Yvan Martel
email: yvan.martel@polytechnique.edu

Pierre Raphaël
email: pierre.raphael@unice.fr

STRONGLY INTERACTING BLOW UP BUBBLES FOR THE MASS CRITICAL NLS

We consider the mass critical two dimensional nonlinear Schrödinger equation (NLS) i∂tu

Let Q denote the positive ground state solitary wave satisfying ∆Q -Q + Q 3 = 0. We construct a new class of multi-solitary wave solutions: given any integer K ≥ 2, there exists a global (for t > 0) solution u(t) of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate as t → +∞ so that the solution blows up in infinite time with the rate

This special behavior is due to strong interactions between the waves, in contrast with previous works on multi-solitary waves of (NLS) where interactions do not affect the blow up rate. Using the pseudo-conformal symmetry of the (NLS) flow, this yields the first example of solution v(t) of (NLS) blowing up in finite time with a rate strictly above the pseudo-conformal one, namely, ∇v(t) L 2 ∼ log |t| t as t ↑ 0.

Such solution concentrates K bubbles at a point x0 ∈ R 2 , i.e.

1. Introduction 1.1. General setting. We consider in this paper the mass critical two dimensional non linear Schrödinger equation (NLS)

i∂ t u + ∆u + |u| 2 u = 0, (t, x) ∈ R × R 2 .
(1.1)

It is well-known (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references therein) that for any u 0 ∈ H 1 (R 2 ), there exists a unique maximal solution u ∈ C((-T ⋆ , T ⋆ ), H 1 (R 2 )) of (1.1) with u(0) = u 0 . Moreover, the following blow up criterion holds

T ⋆ < +∞ implies lim t↑T ⋆ ∇u(t) L 2 = +∞. (1.2)
The mass (i.e. the L 2 norm) and the energy E of the solution are conserved by the flow, where

E(u) = 1 2 ˆR2 |∇u| 2 - 1 4 ˆR2 |u| 4
From a variational argument, the unique (up to symmetry) ground state solution to

∆Q -Q + Q 3 = 0, Q ∈ H 1 (R 2 ), Q > 0, Q is radially symmetric
attains the best constant C in the following Gagliardo-Nirenberg inequality

∀u ∈ H 1 (R 2 ), u 4 L 4 ≤ C u 2 L 2 ∇u 2 L 2
(1.3)

1

(see [START_REF] Berestycki | Nonlinear scalar field equations. I. Existence of a ground state[END_REF][START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF][START_REF] Kwong | Uniqueness of positive solutions of ∆uu + u p = 0 in R n[END_REF]). As a consequence, one has

∀u ∈ H 1 (R 2 ), E(u) ≥ 1 2 ∇u 2 L 2 1 - u 2 L 2 Q 2 L 2
.

(1.4)

Together with the conservation of mass and energy and the blow up criterion (1.2), this implies the global existence of any solution with initial data u 0 2 < Q 2 . Actually it is now known that in this case, the solution scatters i.e. behaves asymptotically in large time as a solution of the linear equation, see [START_REF] Killip | The cubic nonlinear Schrödinger equation in two dimensions with radial data[END_REF][START_REF] Dodson | Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state[END_REF] and references therein. We also know that u L 2 = Q L 2 corresponds to the mass threshold for global existence since the pseudo-conformal symmetry of the (NLS) equation

v(t, x) = 1 |t| u 1 |t| , x |t| e -i |x| 2 4|t|
(1.5) applied to the solitary wave solution u(t, x) = e it Q(x) yields the existence of an explicit single bubble blow up solution S(t) with minimal mass

S(t, x) = 1 |t| Q x |t| e -i |x| 2 4|t| e i |t| , S(t) L 2 = Q L 2 , ∇S(t) L 2 ∼ t∼0 - 1 |t| . (1.6)
We refer to [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] for more properties of the pseudo-conformal transform. From [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF], minimal mass blow up solutions are classified in H 1 (R 2 ): u(t) L 2 = Q L 2 and T * < +∞ imply u ≡ S up to the symmetries of the flow.

Recall also the following well-known general sufficient criterion for finite time blow up: for initial data u 0 ∈ Σ = H 1 ∩ L 2 (|x| 2 dx), the virial identity

d 2 dt 2 ˆR2 |x| 2 |u| 2 = 16E(u 0 ) (1.7)
implies blow up in finite time provided E(u 0 ) < 0 (by (1.4), this implies necessarily u 0 L 2 > Q L 2 ).

1.2. Single bubble blow up dynamics. We focus now on the case of mass slightly above the threshold, i.e.

Q L 2 < u 0 L 2 < Q L 2 + α 0 , 0 < α 0 ≪ 1. (1.8)
We first recall in this context that a large class of finite time blow up solutions was constructed in [START_REF] Bourgain | Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity[END_REF] (see also [START_REF] Krieger | Non-generic blow-up solutions for the critical focusing NLS in 1-D[END_REF], [START_REF] Merle | On collapsing ring blow up solutions to the mass supercritical NLS[END_REF]) as weak perturbation of the minimal mass solution S(t). In particular, these solutions blow up with the pseudo-conformal blow up rate

∇u(t) L 2 ∼ t∼T * 1 T * -t .
(1.9)

Second, recall that the series of works [START_REF] Perelman | On the formation of singularities in solutions of the critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF][START_REF] Raphaël | Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation[END_REF][START_REF] Merle | Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF] provides a thorough study of the stable blow up dynamics under condition (1.8), corresponding to the so called log-log blow up regime

∇u(t) L 2 ∼ t∼T * c * log | log(T * -t)| T * -t . (1.10)
Third, it is proved in [START_REF] Merle | On collapsing ring blow up solutions to the mass supercritical NLS[END_REF] (see also [START_REF] Krieger | Non-generic blow-up solutions for the critical focusing NLS in 1-D[END_REF]) that solutions constructed in [START_REF] Bourgain | Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity[END_REF] are unstable and correspond in some sense to a threshold between the above log-log blow up and scattering.

Finally, recall that under (1.8), a universal gap on the blow up speed was proved in [START_REF] Raphaël | Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation[END_REF]: given a finite time blow up solution satisfying (1.8), either it blows up in the log-log regime (1.10), or it blows up faster than the pseudo-conformal rate ∇u(t) L 2 1 T * -t .

(See also [START_REF] Antonini | Lower bounds for the L 2 minimal periodic blow-up solutions of critical nonlinear Schrödinger equation[END_REF][START_REF] Banica | Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain[END_REF].) However, the existence of solutions blowing up strictly faster than the conformal speed is a long lasting open problem, which is equivalent, by the pseudo-conformal symmetry (1.5), to the existence of global solutions blowing up in infinite time.

1.3. Multi bubbles blow up dynamics. For larger L 2 mass, it is conjectured (see e.g. [START_REF] Merle | Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation[END_REF]) that any finite time blow up solution concentrates at the blow up time universal quanta of mass m j > 0 at a finite number of points x k ∈ R 2 , i.e.

|u(t)| 2 ⇀ K k=1 m k δ x k + |u * | 2 as t ↑ T * ,
where u * ∈ L 2 is a (possibly zero) residual. The first example of multiple point blow up solution is given in [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF]: let K ≥ 1 and let (x k ) 1≤k≤K be K arbitrary distinct points of R 2 , there exist a finite time blow up solution u(t) of (1.1) with

u(t) - K k=1 S(t, . -x k ) H 1 → 0, |u(t)| 2 ⇀ Q 2 L 2 K k=1 δ x k as t ↑ 0.
In particular, u(t) blows up with the pseudo-conformal rate

∇u(t) L 2 ∼ 1 |t| as t ↑ 0.
Other general constructions of multi bubble blow up are provided by [START_REF] Planchon | Existence and stability of the log-log blow-up dynamics for the L 2 -critical nonlinear Schrödinger equation in a domain[END_REF][START_REF] Fan | Propagation of regularity and the localization of log-log blow up[END_REF] in the context of the log-log regime. Observe that these works deal with weak interactions in the sense that the blow up dynamics of each bubble is not perturbed at the main order by the presence of the other (distant) bubbles.

1.4. Main results. In this paper we construct the first example of infinite time blow up solution of (NLS), related to the strong interactions of an arbitrary number K ≥ 2 of bubbles. As a consequence, using the pseudo-conformal transform, we also obtain the first example of solution blowing up in finite time strictly faster than the conformal blow up rate. Such a solution concentrates the K bubbles at one point at the blow up time.

Theorem 1 (Infinite time blow up). Let K ≥ 2 be an integer. There exists a solution u ∈ C([0, +∞), Σ) of (1.1) which decomposes asymptotically into a sum of K solitary waves

u(t) -e iγ(t) K k=1 1 λ(t) Q . -x k (t) λ(t) H 1 → 0, λ(t) = 1 + o(1) log t as t → +∞, (1.11)
where the translation parameters x k (t) converge as t → +∞ to the vertices of a K-sided regular polygon, and where γ(t) is some phase parameter. In particular,

∇u(t) L 2 = K 1 2 ∇Q L 2 (1 + o(1)) log t as t → +∞.
(1.12)

Corollary 2 (Finite time collision). Let u(t) ∈ C([0, +∞), Σ) be given by Theorem 1 and let v ∈ C((-∞, 0), Σ) be the pseudo conformal transform of u(t) defined by (1.5). Then v(t) blows up at T * = 0 with

∇v(t) L 2 = K 1 2 ∇Q L 2 (1 + o(1)) log |t| t , |v| 2 ⇀ K Q 2 L 2 δ 0 as t ↑ 0. (1.13)
Comments on the main results. 1. Dynamics with multiple nonlinear objects. Multiple bubble solutions with weak interactions and asymptotically free Galilean motion have been constructed in various settings, both in stable and unstable contexts, see in particular [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF][START_REF] Mizumachi | Weak interaction between solitary waves of the generalized KdV equations[END_REF][START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg de Vries equations[END_REF][START_REF] Martel | Multi-solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF][START_REF] Perelman | Two soliton collision for nonlinear Schrödinger equations in dimension 1[END_REF][START_REF] Boulenger | Minimal mass blow up for NLS on a manifold[END_REF][START_REF] Fan | Propagation of regularity and the localization of log-log blow up[END_REF]]. As a typical example of weakly interacting dynamics, for the nonlinear Schrödinger equations

i∂ t u + ∆u + |u| p-1 u = 0, x ∈ R d , 1 < p < 1 + 4 d -2 , (1.14)
there exist multi solitary wave solutions satisfying for large t,

u(t) - K k=1 e -i( 1 2 ν k •x-1 4 |ν k | 2 t+ω k t+γ k ) ω 1 p-1 k Q ω 1 2 k (. -ν k t) H 1 e -γt , γ > 0, (1.15)
for any given set of parameters {ν k , ω k , γ k } with the decoupling condition [START_REF] Martel | Multi-solitary waves for nonlinear Schrödinger equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF]).

ν k = ν k ′ if k = k ′ (see
In [START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF], two different regimes with strong interactions related to the two body problem of gravitation are exhibited for the Hartree model (hyperbolic and parabolic asymptotic motions).

We also refer to [START_REF] Mizumachi | Weak interaction between solitary waves of the generalized KdV equations[END_REF][START_REF] Martel | Inelastic interaction of nearly equal solitons for the quartic gKdV equation[END_REF] for works related to sharp interaction problems in the setting of the subcritical (gKdV) equation. We thus see the present work as the first intrusion into the study of strongly interacting non radial multi solitary wave motions for (NLS). Note that the solution given by Theorem 1 is a minimal threshold dynamics and its behavior is unstable by perturbation of the data. An important direction of further investigation is the derivation of stable strongly interacting multiple bubbles blow up dynamics. We observe from the proof of Corollary 2 that the K bubbles of the solution collide at the same point at the blow up time providing the first example of collision at blow up for (NLS).

Note that the geometry of the trajectories of the blow up points (straight lines from the origin to the egde of the K-sided regular polygon) is an essential feature of these solutions. A related one dimensional mechanism is involved in the derivation of degenerate blow up curves in the context of "type I" blow up for the wave equation, see [START_REF] Merle | On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations[END_REF]. For the nonlinear heat equation in one dimension, solutions for which two points of maximum collide at blow up are constructed in [START_REF] Herrero | Flat blow-up in one-dimensional semilinear heat equations[END_REF]. There are also analogies of the present work with the construction of stationary solutions with mass concentrated along specific nonlinear grids, see [START_REF] Musso | Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation[END_REF]. In the context of two dimensional incompressible fluid mechanics, special solutions to the vortex point system are studied as a simplified model for dynamics of interacting and possibly colliding vortex, see for example [START_REF] Miot | Dynamique des points vortex dans une équation de Ginzburg-Landau complexe[END_REF] for an overview of these problems.

2. Construction of minimal mass solutions. The proof of Theorem 1 follows the now standard strategy of constructing minimal dynamics by approximate solutions and compactness, initiated in [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] and extended in various ways and contexts by [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg de Vries equations[END_REF][START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF][START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF]. The proof of Theorem 1 combines in a blow up context the approach developed for multibubble flows in [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg de Vries equations[END_REF][START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF] and a specific strategy to construct minimal blow up solutions for (NLS) type equations introduced in [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF][START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF]. An key ingredient of the proof is the precise tuning of the interactions between the waves. Note that we restrict ourselves to space dimension 2 for simplicity, but similar results hold for the mass critical (NLS) equation in any space dimension with same proof. For the mass subcritical and supercritical nonlinear Schrödinger equations (1.14), we expect using similar approach the existence of bounded strongly interacting multi solitary waves, with logarithmic relative distances, i.e. non free Galilean motion. Interestingly enough, the existence of such solutions is ruled out in the mass critical case by the virial law (1.7). Indeed, the scaling instability direction of the critical case is excited by the interactions which leads to the infinite time concentration displayed in Theorem 1. Note that the construction of of Theorem 1 is performed near t = +∞ (by translation invariance, it is then obvious to obtain a solution on the time interval [0, +∞)). An interesting question is to understand the behaviour of such solutions for t ≤ 0.

Zero energy global solutions.

From the proof of Theorem 1, the solution u has zero energy.

In [START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF], it is proved that any zero energy solution satisfying (1.8) blows up in finite time with the log-log regime. Thus, in the neighborhood of Q, e it Q is the only global zero energy solution. For the critical (gKdV) equation, a similar result holds, though in a stronger topology (see [START_REF] Martel | Blow up for the critical generalized Korteweg de Vries equation. I: Dynamics near the soliton[END_REF]). Note that the existence of global in time zero energy solutions is strongly related to Liouville type theorems and to blow up profile, see [START_REF] Nawa | Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power[END_REF][START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF]. For (NLS), the only known examples of global in positive time zero energy solutions so far were the time periodic solutions e it P where P is any solution to the stationary equation ∆P -P + P 3 = 0. Therefore, the existence of such a non trivial global (for positive time) zero energy solution u(t) is surprising.

For other works related to minimal mass solutions and their key role in the dynamics of the flow, we refer to [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical NLS[END_REF][START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF][START_REF] Banica | Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation[END_REF][START_REF] Merle | On collapsing ring blow up solutions to the mass supercritical NLS[END_REF][START_REF] Duyckaerts | Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg de Vries equation. I: Dynamics near the soliton[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg de Vries equation. II: Minimal mass dynamics[END_REF].

4. Blow up speed for (NLS). The question of determining all possible blow up rates for solutions of nonlinear dispersive equations is in general intricate. For the (NLS) equation (1.14) in the mass supercritical-energy subcritical range, a universal sharp upper bound on the blow up rate has been derived in [START_REF] Merle | On collapsing ring blow up solutions to the mass supercritical NLS[END_REF] for radial data, but no such bound exists for the mass critical problem. For (NLS) with a double power non linearity of the form |u| p-1 u + |u| 2 u where 1 < p < 3, the minimal mass solution has a surprising blow up rate different from the conformal rate, see [START_REF] Le Coz | Minimal mass blow up solutions for a double power nonlinear Schrödinger equations To appear in[END_REF]. For the mass critical (gKdV) equation, solutions arbitrarily close to the solitary wave with arbitrarily fast blow up speed have been constructed in [START_REF] Martel | Blow up for the critical generalized Korteweg de Vries equation. III: Exotic regimes[END_REF]. Recall that constructions of blow up solutions with various blow up rate are also available in the energy critical and super-critical context, see [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF][START_REF] Krieger | Full range of blow up exponents for the quintic wave equation in three dimensions[END_REF][START_REF] Merle | Type II blow up for the energy supercritical NLS to appear in Cambridge Math[END_REF][START_REF] Donninger | Exotic blowup solutions for the u 5 focusing wave equation in R 3[END_REF][START_REF] Collot | Type II blow up manifold for the energy super critical wave equation[END_REF][START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF]. However, such general constructions seem by now out of reach for the mass critical (NLS) problem. In this context, the derivation of the anomalous blow up speed (1.13), in spite of its rigidity, is an interesting new fact. We will see in the proof how such a blow up rate is related to strong coupling between the solitary waves.

1.5. Notation. Let Σ = H 1 ∩ L 2 (|x| 2 dx). The L 2 scalar product of two complex valued functions f, g ∈ L 2 (R 2 ) is denoted by f, g = Re ˆR2 f (x)g(x)dx .
In this paper, K is an integer with K ≥ 2. For brevity, k denotes K k=1 . For k = 1, . . . , K, e k denotes the unit vector of R 2 corresponding to the complex number e i 2π(k-1) K . We define the constant κ = κ(K) by

κ = 1 -e i 2π K = (2 -2 cos(2π/K)) 1/2 > 0.
(1.16)

Recall that we denote by Q(x) := Q(|x|) the unique radial positive ground state of (1.1):

Q ′′ + Q ′ r -Q + Q 3 = 0, Q ′ (0) = 0, lim r→+∞ Q(r) = 0. (1.17)
It is well-known and easily checked by ODE arguments that for some constant c Q > 0,

for all r > 1, Q(r) -c Q r -1 2 e -r + Q ′ (r) + c Q r -1 2 e -r r -3 2 e -r . (1.18)
We set

I Q = ˆQ3 (x)e x 1 dx, x = (x 1 , x 2 ). (1.19)
We denote by Y the set of smooth functions f such that for all p ∈ N, there exists q ∈ N, s.t. for all x ∈ R 2 |f (p) (x)| |x| q e -|x| .

(1.20)

Let Λ be the generator of L 2 -scaling in two dimensions:

Λf = f + x • ∇f.
The linearization of (1.1) around Q involves the following Schrödinger operators:

L + := -∆ + 1 -3Q 2 , L -:= -∆ + 1 -Q 2 .
Denote by ρ ∈ Y the unique radial solution H 1 to

L + ρ = |x| 2 4 Q (1.21) which satisfies on R 2 |ρ(x)| + |∇ρ(x)| (1 + |x| 3 )Q(x).
(1.22) We recall the generalized null space relations (see [START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF])

L -Q = 0, L + (ΛQ) = -2Q, L -(|x| 2 Q) = -4ΛQ, L + ρ = |x| 2 4 Q, L + (∇Q) = 0, L -(xQ) = -2∇Q, (1.23)
and the classical (see e.g. [START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF][START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF][START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF][START_REF] Chang | Spectra of linearized operators for NLS solitary waves[END_REF]) coercivity property: there exists µ > 0 such that for all η ∈ H 1 ,

L + Re η, Re η + L -Im η, Im η ≥ µ η 2 H 1 - 1 µ η, Q 2 + η, |x| 2 Q 2 + η, xQ 2 + η, iρ 2 + η, i∇Q 2 . (1.24)
1.6. Outline of the paper. The main goal of Sect. 2 is to construct a symmetric K-bubble approximate solution to (NLS) and to extract the formal evolution system of the geometrical parameters of the bubbles. The key observation is that this system contains forcing terms due to the nonlinear interactions of the waves, and has a special solution corresponding at the main order to the regime of Theorem 1 (see Sect. 2.2). In Sect. 3, we prove uniform estimates on particular backwards solutions of (NLS) related to the special regime of Theorem 1. We proceed in two main steps. First, we control the residue term by energy arguments in the context of multi-bubbles. Second, a careful adjustment of the final data yields a uniform control of the geometrical parameters. In Sect. 4, we finish the proof of Theorem 1 by compactness arguments on a suitable sequence of backwards solutions of (NLS) satisfying the uniform estimates of Sect. 3.

Acknowledgements. Y.M. was partly supported by ERC 291214 BLOWDISOL. P.R. was supported by the ERC-2014-CoG 646650 SingWave and the Institut Universitaire de France. This work was done while the authors were members of the MSRI (Fall 2015). They warmly thank MSRI for its hospitality.

Approximate solution

In this section, we first construct a symmetric K-bubble approximate solution to (NLS) and extract the evolution system of the geometrical parameters of the bubbles. This system contains forcing terms due to the nonlinear interactions of the waves. Second, we write explicitly a special formal solution of this system that will serve as a guideline for the construction of the special solution u(t) of Theorem 1. Third, we state a standard modulation lemma around the approximate solution. Recall that the integer K ≥ 2 is fixed.

u(t, x) = e iγ(s) λ(s) v(s, y), dt = λ 2 (s)ds, y = x λ(s) , (2.1)
so that

i∂ t u + ∆u + |u| 2 u = e iγ λ 3 i v + ∆v -v + |v| 2 v -i λ λ Λv + (1 -γ)v (2.2)
( v denotes derivation with respect to s). We introduce the following p-modulated ground state solitary waves, for any k ∈ {1, . . . , K},

P k (s, y) = e iΓ k (s,y-z k (s)) Q a(z(s)) (y -z k (s)), (2.3)
for

β k = βe k , z k = ze k , Γ k (s, y) = β k • y - b 4 |y| 2 , (2.4)
and where we have fixed

Q a = Q + aρ, a(z) = -c a z 1 2 e -κz , c a =          κ 1 2 c Q I Q 4 ρ, Q > 0 for K = 2 κ 1 2 c Q I Q 2 ρ, Q > 0 for K ≥ 3 (2.5)
Note that the introduction of such modulated Q a corresponds to the intrinsic instability of the pseudo-conformal blow up regime (a = 0 leads to b(s) = s -1 ). Similar exact Q a (at any order of a) were introduced in [START_REF] Merle | The instability of Bourgain-Wang solutions for the L 2 critical NLS[END_REF]. The explicit above choice of a(z) corresponds to direct integration of the nonlinear interactions at the main order, as explained in Sect. 2.2.

We also refer to (3.26) in the proof of Lemma 7 where this choice of a(z) leads to an almost conservation of the mass for the approximate solution P defined below. Note that the different formula for c a depending on the value of K corresponds to the fact that for K ≥ 3, each given soliton has exactly two closest neighbor solitons. Let P(s, y) = P(y;

(z(s), b(s), β(s))) = k P k (s, y). (2.6)
Then, P is an approximate solution of the rescaled equation in the following sense.

Lemma 3 (Leading order approximate flow). Let the vectors of modulation equations be

m a k =          b + λ λ żk -2β k + λ λ z k γ -1 + |β k | 2 -λ λ (β k • z k ) -(β k • żk ) βk -λ λ β k + b 2 ( żk -2β k + λ λ z k ) ḃ + b 2 -2b(b + λ λ ) -a          , MV =          -iΛV -i∇V -V -yV |y| 2 4 V          . (2.7)
Then the error to the renormalized flow (2.2) at P,

E P = i Ṗ + ∆P -P + |P| 2 P -i λ λ ΛP + (1 -γ)P (2.8)
decomposes as

E P = k e iΓ k Ψ k (y -z k ), Ψ k = m a k • MQ a + i ża ′ (z)ρ + G k + Ψ Qa , (2.9)
where

G k L ∞ z -1 2 e -κz , Ψ Qa L ∞ |a| 2 , (2.10)
and

G k , iQ a + κc a ρ, Q bz 3 2 e -κz (|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z)z -1 2 e -κz + z 3 e -2κz . (2.11)
Proof of Lemma 3. step 1. Equation for P k . Let

E P k = i Ṗk + ∆P k -P k + |P k | 2 P k -i λ λ ΛP k + (1 -γ)P k . Let y z k = y -z k . By direct computations i Ṗk = e iΓ k i ża ′ (z)ρ -( βk • y z k )Q a + ( żk • β k )Q a + ḃ 4 |y z k | 2 Q a - b 2 ( żk • y z k )Q a -i( żk • ∇Q a ) (y z k ), ∆P k = e iΓ k ∆Q a -|β k | 2 Q a - b 2 4 |y z k | 2 Q a -ibQ a + b(β k • y z k )Q a + 2i(β k • ∇Q a ) -ib(y z k • ∇Q a ) (y z k ), ΛP k = e iΓ k ΛQ a + i(β k • y z k )Q a -i b 2 |y z k | 2 Q a + (y z k • ∇Q a ) + i(z k • β k )Q a -i b 2 (z k • y z k )Q a + (z k • ∇Q a ) (y z k ).
Thus,

E P k = e iΓ k -i(b + λ λ )ΛQ a -i( żk -2β k + λ λ z k ) • ∇Q a -( γ -1 + |β k | 2 - λ λ (β k • z k ) -(β k • żk ))Q a -( βk - λ λ β k + b 2 ( żk -2β k + λ λ z k )) • y z k Q a + 1 4 ( ḃ + b 2 -2b(b + λ λ ))|y z k | 2 Q a + i ża ′ (z)ρ + ∆Q a -Q a + |Q a | 2 Q a (y z k ) By ∆Q -Q + Q 3 = 0 and the definition of ρ, L + ρ = -∆ρ + ρ -3Q 2 ρ = 1 4 |y| 2 Q (see (1.21)), we have ∆Q a -Q a + |Q a | 2 Q a = - a 4 |y| 2 Q a + Ψ Qa ,
where

Ψ Qa = |Q a | 2 Q a -Q 3 -3aQ 2 ρ + a 2 4 |y| 2 ρ.
(2.12)

We have thus obtained the P k equation

E P k = e iΓ k m a k • MQ a + i ża ′ (z)ρ + Ψ Qa (y -z k ), (2.13)
where m a k and M are defined in (2.7). step 2. Equation for P. From step 1 and the definition of E P in (2.8), it follows that

E P = k E P k + |P| 2 P - k |P k | 2 P k .
Observe that

|P| 2 P - k |P k | 2 P k = j, k, l P k P j P l - k |P k | 2 P k = k F k , with F k = 2|P k | 2 j =k P j + P 2 k j =k P j + P k j =k, l =k, j =l P j P l = e iΓ k G k (y -z k ),
where we have set

G k = 2G (I) k + G (I) k + G (II) k , (2.14) and G
(I) k (y) = e -iΓ k Q 2 a (y) j =k e iΓ j Q a (y -(z j -z k )), G (II) k (y) = e -2iΓ k Q a (y) j =k, l =k, j =l e iΓ j Q a (y -(z j -z k )) • e iΓ l Q a (y -(z l -z k )) .
Therefore,

E P = k e iΓ k Ψ k (y -z k ) where Ψ k = m a k • MQ a + i ża ′ (z)ρ + G k + Ψ Qa . (2.15)
step 3 Nonlinear interaction estimates. In order to estimate the various terms in (2.15), we will use the following interaction estimates: let ω, ω ∈ R 2 , |ω| ≫ 1, | ω| ≫ 1, let q ≥ 0, then:

ˆ(1 + |y| q )Q 3 (y)Q(y -ω)dy |ω| -1 2 e -|ω| .
(2.16)

ˆ(1 + |y| q )Q 2 (y)Q(y -ω)Q(y -ω)dy e -3 2 |ω| + e -3 2 | ω| .
(2.17)

ˆQ3 (y)Q(y -ω)dy -c Q I Q |ω| -1 2 e -|ω| |ω| -3 2 e -|ω| , (2.18)
with I Q given by (1.19).

Proof of (2.16). From (1.18), observe that

Q(y)Q(y -ω) (1 + |y|) -1 2 (1 + |y -ω|) -1 2 e -|y| e -|ω|+|y| |ω| -1 2 e -|ω| . (2.19)
Thus,

ˆ(1 + |y| q )Q 3 (y)Q(y -ω)dy |ω| -1 2 e -|ω| ˆ(1 + |y| q )Q 2 (y)dy |ω| -1 2 e -|ω| .
Proof of (2.17). From (2.19),

ˆ(1 + |y| q )Q 2 (y)Q(y -ω)Q(y -ω)dy ˆ(1 + |y| q )Q 2 (y)Q 3 4 (y -ω)Q 3 4 (y -ω)dy e -3 4 |ω| e -3 4 | ω| ˆ(1 + |y| q )Q 1 2 (y)dy e -3 4 |ω| e -3 4 | ω| .
Proof of (2.18). First, using (1.18),

ˆ|y|> 3 4 |ω| Q 3 (y)Q(y -ω)dy e -9 4 |ω| ˆQ(y -ω)dy e -9 4 |ω| .
Second, for |y| < 3 4 |ω|, we use (1.18) to write

Q(y -ω) -c Q |y -ω| -1 2 e -|y-ω| |y -ω| -3 2 e -|y-ω| |ω| -3 2 e -|ω|+|y| .
In particular,

ˆ|y|< 3 4 |ω| Q 3 (y)Q(y -ω)dy -c Q ˆ|y|< 3 4 |ω| Q 3 (y)|y -ω| -1 2 e -|y-ω| dy |ω| -3 2 e -|ω| . Still for |y| < 3 4 |ω|, the expansion |y -ω| 2 = |ω| 2 -2y • ω + |y| 2 implies |y -ω| -1 2 -|ω| -1 2 |ω| -3 2 |y| and |y -ω| -|ω| + y • ω |ω| |ω| -1 |y| 2 .
Thus,

e -|y-ω| -e -|ω|+y• ω |ω| |ω| -1 |y| 2 e -|y-ω| + e -|ω|+y• ω |ω| |ω| -1 |y| 2 e -|ω| e |y| .
Therefore,

|y -ω| -1 2 e -|y-ω| -|ω| -1 2 e -|ω|+y• ω |ω| |ω| -3 2 (1 + |y| 2 )e -|ω| e |y| ,
and so

ˆ|y|< 3 4 |ω| Q 3 (y)|y -ω| -1 2 e -|y-ω| dy -|ω| -1 2 e -|ω| ˆ|y|< 3 4 |ω| Q 3 (y)e y• ω |ω| dy |ω| -3 2 e -|ω| ˆ(1 + |y| 2 )e -2|y| dy |ω| -3 2 e -|ω| .
Also, we see that

|ω| -1 2 e -|ω| ˆ|y|> 3 4 |ω| Q 3 (y)e y• ω |ω| dy |ω| -1 2 e -|ω| ˆ|y|> 3
4 |ω| e -2|y| dy e -2|ω| .

Since for all ω = 0 (see (1.19)), ) and the definition of a(z) in (2.5), we have, for some q > 0,

I Q = ˆQ3 (y)e
|Q a | |y| -1 2 e -|y| + |a(z)||y| q e -|y| (1 + |y|) -1 2 e -|y| + z 1 2 e -κz (1 + |y|) q e -|y| .
Moreover, for j = k, from the definition of κ in (1.16),

|z j -z k | = z|e k -e j | ≥ κz.
From this, it follows easily that for j = k,

|Q a (y)||Q a (y -(z k -z j ))| z -1 2 e -κz ,
which in light of the explicit formula (2.14) yields the control of G k in (2.10).

We now turn to the proof of (2.11) which requires a more careful analysis of the interaction terms. We first compute the main order of the contribution of

G (I) 1 to G 1 , iQ . For j = 2, . . . , K, e -iΓ 1 Q 2 a (y) e iΓ j Q a (y -(z j -z 1 )), iQ a = ˆQ3 a (y)Q a (y -z(e j -e 1
)) sin(Γ j (y -z(e j -e 1 )) -Γ 1 (y))dy.

First, by the decay of ρ (see (1.22)), (2.16) and the definition of a(z) in (2.5), we have

ˆ Q 3 a (y)Q a (y -z(e j -e 1 )) -Q 3 (y)Q(y -z(e j -e 1 )) dy |a|z 5 2 e -κz z 3 e -2κz .
Next, note that, since

Γ j = β j • y -b 4 |y| 2 , |sin(Γ j (y -z(e j -e 1 )) -Γ 1 (y)) -(Γ j (y -z(e j -e 1 )) -Γ 1 (y))| |Γ j (y -z(e j -e 1 ))| 2 + |Γ 1 (y)| 2 |β| 2 (|y| 2 + z 2 ) + |b| 2 (|y| 4 + z 4 ), and
(Γ j (y -z(e j -e 1 )) -Γ 1 (y)) + b 4 |z(e j -e 1 )| 2 |β|(|y| + z) + |b|(|y| 2 + |y|z).
Thus, using (2.16), ˆQ3 (y)Q(y -z(e j -e 1 )) sin(Γ j (y -z(e j -e 1 )) -Γ 1 (y)) + b 4 |z(e j -e 1 )| 2 dy

(|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z)z -1 2 e -κz
. Therefore, we have proved

e iΓ 1 Q 2 a (y) e iΓ j Q a (y -z(e j -e 1 )), iQ a + b 4 |z(e j -e 1 )| 2 ˆQ3 (y)Q(y -z(e j -e 1 )) (|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z)z -1 2 e -κz + z 3 e -2κz .
For j = 2 and j = K, we have |z(e j -e 1 ))| = κz, and so using (2.18),

e iΓ 1 Q 2 a (y) e iΓ j Q a (y -z(e j -e 1 )), iQ a + b 4 c Q I Q κ 3 2 z 3 2 e -κz (|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z)z -1 2 e -κz + z 3 e -2κz .
For K ≥ 4 and j = 3, . . . , K -1, we have |e j -e 1 | > κ ′ , for some κ ′ > κ. Thus the following bound follows from similar computations

e iΓ 1 Q 2 a (y) e iΓ j Q a (y -z(e j -e 1 )), iQ a (|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z 2 )z -1 2 e -κ ′ z + z 3 e -2κz .
Note that

2G (I) 1 + G (I) 1 , iQ a = G (I)
1 , iQ a . We finally bound the contribution of

G (II) 1 . For j = 1, l = 1 and l = j, e -2iΓ k Q a (y) e iΓ j Q a (y -(z j -z k )) e iΓ l Q a (y -(z l -z k )), iQ a = ˆQ2 a (y)Q a (y -(z j -z k ))Q a (y -(z l -z k )) × sin(Γ j (y -(z j -z k )) + Γ l (y -(z l -z k )) -2Γ k (y))dy.
By (2.17), the bound on |a| and |Γ j | |β|(|y| + z) + |b|(|y| 2 + z 2 ), this term is bounded by (|β|z + |b|z 2 )e -3 2 κz . Gathering these estimates, using the definition of the constant c a in (2.5) which takes into account the two different cases K = 2 and K ≥ 2 (for K = 2, the soliton P 1 has nonlinear interaction with only one other soliton, while for K ≥ 3, it has exactly two closest neighboring solitons, P 2 and P K ), we obtain finally

| G k , iQ a + κc a ρ, Q bz 3 2 e -κz | (|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z)z -1 2 e -κz + z 3 e -2κz ,
which completes the proof of (2.11).

2.2.

Formal resolution of the modulation system with forcing. From Lemma 3, we derive a simplified modulation system with forcing term and we determine one of its approximate solution that is relevant for the regime of Theorem 1. Moreover, we justify the special choice of function a(z) in (2.5). Formally, i.e. assuming that P is a solution of (1.1) up to error terms of lower order than the ones in (2.9) (making this rigorous will be the object of the bootstrap estimates in Sect. 4), we have the following bounds (

m a k is defined in (2.7)) | m a 1 | z -1 2 e -κz .
(2.20)

Indeed, (2.20) is obtained from (2.9)-(2.10) by projecting E P onto directions related to the generalized null space (1.23) (see Lemma 7 for rigorous computations). To simplify the discussion, we drop the equation of γ, which is not coupled with any other equation and has no influence on the regime. Next, we see that using the first line of m a 1 , i.e. |b + λ λ | z -1 2 e -κz , we can replace λ λ by -b in all the other estimates. Similarly, we insert the estimate on ż from the second line into the estimate for β. We obtain the following simplified system

|b + λ λ | + | ż -2β -bz| + | β + bβ| + | ḃ + b 2 -a| z -1 2 e -κz . (2.21)
It is easy to check the following estimates Lemma 4. Let (z app , λ app (s), β app , b app (s)) be such that

λ app (s) = log -1 (s), z -3 2 app (s)e κzapp(s) = κc a 2 s 2 , |β app (s)| s -1 log -3 2 (s), b app (s) = s -1 log -1 (s).
(2.22)

Then,

z app (s) ∼ 2 κ log(s), |b app + λapp λ app | = 0, | żapp -2β app -b app z app | s -1 log -1 2 (s), |a(z app ) + s -2 log -1 (s)| s -2 log -3 2 (s), | ḃapp + b 2 app -a(z app )| s -2 log -3 2 (s).
(2.23)

The above estimates mean that (2.22) is a reasonnable guess for the first order asymptotics as s → +∞ of some particular solutions of (2.21) (we refer to Sect. 3.4 for a rigorous integration of (2.21)). Note that we do not actually determine the main order of β(s); to do this, more interaction computations would be necessary. However, since | β + bβ| z -1 2 e -κz , formally, we obtain | β| s -2 log -2 (s), which justifies a bootstrap on β(s) of the form |β(s)| ≪ s -1 log -2 (s). Note also that there exist solutions of (2.22) with different asymptotics, corresponding to (NLS) solutions like v(t) of Corollary 2.

To complete this formal discussion, we justify the choice of a(z) in (2.5) in the regime given by (2.22). Indeed, projecting Ψ 1 onto the direction iQ a , from (2.11), we obtain at the leading order

| ża ′ (z) -κc a bz 3 2 e -κz | (|β| 2 z 2 + |b| 2 z 4 + |β|z + |b|z)z -1 2 e -κz + z 3 e -2κz . (2.24)
In the regime suggested by (2.22), since |β| ≪ |b|z, we have bz ∼ ż and thus, simplifying ż, we obtain |a ′ (z) -κc a z 1 2 e -κz | z -1 2 e -κz , which justifies the definition (2.5) by integrating in z.

Modulation of the approximate solution.

We state a standard modulation result around P. We restrict ourselves to the case of solutions invariant by the rotation preserving P. Denote by τ K the rotation of center 0 and angle 2π K in R 2 . Since Q and ρ are radial, by definition of P k and β k , z k in (2.3) and (2.4), we have for k ∈ {1, . . . , K -1}, P k (y) = P k+1 (τ K y) and P K (y) = P 1 (τ K y). In particular, it follows that P(τ K y) = P(y), i.e. P is invariant by the rotation τ K . Note also that equation (1.1) is invariant by rotation. In particular, if a solution of (1.1) is invariant by the rotation τ K at some time, then it is invariant by rotation at any time. In this context, the following modulation result relies on a standard argument based on the Implicit Function Theorem (see e.g. Lemma 2 in [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF]) and we omit its proof.

Lemma 5 (Modulation around P). Let I be some time interval. Let u ∈ C(I, H 1 (R 2 )) be a solution of (1.1) invariant by the rotation τ K and such that

sup t∈I e -iγ(t) λ(t)u(t, λ(t) . ) - k Q( . -e k z(t)) H 1 < δ (2.25)
for some λ(t) > 0, γ(t) ∈ R 2 , z(t) > δ -1 , where δ > 0 is small enough. Then, there exist a

C 1 function p = (λ, z, γ, β, b) : I → (0, ∞) 2 × R 3 ,
such that, for P(t, y) = P(y; z(t), b(t), β(t)) as defined in (2.6), the solution u(t) decomposes on I as

u(t, x) = e iγ(t) λ(t) (P + ε)(t, y), y = x λ(t) , (2.26)
where for all t ∈ I,

|b(t)| + |β(t)| + ε(t) H 1 + |z(t)| -1 δ, (2.27)
and, setting ε(t, y) = e iΓ 1 η 1 (t, y -z 1 ),

η 1 (t), |y| 2 Q = | η 1 (t), yQ | = η 1 (t), iρ = | η 1 (t), i∇Q | + η 1 (t), iΛQ = 0. (2.28)
Moreover, ε is also invariant by the rotation τ K .

Note that the choice of the special orthogonality conditions (2.28) is related to the generalized null space of the linearized equation around Q, (1.23) and to the coercivity property (1.24). See the proof of Lemma 7 for a technical justification of these choices (see also [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF]).

Backwards uniform estimates

In this section, we prove uniform estimates on particular backwards solutions. The key point is to carefully adjust their final data to obtain uniform estimates corresponding to the special regime of Theorem 1 and Lemma 4.

Let (λ in , z in , b in ) ∈ (0, +∞) 2 × R to be chosen with λ in ≪ 1, z in ≫ 1, |b in | ≪ 1. Let u(t)
for t ≤ 0 be the solution of (1.1) with data (see (2.6))

u(0, x) = 1 λ in P in x λ in where P in (y) = P(y; (z in , b in , 0)) (3.1)
(we arbitrarily fix γ in = β in = 0). Note that u(0) satisfies (2.25) and, by continuity of the solution of (1.1) in H 1 , it exists and satisfies (2.25) on some maximal time interval (t mod , 0], where t mod ∈ [-∞, 0). Note also that by invariance by rotation of equation (1.1), u(t) is invariant by the rotation τ K . On (t mod , 0], we consider ( p, ε) the decomposition of u defined from Lemma 5. For s in ≫ 1, we normalize the rescaled time s as follows, for t ∈ (t mod , 0],

s = s(t) = s in - ˆ0 t dτ λ 2 (τ ) . (3.2) Observe from (3.1) that λ(s in ) = λ in , b(s in ) = b in , z(s in ) = z in , γ(s in ) = 0, β(s in ) = 0, ε(s in ) ≡ 0. (3.3)
Proposition 6 (Uniform backwards estimates). There exists s 0 > 10 such that for all s in > s 0 , there exists a choice of parameters (λ in , z in , b in ) with

2 κc a 1 2 (z in ) -3 4 e κ 2 z in -s in < s in log -1 2 (s in ), λ in = log -1 (s in ), b in = 2c a κ 1 2 (z in ) -1 4 e -κ 2 z in , (3.4)
such that the solution u of (1.1) corresponding to (3.1) exists and satisfies (2.25) on the rescaled interval of time [s 0 , s in ], the rescaled time s being defined in (3.2). Moreover, the decomposition of u given by Lemma 5 on [s 0 , s in ] u(s, x) = e iγ(s) λ(s)

(P + ε)(s, y), y = x λ(s) ,
satisfies the following uniform estimates, for all s ∈ [s 0 , s in ],

z(s) - 2 κ log(s) log(log(s)), λ(s) -log -1 (s) log -3 2 (s), b(s) -s -1 log -1 (s) + |β(s)| + ε(s) H 1 s -1 log -3 2 (s), |a(s)| s -2 log -1 (s). (3.5)
The key point in Proposition 6 is that s 0 and the constants in (3.5) are independent of s in as s in → +∞. Observe that estimates (3.5) match the discussion of Sect. 2.2.

The rest of this section is devoted to the proof of Proposition 6. The proof relies on a bootstrap argument, integration of the differential system of geometrical parameters and energy estimates. We estimate ε by standard energy arguments in the framework of multibubble solutions. The particular regime of the geometrical parameters does not create any further difficulty. On the contrary, the special behavior b(s) ∼ s -1 log -1 (s) simplifies this part of the proof (see step 2 of the proof of Proposition 8). We control the geometrical parameters of the bubbles in the bootstrap regime adjusting the final data (λ in , z in , b in ).

3.1. Bootstrap bounds. The proof of Proposition 6 follows from bootstrapping the following estimates, chosen in view of the formal computations in Sect. 2.2,

2 κc a 1 2 z -3 4 e κ 2 z -s ≤ s log -1 2 (s), 1 2 s -1 log -1 (s) ≤ b(s) ≤ 2s -1 log -1 (s), |β(s)| ≤ s -1 log -3 2 (s), ε(s) H 1 ≤ s -1 log -3 2 (s). (3.6)
Note that the estimate on z in (3.6) immediately implies that, for s large

e -κz s -2 log -3 2 (s), z(s) - 2 κ log(s) log(log(s)), |a(s)| s -2 log -1 (s). (3.7)
For s 0 > 10 to be chosen large enough (independent of s in ), and all s in ≫ s 0 , we define

s * = inf{τ ∈ [s 0 , s in ]; (3.6) holds on [τ, s in ]}. (3.8)
3.2. Control of the modulation equations. We claim the following bounds on the modulation system m a 1 and on the error E P given by (2.7), (2.8)-(2.9) in the bootstrap regime (3.6). Lemma 7 (Pointwise control of the modulation equations and the error). The following estimates hold on

[s * , s in ]. | m a 1 (s)| s -2 log -2 (s). (3.9) | η 1 (s), Q | s -2 log -2 (s), (3.10)
| ż -bz| s -1 log -1 (s), | β| + | ḃ -a| s -2 log -2 (s).

(3.11) Moreover, for all s ∈ [s * , s in ], for all y ∈ R 2 ,

|E P (s, y)| + |∇E P (s, y)| s -2 log -2 (s) k Q 1/2 (y -z k (s)).
(3.12)

Proof of Lemma 7. The proofs of the first two estimates are to be combined. Since ε(s in ) ≡ 0, we may define

s * * = inf{s ∈ [s * , s in ]; | η 1 (τ ), Q | ≤ C * * τ -2 log -2 (τ ) holds on [s, s in ]},
for some constant C * * > 0 to be chosen large enough. We work on the interval [s

i ε + ∆ε -ε + |P + ε| 2 (P + ε) -|P| 2 P -i λ λ Λε + (1 -γ)ε + E P = 0. (3.13)
By rotation symmetry (see Lemma 5) it is enough to understand the orthogonality for η 1 . Thus, we change the space variable to match the one of the bubble P 1 . Recall that we have defined ε(s, y) = e iΓ 1 η 1 (s, y -z 1 ). Denote P(s, y) = e iΓ 1 P 1 (s, y -z 1 ), E P (s, y) = e iΓ 1 E P 1 (s, y -z 1 ).

We rewrite the equation of ε into the following equation for η 1 (see also step 1 of the proof of Lemma 3)

i η1 + ∆η 1 -η 1 + (|P 1 + η 1 | 2 (P 1 + η 1 ) -|P 1 | 2 P 1 ) + m 0 1 • Mη 1 + E P 1 = 0, (3.14)
Recall also that η 1 satisfies the orthogonality conditions (2.28). step 2 General null space like computation. Let A(y) and B(y) be two real-valued functions in Y. We claim the following estimate on [s * * , s in ]

d ds η 1 , A + iB -η 1 , iL -A -L + B -m a 1 • MQ, iA -B s -2 log -2 (s) + s -1 | m a 1 |. (3.15) We compute from (3.14), d ds η 1 , A + iB = η1 , A + iB = i η1 , iA -B = -∆η 1 + η 1 -(2Q 2 η 1 + Qη 1 ), iA -B -|P 1 + η 1 | 2 (P 1 + η 1 ) -|P 1 | 2 P 1 -2Q 2 η 1 -Q 2 η 1 , iA -B -m 0 1 • Mη 1 , iA -B -E P 1 , iA -B .
First, since A and B are real-valued, we have

-∆η 1 + η 1 -(2Q 2 η 1 + Qη 1 ), iA -B = η 1 , iL -A -L + B .
Second, note that

|P 1 + η 1 | 2 (P 1 + η 1 ) -|P 1 | 2 P 1 -2Q 2 η 1 -Q 2 η 1 = 2(|P 1 | 2 -Q 2 )η 1 + (P 2 1 -Q 2 )η 1 + 2P 1 |η 1 | 2 + P 1 η 2 1 + 2P 1 |η 1 | 2 + |η 1 | 2 η 1
, and recall the expression of P 1

P 1 = Q + aρ + K k=2 e i(Γ k (y-(z k -z 1 ))-Γ 1 (y)) Q a (y -(z k -z 1 )).
Therefore, using A, B ∈ Y, (3.6)-(3.7) and |z k -z j | ≥ κz, for k = j, we have, for some q > 0,

| (|P 1 | 2 -Q 2 )η 1 , iA -B | + | (P 2 1 -Q 2 )η 1 , iA -B | (|a| + z q e -κz ) η 1 L 2 s -3 log q (s). Next, | 2P 1 |η 1 | 2 + P 1 η 2 1 + 2P 1 |η 1 | 2 , iA -B | ε 2 L 2 s -2 log -3 (s), | |η 1 | 2 η 1 , iA -B | |ε| 3 , |A| + |B| ε 3 H 1 s -3 log -9 2 (
s). Third, we have, using (3.6)-(3.7), integration by parts,

| m 0 1 • Mη 1 , iA -B | m a 1 • Mη 1 , iA -B + ( m a 1 -m 0 1 ) • Mη 1 , iA -B s -1 log -3 2 (s)| m a 1 | + s -3 log -5 2 (s)
. Finally, we claim the following estimate, which is enough to complete the proof of (3.15).

E P 1 , iA -B -m a 1 • MQ, iA -B s -2 log -2 (s) + s -1 | m a 1 |. (3.16)
Indeed, recall the expression of E P 1 (from (2.8)-(2.9))

E P 1 = Ψ 1 + K k=2 e i(Γ k (y-(z k -z 1 ))-Γ 1 (y)) Ψ k (y -(z k -z 1 )). = m a 1 • MQ a + i ża ′ (z)ρ + G 1 + Ψ Qa + K k=2 e i(Γ k (y-(z k -z 1 ))-Γ 1 (y)) Ψ k (y -(z k -z 1 )). First, since Q a = Q + aρ, by (3.6)-(3.7), m a 1 • M(Q a -Q), iA -B |a|| m a 1 | s -2 log -1 (s)| m a 1 |. Second, from (3.6)-(3.7), | ża ′ (z)ρ, iA -B | |a ′ (z)|| ż| s -2 log -1 (s)(| m a 1 | + |β| + | λ λ |z) s -2 log -1 (s)((z + 1)| m a 1 | + |β| + |b|z) s -2 | m a 1 | + s -3 log -1 (s).
(3.17

| G 1 , iA -B | G 1 L ∞ z -1 2 e -κz s -2 log -2 (s). Fourth, from (2.10) and (3.6)-(3.7), | Ψ Qa , iA -B | Ψ Qa L ∞ |a| 2 s -4 log -2 (s).
Last, since A, B ∈ Y, for k ≥ 2, we have and, proceeding as before for the other terms in Ψ k , we obtain

| e i(Γ k (y-(z k -z 1 ))-Γ 1 (y)) ( m a k • MQ a (. -(z k -z 1 ))), iA -B | s -1 | m a k |,
| e i(Γ k (y-(z k -z 1 ))-Γ 1 (y)) Ψ k (y -(z k -z 1 )), iA -B | s -1 | m a k | + s -2 log -2 (s)
. The proof of (3.16) is complete. step 3 Modulation equations. We now use (2.28) and (3.15) to control the modulation vector m a 1 . Using (3.15), we draw one by one the consequences of the orthogonality relations (2.28). Note that the special orthogonality conditions (2.28), related to cancellations (see (1.23)) are crucial in these computations.

η 1 , |y| 2 Q = 0. Let A = |y| 2 Q and B = 0. Since L -(|y| 2 Q) = -4ΛQ, η 1 , iΛQ = 0 and m a 1 • MQ, i|y| 2 Q = -(b + λ λ ) ΛQ, |y| 2 Q = c 1 (b + λ λ ), where c 1 = 0, we obtain b + λ λ s -2 log -2 (s) + s -1 | m a 1 |. (3.18) η 1 , yQ = 0. Let A = yQ and B = 0. Since L -(yQ) = -2∇Q, η 1 , i∇Q = 0 and m a 1 • MQ, iyQ = -( ż -2β + λ λ z) ∇Q, yQ = c 2 ( ż -2β + λ λ z)
, where c 2 = 0, we obtain

ż -2β + λ λ z s -2 log -2 (s) + s -1 | m a 1 |. (3.19) η 1 , iρ = 0. Let A = 0 and B = ρ. Since L + ρ = |y| 2 Q, η 1 , |y| 2 Q = 0 and m a 1 • MQ, |y| 2 Q = c 3 ( γ -1 + β 2 - λ λ βz -β ż) + c 4 ( ḃ + b 2 -2b(b + λ λ ) -a),
where c 3 , c 4 = 0, we obtain, for some c,

( γ -1 + β 2 - λ λ βz -β ż) + c( ḃ + b 2 -2b(b + λ λ ) -a) s -2 log -2 (s) + s -1 | m a 1 |. (3.20) η 1 , i∇Q = 0. Let A = 0 and B = ∇Q. Since L + ∇Q = 0, and m a 1 • MQ, ∇Q = c 5 ( β -λ λ β + b 2 ( ż -2β + λ λ z)
, where c 5 = 0, we obtain

β - λ λ β + b 2 ( ż -2β + λ λ z) s -2 log -2 (s) + s -1 | m a 1 |. ( 3
s * * , | η 1 , Q | C * * s -2 log -2 (s). Moreover, m a 1 • MQ, ∇Q = c 6 ( ḃ + b 2 -2b(b + λ λ ) -a)
, where c 6 = 0, so that we obtain

ḃ + b 2 -2b(b + λ λ ) -a C * * s -2 log -2 (s) + s -1 | m a 1 |. ( 3
P(s) L 2 -P in L 2 s -2 log -2 (s). (3.24)
Note that the implicit constant on the right-hand side does not depend on C * * . By the definition (2.9) of E P , we have 1 2

d ds P 2 L 2 = i Ṗ, iP = E P , iP .
In view of the formula for E P (2.9), and the definition of P = j P j , (3.24) follows by integration of the following estimate: for all j, k ∈ {1, . . . , K},

| e iΓ k Ψ k (y -z k ), i e iΓ j Q a (y -z j ) | s -3 log -2 (s).
(3.25)

Proof of (3.25). We start by proving (3.25) in the case j = k = 1. From (2.9):

e iΓ k Ψ k (y -z k ), i e iΓ k Q a (y -z k ) = Ψ 1 , iQ a = m a 1 • MQ a + i ża ′ (z)ρ + G 1 + Ψ Qa , iQ a . Note that MQ, iQ = 0. Thus, by (3.23), (3.6)-(3.7), | m a 1 • MQ a , iQ a | |a|| m a 1 | C * * s -4 log -3 (s) s -3 log -3 (s)
. Next, we claim the following estimate, which justifies the special choice of a(z) done in (2.5) (see also Sect. 2.2) Finally, since Q a and Ψ Qa given by (2.12) are real-valued, we have the cancellation

| i ża ′ (z)ρ + G 1 , iQ a | s -3 log -2 (s). ( 3
Ψ Qa , iQ a = 0.
The collection of above estimates concludes the proof of (3.25) for j = k = 1. We now prove (3.25) in the case k = 1 and j ∈ {2, . . . , K}. Note that

| e iΓ k Ψ k (y -z k ), i e iΓ j Q a (y -z j ) | = | m a 1 • MQ a + i ża ′ (z)ρ + G 1 + Ψ Qa , ie i(Γ j (y-(z j -z k ))-Γ k (y)) Q a (y -(z j -z k )) | First, by (3.23), for some q > 0, | m a 1 • MQ a , ie i(Γ j (y-(z j -z k ))-Γ k ) Q a (y -(z j -z k )) | | m a 1
|z q e -κz C * * s -4 log q (s) s -3 log -2 (s). Second, using similar arguments, for some q > 0,

| i ża ′ (z)ρ + G 1 + Ψ Qa , ie i(Γ j (y-(z j -z k ))-Γ k ) Q a (y -(z j -z k )) | s -4 log q (s).
The collection of above estimates concludes the proof of (3.24). step 5 Proof of (3.10). The conservation of mass for the solution u and (3.1) imply:

u(s) L 2 = u(s in ) L 2 = P in L 2 . By (2.26), ε(s), P = 1 2 u(s) 2 L 2 -P(s) 2 L 2 -ε(s) 2 L 2 .
Therefore, using (3.6)-(3.7) and (3.24), we obtain | ε(s), P | s -2 log -2 (s). Now, we use the symmetry ε, P k = ε, P j = K -1 ε(s), P for all j, k ∈ {1, . . . , K}. Moreover, by (3.6) (3.10). In particular, choosing C * * large enough, we have s * * = s * . step 6 Conclusion. The estimate (3.11) is a direct consequence of (3.9) and (3.6)-(3.7). We now turn to the proof of (3.12). Using (3.9), (3.6)-(3.7) and (1.22),

-(3.7), ε(s), P 1 = η 1 , Q a = η 1 , Q + O(|a| ε L 2 ) = η 1 , Q + O(s -3 log -5 2 (s)). Gathering this information, we obtain | η 1 , Q | s -2 log -2 (s), i.e. estimate
| ża ′ (z)ρ| Q 1 2 (|b|z + s -1 log -1 (s))s -2 log -1 (s) Q 1 2 s -3 log -1 (s). By (3.9), | m a 1 • MQ a | Q 1 2 s -2 log -2 (s). Next, by the definition of G k in (2.14), the decay |ρ| Q 7 8 (see (1.22)) and |e k -e 1 | ≥ κ for k = 1, |G 1 | Q 1 2 K k=2 Q 5 4 (y)Q(y -z(e k -e 1 )) + |a|Q 5 4 (y)Q 7 8 (y -z(e k -e 1 )) Q 1 2 (z -1 2 e -κz + s -2 e -κ 2 z ) Q 1 2 s -2 log -2 (s).
Finally, by the definition of Ψ Qa in (2.12),

|Ψ Qa | Q 1 2 |a| 2 Q 1 2 s -4 log -2 (s).
The same estimates hold for ∇E, which finishes the proof of (3.12).

Energy functional. Consider the nonlinear energy functional for ε

H(s, ε) = 1 2 ˆ |∇ε| 2 + |ε| 2 - 1 2 |P + ε| 4 -|P| 4 -4|P| 2 Re (εP) .
Pick a smooth function χ : [0, +∞) → [0, ∞), non increasing, with χ ≡ 1 on [0, 1 10 ], χ ≡ 0 on [ 1 8 , +∞). We define the localized momentum:

J = k J k , J k (s, ε) = b Im ˆ(z k • ∇ε)εχ k , χ k (s, y) = χ log -1 (s)|y -z k (s)| .

Finally, set

F(s, ε) = H(s, ε) -J(s, ε). The functional F is coercive in ε at the main order and it is an almost conserved quantity for the problem. Proposition 8 (Coercivity and time control of the energy functional). For all s ∈ [s * , s in ],

F(s, ε(s)) ε(s) 2 H 1 + O(s -4 log -4 (s)), (3.30)
and d ds [F(s, ε(s))] s -2 log -2 (s) ε(s) H 1 + s -1 log -1 (s) ε(s) 2 H 1 . (3.31)
Proof of Proposition 8. step 1 Coercivity. The proof of the coercivity (3.30) is a standard consequence of the coercivity property (1.24) around one solitary wave with the orthogonality properties (2.28), (3.10), and an elementary localization argument. Hence we briefly sketch the argument. First, using the coercivity property (1.24) and the orthogonality properties (2.28), (3.10) and localization arguments, we have

H(s, ε) ε 2 H 1 + O(s -4 log -4 (s)). (3.32)
Note that the error term in O(s -4 log -4 (s)) is due to the bound (3.10). We refer to the proof of Lemma 4.1 in Appendix B of [START_REF] Martel | Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations[END_REF] for a similar proof. Second, we note that by

(3.6)-(3.7), |J(s, ε)| |b|z ε 2 H 1 s -1 ε 2 H 1
, and (3.30) follows. step 2 Variation of the energy. We estimate the time variation of the functional H and claim: for all s ∈ [s * , s in ],

d ds [H(s, ε(s))] - k żk • ∇P k , 2|ε| 2 P k + ε 2 P k s -2 log -2 (s) ε(s) H 1 + s -1 log -1 (s) ε(s) 2 H 1 . (3.33) The time derivative of s → H(s, ε(s)) splits into two parts d ds [H(s, ε(s))] = D s H(s, ε(s)) + D ε H(s, ε(s)), εs ,
where D s denotes differentiation of H with respect to s and D ε denotes differentiation of H with respect to ε. First compute:

D s H = -Ṗ, |P + ε| 2 (P + ε) -|P 2 |P -(2ε|P| 2 + εP 2 ) .
Observe that by the definition of P k in (2.3),

Ṗk = -żk • ∇P k + i( βk • (y -z k ) -ḃ 4 |y -z k | 2 )P k + ża ′ (z)ρ k where ρ k = e iΓ k ρ (y -z k ). By (3.11), (3.6)-(3.7) and (2.5), | βk | + | ḃ| + | ża ′ (z)| s -2 log -2 (s). Since ˆ |P + ε| 2 (P + ε) -|P 2 |P -(2ε|P| 2 + εP 2 ) ε 2 H 1 , we obtain i( βk • (y -z k ) -ḃ 4 |y -z k | 2 )P k + ża ′ (z)ρ k , |P + ε| 2 (P + ε) -|P 2 |P -(2ε|P| 2 + εP 2 ) s -2 log -2 (s) ε 2 H 1 . Next, note that |P + ε| 2 (P + ε) -|P 2 |P -(2ε|P| 2 + εP 2 ) = 2|ε| 2 P + ε 2 P + |ε| 2 ε.
żk • ∇P k , |ε| 3 s -1 ε 3 H 1 s -2 log -3 2 (s) ε 2 H 1 .
For j = k, since e -κz s -2 by (3.6)-(3.7) and the decay properties of P k , P j ,

żk • ∇P k , 2|ε| 2 P j + ε 2 P j |s| -3 ε 2 H 1 .
Gathering these computations, we have obtained

D s H(s, ε) = k żk • ∇P k , 2|ε| 2 P k + ε 2 P k + O(s -2 ε 2 H 1 ). (3.34)
Second,

D ε H(s, ε) = -∆ε + ε -|P + ε| 2 (P + ε) -|P| 2 P , so that the equation (3.13) of ε rewrites i ε -D ε H(s, ε) -i λ λ Λε + (1 -γ)ε + E P = 0.
In particular,

D ε H(s, ε), ε = iD ε H(s, ε), i ε = λ λ D ε H(s, ε), Λε -(1 -γ) iD ε H(s, ε), ε -iD ε H(s, ε), E P .
We recall that

-∆ε, Λε = ∇ε 2 , ε, Λε = 0, |ε| 2 ε, Λε = 1 2 ˆ|ε| 4 ,
and thus, using also (3.6)-(3.7), (1.3), and the decay properties of Q,

| D ε H(s, ε), Λε | ε 2 H 1 + ε 4 H 1 ε 2 H 1 .
In particular, from (3.9) and (3.6)-(3.7), we deduce

λ λ D ε H(s, ε), Λε s -1 log -1 (s) ε 2 H 1 .
Note that the estimate on b in (3.6)-(3.7) implies |b| s -1 log -1 (s) ≪ s -1 which avoids the use of virial localized identities (as in [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF][START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF]) to control the above term. By (3.9) and (3.6)-(3.7), we estimate

|(1 -γ) iD ε H(s, ε), ε | s -2 ε 2 H 1 .
Finally, integrating by parts, using (3.12) and (3.6)-(3.7), we have

| iD ε H(s, ε), E P | |∇ε|, |∇E P | + |ε| + |ε| 3 , |E P | s -2 log -2 (s) ε H 1 .
The collection of above estimates finishes the proof of (3.33).

step 3 Variation of the localized momentum. We now claim: for all s ∈ [s * , s in ],

d ds [J(s, ε(s))] -b k z k • ∇P k , 2|ε| 2 P k + ε 2 P k s -2 log -2 (s) ε(s) H 1 + s -1 log -1 (s) ε(s) 2 H 1 . (3.35)
Indeed, we compute, for any k, d ds

[J k (s, ε(s))] = ḃ Im ˆ(z k • ∇ε)εχ k + b Im ˆ( żk • ∇ε)εχ k + b Im ˆ(z k • ∇ε)ε χk + b i ε, z k • (2χ k ∇ε + ε∇χ k ) .
By (3.9) and (3.6)-(3.7), we have

ḃ Im ˆ(z k • ∇ε)εχ k + b Im ˆ( żk • ∇ε)εχ k s -2 ε 2 H 1 .
Note that by direct computations, (3.9) and (3.6)-(3.7),

| χk | (s -1 log -1 (s)|y -z k | + | żk |) log -1 (s)|χ ′ (log -1 (s)(y -z k (s)))| s -1 log -1 (s)
and so, by

(3.6)-(3.7), b Im ˆ(z k • ∇ε)ε χk s -2 log -2 (s) ε 2 H 1 .
Now, we use the equation (3.13) of ε to estimate b i ε, z k • (2χ k ∇ε + ε∇χ k ) . By integration by parts, we check the following

∆ε, 2(z k • ∇ε)χ k = ˆ|∇ε| 2 (z k • ∇χ k ) -2 (∇ε • ∇χ k ), (z k • ∇ε) , ∆ε, ε(z k • ∇χ k ) = -ˆ|∇ε| 2 (z k • ∇χ k ) + 1 2 ˆ|ε| 2 (z k • ∇(∆χ k )).
Thus,

∆ε, z k • (2χ k ∇ε + ε∇χ k ) = -2 (∇ε • ∇χ k ), (z k • ∇ε) + 1 2 ˆ|ε| 2 (z k • ∇(∆χ k )).
(z k • ∇(∆χ k )) s -1 log -3 (s) ε 2 H 1 .
In conclusion for term ∆ε in the equation of ε, we obtain

|b ∆ε, z k • (2χ k ∇ε + ε∇χ k ) | s -1 log -1 (s) ε 2 H 1 .
For the mass term in the equation of ε, we simply check by integration by parts that

ε, z k • (2χ k ∇ε + ε∇χ k ) = 0.
We also check that

iΛε, z k • (2χ k ∇ε + ε∇χ k ) = 2 iε, (z k • ∇ε)χ k + i(y • ∇ε), ε(z k • ∇χ k ) ,
|b| λ λ | iΛε, z k • (2χ k ∇ε + ε∇χ k ) | s -2 log -1 (s) ε 2 H 1 .
Next, from (3.12),

|b E P , z k • (2χ k ∇ε + ε∇χ k ) | s -3 log -2 (s) ε H 1 . Now,
P + |ε| 2 ε, z k • (2χ k ∇ε + ε∇χ k ) s -1 ε 3 H 1 s -2 log -3 2 (s) ε 2 H 1 . Second, since |b| s -1 log -1 (s), |z k | log(s) and |∇χ k | log -1 (s), b 2ε|P| 2 + εP 2 , ε(z k • ∇χ k ) s -1 log -1 (s) ε 2 H 1 .
Third, by the decay property of Q and the definition of

χ k , b 2ε |P| 2 - j |P j | 2 + ε P 2 - j P 2 j , (z k • ∇ε)χ k s -2 ε 2 H 1 ,
and, for

j = k, b 2ε|P j | 2 + εP 2 j , (z k • ∇ε)χ k s -2 ε 2 H 1 .
Finally, we compute by integration by parts,

2ε|P k | 2 + εP 2 k , (z k • ∇ε)χ k = -z k • ∇P k , 2|ε| 2 P k + ε 2 P k - 1 2 Re ˆ 2|ε| 2 P 2 k + ε 2 |P k | 2 (z k • ∇χ k ) . As before, b Re ˆ 2|ε| 2 P 2 k + ε 2 |P k | 2 (z k • ∇χ k ) s -1 log -1 (s) ε 2 H 1 .
The collection of above bounds concludes the proof of (3.35). step 3 Conclusion. Recall that, by (3.11), | żk -bz k | s -1 log -1 (s), and so

( żk -bz k ) • ∇P k , 2|ε| 2 P k + ε 2 P k s -1 log -1 (s) ε 2
H 1 , and (3.31) now follows from (3.33), (3.35). This concludes the proof of Proposition 8.

3.4.

End of the proof of Proposition 6. We close the bootstrap estimates (3.6) and prove (3.5). step 1 Closing the estimates in ε. By (3.31) in Proposition 8 and then (3.6)-(3.7), we have

d ds [F(s, ε(s))] s -2 log -2 (s) ε H 1 + s -1 log -1 (s) ε 2 H 1 s -3 log -7 2 (s).
Thus, by integration on [s, s in ] for any s ∈ [s * , s in ], using ε(s in ) = 0 (see (3.3)), we obtain

|F(s, ε(s))| s -2 log -7 2 (
s). By (3.30) in Proposition 8, we obtain

ε(s) 2 H 1 s -2 log -7
2 (s). Therefore, for s 0 large enough, for all s ∈ [s * , s in ],

ε(s) 2 H 1 ≤ 1 2 s -2 log -3 (s),
which strictly improves the estimate on ε 2 H 1 in (3.6).

step 2 Closing the parameter estimates. First, note that from (3.11), | β| s -2 log -2 (s). Together with the choice β(s in ) = β in = 0 (see (3.4)), direct integration in time gives, for all s ∈ [s * , s in ], |β(s)| s -1 log -2 (s). For s 0 large enough, we obtain, for all s ∈ [s * , s in ],

|β(s)| < 1 2 s -1 log -3 2 (s),
which strictly improves the estimate on β(s) in (3.6). Second, recall from (3.11), (3.7) and the definition of a(z) in (2.5), for all s ∈ [s * , s in ],

ḃ + c a z

1 2 e -κz s -2 log -2 (s), żz -1 -b s -1 log -2 (s). Since | ḃ| s -2 log -1 (s) and | żz -1 | s -1 log -1 (s), it follows that ḃb + c a żz -1 2 e -κz s -3 log -3 (s).
Integrating on [s, s in ] for any s ∈ [s * , s in ), using the special relation between b in and z in fixed in (3.4)

b 2 (s in ) = 2c a κ z -1 2 (s in )e -κz(s in ) , b(s in ) > 0, we obtain b 2 - 2c a κ z -1 2 e -κz s -2 log -3 (s) + ˆsin s żz -3 2 e -κz ds ′ s -2 log -3 (s), b(s) > 0. (3.36)
From (3.6)-(3.7) and (3.7), we have

2c a κ z -1 2 e -κz -s -2 log -2 (s) s -2 log -5 2 s.
Therefore, the following estimate on b(s) follows from (3.36)

b 2 -s -2 log -2 (s) s -2 log -5 2 (s). This implies, for all s ∈ [s * , s in ], b -s -1 log -1 (s) s -1 log -3 2 (s), (3.37
- 2c a κ 1 2 z -1 4 e -κ 2 z + żz -1 - 2c a κ 1 2 z -1 4 e -κ 2 z s -1 log -2 (s). (3.38)
Using z log -1 (s), we obtain

d ds z -3 4 e κ 2 z - κc a 2 1 2 log -1 (s) + żz -7 4 e κ 2 z log -1 (s). (3.39)
We need to adjust the initial choice of z(s in ) = z in through a topological argument (see [START_REF] Côte | Construction of multi-soliton solutions for the L 2 -supercritical gKdV and NLS equations[END_REF] for a similar argument). We define ζ and ξ the following two functions on According to (3.6), our objective is to prove that there exists a suitable choice of

[s * , s in ] ζ(s) = 2 κc a 1 2 z -3 4 e κ 2 z , ξ(s) = (ζ(s) -s) 2 s -2 log(s). ( 3
ζ(s in ) = ζ in ∈ [s in -s in log -1 2 (s in ), s in + s in log -1 2 (s in )],
so that s * = s 0 . Assume for the sake of contradiction that for all ζ ♯ ∈ [-1, 1], the choice

ζ in = s in + ζ ♯ s in log -1 2 (s in ) leads to s * = s * (ζ ♯ ) ∈ (s 0 , s in ).
Since all estimates in (3.6) except the one on z(s) have been strictly improved on [s * , s in ], it follows from s * (ζ ♯ ) ∈ (s 0 , s in ) and continuity that

|ζ(s * (ζ ♯ )) -s * | = s * log -1 2 s * i.e. ζ(s * (ζ ♯ )) = s * ± s * log -1 2 s * .
We need a transversality condition to reach a contradiction. We compute:

ξ(s) = 2(ζ(s) -s)( ζ(s) -1)s -2 log(s) -(ζ(s) -s) 2 (2s -3 log(s) -s -3 ). At s = s * , this gives | ξ(s * ) + 2(s * ) -1 | (s * ) -1 log -1 2 (s * ). Thus, for s 0 large enough, ξ(s * ) < -(s * ) -1 . (3.42)
Define the function Φ by

Φ : ζ ♯ ∈ [-1, 1] → (ζ(s * ) -s * )(s * ) -1 log 1 2 (s * ) ∈ {-1, 1}.
A standard consequence of the transversality property (3.42) is the continuity of the function

ζ ♯ ∈ [-1, 1] → s * (ζ ♯ ).
In particular, the function Φ is also continuous from [-1, 1] to {1, -1}. Moreover, for ζ ♯ = -1 and ζ ♯ = 1, ξ(s * ) = 1 and ξ(s * ) < 0 by (3.42) and so in these cases s * = s in . Thus, Φ(-1) = -1 and Φ(1) = 1, but this is in contradiction with the continuity. In conclusion, there exists at least a choice of

ζ(s in ) = ζ in ∈ (s in -s in log -1 2 (s in ), s in + s in log -1 2 (s in ))
such that s * = s 0 .

step 3 Conclusion. To finish proving (3.5), we only have to prove the estimate on λ(s). From (3.9) and (3.37), we obtain

λ λ + s -1 log -1 (s) s -1 log -3 2 (s).
By integration on [s, s in ], for any s ∈ [s 0 , s in ), using the value λ(s in ) = λ in = log -1 (s in ) (see (3.4)), we have

|log(λ(s)) + log(log(s))| log -1 2 (s), and thus λ(s) -log -1 (s) log -3 2 (s). (3.43)
This concludes the proof of Proposition 6.

Compactness arguments

The objective of this section is to finish the construction of Theorem 1 by passing to the limit on a sequence of solutions given by Proposition 6.

4.1. Construction of a sequence of backwards solutions. We claim the following consequence of Proposition 6. Lemma 9. There exist t 0 > 1 and a sequence of solutions u n ∈ C([t 0 -T n , 0], Σ) of (1.1), where T n → +∞ as n → +∞,

satisfying the following estimates, for all t ∈ [t 0 -T n , 0],

z n (t) - 2 κ log(t + T n ) log(log(t + T n )), λ n (t) -log -1 (t + T n ) log -3 2 (t + T n ), b n (t) -(t + T n ) -1 log -3 (t + T n ) + |β n (t)| + ε n (t) H 1 (t + T n ) -1 log -7 2 (t + T n ), |a n (t)| (t + T n ) -2 log -1 (t + T n ), ε n (0) ≡ 0, (4.2)
where (λ n , z n , γ n , β n , b n ) are the parameters of the decomposition of u n given by Lemma 5, i.e.

u n (t, x) = e iγn(t) λ n (t) k e iΓ k,n Q an x λ n (t) -z n (t)e k + ε n t, x λ n (t) , (4.3
)

with Γ k,n (t, y) = β n (t)(e k • y) -bn(t) 4 |y| 2 and Q an = Q + a n ρ. Moreover, for all t ∈ [t 0 -T n , 0], ˆ|u n (t, x)| 2 |x| 2 dx 1. (4.4)
Proof. Applying Proposition 6 with s in = n for any large n, there exists a solution u n (t) of (1.1) defined on the time interval [-T n , 0] where

T n = ˆn s 0 λ 2 n (s)ds,
and whose decomposition satisfies the uniform estimates (3.5) on [-T n , 0]. First, we see that (4.1) follows directly from the estimate on λ n (s) in (3.5).

Proof of (4.2). From (3.5) and the definition of the rescaled time s (see (3.2)), for any s ∈ [s 0 , n], we have

t(s) + T n = ˆs s 0 λ 2 n (s ′ )ds ′ where |λ 2 n (s) -log -2 (s)| log -5 2 (s).
Fix s0 > s 0 large enough independent of n so that, for all s0 < s < n,

1 2 s log -2 (s) ≤ ˆs s 0 λ 2 n (s ′ )ds ′ = s log -2 (s) + O(s log -5 2 (s)) ≤ 3 2 s log -2 (s). Fix t 0 = 3 2 s0 log -2 (s 0 ). Then, for all t ∈ [t 0 -T n , 0], t + T n = s log -2 (s) 1 + O(log -1 2 (s)) ≥ 1 2
s log -2 (s), and s = (t + T n ) log 2 (t + T n ) 1 + O(log -1 2 (t + T n )) .

Thus, estimates (4.2) are direct consequences of (3.5).

Proof of (4.4). From (4.3) and ε n (0) ≡ 0, we have u n (0) ∈ Σ. It is then standard (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF], Proposition 6.5.1) that u n ∈ C([t 0 -T n , 0], Σ). We claim the following preliminary estimates. Fix A = Therefore, there exists a subsequence of (u n ) (still denoted by (u n )) and u 0 ∈ Σ such that u n (t 0 -T n ) ⇀ u 0 weakly in H 1 (R 2 ), u n (t 0 -T n ) → u 0 in H σ (R 2 ) for 0 ≤ σ < 1, as n → +∞.

Let u be the solution of (1.1) corresponding to u(t 0 ) = u 0 . By the local Cauchy theory for (1.1) (see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]) and the properties of the sequence u n (t) (recall that T n → ∞), it follows that u ∈ C([t 0 , +∞), Σ). Moreover, for all 0 ≤ σ < 1, for all t ∈ [t 0 , +∞), Next, since for j = k, for some q, λ -2 (t) ˆ|∇Q (y -z(t)e k ) • ∇Q (y -z(t)e j )| dy |z| q e -κz t -1 , we also obtain (1.12). As a final remark, note that by global existence and uniform bound in Σ, the virial identity (1.7) implies the rigidity E(u) = 0. This concludes the proof of Theorem 1.

u n (t -T n ) → u(t)

4.3.

Proof of Corollary 2. For -t -1 0 < t < 0, we set z(t) = z(|t| -1 ), λ(t) = |t|λ(|t| -1 ), ã(t) = a(|t| -1 ), b(t) = b(|t| -1 ), γ(t) = γ(|t| -1 ), β(t) = β(|t| -1 ), ε(t) = ε(|t| Thus, E(v) > 0.

2. 1 .

 1 Approximate solution and nonlinear forcing. Consider a time dependent C 1 function p of the form p = (λ, z, γ, β, b) ∈ (0, +∞) 2 × R 3 , with |b| + |β| ≪ 1 and z ≫ 1. We renormalize the flow by considering

.21) η 1 ,

 1 iΛQ = 0. Let A = 0 and B = ΛQ. Note that L + (ΛQ) = -2Q, and by the definition of

. 26 ) 2 e

 262 Indeed, first by(3.6)-(3.7) and(2.11),| G 1 , iQ a + κc a ρ, Q bz 3 2 e -κz | s -3 log -2 (s).(3.27)Second, we note that by (3.23) and (3.6)-(3.7),| ż -bz| s -1 log -1 (s),(3.28)and that by the definition of a(z) in (2.5),|a ′ (z) -c a κz 1 -κz | z -1 2 e -κz s -2 log -2 (s). (3.29) Gathering (3.27)-(3.29), we obtain (3.26).

By ( 3 .

 3 11) and (3.6)-(3.7), | ż| s -1 and thus by (3.6)-(3.7),

By ( 3 . 6 )

 36 -(3.7), |b| s -1 log -1 (s) and |z k | log(s). Moreover, |∇χ k | log -1 (s). Therefore, |b (∇ε • ∇χ k ), (z k • ∇ε) | s -1 log -1 (s) ε 2 H 1 . Similarly, by |∇(∆χ k )| log -3 (s), we obtain b ˆ|ε| 2

  and thus, by (3.6)-(3.7),

  ) which strictly improves the estimate on b(s) in (3.6). Finally, we address the estimate on z(s). From (3.36), (3.6)-(3.7) and (3.11), we have b

. 40 )

 40 Then,(3.39) writes| ζ(s) -1| log -1 (s). (3.41)

  Minimal mass property of the approximate solution. The proof of the degeneracy estimate (3.10) relies on the following minimal mass property for the ansatz P under the bootstrap assumptions (3.6):

	step 4
		.22)
	Combining (3.18)-(3.22), we have proved, for all s ∈ [s * * , s in ],
	| m a 1 (s)| C * * s -2 log -2 (s).	(3.23)

  we only have to deal with the term b |P + ε| 2 (P + ε) -|P| 2 P, z k • (2χ k ∇ε + ε∇χ k ) . Recall that |P+ε| 2 (P+ε)-|P 2 |P = (2ε|P| 2 +εP 2 )+2|ε| 2 P+ε 2 P+|ε| 2 ε. First, by (3.6)-(3.7), it is clear that b 2|ε| 2 P + ε 2

.

  16 κ ≥ 8. For any k ∈ {1, . . . , K}, for all t ∈ [t 0 -T n , 0],|x| 2 dx = ˆ|Q an (y)| 2 |λ n (t)y + λ n (t)z n (t)e k | 2 dy 1.where we have used from (4.2), λ n (t)z n (t) 1. (4.7) To show (4.6), we see first that by (4.2),∇ e iΓ k,n Q an (y) |∇Q an (y)| 2 + |β n | 2 + b 2 n |y| 2 Q 2 an (y) e -3 2 |y| . Thus, by change of variable (using A ≥ 8), Thus (4.6) is proved. Observe that (4.5)-(4.6) and (4.2) implyxu n (0) L 2 1, ∇u n (t) L 2 (|x|>A) (t + T n ) -1 log -5 2 (t + T n ).(4.8)Define ϕ : R 2 → [0, 1] by ϕ(x) = (|x| -A) 2 for |x| > A and ϕ(x) = 0 otherwise. By elementary computations,d dt ˆ|u n | 2 ϕ = 2 Im ˆ(∇ϕ • ∇u n )u n = 4 ˆ|x|>A x |x| • ∇u n u n ϕ T n ) -1 log -5 2 (t + T n ) ˆ|u n | 2 ϕBy integration and (4.8), the following uniform bound holds on [t 0 -T n , 0],ˆ|u n (t, x)| 2 ϕ(x)dx 1 and thus ˆ|u n (t, x)| 2 x 2 dx 1, which finishes the proof of (4.4). 4.2. Compactness argument. By (4.2)-(4.4), the sequence (u n (t 0 -T n )) is bounded in Σ.

						1 λ 2 n (t)	ˆ	Q an		x λ n (t)	-z n (t)e k	2	|x| 2 dx 1,	(4.5)
				1 λ 2 n (t) ˆ|x|>A	∇ x e iΓ k,n Q an	x λ n (t)	-z n (t)e k	2	dx (t + T n ) -4 .	(4.6)
	Indeed, (4.5) follows from a change of variable and the decay properties of Q and ρ,
	1 λ 2 n (t)	ˆ	Q an	λ n (t) x	-z n (t)e k	2
								2
	1 λ 2 n (t) ˆ|x|>A	∇ x e iΓ k,n Q an	x λ n (t)	-z n (t)e k	2	dx
					=	1 λ 2
	|y + z n (t)e k | >	A λ n (t)	⇒ |y| >	A λ n (t)	-|z n (t)| >	3A 4	-	4 κ	log(t + T n ) ≥	A 2	log(t + T n ).
												1 2 .
	Thus, by (4.8),
	d dt	ˆ|u n | 2 ϕ	ˆ|u n | 2 ϕ	1 2	ˆ|x|>A	|∇u n (t)| 2	1 2	(t + 1 2

n (t) ˆ|y+zn(t)e k |>A/λn(t) ∇ e iΓ k,n Q an (y) 2 dy log 2 (t + T n ) ˆ|y|> A 2 log(t+Tn) e -3 2 |y| dy (t + T n ) -A 2 = (t + T n ) -4 ,

where we have used from (4.2) (possibly taking a larger t 0 ),

  in H σ .By weak convergence in H 1 , u(t) satisfies (2.25) for all t ≥ t 0 . Moreover, the decomposition ( p, ε) of u satisfies, for all t ≥ t 0 ,p n (t -T n ) → p(t), ε n (t -T n ) → ε(t) in H σ , ε n (t -T n ) ⇀ ε(t) in H 1 (4.9)(see e.g.[START_REF] Merle | On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation[END_REF], Claim p.598). In particular, for all t ∈ [t 0 , +∞), u(t) decomposes asH 1 t -1 log -7 2 (t), |a(t)| t -2 log -1 (t), ˆ|u(t, x)| 2 |x| 2 dx 1.Note that by (4.11), we have for all k ∈ {1, . . . , K},x k (t) = λ(t)z(t)e k → 2 κ e k , with x k (t) -Since λ -1 (t) ε(t) H 1 t -1 log -5 2 (t) and, by (4.10) and (4.11), λ -1 (t) e iΓ k Q a -Q H 1 λ -1 (t)(|β(t)| + |b(t)| + |a(t)|) t -1 log -2 (t),

			u(t, x) =	e iγ(t) λ(t)	k	e iΓ k Q a	x -λ(t)z(t)e k λ(t)	+ ε t,	x λ(t)	,	(4.10)
	z(t) -	2 κ	log(t)	log(log(t)),	λ(t) -log -1 (t)	log -3 2 (t),
										(4.11)
									2 κ	e k	log(log(t)) log(t)	.
										(4.12)
	we obtain the following stronger form of (1.11)
			u(t) -e iγ(t)	k	1 λ(t)	Q	. -x k (t) λ(t)	H 1	t -1 log -2 (t).	(4.13)

where

Γ k (t, y) = β(t)(e k • y) -b(t) 4 |y| 2 and b(t) -t -1 log -3 (t) + |β(t)| + ε(t)

  -1 ), Γk (t, y) = β(t)(e k • y) --|t|| log |t|| -3 + | β(t)| + ε(t) H 1 |t|| log |t|| -7 2 , |ã(t)| |t|| log |t|| -1 .We see from (4.10) that the pseudo-conformal transform v(t) of u(t) as defined in (1.5) satisfies Note in particular that λ(t)z(t) ∼ 2 κ |t| as t ↑ 0. From this, it follows that |v(t, x)| 2 ⇀ K Q 2 L 2 δ 0 as t ↑ 0. (t, x), and as t ↑ 0,1 |t| 2 ˆ|x| 2 |w(t, x)| 2 dx Γk Q ã (y -z(t)e k ) + ε (t, y) 2 |y| 2 dy 1, ˆ|∇w(t, x)| 2 dx ∼ K ∇Q 2 L 2 |t| -2 | log |t|| 2 ,we obtain (1.13). Note that ´|x| 2 |v(t, x)| 2 t 2 implies by (1.7) that ´|x| 2 |v(t, x)| 2 = t 2 E(v).

									b(t) 4	|y| 2 ,
	so that from (4.11),
	z(t) -b(t) (4.14) 2 | log |t|| log | log |t||, λ(t) -|t|| log |t|| -1 | log |t|| -3 2 , κ
	v(t, x) = e -i	|x| 2 4|t| w(t, x), w(t, x) =	e iγ(t) λ(t)	k	e i Γk Q ã	x λ(t)	-z(t)e k + ε t,	x λ(t)	.
	Finally, since ∇v(t, x) = e -i	|x| 2 4|t|	∇w -i x 2|t| w λ(t) 2 ˆ
				t

k e i