Stefan Le Coz

AND Yvan Martel
email: yvan.martel@polytechnique.edu

Pierre Rapha Ël
email: praphael@unice.fr

MINIMAL MASS BLOW UP SOLUTIONS FOR A DOUBLE POWER NONLINEAR SCHR ÖDINGER EQUATION

Keywords: Mathematics Subject Classification. 35Q55 (35B44) blow-up, nonlinear Schrödinger equation, double power nonlinearity, minimal mass, critical exponent

We consider a nonlinear Schrödinger equation with double power nonlinearity

 for the construction of minimal blow up elements.

Introduction

We consider the following double power nonlinear Schrödinger equation in R d (NLS)

i∂ t u + ∆u + |u| 4 d u + ǫ|u| p-1 u = 0, u |t=0 = u 0 , 1 < p < 1 + 4 d , ǫ ∈ {-1, 0, 1}. (1)
This model corresponds to a subcritical perturbation of the classical mass critical problem ǫ = 0 which rules out the scaling symmetry of the problem. 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 d ), there exists a unique maximal solution u ∈ C((-T ⋆ , T ⋆ ), H 1 (R d )) ∩ C 1 ((-T ⋆ , T ⋆ ), H -1 (R d )) of [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]. Moreover, the mass (i.e. L 2 norm) and energy E of the solution are conserved by the flow where:

E(u) = 1 2 ∇u 2 2 - 1 2 + 4 d u 2+ 4 d 2+ 4 d -ǫ 1 p + 1 u p+1 p+1 .
Moreover, there holds the blow up criterion:

T ⋆ < +∞ implies lim t↑T ⋆ ∇u(t) 2 = +∞. (2)
In this paper, we are interested in the derivation of a sharp global existence criterion for [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] in connection with the existence of minimal mass blow up solutions of (1).

1.1. The mass critical problem. Let us briefly recall the structure of the mass critical problem ǫ = 0. In this case, the scaling symmetry

u λ (t, x) = λ d 2 u(λ 2 t, λx)
acts on the set of solutions and leaves the mass invariant

u λ (t, •) 2 = u(λ 2 t, •) 2 .
From variational argument [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF], the unique ( [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF][START_REF] Kwong | Uniqueness of positive solutions of ∆uu + u p = 0 in R n[END_REF]) up to symmetry ground state solution to

-∆Q + Q -|Q| 4 d Q = 0, Q ∈ H 1 (R d ), Q > 0, Q radial
Theorem 1 (Existence of a minimal blow up element). Let d = 1, 2, 3 and 1 < p < 1+ 4 d . Then for all energy level E 0 ∈ R, there exist t 0 < 0 and a radially symmetric Cauchy data u(t 0 ) ∈ H 1 (R d ) with u(t 0 ) 2 = Q 2 , E(u(t 0 )) = E 0 , such that the corresponding solution u(t) of [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF] blows up at time T * = 0 with speed:

∇u(t) 2 = C(p) + o t↑0 (1) |t| σ (8)
for some universal constants

σ = 4 4 + d(p -1) ∈ 1 2 , 1 , C(p) > 0.
Comments on the result.

1. On the existence of minimal elements. Since the pioneering work [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF], it has long been believed that the existence of a minimal blow up bubble was related to the exceptional pseudo conformal symmetry [START_REF] Boulenger | Blow-up solutions for the 2-dimensional critical Schrödinger equation on a riemannian manifold[END_REF], or at least to the existence of a sufficiently sharp approximation of it, see [START_REF] Banica | Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation[END_REF][START_REF] Martel | Nonexistence of blow-up solution with minimal L 2 -mass for the critical gKdV equation[END_REF]. However, a new methodology to construct minimal mass elements for a inhomogeneous (NLS) problem, non perturbative of critical (NLS), was developed in [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF], and later successfully applied to problems without any sort of pseudo conformal symmetry, [START_REF] Boulenger | Blow-up solutions for the 2-dimensional critical Schrödinger equation on a riemannian manifold[END_REF][START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF][START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF]. More generally, the heart of the matter is to be able to compute the trajectory of the solution on the soliton manifold, see [START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF][START_REF] Martel | Description of two soliton collision for the quartic gKdV equation[END_REF] for related problems for two solitary waves motion. The present paper adapts this approach which relies on the direct computation of the blow up speed and the control of non dispersive bubbles as in [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF].

Observe that the blow up speed [START_REF] Chang | Spectra of linearized operators for NLS solitary waves[END_REF] is quite surprising since it approaches the self simiar blow up speed |t| -1 2 as p → 1 + 4 d -.

Uniqueness.

A delicate question investigated in [START_REF] Boulenger | Blow-up solutions for the 2-dimensional critical Schrödinger equation on a riemannian manifold[END_REF][START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF][START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF] is the uniqueness of the minimal blow up element. Such a uniqueness statement should involve Galilean drifts since the Galilean symmetry applied to [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF] is an L 2 isometry and automatically induces minimal elements with non trivial momentum. Uniqueness issues lie within the general question of classifying the compact elements of the flow in the Kenig-Merle road map [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]. A more limited question is to determine the global behavior of the minimal element for negative time, which is poorly understood in general. Here, at least in the case E 0 ≥ 0, one can see from Virial type estimates that the solution is global in negative time.

3. Detailed structure of the singular bubble. The analysis provides the following detailed structure of the blow up bubble

u(t, x) = 1 λ d 2 (t) Q x λ(t)
e -iσ |x| 2 4t e iγ(t) + v (t, x)

where Q is the mass critical ground state, and lim t→0 v(t) 2 = 0, λ(t) ∼ C p |t| σ as t → 0 -, for some constant C p > 0. Note also that the dimension restriction d ∈ {1, 2, 3} is for the sake of simplicity but not essential.

The construction of the minimal blow up element for (1) can be viewed as part of a larger program of understanding what kind of blow up speeds are possible for (NLS) type models. Let us repeat that log-log type solutions with super critical mass can be constructed for [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF], but then the question becomes: do these examples illustrate all possible blow up types, at least near the ground state profile? The recent series of works [START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF][START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF][START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton[END_REF] for the mass critical gKdV equation indicate that this is a delicate problem, and that the role played by the topology used to measure the perturbation is essential. More generally, symmetry breaking perturbations are very common in nonlinear analysis, and while they are expected to be lower order for generic stable blow up dynamics, our analysis shows that they can dramatically influence the structure of unstable threshold dynamics such as in our case minimal blow up bubbles.

Aknowldedgments. S. Le Coz is partly supported by the ANR project ESONSE. Y. Martel and P. Raphaël are partly supported by the ERC advanced grant 291214 BLOWDISOL. 1.4. Notation. Let us collect the main notation used throughout the paper. For the sake of simplicity, we work in the radial setting only. The L 2 scalar product and L q norm (q ≥ 1) are denoted by

(u, v) 2 = Re R d u(x)v(x)dx , u q = R d |u| q 1 q
.

We fix the notation:

f (z) = |z| 4 d z; g(z) = |z| p-1 z; F (z) = 1 4 d + 2 |z| 4 d +2 ; G(z) = 1 p + 1 |z| p+1 .
Identifying C with R 2 , we denote the differential of these functions by df , dg, dF and dG. Let Λ be the generator of L 2 -scaling i.e.

Λ = d 2 + y • ∇.
The linearized operator close to Q comes as a matrix

L + := -∆ + 1 -1 + 4 d Q 4 d , L -:= -∆ + 1 -Q 4 d .
and the generalized kernel of 0 L - -L + 0 is non-degenerate and spanned by the symmetries of the problem (see [START_REF] Kwong | Uniqueness of positive solutions of ∆uu + u p = 0 in R n[END_REF][START_REF] Weinstein | Modulational stability of ground states of nonlinear Schrödinger equations[END_REF] for the original results and [START_REF] Chang | Spectra of linearized operators for NLS solitary waves[END_REF] for a short proof). It is completely described in H 1 rad (R d ) by the relations (we define ρ as the unique radial solution to

L + ρ = |y| 2 Q) L -Q = 0, L + ΛQ = -2Q, L -|y| 2 Q = -4ΛQ, L + ρ = |y| 2 Q. ( 10
)
Denote by Y the set of radially symmetric functions

f ∈ C ∞ (R d ) such that ∀α ∈ N d , ∃C α , κ α > 0, ∀x ∈ R d , |∂ α f (x)| ≤ C α (1 + |x|) κα Q(x).
It follows from the kernel properties of L + and L -, and from well-known properties of the Helmholtz kernel (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] for the properties of Helmholtz kernel (i.e. Bessel and Hankel functions) and [9, Appendix A] or proof of Lemma 3.2 in [START_REF] Merle | On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation[END_REF] for related arguments) that

∀g ∈ Y, ∃f + ∈ Y, L + f + = g, (11)
∀g ∈ Y, (g, Q) 2 = 0, ∃f -∈ Y, L -f -= g. (12)
It is also well known (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 | Lyapunov stability of ground states of nonlinear dispersive evolution equations[END_REF]) that L + and L -verify the following coercivity property: there exists µ > 0 such that for all

ε = ε 1 + iε 2 ∈ H 1 rad (R d ), L + ε 1 , ε 1 + L -ε 2 , ε 2 ≥ µ ε 2 H 1 - 1 µ (ε 1 , Q) 2 2 + ε 1 , |y| 2 Q 2 2 + (ε 2 , ρ) 2 2 . ( 13
)
Throughout the paper, C denotes various positive constants whose exact values may vary from line to line but are of no importance in the analysis. When an inequality is true up to such a constant, we also use the notation , or ≈.

Construction of the blow-up profile

In this section, we define the blow-up profile which is relevant to construct the minimal mass solution -see Proposition 3 below.

Blow up profile.

Let us start with some heuristic arguments justifying the construction. As usual in blow up contexts, we look for a solution of the following form, with rescaled variables (s, y):

u(t, x) = 1 λ d 2 (s) w(s, y)e iγ(s)-i b(s)|y| 2 4 , ds dt = 1 λ 2 , y = x λ(s) ,
where the function w, and the time dependent parameters λ > 0, b and γ are to be determined satisfying the following equation

iw s + ∆w -w + f (w) + λ α g(w) -i b + λ s λ Λw + (1 -γ s )w + (b s + b 2 ) |y| 2 4 w -b b + λ s λ |y| 2 2 w = 0, ( 14
)
where

α = 2 - d(p -1) 2 ∈ (0, 2).
Since we look for blow up solutions, the parameter λ(s) should converge to zero as s → ∞. Therefore,

w(s, y) = Q(y), b + λ s λ = b s + b 2 = 1 -γ s = 0 ( 15
)
is a solution of ( 14) at the first order, i.e. when neglecting λ α |w| p-1 w. However, the first order error term λ α Q p cannot be neglected in the minimal mass blow up analysis (while it could be neglected easily in the log-log regime where λ ∼ e -e c b ). Therefore, starting from Q, we need to look for a refined blow up ansatz. Actually, to close the analysis for any α ∈ (0, 2), we need to remove error terms at any order of λ α and b in the equation of w. It is important to note that in the process of constructing the approximate solution, we cannot exactly solve [START_REF] Kwong | Uniqueness of positive solutions of ∆uu + u p = 0 in R n[END_REF] since we need to introduce new terms in the equation (due to degrees of freedom necessary to construct the ansatz) that will modify the modulation equations in [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]. These terms (gathered in the time dependent function θ(s) below) are responsible for the specific blow up law obtained in Theorem 1.

Fix K ∈ N, K ≫ 1 (K > 20/α is sufficient in the proof of Theorem 1), and b 2j λ (k+1)α P + j,k (y) + i

Σ K = {(j, k) ∈ N 2 | j + k ≤ K}.
(j,k)∈ΣK b 2j+1 λ (k+1)α P - j,k (y) ( 16
)
satisfies

i∂ s P + ∆P -P + f (P ) + λ α g(P ) + θ |y| 2 4 P = Ψ K where θ(s) = θ(b(s), λ(s)), θ(b, λ) = (j,k)∈ΣK b 2j λ (k+1)α β j,k and
sup y∈R d e |y| 2 (|Ψ K (y)| + |∇Ψ K (y)|) λ α b + λ s λ + b s + b 2 -θ + (|b| 2 + λ α ) K+2 . (17)
(ii) Rescaled blow up profile. Let

P b (s, y) := P (s, y)e -i b(s)|y| 2 4 . ( 18
)
Then

i∂ s P b + ∆P b -P b + f (P b ) + λ α g(P b ) -i λ s λ ΛP b = -i λ s λ + b ΛP b + (b s + b 2 -θ) |y| 2 4 P b + Ψ K e -i b|y| 2 4 . (19)
(iii) Mass and energy properties of the blow up profile. Let

P b,λ,γ (s, y) = 1 λ d 2 P b s, x λ e iγ .
Then,

d ds |P b,λ,γ | 2 λ α b + λ s λ + b s + b 2 -θ + (|b| 2 + λ α ) K+2 , (20)
d ds E(P b,λ,γ ) 1 λ 2 b + λ s λ + b s + b 2 -θ + (|b| 2 + λ α ) K+2 . ( 21
)
Moreover, for any (j, k) ∈ Σ K , there exist η j,k ∈ R such that

E(P b,λ,γ ) - |y| 2 Q 2 8 E(b, λ) (b 2 + λ α ) K+2 λ 2 , ( 22
)
where

E(b, λ) = b 2 λ 2 - 2β 2 -α λ α-2 + λ α-2 (j,k)∈ΣK ,j+k≥1 b 2j λ kα η j,k . (23)
See a similar construction of a blow up profile at any order of b in [START_REF] Merle | On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation[END_REF]. One sees in [START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF] the impact of the subcritical nonlinearity g(u) on the blow up law b s + b 2 -θ = 0, which differs from the unperturbed equation b s + b 2 = 0, and leads to leading order to λ α ≈ b 2 , see [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF].

Proof of Proposition 3. Proof of (i). For time dependent functions λ(s) > 0, b(s), we set

P = Q + λ α Z where Z = (j,k)∈ΣK b 2j λ kα P + j,k + i (j,k)∈ΣK b 2j+1 λ kα P - j,k , θ(s) = (j,k)∈ΣK b 2j (s)λ (k+1)α (s)β j,k ,
where P + j,k ∈ Y, P - j,k ∈ Y and β j,k are to be determined. Set

Ψ K = i∂ s P + ∆P -P + |P | 4 d P + λ α |P | p-1 P + θ |y| 2 4 P.
The objective is to choose the unknown functions and parameters so that the error term Ψ K is controlled as in [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF]. First,

iP s = i λ s λ (j,k)∈ΣK (k + 1)αb 2j λ (k+1)α P + j,k + ib s (j,k)∈ΣK 2jb 2j-1 λ (k+1)α P + j,k - λ s λ (j,k)∈ΣK (k + 1)αb 2j+1 λ (k+1)α P - j,k -b s (j,k)∈ΣK (2j + 1)b 2j λ (k+1)α P - j,k = -i (j,k)∈ΣK (k + 1)αb 2j+1 λ (k+1)α P + j,k -i   b 2 - (j ′ ,k ′ )∈ΣK b 2j ′ λ (k ′ +1)α β j ′ ,k ′   (j,k)∈ΣK 2jb 2j-1 λ (k+1)α P + j,k + (j,k)∈ΣK (k + 1)αb 2(j+1) λ (k+1)α P - j,k +   b 2 - (j ′ ,k ′ )∈ΣK b 2j ′ λ (k ′ +1)α β j ′ ,k ′   (j,k)∈ΣK (2j + 1)b 2j λ (k+1)α P - j,k + Ψ Ps where Ψ Ps = λ s λ + b (j,k)∈ΣK (k + 1)αb 2j λ (k+1)α iP + j,k -bP - j,k + b s + b 2 -θ (j,k)∈ΣK b 2j-1 λ (k+1)α 2jiP + j,k -(2j + 1)bP - j,k . (24)
We rewrite

iP s = -i (j,k)∈ΣK ((k + 1)α + 2j) b 2j+1 λ (k+1)α P + j,k + i j,k≥0 b 2j+1 λ (k+1)α F Ps,- j,k + j,k≥0 b 2j λ (k+1)α F Ps,+ j,k + Ψ Ps ,
where for j, k ≥ 0, F Ps,± j,k depends on various functions P ± j ′ ,k ′ and parameters β j ′ ,k ′ for (j ′ , k ′ ) ∈ Σ K such that either k ′ ≤ k -1 and j ′ ≤ j + 1 or k ′ ≤ k and j ′ ≤ j -1. Only a finite number of these functions are nonzero.

Next, using ∆Q

-Q + Q 4 d +1 = 0, we get ∆P -P + |P | 4 d P = - (j,k)∈ΣK b 2j λ (k+1)α L + P + j,k -i (j,k)∈ΣK b 2j+1 λ (k+1)α L -P - j,k + f (Q + λ α Z) -f (Q) -λ α df (Q)Z. Let Ψ f = f (Q + λ α Z) - K k=0 d k f (Q)(λ α Z, . . . , λ α Z) = |Q + λ α Z| 4 d (Q + λ α Z) -Q 4 d +1 1 + K+1 n=1 2 d + 1 2 d . . . 2 d -n + 2 n! λ α Z Q n × 1 + K+1 n ′ =1 2 d 2 d -1 . . . 2 d -n ′ + 1 n ′ ! λ α Z Q n ′ . (25)
Then

∆P -P + |P | 4 d P = - (j,k)∈ΣK b 2j λ (k+1)α L + P + j,k -i (j,k)∈ΣK b 2j+1 λ (k+1)α L -P - j,k + i j≥0,k≥1 b 2j+1 λ (k+1)α F f,- j,k + j,k≥0 b 2j λ (k+1)α F f,+ j,k + Ψ f .
where for j, k ≥ 0, F f,± j,k depends on Q and on various functions

P ± j ′ ,k ′ for (j ′ , k ′ ) ∈ Σ K such that k ′ ≤ k -1 and j ′ ≤ j.
Using a similar argument for λ α |P | p-1 P , we obtain

λ α |P | p-1 P = i j≥0,k≥1 b 2j+1 λ (k+1)α F g,- j,k + j≥0,k≥1 b 2j λ (k+1)α F g,+ j,k + Ψ g ,
where

Ψ g = λ α |Q + λ α Z| p-1 (Q + λ α Z) -Q p 2 +1 1 + K+1 n=1 p 2 p 2 -1 . . . p 2 -n + 1 n! λ α Z Q n × 1 + K+1 n=1 p 2 -1 p 2 -2 . . . p 2 -n n! λ α Z Q n ,
and where for j, k ≥ 0, F g,± j,k depends on Q and on various functions

P ± j ′ ,k ′ for (j ′ , k ′ ) ∈ Σ K such that k ′ ≤ k -1 and j ′ ≤ j. Finally, θ |y| 2 4 P =   (j,k)∈ΣK b 2j λ (k+1)α β j,k   |y| 2 4 Q + i j,k≥0 b 2j+1 λ (k+1)α F θ,- j,k + j,k≥0 b 2j λ (k+1)α F θ,+ j,k ,
where F θ,± j,k depends on Q and on various functions

P ± j ′ ,k ′ or parameters β j ′ ,k ′ for (j ′ , k ′ ) ∈ Σ K such that k ′ ≤ k -1 and j ′ ≤ j.
Combining these computations, we obtain

Ψ K = - (j,k)∈ΣK b 2j λ (k+1)α L + P + j,k -F + j,k -β j,k |y| 2 Q -i (j,k)∈ΣK b 2j+1 λ (k+1)α L -P - j,k -F - j,k + ((k + 1)α + 2j)P + j,k + Ψ >K + Ψ Ps + Ψ f + Ψ g , where F ± j,k = F Ps,± j,k + F f,± j,k + F g,± j,k + F θ,± j,k , and Ψ >K = j,k>0, (j,k) ∈ΣK b 2j λ (k+1)α F + j,k + i j,k>0, (j,k) ∈ΣK b 2j+1 λ (k+1)α F - j,k .
(Note that the series in the expression of Ψ >K contains only a finite number of terms.) Now, for any (j, k) ∈ Σ K , we want to choose recursively P ± j,k ∈ Y and β j,k to solve the system (S j,k )

L + P + j,k -F + j,k -β j,k |y| 2 Q = 0 L -P - j,k -F - j,k + ((k + 1)α + 2j)P + j,k = 0
, where F ± j,k are source terms depending of previously determined P ± j ′ ,k ′ and β j ′ ,k ′ . We argue by a suitable induction argument on the two parameters j and k. For (j, k) = (0, 0), we see that the system writes

L + P + 0,0 -Q p -β 0,0 |y| 2 Q = 0 L -P - 0,0 + αP + 0,0 = 0, (the term Q p in
the first line is coming from Ψ g ). By [START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF], for any β 0,0 ∈ R, there exists a unique P + 0,0 ∈ Y so that L + P + 0,0 -Q p -β 0,0 |y| 2 Q = 0. We choose β 0,0 ∈ R so that

P + 0,0 , Q 2 = - 1 2 L + P + 0,0 , ΛQ 2 = - 1 2 Q p + β 0,0 |y| 2 4 Q, ΛQ 2 = 0
(recall from [START_REF] Dodson | Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state[END_REF] that L + ΛQ = -2Q), which gives

β := β 0,0 = - 4 (Q p , ΛQ) 2 (|y| 2 Q, ΛQ) 2 = 2d(p -1) p + 1 Q p+1 p+1 yQ 2 2 > 0. (26)
By [START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF], there exists P - 0,0 ∈ Y (unique up to the addition of cQ) such that L -P - 0,0 + αP + 0,0 = 0. Now, we assume that for some (j 0 , k 0 ) ∈ Σ K , the following assertion is true:

H(j 0 , k 0 ) : for all (j, k) ∈ Σ K such that either k < k 0 , or k = k 0 and j < j 0 , the system (S j,k ) has a solution (P + j,k , P - j,k , β j,k ), P ± j,k ∈ Y.
In view of the definition of F ± j0,k0 , H(j 0 , k 0 ) implies in particular that F ± j0,k0 ∈ Y. We now solve the system (S j0,k0 ) as before. By [START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF], for any β j0,k0 ∈ R, there exists a unique P + j0,k0 ∈ Y so that

L + P + j0,k0 -F + j0,k0 -β j0,k0 |y| 2 Q = 0. We uniquely choose β j0,k0 ∈ R so that -F - j0,k0 + ((k 0 + 1)α + 2j 0 )P + j0,k0 , Q 2 = 0.
By [START_REF] Krieger | Nondispersive solutions to the L 2 -critical half-wave equation[END_REF], there exists P - j0,k0 ∈ Y (unique up to the addition of cQ) such that L -P - j0,k0 -F - j0,k0 + ((k 0 + 1)α + 2j 0 )P + j0,k0 = 0. In particular, we have proved that if j 0 < K, then H(j 0 , k 0 ) implies H(j 0 +1, k 0 ), and H(K, k 0 ) implies H(1, k 0 +1). This is enough to complete an induction argument on the two parameters (j, k). Therefore, system (S j,k ) is solved for all (j, k) ∈ Σ K . It remains to estimate Ψ K and ∇Ψ K . It is straightforward to check that sup

y∈R d e |y| 2 |Ψ Ps (y)| + |∇Ψ Ps (y)| λ α λ s λ + b + b s + b 2 -θ .
Next, we claim

|Ψ f | λ (K+2)α + λ α b 2K+2 Q. ( 27
)
Indeed, first, if y is such that λ α Z(y) Q(y) < 1 2 then the result follows from ( 25) and a order Taylor expansion of order K +1 of (1+

λ α Z Q ) 2 d +1 and (1+ λ α Z Q ) 2 d . Second, if on the contrary, λ α Z(y) Q(y) ≥ 1 2 , then, since Z ∈ Y, we have, for such y, Q(y) ≤ 2λ α |Z(y)| λ α (1 + |y| κ )Q(y) and so Q(y) + |Z(y)| e -1 2 λ α/κ ,
which completes the proof of [START_REF] Merle | On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation[END_REF]. The proofs of estimates for ∇Ψ f , Ψ g and ∇Ψ g are similar.

Finally the following estimates for Ψ >K and ∇Ψ >K are clear:

|Ψ >K | + |∇Ψ >K | λ (K+2)α + λ α |b| 2K+2 Q 1 2 .
The result follows from K ≥ 20 α . Proof of (ii). This is a straightforward computation which is left to the reader. Proof of (iii). To prove [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF], we hit [START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF] with iP b and compute using the critical relation (P, ΛP ) 2 = 0:

1 2

d ds P b 2 2 = (i∂ s P b , iP b ) 2 = (Ψ K e -i b|y| 2 4 , iP b )
and [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF] follows from [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF]. For ( 21), we have from scaling:

E(P b,λ,γ ) = 1 λ 2 1 2 |∇P b | 2 -F (P b ) -λ α G(P b ) =: 1 λ 2 Ẽ(λ, P b )
Therefore,

d ds E(P b,λ,γ ) = 1 λ 2 -2 λ s λ Ẽ(λ, P b ) + Ẽ′ (λ, P b ), ∂ s P b -αλ α λ s λ G(P b ) . ( 28
)
Using the equation [START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF] of P b , we compute:

Ẽ′ (λ, P b ), ∂ s P b = λ s λ Ẽ′ (λ, P b ), ΛP b - λ s λ + b Ẽ′ (λ, P b ), ΛP b + (b s + b 2 -θ) i Ẽ′ (λ, P b ), |y| 2 4 P b + i Ẽ′ (λ, P b ), Ψ K e -i b|y| 2 4 . (29)
We now integrate by parts to estimate

Ẽ′ (λ, P b ), ΛP b = |∇P b | 2 -2 F (P b ) - d(p -1) 2 G(P b ) = 2 Ẽ(λ, P b ) + αλ α G(P b ), ( 30
)
where we have used α = 2 -d(p-1)

2

, from which:

d ds E(P b,λ,γ ) = 1 λ 2 -2 λ s λ Ẽ(λ, P b ) -αλ α λ s λ G(P b ) + λ s λ 2 Ẽ(λ, P b ) + αλ α G(P b ) + 1 λ 2 O λ s λ + b + |b s + b 2 -θ| + (b 2 + λ α ) K+2 .
The estimate [START_REF] Martel | Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton[END_REF] on the time-derivative of the energy then follows from ( 28), ( 29), [START_REF] Raphaël | Existence and stability of a solution blowing up on a sphere for an L 2 -supercritical nonlinear Schrödinger equation[END_REF], and [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF]. Next,

λ 2 E(P b,λ,γ ) = 1 2 |∇P b | 2 -F (P b ) -λ α G(P b ) = 1 2 |∇P | 2 + b 2 8 |y| 2 |P | 2 -F (P ) -λ α G(P ).
Thus, replacing

P = Q + λ α Z, λ 2 E(P b,λ,γ ) = 1 2 |∇Q| 2 -F (Q) + b 2 8 |y| 2 Q 2 -λ α G(Q) + λ α (-∆Q -f (Q))ReZ -λ 2α g(Q)ReZ + b 2 4 λ α |y| 2 QReZ + λ 2α 2 |∇Z| 2 + b 2 λ 2α 8 |y| 2 |Z| 2 -{F (Q + λ α Z) -F (Q) -λ α f (Q)ReZ} -λ α {G(Q + λ α Z) -G(Q) -λ α g(Q)ReZ} .
On the one hand, we recall that from Pohozaev identity,

1 2 |∇Q| 2 -F (Q) = 0,
and from the definition (26) of β 0,0 ,

G(Q) = β 2d(p -1) |y| 2 Q 2 = β 4(2 -α) |y| 2 Q 2
and moreover ∆Q + f (Q) = Q. On the other hand, we observe, since P + 0,0 Q = 0,

λ α ZQ = λ α (j,k)∈ΣK ,j+k≥1 b 2j λ kα η I j,k , for some η I j,k ∈ R; λ 2α Zg(Q) = λ α (j,k)∈ΣK ,k≥1 b 2j λ kα η II j,k , for some η II j,k ∈ R; λ α b 2 |y| 2 QReZ = λ α (j,k)∈ΣK ,j≥1 b 2j λ kα η III j,k ,
for some η III j,k ∈ R;

λ 2α |∇Z| 2 + b 2 λ 2α 8 |y| 2 Z 2 = λ α (j,k)∈ΣK ,j≥1,k≥0 b 2j λ kα η IV j,k ,
for some η IV j,k ∈ R. Moreover, by Taylor expansion as before, for some η

V j,k , η IV j,k ∈ R    F (Q + λ α Z) -F (Q) -λ α f (Q)ReZ -λ α (j,k)∈ΣK ,k≥1 b 2j λ kα η V j,k    λ (K+2)α , λ α    G(Q + λ α Z) -G(Q) -λ α g(Q)ReZ -λ α (j,k)∈ΣK ,k≥2 b 2j λ kα η VI j,k    λ (K+2)α .
Gathering these computations, we obtain [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF].

2.2.

Approximate blow up law. For simplicity of notation, we set

β = β 0,0 = 2d(p -1) p + 1 Q p+1 p+1 yQ 2 2
.

First, we find a relevant solution to the following approximate system

b s + b 2 -βλ α = 0, b + λ s λ = 0. (31)
Indeed, for |b| + λ ≪ 1, βλ α is the main term in θ, and the only term in θ that will modify at the main order the blow up rate.

Lemma 4. Let λ app (s) = α 2 2β 2 -α -2 α s -2 α , b app (s) = 2 αs . ( 32
)
Then (λ app (s), b app (s)) solves [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF] for s > 0.

Proof. We compute:

b 2 λ 2 s = 2 b λ b s + b 2 λ = -2β λ s λ λ α-2 ,
and so

b 2 λ 2 - 2β 2 -α λ α-2 = c 0 . ( 33
)
Taking the constant c 0 = 0, and using b = -λs λ > 0, we find

λ s λ 1+ α 2 = 2β 2 -α .
Therefore,

λ(s) = α 2 2β 2 -α -2 α s -2 α , b(s) = - λ s λ (s) = 2 α 1 s
is solution of (31).

Remark 1. We now express this solution in the time variable t app related to λ app . Let

dt app = λ 2 app ds = α 2 2β 2 -α -4 α s -4 α ds.
Therefore (with the convention that t app → 0 -as s → +∞)

t app = -C s s -4-α α where C s = α 4 -α α 2 2β 2 -α -4 α . ( 34
)
As a consequence, we obtain for t app < 0,

λ app (t app ) = C λ |t app | 2 4-α where C λ = 4 -α α C -α 4-α s 1 2 , ( 35
) b app (t app ) = C b |t app | α 4-α , where C b = 2 α C -α 4-α s . (36)
Now, we choose suitable initial conditions b 1 and λ 1 for b(s) and λ(s) at some large time s 1 , first to adjust the value of the energy of P b,λ,γ (up to the small error term in [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF]) and second to be able to close the perturbed dynamical system of (λ, b) at the end of the proof (see proof of Lemma 16 below). Let E 0 ∈ R and

C 0 = 8E 0 |y| 2 Q 2 . Fix 0 < λ 0 ≪ 1 such that 2β 2-α + C 0 λ 2-α 0 > 0. For λ ∈ (0, λ 0 ], let F (λ) = λ0 λ dµ µ α 2 +1 2β 2-α + C 0 µ 2-α . ( 37
)
Note that the function F is related to the resolution of the system (33) for c 0 = C 0 , see proof of Lemma 16.

Lemma 5. Let s 1 ≫ 1. There exist b 1 and λ 1 such that

λ α 2 1 λ α 2 app (s 1 ) -1 + b 1 b app (s 1 ) -1 s -1 2 1 + s 2-4 α 1 , (38)
F (λ 1 ) = s 1 , E(b 1 , λ 1 ) = C 0 . ( 39
)
Proof. First, we choose λ 1 . Note that F is a decreasing function of λ satisfying F (λ 0 ) = 0 and lim λ↓0 F (λ) = +∞. Thus, there exists a unique λ 1 ∈ (0, λ 0 ) such that F (λ 1 ) = s 1 .

For λ ∈ (0, λ 0 ],

F (λ) - 2
α 2β 2-α λ α 2 1 + λ0 λ dµ µ α 2 +1   1 2β 2-α + C 0 µ 2-α - 1 2β 2-α   1 + λ0 λ dµ µ 1+ α 2 -(2-α) . Thus, F (λ) - 2 α 2β 2-α λ α 2    1 for α ∈ (0, 4 3 ), | log λ| for α = 4 3 , λ 2-3α 2 for α ∈ ( 4 3 , 2
). To simplify, we will use the non sharp but sufficient estimate

F (λ) - 2 α 2β 2-α λ α 2 λ -α 4 + λ 2-3α 2 . ( 40
)
Applied to λ 1 , it gives

s 1 - 2 α 2β 2-α λ α 2 1 λ -α 4 1 + λ 2-3α 2 1
and thus

λ α 2 1 λ α 2 app (s 1 ) -1 s -1 2 1 + s 2-4 α 1
.

Second, we choose b 1 . From the definition of E, we have

h(b) := λ 2 1 E(b, λ 1 ) = b 2 - 2 αs 1 2 - 2β 2 -α λ α 1 -λ α app (s 1 ) + λ α 1 (j,k)∈ΣK , j+k≥1 b 2j λ -kα 1 η j,k = b 2 - 2 αs 1 2 + O(s -5 2
1 ) + O(s

-4 α 1 ).
Observe that

|h(b app (s 1 ))| s -4 α 1 , |h ′ (b app (s 1 ))| ≥ 2b app (s 1 ) + O(s -3 1 ) ≥ s -1 1 . Since λ 2 1 ≈ s -4 α 1 , it follows that there exists a unique b 1 such that |b 1 -b app (s 1 )| s -3 2 1 + s 1-4 α 1 , h(b 1 ) = C 0 λ 2 1 ,
and so E(b 1 , λ 1 ) = C 0 .

Existence proof assuming uniform estimates

This section is devoted to the proof of Theorem 1 by a compactness argument, assuming uniform estimates on specific solutions of [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]. These estimates are given in Proposition 7.

3.1. Uniform estimates in rescaled time variable. The rescaled time depending on a suitable modulation of the solution u(t), we first recall without proof the following standard result (see e.g. [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF]).

Lemma 6 (Modulation). Let u(t) ∈ C(I, H 1 (R d )) for some interval I, be such that

sup t∈I inf λ0>0,γ0 λ d 2 0 u(t, λ 0 y)e iγ0 -Q(y) H 1 ≤ δ, (41)
for δ > 0 small enough. Then, there exist C 1 functions λ ∈ (0, +∞), b ∈ R, γ ∈ R on I such that u admits a unique decomposition of the form

u(t, x) = 1 λ d 2 (t) P b(t) + ε(t, y) e iγ(t) , y = x λ(t) ( 42
)
where ε satisfies the following orthogonality conditions on

I (ρ b (t, y) = ρ(y)e -i b(t)|y| 2 4 ) (ε, iΛP b ) 2 = ε, |y| 2 P b 2 = (ε, iρ b ) 2 = 0. ( 43
)
See [START_REF] Martel | Description of two soliton collision for the quartic gKdV equation[END_REF] for the definition of P b .

Let E 0 ∈ R. Given t 1 < 0 close to 0, following Remark 1, we define the initial rescaled time s 1 as

s 1 := C -1 s t 1 -α 4-α .
Let λ 1 and b 1 be given by Lemma 5 for this value of s 1 . Let u(t) be the solution of (7) for t ≤ t 1 , with data

u(t 1 , x) = 1 λ d 2 1 P b1 x λ 1 . ( 44
)
As long as the solution u(t) satisfies (41), we consider its decomposition (λ, b, γ, ε) from Lemma 6 and we define the rescaled time s by

s = s 1 - t1 t 1 λ 2 (τ ) dτ. ( 45
)
The heart of the proof of Theorem 1 is the following result, giving uniform backwards estimates on the decomposition of u(s) on [s 0 , s 1 ] for some s 0 independent of s 1 .

Proposition 7 (Uniform estimates in rescaled time). There exists s 0 > 0 independent of s 1 such that the solution u of (7) defined by (44) exists and satisfies (41) on [s 0 , s 1 ]. Moreover, its decomposition

u(s, x) = 1 λ d 2 (s)
(P b + ε) (s, y) e iγ(s) , y = x λ(s) ,
satisfies the following uniform estimates on [s 0 , s 1 ],

ε(s) H 1 s -(K+1) , λ α 2 (s) λ α 2 app (s) -1 + b(s) b app (s) -1 s -1 2 + s 2-4 α . ( 46
)
In addition,

|E(P b,λ,γ (s)) -E 0 | ≤ O(s -6 ).
Let us insist again that the key point in Proposition 7 is that s 0 and the constants in the estimates are independent of s 1 → +∞.

3.2.

Proof of Theorem 1 assuming Proposition 7. First, we convert the estimates of Proposition 7 in the original time variable t. We claim: Lemma 8 (Estimates in the t variable). There exists t 0 < 0 such that under the assumptions of Proposition 7,

for all t ∈ [t 0 , t 1 ], b(t) = C b |t| α 4-α (1 + o t↑0 (1)), λ(t) = C λ |t| 2 4-α (1 + o t↑0 (1)) (47) ε(t) H 1 |t| (K+1)α 4-α (48) |E(P b,λ,γ (t)) -E 0 | = o t↑0 (1) (49)
Proof of Lemma 8. Using (46), ( 45), for all large s < s 1 ,

t 1 -t(s) = s1 s λ 2 (σ)dσ = s1 s λ 2 app (σ) 1 + O(σ -1 2 ) + O(σ 2-4 α ) dσ.
Recall that t app given by (34) corresponds to the normalization

t app (s) = - +∞ s λ 2 app (σ), t app (s 1 ) = t 1 ,
from which we obtain

t(s) = t app (s)(1 + o(1)) = -C s s -4-α α [1 + o(1)] .
The estimates of Lemma 8 now follow directly follow from [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF] and Proposition 7 (see the definition of C λ and C b in (35) and ( 36)). Now, we finish the proof of Theorem 1 assuming Proposition 7.

Proof of Theorem 1. Let (t n ) ⊂ (t 0 , 0) be an increasing sequence such that lim n→∞ t n = 0. For each n, let u n be the solution of ( 7) on [t 0 , t n ] with final data at t n u n (t n , x) = 1

λ d 2 (t n ) P b(tn) x λ(t n ) , (50)
where λ(t n ) = λ 1 and b(t n ) = b 1 are given by Lemma 5 for s 1 = |C -1 s t n | -α 4-α , so that u n (t) satisfies the conclusions of Proposition 7 and of Lemma 8 on the interval [t 0 , t n ]. The minimal mass blow up solution for ( 7) is now obtained as the limit of a subsequence of (u n ). In a first step, we prove that a subsequence of (u n (t 0 )) converges to a suitable initial data. Indeed, from Lemma 8, we infer that (u n (t 0 )) is bounded in H 1 (R d ). Hence there exists a subsequence of (u n (t 0 )) (still denoted by (u n (t 0 )) and u

∞ (t 0 ) ∈ H 1 (R d ) such that u n (t 0 ) ⇀ u ∞ (t 0 ) weakly in H 1 (R d ) as n → +∞.
Now, we obtain strong convergence in H s (for some 0 < s < 1) by direct arguments. Let χ : [0, +∞) → [0, 1] be a smooth cut-off function such that χ ≡ 0 on [0, 1] and χ ≡ 1 on [2, +∞). For R > 0, define χ R : R d → [0, 1] by χ R (x) = χ(|x|/R). Take any δ > 0. By the expression of u n (t n ) in (50), we can choose R large enough (independent of n) so that

R d |u n (t n )| 2 χ R dx ≤ δ. (51)
It follows from elementary computations that

d dt R d |u n | 2 χ R dx = 2 Im R d ∇χ R • ∇u n ūn dx.
Hence from the geometrical decomposition

u n (t, x) = 1 λ d 2 n (t) P bn(t) + ε n )(t, y) e iγn(t) , y = x λ n (t) ,
and the smallness (47)-(48) of ε n and λ n we infer

d dt R d |u n (t)| 2 χ R dx ≤ C λ n (t)R e -R 2λn (t) + ε n (t) 2 H 1 ≤ C R |t| (-2 α +K+1) α 4-α .
Integrating between t 0 and t n , we obtain

R d |u n (t 0 )| 2 χ R dx ≤ C R |t 0 | (-2 α +K+1) α 4-α +1 + R d |u n (t n )| 2 χ R dx.
Combined with (51), for a possibly larger R, this implies

R d |u n (t 0 )| 2 χ R dx ≤ 2δ.
We conclude from the local compactness of Sobolev embeddings that for 0 ≤ s < 1:

u n (t 0 ) → u ∞ (t 0 ) strongly in H s (R d ), as n → +∞.
Let u ∞ (t) be the solution of ( 7) with u ∞ (t 0 ) as initial data at t = t 0 . From [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF] there exists 0 < s 0 < 1 such that the Cauchy problem for ( 7) is locally well-posed in H s0 (R d ). This implies that u ∞ exists on [t 0 , 0) and for any t ∈ [t 0 , 0),

u n (t) → u ∞ (t) strongly in H s0 (R d ), weakly in H 1 (R d ), as n → +∞. Moreover, since lim n→∞ u 2 n (t n ) = Q 2 , we have u 2 ∞ = Q 2 .
By weak convergence in H 1 (R d ) and the estimates from Lemma 8 applied to u n , u ∞ (t) satisfies (41), and denoting (ε ∞ , λ ∞ , b ∞ , γ ∞ ) its decomposition, we have by standard arguments (see e.g. [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF]), for any t ∈ [t 0 , 0),

λ n (t) → λ ∞ (t), b n (t) → b ∞ (t), γ n (t) → γ ∞ (t), ε n (t) ⇀ ε ∞ (t) H 1 (R d ) weak, as n → ∞.
The uniform estimates on u n from Lemma 8 give, on [t 0 , 0),

b ∞ (t) = C b |t| α 4-α (1 + o t↑0 (1)) , λ ∞ (t) = C λ |t| 2 4-α (1 + o t↑0 (1)) , ε ∞ (t) H 1 |t| (K+1)α 4-α , (52) b ∞ (t) λ 2 ∞ (t) = C b C 2 λ |t| α 4-α -4 4-α (1 + o t↑0 (1)) = 2 4 -α 1 |t| (1 + o t↑0 (1)) = σ |t| (1 + o t↑0 (1)) , (53)
which justifies the form ( 9) and the blow up rate [START_REF] Chang | Spectra of linearized operators for NLS solitary waves[END_REF]. Finally, we prove that E(u ∞ ) = E 0 . Let t 0 < t < 0. We have by ( 49) and ( 22),

E(b n (t), λ n (t)) - 8E 0 |y| 2 Q 2 = o t↑0 (1)
where the o t↑0 (1) is independent of n, and thus

E(b ∞ (t), λ ∞ (t)) - 8E 0 |y| 2 Q 2 = o t↑0 (1)
Using ( 22), we deduce

E(P b∞,λ∞,γ∞ (t)) -E 0 = o t↑0 (1)
and thus, by ( 52),

E(u ∞ (t)) -E 0 = o t↑0 (1).
Thus, by conservation of energy and passing to the limit t ↑ 0, we obtain E(u ∞ (t)) = E 0 .

3.3. Bootstrap estimates. The rest of the paper is devoted to the proof of Proposition 7. We use a bootstrap argument involving the following estimates:

ε(s) H 1 < s -K , λ α 2 (s) λ α 2 app (s) -1 + b(s) b app (s) -1 < s -δ(α) (54)
for some small enough universal constant δ(α) > 0. The following value is suitable in this paper

δ(α) = min 1 4 , 2 α -1 > 0. ( 55
)
For s 0 > 0 to be chosen large enough (independently of s 1 ), we define

s * = inf{τ ∈ [s 0 , s 1 ]; (54) holds on [τ, s 1 ]}. ( 56
)
Observe from (38) that

λ α 2 1 λ α 2 app (s 1 ) -1 + b 1 b app (s 1 ) -1 s -1 2 1 + s 2-4 α 1 ≪ s -δ(α) 1 ,
for s 1 large, and hence by the definition (44) of u(s 1 ), s * is well-defined and s * < s 1 . In §5, §6 and §7, we prove that (46) holds on [s * , s 1 ]. By a standard continuity argument, provided that s 0 is large enough, we obtain s * = s 0 which implies Proposition 7. The main lines of the proof are as follows: first, we derive modulation equations from the construction of P b , second we control the remaining error using a mixed Energy/Morawetz functional first derived in [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF].

Modulation equations

In this section, we work with the solution u(t) of Proposition 7 on the time interval [s * , s 1 ] (see ( 54)-( 56)). We justify that the dynamical system satisfied by the modulation parameters λ, b is at the main order given by [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF]. Define

Mod(s) =   b + λs λ b s + b 2 -θ 1 -γ s   .
Lemma 9 (Modulation equations and additional orthogonality). For all s ∈ [s * , s 1 ],

| Mod(s)| 1 s K+2 , (57)
| (ε(s), Q) 2 | 1 s K+1 . ( 58
)
Proof of Lemma 9. The proofs of the two estimates are combined. Since ε(s 1 ) ≡ 0, we may define

s * * = inf{s ∈ [s * , s 1 ]; | (ε(τ ), P b ) 2 | < τ -(K+2) holds on [s, s 1 ]}.
We work on the interval [s * * , s 1 ].

Since P b verifies equation [START_REF] Martel | Blow up for the critical gKdV equation II: minimal mass dynamics[END_REF], we obtain the following equation for ε:

iε s + ∆ε -ε + ibΛε + (f (P b + ε) -f (P b )) + λ α (g(P b + ε) -g(P b )) -i b + λ s λ Λ(P b + ε) + (1 -γ s )(P b + ε) + (b s + b 2 -θ) |y| 2 4 P b = -Ψe -i b|y| 2 4 . ( 59
)
where Ψ := Ψ K . Recall that equation ( 59) combined with the orthogonality conditions chosen on ε -see (43) -contains the equations of the modulation parameters. Technically, one differentiates in time the orthogonality conditions for ε, then uses the equation ( 59) on ε and the estimate ( 17) on the error term Ψ. Here, as in [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF], the orthogonality conditions are chosen to obtain quadratic control in ε. Since it is a standard argument (see e.g. [START_REF] Merle | On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation[END_REF][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] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF]), we only sketch relevant computations.

Consider for example the orthogonality condition (ε, iΛP b ) 2 = 0. Differentiating in s, we obtain

ε s , iΛP b + ε, i∂ s (ΛP b ) = 0. Since d ds (ΛP b ) = (ΛP ) s -i b s 4 |y| 2 ΛP e -i b 4 |y| 2 ,
and

(ΛP ) s = λ α α λ s λ Z + (j,k)∈ΣK kb 2j λ kα-1 (P + j,k + bP - j,k ) + b s (j,k)∈ΣK 2jb 2j-1 λ kα P + j,k + (j,k)∈ΣK (2j + 1)b 2j λ kα P - j,k
, proceeding as in the proof of Proposition 3, and using the properties of the functions P ± j,k , we note that

sup y∈R e y 2 d ds (ΛP b )(y) | Mod(s)| + b 2 (s) + λ α (s).
Thus, by (54),

| (ε, i∂ s (ΛP b )) 2 | ε(s) 2 | Mod(s)| + b 2 (s) + λ α (s) s -2 | Mod(s)| + s -(K+2) .
Next, we write ε s , iΛP b = -iε s , ΛP b and we use the equation of ε. We start by the contribution of the first line of (59). Remark that by (54), Therefore, using (54) and P = Q + O H 1 (s -2 ) (see the definition of P in ( 16)), we have

f (P b + ε) -f (P b ) = e -ib |y| 2 4 f P + e ib |y| 2 4 ε -f (P ) = e -ib
-∆ε + ε -ibΛε -(f (P b + ε) -f (P b )) + λ α (g(P b + ε) -g(P b )) , ΛP b = -∆ e ib |y| 2 4 ε + e ib |y| 2 4 ε -pQ p-1 e ib |y| 2 4 ε , ΛQ -ib |y| 2 2 Q + O(s -2 ε 2 ) = L + e ib |y| 2 4 ε , ΛQ - b 2 L -e ib |y| 2 4 ε , i|y| 2 Q + O(s -2 ε 2 ) = e ib |y| 2 4 ε, L + (ΛQ) - b 2 e ib |y| 2 4 ε, iL -(|y| 2 Q) + O(s -2 ε 2 ) = -2 ε, e -ib |y| 2 4 Q 2 + 2b ε, ie -ib |y| 2 4 ΛQ 2 + O(s -2 ε 2 ) = -2 (ε, P b ) 2 + 2b (ε, iΛP b ) 2 + O(s -2 ε 2 ) = O(s -(K+2) ).
Note that we have used algebraic relations from [START_REF] Dodson | Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state[END_REF], then (54), (ε, iΛP b ) 2 = 0 and the definition of s * * .

The part corresponding to the second line of (59) gives

-i b + λ s λ Λ(P b + ε) + (1 -γ s )(P b + ε) + (b s + b 2 -θ) |y| 2 4 P b , ΛP b 2 = -(b s + b 2 -θ) yP b 2 2 + O(| Mod(s)| ε 2 ) = -(b s + b 2 -θ)( yQ 2 2 + O s -2 ) + O(s -2 | Mod(s)|).
Finally, from the estimate (17) on Ψ, we have

Ψ, ΛP -ib |y| 2 2 P 2 s -2 | Mod(s)| + s -2(K+2) .
Combining the previous estimates, we find

|b s + b 2 -θ| s -2 | Mod(s)| + s -(K+2) .
Using the other orthogonality conditions in (43) in a similar way, together with [START_REF] Dodson | Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state[END_REF], we find

| Mod(s)| s -2 | Mod(s)| + s -(K+2) .
We deduce that for all s ∈ [s * * , s 1 ],

| Mod(s)| s -(K+2) . (60)
By conservation of the L 2 norm and (44), we have

u(s) 2 2 = u(s 1 ) 2 2 = P b (s 1 ) 2 2
. Thus, by (42),

(ε(s), P b ) 2 = 1 2 u(s) 2 2 -P b (s) 2 2 -ε(s) 2 2 = - 1 2 ε(s) 2 2 + 1 2 P b (s 1 ) 2 2 -P b (s) 2 2 .
Moreover, by [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF], ( 54) and (60),

d ds |P b | 2 s -(K+4) .
Integrating and combining the previous estimates with (54), we obtain, for all s ∈ [s * * , s 1 ],

| (ε(s), P b ) 2 | s -(K+3) . (61)
Therefore, s * * = s * and the estimates (60) and (61

) are proved on [s * , s 1 ]. Since |P b -Q| Q 1 2 s -1
, we obtain (58).

The mixed energy Morawetz monotonicity formula

In this section, following [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF], we introduce a mixed Energy/Morawetz functional to control the remaining part of the solution in H 1 (R d ). First, define the energy of ε

H(s, ε) := 1 2 ∇ε 2 2 + 1 2 ε 2 2 - R d (F (P b + ε) -F (P b ) -dF (P b )ε)dy -λ α R d (G(P b + ε) -G(P b ) -dG(P b )ε)dy.
Note that as in [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF], the time derivative of the linearized energy H for ε cannot be controlled alone, and one has to add a virial type functional such as

b 2 Im R d ∇ |y| 2 2
∇εεdy. In practice, due to the lack of control on yε 2 , we replace 1 2 |y| 2 by a function whose gradient is bounded, which we introduce now. Let φ : R → R be a smooth even and convex function, nondecreasing on R + , such that

φ(r) =    1 2 r 2 for r < 1,
3r + e -r for r > 2, and set φ(x) = φ(|x|). Let A ≫ 1 to be fixed. Define φ A by φ A (y) = A 2 φ y A and

J(ε) = 1 2 Im R d ∇φ A • ∇εεdy.
Finally, set

S(s, ε) = 1 λ 4 (s) (H(s, ε) + b(s)J(ε(s))).
The relevance of the functional S lies on the following two properties.

Proposition 10 (Coercivity of S). For any s ∈ [s * , s 1 ],

S(s, ε(s))

1 λ 4 (s) ε(s) 2 H 1 + O(s -2(K+1) . Proposition 11. For any s ∈ [s * , s 1 ], d ds [S(s, ε(s))] b λ 4 (s) ε(s) 2 H 1 + O(s -2(K+1) .
The rest of this section is organized as follows. We first prove Proposition 10 in §5.1. In §5.2 we compute the time derivative of H and in §5.3, the time derivative of J. We finish the proof of Proposition 11 in §5.4.

Coercivity of S.

We prove Proposition 10. We first claim a coercivity property for H, consequence of the properties of L + and L -(see [START_REF] Krieger | Two-soliton solutions to the three-dimensional gravitational Hartree equation[END_REF]) and of the orthogonality conditions of ε (see (43)).

Lemma 12 (Coercivity of H). For all

s ∈ [s * , s 1 ], H(s, ε) ε 2 H 1 + O(s -2(K+1)
). Proof. From the orthogonality conditions (43), (58), and estimates (54), the following holds:

ε, |y| 2 Q 2 = ε, |y| 2 P b 2 + O(|b| ε 2 ) + O(λ α ε 2 ) = O(s -1 ε H 1 ), (ε, iρ) 2 = (ε, iρ b ) 2 + O(|b| ε 2 ) = O(s -1 ε H 1 ), (ε, Q) 2 = O(s -(K+1) ).
From (54), we have

λ α R d (G(P b + ε) -G(P b ) -dG(P b )ε)dx = O(s -2 ε 2 H 1 ). Next, (denoting ε = ε 1 + iε 2 ), F (P b + ε) -F (P b ) -dF (P b )ε -1 + 4 d Q 4 d ε 2 1 - 1 2 Q 4 d ε 2 2 e -1 2 |y| |ε| 3 + |ε| 2+ 4 d + |ε| 2 (|b| + λ α ).
Thus, from (54),

F (P b + ε) -F (P b ) -dF (P b )ε -1 + 4 d Q 4 d ε 2 1 - 1 2 Q 4 d ε 2 2 O(s -1 ε 2 H 1 ), and
H(s, ε) - 1 2 L + ε 1 , ε 1 - 1 2 L -ε 2 , ε 2 O(s -1 ε 2 H 1 ).
Combining these estimates with the coercivity properties of L + , L -(see ( 13)), we obtain the result.

Since 54)), Lemma 12 implies Proposition 10.

|bJ(ε)| ≤ |b| ∇φ A ∞ ε 2 H 1 O(s -1 ε 2 H 1 ) (from (
For future reference, we also claim the following localized coercivity property (see similar statement in [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF] and [START_REF] Raphaël | Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS[END_REF]).

Lemma 13. There exists A 0 > 1 such that for any A > A 0 ,

1 2 R d ∇ε T ∇ 2 φ A ∇εdy + 1 2 ε 2 2 - R d (F (P b + ε) -F (P b ) -dF (P b )ε)dx ε 2 2 + O(s -2(K+1) ).
For now on, we consider A > A 0 .

5.2. Time variation of the energy of ε.

Lemma 14. For all s ∈ [s * , s 1 ], d ds [H(s, ε(s))] = λ s λ ∇ε 2 2 -f (P b + ε) -f (P b ), Λε + O(s -(2K+3) ) + O(s -2 ε 2 H 1 ).
Proof of Lemma 14. The time derivative for H separates into two parts:

d ds [H(s, ε(s))] = D s H(s, ε) + D ε H(s, ε), ε s ,
where D s (respectively, D ε ) denotes differentiation of the functional with respect to s (respectively, ε). In particular,

D s H(s, ε) = -(P b ) s (f (P b + ε) -f (P b ) -df (P b )ε) -λ α (P b ) s (g(P b + ε) -g(P b ) -dg(P b )ε) -α λ s λ λ α (G(P b + ε) -G(P b ) -dG(P b )ε) .
Note that

e i b|y| 2 4 (P b ) s = P s -ib s |y| 2 4 P = P s -i b s + b 2 -βλ α |y| 2 4 P + i b 2 -βλ α |y| 2 4 P.
By ( 24), (54) and Lemma 9, we obtain

|(P b ) s | s -2 e -|y| 2 and λ s λ λ α s -3 . Thus, |D s H(s, ε)| s -2 ε 2 H 1 . Now, we compute D ε H(s, ε), ε s . Note that (59) rewrites iε s -D ε H(s, ε) + Mod op (s)P b -i λ s λ Λε + (1 -γ s )ε + e -ib |y| 2 4 Ψ = 0, (62)
where

Mod op (s)P b := -i b + λ s λ ΛP b + (1 -γ s )P b + (b s + b 2 -θ) |y| 2 4 P b .
Using (62), since iD ε H(s, ε), D ε H(s, ε) = 0, we have

D ε H(s, ε), ε s = iD ε H(s, ε), iε s = -iD ε H(s, ε), Mod op (s)P b + λ s λ iD ε H(s, ε), iΛε -(1 -γ s ) iD ε H(s, ε), ε -iD ε H(s, ε), e -ib |y| 2 4 Ψ . ( 63
)
From the proof of Lemma 9

D ε H(s, ε) = -∆ε + ε -(f (P b + ε) -f (P b )) -λ α (g(P b + ε) -g(P b )) = e -ib |y| 2 4 L + Re e ib |y| 2 4 ε + iL -Im e ib |y| 2 4 ε + ibΛε + b 2 |y| 2 4 ε + O(s -2 |ε|).
Therefore, using the orthogonality conditions (43), (58) and estimates (54), we have (see also proof of Lemma 9),

D ε H(s, ε), ΛP b = -2 (ε, P b ) 2 + b (ε, iΛP b ) 2 + O(s -2 ε 2 ) = O(s -(K+1) ).
Thus, from Lemma 9,

λ s λ + b | D ε H(s, ε), ΛP b | O(s -(2K+3) ).
Using similar arguments we get

D ε H(s, ε), iP b = -4 (ε, ΛP b ) 2 + O(s -1 ε 2 ) = O(s -1 ε 2 ) = O(s -(K+1) )
and

D ε H(s, ε), i |y| 2 4 P b = (ε, ρ b ) 2 + O(s -1 ε 2 ) = O(s -1 ε 2 ) = O(s -(K+1) ).
Using Lemma 9, we obtain in conclusion for this term

iD ε H(s, ε), Mod op (s)P b = O(s -(2K+3) ).
Next, we have

iD ε H(s, ε), iΛε = D ε H(s, ε), Λε = -∆ε + ε -(f (P b + ε) -f (P b )) -λ α (g(P b + ε) -g(P b )), Λε .
Note that (by direct computations)

-∆ε, Λε = ∇ε 2 2 , ε, Λε = 0,
and by (54),

| λ α (g(P b + ε) -g(P b )), Λε | O(s -2 ε 2 H 1 ). Thus, λ s λ iD ε H(s, ε), iΛε = λ s λ ∇ε 2 2 -f (P b + ε) -f (P b ), Λε + O(s -3 ε 2 H 1 ).
For the third term in the right-hand side of (63), we claim

|(1 -γ s ) iD ε H(s, ε), ε | = (1 -γ s ) (f (P b + ε) -f (P b )) + λ α (g(P b + ε) -g(P b )), ε | Mod(s)| ε 2 2 + ε 2+ 4 d H 1 = O(s -4 ε 2 H 1 ).
Finally, the fourth term in the right-hand side of (63) is estimated by [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF] combined with Lemma 9 and (54)

| iD ε H(s, ε), Ψ | ≤ O(s -(K+4) ε H 1 ) ≤ O(s -(2K+3) ) + O(s -5 ε 2 H 1 ).
Gathering these estimates, we have proved the lemma.

d ds [J(ε(s))] = R d ∇ε T ∇ 2 φ A ∇εdy - 1 4 R d |ε| 2 ∆ 2 φ A dy -f (P b + ε) -f (P b ), 1 2 ∆φ A ε + ∇φ A ∇ε + O(s -(2K+2) ) + O(s -2 ε 2 H 1 ).
Proof. From the definition of J(ε), we have

d ds [J(ε(s))] = Re R d iε s 1 2 ∆φ A ε + ∇φ A ∇ε dy.
We replace iε s using (59). First, from standard computations Re

R d -∆ε 1 2 ∆φ A ε + ∇φ A ∇ε dy = R d ∇ε T ∇ 2 φ A ∇εdy - 1 4 R d |ε| 2 ∆ 2 φ A dy, Re R d ε 1 2 ∆φ A ε + ∇φ A ∇ε dy = 0, λ s λ Re R d iΛε 1 2 ∆φ A ε + ∇φ A ∇ε dy = 0.
Next,

λ α Re R d (g(P b + ε) -g(P b )) 1 2 ∆φ A ε + ∇φ A ∇ε dy = O(λ α ε 2 H 1 ) = O(s -2 ε 2 H 1 ).
The term corresponding to the second line of ( 59) is estimated as follows.

-

i(b + λ s λ )Λ(P b + ε) + (1 -γ s )(P b + ε) -(b s + b 2 -θ) |y| 2 4 P b , 1 2 ∆φ A ε + ∇φ A ∇ε | Mod(s)| ε H 1 O(s -(2K+2) ).
Finally, by [START_REF] Martel | Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation[END_REF] and Lemma 9,

Ψe -i b|y| 2 4 , 1 2 ∆φ A ε + ∇φ A ∇ε ≤ O(s -(K+4) ε H 1 ) ≤ O(s -(2K+4) ).
The result follows.

The Lyapunov property.

Proof of Proposition 11. By definition of S, we have

d ds [S(s, ε(s))] = 1 λ 4 -4 λ s λ (H(s, ε) + bJ(ε)) + d ds [H(s, ε(s))] + b d ds [J(ε(s))] + b s J(ε)
First, we claim the following estimate

d ds [H(s, ε(s))] + b d ds [J(ε(s))] = b R d ∇ε T ∇ 2 φ A ∇εdy -b ∇ε 2 2 + b A O( ε 2 H 1 ) + O(s -(2K+3) ). ( 64
)
Proof of (64). It is essential to see from Lemmas 14 and 15 that the main nonlinear terms are cancelling. Indeed, by integration by parts,

-Re R d (f (P b + ε) -f (P b ))Λεdy = - d 2 Re R d (f (P b + ε) -f (P b ))εdy -Re R d y∇(F (P b + ε) -F (P b ) -dF (P b )ε)dy + Re R d (f (P b + ε) -f (P b ) -df (P b )ε)y∇ Pb dy = - d 2 Re R d (f (P b + ε) -f (P b ))εdy + d Re R d (F (P b + ε) -F (P b ) -dF (P b )ε)dy + Re R d (f (P b + ε) -f (P b ) -df (P b )ε)y∇ Pb dy, -Re R d (f (P b + ε) -f (P b )) 1 2 ∆φ A ε + ∇φ A ∇ε dy = - 1 2 Re R d (f (P b + ε) -f (P b ))∆φ A εdy + Re R d ∆φ A (F (P b + ε) -F (P b ) -dF (P b )ε)dy + Re R d (f (P b + ε) -f (P b ) -df (P b )ε)∇φ A ∇ Pb dy.
Writing these two terms as above, it becomes clear that when y or ∇φ A appear, they are multiplied by ∇P b , which is exponentially decaying in space (see Proposition 3). Therefore, such terms are controlled by expressions involving only ε H 1 .

Therefore, combining Lemma 14 and Lemma 15, we have

d ds [H(s, ε(s))] + b d ds [J(ε(s))] = b R d ∇ε T ∇ 2 φ A ∇εdy -b ∇ε 2 2 + b + λ s λ ∇ε 2 2 - d 2 Re R d (f (P b + ε) -f (P b ))εdy + d Re R d (F (P b + ε) -F (P b ) -dF (P b )ε)dy + Re R d (f (P b + ε) -f (P b ) -df (P b )ε)y∇ Pb dy + b - 1 2 Re R d (f (P b + ε)-f (P b ))(∆φ A -d)εdy + Re R d (F (P b + ε)-F (P b )-dF (P b )ε)(∆φ A -d)dy + Re R d (f (P b + ε) -f (P b ) -df (P b )ε)(∇φ A -y)∇ Pb dy -b 1 4 R d |ε| 2 ∆ 2 φ A dy + O(s -(2K+3) ) + O(s -2 ε 2 H 1 ). By b + λs λ O(s -4 ), we have b + λ s λ ∇ε 2 2 - d 2 Re R d (f (P b + ε) -f (P b ))εdy + d Re R d (F (P b + ε) -F (P b ) -dF (P b )ε)dy +Re R d (f (P b + ε) -f (P b ) -df (P b )ε)y∇ Pb dy s -4 ε 2 H 1 . Next, |b| - 1 2 Re R d (f (P b + ε) -f (P b ))∆(φ A -d)εdy 1 s R d |P | 4 d |ε| 2 |∆φ A -d| + |ε| 2+ 4 d dy e -A 2 s ε 2 2 + O s -1 ε 2+ 4 d H 1
, and similarly for bRe

R d (F (P b + ε)-F (P b )-dF (P b )ε)(∆φ A -d)dy and bRe R d (f (P b + ε)-f (P b )- df (P b )ε)(∇φ A -y)∇ Pb dy. Next, -b R d |ε| 2 ∆ 2 φ A dy b A 2 ε 2 2 .
In conclusion for this term, we have obtained (64)

Using -λs λ = b + O(s -2 ) and the expression of H we have

-4 λ s λ H(s, ε) + d ds [H(s, ε(s))] + b d ds [J(ε(s))] 4bH(s, ε) + b R d ∇ε T ∇ 2 φ A ∇εdy -b ∇ε 2 2 + O(s -2 ε 2 H 1 ) + b A O( ε 2 H 1 ) + O(s -(2K+3) ) b R d ∇ε T ∇ 2 φ A ∇εdy + ε 2 2 -2 R d (F (P b + ε) -F (P b ) -dF (P b )ε)dx + 2bH(s, ε) + b A O( ε 2 H 1 ) + O(s -(2K+3)
) Thus, and the coercivity properties Lemma 12 and Lemma 13, we obtain (for A large enough) ) ) . This finishes the proof.

-4 λ s λ H(s, ε) + d ds [H(s, ε(s))] + b d ds [J(ε(s))] b ε 2 H 1 + O(s -(2K+3) ). Since b = O(s -1 ), b s = O(s -2 ) and J(ε) = O( ε 2 H 1 ), we have λ s λ b + |b s | |J(ε)| s -2 O( ε 2 H 1 ) and thus d ds [S(s, ε(s))] b λ 4 ε 2 H 1 + O(s -( 2K+2

End of the proof of Proposition 7

In this section, we finish the proof of Proposition 7. Recall from §3.3 that our objective is to prove s * = s 0 by improving estimates (54) into (46). Therefore, it is sufficient to prove the following lemma which closes the bounds (54) provided δ(α) > 0 has been chosen small enough (e.g. as in (55)).

Lemma 16 (Refined estimates). For all

s ∈ [s * , s 1 ], ε(s) H 1 s -(K+1) , (65)
λ α 2 (s) λ α 2 app (s) -1 + b(s) b app (s) -1 s -1 2 + s 2-4 α . ( 66
)
Proof. First, we prove (65). From Proposition 10, and the expression of S, there exists a universal constant κ > 1 such that for any s ∈ [s * , s 1 ],

1 κ

1 λ 4 ε 2 H 1 -κ 2 s -2(K+1) ≤ S(s, ε) ≤ κ λ 4 ε 2 H 1 . (67)
From Proposition 11, possibly taking a larger κ, d ds ) for all τ ∈ [s, s 1 ]}. Since ε(s 1 ) = 0, by continuity s † is well-defined and s † < s 1 . For the sake of contradiction, assume that s † > s * . In particular, ε(s

[S(s, ε(s))] ≥ 1 κ b λ 4 ε 2 H 1 -κ 2 s -2(K+1) . ( 68
) Define s † := inf{s ∈ [s * , s 1 ], ε(τ ) H 1 ≤ 2κ 2 τ -(K+1
† ) H 1 = 2κ 2 s -(K+1) † . Define s ‡ := sup{s ∈ [s † , s 1 ], ε(τ ) H 1 ≥ κτ -(K+1) for all τ ∈ [s † , s]}. In particular, s † < s ‡ < s 1 and ε(s ‡ ) H 1 = κs -(K+1) ‡
, and from (68), S is nondecreasing on [s † , s ‡ ]. From equations (67)-(68) and the estimates on λ (see ( 46)), we obtain , which is a contradiction. Hence s † = s * and (65) is proved. Now, we prove (66). The main idea is to use a conservation law on (b, λ) which can be found from the differential system satisfied by (b, λ), but that we rather derive from energy properties of the blow up profile. Recall that λ(s 1 ) = λ 1 and b(s 1 ) = b 1 are chosen in Lemma 5 so that F (λ(s 1 )) = s 1 and E(b(s 1 ), λ(s 1 )) = 8E0 |y| 2 Q 2 . In particular, we deduce from [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF] that |E(P b1,λ1,γ1 ) -E 0 | s -6 1 . Using ( 21) and ( 54), (57), for all s ∈ [s * , s 1 ],

ε(s † ) 2 H 1 -κ 2 s -2(K+1) † ≤ κλ 4 (s † )S(s † , ε(s † )) ≤ κλ 4 (s † )S(s ‡ , ε(s ‡ )) ≤ κ 2 λ 4 (s † ) λ 4 (s ‡ ) ε(s ‡ )
d ds E(P b,λ,γ ) s -(K+2)+ 4 α .

In particular, by integration, we find, for all s ∈ [s * , s 1 ], |E(P b,λ,γ (s))-E 0 | s -6 (recall K > 20/α) and using [START_REF] Merle | Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power[END_REF] at s,

E(b(s), λ(s)) - 8E 0 |y| 2 Q 2 s -6 .
We obtain from the expression ( 23) of

E with C 0 = 8E0 |y| 2 Q 2 : b 2 - 2β 2 -α λ α -C 0 λ 2 λ α s 2
where the error term O( λ α s 2 ) comes from θ and cannot be improved. In this estimate, since λ 2 ≈ s -4 α and λ α s 2 ≈ s -4 , whether or not C 0 λ 2 is controled by the error term depends on the value of α. We address both cases at once in what follows. Since b ≈ λ and thus, by the choice F (λ(s 1 )) = s 1 , we obtain F (λ(s)) = s + O(s -1 ).

Therefore, using (40) and the definition of λ app (s) in [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF],

λ α 2 app (s) λ α 2 (s) -1 s -1 2 + s 2-4 α .
We reinject this estimate into (69) and use the definition of b app to conclude:

b(s) = b app (s) + O(s -3 2 + s -4-α α ).
This finishes the proof.

Appendix A. Proof of Lemma 1

By contradiction, assume that there exists a blow up solution u(t) of ( 1) with ǫ = -1 and u(t) 2 = Q 2 . Let a sequence t n → T * ∈ (0, +∞] with ∇u(t n ) 2 → +∞ and consider the renormalized sequence

v n (x) = λ(t n ) d 2 u(t n , λ(t n )x), λ(t n ) = ∇Q 2 ∇u(t n ) 2 .
Then, by conservation of mass, v n 2 = Q 2 and conservation of energy and ǫ < 0,

E 0 = E(u n ) ≥ E crit (u n ) = E crit (v n ) λ 2 (t n ) .
Therefore, the sequence v n satisfies:

v n 2 = Q 2 , ∇v n 2 = ∇Q 2 , lim sup n→+∞ E crit (v n ) ≤ 0.
From standard concentration compactness argument, see [START_REF] Merle | The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation[END_REF][START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF], there holds, up to a subsequence, for some

x n ∈ R d , γ n ∈ R, v n (. -x n )e iγn → n→+∞ Q in H 1 (R d ).
In particular,

u(t n ) p+1 = v n p+1 λ d(p-1)
2(p+1) (t n )

→ +∞ as n → ∞, which contradicts the a priori bound from the energy conservation law and ( 3):

E 0 = E(u) ≥ E crit (u) + 1 p + 1 |u| p+1 ≥ 1 p + 1 |u| p+1 . Appendix B. Proof of Lemma 2
For the sake of simplicity, we give the proof only for d ≥ 2. The case d = 1 would require an additionnal (standard) concentration compactness argument (see [START_REF] Weinstein | Nonlinear Schrödinger equations and sharp interpolation estimates[END_REF]). For M < Q 2 , set In particular, for 0 < λ ≪ 1 and u ≡ 0, E(v λ ) < 0 and (71) follows.

A M = {u ∈ H 1 rad (R d
Second, let u λ = λ 2 p-1 u(λx), so that (73)

E(u λ ) = λ 4 p-1 +2-d 1 2 |∇u| 2 - 1 p + 1 |u| p+1 - λ 2 p-1 (2+
To finish, let (u n ) be a minimizing sequence. Up to a subsequence and from the standard radial compactness of Sobolev embeddings (see [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF])

u n ⇀ u in H 1 (R d ), u n → u in L q , 2 < q ≤ 2 + 4 d .

Proposition 3 .

 3 Let λ(s) > 0 and b(s) ∈ R be C 1 functions of s such that λ(s) + |b(s)| ≪ 1. (i) Existence of a blow up profile. For any (j, k) ∈ Σ K , there exist real-valued functions P + j,k ∈ Y, P - j,k ∈ Y and β j,k ∈ R such that P (s, y) = PK (y; b(s), λ(s)), where PK is defined by PK (y; b, λ) := Q(y) + (j,k)∈ΣK

|y| 2 4 4 ΛP -ib |y| 2 2

 442 df (P ) e ib |y| 2 4 ε + O(|ε| 2 ) = e -ib |y| 2 4 df (P ) e ib |y| 2 4 ε + O(s -2 |ε|), λ α (g(P b + ε) -g(P b )) = O(λ α |ε|) = O(s -2 |ε|), and ∆ε + ibΛε = e -ib |y| 2 4 ∆ e ib |y| 2 4 ε + b 2 |y| 2 4 ε, ΛP b = e -ib |y| 2 P .

5. 3 .

 3 The time derivative of the Morawetz part. Lemma 15. For all s ∈ [s * , s 1 ],

+ 1 =

 1 K+1) , we obtain (see (37) for the definition of F ) |F ′ (s) -1| s -2 . (70) Integrating (70) on [s, s 1 ], we obtain|F (λ(s 1 )) -F (λ(s)) -(s 1 -s)| s -1

  ) with u 2 = M } and consider the minimization problemI M = inf u∈AM E(u).

First 1 1 λ 2 -

 112 < p < 1 + 4 d , we note that I M > -∞ and that any minimizing sequence is bounded inH 1 (R d ). Let u ∈ A M and v λ (x) = λ d 2 u(λx), then v λ ∈ A M and E(v λ ) = λ 2 E crit (u) -

Together with u λ 2 = λ 2 p- 1 -d 2 u 2 ,

 212 which implies d dλ u λ 2 |λ=1 > 0, this proves that I(M ) is decreasing in M.

  2 H 1 ≤ κ 4 λ 4 (s † ) λ 4 (s ‡ ) s

	since K > 4/α. Therefore ε(s † )	2 H 1 ≤ 3κ 4 s	-2(K+1) †
			-2(K+1) ‡	≤ 2κ 4 s ‡ s †	8 α	s	-2(K+1) ‡	≤ 2κ 4 s	-2(K+1) †	,