand to obtain the expression,

$$\begin{aligned}\begin{aligned} \frac{r^2}{k^2} {\text {D}}E ( \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }))&= \frac{1}{2} \sin \left( 2\sum _{ i =2}^M \iota _i Q_{\lambda _i} + 2 \iota _1 Q_{\lambda _1} \right) - \frac{1}{2}\sum _{ i = 1}^M \iota _i \sin 2 Q_{\lambda _i}\\&= - \sin \left( 2\sum _{ i =2}^M \iota _i Q_{\lambda _i}\right) \sin ^2 Q_{\lambda _1} - \iota _1\sin ^2 \left( \sum _{ i =2}^M \iota _i Q_{\lambda _i}\right) \sin 2 Q_{\lambda _1} \\&\quad + \frac{1}{2} \sin \left( 2\sum _{ i =2}^M \iota _i Q_{\lambda _i}\right) - \frac{1}{2} \sum _{i =2}^M \iota _i \sin 2 Q_{\lambda _i} \end{aligned} \end{aligned}$$

Iterating this expansion in the last line above and using the identity $k\sin Q = \Lambda Q$ we obtain the pointwise estimates,

$$\begin{aligned} \begin{aligned} |{\text {D}}E( \mathcal Q(m, \mathbf {\iota }, \mathbf {\lambda }))|&\lesssim \frac{1}{r^2} \sum _{i,j, \ell \, \, \text {not all equal}} \Lambda Q_{\lambda _i} \Lambda Q_{\lambda _j} \Lambda Q_{\lambda _\ell } \end{aligned} \end{aligned}$$

(5.27)

from which the estimate for $f_{\textbf{i}}$ in (5.26) follows by way of Lemma 2.8. The estimate for $f_{\textbf{q}}$ in (5.26) is straightforward.

For each $j \in \{1, \dots , N\}$ we pair (5.25) with $\mathcal Z_{\underline{\lambda _j}}$ and use (5.24) to obtain the following system

$$\begin{aligned}{} & {} \iota _j \lambda _j' \Big ( \big \langle \Lambda Q \mid \mathcal Z\big \rangle - \frac{\iota _j}{\lambda _j} \big \langle \mathcal Z_{\underline{\lambda _j}} \mid g\big \rangle \Big ) + \sum _{i \ne j} \iota _i \lambda _i'\big \langle \Lambda Q_{\underline{\lambda _i}} \mid \mathcal Z_{\underline{\lambda _j}} \big \rangle \\{} & {} \quad = \big \langle \mathcal L_{\mathcal Q} g \mid \mathcal Z_{\underline{\lambda _j}} \big \rangle - \big \langle f_{\textbf{i}}( m_\Delta , \mathbf {\iota }, \mathbf {\lambda })\mid \mathcal Z_{\underline{\lambda _j}} \big \rangle - \big \langle f_{\textbf{q}}(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }, g)\mid \mathcal Z_{\underline{\lambda _j}} \big \rangle - \big \langle \phi (u, \alpha ) \mid \mathcal Z_{\underline{\lambda _j}} \big \rangle . \end{aligned}$$

The above is diagonally dominate for all sufficiently small $\eta >0$, hence invertible. We note the brutal estimates,

$$\begin{aligned} \begin{aligned} \Big | \big \langle \mathcal L_{\mathcal Q} g \mid \mathcal Z_{\underline{\lambda _j}} \big \rangle \Big |&\lesssim \frac{1}{\lambda _j} \Vert g \Vert _{\mathcal {E}} \\ \Big | \big \langle f_{\textbf{i}}( m_\Delta , \mathbf {\iota }, \mathbf {\lambda })\mid \mathcal Z_{\underline{\lambda _j}} \big \rangle \Big |&\lesssim \frac{1}{\lambda _j} \sum _{j=1}^{N-1}\Big ( \frac{\lambda _j}{\lambda _{j+1}} \Big )^k \\ \Big | \big \langle f_{\textbf{q}}(m_\Delta , \mathbf {\iota }, \mathbf {\lambda }, g)\mid \mathcal Z_{\underline{\lambda _j}} \big \rangle \Big |&\lesssim \frac{1}{\lambda _j} \Vert g \Vert _{\mathcal {E}}^2 \\ \Big |\big \langle \phi (u, \alpha ) \mid \mathcal Z_{\underline{\lambda _j}} \big \rangle \Big |&= \frac{1}{\lambda _j} o_n(1) \end{aligned} \end{aligned}$$

(5.28)

We remark that to prove the second inequality in (5.28) we may use (5.27) and the definition of $f_{\textbf{i}}$. The estimates of the remaining estimates are straightforward and we omit the proofs. It follows that,

$$\begin{aligned} \left|{\lambda _j'}\right| \lesssim \frac{1}{\lambda _j} \Big ( \textbf{d}(t) + \zeta _{3, n} \Big ) \end{aligned}$$

for some sequence $\zeta _{3, n} \rightarrow 0$ as $n \rightarrow \infty$. Then (5.9) follows by enlarging $\epsilon _n$ (note that because of Remark 5.8, the act of “enlarging” $\epsilon _n$ does not affect the sequence $\zeta _{3, n}$). This completes the proof.$\square$

6 Conclusion of the proof

For the remainder of the paper, when we write $[a_n, b_n] \in \mathcal C_K(\epsilon _n, \eta )$ we we always assume that $\textbf{d}(a_n) = \epsilon _n$, $\textbf{d}(b_n) = \eta$ and $\textbf{d}(t) \in [\epsilon _n, \eta ]$ for all $t \in [a_n, b_n]$. This assumption is valid by Remark 5.8.

Lemma 6.1

If $\eta _0>0$ is small enough, then for any $\eta \in (0, \eta _0]$ there exist $\epsilon \in (0, \eta )$ and $C_u>0$ with the following property. If $[c, d] \subset [a_n, b_n]$, $\textbf{d}(c) \le \epsilon$ and $\textbf{d}(d) \ge \eta$, then,

$$\begin{aligned} (d- c)^{\frac{1}{2}} \ge C_u^{-1} \lambda _K(c) \end{aligned}$$

Proof

If not, there exists $\eta >0$, sequences $\epsilon _n \rightarrow 0$, $[c_n, d_n] \subset [a_n, b_n]$, and $C_n \rightarrow \infty$ so that $\textbf{d}(c_n) \le \epsilon _n$, $\textbf{d}(d_n) \ge \eta$ and

$$\begin{aligned} \begin{aligned}{ (d_n - c_n)^{\frac{1}{2}} \le C_n^{-1} \lambda _K(c_n) } \end{aligned} \end{aligned}$$

(6.1)

We show that in this case $[c_n, d_n] \in \mathcal C_{K-1}(\epsilon _n, \eta )$, which contradicts the minimality of K.

First, using (5.9) we see for all j,

$$\begin{aligned} \begin{aligned}{ \left|{\lambda _j(t)^2 - \lambda _j(c_n)^2}\right| \le C_0 (t- c_n) } \end{aligned} \end{aligned}$$

(6.2)

for all $t \in [c_n, d_n]$. Hence, using the contradiction assumption (6.1) we can ensure that for large enough n,

$$\begin{aligned} \frac{3}{4} \le \frac{ \lambda _j(t)}{\lambda _j(c_n)} \le \frac{5}{4} \end{aligned}$$

for all $j = K, \dots , N$ and all $t \in [c_n, d_n]$. Since $\textbf{d}(c_n) \rightarrow 0$, it follows that,

$$\begin{aligned} \begin{aligned}{ \lim _{n \rightarrow \infty } \sup _{t \in [c_n, d_n]} \sum _{j=K}^{N} \Big ( \frac{ \lambda _j(t)}{\lambda _{j+1}(t)} \Big )^k = 0. } \end{aligned} \end{aligned}$$

(6.3)

Next, since $\textbf{d}(c_n) \rightarrow 0$ we can find a sequence $r_n$ such that

$$\begin{aligned} \begin{aligned}{ \lambda _{K-1}(c_n) + (d_n - c_n)^{\frac{1}{2}} \ll r_n \ll \lambda _K(c_n) {\ \ \text {and} \ \ }\lim _{n \rightarrow \infty } E( u(c_n) - u^*(c_n); \frac{1}{8} r_n, 8 r_n) = 0. } \end{aligned}\nonumber \\ \end{aligned}$$

(6.4)

Since $r_n \ll \alpha (t)$ we see that $u(t, r) - u^*(t, r) = \chi _{\alpha (t)} u(t, r) + (1- \chi _{\alpha (t)}) m_{\Delta } \pi = u(t, r)$ for all $r \in (1/8 r_n, 8 r_n)$. Letting $\phi (r)$ be a smooth bump equal to 1 for $r \in (1/4, 4)$ and supported for $r \in (1/8, 8)$ with $|\phi '(r)| \le 16$, we apply (2.5) with such a $\phi$ and deduce that for any $t \in [c_n, d_n]$,

$$\begin{aligned} E( u(t); \frac{1}{4} r_n, 4 r_n) \le E( u(c_n); 1/8 r_n, 8 r_n) + C_0 \frac{ d_n - c_n}{r_n^2} \end{aligned}$$

and hence,

$$\begin{aligned}\begin{aligned}{ \lim _{n \rightarrow \infty } \sup _{t \in [c_n, d_n]} E( u(t)- u^*(t); \frac{1}{4} r_n, 4 r_n) = 0. } \end{aligned} \end{aligned}$$

Next we claim that

$$\begin{aligned} \begin{aligned}{ \sup _{t \in [c_n, d_n]} E( u(t) - u^*(t); \frac{1}{4} r_n, \infty ) \le (N-(K-1)) E(Q) + o_n(1) } \end{aligned} \end{aligned}$$

(6.5)

In the case $T_+< \infty$ we recall that $\alpha (t) = \alpha _n (T_+-t)^{\frac{1}{2}}$ and we write,

$$\begin{aligned} E( u(t) - u^*(t); \frac{1}{4} r_n, \infty ) = E( u(t) - u^*(t); \frac{1}{4} r_n, \frac{1}{4}\alpha (t)) + E( u(t) - u^*(t); \frac{1}{4} \alpha (t), \infty ) \end{aligned}$$

Since $\alpha (t) \ge \rho (t)$ we have,

$$\begin{aligned} \lim _{t \rightarrow \infty } E( u(t) - u^*(t); \frac{1}{4} \alpha (t), \infty ) = 0 \end{aligned}$$

Recalling that $u(t, r) - u^*(t, r) = u(t, r)$ for all $r \le \alpha (t)$ we again apply (2.5) with the cut-off function $\phi (t, r) = (1- \chi _{4 r_n}(r)) \chi _{\frac{1}{4}\alpha (t)}(r)$. Since $\frac{\textrm{d}}{\textrm{d}t} \phi (t, r) \le 0$ we use  (2.5) to deduce that for all $t \in [c_n, d_n]$,

$$\begin{aligned} E\left( u(t) - u^*(t); \frac{1}{4} r_n, \frac{1}{4}\alpha (t)\right) \le E\left( u(c_n) - u^*(c_n); \frac{1}{8} r_n, \frac{1}{2}\alpha (t)\right) + C_0\frac{ d_n - c_n}{r_n^2} \end{aligned}$$

and the right hand side tends to zero as $n \rightarrow \infty$, proving (6.5) in the case $T_+< \infty$. If $T_+ = \infty$, we use the same argument, but without the need to truncate at $\alpha (t)$ since we have $u^*(t):= 0$.

Next, using (6.2) with $j = K-1$ gives,

$$\begin{aligned} \sup _{t \in [c_n, d_n]} \left|{ \lambda _{K-1}(t)^2 - \lambda _{K-1}(c_n)^2}\right| \lesssim d_n - c_n, \end{aligned}$$

and hence

$$\begin{aligned} \sup _{t \in [c_n, d_n]} \frac{\lambda _{K-1}(t)}{ r_n} \lesssim \frac{\lambda _{K-1}(c_n)}{ r_n} + \frac{(d_n - c_n)^{\frac{1}{2}}}{r_n} \rightarrow 0 {\ \ \text {as} \ \ }n \rightarrow \infty \end{aligned}$$

given our choice of $r_n$ in (6.4). Using all of the above, we can find $m_n \in \mathbb {Z}$ so that defining,

$$\begin{aligned} v(t):= (1- \chi _{r_n}) (u(t) - u^*(t)) + \chi _{r_n} m_n \pi \end{aligned}$$

we have $v(t) \in \mathcal {E}_{m_n, m_\Delta }$ for $t \in [c_n, d_n]$ and such that

$$\begin{aligned} \Vert v(t) - \mathcal Q( m_\Delta , \iota _{K}, \dots , \iota _{N}, \lambda _K(t), \dots \lambda _{N}(t)) \Vert _{\mathcal {E}} + \sum _{j = K}^{N-1} \Big (\frac{ \lambda _{j}(t)}{ \lambda _{j+1}(t)} \Big )^{\frac{k}{2}} \lesssim \eta \end{aligned}$$

It follows that $\textbf{d}_{m_\Delta , N-K+1}(v(t)) \lesssim \eta$ and we can apply Lemma 2.12 to find modulation parameters $\mathbf {\widetilde{\iota }} \in \{-1, 1\}^{N-K+1}$, $\widetilde{\lambda }_{K}(t), \dots , \widetilde{\lambda }_{N}(t)$ and h(t) defined by

$$\begin{aligned} h(t) = v(t) - \mathcal Q( m_\Delta , \iota _{K}, \dots , \iota _{N}, \widetilde{\lambda }_K(t), \dots , \widetilde{\lambda }_{N}(t)) \end{aligned}$$

so that

$$\begin{aligned} 0= \big \langle \mathcal Z_{\underline{\widetilde{\lambda }_j(t)}} \mid h(t) \big \rangle , {\ \ \text {and} \ \ }\Vert h(t) \Vert _{\mathcal {E}} + \sum _{j=K}^{N-1} \Big (\frac{\widetilde{\lambda }_{j}(t)}{\widetilde{\lambda }_{j+1}(t)} \Big )^{\frac{k}{2}} \lesssim \eta \end{aligned}$$

In fact, using (6.3) and the fact that the $\widetilde{\lambda }_j(t)$ satisfy $| \widetilde{\lambda }_j(t)/ \lambda _j(t) - 1| \lesssim \eta$, we have,

$$\begin{aligned} \lim _{n \rightarrow \infty } \sup _{t \in [c_n, d_n]} \sum _{j=K}^{N-1} \Big (\frac{\widetilde{\lambda }_{j}(t)}{\widetilde{\lambda }_{j+1}(t)} \Big )^{\frac{k}{2}} = 0. \end{aligned}$$

And, thus, using (2.13) along with (6.5) we have the bound,

$$\begin{aligned} \Vert h(t) \Vert _{\mathcal {E}} \lesssim \sum _{j=K}^{N-1} \Big (\frac{\widetilde{\lambda }_{j}(t)}{\widetilde{\lambda }_{j+1}(t)} \Big )^{\frac{k}{2}} + o_n(1) \end{aligned}$$

and thus $\lim _{n \rightarrow \infty } \sup _{t \in [c_n, d_n]} \Vert h(t) \Vert _{\mathcal {E}} = 0$ as well. Letting $\rho _{K-1}(t):= r_n$ for $t \in [c_n, d_n]$ we have proved that

$$\begin{aligned} \lim _{n \rightarrow \infty } \sup _{t \in [c_n, d_n]} \textbf{d}(t; \rho _{K-1}(t)) = 0 \end{aligned}$$

which means we can find $\widetilde{\eta }>0$, $\widetilde{\epsilon }_n \rightarrow 0$ such that $[c_n, d_n] \in \mathcal C_{K-1}( \widetilde{\epsilon }_n, \widetilde{\eta })$ contradicting the minimality of K.$\square$

Lemma 6.2

Let $\eta _0>0$ be as in Lemma 6.1, $\eta \in (0, \eta _0]$, $\epsilon _n \rightarrow 0$ be some sequence, and let $[a_n, b_n] \in \mathcal C_K(\epsilon _n, \eta )$. Then, there exist $\epsilon \in (0, \eta )$, $n_0 \in \mathbb {N}$, and $c_n, d_n \in (a_n, b_n)$ such that for all $n \ge n_0$, we have

$$\begin{aligned}{} & {} \textbf{d}(t) \ge \epsilon , \quad \forall \, \, t \in [c_n, d_n], \end{aligned}$$

(6.6)

$$\begin{aligned}{} & {} d_n - c_n = \frac{1}{n} \lambda _K(c_n)^2, \end{aligned}$$

(6.7)

and

$$\begin{aligned} \begin{aligned}{ \frac{1}{2} \lambda _K(c_n) \le \lambda _K(t) \le 2\lambda _K(c_n) \quad \forall \, \, t \in [c_n, d_n]. } \end{aligned} \end{aligned}$$

(6.8)

Proof

Choose $\epsilon >0$ so that Lemma 6.1 holds and define $c_n:= \sup \{t \in [a_n, b_n] \mid \textbf{d}(t) \le \epsilon \}$. Then $\textbf{d}(c_n) = \epsilon$ and by Lemma 6.1 we have

$$\begin{aligned} b_n - c_n \ge C_u^{-1} \lambda _K(c_n). \end{aligned}$$

We then let $d_n:= c_n + \frac{1}{n} \lambda _K(c_n)^2$ and for n sufficiently large we have $d_n < b_n$. Then by (5.9) we have,

$$\begin{aligned} \left| \frac{ \lambda _K(t)^2}{\lambda _K(c_n)^2} - 1 \right| \lesssim \frac{ d_n- c_n}{\lambda _K(c_n)} = \frac{1}{n}. \end{aligned}$$

from which (6.8) follows.$\square$

Lemma 6.3

There exists $\eta _1>0$ with the following property. Let $\eta \in (0, \eta _1]$, $\epsilon _n \rightarrow 0$ and let $[a_n, b_n] \in \mathcal C_K(\epsilon _n, \eta )$. If $\{s_n \}_{n}$ and $\{r_n\}_n$ are any sequences such that $s_n \in [a_n, b_n]$ for all n, $1 \ll r_n \ll \lambda _{K+1}(s_n)/\lambda _{K}(s_n)$, and $\lim _{n \rightarrow \infty } \varvec{\delta }_{r_n \lambda _K(s_n)}(u(s_n)) = 0$, then $\lim _{n \rightarrow \infty } \textbf{d}(s_n) = 0$.

Proof

Let $R_n$ be a sequence such that $r_n\lambda _K(s_n) \ll R_n \ll \lambda _{K+1}(s_n)$. Without loss of generality, we can assume $\nu (s_n) \le R_n \le \alpha (s_n)$, since it suffices to replace $R_n$ by $\nu (s_n)$ for all n such that $R_n < \nu (s_n)$. If $K = N$ we can similarly ensure that $R_n \le \alpha (s_n)$. Let $M_n, m_n, \mathbf {\sigma }_n \in \{-1, 1\}^{M_n}, \mathbf {\mu }_n \in (0, \infty )^{M_n}$ be parameters such that

$$\begin{aligned} \Vert u(t_n) - \mathcal Q( m_n, \mathbf {\sigma }_n, \mathbf {\mu }_n) \Vert _{H( r \le r_n\lambda _K(s_n))}^2 + \sum _{j = 1}^{M_n} \Big (\frac{ \mu _{n, j}}{ \mu _{n, j+1}}\Big )^k + \frac{\mu _{n, M_n}}{r_n \lambda _{K}(s_n)} \rightarrow 0, \end{aligned}$$

(6.9)

which exist by the definition of the localized distance function (3.1). Since $\textbf{d}(t) \le \eta$ on $[a_n, b_n]$ we can choose $\eta _1>0$ sufficiently small so that,

$$\begin{aligned} \Big (K - \frac{1}{2}\Big )E( Q)\le & {} \liminf _{n\rightarrow \infty }E( u(s_n); 0, r_n\lambda _K(s_n)) \\\le & {} \limsup _{n\rightarrow \infty }E(u(s_n); 0, r_n\lambda _K(s_n)) \le \Big (K+\frac{1}{2}\Big ) E( Q), \end{aligned}$$

after noting that the radiation $u^*$ is negligible on the region $r \le r_n\lambda _K(s_n)$. Hence, $M_n = K$ for n large enough. We set $\mu _{n, j}:= \lambda _j(s_n)$ and $\sigma _{n, j}:= \iota _j$ for $j > K$. We claim that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left( \Vert u(s_n) - u^* - \mathcal Q(m_\Delta , \mathbf {\sigma }_n, \mathbf {\mu }_n) \Vert _{\mathcal {E}}^2 + \sum _{j=1}^{N}\Big (\frac{ \mu _{n, j}}{\mu _{n, j+1}}\Big )^{k}\right) = 0. \end{aligned}$$

By the definition of $\textbf{d}$, the proof will be finished. First, recall that $\mu _{n, K} \ll r_n\mu (t_n)$, so $\mu _{n, K} / \mu _{n, K+1} \rightarrow 0$. In the region $r \le r_n\lambda _K(s_n)$, convergence follows from (6.9), since the energy of the exterior bubbles asymptotically vanishes there. In the region $r \ge R_n$, the energy of the interior bubbles vanishes, hence it suffices to apply (5.11). In particular, by the above and (5.10),

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty }E( u(s_n); 0, r_n\lambda _K(s_n)) = KE(Q), \qquad \\{} & {} \lim _{n\rightarrow \infty } E( u(s_n); R_n, \infty ) = (N-K)E(Q) + E( u^*), \end{aligned}$$

which implies

$$\begin{aligned} \lim _{n\rightarrow \infty } E( u(s_n); r_n\lambda _K(s_n), R_n) = 0, \end{aligned}$$

and (2.3) yields convergence of the error also in the region $r_n\lambda _K(s_n) \le r \le R_n$.$\square$

Proof of Theorem 1

Assume the theorem is false and let $[a_n, b_n] \in \mathcal C_K(\epsilon _n, \eta )$ be a sequence of disjoint collision intervals given by Lemma 5.9, and $\eta >0$ is sufficiently small so that Lemmas 6.1 and 6.3 hold. Let $\epsilon >0$, $n_0$, and $[c_n, d_n]$ be as in Lemma 6.2.

We claim that there exists $c_0>0$ such that for every $n \ge n_0$,

$$\begin{aligned} \begin{aligned}{ \inf _{t \in [c_n, d_n]} \lambda _K(t)^2 \Vert \partial _t u(t) \Vert _{L^2}^2 \ge c_0. } \end{aligned} \end{aligned}$$

(6.10)

If not, we could, after passing to a subsequence, find a sequence $s_n \in [c_n, d_n]$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \lambda _K(s_n) \Vert \partial _t u(s_n) \Vert _{L^2} = 0 \end{aligned}$$

But then an application of Lemma 3.1 gives a sequence $r_n \rightarrow \infty$ such that, after passing to a further subsequence, $\lim _{n \rightarrow \infty } \varvec{\delta }_{r_n \lambda _K(s_n)} ( u(s_n)) = 0$. But then Lemma 6.3 gives that $\lim _{n \rightarrow \infty } \textbf{d}(s_n) = 0$, which contradicts (6.6). Thus (6.10) holds.

Therefore, using (6.10), (6.8), and (6.7) we have

$$\begin{aligned} \sum _{n \ge n_0} \int _{c_n}^{d_n} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t \ge \frac{c_0}{4} \sum _{n \ge n_0} \int _{c_n}^{d_n} \lambda _K(c_n)^{-2} \, \textrm{d}t \ge \frac{c_0}{4} \sum _{n \ge n_0} n^{-1} = \infty . \end{aligned}$$

On the other hand, by (2.2) and the fact that the $[c_n, d_n]$ are disjoint, we have,

$$\begin{aligned} \sum _{n \ge n_0} \int _{c_n}^{d_n} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t \le \int _0^{T_*} \Vert \partial _t u(t) \Vert _{L^2}^2 \, \textrm{d}t < \infty , \end{aligned}$$

which is a contradiction.$\square$