Raphaël Côte
email: cote@math.polytechnique.fr

ON THE SOLITON RESOLUTION FOR EQUIVARIANT WAVE MAPS TO THE SPHERE

Keywords: 2010 Mathematics Subject Classification. 35L05, 35L71 wave maps, equivariant, classification, profile, soliton resolution

We consider finite energy corotationnal wave maps with target manifold S 2 . We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow up case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space.

1. Introduction 1.1. Statement of the main results. Let (M, g) be a Riemannian manifold, and R 1+d be endowed with the Minkowski metric η = diag(-1, 1, . . . , 1). Wave maps U : (R 1+d , η) → (M, g) are defined formally as critical points of the Lagrangian

L (U, ∂U ) = 1 2 R 1+d η αβ ∂ α U, ∂ β U g dxdt.
In local coordinates, they satisfy the Euler-Lagrange equation

U k = -η αβ Γ k ij (U )∂ α U i ∂ β U j (U, ∂ t U )| t=0 = (U 0 , U 1 ), (1.1)
where Γ k ij are the Christoffel symbols on T M . We consider the case where d = 2 and M is a 2 dimensional surface of revolution with metric ds 2 = dρ 2 + g(ρ) 2 dθ 2 , where (ρ, θ) are the polar coordinates on M , and g ∈ C 3 (R). We assume that U has corotationnal equivariant symmetry, that is, denoting (r, ω) the polar coordinates on R 2 , it takes the form U (t, r, ω) = (ψ(t, r), ω).

for some function ψ. System (1.1) then simplifies to the following equation on ψ :

   ∂ tt ψ -∂ rr ψ - 1 r ∂ r ψ + f (ψ) r 2 = 0 (ψ, ∂ t ψ)| t=0 = (ψ 0 , ψ 1 )
where f = gg .

(WM)

We say that such a solution ψ = (ψ, ∂ t ψ) to (WM) is a wave map. We define the energy space H × L 2 and similarly the Hilbert space H × L 2 as follows: given a couple of function φ = (φ 0 , φ 1 ), and for 0 r 1 < r 2 ∞, E( φ; r 1 , r 2 ) := We omit r 1 , r 2 in the case r 1 = 0 and r 2 = ∞: the energy is E( φ) := E( φ; 0, ∞), and H × L 2 = { φ | E( φ) < +∞}.

If ψ = (ψ(t), ∂ t ψ(t)) is a finite energy wave map, then at least formally its energy is preserved: for all t where defined, E( ψ(t)) = E( ψ(0)). (1.2) (WM) is energy critical in the following sense. Consider the scaling (for λ > 0)

Λ[λ] φ(t, r) := φ 0 t λ , r λ , 1 λ φ 1 t λ , r λ .
Then ψ is a wave map if and only if Λ[λ] ψ is a wave map, and the energy is scaling invariant: E( ψ) = E(Λ[λ] ψ). Notice that the H × L 2 norm is also scaling invariant. Recall that if φ ∈ H , then φ continuous and bounded, and has well defined limits at 0 and +∞, which cancel g: we denote them φ(0) and φ(∞). If φ is a wave map, these limits do not depend on time. This motivates the introduction of the set of points where g vanishes Let ∈ V and make the non degeneracy assumption that g ( ) = 0. Then one can see that φ -∈ H if and only if φ ∈ H and φ(0) = φ(∞) = (cf. Lemma A.2 in the appendix). H can then be seen as the tangent (Hilbert) space to H at point . We recall the local well-posedness result in the energy space, due to Shatah and Tahvildar-Zadeh.

V := { ∈ R | g( ) = 0}.
Theorem ( [START_REF] Shatah | On the Cauchy problem for equivariant wave maps[END_REF]). Let (ψ 0 , ψ 1 ) ∈ H × L 2 . Then there exists a unique wave map ψ = (ψ, ∂ t ψ) ∈ C (I, H × L 2 ) solution to (WM), defined on a maximal interval I =: (T -( ψ), T + ( ψ)), and which preserves the energy (1.2).

The wave map equation (WM) has been intensively studied as a model for geometric wave equations. It has been long understood that the geometry of the target M , i.e. the metric g, plays a crucial role in the long time behavior of wave maps. We recall the result by Struwe [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF]: a wave map that blows up in finite time must bubble up a harmonic map at blow up time. In particular, if M does not admit non constant harmonic maps, then any wave map is global in time.

Let us mention a few results on the general case (1.1), where the proofs are much more technical. The first breakthrough was concerned with global well posedness for small data in the critical Sobolev space Ḣd/2-1 × Ḣd/2 with target N = S d-1 , and due to Tao [START_REF] Tao | Global regularity of wave maps I. Small critical Sobolev norm in high dimension[END_REF][START_REF] Tao | Global regularity of wave maps II. Small critical Sobolev norm in high dimension[END_REF]; it was then generalized by Tataru [START_REF] Tataru | Rough solutions for the wave maps equation[END_REF] to general targets. The result by Struwe [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF] was extended to non symmetric data by Sterbenz and Tataru [START_REF] Sterbenz | Energy dispersed large data wave maps in 2 + 1 dimensions[END_REF][START_REF] Sterbenz | Regularity of wave-maps in dimension 2 + 1[END_REF] (see also the works by Tao [START_REF] Tao | Global regularity of wave maps III-VII[END_REF] and Krieger and Schlag [START_REF] Krieger | Full range of blow up exponents for the quintic wave equation in three dimensions[END_REF] when the target is the hyperbolic space). We refer to the review article [START_REF] Krieger | Global regularity and singularity development for wave maps[END_REF] and the reference therein for more details, and from now on we focus on the equivariant case (WM) only.

Actual examples of wave maps blowing up in finite time were constructed by Rodnianski and Sterbenz [START_REF] Rodnianski | On the formation of singularities in the critical O(3) σ-model[END_REF] and Raphaël and Rodnianski [START_REF] Raphaël | Stable blow up dynamics for the critical corotational Wave map and equivariant Yang-Mills problems[END_REF] (as a perturbation of the self similar regime), and by Krieger, Schlag and Tataru [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF] (with prescribed, polynomial blow up rate). On a different side, together with Kenig, Lawrie and Schlag [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF][START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF], we classified the asymptotic behavior of wave maps with energy less than 3 times the energy of a harmonic map, for large time or near blow up time.

Our goal in this paper is to obtain a similar classification for wave maps of arbitrarily large energy, that is to relax the bound on the energy. We provide a description of a wave map into decoupled profiles, a so called soliton resolution. It turns out that these profiles are harmonic maps and linear scattering terms. Recall that a harmonic map is a solution Q of finite energy of

∂ rr Q + 1 r ∂ r Q = f (Q) r 2 .
(Hence (Q, 0) is a finite energy stationary wave map). From [START_REF] Côte | Instability of nonconstant harmonic maps for the (1 + 2)-dimensional equivariant wave map system[END_REF], they are classified as follows: a non constant harmonic map is monotonic, satisfies one of the ODEs

r∂ r Q = g(Q) or r∂ r Q = -g(Q),
and joins two consecutive points of V , that is for some , m ∈ V , < m,

{Q(0), Q(∞)} = { , m} and V ∩ ( , m) = ∅. It has energy E(Q) := E(Q, 0) = 2(G(m) -G( )).
In particular if #V = 1, there exists no non constant harmonic map (if V is empty, there is no finite energy map).

On the other hand, given ∈ V , we define the linearized wave map flow around :

∂ tt φ -∂ rr φ - 1 r ∂ r φ + g ( ) 2 r 2 φ = 0. (LW )
Solutions to this linear wave equation preserve the following H ×L 2 related quantity φ(t) 2 H ×L 2 := |∂ t φ(t, r)| 2 + |∂ r φ(t, r)| 2 + g ( ) 2 φ(t, r) 2 r 2 rdr = φ(0) 2 H ×L 2 .

We now state the main result of this paper. For this, we make the following assumptions on the metric g:

(A1) G(x) → ±∞ as x → ±∞. (A2) V is discrete, (A3) For all ∈ V , g ( ) ∈ {-1, 1}.

Assumption (A1) prevents the formation of bubbles at infinity, and is a very natural assumption. (A2) is also a natural assumption of non degeneracy of g, which prevent a decomposition with harmonic maps of arbitrarily small energy. The physically relevant metrics g are (1) g(ρ) = sin(ρ) (wave maps to the sphere S 2 ), and

(2) g(ρ) = 1 -ρ 2 (radial 4D Yang-Mills equation).

Hence (A3) allows to handle wave maps to the sphere S 2 ; however, dealing with the radial 4D Yang-Mills equation requires to relax (A3) to (A3') For all ∈ V , g ( ) ∈ {-2, -1, 1, 2}.

It should be noted that most of the results in this article hold under (A3') instead of (A3). One could even consider the natural condition g (V ) ⊂ Z \ {0}, which makes the linearized problem (LW ) around to be of wave type (in dimension 2|g ( )| + 2). However large |g ( )| raise technical issues for the Cauchy problem as noted in [5, Theorem 2], which is restricted to the case (A3'); if these issues could be dealt with, most results here hold under this last condition.

Theorem 1.1. We make assumptions (A1)-(A2)-(A3'). Let ψ(t) be a finite energy wave map. Then there exist a sequence of times t n ↑ T + ( ψ), an integer J 0, J sequences of scales λ J,n

• • • λ 2,n λ 1,n and J harmonic maps Q 1 , . . . , Q J such that Q J (0) = ψ(0), Q j+1 (∞) = Q j (0) for j = 1, . . . , J -1,
and that the following holds.

(1) (Blow up case

) If T + ( ψ) = +∞, denote = ψ(∞). Then Q 1 (∞) = , λ 1,n t n and there exists a solution φ L (t) ∈ C (R, H × L 2 ) to the linear wave equation (LW ) such that ψ(t n ) = J j=1 (Q j (•/λ j,n ) -Q j (∞), 0) + ( , 0) + φ L (t n ) + b n . (1.3) (2) (Global case) If T + ( ψ) < +∞, denote = lim t↑T + ( ψ) ψ(t, T + ( ψ) -t) (it is well defined). Then J 1, λ 1,n T + ( ψ) -t n and there exists a function φ ∈ H × L 2 of finite energy such that Q 1 (∞) = φ(0) = and ψ(t n ) = J j=1 (Q j (•/λ j,n ) -Q j (∞), 0) + φ + b n . (1.4) In both cases, b n vanishes in the following sense b n,0 L ∞ + b n,1 L 2 → 0 as n → +∞,
and more precisely, for any sequence λ n > 0, and A > 0 b n,0 H(A/λn r Aλn) → 0.

If we furthermore assume (A3), then (in both cases) b n H×L 2 → 0 as n → +∞.

Remark 1.2. Assume (A3). In the global case, E( ψ)

= J j=1 E(Q j ) + φ L 2 H ×L 2 ,
and in the blow up case, E( ψ) = J j=1 E(Q j ) + E( φ). This gives a bound on J. (when assuming solely (A3'), only an inequality holds, but still provides a bound on J).

Also, if T + ( ψ) = +∞, then J #(V ∩ [ψ(0), ψ(∞)]) -1.
This last number can be made arbitrarily large when V is infinite (as for g = sin).

Remark 1.3. Many possibilities are left open regarding the behavior of the λ j,n : for example, in the global case, one could have λ J,n → 0 (infinite time blow-up), or λ J,n → +∞ (infinite time flattening). Although no such solutions were constructed for (WM), let us refer to [START_REF] Donninger | Nonscattering solutions and blowup at infinity for the critical wave equation[END_REF] in the context of the semilinear wave equation. The construction of solutions with multiple profiles (J 2) having various asymptotic behavior is an important open problem.

The question whether the decomposition holds for all times and not merely for a sequence is open. However there are some cases where it can be proved.

Corollary 1.4. Assume (A1)-(A2)-(A3).

We use the notations of Theorem 1.1: we dispose of a sequence of harmonic maps

Q 1 , . . . , Q J . Assume that 2|G( ) -G(ψ(0))| = J j=1 E(Q j ), or, equivalently, that the sequence = Q 1 (∞), Q 1 (0), Q 2 (0), . . . , Q J (0) = ψ(0) is monotonic.
Then the decomposition holds for all times, that is, there exists J functions λ J (t)

• • • λ 1 (t), such that (1) If T + ( ψ) = +∞, then λ 1 (t) t and
ψ(t) = J j=1 (Q j (•/λ j (t)) -Q j (∞), 0) + ( , 0) + φ L (t) + b(t), (1.5) where b(t) → 0 in H × L 2 as t → ∞. (2) If T + ( ψ) < +∞, then λ 1 (t)
T + ( ψ) -t and

ψ(t) = J j=1 (Q j (•/λ j (t)) -Q j (∞), 0) + φ + b(t), (1.6) where b(t) → 0 in H × L 2 as t → T + ( ψ).
Remark 1.5. Notice the condition of Corollary 1.4 always holds if J = 0 or J = 1. Also, the decomposition also holds for all times when the excess energy of ψ with respect to the energy to connect ψ(0) to ψ(∞) is not enough to bubble more harmonic maps: that is

E( ψ) < 2|G(ψ(∞)) -G(ψ(0))| + 4δ, where δ = min ∈V \{ψ(0),ψ(∞)} {|G( ) -G(ψ(0))|, |G( ) -G(ψ(∞))|}.
Theorem 1.1 is an extension of [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF][START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF] where only one profile was allowed (i.e J = 1) through the bound on the energy. It is in the spirit of the seminal papers by Duyckaerts, Kenig and Merle [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF][START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF][START_REF] Duyckaerts | Profiles of bounded radial solutions of the focusing, energy-critical wave equation[END_REF][START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF] where large solutions of the radial energy critical (focusing) wave equation in 3D were described. It is actually very similar in nature to [START_REF] Duyckaerts | Profiles of bounded radial solutions of the focusing, energy-critical wave equation[END_REF]. In [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF], the authors show the strong soliton resolution; i.e they give a decomposition for all times and not only for a well chosen sequence of times. The improvement from [START_REF] Duyckaerts | Profiles of bounded radial solutions of the focusing, energy-critical wave equation[END_REF] is involved and relies on refined version of the "channels of energy" methods, which currently are known for the 3D linear wave equation only. For wave maps, the relevant linearized operator is of wave type in even dimensions, where only weak versions of the channels of energy are available (see [START_REF] Côte | Energy partition for the linear radial wave equation[END_REF]). Proving the analog of [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF] for equivariant wave maps is hence a challenging open question. Let us however observe that their analysis, even in [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF], can not provide a description of type I blow up solutions (i.e when lim sup t↑T + ( u) u(t) Ḣ1 ×L 2 = +∞): it is clear that a decomposition into a finite sum of rescaled profiles and a scattering term cannot hold in that case. This phenomenon does not occur in the wave map case (mainly because the energy is coercive, even if it doesn't bound H × L 2 ), and we give a description of any wave map, without any further assumption. 1.2. Outline of the proof. Let us mention two delicate issues. First, geometry has to be taken into account: the harmonic maps do never belong to H × L 2 . This means we must derive a procedure to extract them without relying on a linear profile decomposition, as for the wave equation.

Second, the linearized operator of the wave map flow at spatial infinity is of wave type in even dimension: most of the delicate linear estimates available in the radial 3D case break down for wave maps, in particular the so-called "energy channels". When g ( ) is odd, the relevant linearized equation (LW ) corresponds to a wave equation in R d with d ≡ 0 mod 4, and the linear estimate obtained in [START_REF] Côte | Energy partition for the linear radial wave equation[END_REF] in this case suffices to conclude. When g ( ) is even, then d ≡ 2 mod 4 and the desired linear estimate fails. This is the reason why we must restrict ourselves to (A3) instead of (A3').

The first step in the proof is to choose a sequence of times t n → T + ( ψ) on which the space-time kinetic energy inside the light cone vanishes. This is a reformulation that the averaged kinetic energy inside the light cone vanishes, which is a well known result, and is the content of Section 2.1. Section 2.2 focuses on various aspects of the profile decomposition in H × L 2 to be used later in the paper.

The second step, Section 3, is concerned with sequence of wave maps whose spacetime kinetic energy vanishes, and shows, in Theorem 3.5, that up to a subsequence, one can construct a bubble decomposition i.e extract the harmonic maps. This decomposition holds up to an error which tends to 0 in L ∞ . Notice that this result does not make use of assumption (A3) or (A3'), but only (A1) and (A2).

The bound on the error is insufficient to capture the linear scattering term for example, but it is enough to derive a sharp scattering theorem below the threshold in L ∞ . As linear scattering is involved, we do need assumption (A3') here. This result has its own interest: let us state it here, and postpone the proof to Section 4.

For ∈ V , define d as the distance of to the closest (distinct) element in V :

d = inf{| -k| | k ∈ V \ { }}. d > 0 due to assumption (A2).
Theorem 1.6. Let ∈ V and assume (A1)-(A2), and

g ( ) ∈ {-2, -1, 1, 2}.
Let ψ be a wave map such that ψ(∞) = , and that for some c < d ,

(1.7) ∀t ∈ [0, T + ( ψ)), ψ(t) -L ∞ c < d .
Then T + ( ψ) = +∞ and ψ scatters at +∞, in the sense

ψ -S ([0,+∞)) < +∞.
(S is an adequate Strichartz space, defined below on (2.4)). It follows that there exists a (unique) solution φ L to (LW ) such that

ψ(t) -( , 0) -φ L (t) H×L 2 → 0 as t → +∞.
Remark 1.7. Observe that if g vanishes in at most one point , then d = +∞, and so Theorem 1.6 proves that all wave maps are global and scatter in this case. This strengthens the global well posedness result by Struwe [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF] mentioned above.

As a consequence of Theorem 1.6, we can extract the scattering term (for all times, not merely a sequence) in the global case. In an analogous way, we can define the regular part φ in the blow up case. This is the content of Propositions 5.1 and 5.2 of Section 5. Let us emphasize that this step only requires (A3').

In Section 6, we revisit Theorem 3.5. Under the additional assumption (A3) -crucial but used only on this step, we show that the error term tends to 0 in H × L 2 . This section is independent of Sections 4 and 5.

Finally, we gather all the previous results together in Section 7 and prove Theorem 1.1 and Corollary 1.4.

1.3. Acknowlegdment. The author would like to thank the anonymous referee for very helpful comments and remarks.

Preliminaries

The purpose of this section is to recall or adapt a few important earlier results, and derive some consequences.

2.1. The self-similar region. Proof. The argument in [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF] is done for g = sin, after ideas of [START_REF] Christodoulou | On the asymptotic behavior of spherically symmetric wave maps[END_REF], and extends seamlessly for any non linearity g.

We derive a few consequences from this. One fundamental outcome of Proposition 2.1 is that the kinetic part of the energy vanishes in an averaging sense. More precisely, we have:

Corollary 2.2 ([7, Corollary 2.2]). lim sup T →+∞ 1 T T A t-A 0 |∂ t ψ(t, r)| 2 rdrdt → 0 as A → +∞.
From there, we find a sequence of times for which the condition (3) in Theorem 3.5 holds.

Corollary

T →+∞ 1 T T A f (t)dt → 0 as A → +∞.
We now argue by contradiction. Assume that the conclusion is not correct, then this means that for some δ > 0, lim inf

T →+∞ sup s,0 s T /2 1 s T +s T -s f (t)dt 41δ.
Fix A and T 0 2A be large enough such that for all T T 0 ,

• 1 T T A f (t)dt δ • there exists s(T ) ∈ [0, T /2] such that 1 s(T ) T +s(T ) T -s(T ) f (t)dt 40δ.
Consider the sets (T -s(T ), T + s(T )) for T ∈ [T 0 , 2T 0 ]. Their diameter is bounded by 2T 0 , hence Vitali covering lemma applies: there exist a sequence (T n ) n such that the intervals (T n -s(T n ), T n + s(T n )) are disjoints and

[T 0 , 2T 0 ] ⊂ T ∈[T0,2T0] (T -s(T ), T + s(T )) ⊂ n (T n -5s(T n ), T n + 5s(T n )).
From this last condition, it follows that

T 0 10 n s(T n ).
On the other hand, by definition of the s(T n ), we get that

T n +s(T n ) T n -s(T n ) f (t)dt 40δs(T n ),
and as the intervals under consideration are disjoint, we infer

3T0 T0/2 f (t)dt n T n +s(T n ) T n -s(T n ) f (t)dt 40δ n s(T n ) 4δT 0 .
But as T 0 /2 A, we also have

1 3T 0 3T0 T0/2 f (t)dt δ,
and we reached a contradiction.

Finally we recall that the L ∞ norm outside vanishes in the self similar region and outside the light cone.

ψ(t) -ψ(∞) L ∞ (r λt) → 0 as t → ∞.
Proof. The argument in [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF] is done for g = sin and can be extended seamlessly for any non linearity g.

2.1.2.

Blow up wave maps. Throughout this Subsection, ψ is a finite energy wave map such that T + ( ψ) < +∞. The results here very similar to those in the global case, and in fact simpler (integration can be done up to the light cone). Proposition 2.5 ([27, Lemma 2.2]). For all λ ∈ (0, 1),

E( ψ(t); λ(T + ( ψ) -t), T + ( ψ) -t) → 0 as t ↑ 1. Corollary 2.6 ([27, Corollary 2.2]). 1 T + ( ψ) -T T + ( ψ) T T + ( ψ)-t 0 |∂ t ψ(t, r)| 2 rdrdt → 0 as T → T + ( ψ).
Corollary 2.7. There exists a sequence t n ↑ T + ( ψ) such that sup s,0<s T + ( ψ)-tn

1 s tn+s tn-s T + ( ψ)-t 0 |∂ t ψ(t, r)| 2 rdrdt → 0 as n → +∞.
Proof. It is very similar to the proof of Corollary 2.3. Let

f (t) = T + ( ψ)-t 0 |∂ t ψ(t, r)| 2 rdr → 0.
f (t) 0 and we know that

1 T + ( ψ) -T T + ( ψ) T f (t)dt → 0 as T → T + ( ψ).
Assume that the conclusion fails, for the sake of contradiction: then there exists δ > 0 and a function s defined on [T + ( ψ) -δ, T + ( ψ)) such that 0 s(T ) T + ( ψ) -T and

1 s(T ) T +s(T ) T -s(T ) f (t)dt 40δ. (2.1) Corollary 2.6 yields T 0 < T + ( ψ) such that for all T ∈ [2T 0 -T + ( ψ), T + ( ψ)), 1 T + ( ψ) -T T + ( ψ) T f (t)dt δ. (2.2)
We apply Vitali covering lemma to the intervals (T -s(T ), T + s(T )) where T ∈ [T 0 , T + ( ψ)) to find a sequence T n such that (T n -s(T n ), T n + s(T n )) are disjoints and

[T 0 , T + ( ψ)) ⊂ n (T n -5s(T n ), T n + 5s(T n )).
Hence taking the length:

T + ( ψ) -T 0 10 n s(T n ).
Therefore, as the intervals are disjoint, and using (2.1),

T + ( ψ) 2T0-T + ( ψ) f (t)dt n T n +s(T n ) T n -s(T n ) f (t)dt 4δ(T + ( ψ) -T 0 ). Now T + ( ψ) -(2T 0 -1) = 2(T + ( ψ) -T 0 )
, and we reached a contradiction with (2.2).

2.2.

Energy concentration on the light cone for (LW ). For the rest of this Subsection, we fix ∈ V and focus on the linear equation (LW ).

We define the transformation T by

(T φ)(r) = φ(r)/r |g ( )| .
Then φ is a solution of (LW ) if and only if ϕ = (T φ, T ∂ t φ) solves the radial wave equation in 2 + 2|g ( )| dimensions:

(2.3) ∂ tt ϕ -∂ rr ϕ - 1 + 2|g ( )| r ∂ r ϕ = 0.
Observe that

φ H = T φ Ḣ1 (r 1+2|g ( )| dr) .
The norms H and H are equivalent. It follows that T is a bicontinuous bijective linear map L 2 (rdr) → L 2 (r 1+2|g ( )| dr) and H → Ḣ1 (r 1+2|g ( )| dr).

We start by recalling a result regarding equipartition of energy and concentration of energy on the light cone for linear solutions.

Proposition 2.8. Let φ be a solution to (LW ). Then Also,

∂ t φ(t) 2 L 2 → 1 2 φ(0) 2 H ×L 2 and φ(t) 2 H → 1 2 φ(0) 2 H ×L 2 as t → +∞.
Proof. The first statement is the content of [START_REF] Côte | Energy partition for the linear radial wave equation[END_REF]Theorem 4]. The second is equipartition of the energy, and is classical for the linear wave equation.

We will sometimes use, in the context of a profile decomposition, the following weaker form, namely all the energy concentrates on one scale.

Corollary 2.9. Let φ be a solution to (LW ), and (t n , λ n ) be two sequences with λ n > 0, and such that

|t n | λ n → +∞.
Then for any c > 1, as n → +∞,

Λ[λ n ] φ (-t n , r) H ×L 2 ( 1 c tn r ctn) → φ(0) H ×L 2 .
Now we recall a result giving some condition so that some energy of a linear solution remains outside of the light cone. It was already crucial in [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF][START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF]. Here is the only place in the argument where we need to restrict to odd g ( ) (and hence (A3) to have the nonlinear argument run).

Proposition 2.10 ([8, Theorem 1]). Assume g ( ) is an odd integer. There exists β( ) > 0 such that the following holds. Let φ be a solution to (LW ), such that

∂ t φ(0) = 0.
Then for all t ∈ R,

φ(t) 2 H×L 2 (r |t|) β( ) φ(0) 2 H×L 2 .
Proof. As mentioned, this is an easy consequence of [8, Theorem 1] and the remark that follows. We refer to [6, Corollary 2.3] for the complete argument to pass from the linear wave equation to (LW ).

2.3.

Profile decomposition for (LW ) in H × L 2 . Again, we fix ∈ V for the rest of this subsection.

Our goal is to derive a suitable notion of linear profile decomposition, adapted to our setting, in particular L ∞ bounds. The notion of profile decomposition was first introduced by Bahouri and Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] in the context of the energy critical wave equation and Merle and Vega [START_REF] Merle | Compactness at Blow-up time for L 2 Solutions of the Critical Nonlinear Schrödinger Equation in 2D[END_REF] in the context of the mass critical Schrödinger equation.

Using transformation T , a notion of profile decomposition for the wave equation will immediately translate to a similar decomposition for (LW ), but we will in fact improve it. We elaborate on this in what follows.

For I an interval of R, we define

φ S (I) := t∈I ∞ r=0 |φ(t, r)| 2+3/g ( ) drdt r 2 1 2+3/g ( ) . (2.4) It is simply the norm of T φ in the Strichartz space L 2+3/g ( ) t,r
(r 1+2g ( ) drdt), adapted to the Ḣ1 critical wave equation in dimension 2 + 2g ( ) (we refer to [5, Section 3] for further details).

Lemma 2.11. Let θ( ) = 3 5 + 3/g ( ) ∈ (0, 1). There exist C > 0 such that for any finite energy solution γ to the linear wave equation (LW ).

γ L ∞ t (R,L ∞ r ) C γ(0) θ( ) H×L 2 γ 1-θ( ) S (R) .
Proof. As γ is a finite energy solution to the linear equation (LW ), all terms in the desired estimate are finite.

Denote M = γ L ∞ t (R,L ∞ r ) and A = γ(0) H×L 2 .
The norm γ(t) H×L 2 is bounded below and above by γ(0) H×L 2 (indeed, it is equivalent to the conserved quantity γ(t) H ×L 2 ). Due to Lemma A.1, it follows that M KA for some K only depending on . As all the functional spaces under consideration are invariant under scaling and time translation, we can assume that |γ(0, 1)| 2M/3. Let s, t ∈ R, and r q 0. Then for any ρ ∈ [q, r] we have

|γ(t, r) -γ(s, q)| 2 ρ q |∂ r γ(s, r )|dr + [s,t] |∂ t γ(t , ρ)|dt + r ρ |∂ r γ(t, r )|dr 2 3 ln ρ q ρ q |∂ r γ(s, r )| 2 r dr + 3 |t -s| ρ [s,t] |∂ t γ(t , ρ)| 2 ρdt + 3 ln r ρ r ρ |∂ r γ(t, r )| 2 r dr 3 ln r q γ(s) 2 H + 3 |t -s| q [s,t] |∂ t γ(t , ρ)| 2 ρdt
After averaging in ρ ∈ [q, r], we obtain

|γ(t, r) -γ(s, q)| 2 C ln r q γ(0) 2 H×L 2 + C |t -s| q|r -q| r q [s,t] |∂ t γ(t , ρ)| 2 ρdt dρ C 1 ln r q + |t -s| 2 q|r -q| γ(0) 2 H×L 2 .
Pick q = 1, s = 0. Observe that for all y 0, 0 ln(1 + y) + y 2y. Then fix B = M 2 /(18C 1 A 2 ). We deduce, for all r ∈ [1, 1 + B] and |t| B(r -1),

|γ(t, r)| 2M/3 -A C 1 (ln(1 + B) + B) 2M/3 -A 2C 1 B M/3. Hence (notice that 0 B K 2 /(18C 1 ) is uniformly bounded in M , A), γ 2+3/g ( ) S(R) 1+B 1 |t| √ B(r-1) (M/3) 2+3/g ( ) drdt r 2 (M/3) 2+3/g ( ) 1+B 1 B(r -1) r 2 dr C B 3/2 (1 + B) 2 (M/3) 2+3/g ( ) C M 5+3/g ( ) A 3 .
We can conclude, with θ = 3 5 + 3/g ( )

, M CA θ γ 1-θ S(R) .
From this, the profile decomposition for (LW ) takes the following form.

Theorem 2.12 (Profile decomposition). Let ψ n a bounded sequence of H × L 2 . Then there exists a sequence of scales (t j,n , λ j,n ) n and linear profiles V j,L solutions to (LW ), such that, up to a subsequence that we still denote ψ n , we have for all J 1

ψ n (r) = J j=1 V j,L - t j,n λ j,n , r λ j,n , 1 λ j,n ∂ t V j,L - t j,n λ j,n , r λ j,n + γ J,n (0),
where γ J,n is a solution to (LW ) which satisfies

lim sup n γ J,n S (R) + γ J,n L ∞ t,r → 0 as J → +∞,
and for all j, ∀n, t j,n = 0 or t j,n λ j,n has a limit which is + ∞ or -∞ ,

and if j = k, λ j,n λ k,n + λ k,n λ j,n → +∞, or ∀n, λ j,n = λ k,n and |t j,n -t k,n | λ j,n → +∞ .
Furthermore, there hold (1) Pythagorean expansion of the H × L 2 norm: for all fixed J, we have as

n → +∞ ψ n 2 H ×L 2 = J j=1 Λ[λ j,n ] V j,L (-t j,n ) 2 H ×L 2 + γ J,n (0, r) 2 H ×L 2 + o n (1).
(2) Pythagorean expansion of the energy: for all J fixed, we have as n → +∞

E( ψ n ) = J j=1 E Λ[λ j,n ] V j,L (-t j,n ) + E( γ J,n (0)) + o(1).
(3) L ∞ profile selection: ψ n L ∞ has a limit as n → +∞ and exists

(2.5) lim n ψ n L ∞ = sup j∈J0 V j (0) L ∞ where J 0 := {j 1 | ∀n, t j,n = 0}.
(with the convention that the sup is 0 if J 0 is empty).

Proof. This is essentially contained in [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF]Main Theorem]. We refer to [6, Corollary 2.15] for the profile decomposition in the wave map context and the Pythagorean expansion of the H × L 2 norm, and to [6, Lemma 2.16] for the Pythagorean expansion of the energy.

The only extra points with respect to the usual profile decomposition are γ J,n L ∞ t,r → 0 and the L ∞ profile selection. For the former, the Pythagorean expansion ensures that γ J,n (0) H×L 2 is a bounded sequence, hence this follows from the previous Lemma and lim sup

n γ J,n S (R) → 0.
For the latter, let r n such that

|ψ n (r n ) -ψ n L ∞ | 1 n . First assume lim inf n ψ n L ∞ > 0, and let ε > 0 such that 2ε < lim inf n ψ n L ∞ . Choose J so large that γ J,n L ∞ t,r ε for n large enough. Then consider j J. If j / ∈ J 0 , then |t j,n |/λ j,n → +∞ so that V j - tj,n λj,n L ∞ → 0. Hence |ψ n (r n )| j J,j∈J0 V j 0, r n λ j,n + o n (1) + ε.
In particular J 0 = ∅. As ln λj,n λ k,n → +∞, we see that

lim n j J,j∈J0 V j 0, r n λ j,n L ∞ = max{ V j (0) L ∞ | j J, j ∈ J 0 }.
This shows that

lim sup n ψ n L ∞ sup { V j (0) L ∞ | j ∈ J 0 } .
For the reverse inequality, first notice that as j → +∞, V j (0) H → 0 (due to the Pythagorean expansion of the energy), and so V j (0) L ∞ → 0. Hence there exists

j 0 such that sup { V j (0) L ∞ | j J, j ∈ J 0 } = V j0 L ∞ .
Also, as V j0 (0) is continuous and tend to 0 at 0 and +∞, there exists

r 0 > 0 such that |V j0 (0, r 0 )| = sup { V j (0) L ∞ | j ∈ J 0 } .
Consider the sequence ψ n (λ j0,n r 0 ). Then we have the expansion

ψ n (λ j0,n r 0 ) = V j0 (0, r 0 ) + j J,j∈J0\{j0} V j 0, r n λ j,n + γ J,n (0, λ j0,n r 0 ).
Again due to orthogonality of the profiles, we see that

lim inf n ψ n L ∞ lim inf n |ψ n (λ j0,n r 0 )| → |V j0 (0, r 0 )|. If lim inf n ψ n L ∞ = 0, then choosing a subsequence such that ψ σ(n) → 0 in L ∞ ,
and arguing as previously, we see that J 0 = ∅. Therefore ψ n -γ J,n L ∞ → 0 for all J, and so ψ n L ∞ → 0. The desired equality also holds in this case.

Proposition 2.13 (Pythagorean expansion with cut-off, [START_REF] Côte | Energy partition for the linear radial wave equation[END_REF]Corollary 8]). We use the notation of the previous Proposition. Fix J 1 and let 0 r n s n +∞ be two sequences. Then we have the expansion:

ψ n 2 H ×L 2 (rn r sn) = J j=1 Λ[λ j,n ] V j,L (-t j,n , r) 2 H ×L 2 (rn r sn) + γ J,n (0, r) 2 H ×L 2 (rn r sn) + o n (1).
Proof. This is the content of [8, Corollary 8]. The proof there is done for one sequence i.e. r n = 0. To derive the above expansion, it suffices to do the difference between the expansions with cut-off r s n and r r n .

We will use several times the following simple remark.

Corollary 2.14. Let ψ n be a bounded sequence in H × L 2 with no energy at all scales, that is, for any sequence λ n > 0,

∀A > 1, ψ n H×L 2 (λn/A r Aλn) → 0 as n → +∞.
Denote ψ n,L the linear evolution (to (LW )) with data ψ n at time 0, then

ψ n,L S(R) → 0.
Proof. Let ( V j,L ; (t j,n , λ j,n ) n ) j be a profile decomposition of ( ψ n ) n in the sense of Theorem 2.12. Let us show that all the profiles V j,L are trivial. Indeed, first consider the case of a profile V k,L where t k,n /λ k,n → +∞ or -∞. Then by Proposition 2.13 and Corollary 2.9 (with c = 2), we have for all c 1

o n (1) = ψ n 2 H×L 2 (t k,n /2 r 2t k,n ) = k j=1 Λ[λ j,n ] V j,L (-t j,n , r) 2 H ×L 2 (t k,n /2 r 2t k,n ) + γ J,n (0, r) 2 H ×L 2 (t k,n /2 r 2t k,n ) + o n (1) Λ[λ k,n ] V k,L (-t k,n , r) 2 H ×L 2 (t k,n /2 r 2t k,n ) + o n (1) V k,L (0) 2 H ×L 2 + o n (1). Therefore, V k,L (0) H ×L 2 = 0 and V k,L = 0.
In the other case where t k,n = 0 for all n, there holds similarly for all A 1

o n (1) = ψ n 2 H×L 2 (λ k,n /A r Aλ k,n ) = k j=1 Λ[λ j,n ] V j,L (-t j,n , r) 2 H ×L 2 (λ k,n /A r Aλ k,n ) + γ J,n (0, r) 2 H ×L 2 (λ k,n /A r Aλ k,n ) + o n (1) Λ[λ k,n ] V k,L (0, r) 2 H ×L 2 (λ k,n /A r Aλ k,n ) + o n (1) V k,L (0) 2 H ×L 2 (1/A,A) + o n (1).
This means that V k,L (0) H ×L 2 (1/A,A) = 0, and as this is true for all A 1, it implies that V k,L (0) = 0. As claimed, we proved that all profiles V j,L are trivial. As a consequence, ψ n is the remainder term γ J,n which (does not depend on J and) satisfies the desired dispersion property.

We recall the notion of nonlinear profile. If V is a solution of the linear equation (LW ) and T ∈ R, there exists a unique wave map U solution to (WM), defined on a neighborhood of T and satisfying U (0) = U (∞) = and

U (t) -V (t) -( , 0) H×L 2 → 0 as t → T.
Notice that if T = +∞, then U is defined on some interval [T 0 , +∞) and scatters at +∞, i.e U -S([T0,+∞)) < +∞.

This notion follows from local well posedness [5, Theorem 2]; we refer to [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF][START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation[END_REF] for further details.

Proposition 2.15 (Evolution of the decomposition). Let ψ n be a sequence of wave maps such that

( ψ n (0) -( , 0
)) n admits a profile decomposition in the sense of Theorem 2.12, from which we use the notations. Denote U j nonlinear profiles associated to V j , -lim n tj,n λj,n . Let t n be such that

∀J 1, t n -t j,n λ j,n < T + (U j ), and sup n U j -S([-t j,n λ j,n , tn -t j,n λ j,n ]) < +∞.
Then for n large enough, T + ( ψ n ) > t n , and for all t ∈ [0, t n ],

ψ n (t) -( , 0) = J j=1 U j t -t j,n λ j,n , r λ j,n , 1 λ j,n ∂ t U j t -t j,n λ j,n , r λ j,n -( , 0) + γ J,n (t, r) + r n,J (t, r),
where lim sup n→+∞ r J,n S ([0,tn]) + r J,n L ∞ ([0,tn],H×L 2 ) → 0 as J → +∞.

Proof. It is the translation of [10, Proposition 2.8] via the transformation T .

Notice that

ψ n (0) -( , 0)) n is a bounded sequence in H × L 2 , in particular ψ n (0) = ψ n (∞) = .
Let us emphasize that we can not evolve a decomposition where harmonic maps appear: this is a fundamental difference with the semi linear wave equation, where such stationary profile were allowed (they belong to Ḣ1 × L 2 ). We will often use this result in the following particular case.

Corollary 2.16. Let ψ n be a sequence of wave maps such that ψ n (∞) = ψ n (0) = , and (1)

ψ n (0) -( , 0) is a bounded sequence in H × L 2 .
(2) If ψ n,L denotes the linear solution to (LW ) with initial data ψ n (0) -( , 0), then ψ n,L S(R) → 0.

Then for n large enough ψ n is defined globally on R and

sup t∈R ψ n (t) -( , 0) -ψ n,L (t) H×L 2 → 0 as n → +∞.
Remark 2.17. Notice we can combine Corollary 2.14 with this last result, as the hypothesis of the latter are the conclusions given by the former.

Proof. The second condition means that any profile decomposition is trivial (i.e) does not contain any non trivial profile. Hence with the notation of the previous Proposition, ψ n (0) -( , 0) = γ J,n (0) =: ψ n,L (0) (does not depend on J) and

ψ n (t) + ( , 0) = ψ n,L (t) + r n (t).
To conclude, we argue by contradiction:

(1) if ψ n blow-up in finite time, consider

t n = T + ( ψ n ) + 1 (and similarly for T -( ψ n )).
(2) if the convergence does not hold, there exists η > 0 and t n be such that r n (t n ) H×L 2 η. Each hypothesis contradicts Proposition 2.15.

Bubble decomposition for a sequence of wave maps

Our goal here is to study a sequence of wave maps with vanishing (space-time) kinetic energy. First we prove that at any scale, there is local (strong) limit which is a harmonic map (possibly constant). Proposition 3.1 (Profile at any scale). We assume (A1)-(A2). Let A > 0 and ψ n be wave maps be defined on the time interval [-A, A], such that (1) ψ n have uniformly bounded energy E( ψ n ) E.

(2) ψ n (0) is bounded.

(3) For some sequence r n → +∞,

∂ t ψ n L 2 ((-A,A),L 2 (r rn)) → 0 as n → +∞.
Then ψ σ(n) , up to a subsequence σ(n), converges to some harmonic map (Q, 0) locally strongly in the following sense: for any R > 0,

(3.1) sup t∈[-A,A] ψ σ(n) (t) -(Q, 0) H×L 2 ([1/R,R]) → 0 as n → +∞.
Remark 3.2. Assume furthermore that for some sequences

t n ⊂ [-A, A] and r n → r ∞ ∈ (0, +∞), ψ σ(n) (t n , r n ) → l. It follows from the convergence that Q(r ∞ ) = l, hence from classification of har- monic maps, Q is non constant if and only if g(l) = 0.
Remark 3.3. A result of this type was already obtained in [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF], but only for the first scale (and in the L 2 loc (H × L 2 ([0, +∞) loc ) topology), i.e. under the additional assumption that sup

t∈[-A,A] E( ψ n ; 0, 1) δ 0 .
for some small δ 0 > 0.

Proof. Up to rescaling, we can assume that A = 1. As E( ψ n ) and ψ n (0) are bounded,

ψ n is bounded in C ([-1, 1], ( Ḣ1 ∩ L ∞ ) × L 2
). Therefore, up to a subsequence, there exists ψ ∞ such that ψ n → ψ ∞ a.e. (-1, 1) × (0, +∞), and

ψ n → ψ ∞ in C loc ([-1, 1] × [0, +∞)) and ψ n * ψ ∞ L ∞ ((-1, 1), Ḣ1 × L 2 ) star-weakly.
From Fatou Lemma, we deduce that

∀t ∈ [-1, 1] -a.e., g 2 (ψ ∞ (t, r)) r 2 rdr lim inf n g 2 (ψ n (t, r)) r 2 rdr.
It follows that ψ ∞ satisfies the wave map equation (WM) in a weak sense, and has finite energy

E( ψ ∞ ) lim inf n E( ψ n ). Now, as ∂ t ψ n L 2 ((-1,1),L 2 (r rn)) → 0, then for all R 0, ∂ t ψ ∞ L 2 ((-1,1),L 2 (r R)) = 0,
and hence ∂ t ψ ∞ = 0, that is ψ ∞ is a weak harmonic map: r∂ r (r∂ r ψ) = f (ψ). It follows by elliptic regularity that ψ ∞ = (Q, 0) a smooth harmonic map.
Let us prove as a first step that there holds strong local convergence in space-time

L 2 loc ((-1, 1), H × L 2 ) loc ). Let K be a compact of (-1, 1) × (0, +∞) (in space time). Then (3.2) ψ n (t, r) -ψ ∞ (t, r) L 2 (K,rdrdt) + ∂ t ψ n (t, r) L 2 (K,rdrdt) → 0 due the compact embedding Ḣ1 ∩L ∞ (rdrdt) → L 2 (K, drdt/r
) and to the vanishing of the kinetic energy. Fix now ϕ ∈ D((-1, 1) × (0, +∞)). We compute

(∂ r ψ n -∂ r ψ ∞ ) 2 (t, r)ϕ(t, r)rdrdt = -∆(ψ n -ψ ∞ )(t, r)(ψ n -ψ ∞ )(t, r)ϕ(t, r)rdrdt -∂ r (ψ n -ψ ∞ )(t, r)(ψ n -ψ ∞ )(t, r)∂ r ϕ(t, r)rdrdt = -∂ tt ψ n (t, r)(ψ n -ψ ∞ )(t, r)ϕ(t, r)rdrdt - f (ψ n (t, r)) -f (ψ ∞ (t, r)) 2r (ψ n -ψ ∞ )(t, r)ϕ(t, r)drdt -∂ r (ψ n -ψ ∞ )(t, r)(ψ n -ψ ∞ )(t, r)∂ r ϕ(t, r)rdrdt = |∂ t ψ n (t, r)| 2 ϕ(t, r)rdrdt + ∂ t ψ n (t, r)(ψ n -ψ ∞ )(t, r)∂ t ϕ(t, r)rdrdt - f (ψ n (t, r)) -f (ψ ∞ (t, r)) 2r (ψ n -ψ ∞ )(t, r)ϕ(t, r)drdt -∂ r (ψ n -ψ ∞ )(t, r)(ψ n -ψ ∞ )(t, r)∂ r ϕ(t, r)rdrdt
Now, using again the compact embedding H loc (rdrdt) → L 2 loc (drdt/r), and the Cauchy-Schwarz inequality, we have

∂ t ψ n (ψ n -ψ ∞ )(t, r)∂ t ϕ(t, r)rdrdt ∂ t ψ n |∂ t ϕ| L 2 (ψ n -ψ ∞ ) |∂ t ϕ| L 2 → 0,
and similarly

f (ψ n (t, r)) -f (ψ ∞ (t, r)) 2r (ψ n (t, r) -ψ ∞ (t, r))ϕ(t, r)drdt (f (ψ n ) -f (ψ ∞ ))ϕ/r L 2 (ψ n -ψ ∞ ) ϕ/r L 2 ω f (sup n ψ n L ∞ ) (ψ n -ψ ∞ ) ϕ/r 2 L 2 → 0 (We recall that f is C 1 , hence we can define ω f (A) := sup x,y∈[-A,A] |f (x)-f (y)| |x-y|
; we already noticed earlier that ψ n was a bounded sequence of L ∞ t,x ). Finally, as

∂ r (ψ n -ψ ∞ ) L 2 (rdrdt) is bounded, ∂ r (ψ n -ψ ∞ )(t, r)(ψ n -ψ ∞ )(t, r)∂ r ϕ(t, r)rdrdt ∂ r (ψ n -ψ ∞ )(t, r) L 2 (ψ n -ψ ∞ )(t, r)|∂ r ϕ(t, r)| L 2 → 0.
This proves that

(∂ r ψ n -∂ r ψ ∞ ) 2 (t, r) + (ψ n -ψ ∞ ) 2 (t, r) r 2 ϕ(t, r)rdrdt → 0.
Let us now prove (3.1). Due to the previous convergence, the set

P := {t ∈ [-1, 1] | ∀R > 0, ψ n (t) -(Q, 0) H×L 2 ([1/(2R),R+1]) → 0} is dense in [-1, 1].
We now need the following version of uniform continuity of the flow around the harmonic map (Q, 0). Lemma 3.4. Let Q be a harmonic map, T 0 and ε > 0. There exist δ > 0 such that for all 0 r 1 2r 1 < r 2 +∞ and wave map ψ such that

ψ(0) -(Q, 0) H×L 2 ([r1,r2]) δ,
Then for all time t ∈ (T -( ψ), T + ( ψ)) such that |t| min{T, (r 2 -r 1 )/2},

ψ(t) -(Q, 0) H×L 2 ([r1+|t|,r2-|t|]) ε.
Proof. We postpone it to the Appendix.

Fix R > √ 2 (so that 2/R < R) and let ε > 0. Define the integer K to be the integer part of 2R + 1 so that K 2R. Then for all k ∈ -K, K -1 , there exist t k ∈ [k/K, (k + 1)/K] such that t k ∈ P. Let δ > 0 be provided by the previous Lemma 3.4 with T = 1 (and our previously fixed ε > 0).

Let N be such that

∀k ∈ -K, K -1 , ∀n N, ψ n (t k ) -(Q, 0) H×L 2 ([1/(2R),R+1]) δ. Let t ∈ [-1, 1] and n N . There exists k ∈ -K, K -1 such that |t -t k | 1/K.
Hence by 3.4 (as 1/K 1),

ψ n (t) -(Q, 0) H×L 2 ([1/(2R)+1/K,R+1-1/K]) ε. Now R + 1 -1/K R and 1/(2R) + 1/K 1/R; hence Lemma 3.4 yields ∀t ∈ [-1, 1], ψ n (t) -(Q, 0) H×L 2 ([1/R,R]) ε, that is ∀n N, sup t∈[-1,1] ψ n (t) -(Q, 0) H×L 2 ([1/R,R]) ε.
This is the desired convergence.

We now prove a bubble decomposition result, which is the main result of this Section. It will be central in the soliton resolution in both the blow-up and global cases, and that we will also crucially use for the sharp scattering result (Theorem 1.6).

Theorem 3.5 (Bubble decomposition). We assume (A1)-(A2).

Let T > 0 and

(ψ n ) n ∈ C ([-T, T ], H × L 2
) be a sequence of wave maps. Assume that for some R > 0, (1) The ψ n have uniformly bounded energy sup n E(ψ n ) = E < +∞.

(2) For some ∈ V , ψ n (0, R) → as n → +∞.

(3) At t = 0, the ∂ t ψ n have vanishing L 2 -norm on [0, R]:

sup λ 1 1 λ λ -λ R 0 |∂ t ψ n (t, r)| 2 rdrdt → 0 as n → +∞.
(4) ψ n (0) has vanishing energy on scale 1:

∀t ∈ [-T, T ], ∀r > 0, E( ψ n (t); r, R) → 0 as n → +∞.
Then there exists an integer J 0, J scales (λ j,n ) n verifying

0 < λ J,n • • • λ 2,n λ 1,n 1,
and J harmonic maps Q j ∈ H such that, up to a subsequence ψ σ(n) ,

ψ σ(n) (t) -( , 0) = J j=1 (Q j (•/λ j,n ) -Q j (∞), 0) + b n (t)
, b n (t) H×L 2 (λn/A r Aλn) → 0 as n → +∞.

where b n ∈ C ([-T, T ], H × L 2 ([0, R])) satisfies the following convergences. For all A > 0, (1)
(2) (No energy up to the last scale) If J 1, then

sup t∈[-Aλ J,n ,Aλ J,n ]
b n (t) H×L 2 (r Aλ J,n ) → 0 as n → +∞.

If J = 0, then sup t∈[-T /2,T /2] b n (t) H×L 2 (r R) → 0. (3) Let C A = {(t, r) | |t| min(Ar, T ), r R} be a truncated cone. Then sup (t,r)∈C A |b n (t, r)| → 0. Also (1) J j=1 E(Q j , 0) E, (2) For all 1 j < J, Q j+1 (∞) = Q j (0), and Q 1 (∞) = .
Proof. Up to rescaling, we can always assume T = 1. We first need to introduce some notation. As E(ψ n ) E and ψ n (0, R) → , hence is bounded, there hold an L ∞ bound on ψ n : for some K > 0, ∀n, ∀t, ∀r, |ψ n (t, r)| K. By (A1), V ∩ [-K, K] is finite. For ∈ V , denote -, + ∈ V the preceding and following points in V (respectively), that is

-< < + ( -, + ) ∩ V = { }. Then define η = min{sup{|g(x)| | x ∈ ( , + )}, sup{|g(x)| | x ∈ ( -, )}} > 0, and δ 0 = 1 2 inf{η | ∈ V ∩ [-K, K]} > 0.
Finally, for Q ∈ H a non constant harmonic map such that Q(∞) = and normalized in the sense that

Q(1) = 1 2 (Q(0) + Q(∞)), let ε > 0 be such that ∀r √ ε , |g(Q(r)| δ 0 /2, and |g(Q(1/r)| δ 0 /2,
and ε 0 = inf{ε | ∈ V ∩ [-K, K]} > 0. Notice that by definition of ε 0 , if Q is a normalized harmonic map such that Q(∞) ∈ [-K, K] and r 0 is such that |g(Q(r 0 ))| δ 0 /2, then if r ε 0 r 0 or r r 0 /ε 0 , |g(Q(r))| δ 0 /2.
Also notice that due to monotonicity of the energy along light cones, hypothesis (2) and (4) are in fact uniform in t ∈ [-1, 1]. More precisely, there holds

(2') for all r > 0, sup t∈[-1,1] ψ n (t) -L ∞ ([r,R]) → 0 as n → +∞. ( 4
') for all r > 0, sup t∈[-1,1] E( ψ n (t); r, R) → 0 as n → +∞
Let us prove (4') first. Let r > 0 and δ > 0. For r r/2 be such that 1/r ∈ N, apply (4) with t = kr , k ∈ -1/r ; 1/r . This gives N such that for all n N and all k ∈ -1/r ; 1/r , E( ψ n (kr ); r , R) δ. By monotonicity of the energy, with τ ∈ [-r , r ] we get

E( ψ n (kr + τ ); r + |τ |, R) E( ψ n (kr ); r , R) δ.
The kr + τ cover all [1, 1], hence (4'). We now turn to (2'). Notice that for φ ∈ H,

g(φ) 2 H (1 + g (φ) 2
L ∞ )E(φ, 0), so that for all t and r > 0,

g(ψ n (t)) 2 H([r,R]) (1 + g 2 L ∞ ([-K,K]
) )E( ψ n (t); r, R) → 0, where the convergence is uniform in t due to (4'). Due to Lemma A.3, we deduce that (for all 0 < r R/2)

sup t∈[-1,1] g(ψ n (t)) L ∞ ([r,R]) → 0. As V ∩[-K, K] is finite and ψ n (0, R) → ∈ V , it follows from a continuity argument that ∀r > 0, sup t∈[-1,1] ψ n (t) -L ∞ ([r,R]) → 0 as n → +∞,
as desired.

Step 1: Extraction of the profiles, and definition of λ j,n and b n . We recall that an extraction is a function φ : N → N which is (strictly) increasing. We now define the set of scales S made of couples λ := ((r n ) n , φ) where

(r n ) n ∈ [0, R] N , and φ : N → N is an extraction, such that (3.3) ψ φ(n) (0, r n ) has a limit l λ such that |g(l λ )| = δ 0 .
We denote by S 0 the subset of S made of scales ((r n ) n , φ) such that for some non constant harmonic map Q, it satisfies furthermore

(3.4) ∀A > 0, ψ φ(n) (0) -(Q(r n •), 0) H×L 2 ([rn/A,Arn]) → 0.
Notice that for all r > 0, ψ n (0) -( , 0) H×L 2 ([r,R]) → 0 due to our hypothesis (4).

Hence the pointwise bound gives

∀r > 0, g(ψ n (0)) L ∞ ([r,R]) → 0.
It transpires that for any λ = ((r n ) n , σ) ∈ S, then r n → 0. Also, in this case, if we define φ n (t, r) = ψ σ(n) (r n t, r n r), then φ n is a wave map defined for t ∈ [-1/r n , 1/r n ] and for all A > 0,

A -A R/rn 0 |∂ t φ n (t, r)| 2 rdrdt = 1 r n Arn -Arn R 0 |∂ t φ σ(n) (r n t, r n r)| 2 rdrdt → 0, and
g(φ n (1)) = lim n g(ψ σ(n) (r n )) = g(l λ ) = 0.
It follows from Proposition 3.1 (by rescaling again by a fixed factor A and using a diagonal argument) that there exists an extraction π and a non constant harmonic map Q, such that

∀A > 0, sup t∈[-A,A] φ π(n) (t) -(Q, 0) H×L 2 ([1/A,A]) → 0.
In particular, due to continuity of the flow, we have that

∀t ∈ R, lim A→+∞ lim n E( φ π(n) (t); 1/A, A) = E(Q, 0).
Unscaling, this can be rewritten as

ψ φ•π(n) (t)-(Q(r π(n) •), 0) H×L 2 ([r π(n) /A,Ar π(n) ]) → 0.
Notice that the scale λ := ((r π(n) ) n , φ • π) ∈ S 0 , we say it is adapted to λ ∈ S, and we call Q the local limit at scale λ.

We now proceed with the extraction of the profiles. First assume that S 0 is empty. In this case, let us prove that for n large enough, g(ψ n (0)) ∞ δ 0 . Indeed, recall that g(ψ n (0, R)) → 0. If there exist an extraction φ such that for all n, g(ψ σ(n) (0)) ∞ > δ 0 , then by continuity of g(ψ φ(n) )(0), for all n there exist r n such that |g(ψ φ(n) )(0, r n )| = δ 0 . As g -1 ({±δ 0 }) ∩ [-K, K] is compact, up to extracting a subsequence, we can assume that g(ψ φ(n) )(0, r n ) → where |g( )| = δ 0 . Hence ((r n ) n , σ) ∈ S and we saw at the previous paragraph how to construct an adapted scale to it: it follows that S 0 = ∅, a contradiction. Hence for n N ,

g(ψ n (0)) ∞ δ 0 .
Then we choose J = 0 and b n = ψ n . If S 0 is not empty, we proceed by induction and construct a (finite) sequence of scales λ j = (r j,n ) n , σ j ) ∈ S 0 for j = 1, . . . J, such that (1) There exist an extraction π j such that σ j+1 = σ j • π j .

(2) Orthogonality: r j+1,n r j,πj+1(n) → 0 as n → +∞.

(3) For j ∈ 1, J -1 and all n ∈ N and r ∈ (r j+1,n , ε 0 r j,πj (n) ), there holds

|g(ψ σj+1(n) (0, r))| < δ 0 . (4) For all r ∈ [0, ε 0 r J,π J (n) ), |g(ψ σ J (n) (0, r))| < δ 0 .
The first two conditions give an order on S 0 , the third one ensures that the scale λ j and λ j+1 are "consecutive", and the fourth one is the stopping condition.

Let φ 1 be an extraction such that for all n ∈ N, g(ψ σ1(n) (0)) L ∞ ([0,R]) δ 0 . We can furthermore choose φ 1 such that ψ σ1(n) (r 1,n ) has a limit l 1 ∈ g -1 ({±δ 0 }), where r 1,n is defined as follows: r 1,n is such that |g(ψ n (r 1,n )| = δ 0 and ∀r ∈ (r 1,n , R], |g(ψ n (0, r))| < δ 0 . φ 1 and r 1,n are well defined because ψ n is continuous, g(ψ n (R)) → 0 and S 0 is not empty. Then (r 1,n , σ 1 ) ∈ S, and let π 1 be adapted. This yields the first scale

λ 1 := ((r 1,π1(n) ) n , σ 1 • π 1 ) ∈ S 0 ,
and Q 1 is the local limit at scale λ 1 . Now assume that ((r j,n ) n , σ j ) is constructed. For some non constant harmonic map Q j we have the convergence

∀A > 0, ψ σj (n) (0) -(Q j (r j,n •), 0)) H×L 2 (rj,n/A,Arj,n) → 0 as n → +∞.
For A 2, the convergence also holds in L ∞ (r j,n /A, Ar j,n ), due to Lemma A.3. Now, as |g(Q j (ε 0 ))| δ 0 /2, we see (with A = 1/ε 0 ) that for large n,

|g(ψ σj (n) (0, ε 0 r j,n ))| 2δ/3. If for n large enough, sup{|g(ψ σj (n) (0, r)| | r ∈ [0, ε 0 r j,n ]}
δ 0 , we stop here. Otherwise, we construct ((r j+1,n ) n , σ j+1 ) as follows. First choose an extraction ρ j such that denoting

φ j+1 = σ j • ρ j , we have ∀n, sup r∈[0,ε0r j,ρ j (n) ] |g(ψ φj+1(n) (0, r))| > δ 0 .
This allows to define rj+1,n such that |g(ψ φj+1(n) (0, rj+1,n )| = δ 0 and ∀r ∈ (r j+1,n , ε 0 r j,ρj (n) ), |g(ψ φj+1(n) (r)| < δ 0 .

Extracting further, we can assume without loss of generality that ψ φj+1(n) (0, rj+1,n ) has a limit l j+1 , i.e. ((r j+1,n ) n , φ j+1 ) ∈ S. Notice that due to the local convergence

ψ σj (πj (n)) (0)) -(Q j (r j,ρj (n) •), 0)
on the scale r j,n , it follows that rj+1,n r j,ρj (n) → 0 as n → +∞.

Let finally choose an extraction j adapted such that ((r j+1,n ) n , σ j+1 ) ∈ S 0 , where σ j+1 := φ j+1 • j and r j+1,n := rj+1, j (n) . Finally let Q j+1 be the local limit at scale ((r j+1,n ) n , σ j+1 ). Then π j := ρ j • j is an extraction such that

σ j+1 = σ j • π j .
Also, we see that

r j+1,n r j,πj (n) = rj+1, j (n) r j,ρj ( j (n)) → 0 as n → +∞.
Finally, by the definition of rj,+1,n , we have

∀r ∈ (r j+1,n , ε 0 r j,πj (n) ), |g(ψ σj+1(n) (r)| < δ 0 .
Hence (r j+1,n ) n , σ j+1 ) is as desired.

Notice that from our construction, we have Q j+1 (∞) = Q j (0).

We now claim that this process has to stop after a finite number of steps. Indeed, fix J 1 consider the sequence ( ψ σ J (n) ) n . Then for 1 j J, we have for all A > 0,

E( ψ σj (n) ; 1/Ar πj •••••πj (n),j , Ar πj •••••π J (n),j ) → E(Q j , 0; 1/A, A). Now for j < j , if we denote = π j • • • • • π j -1 , and π = π j • • • • • π k r π j •••••π k (n),j r πj •••••π k (n),j = r π(n),j r •π(n),j
→ 0 due to the fact that the scales r n,j and r (n),j are orthogonal. Summing this for 1 j J, we thus get

E lim inf n E( ψ σ k (n) ) J j=1 E(Q j , 0; 1/A, A).
Letting A → +∞, we get

E J j=1 E(Q j , 0). As Q j (0), Q j (∞) ∈ [-K, K] for all j, we have that E(Q j , 0) E K := inf{G(k) -G(k ) | k, k ∈ V ∩ [-K, K], k > k }.
E K > 0 due to assumption (A2). Hence E JE K and this prove that the process has to stop after at most E/E K steps.

Thus we have constructed a sequence of J scales ((r 1,n ) n , σ 1 ), . . . , ((r J,n ) n , σ J ) ∈ S 0 and of non constant harmonic maps Q j . We can now define for j = 1, . . . J,

σ = σ J , λ j,n = r j,σj+1•••••σ J (n)
It will be convenient to write λ 0,n = R, λ J+1,n = 0.

From our construction, we have that for all j = 1, . . . , J, and A > 0, sup t∈[-Aλj,n,Aλj,n]

ψ σ(n) (t) -Q j (•/λ j,n ) H×L 2 (λj,n/A r Aλj,n) → 0 as n → +∞, and sup t∈[-1,1] ψ σ(n) (t) -H×L 2 (λ0,n/A r λ0,n) → 0.
Also, for all n ∈ N and j = 0, . . . , J,

λ j+1,n λ j,n → 0 as n → +∞ (3.6) g(ψ σ(n) (0)) L ∞ ([λj+1,n/ε0,ε0λj,n]) δ 0 , (3.7)
Define the error term: for t ∈ [-1, 1]

(3.8) b n (t) = ψ σ(n) (t) -( , 0) - J j=1 (Q j (•/λ j,n ) -Q j (∞)x, 0).
Step 2 : Convergence at all scales. Let A > 0 and a sequence λ n ⊂ [0, R/A] be given. We now prove conclusion (1), that is

(3.9) sup t∈[-Aλn,Aλn] b n (t) H×L 2 ([λn/A,Aλn]) → 0 as n → +∞.
This means that b n has no profile in the cone C A . First let us prove

(3.10) b n (0) L ∞ ([0,R]) → 0 as n → +∞
From assumption (4), for all A > 0, b n (0) H×L 2 (R/A r R) → 0 as n → +∞.

And it follows that from Step 1 and an easy induction that for all j = 1, . . . , J, and

A > 0, b n (0) H×L 2 (λj,n/A r Aλj,n) → 0 as n → +∞. Hence due to Lemma A.3, b n (0) L ∞ (R/A r R) + b n (0) L ∞ (rj,n/A r Aλj,n) → 0 as n → +∞.
We now argue by contradiction. Let r n ∈ [0, R] be such that lim sup n |b n (0, r n )| > 0.

From convergence on the scales λ j,n of b n (0), we have that

∀j = 0, . . . J, r ρ(n) λ j,ρ(n) + λ j,ρ(n) r ρ(n) → +∞.
We can assume that for some extraction ρ, and for some j 0 ∈ 0, J ,

λ j0+1,n r ρ(n) λ j0,n . For j j 0 , Q j (r ρ(n) /λ j,n ) → Q j (0) = Q j+1 (∞) and for j > j 0 , Q j (r ρ(n) /λ j,n ) → Q j (∞) = Q j-1 (0), hence J j=1 Q j (r ρ(n) /λ j,n ) -Q j (∞) → j0 j=1 Q j+1 (∞) -Q j (∞) = Q j0 (0) -Q 1 (∞). Now, as Q 1 (∞) = , we deduce that b n (0, r ρ(n) ) = ψ σ•ρ(n) (r n ) -Q j0 (0) + o n (1).
Up to extracting further we can assume, (recall b n (0) is continuous and b n (0, R) → 0), b ρ(n) (0, r ρ(n) ) = ε, where ε = 0 is small so that we also have g(ε + Q j0 (0)) = 0. It follows that

ψ σ•ρ(n) (0, r n )) = b ρ(n) (0, r ρ(n) ) + Q j0 (0) + o n (1) → ε + Q j0 (0).

Arguing as in

Step 1, (and relying on Proposition 3.1), we deduce that there exists a harmonic map Q such that Q(1) = Q j0 (0) + ε (in particular, Q is not constant) and an extraction such that

∀A > 0, ψ σ•ρ• (n) (0) -Q H×L 2 ([r ρ• (n) /A,Ar ρ• (n) ]) → 0 as n → +∞.
But then convergence also holds in L ∞ and as g(Q) L ∞ 2δ 0 , and we deduce

lim inf n→+∞ g(ψ σ(n) (0)) L ∞ ([λj 0 +1,n ,λj 0 ,n]) 2δ 0 ,
and we reached a contradiction with (3.7). This proves that for all sequences (r n

) n ⊂ [0, R], lim n b n (0, r n ) = 0, and hence b n (0) L ∞ ([0,R]) → 0 as n → +∞.
which is (3.10).

We now prove (3.9) arguing by contradiction. Up to extracting a subsequence, we can assume without loss of generality the existence of A > 0 and of a sequences t n such that 0 |t n | Aλ n AR, and for some ρ 0 and ε > 0 b n (t n ) H×L 2 (λn/A r Aλn) ε, and

t n λ n → ρ.
Also (2) and (4) show that t n , λ n → 0 as n → +∞.

Similarly, due to (3.5), for all j = 0, . . . , J,

λ n λ j,n + λ j,n λ n → ∞.
Up to extracting further we can assume that there is j 0 ∈ 0, J such that for all n, λ j0+1,n λ n λ j0,n .

Now consider the wave map

φ n (t, r) = ψ σ(n) (λ n t, λ n r).
By inspection, φ n is a finite energy wave map defined for times t ∈ [-A, A], and for n large enough,

(3.11) φ n t n λ n -(Q j0 (0), 0) H×L 2 (1/A r A)
ε/2, and

1 A A -A R/λn 0 |∂ t φ n (t, r)| 2 rdrdt = 1 Aλ n Aλn -Aλn R 0 |∂ t ψ σ(n) (t, r)| 2 rdrdt → 0.
Hence for some harmonic map Q, φ n → Q in the sense of Proposition 3.1. Q is not constant due to (3.11). Unscaling, we have

∀R, sup t∈[-Aλn,Aλn] b n (t) -(Q(•/λ n ), 0) H×L 2 ([λn/A,Aλn]) → 0 But then, for any t ∈ [-Aλ n , Aλ n ], b n (t, λ n ) → Q(1)
= 0 as n → +∞: this contradicts (3.10). Hence (3.9) is proved.

Notice that it follows immediately from (3.9) and Lemma A.3 that (3.12)

sup{|b n (t, r)| | 0 r R, 0 |t| min{1, Ar}} → 0 as n → +∞.
As b n (0) L ∞ ([0,R]) → 0, and ψ σ(n) (0) ∈ V , we see that for n large enough,

ψ σ(n) (0) = Q J (0) =: l.
Let us drop the first terms of the sequence, and assume that this holds for all n. Then (3.8) can be rewritten

(3.13) b n (t) = ψ n (t) -(l, 0) - J j=1 (Q j (•/λ j,n ) -Q j (0), 0).
Step 3: (Convergence up to the last scale) We do the proof for J 1, the proof in the case J = 0 being completely similar. Fix A > 0, we now prove that

(3.14) sup t∈[-Aλ J,n ,Aλ J,n ] b n (t) H×L 2 ([0,Aλ J,n ]) → 0.
From Step 2, we know that there exists α n ↑ +∞ such that

sup t∈[-αnλ J,n ,αnλ J,n ] b n (t) H×L 2 ([λ J,n /αn,αnλ J,n ]) → 0.
Let us first prove that

(3.15) E( ψ σ(n) (0); 0; α n λ J,n ) → 0.
We argue by contradiction. If the above convergence does not holds, there exists ε > 0 and a subsequence that we still denote ψ σ(n) such that for some µ n λ J,n /α n , we have

E( ψ σ(n) (0); 0; 2µ n ) = ε > 0.
By decreasing ε > 0 if necessary, we can furthermore assume that

(3.16) ∀x ∈ R, |G(x) -G(l)| ε/2 =⇒ |f (x) -f (l)(x -l)| f (l)|x -l| 2 .
(we recall that l := Q J (0) ∈ V , so that f (l) = 0 and f (l) = g (l) 2 > 0). By monotonicity of the energy, we see that for t ∈ [-µ n , µ n ],

E( ψ σ(n) (t); 0; µ n ) ε and E( ψ σ(n) (t); 0; 3µ n ) ε. Let u n (t, r) = (ψ σ(n) (µ n t, µ n r), µ n ∂ t ψ σ(n) (µ n t, µ n r)).
Then u n (t, r) is a wave map defined on the time interval [-1, 1], and for all t ∈ [-1, 1],

(3.17) E( u n (t); 0; 1) ε and E( u n (t); 0; 3) ε.

The definition of ε shows that for all t ∈ [-1, 1], r ∈ [0, 1]

(3.18) |f (u n (t, r)) -f (l)(u n (t, r) -l)| f (l)|u n (t, r) -l| 2 .
Also the condition (3) yields

(3.19) 1 -1 R/µn 0 |∂ t u n (t, r)| 2 rdrdt → 0.
This allows to apply Proposition 3.1. As u n (0) -Q J (0) L ∞ ([0,λ J,n /(αnµn)) → 0, the local limit is constant, i.e. for all B > 0,

(3.20) sup t∈[-1,1] u n (t) -(l, 0) H×L 2 ([1/B,B]) → 0.
Let ϕ : R → R be an even cutoff function such that ϕ(x) = 1 for |x| 1/2 and ϕ(x) = 0 for |x| 1. We can compute

|∂ r u n (t, r)| 2 ϕ(r)ϕ(t)rdrdt = - (u n (t, r) -l)∂ rr u n (t, r)ϕ(r)ϕ(t)rdrdt - (u n (t, r) -l)∂ r u n (t, r)(rϕ (r) + ϕ(r))ϕ(t)drdt = - (u n (t, r) -l)∂ tt u n (t, r)ϕ(r)ϕ(t)rdrdt - (u n (t, r) -l)f (u n (t, r)) r 2 ϕ(r)ϕ(t)rdrdt - (u n (t, r) -l)∂ r u n (t, r)ϕ (r)ϕ(t)rdrdt = |∂ t u n (t, r)| 2 ϕ(r)ϕ(|t|)rdrdt - (u n (t, r) -l)f (u n (t, r)) r 2 ϕ(r)ϕ(t)rdrdt + (u n (t, r) -l)∂ t u n (t, r)ϕ(r)ϕ (t)rdrdt - (u n (t, r) -l)∂ r u n (t, r)ϕ (r)ϕ(t)rdrdt
Now from estimate (3.19),

|∂ t u n (t, r)| 2 ϕ(r)ϕ(t)rdrdt = o(1).
Combining (3.19) with the Cauchy-Schwarz inequality, we also have

(u n (t, r) -l)∂ t u n (t, r)ϕ(r)ϕ (t)rdrdt = o(1).
Also, as ϕ has support on [1/2, 1], estimate (3.20) and the Cauchy Schwarz inequality, we have

(u n (t, r) -l)∂ r u n (t, r)ϕ (r)ϕ(t)rdrdt = o(1)
.

We now use that f (l) = g (l) 2 > 0. Then it follows from (3.18) that

(u n (t, r) -l)f (u n (t, r)) r 2 ϕ(r)ϕ(t)rdrdt f (l) 2 |u n (t, r) -l| 2 r 2 ϕ(r)ϕ(t)rdrdt.
Hence

0 |∂ r u n (t, r)| 2 ϕ(r)ϕ(t)rdrdt - f (l) 2 |u n (t, r) -l| 2 r 2 ϕ(r)ϕ(t)rdrdt + o(1)
.

From this we deduce first that

|u n (t, r) -l| 2 r 2 ϕ(r)ϕ(t)rdrdt → 0, then |∂ r u n (t, r)| 2 ϕ(r)ϕ(t)rdrdt → 0.
Adding up the last 2 results along with (3.19), we get

1/2 -1/2 u n (t) -(l, 0) 2 H×L 2 (r 1/2) dt → 0.
Now recalling (3.20), we get

∀B > 0, 1/2 -1/2 u n (t) -(l, 0) 2 H×L 2 (r B) dt → 0.
This shows that

∀B > 0, 1/2 -1/2 E( u n (t); 0, B)dt → 0.
However, this contradicts the second estimate in (3.17), and from there, estimate (3.15) holds true.

From (3.15) it is now easy to prove (3.14). Let A > 0. For n large enough, α n 2A.

By finite speed of propagation, we deduce that

sup t∈[-Aλ J,n ,Aλ J,n ] E( ψ σ(n) (t); 0, Aλ J,n ) → 0.
By coercivity of the energy around l = ψ σ(n) (0), we deduce

sup t∈[-Aλ J,n ,Aλ J,n ] ψ σ(n) (t) -(l, 0) H×L 2 ([0,Aλ J,n ]) → 0.
Notice that for all 1 j < J, as λ j,n λ j,n , we have

(Q j (•/λ j,n ) -Q j (0), 0) H×L 2 ([0,Aλ J,n ]) → 0.
The last two statements and (3.13) yield (3.14).

In the case J = 0, the same proof shows that

sup t∈[-1/2,1/2] b n (t) H×L 2 ([0,1/2]) → 0.
Now (4') reads: for all r ∈ (0, R),

sup t∈[-1,1] b n (t) H×L 2 ([r,R]) → 0.
We add up these last two statements to conclude the case J = 0.

Scattering for wave maps below the L ∞ threshold

Proof of Theorem 1.6. Let ψ be a finite energy wave map, with ψ(∞) = and such that it satisfies (1.7). Notice that ψ(0) = and there exists c > 0 (depending only on δ 0 and ) such that

(4.1) ∀t ∈ [0, T + ( ψ)), ∀r 0, c|ψ(t, r) -| |g(ψ(t, r)| 1 c |ψ(t, r) -|.
From this point wise bound, we derive that

(4.2) ∀t ∈ [0, T + ( ψ)), ψ(t) -( , 0) H×L 2 CE( ψ).
Also notice that T + ( ψ) = +∞. Assume indeed for the sake of contradiction that T + ( ψ) < ∞. Due to [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF], a bubble would form: hence for a sequence of times

t n ↑ T + ( ψ), and of points r n , ψ(t n , r n ) → k where k is such that , k are two consecutive elements of V . Thus lim inf n ψ(t n ) -ψ(∞) L ∞ |k -| d > c, a contradiction.
We now do an induction on the energy in the spirit of the Kenig Merle concentration compactness argument [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF][START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation[END_REF]. Define E c to be the supremum of all E 0 such that all wave maps ψ of energy E( ψ) E, which satisfies ψ(∞) = and (1.7), are global and scatters. Then [5, Theorem 2] shows that E c > 0 (recall |g ( )| ∈ {1, 2}). We argue by contradiction and assume that E c is finite.

Step 1. We first construct a critical element, that is a wave map V defined on [0, +∞), that satisfies (1.7), but V -S ([0,+∞)) = +∞. Let ψ n be a minimizing sequence of wave maps, i.e. ψ n satisfies (1.7) (hence

T + ( ψ n ) = +∞)), and
E( ψ n ) E c + 1 n , and
ψ n -S ([0,+∞)) = +∞.
Up to rescaling, we can also assume that for all n E( ψ n (0); 1, +∞) = E c /100.

Choose a sequence of times t n with vanishing L 2 norm of ∂ t ψ n (t n ). More precisely, we claim that for all n, there exist a sequence of times t n,m such that (

) E( ψ n (t n,m ); t n,m /m; t n,m (1 -1/m)) 1 m , (3) ψ n (t n,m ) -L ∞ (r tn,m/m) 1 m , 2
Indeed Corollary 2.3 provides us with a sequence satisfying the first condition, and then, up to extracting, Proposition 2.1 and Corollary 2.4 allows to satisfy the second and third conditions (we emphasize that these last two results hold for any sequence). Now choose t n = t n,n . Then Theorem 3.5 applies to the sequence of wave maps ϕ n (t, r) = ψ n (t n + t n t, t n r), with R = 1/2: indeed, we have by scaling

sup 0<λ 1 1 λ λ -λ 1/2 0 |∂ t ϕ n (t, r)| 2 rdrdt → 0,
and for t ∈ [-1, 1], and r > 0, we have due to our second condition

E( ϕ n (t); r, 1/2) = E( ψ n (t n ); rt n , t n /2) → 0.
Also notice that due to finite speed of propagation, we have

(4.3) ∀α > 1, ψ n (t n ) -( , 0) H×L 2 (r αtn) → 0.
We can therefore apply Theorem 3.5. It yields a bubble decomposition which is trivial: it can not contain any harmonic map profile Q j due to (1.7), and we see that for all A > 0 and sequence 0 < µ n t n , (4.4)

ψ n (t n ) -( , 0) H×L 2 ([µn/A,Aµn]) → 0, and ψ n (t n ) -L ∞ (0,tn/2) → 0.
Condition (2) then translate into absence of profile on scale 1:

(4.5) ψ n (t n ) -( , 0) H×L 2 ([tn/n,(1-1/n)tn]) → 0.
Using the third condition, we also get

(4.6) ψ n (t n ) -L ∞ → 0.
In view of (4.2)

ψ n (t n ) -( , 0) is bounded in H × L 2
, hence admits (up to a subsequence) a profile decomposition in the sense of Theorem 2.12. Denote V j,L the linear profiles, and (t j,n , λ j,n ) the parameters. We claim that there are no nontrivial profiles V j such that t j,n = 0. Consider indeed such a profile V j , for the sake of contradiction. Up to extracting, and changing scale by a fixed factor, it suffices to rule out three cases:

1) λ j,n t n , 2) λ j,n = t n , 3) λ j,n t n .
In case 1), from (4.4), for any A > 1, we have

ψ n (t n ) -( , 0) H×L 2 (tλj,n/A r Aλj,n) → 0.
Recall the Pythagorean expansion with cut-off Proposition 2.13, it implies in particular that

V j,L 0, r λ j,n , 1 λ j,n ∂ t V j,L 0, r λ j,n H×L 2 (λj,n/A r Aλj,n) → 0. Unscaling, we get V j,L (0) H×L 2 ([1/A,A]) → 0.
As this is true for all A > 1, we get V j,L (0) = 0. In case 3), from (4.3) and Proposition 2.13, we similarly get with α = 2

V j,L 0, r λ j,n , 1 λ j,n ∂ t V j,L 0, r λ j,n H×L 2 (r 2tn) → 0.
Unscaling, we deduce V j,L (0) H×L 2 (r 2tn/λj,n) → 0.

As t n /λ j,n → 0, we get V j,L (0) = 0.

In case 2), from (4.5) and Proposition 2.13, we get

V j,L 0, r t n , 1 λ j,n ∂ t V j,L 0, r t n H×L 2 (tn/n r tn(1-1/n))
→ 0, which after unscaling yields

V j,L (0) H×L 2 ([1/n,1-1/n]) → 0.
Similarly, (4.3) and Proposition 2.13 give after unscaling

V j,L (0) H×L 2 ([α,+∞)) → 0. Hence V j,L (0) = 0.
We ruled out all three cases, and this establishes our claim that there is no profile with t j,n = 0.

Consider now the set J of indices j such that t j,n λ j,n → +∞ and the nonlinear profile V j does not scatter at +∞ (or blows up in finite time). Any of these profile have energy greater or equal to E c > 0, hence (by Pythagorean expansion of the energy) there only is a finite number of them.

If there is no such profiles then for all j, V j S(t -tj,n/λj,n) is bounded (it tends to 0 if t j,n /λ j,n → -∞). Then Proposition 2.15 shows that there is a uniform bound M such that, for all sequence τ n t n ,

ψ n -S ([tn,τn]) M.
This in turn implies that ψ n scatters at +∞: it is a contradiction. Hence J = ∅. We can assume without loss of generality that J is indexed by 1, . . . , J 0 . Among such j J 0 , choose λ j,n slowest, then among such j, we consider j 0 such that t j0,n is lowest, i.e ∀j J 0 , λ j,n λ j0,n → +∞ or λ j,n = λ j0,n and t j0,n -t j,n λ j0,n → -∞ .

V := V j0 will be our critical element. First let us show that V is global and satisfies (1.7). Let t ∈ [0, T + ( V )) and r 0. Define τ n = t j0,n + λ j0,n t. Then

V j0,L S ([-t j,n λ j,n , τn -t j,n λ j,n ]) → V j0,L S ((-∞,t]) < +∞.
By construction, for j ∈ J \ {j 0 },

τ n -t j,n λ j,n → -∞ hence V j,L S ([-t j,n λ j,n , τn-t j,n λ j,n
] → 0.

For j / ∈ J , notice that τ n 0 for n large enough (t n /λ j,n → +∞) and

• if t j,n λ j,n → +∞, V j,L S (R) < +∞, and
• if t j,n λ j,n → -∞, V j,L S ([-t j,n λ j,n
,+∞)) → 0. Hence Proposition 2.15 applies, we can evolve the profile decomposition up to τ n :

ψ n (τ n , λ j0,n r) -= V (t, r) -+ j =j0,j J V j τ n -t j,n λ j,n , λ j0,n λ j,n r - + γ J,n (τ n , λ j0,n r) + r J,n (τ n , λ j0,n r).
By inspection (arguing as in the proof of equation (2.5)), we deduce that

|V (t, r) -| c.
It follows that V j satisfies (1.7); as noticed above, we then have T + ( V ) = +∞. Also, due to the Pythagorean expansion of the energy, we see that

E( V ) E c .
As we chose it so that it does not scatter at +∞, we must have E( V ) = E c by definition of the critical energy: therefore V is a critical element.

Step 2. We reach a contradiction. For this, we can repeat the argument of Step 1 on V : there exist a sequence of times t n ↑ +∞ such that

(1) sup

0<s tn/2 1 s tn+s tn-s tn/4 0 |∂ t V (t, r)| 2 rdrdt → 0 as n → +∞, (2)
V (t n ) -L ∞ → 0, (3)
V (t n ) -( , 0) admits a profile decomposition, with profiles U j,L . Also, as the bubble decomposition yield no bubble, convergence up to the last scale shows that (4.7)

V (t n ) -( , 0) H×L 2 ([0,tn/2]) → 0.

Arguing as in

Step 1, we see that one of the nonlinear profiles

U j0 is critical, in particular E( U j0 ) = E c = E( V )
. By Pythagorean expansion of the energy, it follows that there are no other nontrivial linear profiles in the profile decomposition of V (t n ), and that the dispersion term tends to 0 in H × L 2 . In short, there holds

V (t n , r)-( , 0) = U j0,L - t j0,n λ j0,n , r λ j0,n , 1 λ j0,n ∂ t U j0,L - t j0,n λ j0,n , r λ j0,n +o n (1)
,

where the o n (1) is in H × L 2 .
Assume t j0,n = 0. Observe that V has energy on the light cone: from monotonicity of the energy along light cones, Proposition 2.1 and (4.7), we have

lim sup n→+∞ V (t n ) -( , 0) H×L 2 (|r-tn| A) → 0 as A → +∞.
In particular,

V (t n ) -( , 0) H×L 2 (|r-tn| tn/2) → 0. Now, as U j0,L (0) = 0, let ρ > 0 such that U j0,L (0) H×L 2 ([ρ,2ρ]) =: α 0 > 0. Then U j0,L 0, r λ j0,n , 1 λ j0,n ∂ t U j0,L 0, r λ j0,n H×L 2 ([ρλj 0 ,n,2ρλj 0 ,n ]) = α 0 > 0.
Comparing with V (t n ) we must have for n large enough 2ρλ j0,n t n /2 and ρλ j0,n 3t n /2.

Up to extracting, we can furthermore assume that λ j0,n /t n → λ ∈ (0, +∞). But then, unscaling the concentration on V (t n ) we have lim sup n→+∞ U j0,L (0) H×L 2 (|r-tn|/λj 0 ,n A/λj 0 ,n ) → 0 as A → +∞.

But this implies U j0,L (0) = 0, a contradiction. Assume now that t j,n λ j,n → +∞. Then U j0,L S ((-∞;-tj,n/λj,n]) → 0 so the same holds for the non linear profile: U j0 S ((-∞;-tj,n/λj,n]) → 0. Then applying Proposition 2.15 backward in time up to time t = 0 t n , we get V -S (0,tn) U j0 -S ((-∞;-tj,n/λj,n]) + o(1) → 0.

Hence by monotone convergence, we deduce U S ([0,+∞)) = 0, and V = 0, a contradiction.

Assume finally that t j,n λ j,n → -∞. Then U j0,L S ([-tj,n/λj,n,+∞)) → 0 so the same holds for the non linear profile: U j0 -S ([-tj,n/λj,n,+∞)) → 0. Then we can use Proposition 2.15 to get that for any sequence τ n t n , V -S ([tn,τn)) → 0.

This implies V -S ([tn,+∞)) → 0: in particular V scatters at +∞, a contradiction. We reached a contradiction in all cases, hence E c = +∞.

Outside the light cone

Proposition 5.1 (Scattering state). We assume (A1)-(A2)-(A3').

Let ψ be a finite energy wave map such that T + ( ψ) = +∞. Denote = ψ(∞). There exist a map φ L solution to linear problem (LW ) and an increasing non-negative continuous function α(t) such that α(t) = o(t) and

ψ(t) -( , 0) -φ L (t) H×L 2 (r α(t)) → 0 as t → +∞.
Proof. First we recall Proposition 2.1, hence it suffices to construct φ L such that for all A 0, (5.1)

ψ(t) -( , 0) -φ L (t) H×L 2 (r t-A) → 0 as t → +∞.
The proof follows the scheme of [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: II[END_REF]Proposition 2.8], except now it is more involved to obtain scattering for the approximations of ψ around the light cone. We crucially rely on our new scattering result Theorem 1. (Recall that = ψ(∞)). Then φ n (0) = φ n (∞) = and from Lemma A.4, (5.2)

φ n (t n ) -( , 0) H×L 2 (r tn/2) → 0.
By construction ψ and φ n coincide at time t n on [t n /2, +∞), hence by finite speed of propagation, as long as they are defined at time t,

(5.3) ∀r t n /2 + |t -t n |, φ n (t, r) = ψ(t, r).
Step 1: Let us show that for n large enough,

(1) φ n is defined on [A 0 , +∞) for some A 0 not depending on n.

(2) φ n scatters at +∞.

Choose λ = 1/2 in Proposition 2.1, and apply it to ψ:

lim sup t→+∞ E( ψ(t); t/2, t -A) → 0 as A → +∞.
We then deduce that

lim sup n E( φ n (t n ); 0, t n -A) → 0 as A → +∞.
Let E > 0 be the minimal energy of a non constant harmonic map Q with Q(∞) = (equivalently, E = 2 min{G( + ) -G( ), G( ) -G( -)}, where ± ∈ V \ { } are the closest elements to , with -< < + ). Choose A 0 large enough, so that for n large enough, E( ψ(t n ); 0, t n -A 0 ) E /2, By finite speed of propagation, we deduce that for all τ such that φ n (t+τ ) is defined

E( φ n (t n + τ ); 0, t n -A 0 -|τ |) E( ψ n (t n ); 0, t n -A 0 ) E /2.
We recall the blowup criterion derived in [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF]: blow up concentrates in the light cone an energy at least E . Hence for |τ | t n -A 0 , this blow up criterion shows that ψ n (t n + τ ) is well defined: it then suffices to choose τ = A 0 -t n , and

T -( φ n ) < A 0 .
Of course we can drop the first terms, so that is holds for all n.

We now turn to scattering at +∞. For this, we will prove that as n → +∞,

(5.4) sup t∈[tn,T + (φn)) φ n (t) -L ∞ → 0.
By monotonicity of the energy outside cones, we see that for all A ∈ R, the limit lim t→+∞ E( ψ(t), t -A, +∞) exists, let us denote it E (A). As the energy density is non-negative, E (A) is an increasing function of A, and is also bounded by E( ψ).

Denote E = lim A→+∞ E (A) E( ψ). Let us show that (5.5) E( ψ(t); t/2, +∞) → E .

Indeed, let ε > 0, and choose A large so that

E -ε E (A) E and lim sup t→+∞ E( ψ(t); t/2, t -A) ε.
There exist T large such that for all t T , E (A) E( ψ(t); t -A, +∞) E (A) + ε and E( ψ(t); t/2, t -A) 2ε.

Then for all t T , we have

E -ε E (A) E( ψ(t); t -A, +∞) E (A) + ε E + ε.
Hence for t max{T, 2A},

E -ε E( ψ(
(φ n )), |E( φ n (t); t -A 1 , +∞) -E | ε.
By conservation of the energy, ∀t ∈ [t n , T + (φ n )), E( φ n (t); 0, t -A 1 ) 2ε.

Due to the point wise bound, we get as

φ n (0) = ∀t ∈ [t n , T + (φ n )), φ n (t) -L ∞ (r t-A1) Cε.
Also if t t n , then t T 2 and again by (5.3) and

∀r t -A 1 , |φ n (t, r) -| = |ψ(t, r) -| ε.
This proves that for n N ,

∀t ∈ [t n , T + ( φ n )), φ n (t) -L ∞ 2ε,
which is exactly (5.4). Then Theorem 1.6 applies for all n N , and shows that T + ( φ n ) = +∞ and φ n scatters at +∞. Also notice that (5.4) and (5.7) show that for some C > 0 and for n large enough, (5.8) sup

t tn φ n (t) -( , 0) H×L 2 CE .
Step 2: Construction of φ L and end of the proof. Let φ n,L be the linear solution of (LW ) which is the scattering state of φ n , that is, for all n,

(5.9) φ n (t) -( , 0) -φ n,L (t) H×L 2 → 0 as t → +∞.

Recall that the flow of (LW ) preserves the H × L 2 norm. Along with the bound on φ n (t) H×L 2 , this shows that φ n,L (0) H×L 2 CE .

Up to extracting, we can assume that φ n,L (0) has a weak limit φ L (0) in H × L 2 . We define φ L (t) as the linear solution of (LW ) with initial data φ L (0) at time 0. Let τ n be such that

φ n (τ n ) -( , 0) -φ n,L (τ n ) H×L 2 1 n .
Up to extracting further, we can assume that the sequence φ n (τ n ) -( , 0) -φ L (τ n ), which is bounded in H ×L 2 , admits a profile decomposition in the sense of Theorem 2.12:

φ n (τ n , r) -( , 0) = φ L (τ n , r) + J j=2 1 λ d-1 j,n V j,L - t j,n λ j,n , r λ j,n , 1 λ d j,n ∂ t V j,L - t j,n λ j,n , r λ j,n + γ J,n (0, r)
Notice that this appears as a profile decomposition for the sequence ( φ n,L (τ n , r)) with first profile φ L and parameter t 1,n = τ n , λ 1,n = 1. Indeed, the profile decomposition is constructed of via taking weak limits, that is:

V j,L (0) is the weak limit of Λ[λ j,n ]S (t j,n )( φ n (τ n ) -( , 0)),
where S(t) is the linear flow of (LW ) and Λ is the scaling operator

Λ[µ](ϕ 0 , ϕ 1 )(t, r) := ϕ 0 t µ , r µ , 1 µ ϕ 1 t µ , r µ . Also φ L (0) is the weak limit of S(-τ n )(Λ[1] φ n (τ n ) -( , 0)) = S(-τ n )( φ n (τ n ) - ( , 0
)). Indeed endow H × L 2 with the natural scalar product derived from the H × L 2 norm:

ρ, σ = ρ 1 σ 1 + ∂ r ρ 0 ∂ r σ 0 + g ( ) 2 ρ 0 σ 0 r 2 rdr
Then S(t) is an isometry for , and if

ϕ ∈ H × L 2 , S(-τ n )( φ n (τ n ) -( , 0)), ϕ = φ n (τ n ) -( , 0), S(-τ n ) ϕ = φ n,L (τ n ), S(-τ n ) ϕ + O(1/n) = φ n,L (0), ϕ + O(1/n) → φ L (0), ϕ .
We now proceed to prove (5.1); it suffices to show

∀A 0, ψ(t) -( , 0) -φ L (t) H×L 2 (r t-A) → 0.
Let A 0 and N be such that for t N 2A. Notice that in a similar fashion as previously, there hold the following profile decomposition for the sequence

( φ n (τ n )- ( , 0) -φ N,L (τ n )) n φ n (τ n , r) -( , 0) -φ N,L (τ n , r) = φ L (τ n , r) -φ N,L (τ n , r) + J j=2 1 λ d-1 j,n V j,L - t j,n λ j,n , r λ j,n , 1 λ d j,n ∂ t V j,L - t j,n λ j,n , r λ j,n + γ J,n (0, r)
Now recall (5.9) and (5.3), so that (5.10)

ψ(t) -( , 0) -φ N,L (t) H×L 2 (r t-t N /2) → 0 as t → +∞.
As φ n and ψ coincide for r t -t n /2, hence for r t -t N /2, we get

φ n (t) -( , 0) -φ N,L (t) H×L 2 (r t-t N /2) → 0 as t → +∞.
Using the Pythagorean expansion with cut off (Proposition 2.13), we deduce that for all profiles in the above profile decomposition, the H × L 2 (r t -t n /2) (semi-) norm tends to 0, and more specifically for the first profile, we get

φ L (τ n ) -φ N,L (τ n ) H×L 2 (r τn-t N /2) → 0 as n → +∞.
As φ L (t) -φ N,L (t) is solution to the linear solution (LW ), using monotonicity of the H × L 2 norm on outside cones, we deduce φ L (t) -φ N,L (t) H×L 2 (r t-t N /2) → 0 as t → +∞.

Combined with (5.10), we get

ψ(t) -( , 0) -φ L (t) H×L 2 (r t-t N /2) → 0 as t → +∞.
As t N /2 A, this proves (5.1), and the proof is complete.

We now turn to the analogous result regarding blow up wave maps.

Proposition 5.2. We assume (A1)-(A2). Let ψ be a finite energy wave map that blows up at time T + ( ψ). Then there exist := lim t↑T + ( ψ) ψ(t, T + ( ψ) -t) ∈ V and a wave map φ defined on a neighborhood of T + ( ψ) such that for t < T + ( ψ) and t max{0,

T -( φ)} ∀r T + ( ψ) -t, ψ(t, r) = φ(t, r), and
φ(t) -( , 0) H×L 2 ([0,T + ( ψ)-t]) → 0 as t ↑ T + ( ψ).
Proof.

Step 1. Let us first construct φ.

Claim 5.3. ψ(t, T + ( ψ) -t) has a limit ∈ V as t ↑ T + ( ψ).

Proof. Indeed, we recall Proposition 2.1 with λ = 1/2:

E( ψ(t); (T + ( ψ) -t)/2, T + ( ψ) -t) → 0 as t ↑ T + ( ψ).
In particular, we get

sup r∈[(T + ( ψ)-t)/2,T + ( ψ)-t] |G(ψ(t, r)) -G(ψ(t, T + ( ψ) -t))| → 0.
Also recall that for all t, ψ(t)

L ∞ K. As G is a homeomorphism R → R from As- sumptions (A1)-(A2), we get that G -1 is uniformly continuous on [G(-K), G(K)]
and from there,

sup r∈[(T + ( ψ)-t)/2,T + ( ψ)-t] |ψ(t, r) -ψ(t, T + ( ψ) -t)| → 0.
Now we use again the vanishing of the energy in the self-similar region:

T + ( ψ)-t (T + ( ψ)-t)/2 |g(ψ(t, r)| 2 dr r → 0, so that there exist r(t) ∈ [(T + ( ψ) -t)/2, T + ( ψ) -t] such that g(ψ(t, r(t)) → 0 as t ↑ T + ( ψ).
Due to the previous uniform convergence and continuity of g, we deduce that g(ψ(t, T

+ ( ψ) -t)) → 0. Now t → ψ(t, T + ( ψ) -t) is continuous on [0, T + ( ψ)). As V = g -1 ({0}) is discrete, this implies that ψ(t, T + ( ψ) -t) has limit ∈ V as t → T + ( ψ), as desired.
Now consider any sequence of times τ n ↑ T + ( ψ) and the wave maps φ n defined at time τ n as follows

φ n (τ n , r) =      + ψ(τ n , T + ( ψ) -τ n ) - T + ( ψ) -τ n r if 0 r T + ( ψ) -τ n ψ(τ n , r) if r T + ( ψ) -τ n ∂ t φ n (τ n , r) = 0 if 0 r T + ( ψ) -τ n ∂ t ψ(τ n , r) if r T + ( ψ) -τ n .
Let m > n. By definition and finite speed of propagation, we have

∀r T + ( ψ) -τ n -|τ n -τ m |, φ n (τ m , r) = φ m (τ m , r),
hence, using again finite speed of propagation,

∀r 2T + ( ψ) -2τ n , φ n (T + ( ψ), r) = φ m (T + ( ψ), r),
It is then meaningful to define φ to be the wave map with initial data at time T + ( ψ):

(5.11) φ(T + ( ψ), r) = φ n (T + ( ψ), r) for r 2T + ( ψ) -2τ n .
Step 2. Properties of φ.

Let E (t) := E( ψ(t); T + ( ψ) -t, ∞). This is a non-decreasing function, so let us define here

E := lim t↑T + ( ψ) E (t).
Of course E E( ψ). Then by construction of φ n and as φ n (τ n , T + ( ψ) -τ n ) → , there holds

E( φ n ) → E .
Hence by monotone convergence, we deduce that

E( φ) = lim n E( φ(T + ( ψ)); T + ( ψ) -2τ n , +∞) lim n E( φ n (T + ( ψ) -τ n ); T + ( ψ) -τ n , +∞) = E ,
that is E( φ) E . On the other side,

E( φ) = E( φ(T + ( ψ)); 0, +∞) lim n E( φ n (T + ( ψ) -τ n ); T + ( ψ) -τ n , +∞) = E ,
and finally we obtain that φ has finite energy (5.12)

E( φ) = E .
By the definition (5.11) of φ and finite speed of propagation, we have for all t ∈ (T -( φ), T + ( ψ)):

∀r 3T + ( ψ) -t -2τ n , φ(t, r) = φ n (t, r), which yields for t = τ n ∀r 3T + ( ψ) -3τ n , φ(τ n , r) = φ n (τ n , r) = ψ(τ n , r),
Hence, again by finite speed of propagation, we conclude that for t

∈ (T -( φ), τ n ] ∀r 3T + ( ψ) -t -2τ n , φ(t, r) = ψ(t, r),
Letting n → +∞, we finally obtain that for t ∈ (T -( φ), T + ( ψ)):

(5.13) ∀r > T + ( ψ) -t, φ(t, r) = ψ(t, r).

By continuity, for t < T + ( ψ), it also holds for r = T + ( ψ) -t. In particular,

φ(t, T + ( ψ) -t) → as t ↑ T + ( ψ).
Now by definition of E , (5.13) implies that

E( φ(t, r); T + ( ψ) -t, +∞) → E = E( φ),
and by difference E( φ(t, r); 0, T + ( ψ) -t) → 0.

As φ(t, T + ( ψ) -t) → , this implies

sup r∈[0,T + ( ψ)-t] |φ(t, r) -| → 0 as t ↑ T + ( ψ).
Therefore φ(0) = and

φ(t) -( , 0) H×L 2 ([0,T + ( ψ)-t]) → 0.

H × L 2 convergence for the bubble decomposition

In this Section we consider again a sequence of wave maps, and improve the result of Theorem 3.5 under the extra assumption (A3), that is

∀ ∈ V , g ( ) ∈ {-1, 1}.
We show that that the error term in the bubble decomposition does in fact converge to 0 in H × L 2 . This is the only step in this paper where we use (A3): it guaranties that Proposition (2.10) holds for the linearized problem (LW ). This Section is independent of Section 4 and 5.

Hence, we get from (6.12)

n (τ 2,n ) -(m, 0) -b n,L (τ 2,n ) H×L 2 ([τ2,n,2τ2,n]) → 0.
To complete the proof, it suffices to notice that n and ψ n coincide at time τ 1,n on the interval [τ 1,n , 3τ 2,n ], so that at time τ 2,n , they coincide on the interval [τ 2,n , 2τ 2,n ].

We can now easily reach a contradiction. Indeed from ( 2)-( 3)-( 4), equivalence of energy and H × L 2 for L ∞ perturbation (Lemma A.1) and the definition of β n , we have

ψ(τ 2,n ) -(m, 0) 2 H×L 2 ([τ2,n,2τ2,n]) CE( ψ(τ 2,n ) -(m, 0); τ 2,n , 2τ 2,n ) C J k=1 E((Q k (•/λ k,n ), 0); τ 2,n , 2τ 2,n ) + o(1) C J k=1 E (Q k , 0); λ j,n β n λ k,n , 2 λ j,n β n λ k,n + o(1)
→ 0. (6.13) However, due to Proposition 2.10, we have

b n,L (τ 2,n ) H×L 2 ([τ2,n,+∞)) β(1) b n,L (0) H β(1)δ 1 .
From the definition of b n,L (0), we see that Then (6.13) and (6.14) are in contradiction with Claim 6.3. This shows that b n (0) H([0,R]) → 0, and so b n (0) H×L 2 ([0,R]) → 0, as desired.

Proofs of the main results

Proof of Theorem 1.1. We start by working under assumptions (A1)-(A2)-(A3) and consider the global case. Let ψ be a finite energy wave map such that T + ( ψ) = +∞ and let := ψ(∞) ∈ V . Then Proposition 5. Then for R = 1/2 and T = 1/4, the sequence ψ n satisfies the conditions of Theorem 3.5, which we can combine with Proposition 6.1. Hence there exists harmonic maps Q 1 , . . . , Q J and scales λ J,n

• • • λ 1,n 1 such that ψ n (0) -( , 0) - J j=1 (Q j (•/λ j,n ) -Q j (∞), 0) H×L 2 ([0,1/2]) → 0.
Unscaling, and denoting µ j,n = λ j,n /t n , we have

(7.3) ψ(t n ) -( , 0) - J j=1 (Q j (•/µ j,n ) -Q j (∞), 0)
H×L 2 ([0,tn/2]) → 0.

Also, for all j, as µ j,n t n , Q j (•/µ j,n ) -Q j (∞) H([tn/2,∞]) → 0. Combining this with (7.1), (7.2) and (7.3) yields

ψ(t n ) -( , 0) - J j=1 (Q j (•/µ j,n ) -Q j (∞), 0) -φ L (t) H×L 2 → 0 as n → +∞.
This concludes the proof in the global case.

We now turn to the blow-up case. Let ψ be a wave map which blow-up in finite time T + ( ψ). Proposition 5.2 show that there is a regular wave map φ(t) defined on a neighborhood of T + ( ψ) such that for t < T + ( ψ) (and t near enough T + ( ψ)) there holds (7.4) ∀r T + ( ψ) -t, ψ(t, r) = φ(t, r).

Also ψ(t, T + ( ψ) -t) → ∈ V and Then sequence of wave maps ψ n (t, r) := (ψ(t n + θ n t, θ n r), θ n ∂ t ψ(t n + θ n t, θ n r), satisfies the conditions of Theorem 3.5 with R = 1 and T = 1 which we can combine with Proposition 6.1. Hence there exists harmonic maps Q 1 , . . . , Q J and scales λ J,n

• • • λ 1,n 1 such that ψ n (0) -( , 0) - J j=1 (Q j (•/λ j,n ) -Q j (∞), 0) H×L 2 ([0,1]) → 0.
Let us unscale, and denote µ j,n = λ j,n /θ n , and recall (7.5) to deduce (7.6)

ψ(t n ) -φ(t n ) - J j=1 (Q j (•/µ j,n ) -Q j (∞), 0)
H×L 2 ([0,θn]) → 0.

Hence combining with (7.4), this yields

ψ(t n ) -φ(t n ) - J j=1 (Q j (•/µ j,n ) -Q j (∞), 0) H×L 2 → 0 as n → +∞.
This settles the proof of Theorem 1.1 in the blow-up case.

If we relax our assumptions to (A1)-(A2)-(A3'), the same argument apply, except that we do no longer longer dispose of Proposition 6. Define, for j = 1, . . . , J, λ j (t) and ν j (t) by ν j (t) := inf{r | ψ(t, r) = Q j (0)}, λ j (t) := inf {r | ψ(t, r) = Q j (1)} .

(so that ν J (t) = 0). Also let ν 0 (t) = t/2. For all t large enough, λ j (t) and ν j (t) are well defined, because ψ(t) is continous, ψ(0) = Q J (0), ψ(t, t/2) → = Q 1 (∞) and our hypothesis regarding the ordering of the sequence Q j (0). By construction, ∀j = 1, . . . J, ν j (t) λ j (t) ν j-1 (t).

Recall that any harmonic map minimize the energy to join two consecutive points k, in V . For j = 2, . . . , J, we extend the restrictions of ψ(t)| [νj (t),νj-1(t)] as follows ψ j,0 (t, r) :=      Q j (0) if r ν j (t), ψ(t, r) if ν j (t) r ν j-1 (t), Q j (∞) if ν j-1 (t) r, ψ 1,0 (t, r) := . By construction, |E( ψ j (t)) -E( ψ(t); ν j (t), ν j-1 (t))| → 0, and as ψ j (0) = Q j (0) and ψ j (∞) = Q j (∞),

           Q 1 (0) if
E( ψ j (t)) E(Q j ).
It follows from this and (7.7) that for all j = 1, . . . , J, E( ψ j (t)) → E(Q j ) = E(Q j ), and ψ j (t, λ j (t)) = Q j (1).

(and also ∂ t ψ L 2 ([0,t/2]) → 0). But Q j is the unique minimizer (up to scaling) of the energy of functions joining Q j (0) to Q j (∞); more precisely, [START_REF] Côte | Instability of nonconstant harmonic maps for the (1 + 2)-dimensional equivariant wave map system[END_REF]Proposition A.1] implies that for all j = 1, . . . , J, ψ j (t) -(Q j (•/λ j (t)), 0) H×L 2 → 0 as t → +∞.

From there, we conclude that for j = 1, . . . J, as t → +∞, ν j (t) λ j (t) ν j-1 (t), Proof. We focus on the case 0 < r 1 and r 2 < ∞, as the other cases are well-known and simpler. We recall that for r s, Proof. Again, we only consider here the case 0 < r 1 < r 2 < +∞ and leave the other simpler cases to the reader. We extend φ in the following way: let r 1 = r 1 /2 and r 2 = 2r 2 and

ψ(r) =                0 if r r 0 φ(r 1 )
r-r 1 r1-r 1 if r 1 r r 1 (affine extension)

φ(r) if r 1 r r 2 φ(r 2 ) r 2 -r r 2 -r2
if r 2 r r 2 (affine extension)

0 if r 2 r
.

r2 r1 |φ 1 |φ 1

 11 (t, r)| 2 + |∂ r φ 0 (t, r)| 2 + |g(φ 0 (t, r))| 2 (r)| 2 + |∂ r φ 0 (r)| 2 + |φ 0 (r)| 2 r 2 rdr.

Corollary 2 . 4 ([ 7 ,

 247 Corollary 2.3]). For any λ > 0 we have

  ) H×L 2 (|r-t| A) → 0 as A → +∞.

(

  No energy at all scales) Let λ n be such that 0 λ n min{R, T }/A. Then sup t∈[-Aλn,Aλn]

( 3 . 5 )

 35 ∀A > 0, sup t∈[-Arn,Arn]

0

  |∂ t ψ n (t, r)| 2 rdrdt 1/m.

  b n,L (0) H ×L 2 ([τ2,n,+∞)) → 0, hence by monotonicity of the H m × L 2 norm along cones, we get b n,L (τ 2,n ) H×L 2 ([2τ2,n,+∞)) → 0. We can then conclude: (6.14) b n,L (τ 2,n ) H×L 2 ([τ2,n,2τ2,n]) β(1)δ 1 > 0.

(7. 5 )|∂

 5 φ(t) -( , 0) H×L 2 ([0,T + ( ψ)-t]) → 0 as t ↑ T + ( ψ). Now Corollary 2.7 provides us with a sequence t n ↑ T + ( ψ) such that (we denoteθ n = T + ( ψ) -t n ) t ψ(t, r)| 2 rdrdt → 0 as n → +∞.

  1 and rely solely on Theorem 3.5 and Proposition 5.1 (in the global case) and Proposition 5.2 (in the blow-up case). These yield the desired convergence on b n . Proof of Corollary 1.4. Consider the global case. From Proposition 5.1, we have that as t → +∞ E( ψ(t); 0, t/2) → E( ψ) -φ L H×L 2 . Considering specifically the sequence t n → +∞ provided by Theorem 1.1, we infer E( ψ(t); 0, t/2) → J j=1 E(Q j ) = 2|G( ) -G(ψ(0))|. (7.7)

Lemma A. 3 .

 3 There exists c > 0 such that for any 0r 1 < r 2 +∞ with r 2 2r 1 , and φ ∈ H([r 1 , r 2 ]), then φ ∈ C ([r 1 , r 2 ]) and φ L ∞ ([r1,r2]) c φ H([r1,r2]) .

1 5 φ 2 Lφ√ 2 . 2 -r 0 r 0 √ 2 2 L

 15222022 This proves continuity. Then let r0 ∈ [r 1 , r 2 ] be such that φ L ∞ ([r1,r2]) = |φ(r 0 )|. If φ L ∞ ([r1,r2]) φ H([r1,r2]) , there is nothing to prove. Assume the opposite, that is φ L ∞ ([r1,r2]) φ H([r1,r2]) . Then if r ∈ [r 1 , r 2 ] is such that |r -r 0 | r 0 ∞ ([r1,r2]) L ∞ ([r1,r2]) 2 φ H([r1,r2])(we used ln(4/5) -1/4) and|φ(r)| |φ(r 0 )| -|φ(r) -φ(r 0 )| |φ(r 0 ))| -φ H([r1,r2]) φ L ∞ 2 φ H([r1,r2]) φ L ∞ ([r1,r2]) /2.Let us assume the latter (the former would be treated accordingly), that isr ∞ ([r1,r2]) .Lemma A.4 (Extension in H). Let c be as in Lemma A.3, and 0 r 1 < r 2 +∞. Let φ ∈ H([r 1 , r 2 ]), then there exists ψ ∈ H that extends φ, that is,∀r ∈ [r 1 , r 2 ], ψ(r) = φ(r), and ψ H φ H([r1,r2]) + 3 φ L ∞ ([r1,r2]) . If r 2 2r 1 , the previous Lemma ensures ψ H (3c + 1) φ H([r1,r2]) .

  [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF]. Let t n ↑ +∞ and define the sequence of wave maps φ n with data at time t n as a suitable extension in H × L 2 of ψ(t n )| r tn/2 , as in Lemma A.4. Specifically let φ n (t n ) = (φ n,0 , φ n,1 ) where

	φ n,0 (r) =	 	2	ψ(t n , t n /2) -t n	r +	if r t n /2,
			ψ(t n , r)		if r t n /2,
	φ n,1 (r) = ∂ t ψ(t n , r).

  Define now N such that t N max{T 1 , 2A 1 , T 2 } and for n N ,|E( φ n (t n )) -E | ε. ,we have t n t N so that if t t n , then t -A 1 t -t n /2 t n /2 + |t -t n |. By (5.3), it transpires ∀t ∈ [t n , T +

	Then for n	N
		t); t -A, +∞) E( ψ(t); t/2, +∞)
				E( ψ(t); t/2, t -A) + E + ε E + 3ε.
	(5.5) follows.
	Now, from (5.5) and 2.1, we have
	(5.6)	lim sup	|E( ψ(t); t -A, +∞) -E | → 0 as A → +∞.
		t→+∞
	From (5.5) and (5.2), we deduce that
	(5.7)		E( φ n (t n )) → E as n → +∞.
	We now prove (5.4). Let ε > 0. From (5.6), there exist A 1 and T 1 such that
		∀t T 1 , |E( ψ(t); t -A 1 , +∞) -E | ε.
	We also use Corollary 2.4. Let T 2 such that
			∀t T 2 ,	ψ(t) -L ∞ (r t/2) ε.

  1 (and Proposition 2.1) provides us with a scattering state φ L and α(t) = o(t) such that:(7.1) ψ(t) -( , 0) -φ L (t) H×L 2 (r α(t)) → 0.|∂ t ψ(t, r)| 2 rdrdt → 0 as n → +∞.

	Recall Proposition 2.8, from which a weak version yields
	(7.2)			φ L (t) H×L 2 ([0,t/2]) → 0.
	On the other side, Corollary 2.3 yields a sequence t n such that
	sup s,0<s tn/4	1 s	tn+s tn-s	0	t/2

Consider now the sequence of wave maps ψ n (t, r) = (ψ(t n + t n t, t n r), t n ∂ t ψ(t n + t n t, t n r),

European Research Council Advanced Grant no. 291214, BLOWDISOL.

Proposition 6.1. We assume (A1)-(A2)-(A3). Let ψ n be a sequence of wave maps as in Theorem 3.5, and we use its notations. We recall the existence of harmonic maps Q 1 , . . . Q J and scales λ J,n

• • • λ 1,n 1 and denote b n (t, r) := ψ n (t, r) -( , 0) -J j=1 (Q j (r/λ j,n ), 0).

Then

b n (0) H×L 2 ([0,R]) → 0 as n → +∞.

Proof. By Theorem 3.5, we recall that b n has vanishing energy at all scales on [0, R], that is for any A 1 and sequence (λ n ) n ⊂ [0, min{R, T }/A],

b n (t) L ∞ ([-Aλn,Aλn],H×L 2 (λn/A r Aλn)) → 0. (6.1)

We will mainly use this property for the scales λ 0,n , . . . λ J,n here. More precisely, we will use the following convergence: there exists an increasing sequence α n → +∞ (and we also denote λ 0,n = 1), such that (1) b n (t) L ∞ ([-αnλ J,n ,αnλ J,n ],H×L 2 (r αnλ J,n )) → 0, (2) b n (t) L ∞ ([-αnλj,n,αnλj,n],H×L 2 (λj,n/αn r αnλj,n)) → 0 for all j = 1, . . . , J -

We now argue by contradiction. Assume that b n (0) H×L 2 ([0,R]) → 0 as n → +∞. Up to extracting, we can assume that for some δ 0 > 0,

Due to convergences (1)-( 2)-(3), and up to extracting further, there exists j ∈ 1, . . . , J and δ 1 > 0 such that (6.3) b n (0) H×L 2 ([0,αnλj,n]) → 0, and ∀n ∈ N, b n (0)

) and for all k = 0, . . . , J,

as n → +∞. We consider two times:

We denote m = Q j (∞): the linearized flow which will interest us is now given by (LW m ). Let b n,L be the linear solution to (LW m ) with initial data at time 0:

Claim 6.2. We have (6.4)

Proof. By definition of b n and (2), notice that

Hence we are led to compare b n and b n,L . First consider the interval [τ 1,n , 2τ

We now work on [2τ 1,n , 3τ 2,n ]. Consider the wave maps ϑ n with initial data defined as follows

ϑ n (0) is an extension of ψ n (0) with adequate affine reconnection. ϑ n coincide with ψ n on [τ 1,n , 4τ 2,n ] at time 0, and hence at time τ 1,n τ 2,n we have

Observe that ϑ n (0) = ϑ n (∞) = m. Because ψ n (0, τ 1,n ) and ψ n (0, 4τ 2,n ) tend to m, and with the equivalence of energy and H × L 2 norm for L ∞ -small perturbation of m ∈ V (Lemma A.1), we infer that

From the definition of τ 1,n and τ 2,n , and as b n (0) has no energy at all scales on [0, R] (statement (6.1)), we see that ϑ n (0) has no enery at all scales, that is for any A 1 and sequence λ n > 0,

Hence Corollary 2.14 combined with Corollary 2.16 allows to conclude that ϑ n is globally defined on R and (6.7)

where ϑ n,L is the linear solution to (LW m ) with initial data ϑ n,L (0) = ϑ n (0)-(m, 0) at time 0. Now notice that ϑ n,L -b n,L is a solution to (LW m ) which also satisfies

By conservation of the H m × L 2 norm, we deduce that (6.8)

Combining (6.5), (6.7) and (6.8) yields

This proves the claim.

We now evolve up to time τ 2,n .

Claim 6.3. We have

Proof. Let n be the wave map with initial data at time τ 1,n as follows:

Notice that E( n (τ 1,n ); 0, 3τ 1,n ) → 0 (consider separately the intervals [0, τ 1,n ] and [τ 1,n , 3τ 1,n ]), hence n is defined at least on the time interval (-2τ 1,n , 4τ 1,n ), and by monotonicity of the energy along cones (6.9)

Similarly,

, and [4τ 2,n , ∞)). Hence, again by monotonicity of the energy along light cones, (6.10)

By equivalence of the energy and the H × L 2 norm under the assumption of small energy (Lemma A.1), it follows that

n (0, r) = ψ n (0, r). Then it follows easily from (6.1) (and the definition of τ 1,n , τ 2,n , and m) that n (0)-(m, 0) has no energy at all scales, that is, for any A 1 and sequence λ n > 0, (6.11) n (0) -(m, 0) H×L 2 (λn/A r Aλn) → 0. From there, we conclude by Corollary 2.14 combined with Corollary 2.16 that n is global and if we denote n,L the linear solution to (LW m ) with initial data n (0) -(m, 0) at time 0 : (6.12)

n,L S(R) + sup

Notice that at time τ 1,n , we have

→ 0 as t → +∞.

Recalling Proposition 5.1, the conclusion follows. For blow-up wave maps, it suffices to reproduce the argument, considering in this case ψ(t) on the interval [0, T * ( ψ) -t].

Appendix A. Proof of Lemma 3.4

Lemma A.1 (Energy and H norm). Let ∈ V such that g ( ) = 0; and let ψ be a function of finite energy on [r 1 , r 2 ], with 0 r 1 < r 2 ∞.

(

(2) There exists δ > 0 and C > 0 independent of ψ such that if

(3) If ψ(r 1 ) = , and E(ψ, 0; r 1 , r 2 ) δ then the hypothesis and the conclusion of (2) above hold.

Proof. These bounds are elementary, based on the Taylor expansion of g and G around . We refer to [START_REF] Côte | Instability of nonconstant harmonic maps for the (1 + 2)-dimensional equivariant wave map system[END_REF] for the first result, and to [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF] for the second and third ones.

Lemma A.2 (Relating H and H ). Let ∈ V be such that g ( ) = 0. Then

Proof. First suppose φ -∈ H. Then φ -is bounded say by M , and as g is C 1 with g( ) = 0, there exists C 1 such that

Then taking square and integrating, we get E(φ, 0) C φ 2 H and φ ∈ H . Also φ is continuous and tends to at 0 and at +∞: hence φ(0) = φ(∞) = . We now prove the converse and suppose φ ∈ H and φ(0) = φ(∞) = . There exists A > 0 large such that E((φ, 0); 0, 1/A) δ and E((φ, 0); A, +∞) δ , where δ appears in Lemma A. 1 (3). Its conclusion gives that Therefore, φ -2 H([1/A,A]) < +∞. Summing up, we get φ -2 H < ∞ and φ -∈ H.

And of course

Our goal is now to prove Lemma 3.4. We start by a claim regarding the control of L ∞ norm by H even locally outside 0 or ∞.

Then ψ is continuous, hence we deduce ψ ∈ H and

From this, we see that

hence, if we combine it with the previous Lemma when r 2 2r 1 , we get

Proof of Lemma 3. Combining the last two properties proves Lemma 3.4.