Thomas Duyckaerts

Phan Van Tin

PROFILE DECOMPOSITION AND SCATTERING FOR GENERAL NONLINEAR SCHRÖDINGER EQUATIONS

Keywords: February 10, 2024. 2020 Mathematics Subject Classication. 35Q55, 35A01, 35B40 Nonlinear Schrödinger equations, prole decomposition, stability theory, scattering

We consider a Schrödinger equation with a nonlinearity which is a general perturbation of a power nonlinearity. We construct a prole decomposition adapted to this nonlinearity. We also prove global existence and scattering in a general defocusing setting, assuming that the critical Sobolev norm is bounded in the energy-supercritical case. This generalizes several previous works on double-power nonlinearities.

1.1. General setting. This article concerns the nonlinear Schrödinger (NLS) equation (1.1) i∂ t u + ∆u = g(u), in space dimension d ⩾ 1, where g(u) is a L 2 -supercritical nonlinearity of the form (1.2) g(u) = ι 0 |u| p0 u + g 1 (u), ι 0 ∈ {±1} and g 1 : C → C is a L 2 -supercritical lower-order term, i.e. g 1 is C k0 and ∃p 1 , p 2 ,

4 d < p 1 ⩽ p 2 < p 0 , ∀u ∈ C, ∀k ∈ 0, k 0 , |D k g 1 (u)| ≲ |u| p2-k + |u| p1-k ,
for some k 0 that we will be specied later. The model case for g is a sum of k + 1 powers, k ⩾ 1

(1.3) g(u) = k j=0 ι j |u| pj u, ι j ∈ R \ {0}, 4 d < p k < . . . < p 1 < p 0 .
With the regularity assumption:

(1.4) ∀j ∈ {0, . . . , k}, p j is an even integers, or ⌈s 0 ⌉ < p j ,

where s 0 = d 2 -2 p0 (see Subsection 2.1 for the notation ⌈s 0 ⌉ and other notations that will be used in this introduction). The case of a double-power, energy-subcritical (i.e. s 0 ⩽ 1) nonlinearity was studied in many work. Our goal is give a general setting for the study of (1.1) which includes also the supercritical case s 0 > 1 and more general lower order nonlinearity g 1 .

We are interested in the global well-posedness and scattering for solutions of (1.1). Neglecting the lower order term in g, we obtain the usual (NLS) equation with a single power nonlinearity (1.5) i∂ t u + ∆u = ι 0 |u| p0 u,

The equation (1.5) is invariant by scaling: if u is a solution of (1.5) and λ > 0, then u λ (t, x) = λ 2/p0 u(λ 2 t, λx) is also a solution of (1.5). The critical Sobolev exponent s 0 for (1.5) is the unique s 0 such that ∥u∥ Ḣs 0 = ∥u λ (0)∥ Ḣs 0 for all λ > 0. The equation (1.5) is well-posed in Ḣs0 (with additional technical conditions in high dimensions ensure a minimal regularity of the nonlinearity) see [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF].

1.2. Well-posedness and prole decomposition. In Section 2 we prove that (1.1) is locally well-posed in the inhomogenous space H s , for any s ⩾ s 0 , assuming that g ∈ C ⌈s⌉+1 and L 2supercritical (see Assumption A p.8 for the precise assumptions). We also develop a full stability/long time perturbation theory for (1.1). The existence and uniqueness of solutions yields for all u 0 ∈ H s a maximal interval of existence I max (u 0 ) = (T -(u 0 ), T + (u 0 )). Assuming g(u) = G ′ (|u| 2 )u for some C 1 function G, with G(0) = 0 we also have conservation of the mass: Our rst main result is the construction of a prole decomposition adapted to bounded sequences of H s0 solutions of equation (1.1), which builds up on the stability theory developed in Section 2. This amounts to expressing such a sequence as a sum of three distinct types of objects: a dispersive behaving as a solution of the linear Schrödinger equation, concentrating (nonlinear) proles that are solutions of (1.5) rescaled with a scaling parameter going to 0, and nonconcentrating proles, that are solutions of (1.1). We refer to Section 3, and in particular Subsection 3.3 and Theorem 3.15 for the detailed statements. This prole decomposition is valid in the general setting described above, and generalizes various previous constructions on double power nonlinearities (see Subsection 1.5 below for references).

M (u(t)) = R d |u(t, x)| 2 dx,
1.3. Global well-posedness. Solutions of (1.1) are not always global. Indeed, in the case of a double power nonlinearity, if ι 0 = -1 (the higher-order nonlinearity is focusing), a standard convexity argument (see [START_REF] Zakharov | Collapse of langmuir waves[END_REF][START_REF] Glassey | On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations[END_REF]) shows that any solution with negative energy and nite variance blows up in nite time (at least in the case of a double-power nonlinearity, (1.3) with k = 2).

More surprisingly, Merle, Raphaël, Rodnianski and Szeftel [START_REF] Merle | On blow up for the energy super critical defocusing nonlinear Schrödinger equations[END_REF] have constructed solutions of the homogeneous equation (1.5) with a defocusing, energy-supercritical nonlinearity s 0 > 1, ι 0 = 1 that blow up in nite time. It is known however in the defocusing case ι 0 = 1, for many values of p 0 , that solution of (1.5) that remains bounded in the critical Sobolev space are global and scatter.

We will prove that this property implies that solutions of (1.1), with g of the form (1.2) satisfying the same boundedness condition are global. We will thus consider: Property 1.1. Let A 0 ∈ (0, ∞]. For any solution u of (1.5) with initial data in Ḣs0 , if (1.6) lim sup t→T+(u) ∥u(t)∥ Ḣs 0 < A 0 .

Then T + (u) = +∞ and u scatters for positive times in Ḣs0 , i.e. there exists v 0 ∈ Ḣs0 such that lim t→∞ u(t) -e it∆ v 0 Ḣs 0 = 0.

It is conjectured that Property 1.1 always holds for A 0 = ∞ in the defocusing case ι 0 = 1. In the defocusing energy-critical case d ⩾ 3, p 0 = 4 d-2 , Property 1.1 is unconditional (the bound (1.6) is given by conservation of the energy), and was proved in [START_REF] Colliander | Global well-posedness and scattering for the energy-critical Schrödinger equation in R 3[END_REF], [START_REF] Ryckman | Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4[END_REF] and [START_REF] Visan | The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions[END_REF],

The study of Property 1.1 in the defocusing case for other critical exponents was initiated in [START_REF] Kenig | Scattering for H 1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions[END_REF] where it was proved when d = 3, p 0 = 2 (thus s 0 = 1/2), for radial solutions. It was later proved in many other cases: see [START_REF] Killip | Energy-supercritical NLS: critical Ḣs -bounds imply scattering[END_REF], [START_REF] Miao | The defocusing energy-supercritical NLS in four space dimensions[END_REF], [START_REF] Lu | The radial defocusing energy-supercritical NLS in dimension four[END_REF], [START_REF] Dodson | The defocusing quintic NLS in four space dimensions[END_REF] for supercritical nonlinearities in dimension d ⩾ 4 (with technical restriction if d ⩾ 7), and [START_REF] Xie | Global well-posedness and scattering for the defocusing Ḣs -critical NLS[END_REF], [START_REF] Murphy | The defocusing Ḣ1/2 -critical NLS in high dimensions[END_REF], [START_REF] Murphy | The radial defocusing nonlinear Schrödinger equation in three space dimensions[END_REF], [START_REF] Gao | On scattering for the defocusing high dimensional inter-critical NLS[END_REF], [START_REF] Yu | Global well-posedness and scattering for the defocusing Ḣ1/2 -critical nonlinear schrödinger equation in R 2[END_REF] and [START_REF] Bellazzini | Scattering for non-radial 3d nls with combined nonlinearities[END_REF] for several energy-subcritical nonlinearities.

In the focusing case ι 0 = -1, Property 1.1 is only known to hold when A 0 is small, from the small data theory for equation (1.5). In the focusing energy critical case, p 0 = 4 d-2 , it follows from [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF], [START_REF] Killip | The focusing energy-critical nonlinear Schrödinger equation in dimensions ve and higher[END_REF] and [START_REF] Dodson | Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension d = 4[END_REF] that Property 1.1 holds with A 0 = ∥∇W ∥ 2 L 2 in dimension d ⩾ 4, and d = 3 in the radial case, where W is the ground state of equation (1.5). This is optimal, since the existence of W shows that Property 1.1 does not hold for A 0 > ∥∇W ∥ 2

L 2 .
When ι 0 = -1 and s 0 ∈ (0, 1), there exists standing wave solutions of (1.5), so that Property 1.1 does not hold for large A 0 . When ι 0 = -1, s 0 > 1, travelling wave solutions in Ḣs0 do not exist, and the validity of Property 1.1 for large A 0 is an open question. Let us also mention that the analogue of this property was proved for radial focusing nonlinear wave equation in the energy supercritical and subcritical settings (see e.g. [START_REF] Duyckaerts | Scattering for radial, bounded solutions of focusing supercritical wave equations[END_REF] for supercritical p 0 in space dimension 3).

Our result on global well-posedness is as follows: Theorem 1.2. Let ι 0 , s 0 , g such that Assumption B holds, and such that Property 1.1 is valid for some A 0 ∈ (0, ∞]. Let u be a solution of (2.1), with initial data in H s0 , such that (1.6) holds. Then T + (u) = +∞.

We refer to Denition 2.7 and Section 3, p. [START_REF] Gao | On scattering for the defocusing high dimensional inter-critical NLS[END_REF] for the details of Assumption B. Let us mention that a multi-power non-linearity as in (1.3) with the additional technical assumption (1.4) satises this assumption.

If Property 1.1 holds only for radial functions, then the conclusion of Theorem 1.2 is also valid when restricted to radial solutions of (1.1). Theorem 1.2 is new in the energy supercritical case s 0 > 1. In the energy-subcritical and energycritical cases 0 < s 0 ⩽ 1, it was proved in [START_REF] Tao | The nonlinear Schrödinger equation with combined power-type nonlinearities[END_REF] for a double power non-linearity with the stronger assumptions ι 0 = 1, u 0 ∈ H 1 , without assuming Property 1.1.

The proof of Theorem 1.2 uses the prole decomposition mentioned above. The Property 1.1 is used to deal with the concentrating proles.

In the defocusing energy-critical case, by conservation of the energy and the scattering result for the energy-critical Schrödinger defocusing equation, the global well-posedness is unconditional: Corollary 1.3. Assume ι 0 = 1. Let d ∈ {3, 4, 5}, and g such that Assumption B holds with

p 0 = 4 d-2 .
Let u be a solution of (2.1), with initial data in H 1 . Then u is global. Corollary 1.3 was proved in the case of a double-power nonlinearity in [START_REF] Zhang | On the Cauchy problem of 3d energy-critical Schrödinger equations with subcritical perturbations[END_REF] (d = 3) and in [START_REF] Tao | The nonlinear Schrödinger equation with combined power-type nonlinearities[END_REF] (for general d ⩾ 4). Corollary 1.3 generalizes these results to more general perturbations of the energy-critical nonlinearity, in low dimension. The restriction on the dimension is due to the regularity assumption g ∈ C 2 in Assumption B. This restriction could be weakened using a rened well-posedness/stability theory as in [START_REF] Bulut | Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions[END_REF].

1.4. Scattering. Our next goal is to give sucient conditions for scattering of solutions of (1.1).

We recall that a solution of (1.1) with initial data in H s0 is said to scatter (in H s0 , forward in time) when T + (u) = +∞ and there exists v 0 ∈ H s0 such that lim t→∞ e it∆ v 0 -u(t) H s 0 = 0.

We will prove scattering for a general defocusing nonlinearity dened as follows:

Denition 1.4. The nonlinearity g is defocusing when it is of the form g(u) = G ′ (|u| 2 )u for some C 1 function G such that for almost all a > 0, aG ′ (a) -G(a) > 0.

Note that any power nonlinearity with a positive coecient is defocusing in the sense of Definition (1.4). For a multi-power nonlinearity g as in (1.3) the assumption that g is defocusing is equivalent to

(1.7) ∀s > 0, k j=0 ι j p j p j + 2 s p j 2 -1 > 0.
Note that (1.7) holds when all the ι j are positive. Also, (1.7) implies ι 0 > 0, ι k > 0.

For our scattering results, we distinguish between the energy-supercritical case and the energysubcritical case:

Theorem 1.5. Let d ⩾ 3 and assume s 0 > 1. Let g be a defocusing nonlinearity that satises Assumption B. Assume that Property 1.1 holds for some A 0 ∈ (0, ∞]. Let u be a solution of (1.1) with initial data u 0 ∈ H s0 and that satises (1.6). Then u is global and scatter forward in time.

When s 0 ⩽ 1, we must further assume that the initial data has nite energy.

Theorem 1.6. Assume 0 < s 0 ⩽ 1. Let g be a defocusing nonlinearity that satises Assumption B. Let u be a solution of (1.1) with initial data u 0 ∈ H 1 . Then u is global and scatter in both time directions.

Note that the assumption that g is defocusing together with the fact that G(a)/a goes to 0 as a goes to 0 (which is a consequence of Assumption B) implies that G ⩾ 0. Thus the assumptions of Theorem 1.6 and the conservation of mass and energy imply that any H 1 solution of u is bounded in H 1 . Theorems 1.5 and Theorem 1.6 show that scattering holds for a multi-power nonlinearity satisfying (1.3), (1.4) and (1.7).

In the case of a double-power nonlinearity the condition (1.7) is equivalent to ι 0 > 0, ι 1 > 0. In this case, Theorem 1.5 is new. Theorem 1.6 is proved in [START_REF] Tao | The nonlinear Schrödinger equation with combined power-type nonlinearities[END_REF]Theorem 1.3] and [START_REF] Zhang | On the Cauchy problem of 3d energy-critical Schrödinger equations with subcritical perturbations[END_REF]).

For a double-power nonlinearity with (ι 0 > 0, ι 1 < 0) or (ι 0 < 1, 0 < s 0 < s 1 < 1), there are solitary wave solutions and thus it is impossible to prove an analog of Theorems 1.5 and 1.6. This is an open question for other double power nonlinearities.

When only the main order term of the nonlinearity is defocusing, i.e. when ι 0 > 0, scattering holds for initial data with small mass. We again distinguish between s 0 > 1 and s 0 ∈ (0, 1] Theorem 1.7. Let d ⩾ 3 and assume s 0 > 1. Let g be a nonlinearity that satises Assumption B with ι 0 > 0. Assume that Property 1.1 holds for some A 0 ∈ (0, ∞]. There exists m c > 0 such that any solution of (1.1) with initial data in H s0 such that M (u 0 ) < m c and that satises (1.6) is global and scatters forward in time.

Theorem 1.8. Let g be nonlinearity that satises Assumption B with ι 0 > 0 and s 0 ∈ (0, 1]. There exists m c > 0 such that any solution of (1.1) with initial data in H 1 and such that M (u 0 ) < m c is global and scatters in both time directions. Remark 1.9. In Theorem 1.5 and Theorem 1.7, if the analog of (1.6) as t goes to T -(u) is valid then the conclusions hold backward in time.

Theorem 1.8 is new, even for double power nonlinearities. Theorem 1.8 generalizes [50, Theorem 1.3] which concerns a double power nonlinearity with ι 0 > 0 and ι 1 < 0. In particular the case where d = 3, p 0 = 4 (thus s 0 = 1), p 1 = 2 scattering was proved to hold for a larger set of initial data in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF], [START_REF] Killip | Scattering for the cubic-quintic NLS: crossing the virial threshold[END_REF]. In a subsequent work, we will use the material of this article, together with [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF] to obtain an improvement of Theorems 1.7 and 1.8 in the same spirit.

The proofs of Theorem 1.5 and Theorem 1.7 follow the by now classical rigidity-compactness roadmap (see [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]), using the prole decomposition constructed in Section 3. This provides, in a contradiction argument, a global critical solution u c of (1.1) such that there exists x(t) such that

K = {u c (t, • + x(t)), t ∈ R} has compact closure in H s0 .
To exclude this critical element and obtain a contradiction, we use the virial identity

(1.8) d dt Im x • ∇u u = 2Φ(u),
where

Φ(u) = |∇u| 2 + ι 1 dp 1 2(p 1 + 2) |u| p1+2 + ι 0 dp 0 2(p 0 + 2) |u| p0+2 ,
and the center of mass identity:

(1.9)

d dt x|u(t)| 2 = 2P (u).
which are valid for solutions of u with enough decay at innity.

Using a localized version of (1.9), one can prove:

(1.10)

lim t→+∞ 1 t |x(t) -X(t)| = 0 where X(t) = 2 P (u c ) M (u c ) t.
When the momentum of u c is zero, this allows to control the growth of x(t). A localized version of (1.8) using the relative compactness K, gives a contradiction if inf t∈R Φ(u c (t)) -|P (uc)| 2 M (uc) > 0. In the defocusing/defocusing case (as in Theorem 1.5), this property is true whenever u c is not identically zero. In the defocusing/focusing case, we show, using the quite complete study of the elliptic problem in [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF], and some of the ideas in [START_REF] Killip | Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on R 3[END_REF], that Φ is positive in the region described in the assumptions of Theorem 1.7, yielding again the desired contradiction.

One must adapt this argument when the momentum of u c is not zero. The standard strategy, going back to [START_REF] Duyckaerts | Scattering for the non-radial 3D cubic nonlinear Schrödinger equation[END_REF] is to use the Galilean transformation to reduce to the case to a critical solution with zero momentum. However the eect of the Galilean transform on the Sobolev norm Ḣs0 of the solution is not explicit, and thus the strategy breaks down in the case s 0 > 1, where our proof relies on an induction-type argument on this norm. To tackle this diculty, we observe that (1.8), (1.9) and the conservation of momentum imply:

d dt Im (x -X(t))∇u u = 2Φ(u(t)) - 2|P (u)| 2 M (u) ,
which we localize with a time-dependant localization around X(t). This gives again a contradiction using an improved Cauchy-Schwarz inequality going back to [START_REF] Banica | Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain[END_REF] to prove that Φ(u)

-|P (u)| 2 M (u) is still
positive for the solutions that we consider.

1.5. Previous works. To our knowledge, the prole decomposition for a general nonlinearity of the form (1.2) was not considered before.

Dynamics of nonlinear Schrödinger equations with general nonlinearities were treated in very few works. In [START_REF] Liu | The large time asymptotic solutions of nonlinear schrödinger type equations[END_REF], [START_REF] Soer | Soliton resolution for nonlinear schrödinger type equations in the radial case[END_REF], the authors prove that any radial, bounded, global solutions of a nonlinear Schrödinger equation with a general energy-subcritical or energy-critical nonlinearity is asymptotically the sum of a radiation term, solution of the linear Schrödinger equation, and a localized term.

We refer to [START_REF] Soer | On the large time asymptotics of schrödinger type equations with general data[END_REF] for results in a nonradial setting.

Let us mention a few works on NLS equation with a double-power nonlinearity. The study of this type of equation was initiated in [START_REF] Zhang | On the Cauchy problem of 3d energy-critical Schrödinger equations with subcritical perturbations[END_REF], in dimension 3, where the author investigated the global well-posedness, scattering and blow-up phenomena in the case p 0 = 4. This includes in particular a scattering result for small mass, in the spirit of Theorem 1.7. Similar results were obtained in [START_REF] Tao | The nonlinear Schrödinger equation with combined power-type nonlinearities[END_REF], in general dimension d in the energy-critical and subcritical setting s 0 ⩽ 1.

The problem with p 1 = 4 d , ι 1 = -1, p 0 < 4 d-2 was considered in [11], [43], where the author investigated scattering below or at the mass of the ground-state for the mass-critical homogeneous equation. See also [START_REF] Carles | Orbital stability vs. scattering in the cubic-quintic Schrödinger equation[END_REF] which considers the case (p 0 , p 1 , ι 0 , ι 1 ) = (4, 2, 1, -1) in space dimensions 1, 2 and 3.

The problem with a focusing dominant nonlinearity ι 0 = -1 was considered in many works, always in the energy-critical or subcritical cases. Let us mention in particular [START_REF] Akahori | Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equations with energy-critical growth at low frequencies[END_REF] where a nine-set theorem in the spirit of [START_REF] Nakanishi | Global dynamics above the ground state energy for the focusing nonlinear Klein Gordon equation[END_REF], [START_REF] Nakanishi | Global dynamics above the ground state energy for the cubic NLS equation in 3D[END_REF] was proved. We also refer to [START_REF] Miao | The dynamics of the 3d radial NLS with the combined terms[END_REF], [START_REF] Xie | Scattering for focusing combined power-type NLS[END_REF], [START_REF] Huang | Sharp conditions of global existence and scattering for a focusing energy-critical Schrödinger equation[END_REF], [START_REF] Xu | Long time dynamics of the 3d radial NLS with the combined terms[END_REF], [START_REF] Cheng | Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case[END_REF], [START_REF] Miao | On the 4d nonlinear Schrödinger equation with combined terms under the energy threshold[END_REF], [START_REF] Luo | Sharp scattering for the cubic-quintic nonlinear Schrödinger equation in the focusing-focusing regime[END_REF] for scattering result or scattering blow-up dichotomy in the case ι 0 = -1.

Let us mention that in several of the preceding works, the authors construct and use a prole decomposition adapted to NLS equation with a particular double-power nonlinearity. The prole decomposition in Section 3 generalises these proles decompositions to the large class of nonlinearities described above.

1.6. Outline. We conclude this introduction by an outline of the article. Section 2 is devoted to some preliminaries, well-posedness and perturbation theory for the NLS equation (1.1) with a general nonlinearity g. In Section 3, we construct a prole decomposition for sequences that are bounded in H s0 adapted to (1.1) with a nonlinearity of the form (1.2). This prole decomposition is based on the linear prole decomposition of Shao [START_REF] Shao | Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation[END_REF] (which relies ultimatly on the result by Merle-Vega [START_REF] Merle | Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2[END_REF]) and the long-time perturbation result proved in Subsection 2.4. In Section 4, we use this prole decomposition to prove our global well-posedness result, Theorem 1.2. In Section 5 we prove a general rigidity result for solutions of (1.1) with a relatively compact trajectory in H s0 . In Section 6, we prove our scattering result Theorems 1.5 and 1.7, using the material of the preceding sections.

Acknowledgment

The second author is supported by Post-doc fellowship from Labex MME-DII: SAIC/2022 No 10078.

Preliminaries and Cauchy theory

This section is concerned with the Cauchy and stability theory for the equation:

(2.1)

i∂ t u + ∆u = g(u).
for a general L 2 supercritical nonlinearity g(u) in space dimension d ⩾ 1. Our assumptions on g will of course include the case of double power nonlinearities that we are interested in. We start by introducing some notations and functions spaces (see 2.1) and recalling some nonlinear estimates (see 2.2). In 2.3 we prove estimates on the nonlinearity g(u) that are crucial for the well-posedness theory. In 2.4 we prove our main results.

2.1. Notations and function spaces. For s ∈ R, ⌈s⌉ is the smallest integer number larger or equal s, and ⌊s⌋ is the largest integer smaller or equal s (the integer part of s). If j and k are integers with j ⩽ k, we denote by j, k = {j, j + 1, . . . , k -1, k}.

When A and B are two positive quantities depending on some parameters, we denote A ≲ B when there is a constant C > 0 such that A ⩽ CB and A ≈ B when A ≲ B and B ≲ A.

For each q ⩾ 1, we dene q ′ such that

1 q + 1 q ′ = 1.
If X is a vector space, (u, v) ∈ X 2 , we will make a small abuse of notation, denoting ∥(u, v)∥ X = ∥u∥ X + ∥v∥ X . We x d ⩾ 1. If m is a complex valued function on R d , we dene by m(∇) the Fourier multiplier with symbol m(ξ), i.e. m(∇)u = m(ξ)û(ξ), where û is the Fourier transform of u. For s ⩾ 0, p ⩾

∥u∥ H s,p ≈ ∥u∥ L p + ∥|∇| s u∥ L p , 1 < p < ∞.
(see the proof in the appendix). For a multi-index α = (α 1 , α 2 , ..., α d ), denote

D α = ∂ α1 x1 • • • ∂ α d x d , |α| = d i=1 |α i |.
For s > 0 and 1 < p < ∞ and v = s -⌊s⌋, we have (see [START_REF] An | Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in H s (R n ). Nonlinear Anal[END_REF]Lemma 3.2]),

(2.3) |α|=⌊s⌋ ∥D α f ∥ Ḣv,p ≈ ∥f ∥ Ḣs,p .
We recall that a pair (q, r) is Strichartz-admissible for the Schrödinger equation when 2 ⩽ q ⩽ ∞,

2 ⩽ r ⩽ ∞, 2 q + d r = d 2 ,
and (q, r, d) ̸ = (2, ∞, 2). We recall that if (q, r) and (a, b) are Strichartz admissible, we have,

∥u∥ L q t L r x ≲ ∥u 0 ∥ L 2 + ∥f ∥ L a ′ t L b ′
x for any solution of the Schrödinger equation i∂ t u + ∆u = f with initial data u 0 . We denote:

S 0 (I) =        L ∞ I, L 2 (R d ) ∩ L 2 I, L 2d d-2 (R d ) if d ⩾ 3 L ∞ I, L 2 (R 2 ) ∩ L q2 I, L r2 (R 2 ) if d = 2 L ∞ I, L 2 (R) ∩ L 4 (I, L ∞ (R)) if d = 1,
where when d = 2, (q 2 , r 2 ) is an admissible pair with q 2 > 2 close to 2, and

N 0 (I) =        L 1 I, L 2 (R d ) + L 2 I, L 2d d+2 (R d ) if d ⩾ 3 L 1 I, L 2 (R 2 ) + L q ′ 2 I, L r ′ 2 (R 2 ) if d = 2 L 1 I, L 2 (R) + L 4/3 I, L 1 (R) if d = 1,
Note that the norm of S 0 (I) is equivalent to the supremum of all L q (I, L r ) norms, and that the norm of N 0 (I) is smaller than the inmum of all L q ′ (I, L r ′ ) norms, where in each case, (q, r) is taken over all Strichartz admissible pairs (with q ⩾ q 2 if d = 2). We also dene the following Strichartz spaces, and dual Strichartz spaces:

W 0 (I) = L 2(2+d) d I × R d Z 0 (I) = W 0 (I) ′ = L 2(d+2) d+4 (I × R d ),
so that we have S 0 (I) ⊂ W 0 (I) and Z 0 (I) ⊂ N 0 (I) (with continous embedding). We denote

X p (I) = L p(d+2) 2
(I × R d ).
For s ⩾ 0, we denote ∥u∥ S s (I) = ∥⟨∇⟩ s u∥ S 0 (I) , ∥u∥ Ṡs (I) = ∥|∇| s u∥ S 0 (I)

and dene similarly W s (I), Ẇ s (I), N s (I), Ṅ s (I), Z s (I) and Żs (I).

2.2. Preliminary nonlinear estimates. Lemma 2.1. (Product Rule 1) Let s ⩾ 0 and 1 < r, r 1 , r 0 , q 1 , q 0 < ∞ such that 1

r = 1 ri + 1 qi , for i = 0, 1. Then, ∥|∇| s (f φ)∥ L r x ≲ ∥f ∥ L r 1 x ∥|∇| s φ∥ L q 1 x + ∥|∇| s f ∥ L r 0 x ∥φ∥ L q 0 x (2.4) ∥⟨∇⟩ s (f φ)∥ L r x ≲ ∥f ∥ L r 1 x ∥⟨∇⟩ s φ∥ L q 1 x + ∥⟨∇⟩ s f ∥ L r 0 x ∥φ∥ L q 0 x .
(2.5)

Proof. See e.g [13, Proposition 3.3]. For (2.4), see e.g [2, Lemma 2.2] for the statement and [START_REF] Christ | Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation[END_REF] for proof in dimension 1 and s ∈ (0, 1).

By (2.2), (2.4) and Hölder inequality, we have

∥⟨∇⟩ s (f φ)∥ L r ≈ ∥f φ∥ L r +∥|∇| s (f φ)∥ L r ≲ ∥f ∥ L r 1 ∥φ∥ L q 1 +∥f ∥ L r 1 ∥|∇| s φ∥ L q 1 +∥|∇| s f ∥ L r 0 ∥φ∥ L q 0 ≲ ∥f ∥ r1 ∥⟨∇⟩ s φ∥ q1 + ∥⟨∇⟩ s f ∥ r0 ∥φ∥ q0 .
Hence (2.5).

□

Lemma 2.2 (Product rule 2). (see e.g [2, Corollary 2.3]). Let s ⩾ 0, n ∈ N (n ⩾ 1), and, for

i, j ∈ 1, n , 1 < r, r i k < ∞, such that for all k ∈ 1, n , 1 r = n i=1 1 r i k . Then n i=1 f i Ḣs,r ≲ n k=1   ∥f k ∥ Ḣs,r k k i̸ =k ∥f i ∥ L r i k   . Lemma 2.3 (Fractional chain rule). (see e.g [2, Lemma 2.4]). Let G ∈ C 1 (C, C), s ∈ (0, 1), 1 < r, r 2 < ∞, and 1 < r 1 ⩽ ∞ satisfying 1 r = 1 r1 + 1 r2 , ∥G(u)∥ Ḣs,r ≲ ∥G ′ (u)∥ L r 1 ∥u∥ Ḣs,r 2 .
Lemma 2.4 (Gagliardo-Nirenberg inequality [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: the full story[END_REF]). Let

s 1 ⩽ s 2 , p 2 > 1, s = θs 1 + (1 -θ)s 2 , 1 p = θ p1 + 1-θ p2 . Then ∥u∥ H s,p ≲ ∥u∥ θ H s 1 ,p 1 ∥u∥ 1-θ H s 2 ,p 2 .
Lemma 2.5 (Homogeneous Gagliardo-Nirenberg inequality). Let

s 1 ⩽ s 2 , p 2 > 1, s = θs 1 + (1 - θ)s 2 , 1 p = θ p1 + 1-θ p2 . Then ∥u∥ Ḣs,p ≲ ∥u∥ θ Ḣs 1 ,p 1 ∥u∥ 1-θ Ḣs 2 ,p 2 .
Proof. Let φ ∈ H s Dene φ λ (x) = φ(λx). Applying Lemma 2.4 for φ λ , we have

λ -d ∥φ∥ p L p + λ ps-d ∥φ∥ p Ḣs,p 1 p ≲ λ -d ∥φ∥ p1 L p 1 + λ p1s1-d ∥φ∥ p1 Ḣs 1 ,p 1 1 p 1 λ -d ∥φ∥ p2 L p 2 + λ p2s2-d ∥φ∥ p2 Ḣs 2 ,p 2 1 p 2 .
By dividing both sides by λ s-

d p = λ θs1-dθ p 1 λ (1-θ)s2-d(1-θ) p 2
, we have

λ -ps ∥φ∥ p L p + ∥φ∥ p Ḣs,p 1 p ≲ λ -p1s1 ∥φ∥ p1 L p 1 + ∥φ∥ p1 Ḣs 1 ,p 1 1 p 1 λ -p2s2 ∥φ∥ p2 L p 2 + ∥φ∥ p2 Ḣs 2 ,p 2 1 p 2 .
Let λ → +∞, we obtain the desired result.

□ Lemma 2.6 (Leibniz rule). Let f ∈ C k (C, C) and α = (α 1 , α 2 , • • •, α d ) ∈ N d such that |α| ⩽ k.
Then D α (f (u)) is a linear combinations of terms of the form

(∂ h1 z ∂ h2 z f )(u) h1 k=1 D β k u h2 k=1 D γ k u,
where

h1 k=1 β k + h2 k=1 γ k = α, 1 ⩽ h 1 + h 2 ⩽ |α|, |β k | ⩾ 1, |γ k | ⩾ 1 for all k.
Proof. The proof is by induction, using the formula:

∂ xi f (u) = ∂ z f (u)∂ xi u + ∂ z f (u)∂ xi u. □ 2.3.
Local Lipschitz continuity of the nonlinearity. In this subsection we consider the following general classes of nonlinearities: Denition 2.7. Let 0 < p 1 ⩽ p 0 , s ⩾ 0 be real numbers. We denote by N (s, p 0 , p 1 ) the vector

space of functions g ∈ C ⌈s⌉+1 (C, C) such that ∃C > 0, ∀k ∈ 0, ⌈s⌉ + 1 , ∀z ∈ C, g (k) (z) ⩽ C |z| p0+1-k + |z| p1+1-k . (2.6)
In the denition |g (k) (z)| denotes the supremum of all the derivatives (in z, z) of order k of g.

Assumption A. d ⩾ 1, g ∈ N (s, p 0 , p 1 ) with 0 ⩽ s, 4 d ⩽ p 1 ⩽ p 0 , 1 < p 1 and ⌈s⌉ ⩽ p 0 or g is a polynomial in u, u. We recall the notation X p (I) = L p(d+2) 2 (I × R d ). We will prove: Proposition 2.8. If Assumption A is satised, we let (2.7) X(I) = X p0 (I) ∩ X p1 (I) = L p0(d+2)/2 t,x (I × R d ) ∩ L p1(d+2)/2 t,x (I × R d ). Then (2.8) ∥g(u) -g(v)∥ N s (I) ≲ ∥(u, v)∥ p1-1 X(I) + ∥(u, v)∥ p0-1 X(I) × ∥u -v∥ Ẇ s (I) ∥(u, v)∥ X(I) + ∥u -v∥ X(I) ∥(u, v)∥ W s (I) ,
In particular

(2.9)

∥g(u)∥ N s (I) ≲ ∥u∥ W s (I) ∥u∥ p0 X(I) + ∥u∥ p1 X(I) .
Let us insist on the important fact that the rst norm of u -v in the second line of (2.8) is the norm in the homogeneous space Ẇ s .

We start with a few lemmas.

Lemma 2.9. Let p ⩾ 1 be an integer, and g(u) be a homogeneous polynomial of degree p + 1 in u, u. Let s ⩾ 0, u, v ∈ S s (I). Then

(2.10)

∥|∇| s (g(u) -g(v))∥ N 0 (I) ≲ ∥u∥ p-1 Xp(I) + ∥v∥ p-1 Xp(I) × ∥|∇| s (u -v)∥ W 0 (I) ∥u∥ Xp(I) + ∥v∥ Xp(I) + ∥u -v∥ Xp(I) ∥|∇| s u∥ W 0 (I) + ∥|∇| s v∥ W 0 (I) ,

and

(2.11)

∥g(u) -g(v)∥ N 0 (I) ≲ ∥u -v∥ Xp(I) ∥(u, v)∥ p-1 Xp(I) ∥(u, v)∥ W 0 (I) .
Proof. It is sucient to prove (2.10) and (2.11) for g(u) = u p+1-j u j , where j ∈ 0, p + 1 . We have (2.12)

g(u) -g(v) = u p+1-j -v p+1-j u j + v p+1-j (u j -v j ).
We treat the contribution of the rst term in the right-hand side of (2.12). The contribution of the second term is similar. We have

(2.13) u p+1-j -v p+1-j u j = (u -v)u j p-j k=0 u p-j-k v k .
We work on the interval I for all norms. We note the following relations between the exponents dening the W 0 , Z 0 = (W 0 ) ′ and X p norms:

(2.14)

d + 4 2(d + 2) = d 2(d + 2) + p × 2 p(d + 2)
.

By Hölder inequality, the denitions of X p and Z 0 and the fact that Z 0 is continuously embedded into Ṅ s , we obtain (2.11).

We next prove (2.10). Using (2.13), Lemma 2.1 and the denitions of X p and Z 0 , we have

u p+1-j -v p+1-j u j Ṅ s ≲ |∇| s u p+1-j -v p+1-j u j Z 0 ≲ p-j k=0 ∥|∇| s (u -v)∥ W 0 ∥u∥ p-k Xp ∥v∥ k Xp + p-j k=0 ∥u -v∥ Xp ∥|∇| s u∥ W 0 ∥u∥ p-k-1 Xp ∥v∥ k Xp + ∥|∇| s v∥ W 0 ∥v∥ k-1 Xp ∥u∥ p-k Xp ≲ ∥u -v∥ W s (∥u∥ p Xp + ∥v∥ p Xp ) + ∥u -v∥ Xp (∥u∥ W s + ∥v∥ W s )(∥u∥ p-1 Xp + ∥v∥ p-1 Xp ).
Combining with the same bound for the second term in (2.12), we obtain (2.10), concluding the proof.

□

Lemma 2.10. Let s, p be real numbers such that 0 < s, ⌈s⌉ ⩽ p, 1 < p and 4

d ⩽ p. Let g ∈ C ⌈s⌉+1 (C, C) such that (2.15) ∀k ∈ 0, ⌈s⌉ + 1 , g (k) (u) ⩽ C|u| p+1-k .
Then (2.10) and (2.11) hold.

Proof. We have

g(u) -g(v) = (u -v) 1 0 g z (v + θ(u -v)) dθ + u -v 1 0 g z (v + θ(u -v)) dθ, where g z = ∂g ∂z , g z = ∂g ∂z . Since ∂g ∂z , ∂g ∂z ⩽ C|z| p , we have |g(u) -g(v)| ⩽ C|u -v|(|u| p + |v| p ).
Thus, by Hölder inequality and (2.14), (2.16)

∥g(u) -g(v)∥ N 0 ≲ ∥g(u) -g(v)∥ Z 0 ≲ ∥u -v∥ Xp ∥(u, v)∥ p-1 Xp ∥(u, v)∥ W 0 .
This yields (2.11). We are left with proving (2.10) when s > 0 . We dene a by

1 a = d + 4 2(d + 2) - 2 p(d + 2)
.

Note that the assumption p ⩾ 4

d implies 2(d+2) d+4 ⩽ a ⩽ d+2 2 . By using product rule Lemma 2.1, we have ∥g(u) -g(v)∥ Ṅ s ⩽ ∥|∇| s (g(u) -g(v))∥ Z 0 (2.17) ≲ ∥|∇| s (u -v)∥ W 0 1 0 g z (v + θ(u -v)) dθ L d+2 2 t,x + 1 0 g z (v + θ(u -v)) dθ L d+2 2 t,x (2.18) + ∥u -v∥ Xp 1 0 |∇| s g z (v + θ(u -v)) dθ L a t,x + 1 0 |∇| s g z (v + θ(u -v)) dθ L a t,x . (2.19)
By the assumption (2.15), the term (2.18) is bounded as follows

(2.20) (2.18) ≲ ∥u -v∥ Ẇ s (∥u∥ p L p(d+2)/2 t,x + ∥v∥ p L p(d+2)/2 t,x ) ≈ ∥u -v∥ Ẇ s ∥(u, v)∥ p Xp .
We now consider the term (2.19). We will prove the following bound

(2.21) (2.19) ≲ ∥u -v∥ Xp ∥(u, v)∥ Ẇ s ∥(u, v)∥ p-
∥|∇| s g z (v + θ(u -v))∥ L a t,x + sup θ∈[0,1] ∥|∇| s g z (v + θ(u -v))∥ L a t,x
).

Letting f = g z or f = g z , we will prove, for a general function u ∈ Ẇ s ∩ X p , (2.22)

∥|∇| s (f (u))∥ L a t,x ≲ ∥u∥ Ẇ s ∥u∥ p-1
Xp .

Note that this will conclude the proof of (2.21).

The function f belongs to C ⌈s⌉ and satises (2.23)

|f (k) (z)| ≲ |z| p-k , for each 0 ⩽ k ⩽ ⌈s⌉.
∥|∇| s f (u)∥ L a t,x ≲ |α|=⌊s⌋ ∥D α f (u)∥ L a Ḣv,a , where v = s -⌊s⌋. By Lemma 2.6, D α (f (u)) is a linear combinations of terms of the form (∂ h1 z ∂ h2 z f )(u) h1 k=1 D β k u h2 k=1 D γ k u, where h1 k=1 β k + h2 k=1 γ k = α, 1 ⩽ h 1 +h 2 ⩽ |α|, |β k | ⩾ 1, |γ k | ⩾ 1 for all k.
To simplify notations we will only consider terms of the form

f (h) (u) h k=1 D β k u, h k=1 β k = α, 1 ⩽ h ⩽ |α|, |β k | ⩾ 1,
where f (h) = ∂ ∂z h f . The proof is the same for the other terms. We distinguish between the cases h < p and h = p.

Case h < p.

Using Lemma 2.2 and (2.23), we have

f (h) (u) h i=1 D βi u L a Ḣv,a (2.25)
≲ f (h) (u)

L q 1 Ḣv,q 1 h i=1 D βi u L r i t,x (2.26)
+ |u| p-h L q 2 t,x h k=1 h i=1,i̸ =k D βi u L r i t,x D β k u L ρ k Ḣv,ρ k . (2.27)
where, for i ∈ 1, h , (2.28)

1 r i = |β i | s d 2(d + 2) + 1 - |β i | s 2 p(d + 2) , 1 ρ i = |β i | + v s d 2(d + 2) + 1 - |β i | + v s 2 p(d + 2) and (2.29) 1 q 1 = v s d 2(2 + d) + 2 p(d + 2) ⌊s⌋ s + 2(p -h -1) p(d + 2) , 1 q 2 = 2(p -h) p(d + 2) . Since h k=1 |β k | = |α| = ⌊s⌋,
we see that the right-hand sides of the equalities in (2.28) and (2.29) are positive, and thus that q 1 , q 2 , and, for i = 1, h , r i and ρ i are nite and positive. Using also that v = s -⌊s⌋, we obtain

1 a = 1 q 1 + h i=1 r i = 1 q 2 + 1⩽i⩽h i̸ =k 1 r i + 1 ρ k ,
which proves that q 1 , q 2 , and, for i = 1, h , r i and ρ i are all greater than a, and that the assumptions of Lemma 2.2 are satised.

By Gagliardo-Nirenberg inequality Lemma 2.5 and the denition of r i , ρ i , we see that

D βi u L r i x ≲ ∥|∇| s u∥ |β i | s L 2(2+d) d x ∥u∥ 1- |β i | s L p(d+2) 2 x , D βi u Ḣv,ρ i ≲ ∥|∇| s u∥ |β i |+v s L 2(2+d) d x ∥u∥ 1- |β i |+v s L p(d+2) 2 x .
Integrating in time and using Hölder inequality, we obtain (2.30)

D βi u L r i t,x ≲ ∥u∥ |β i | s Ẇ s ∥u∥ 1- |β i | s Xp , D βi u L ρ i t Ḣv,ρ i ≲ ∥u∥ |β i |+v s Ẇ s ∥u∥ 1- |β i |+v s
Xp By (2.30), we obtain:

(2.27) ≲ ∥u∥ p-1 Xp ∥u∥ Ẇ s .
Using the rst inequality in (2.30), (2.26) is estimated by

(2.26) ≲ f (h) (u) L q 1 Ḣv,q 1 h i=1 ∥u∥ |β i | s Ẇ s ∥u∥ 1- |β i | s Xp = f (h) (u) L q 1 Ḣv,q 1 ∥u∥ ⌊s⌋ s Ẇ s ∥u∥ h- ⌊s⌋ s Xp . (2.31)
In (2.31), we will estimate

f (h) (u) L q 1 Ḣv,
f (h) (u) L q 1 Ḣv,r 1 ≲ f (h+1) (u) L q 3 t,x ∥u∥ L q 4 Ḣv,q 4 ≲ |u| p-h-1 L q 3 t,x ∥u∥ L q 4 Ḣv,q 4 = ∥u∥ p-h-1 L (p-h-1)q 3 t,x
∥u∥ L q 4 Ḣv,q 4 , where

1 q 3 = 2(p -h -1) p(d + 2) , 1 q 4 = 1 q 1 - 1 q 3 = v s d 2(2 + d) + ⌊s⌋ s 2 p(d + 2)
.

For (q 4 , r 4 ) as above, using Gagliardo-Nirenberg inequality Lemma 2.

≲ |u| p-h L q 1 t,x ∥u∥ Ẇ s ∥u∥ h-1 Xp = ∥u∥ p-1 Xp ∥u∥ Ẇ s ,
which proves (2.22) in this case also.

Case h = p.

In this case, we have p = |α|, and thus, since |α| = ⌊s⌋ ⩽ ⌈s⌉ ⩽ p, s is an integer, v = 0 and s = p.

By the assumptions on f , we have |f

(h) (u)| ≲ 1. Thus (2.25) ≲ f (h) (u) h i=1 D βi u L a Ḣv,a ≲ h i=1 D βi u L a Ḣv,a ≲ h i=1 D βi u L r i t,x
, where the r i are dened as above. Using (2.30) and since

□

Proof of Proposition 2.8. Proposition 2.8 follows easily from Lemmas 2.9 and 2.10. We x g ∈ N (s, p 0 , p 1 ) with 0 ⩽ s, p 0 ⩾ q d and p 0 ⩾ ⌈s⌉. All the norms used are over the interval I.

The estimate (2.9) is exactly (2.8) with v = 0. To prove (2.8), we decompose g(u) as follows:

g(u) = P (u) + g(u).

Where P (u) is the Taylor expansion of g at u = 0 up to order ⌈s⌉. As a consequence P is a polynomial of the form

P (u) = p1+1⩽k1+k2⩽⌈s⌉ a k1,k2 u k1 u k2 .
By Lemma 2.9, using that by the denition of X and the assumptions p 1 ⩾ 4 d and p 0 ⩾ ⌈s⌉ we have

p 1 ⩽ q ⩽ ⌈s⌉ -1 =⇒ ∥u∥ Xq ⩽ ∥u∥ X , we obtain (2.33) ∥P (u) -P (v)∥ N s ≲ ∥u∥ p0-1 X + ∥v∥ p0-1 X + ∥u∥ p1-1 X + ∥v∥ p1-1 X × ∥u -v∥ Ẇ s (∥u∥ X + ∥v∥ X ) + ∥u -v∥ X (∥u∥ W s + ∥v∥ W s ) .
Next, we notice that since g ∈ N (s, p 0 , p 1 ), the denition of P and the inequalities p 1 < p 0 and ⌈s⌉ ⩽ p 0 imply, for k ∈ 0, ⌈s⌉

+ 1 , z ∈ C, |z| ⩾ 1 =⇒ g(k) (z) ⩽ C|z| p0+1-k , |z| ⩽ 1 =⇒ g(k) (z) ⩽ C|z| ⌈s⌉+1-k .
We let χ be a smooth function such that

χ = 1 if |x| < 1, 0 if |x| > 2. Set g 0 (u) = (1 -χ(u))g(u), g 1 (u) = χ(u)g(u)
, so that g 0 satises the assumptions of Lemma 2.10 with p = max(p 0 , ⌈s⌉) = p 0 , and g 1 the same assumptions with p = ⌈s⌉. Combining the conclusion of this Lemma for g 0 , g 1 with the estimate (2.33), we obtain the conclusion of the proposition. □ 2.4. Cauchy and stability theory for general nonlinearities. In all this subsection, we x g, p 0 , p 1 , s such that Assumption A holds. We assume furthermore (2.34)

s ⩾ s 0 = d 2 - 2 p 0 .
Note that these assumptions are satised for a sum of powers:

g(u) = k j=0 λ j |u| pj u,
where k ⩾ 0, 4 d ⩽ p 1 < . . . < p k-1 < p 0 , λ j ∈ R for all j, provided g ∈ C ⌈s⌉+1 (which is the case, for example, when all the p j are even integers), s ⩾

u(t) = e it∆ u 0 -i t 0 e i(t-τ )∆ g(u)(τ ) dτ,
for all t ∈ I.

We recall from (2.7) the denition of X(I). Noting that the assumption p 0 ⩾ 4 d implies p 0 (d + 2)/2 > 2 (and p 0 (d+2)/2 ⩾ 6 if d = 1), we can choose q 0 such that (p 0 (d+2)/2, q 0 ) is an admissible pair. By Sobolev inequality and the denitions of s 0 , q 0 , one can check

(2.36) ∥u∥ L p 0 (d+2) 2 x ≲ ∥|∇| s0 u∥ L q 0 and thus (2.37) ∥u∥ L p 0 (d+2) 2 (I×R d ) ≲ ∥|∇| s0 u∥ S 0 (I) , ∥u∥ X(I) ≲ ∥u∥ S s 0 (I) .
Proposition 2.12. Let u 0 ∈ H s . Let g, p 0 , p 1 , s such that Assumption A holds and s ⩾ d 2 -2 p0 . Let 0 ∈ I be an interval of R, and A > 0. Assume that ∥u 0 ∥ H s ⩽ A and

e i•∆ u 0 X(I) = δ ⩽ δ 0 (A) small.
Then there exists a unique solution u to (1.1) such that

u(0) = u 0 , ∥u∥ S s (I) ≲ ∥u 0 ∥ H s , ∥u∥ X(I) ⩽ 2δ.
Moreover, if u 0,k → u 0 in H s (so that, for k large, e it∆ u 0,k X(I) < δ) then the corresponding solution u k → u in C(I, H s ).

Remark 2.13. By Strichartz estimates, (2.37) and Proposition 2.12, if ∥u 0 ∥ H s is small then u is global and

∥u∥ S s ≲ ∥u 0 ∥ H s .
Proof. We use similar argument as in [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]Lemma 2.5]. We work on the interval I for all norms.

Consider B = u : ∥u∥ X ⩽ 2δ, ∥u∥ S s ⩽ M A ,
for some large universal contant M , with the topology induced by the norm in S s (I). We denote by Ψ(u) the right hand side of (2.35). We show that if δ ⩽ δ 0 (A) small enough, and M large enough (independently of A), Ψ is a contraction map on B.

For u ∈ B, we have, by Strichartz estimates and Proposition 2.8

∥Ψ(u)∥ X(I) ⩽ e it∆ u 0 X(I) + t 0 e i(t-τ )∆ g(u)(τ ) dτ X(I) ⩽ δ + C t 0 e i(t-τ )∆ g(u)(τ ) dτ S s (I) ⩽ δ + C∥g(u)∥ N s (I) ⩽ δ + C∥u∥ S s (∥u∥ p1 X + ∥u∥ p0 X ) Thus, ∥Ψ(u)∥ X(I) ⩽ δ + CM Aδ p1 ⩽ 2δ, if δ is small enough (so that CM Aδ p1-1 ⩽ 1). Similarly, ∥Ψ(u)∥ S s ⩽ e i•∆ u 0 S s + C ∥g(u)∥ N s ⩽ CA + CM Aδ p1 ⩽ M A, choosing M ⩾ 2C,
∥Ψ(u) -Ψ(v)∥ S s ⩽ C ∥g(u) -g(v)∥ N s ⩽ ∥u -v∥ S s (∥u∥ S s + ∥v∥ S s ) ∥u∥ p1-1 X + ∥v∥ p1-1 X ⩽ Cδ p1-1 M A∥u -v∥ S s
Thus, taking δ small enough, we obtain that Ψ is a contraction map on B. By the xed point theorem, there exists a unique u ∈ B such that u = Ψ(u). Thus, u is a solution to (1.1). Also, since u ∈ B (and since we can take A = ∥u 0 ∥ H s ) we have as anounced ∥u∥ S(I)

≲ ∥u 0 ∥ H s , ∥u∥ X(I) ⩽ 2δ. If ∥u 0 ∥ H s is small then e i•∆ u 0 X(R) ⩽ C ∥u 0 ∥ H s is small. Thus, u is global and ∥u∥ S s (R) ⩽ M ∥u 0 ∥ H s . □ Remark 2.14. We have e i•∆ u 0 X(R) ⩽ C ∥u 0 ∥ H s < ∞.
Thus, for each ε > 0, there exists a small interval I around 0 such that e i•∆ u 0 X(I) < ε. This implies local existence of solution.

Remark 2.15. If u, v are two solutions to (1.1) on 0 ∈ I such that u(0) = v(0) then u ≡ v. We rst show this assertion when I ′ ⊂ I is a small interval around 0. Shrinking I ′ if necessary, we assume

∥u∥ X(I ′ ) ⩽ δ 0 (A), ∥v∥ X(I ′ ) ⩽ δ 0 (A),
with δ 0 (A) as in Proposition 2.12, and

A = max t∈I ′ ∥u(t)∥ H s + ∥v(t)∥ H s .
Thus, u, v ∈ B (where B is as in the preceding proof ), which shows that u = v on I ′ . Repeating this argument, we deduce that u ≡ v on all I. This allows us to dene a maximal interval

I max = (T -, T + ) with T -< 0, T + > 0.
Lemma 2.16. Let u be a maximal solution on I max as in Proposition 2.12, Remark 2.15. Assume that u ∈ X([0, T + )).

Then T + = ∞, and u scatters in the future in in H s . A similar result holds in the past, if

u ∈ X((T -, 0]).
Remark 2.17. Lemma 2.16 implies the blow-up criterion T + < ∞ =⇒ ∥u∥ X(0,T+) = ∞ (and similarly for T -). In the case s > d 2 -2 p0 we can also show (see e.g. [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]) that if T + (u) is the maximum time of existence of u and T + (u) < +∞ then

∥u(t)∥ H s → ∞, as t → T + (u).
Proof. We only work on [0, T + ). In (T -, 0], we use similar argument. We only need to prove that (2.38) ∥u∥ S s (0,T+) < ∞.

Indeed, if (2.38) holds and T + is nite, then Proposition 2.12 shows that one can extend the solution u beyond T + , a contradiction with the denition of T + . Thus T + = ∞, and Proposition 2.8 implies ∥g(u)∥ N ([0,∞)) < ∞, which implies by standard arguments scattering in H s .

We divide [0, T + ) into nite intervals I k such that ∥u∥ X(I k ) < ε.

On each I k = [t k , t k+1 ], we have ∥u∥ S s (I k ) ⩽ C ∥u(t k )∥ H s + C ∥u∥ S s (I k ) (∥u∥ p1 X(I k ) + ∥u∥ p0 X(I k ) ) ⩽ C ∥u(t k )∥ H s + C ∥u∥ S s (I k ) (ε p1 + ε p0 ).
Choosing ε small and using that ∥u(t k+1 )∥ H s ≲ ∥u∥ S s (I k ) , we obtain by induction on k that ∥u∥ L ∞ ([0,T k max ),H s ) < ∞ then (2.38), concluding the proof. □

One can also prove the existence of wave operators for equation (

u ∈ C 0 ((T -(u), ∞)) of (1.1) such that lim t→∞ ∥u(t) -v L (t)∥ H s = 0.
We omit the proof which is similar to the proof of Proposition 2.12.

Theorem 2.19. (Long time perturbation theory) Let A > 0, s ⩾ s 0 . There exists constants ε(A, s) ∈ (0, 1], C(A, s) > 0 with the following properties. Let 0 ∈ I be an compact interval of R and w be a solution of the following equation

Lw = g(w) + e,
and u 0 ∈ H s such that

∥w 0 ∥ H s + ∥w∥ X(I) ⩽ A, ∥e∥ N s (I) + ∥u 0 -w(0)∥ H s ) = ε ⩽ ε(A, s).
There the solution u of (1.1) with initial data u 0 is dened on I and satises (2.39) ∥u -w∥ S s (I) ⩽ C(A, s)ε, ∥u∥ S s (I) ⩽ C(A, s).

Proof. Without loss of generality, using the reversibility of the equation, we can assume I = [0, T ).

We can also assume that I ⊂ I max (u) (indeed, if T + (u) < T , the proof below will show that (2.39) holds with I replaced by [0, T + ), contradicting the blow-up criterion).

Divide

I into J = J(A, s) subintervals I j = [t j , t j+1 ], j ∈ [0, J -1], with t 0 = 0, such that (2.40) ∥w∥ X(Ij ) ⩽ δ,
where δ is a small constant (depending only on s), to be specied. On each I j , by Strichartz and Proposition 2.8, we have

∥w∥ S s (Ij ) ⩽ C ∥w(t j )∥ H s + ∥g(w)∥ N s (Ij ) + ∥e∥ N s (Ij ) ≲ ∥w(t j )∥ H s + ∥w∥ S s (Ij ) (∥w∥ p1 X(Ij ) + ∥w∥ p0 X(Ij ) ) + ε ≲ ∥w(t j )∥ H s + ε + ∥w∥ S s (Ij ) (δ p1 + δ p0 ).
Thus, for δ small enough, ∥w∥ S s (Ij ) ⩽ ∥w(t j )∥ H s + ε. Since ∥w(t j+1 )∥ H s ⩽ ∥w∥ S s (Ij ) , ∥w(0)∥ H s ⩽ A, we obtain by a nite induction ∥w(t j )∥ H s ⩽ C(A, s), and thus (2.41) ∥w∥ S s (I) ⩽ C(A, s).

Thus, we may divide I into J = J(A, s) subintervals, which we still denote by I j = [t j , t j+1 ], j ∈ 0, J -1 , such that for all j, (2.42)

∥w∥ W s (Ij ) + ∥w∥ X(Ij ) ⩽ δ.
Taking δ small (independently of A and ε), and ε ⩽ ε(A, s) small, we prove by induction

(2.43) ∀j ∈ 0, J -1 , ∥u -w∥ S s (Ij ) ⩽ (Kj + K + 1)ε,
for some large constant K independent of A and ε. This will imply the desired conclusion (2.39). More precisely, we will prove, for j ∈ 0, J -1 , (2.44)

∥u(t j ) -w(t j )∥ H s ⩽ (Kj + 1)ε =⇒ ∥u -w∥ S s (Ij ) < (Kj + K + 1)ε, which will yield (2.43), since ∥u(t 0 ) -w(t 0 )∥ H s ⩽ ε.
Assuming j ∈ 0, J -1 , ∥u(t j ) -w(t j )∥ H s ⩽ (Kj + 1)ε, we argue by contradiction, assuming also that there exists T ∈ [t j , t j+1 ] such that ∥u -w∥ S s ([tj ,T ]) = (Kj + K + 1)ε. Then by the equation satised by u -w and Strichartz estimates, we obtain

(2.45) ∥u -w∥ S s ([tj ,T ]) ⩽ ∥u(t j ) -v(t j )∥ H s + C∥e∥ N s ([tj ,T ]) + C∥g(u -w)∥ N s ([tj ,T ])
We have ∥u∥ W s ([tj ,T ]) ⩽ ∥u -w∥ W s ([tj ,T ]) + ∥w∥ W s ([tj ,T ]) ⩽ (Kj + K + 1)ε + δ, and similarly ∥u∥ X s ([tj ,T ]) ⩽ C(Kj + K + 1)ε + δ. Thus by Proposition 2.8, taking ε small enough, so that

(Kj + K + 1)ε ⩽ 1. ∥g(u -w)∥ N s ([tj ,T ]) ⩽ C(Kj + K + 1) (Kj + K + 1) p1 ε p1 + δ p1 .
Going back to (2.45), we deduce

(Kj + K + 1)ε = ∥u -w∥ S s ([tj ,T ]) ⩽ Cε + (Kj + 1)ε + C(Kj + K + 1)ε (Kj + K + 1) p1 ε p1 + δ p1 .
Taking K large enough, ε ⩽ ε(s, A) small (so that (KJ + K + 1)ε ⩽ δ), and δ small, we obtain.

(Kj + K + 1)ε ⩽ (Kj + K/2 + 1)ε, a contradiction which concludes the proof. □ Remark 2.20. Assume d ⩾ 1, g(u) = |u| p0 u, s ⩾ s 0 = d 2 -2
p0 , p 0 ⩾ 4 d and g ∈ C ⌈s⌉+1 (i.e. p 0 is an even integer or p 0 > ⌈s⌉). Then analogs of Proposition 2.12 and Theorem 2.19 where all the spaces are replaced by homogeneous spaces hold. Precisely, in the statement of Proposition 2.12, one can replace in this case X(I) by X p0 (I) = L p 0 (d+2) 2

(I × R d ),
(z) = G ′ (|z| 2 )z for a C 1 function G : [0, ∞) → R with G(0) = 0. Let u ∈ C 0 (I, H s (R d )) be a solution of (2.1), where s ⩾ d 2 -2 p0 .
Then the mass

M (u) := |u(t, x)| 2 dx
is conserved on I. Furthermore, if s ⩾ 1, then the momentum:

P (u) := Im ∇u(t, x)u(t, x)dx
and the energy

E(u) := |∇u(t, x)| 2 dx + G(|u(t, x)| 2 )dx.
are well-dened and conserved on I.

We omit the classical proof. See e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. Observe that the assumptions of Proposition

i∂ t u L + ∆u L = 0, with initial data (3.2) u L↾t=0 = u 0 ∈ Ḣs . Denition 3.1. Let p > 4 d such that s = d 2 -2
p . A linear Ḣs -prole, in short prole, is a sequence (φ Ln ) n , of solutions of (3.1), of the form

(3.3) φ Ln (t, x) = 1 λ 2 p n φ L t -t n λ 2 n , x -x n λ n ,
where φ L is a xed solution of (3.1), (3.2) and

Λ n = (λ n , t n , x n ) n is a sequence in (0, ∞) × R × R d (called sequence of transformations) such that (3.4) lim n→∞ -t n λ 2 n = τ ∈ R ∪ {±∞}.
Denition 3.2. We say that two sequence of transformations Λ n = (λ n , t n , x n ) and M n = (µ n , s n , y n ) are orthogonal when they satisfy

lim n→∞ |t n -s n | λ 2 n + |x n -y n | λ n + log λ n µ n = ∞.
We say that two Ḣs -proles φ Ln and ψ Ln are equivalent (in Ḣs ) when lim n→∞ ∥φ Ln (0) -ψ Ln (0)∥ Ḣs = 0.

We say that they are orthogonal when one of the two proles is identically 0 or when the corresponding sequence of transformations are orthogonal.

Remark 3.3. For a given Ḣs -prole φ Ln , the choice of the solution φ L and the sequence of transformations (Λ n ) n are not unique. However the denitions of equivalent and orthogonal proles do not depend on these choices. Also, two equivalent Ḣs -proles are orthogonal to the same Ḣs -proles.

Denition 3.4. Let (u 0,n ) n be a bounded sequence in Ḣs , and u Ln = e it∆ u 0,n . We say that the sequence φ j

Ln n j⩾1

of Ḣs -proles is a prole decomposition of (u 0,n ) n if these proles are pairwise orthogonal, and satisfy

(3.5) lim J→∞ lim sup n→∞ w J Ln Xp(R) + |∇| s w J Ln W 0 (R) = 0, where (3.6)
w J Ln = u Ln -J j=1 φ j Ln .

Proposition 3.5. For any bounded sequence (u 0,n ) n in Ḣs , there exists a subsequence (that we still denote by (u 0,n ) n ) that admits a prole decomposition (φ j Ln ) n j

. Furthermore, we have the Pythagorean expansion:

(3.7) ∀J ⩾ 1, ∥u 0,n ∥ 2 Ḣs = J j=1 φ j L (0) 2 Ḣs + w J Ln (0) 2 Ḣs + o(1), n → ∞,
where w J n is as in Denition 3.4.

Proposition 3.5 is proved by Shao in [START_REF] Shao | Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation[END_REF], using the L 2 -critical prole decomposition of Merle and Vega [START_REF] Merle | Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2[END_REF], generalized to higher dimension by Bégout and Vargas [START_REF] Bégout | Mass concentration phenomena for the L 2 -critical nonlinear Schrödinger equation[END_REF]. See also [START_REF] Keraani | On the defect of compactness for the Strichartz estimates of the Schrödinger equations[END_REF] for the energy-critical case p = 4 d-2 .

Remark 3.6. In the notations of Denition 3.4, if (u 0,n ) admits a prole decomposition, then By (3.9),(3.10), one sees that the initial data of the nonzero proles are exactly the nonzero weak limits of the form (3.10). This implies in particular that the prole decomposition is unique, up to reordering and equivalent proles, if one ignore the null proles.

(λ j n ) 2 p φ k Ln (t j,n , x j,n + λ j n •) n→∞ ----⇀ 0 in Ḣs , j ̸ = k (3.8) (λ j n ) 2 p w J Ln (t j,n , x j,n + λ j n •) n→∞ ----⇀ 0 in Ḣs , J ⩾ j (3.9) (λ j n ) 2 p u Ln (t j,n , x j,n + λ j n •) n→∞ ----⇀ φ j L (0) in Ḣs .
Let q = 2d d-2s = d 2 p be the Lebesgue exponent such that the Sobolev embedding Ḣs ⊂ L q holds. Let (u 0,n ) n be a sequence that has a prole decomposition as above. We next prove the analog of the property (3.5) of the remainder w J n in the space L ∞ (R, L q ) and obtain a Pythagorean expansion of the L q norm of u 0,n . Extracting subsequences, we can assume that for all j, the following limit exists in R ∪ {±∞}:

(3.11) τ j = lim n→∞ -t j n (λ j n ) 2
.

We can also assume (translating the proles in time if necessary), τ j ∈ {0, -∞, +∞}. If τ j ∈ {±∞}, we have (3.12) lim n→∞ ∥φ j Ln (0)∥ L q = 0.

Lemma 3.7. Let (u 0,n ) n be as in Proposition 3.5. Then

lim J→+∞ lim sup n→+∞ w j Ln L ∞ (R,L q ) = 0 (3.13) lim n→∞ ∥u 0,n ∥ q L q = j⩾1 τ j =0
∥φ j Ln (0)∥ q L q .

(3.14)

Proof. This follows from the elliptic prole decomposition of Patrick Gérard [START_REF] Gérard | Description du défaut de compacité de l'injection de sobolev[END_REF]. We rst prove (3.13) by contradiction, assuming that there exists η > 0 and a sequence J k → ∞ such that

(3.15) ∀k, lim J→∞ lim sup n→∞ ∥w J k Ln ∥ L ∞ (R,L q ) ⩾ η. Let ε k = lim sup n→∞ ∥w J k Ln ∥ Xp(R) ----→ k→∞ 0,
where the convergence to 0 follows from (3.5). For all k, there exists N k such that

n ⩾ N k =⇒ ∥w J k Ln ∥ Xp(R) ⩽ 2ε k . By (3.15), we can nd n k ⩾ N k , t k ∈ R such that ∥w J k Ln k (t k )∥ L q ⩾ 1 2 η.
As a consequence, by the main result of [START_REF] Gérard | Description du défaut de compacité de l'injection de sobolev[END_REF], we can nd φ ∈ Ḣs \ {0} such that

w J k Ln k (t k ) ----⇀ k→∞ φ.
By Strichartz estimates, this implies that the sequence w J k Ln k (• + t k ) k converges weakly to e i•∆ φ in X p (R), contradicting (3.5). This concludes the proof of (3.13). As a consequence (using also (3.12)), we have the elliptic prole decomposition at xed time:

u 0,n = 1⩽j⩽J τ j =0 φ j Ln (0) + wJ Ln (0), lim J→∞ lim sup n→∞ ∥ wJ Ln (0)∥ L q = 0.
The pseudo-Pythagorean expansion (3.14) then follows from [START_REF] Gérard | Description du défaut de compacité de l'injection de sobolev[END_REF]: see the expansion (1.11) there. □ 3.2. Proles in non-homogeneous Sobolev spaces. In all this subsection, we x s 0 > 0 and

p 0 > 4 d with s 0 = d 2 -2
p0 . We consider a sequence (u 0,n ), which is bounded in the inhomogeneous Sobolev space H s0 (R d ). We assume that u 0,n admits a Ḣs0 prole decomposition (φ j Ln ) n,j , where

φ j Ln (t, x) = 1 (λ j n ) 2 p 0 φ j L t -t j n λ j n , x -x j n λ j n .
Claim 3.8. For all j, the sequence (λ j n ) n is bounded.

Proof. We have

(3.16) (λ j n ) 2 p 0 u Ln (t j n , λ j n • +x j n ) n→∞ ----⇀ φ j L (0) weakly in Ḣs0 . Furthermore, (λ j n ) 2 p 0 u Ln t j n , λ j n • +x j n L 2 = (λ j n ) 2 p 0 -N 2 ∥u 0,n ∥ L 2 .
Assume that λ j n → ∞ along a subsequence in n. As a consequence, since ∥u 0,n ∥ L 2 is bounded, we obtain, along the same subsequence, that (λ j n ) 2 p 0 u Ln t j n , λ j n • +x j n converges strongly to 0 in L 2 . By uniqueness of the distributional limit, φ j L (0) = 0 as announced. □

Using the claim and extracting subsequences we obtain that for all j, one of the following holds:

• φ j L ≡ 0. In this case we say that (φ j Ln ) n is a null prole, and denote j ∈ J 0 .

• φ j L ̸ ≡ 0 and lim n→∞ λ j n = λ j ∈ (0, ∞). In this case we say that (φ j Ln ) n is a nonconcentrating prole, and denote j ∈ J N C . The weak limit (3.16) and the fact that the sequence (u 0,n ) is bounded in H s0 proves that φ j 0 = φ j L (0) ∈ H s0 . Replacing φ j 0 by 1 (λ j ) N 2 φ j 0 x λ j and λ j n by λ j n /λ j , we see that we can assume λ j = 1. As a consequence, we can assume λ j n = 1 for all n (this will modify the prole φ j Ln only by a term which goes to 0 in H s0 as n → ∞). • φ j L ̸ ≡ 0 and lim n→∞ λ j n = 0. In this case we say that (φ j Ln ) n is a concentrating prole, and denote j ∈ J C . Remark 3.9. Assume that the sequence (u 0,n ) is bounded in H s2 for some s 2 > s 0 . Then it is easy to see that J C is empty, i.e. that there is no concentrating prole.

Remark 3.10. Let s such that 0 < s < s 0 . Then (φ j

Ln ) n j∈J N C
is a Ḣs prole decomposition of (u 0,n ) n . Indeed, by Remark 3.9 the Ḣs -prole decomposition of this sequence has no concentrating prole. By the preceding subsection, we have the Pythagorean expansion (3.17)

∀J ⩾ 1, ∥u 0,n ∥ 2 Ḣs = j∈J N C 1⩽j⩽J φ j L (0) 2 Ḣs + w J Ln (0) 2 Ḣs + o(1), n → ∞, where w J Ln = u Ln -j∈J N C 1⩽j⩽J φ j
Ln . One can also prove, as a consequence of the orthogonality of the proles,

(3.18) ∀J ⩾ 1, ∥u 0,n ∥ 2 L 2 = j∈J N C 1⩽j⩽J φ j L (0) 2 L 2 + w J Ln (0) 2 L 2 + o(1), n → ∞.
Remark 3.11. Let 2 < q < q 0 = 2d d-2s0 , and τ j be dened by (3.11). Assume as before τ j ∈ {0, ±∞}. Then if τ j ∈ {±∞}, one has, by standard properties of the linear Schrödinger equation, (3.19) lim n→∞ ∥φ j Ln (0)∥ L q = 0.

Moreover, if j ∈ J C , then (3.19) holds by a simple scaling argument. Finally, using Lemma 3.7

and the same argument as in Remark 3.10, one obtain

lim n→∞ ∥u 0,n ∥ q L q = j∈J N C τ j =0
∥φ j L (0)∥ q L q .

3.3. Nonlinear prole decomposition. We now construct a nonlinear prole decomposition, based on the preceding linear prole decomposition, and adapted to the equation (2.1), where g satisfy the following assumptions.

Assumption B.

(3.20)

g(u) = g 0 (u) + g 1 (u), g 0 (u) = ι 0 |u| p0 u, ι 0 ∈ {±1}
and g 1 (u) is a nonlinearity of lower order. Precisely, we will assume (3.21)

g 1 ∈ N (s 0 , p 2 , p 1 ), 4 d < p 1 ⩽ p 2 < p 0 , 1 < p 1 ,
where as usual s 0 = d 2 -2 p0 , and

(3.22)
⌈s 0 ⌉ ⩽ p 0 , or g is a polynomial in u, u.

Furthermore, the nonlinearity is of the form g

(z) = G ′ (|z| 2 )z, G ∈ C 1 ([0, ∞), R) with G(0) = 0.
We note that the last condition on g implies conservation of the mass, and, if s 0 ⩾ 1, of the energy and the momentum. Note that Assumption B implies Assumption A with s = s 0 .

We will also consider the equation (2.1) with g = g 0 , that is the homogeneous nonlinear Schrödinger equation (1.5) with initial data in Ḣs0 .

To each (linear) prole φ j Ln , we associate a nonlinear prole φ j n and a modied nonlinear prole φj n in the following way:

• If j ∈ J 0 , φj n and φ j n are both equal to the constant null function.

• If j ∈ J N C , the modied nonlinear prole and the nonlinear prole are equal, and dened by φj n (t, x) = φ j n (t, x) = φ j t -t j n , x -x j n , where φ j is the unique solution of (2.1) such that (3.23) lim t→τ j φ j (t) -φ j L (t)

H s 0 = 0, τ j = lim n→∞ -t j n .
This solution is given by the well-posedness theory, Proposition 2.12 (if τ j is nite) or by the existence of wave operators, Proposition 2.18 (if τ j ∈ {±∞}). • If j ∈ J C , the nonlinear prole φ j n is dened by

(3.24) φ j n (t, x) = 1 (λ j n ) 2 p 0 φ j t -t j n (λ j n ) 2 , x -x j n λ j n
where φ j is the unique solution of the homogeneous equation (1.5) such that lim t→τ j φ j (t) -φ j L (t)

Ḣs 0 = 0, τ j = lim n→∞ -t j,n (λ j n ) 2
.

By denition of φ j , we see that φ j (τ ) is in Ḣs0 for all τ in the domain of existence of φ j . However it is not necessarily in H s0 . To tackle with this diculty, we x σ j in the maximal interval of existence of φ j (if τ j is nite, we can take σ j = τ j , if τ j = ±∞, |σ j | large and with the same sign than τ j ). We let

s j n = (λ j n ) 2 σ j + t j n
and denote by φj n the solution of (1.5) such that (3.25)

φj n (s j n ) = χ x -x j n φ j n (s j n ) = χ x -x j n 1 (λ j n ) 2 p 0 φ j σ j , x -x j n λ j n , where χ ∈ C ∞ 0 (R N ) is radially symmetric, χ(x) = 1 for |x| < 1, χ(x) = 0 for |x| > 2.
Lemma 3.12. Let j ∈ J C . Then More precisely, let 0 ∈ J n be a sequence of interval. Let

I j n = t-t j n (λ j n ) 2 , t ∈ J n .
Assume that there exists an interval I in the domain of existence of φ j such that ∥φ j ∥ Xp 0 (I) < ∞ and for large n, I j n ⊂ I. Then for large n, J n is included in the domain of existence of φj n and

sup t∈Jn φ j n (t) -φj n (t) Ḣs 0 + ∥φ j n -φj n ∥ Ṡs 0 (Jn) -→ n→∞ 0 (3.27) ∀s ∈ [0, s 0 ), sup t∈Jn φj n (t) H s + ∥ φj n ∥ S s (Jn) -→ n→∞ 0 (3.28)
Proof. We rst prove (3.27). Without loss of generality, we can assume σ j ∈ I. Let

Φ j n (τ, y) = (λ j n ) 2/p φj n t j n + (λ j n ) 2 τ, x j n + λ j n y . Then Φ j n (σ j , y) = (λ j n ) 2
p 0 φj n s j n , x j n + λ j n y = χ(λ j n y)φ j (σ j , y). Thus, using that λ j n → 0 as n → ∞, we obtain that I is included in the domain of existence of Φ j To prove (3.28), we rst notice that using (3.25) and λ j n → 0, lim It remains to prove the second limit in (3.28). We let s ∈ [0, s 0 ). To any J ⊂ J n such that φn j X(J) ⩽ ε (where ε is a small constant), and a n ∈ J, we have by Strichartz estimates and Proposition 2.8,

φj n S s (J) ≲ φj n (a n ) H s + g 0 ( φj n ) N s (J) ≲ φj n (a n ) H s + φj n S s (J) φj n p0 X(J) , and thus (if ε is small enough), φj n S s (J) ≲ φj n (a n ) H s . Since lim sup n→∞ φj n S s 0 (Jn) < ∞,
and S s0 (J) is continuously embedded in X(J), we can divide the interval

J n in N subintervals J k n , k ∈ 1, N (N independent of n), such that φj n X(J k n ) ⩽ ε.

□

When j ∈ J C , the modied proles φj n are approximate solutions of (2.1): Lemma 3.13. Let j ∈ J C , and J n be as in Lemma 3.12. Let (3.30)

ẽj n = i∂ t φj n + ∆ φj n -g( φj n ) = -g 1 ( φj n ).
Then lim n→∞ ẽj n N s 0 (Jn) = 0.

Proof. Since φj

n is a solution of (1.5) with p = p 0 , we indeed have ẽj n = -g 1 ( φj n ). As a consequence, by Proposition 2.8, using that g 1 ∈ N (s 0 , p 2 , p 1 ),

□

We next give the announced approximation result. For this we must also modify the linear remainder w J L,n : we let wJ L,n be the solution of the linear wave equation with initial data

(3.32) wJ L,n (0) = u 0,n - J j=1 φj n (0).
Claim 3.14. For all s with 0 < s ⩽ s 0 ,

lim J→∞ lim sup n→∞ wJ L,n X(R) + wJ L,n Ẇ s (R) = 0.
Proof. We have

wJ L,n (0) = w J L,n (0) + 1⩽j⩽J (φ j Ln (0) -φj n (0)).
By Lemma 3.12 and linear prole decomposition, we have

lim n→∞ φ j Ln (0) -φj n (0) Ḣs 0 ⩽ lim n→∞ φ j Ln (0) -φ j n (0) Ḣs 0 + lim n→∞ φ j n (0) -φj n (0) Ḣs 0 → 0.
Combining with (3.5), we obtain

(3.33) lim J→∞ lim sup n→∞ wJ L,n Xp 0 (R) + |∇| s0 wJ L,n W 0 (R) = 0.
By Lemma 3.12, we also have

wJ L,n (0) = u 0,n - j∈J N C 1⩽j⩽J φ j n (0) + o n (1), in H s , 0 ⩽ s < s 0 ,
which shows by the Pythagorean expansion (3.18) that sup

J lim sup n→∞ ∥ wJ L,n (0)∥ L 2 < ∞.
Using Strichartz estimates, we obtain

(3.34) lim J→∞ lim sup n→∞ wJ L,n W 0 (R) < ∞,
which yields, combining with (3.33), the conclusion of the claim.

□

Theorem 3.15 (Approximation by proles). Let (u 0,n ) n be a sequence bounded in H s0 that admits a prole decomposition (φ j Ln ) n j . Dene as above the nonlinear proles φ j n , the modied nonlinear proles φj n and the modied remainder wJ L,n . Let I n be a sequence of intervals such that 0 ∈ I n , and assume that for each j ⩾ 1, for large n,

0 ∈ I n ⊂ I max (φ j n ), j ∈ J C =⇒ lim sup n→∞ |∇| s0 φ j n S 0 (In) < ∞, (3.35)
j ∈ J N C =⇒ lim sup n→∞ φ j n S s 0 (In) < ∞. (3.36)
Let u n be the solution of (2.1) with initial data u 0,n . Then for large n, I n ⊂ I max (u n ), We will also need the fact that Pythagorean expansions of the Sobolev norms hold in the setting of the preceding Theorem: Lemma 3.16. With the same assumptions and notations as in Theorem 3.15, if (t n ) n is a sequence of time with t n ∈ I n for all time, then for all J ⩾ 1,

u n (t) = 1⩽j⩽J φj n (t, x) + wJ L,n (t) + r J n (t), t ∈ I n ,
∥u n (t n )∥ 2 Ḣs 0 = J j=1 ∥ φj n (t n )∥ 2 Ḣs 0 + ∥ wJ n (t n )∥ 2 Ḣs 0 + o J,n (1),
where lim J→∞ lim sup n→∞ o J,n (1) = 0. Furthermore, for all s with 0 ⩽ s < s 0 ,

∥u n (t n )∥ 2 Ḣs = 1⩽j⩽J j∈J N C ∥ φj n (t n )∥ 2 Ḣs + ∥ wJ n (t n )∥ 2 Ḣs + o J,n (1),
Before proving Theorem 3.15 and Lemma 3.16, we need two technical lemmas.

Lemma 3.17. Let (q, r) be a Schrödinger admissible pair with q, r nite, φ such that |D| s0 φ ∈

L q L r (R × R d ), ψ ∈ C ∞ 0 (R × R d )). Let Λ n = (λ n , t n , x n )
and M n = (µ n , s n , y n ) be two sequences of transformations that are orthogonal in the sense of Denition 3.2. Let

φ n (t, x) = 1 λ 2 p 0 n φ t -t n λ 2 n , x -x n λ n , ψ n (t, x) = ψ t -s n µ 2 n , x -y n µ n .
Assume that (µ n /λ n ) n is bounded. Then,

lim n→∞ |∇| s0 (φ n ψ n ) L q L r = 0.
Proof. By density (since (q, r) are nite), we can assume φ ∈ C ∞ 0 (R × R d ). Rescaling and translating φ n ψ n , we can assume (since µ n /λ n is bounded) that one of the following holds:

• lim n→∞ λ n = lim n→∞ µ n = 1. In this case, the fact that the two sequences are orthogonal implies that φ n ψ n = 0 for large n. • lim n λ n = ∞ and ∀n, µ n = 1, s n = 0, y n = 0. In this case, we have

|φ n ψ n (t, x)| ⩽ 1 λ 2/p0 n ∥φ∥ ∞ |ψ(t, x)|
and thus φ n ψ n goes to 0 in L q L r as n → ∞. The same argument proves that for all α ∈ N,

β ∈ N d , ∂ α t ∂ β x (φ n ψ n ) goes to 0 in L q L r as n → ∞.
Interpolating we obtain the conclusion of the lemma.

□ Lemma 3.18. Let (Λ j n ) n = (λ j n , t j n , x j n ) n , 1 ⩽ j ⩽ J be
a family of sequences of transformations that are pairwise orthogonal. For j ∈ 1, J , we let φ j such that |∇| s0 φ j ∈ S 0 (R), and let φ j n be dened by (3.24). Then

lim n→∞ J j=1 φ j n p 0 (d+2) 2 Xp 0 (R) = J j=1 φ j p 0 (d+2) 2
Xp 0 (R)

(3.38) lim n→∞ J j=1 |∇| s0 φ j n 2(d+2) d W 0 (R) = J j=1 |∇| s0 φ j 2(d+2) d W 0 (R) (3.39)
lim n→∞ g 0   J j=1 φ j n   - J j=1 g 0 (φ j n ) Ṅ s 0 (R) = 0. (3.40)
Furthermore, assuming that λ j n = 1 and that φ j ∈ S s0 (R) for all n, one has

lim n→∞ J j=1 φ j n p 1 (d+2) 2 Xp 1 (I) - J j=1 φ j n p 1 (d+2) 2
Xp 1 (I) = 0

(3.41) lim n→∞ J j=1 φ j n 2(d+2) d W 0 (R) - J j=1 φ j n 2(d+2) d W 0 (R) = 0 (3.42) lim n→∞ g   J j=1 φ j n   - J j=1 g(φ j n ) N s 0 (R) = 0. (3.43)
Proof. Proof of (3.41), (3.42) and (3.43). By a density argument (and Proposition 2.8 for the proof of (3.43)), we can assume that

φ j ∈ C ∞ 0 (R × R d ).
As a consequence of the orthogonality of the sequences Λ j n , we deduce that for large n, the supports of the functions φ j n , j ∈ 1, J are two-by-two disjoint. The three estimates follow immediately.

Proof of (3.40). By a density argument and (2.10), we can assume φ j ∈ C ∞ 0 R × R d . We argue by induction on J. We x J ⩾ 2. Arguing by contradiction, reordering the proles and extracting subsequences, we see that we can assume ∀n, λ J n = min j∈ 1,J λ j n .

It is sucient to prove:

(3.44) lim n→∞ g 0 J j=1 φ j n -g 0 J-1 j=1 φ j n -g 0 (φ J n ) Ṅ s 0 (R) = 0. We let χ ∈ C ∞ 0 R × R d , with χ = 1 on the support of φ J . We let χ n (t, x) = χ t-t J n (λ J n ) 2 , x-x J n λ J n
. By Lemma 3.17, and Lemma 2.9 or Lemma 2.10 (see (2.10)),

g 0 J j=1 φ j n = g 0 (1 -χ n ) J-1 j=1 φ j n + φ J n + o n (1) in Ṅ s0 g 0 J-1 j=1 φ j n = g 0 (1 -χ n ) J-1 j=1 φ j n + o n (1) in Ṅ s0 .
Next, we note that, since g 0 (0) = 0 and the supports of (1

-χ n ) J-1 j=1 φ j n and φ J n are disjoint, one has g 0 (1 -χ n ) J-1 j=1 φ j n + φ J n = g 0 (1 -χ n ) J-1 j=1 φ j n + g 0 φ J n .
Combining the preceding estimates, we obtain (3.44), and hence (3.40). We omit the similar proofs of (3.38) and (3.39).

□

Proof of Theorem 3.15. We let, for t ∈ I n ,

(3.45) v J n (t) = 1⩽j⩽J φJ n (t, x) + wJ L,n (t, x), e J n = Lv J n -g(v J n ).
We will prove that there exists a constant C > 0 such that

(3.46) ∀J ⩾ 1, lim sup n→∞ ∥v J n ∥ X(In) + ∥v J n ∥ W s 0 (In) ⩽ C,
and that for all δ > 0, there exists J δ such that

(3.47) ∀J ⩾ J δ , lim sup n→∞ ∥e J n ∥ N s 0 (In) ⩽ δ.
By the denitions (3.6) and (3.32) of w J L,n and wJ L,n , we have v J n (0) = u 0,n . This implies, together with (3.46) and the assumption that (u 0,n ) n is bounded in H s0 that there exists a constant C > 0 independent of J such that ∀J, lim sup

n→∞ ∥v J n (0)∥ H s 0 ⩽ C.
Thus we see that for J ⩾ J δ , and n large enough, the assumptions of Theorem 2.19 are satised.

The conclusion of Theorem 3.15 follows. We are left with proving (3.46) and (3.47). In all the proof, we will denote by C a large positive constant that may change from line to line, might depend on the sequence (u n ) n , but is independent of J.

Proof of (3.46). We note that it is sucient to prove the bound in (3.46) for J large.

By the Pythagorean expansion (3.7) we have,

∀J ⩾ 1, J j=1 ∥φ j L (0)∥ 2 Ḣs 0 ⩽ lim sup n→∞ ∥u n (0)∥ 2 Ḣs 0 ⩽ C. Hence ∞ j=1 ∥φ j L (0)∥ 2 Ḣs 0 < ∞.
Letting ε 0 be a small positive number, we see that there exists

J 0 ⩾ 1 such that (3.48) ∞ j=J0 φ j L (0) 2 Ḣs 0 ⩽ ε 0 .
Using the small data theory for equation (1.5) (see Proposition 2.12 and Remark 2.20), we deduce that if j ∈ J C and j ⩾ J 0 , φ j is global and

(3.49) j⩾J0 j∈J C φ j 2 Ṡs 0 (R) < ∞
Arguing similarly with the Pythagorean expansion of the H s0 norm given by (3.7), (3.18), we obtain (taking a larger J 0 if necessary) that if j ∈ J N C and j ⩾ J 0 , φ j is global and

(3.50) j⩾J0 j∈J N C φ j 2 S s 0 (R) < ∞.
Next, we see that the assumptions of the theorem implies that for all 1 ⩽ j ⩽ J 0 -1, there exists an interval I j ⊂ I max (φ j ) such that for large n,

t-t j n (λ j n ) 2 , t ∈ I n ⊂ I j and ∥φ j ∥ Ṡs 0 (I j ) < ∞ if j ∈ J C ∥φ j ∥ S s 0 (I j ) < ∞ if j ∈ J N C .
We let φ j (t, x) = φ j (t, x), if t ∈ I j , φ j (t, x) = 0 if t ∈ R \ I j . We denote by φ j n the corresponding modulated proles, dened similarly as in (3.24). By (3.49), (3.50), the denition of φj n and Lemma 3.18, we obtain, for J ⩾ J 0 . Proof of (3.47). We have, by (3.45),

φ j n + J j=J0 φ j n 2(2+d) d Ẇ s 0 (R) = J0-1 j=1 ∥φ j ∥ 2(2+d) d Ẇ s 0 (R) + J j=J0 ∥φ j ∥ 2(2+d) d Ẇ s 0 (R) ⩽ C.
e J n = 0 1⩽j⩽J j∈J N C L φj n -g( φj n ) + 0 1⩽j⩽J j∈J C L φj n -g 0 ( φj n ) + 0 L wJ L,n -g(v J n ) + 1⩽j⩽J j∈J C g 0 ( φj n ) +
g(v J n ) -g J j=1 φj n N s 0 (In) ⩽ C ∥ wJ L,n ∥ X(In) + ∥ wJ L,n ∥ Ẇ s 0 (In)
By Claim 3.14, there exists J δ such that, for all J ⩾ J δ ,

(3.53) ∀J ⩾ J δ , lim sup n→∞ g(v J n ) -g J j=1 φj n N s 0 (In) ⩽ δ 2 .
Using Lemma 2.9 or Lemma 2.10 together with Lemma 3.18:

(3.54)

g 0 J j=1 φj n = g 0 J j=1 φ j n + o n (1) = J j=1 g 0 φ j n + o n (1) in Ṅ s0 (I n ),
and similarly (using also Lemma 3.12), (3.55)

g 0 J j=1 φj n = g 0 1⩽j⩽J j∈J N C φj n + o n (1) = 1⩽j⩽J j∈J N C g 0 φj n + o n (1) = 1⩽j⩽J g 0 φj n + o n (1) in N 0 (I n )
By Proposition 2.8 and Lemma 3.12, using that g 1 ∈ N (s 0 , p 0 , p 1 ), we obtain

g 1 J j=1 φj n -g 1 1⩽j⩽J j∈J N C φj n N 0 (In) ≲ j∈J C ,1⩽j⩽J φj n Xp 1 ( 1⩽j⩽J φj n Xp 1 ) p1-1 1⩽j⩽J φj n W 0 -→ n→∞ 0.
Combining with Lemma 3.13 and Lemma 3.18, we obtain (3.56)

g 1 J j=1 φj n = g 1 1⩽j⩽J j∈J N C φj n + o n (1) = 1⩽j⩽J j∈J N C g 1 φj n + o n (1) in N s0 (I n ).
u n (t n ) = J j=1 φj n (t n ) + wJ L,n (t n ) + r J n (t n ).
We will interpret this expansion as a prole decomposition of u n (t n ). Using the property of wJ L,n

given by Claim 3.14, the property (3.37) of r J n and the bound of φ j n -φj n , j ∈ J C given by Lemma 3.12, we obtain

u n (t n , x) = J j=1 φ j n (t n , x) + R J n (x),
where lim J→∞ lim sup n→∞ e i•∆ R J n Ẇ s 0 (R)∩Xp 0 (R) = 0.

We have

φ j n (t n , x) = 1 (λ j n ) 2 p 0 φ j t n -t j n (λ j n ) 2 , x -x j n λ j n .
Extracting subsequences, we can assume that tn-t j n (λ j n ) 2 has a limit σ j as n → ∞. We dene a new linear prole ψ j Ln by

ψ j Ln (t, x) = 1 (λ j n ) 2 p 0 ψ j L t + t n -t j n (λ j n ) 2 , x -x j n λ j n ,
where ψ j L is the solution of the linear Schrödinger equation such that lim t→σ j ψ j L (t) -φ j (t)

Ḣs 0 = 0.
With these choice of ψ j L , we see that (ψ j L,n ) n j⩾1 is a Ḣs0 prole decomposition for the sequence (u n (t n )) n . The conclusion of the lemma follows from the Pythagorean expansions (3.7), (3.17 Then for all A ∈ (0, A 0 ), there exists F(A) > 0 such that for all interval 0 ∈ I, for all solution u ∈ C 0 (I, Ḣs0 ) of (1.5) such that Sketch of proof. The proof is by contradiction. Let us denote by P(A) the property that there exists F(A) ∈ (0, ∞) such that (4.1) implies (4.2). By the small data theory for equation (1.5), P(A) holds for small A > 0. Assuming that it does not hold for all A ∈ (0, A 0 ), we obtain the existence of a critical A c ∈ (0, A 0 ) such hat P(A) holds for A < A c , but P(A c ) does not hold. Thus there exists a sequence of intervals I n = (a n , b n ) ∋ 0, and of solutions u n ∈ C 0 ((a n , b n ), Ḣs0 ) such • -

x n λ n -φ 0 Ḣs 0 = 0,
and the solution φ of (1.5) with initial data φ 0 satises that for all t ∈ R there exist x(t) ∈ R N , λ(t) > 0 such that

j⩾1 ∥φ j L (0)∥ 2 L 2 < ∞.
We denote by φ j n the nonlinear proles associated to the preceding prole decomposition. We will prove that for every j ∈ J c , (

sup 0⩽τ <T+-tn φ j n (τ ) Ḣs 0 ⩽ m + 2ε 0 , n ≫ j 1.

Let ε > 0 be a small constant. By (4.4) , (4.5) and the small data theory for equations (1.5) and (2.1) there exists J 0 ⩾ 1 such that for j ⩾ J 0 + 1, φ j is global and

(4.7) ε ⩾ ∥φ j ∥ S s 0 (R) if j ∈ J N C ∥φ j ∥ Ṡs 0 (R) if j ∈ J C .
In particular (4.6) is satised for j ∈ J C , j ⩾ J 0 + 1. We next prove by contradiction that (4.6) holds for j ∈ J C ∩ 1, J 0 . If not, by (4.4), there exists τ ′ n ∈ [0, T + -t n ) such that (4.8)

sup j∈ 1,J0 ∩J C sup 0⩽τ ⩽τ ′ n φ j n (τ ) Ḣs 0 = m + ε 0 ∈ (0, A 0 ).
By the local well-posedness theory for equation (2.1) and the fact that λ j n = 1 for j ∈ J N C , there exists τ 0 > 0 such that (4.9)

lim sup n→∞ sup j∈ 1,J0 ∩J N C ∥φ j n ∥ S s 0 (0,τ0) < ∞. Since m + ε 0 < A 0 ,
(σ n ) n with 0 ⩽ σ n ⩽ τ ′ n , for all J, lim sup n j∈ 1,J ∩J C φ j n (σ n ) Ḣs 2 ⩽ m.
This clearly contradicts (4.8), proving (4.6).

Next, we observe that (4.6) implies by Proposition 4.1 ∀j ∈ 1, J 0 ∩ J c , lim sup n→∞ φ j n Ṡs 0 (0,T+-tn) < ∞ Combining this information with (4.7) and (4.9), we see that the Assumptions of Theorem 3.15 are satised on the interval I n = [0, T + -t n ). By the conclusion of the theorem, we obtain that for large n, u(• + t n ) ∈ S s0 ((0, T + -t n )). This implies u ∈ S s0 ((0, T + )), contradicting the blow-up criterion for equation (2.1). The proof is complete.

□

General rigidity result

In this section, we consider equation (2.1), where g satises Assumption A p. 8, and is of the form g(u) = G ′ (|u| 2 )u for some C 1 function G. We recall that with these assumptions, the mass M (u), the energy E(u) and the momentum P (u) are conserved for H 1 ∩ H s0 solutions of (2.1), where as usual s 0 = d 2 -2 p0 . We will also consider the virial functional:

(5.1)

Φ(u) = |∇u| 2 + d 2 (G ′ (|u| 2 )|u| 2 -G(|u| 2 ).
Since the assumptions on g imply G ′ (|u|

min t⩾0 Φ(u(t)) - |P (u)| 2 M (u) = 0. Furthermore, we have z Rn (0) = R d Rn φ x Rn |u 0 (x)| 2 dx = |x|<R0(ε) Rn φ x Rn |u 0 (x)| 2 dx + |x|>R0(ε) Rn φ x Rn |u 0 (x)| 2 dx.
Thus, for some constant C > 0, |I| ≲ Rn ε.

Furthermore, in the integral dening II, we have:

|x -X( tn )| ⩽ |x -x( tn )| + |x( tn ) -X( tn )| ⩽ R 0 (ε) + R n = Rn ,
where we have used the denition of Rn .

We next write

II = |x-x( tn)|⩽R0(ε) (x -X( tn ))|u( tn , x)| 2 dx = (x( tn ) -X( tn )) R d |u( tn , x)| 2 dx -(x( tn ) -X( tn )) |x-x( tn)|⩾R0(ε) |u( tn , x)| 2 dx + |x-x( tn)|⩽R0(ε) (x -x( tn ))|u( tn , x)| 2 dx.
By (5.9), (5.11), we have (5.17)

|II| ⩾ R n (M (u) -Cε) -R 0 (ε)M (u).
By (5.15), (5.16) and (5.17), we obtain

|z Rn ( tn )| ⩾ R n (M (u) -Cε) -R 0 (ε)M (u) -Rn ε.
Thus, (5.18) |z Rn ( tn

)| ⩾ R n (M (u) -Cε) -2R 0 (ε)M (u),
where we chose ε ≪ M (u).

By (5.13), (5.14) and (5.18), we have

R n M (u) ≲ R 0 (ε)M (u) + ε tn .
By (5.10), we deduce:

R n M (u) ≲ R 0 (ε)M (u) + ε R n ε 0 . Choosing ε ≪ ε 0 M (u), we obtain R n ≲ R 0 (ε).
Letting n → +∞, we obtain a contradiction. This completes the proof. □

The second lemma concerns the derivative of the localized virial functional. We consider

W R (t) = RIm R d φ x -X(t) R ∇uu dx,
where X(t) is dened by (5.4) and φ is as in the proof of Lemma 5.2. By the relative compactness of K and the continuous embedding of H 1 ∩ H s0 into L p0+2 ∩ L p1+2 , we have that for any ε > 0, there exists R 1 (ε)

W ′ R (t) = j RIm θ ′ x j -X j (t) R -X ′ j (t) R u j u dx A1 + j RIm θ x j -X j (t) R ∂ t (u j u) dx A2 .
We will prove the following result on the terms A 1 and A 2 :

Lemma 5.3. Let ε, ε > 0 be small. There exist L(ε) depending on ε such that for R ⩾ 2R 1 (ε),

L(ε) < t < R 2ε , A 1 = -2|P (u)| 2 M (u) + O(ε), A 2 = 2Φ(u) + O(ε).
Proof. Writing ∂ t (u j u) = (u tj u + u j u t ) and integrating by part, we obtain

A 2 = j -2RIm θ x j -X j (t) R u j u t - j Im θ ′ x j -X j (t) R uu t = j C j + j D j = I + II.
Using the equation (2.1), we obtain

C j = 2RRe θ x j -X j (t) R u j (-∆u + G ′ (|u| 2 )u) dx = 2RRe ∇ θ x j -X j (t) R u j ∇u + θ x j -X j (t) R 1 2 ∂ j G(|u| 2 ) = 2R θ x j -X j (t) R ∂ j |∇u| 2 2 dx + 2 θ ′ x j -X j (t) R |u j | 2 dx + 2R θ x j -X j (t) R 1 2 ∂ j G(|u| 2 ) dx.
Summing up, we obtain

I = -2 j θ ′ x j -X j (t) R |∇u| 2 2 + 1 2 G(|u| 2 ) dx + 2 j θ ′ x j -X j (t) R |u j | 2 dx = -d |∇u| 2 + G(|u| 2 ) dx -2 j θ ′ x j -X j (t) R -1 |∇u| 2 2 + 1 2 G(|u| 2 ) dx + 2 |∇u| 2 dx + 2 j θ ′ x j -X j (t) R -1 |u j | 2 dx. Moreover D j = Re θ ′ x j -X j (t) R uiu t dx = Re θ ′ x j -X j (t) R u(-∆u + G ′ (|u| 2 )u) dx = θ ′ x j -X j (t) R G ′ (|u| 2 )|u| 2 dx -θ ′ x j -X j (t) R ∆ |u| 2 2 -|∇u| 2 dx = θ ′ x j -X j (t) R (G ′ (|u| 2 )|u| 2 + |∇u| 2 ) dx - 1 2R 2 θ ′′′ x j -X j (t) R |u| 2 dx.
where we have used

θ ′ xj -Xj (t) R ∂ kk |u| 2 2 = 0 for all k ̸ = j. Thus, II = d |∇u| 2 + G ′ (|u| 2 )|u| 2 dx + j θ ′ x j -X j (t) R -1 (|∇u| 2 + G ′ (|u| 2 )|u| 2 ) dx - 1 2R 2 θ ′′′ x j -X j (t) R |u| 2 dx.
Combining the above, we have

A 2 = 2Φ(u) - j 1 2R 2 θ ′′′ x j -X j (t) R |u| 2 dx (5.21) + j θ ′ x j -X j (t) R -1 2|u j | 2 + G ′ (|u| 2 )|u| 2 -G(|u| 2 ) . (5.22)
Applying Lemma 5.2, for each ε, there exists L(ε) such that

(5.23) |x(t) -X(t)| ⩽ εt, ∀t ⩾ L(ε). Assume L(ε) < t < R 2ε and R ⩾ 2R 1 (ε). Then |(5.21)| + |(5.22)| ≲ |x-X(t)|⩾R 1 2R 2 |u| 2 + 2|∇u| 2 + |G ′ (|u| 2 )||u| 2 + |G(|u| 2 )| dx,
where we have used the fact that for |x -

X(t)| ⩽ R then |x j -X j (t)| ⩽ R, for each 1 ⩽ j ⩽ d. Thus, θ ′ xj -Xj (t) R = 1 and θ ′′′ xj -Xj (t) R = 0 for each 1 ⩽ j ⩽ d. Moreover, for |x -X(t)| ⩾ R and L(ε) < t < R 2ε , we have |x -x(t)| ⩾ R -|x(t) -X(t)| ⩾ R -εt > R/2 ⩾ R 1 (ε).
This implies that

A 2 = 2Φ(u) + O(ε). Similarly, for L(ε) < t < R 2ε , R ⩾ 2R 1 (ε), we have A 1 = j -X ′ j (t)Im θ ′ x j -X j (t) R u j u dx = j -2P j (u) M (u) Im u j u dx - 2P j (u) M (u) Im θ ′ x j -X j (t) R -1 u j u dx = j -2P j (u) M (u) P j (u) + O(ε) = -2|P (u)| 2 M (u) + O(ε).
This completes the proof of Lemma 5.3.

□

We are now ready to prove Proposition 5.1.

Theorem 6.5. Let g satisfy Assumption B. Assume that s 0 ⩾ 1, Properties 1.1 and 6.1 hold.

Then for all A ∈ (0, A 0 ) there exists F(A, η) > 0 such that for any interval I, for any solution u ∈ C 0 (I, H s0 ) of (2.1) such that (6.2)

M (u) ⩽ m c -η and sup t∈I ∥u(t)∥ 2 Ḣs 0 + η∥u(t)∥ 2 2 ⩽ A 2 ,
one has u ∈ S s0 (I) and ∥u∥ S s 0 (I) ⩽ F(A, η).

Theorem 6.5 implies Theorem 6.4 by the scattering criterion (Lemma 2.16) and letting η go to zero.

Proof of Theorem 6.5. We argue by contradiction, following the compactness/rigidity scheme as in [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]. We x η > 0 throughout the argument. In all the proof, we will endow H s0 with the norm dened by (6.3) ∥u∥ 2 H s 0 = ∥u∥ 2 Ḣs 0 + η∥u∥ 2 2 . We will denote by P(A) the property that there exists F(A) such that for any interval I, for any solution u ∈ C 0 (I, H s0 ) such that (6.2) holds, one has u ∈ S s0 and ∥u∥ S s 0 (I) ⩽ F(A).

By the small data theory for (2.1) (see Proposition 2.12), if A > 0 is small and ∥u(t)∥ H s 0 ⩽ A for some t ∈ I max (u), then u is globally dened, scatters and ∥u∥ S s 0 (R) ≲ A. This implies that P(A) holds for small A > 0.

Thus if the conclusion of Theorem 6.5 does not hold, there exists A c ∈ (0, A 0 ) such that for all A < A c , P(A) holds, and P(A c ) does not hold, i.e. there exists a sequence of intervals ((a n , b n )) n , a sequence (u n ) n of solutions of (2.1) on (a n , b n ), We rst assume the claim and conclude the proof of Theorem 6.5. By the claim, there exist (after extraction of subsequences) φ ∈ H s0 and (x n ) n such that (6.6) holds. Let u be the solution of (2.1) such that u(0) = φ. Since R η is closed in H s0 , we have φ ∈ R η .

We next prove by contradiction ∥u∥ S s 0 ((-∞,0)) = ∥u∥ S s 0 ((0,+∞)) = +∞.

If (t n ) n is any sequence of times, Claim 6.6 and the preceding properties imply that one can extract subsequence such that u(t n , • -x n ) converges in H s0 for some sequence (x n ) n ∈ (R d ) N . This is classical that it implies that one can nd a function x(t), t ∈ R such that K dened by (5.2) has compact closure in H s0 . We give a sketch of proof of this fact. Using the compactness and the fact that the solution u is not identically 0, we rst notice that there exists R 0 > 0, η > 0 By the claim, extracting subsequences, there exists (x n ) n such that u(t n , • -x n ) convergences to φ 0 in H s0 (up to extract subsequence). By (6.9), (6.10)

Φ(φ 0 ) - |P (φ 0 )| 2 M (φ 0 ) = 0.
Since R η is closed in H s0 , φ 0 ∈ R η . This implies, by Lemma 6.3, that Φ(φ 0 ) -|P (φ 0 )| 2 M (φ 0 ) > 0.

This contradicts to (6.10). This completes the proof.

□

We are left with proving Claim 6.6.

Proof of Claim 6.6. Step 1. Prole decomposition. Extracting subsequences, we can assume that u 0,n = u n (0) has a prole decomposition as in Section 3: . We denote by φ j n the corresponding nonlinear proles, and φj n the modied nonlinear proles.

Our goal is to prove that there is a unique j 0 ⩾ 1 such that φ j0 is not identically zero, and that j 0 ∈ J c . We rst note that there is at least one j such that φ j is not identically zero. If not, by the small data local well-posedness, we would have lim n→∞ ∥u n ∥ S s 0 (In) = 0, a contradiction with our assumptions.

In the remaining step, we will prove that there is at most one nonzero prole. Arguing by contradiction, we assume that there is at least two nonzero proles, say φ 1 and φ 2 . By the small data theory, there exists ε 0 > 0 such that (6.11) inf t∈In φj n H s 0 ⩾ ε 0 , j ∈ {1, 2}.

Step 2. Bound of the H s0 norm. We prove that for all j ⩾ 1 we have I n ⊂ I max ( φj n ) and, for large n, By (6.4), and the Pythagorean expansions (3.7), (3.18), we obtain the bounds, for J ⩾ 1

j⩾1 ∥φ j L (0)∥ 2 Ḣs 0 ⩽ A 2 c , j∈J N C ∥φ j L (0)∥ 2 H s 0 ⩽ A 2 c
Fixing a small ε > 0, we obtain, by the small data theory for equations (2.1) and (1.5) and Lemma 3.12, that there exists J 0 ⩾ 1 such that, for j ⩾ J 0 + 1, I max (φ j ) = R, (6.13)

∀j ⩾ J 0 + 1, ∥φ j ∥ Ṡs 0 (R) < ∞ if j ∈ J C ∥φ j ∥ S s 0 (R) < ∞ if j ∈ J N
H s 0 = A 2 c - 1 2 ε 2 0 .
This implies that for large n ∀j ∈ J C ∩ 1, J 0 , sup 0⩽t∈b ′ n ∥φ j n (t)∥ Ḣs 0 < A c < A 0 .

Thus by Proposition 4.1, we obtain a constant C > 0 such that for large n, Going back to (6.15), we see also that for large n ∀j ∈ 1, J 0 ∩ J N C , sup

0⩽t⩽b ′ n ∥φ j n (t)∥ H s 0 ⩽ A 2 c - 1 4 ε 2 0 .
Also, using the Pythagorean expansion of the mass we see that ∀j ∈ J N C , M (φ j ) ⩽ m c -η.

Using that P(A) holds for A = A 2 c -1 4 ε 2 0 we obtain that there exists a constant C > 0 such that for large n (6.17)

sup j∈J N C ∩ 1,J0 ∥φ j n ∥ S s 0 ([0,b ′ n ]) ⩽ C.
Combining (6.13), (6.16) and (6.17 contradicting (6.15). This proves that (6.12) holds for all j ⩾ 1, for large n.

Step 3. Uniqueness of the nonzero prole.

In this step we still assume that φ 1 and φ 2 are nonzero proles. Using (6.4) and (6.12), and arguing as in Step 2, we see that the assumptions of Theorem 3.15 are satised on [a n , b n ]. This

and, if s ⩾ 1 ,

 1 of the energyE(u(t)) = R d |∇u(t, x)| 2 dx + R d G(|u(t, x)| 2 )dx.and the momentum P (u(t)) = Im R d ∇u(t, x) u(t, x)dx of a solution.

h

  i=1 |β i | = s, and h = p, we obtain(2.22), which concludes the proof of the Lemma.

(3. 10 )

 10 Indeed,(3.8) follows from the orthogonality of the proles. The property (3.9) follows from(3.5) and the orthogonality of the proles. Finally, (3.10) follows easily from the two other properties.

  -φ j n (0) Ḣs 0 = 0.

  Using the long time perturbation theory for equation (1.5) (see Theorem 2.19 and Remark 2.20), we obtain sup τ ∈I φ j (τ ) -Φ j n (τ ) Ḣs 0 + φ j -Φ j n Ṡs 0 (I) By the change of variable τ = t-tj,n (λ j n ) 2 , y = x-xj,n λ j n , we obtain (3.27). Applying (3.27), we obtain (3.26).

  n→∞ ∥ φj n (s j n )∥ L 2 = 0,By conservation of the L 2 norm, and interpolation with the bound of the Ḣs norm which follows from

( 3 .

 3 31) ẽj n N s 0 (Jn) ≲ φj n S s 0 (Jn) φj n p1 L p 1 (d+2)/2 (Jn×R d ) + φj n p2 L p 2 (d+2)/2 (Jn×R d ) ≲ φj n S s 0 (Jn) φj n p1 S s 1 (Jn) + φj n p2 S s 2 (Jn) , where s k = d 2 -2 p k , k ∈ {1, 2}. Since s 1 < s 2 < s 0 , the conclusion of the Lemma follows from (3.28).

A,

  similar argument yields lim sup n→∞ J j=1 φj n Xp 0 (In) ⩽ C. We also have, for 0 ⩽ s < s 0 , by (3.28) in Lemma 3.12, and the same argument as above, using (3.50) and Lemma 3.18, yields lim sup n→∞ J j=1 φj n W s (In) ⩽ C. Combining the estimates above with Claim 3.14 and (3.34), we obtain (3.46).

□ 4 .

 4 Global well-posednessIn this section we prove our theorem on global well-posedness, Theorem 1.2.We rst observe that Property 1.1 is equivalent to the existence of uniform space-time bound for solutions of equation (1.5) that are bounded in critical norm. Proposition 4.1. Let d ⩾ 2, p 0 > 4 d , s 0 = d 2 -2 p0 . Assume Property 1.1.

(4. 1 )

 1 sup t∈I ∥u(t)∥ Ḣs 0 ⩽ A we have (4.2)∥u∥ Ṡs 0 (I) ⩽ F(A).

  sup an<t<bn ∥u n (t)∥ Ḣs 0 -→ n→∞ A c , and lim n→∞ ∥u n ∥ Ṡs 0 (an,0) = lim n→∞ ∥u n ∥ Ṡs 0 (0,bn) = +∞. By a standard compactness argument, using the homogeneous prole decomposition of Subsection 3.1, with the analog, for the homogeneous equation, of Theorem 3.15, we obtain, after extraction of subsequences, that there exists x n ∈ R N , λ n > 0 and φ 0 ∈ Ḣs0 \ {0} such that lim

(5. 14 )

 14 |z Rn (0)| ⩽ R 0 (ε)M (u) + C Rn ε ⩽ 2R 0 (ε)M (u) + CR n ε,where we have used for the rst bound that |φ(x)| ≲ |x| and for the second bound that φ ∈ L ∞ . Furthermore, z Rn ( tn ) = |x-x( tn)|⩾R0(ε) Rn φ x -X( tn ) Rn |u( tn , x)| 2 dx + |x-x( tn)|⩽R0(ε) Rn φ x -X( tn ) Rn |u( tn , x)| 2 dx = I + II.

( 6 . 4 )

 64 u n ∈ C 0 ((a n , b n ), H s0 ), M (u n ) ⩽ m c -η, lim n→∞ sup an<t<bn ∥u n (t)∥ H s 0 = A c .and lim n→∞ ∥u n ∥ S s 0 ((an,bn)) = ∞. Time translating u n , we can assume (6.5)a n < 0 < b n , lim n→∞ ∥u n ∥ S s 0 ((an,0)) = lim n→∞ ∥u n ∥ S s 0 ((0,bn)) = +∞.We will prove Claim 6.6. For any sequences (a n ) n , (b n ) n with a n < 0 < b n , for any sequence (u n ) n of solutions of (2.1) satisfying (6.4), (6.5), there exist, after extraction of subsequences, a sequence (x n ) n ∈ (R d ) N and φ ∈ H s0 such that (6.6) lim n→∞ ∥u n (• -x n ) -φ∥ H s 0 = 0.

  Assume to x ideas, and after extraction of subsequences limn→∞ b n = b ∈ [0, ∞). Using that lim n ∥u n ∥ S s 0 ([0,bn)) = ∞, we must have T + (u) < ∞ and b ⩾ T + (u). By the last assertion of (6.4), we obtain sup 0⩽t<T+(u) ∥u(t)∥ H s 0 < ∞.

  |u(t, x -X)| 2 dx ⩾ η.

  Ṡs 0 ([0,b ′ n ]) ⩽ C.

2 H

 2 ), we obtain that the assumptions of Theorem 3.15 are satised on [0, b ′ n ]. Using the Pythagorean expansion of Lemma 3.16 together with the limit in (6s 0 ⩽ A 2 c -ε 2 0 ,

  1, ∥⟨∇⟩ s u∥ L p , ∥u∥ Ḣs,p = ∥|∇| s u∥ L p

	we dene
	∥u∥ H s,p = where ⟨ξ⟩ = (1 + |x| 2 ) s/2 . It follows from Mikhlin multiplier theorem that
	(2.2)

  q 1 by ∥u∥ Xp and ∥u∥ Ẇ s using fractional chain rule Lemma 2.3 and Gagliardo-Nirenberg inequality Lemma 2.5.

If s /

∈ N then ⌈s⌉ ⩾ h + 1 (because h ⩽ |α| = ⌊s⌋). Using Fractional chain rule Lemma 2.3, we have

  5, we have ∥u∥ L q 4 Ḣv,r 4 ≲ ∥u∥

					v	⌊s⌋
					s Ẇ s ∥u∥	s Xp .
	Thus, for s / ∈ N,
	(2.32)	(2.26) ≲ (2.31) ≲ ∥u∥ p-h-1 Xp	∥u∥	v s Ẇ s ∥u∥ Xp ∥u∥ ⌊s⌋ s	⌊s⌋ s Ẇ s ∥u∥ h-Xp	⌊s⌋ s	= ∥u∥ Ẇ s ∥u∥ Xp , p-1
	yielding (2.22). If s ∈ N then v = 0,	k i=1 |β i | = s, and 1 q1 = p-h p(d+2)/2 . Using (2.30), we obtain
		(2.26) ≲ (2.31)

  d 2 -2 p0 , and ⌈s⌉ ⩽ p 0 if g is not a polynomial. Denition 2.11. Let 0 ∈ I be an interval. By denition, a solution u to (1.1) on I, with initial data in H s (s ⩾ s 0 ) is a function u ∈ C(I, H s ) such that for all K ⊂ I compact, u ∈ S s (K) and u

	satises the following Duhamel formula
	(2.35)

  and δ ⩽ δ 0 (A) small enough. Thus Ψ(u) ∈ B. Now, let u, v ∈ B. We have, by Strichartz estimates and Proposition 2.8,

  1.1): Proposition 2.18. Let u 0 ∈ H s . Let g, p 0 , p 1 , s such that Assumption A holds and s ⩾ d 2 -2 p0 . Let v 0 ∈ H s and v L (t) = e it∆ v 0 . Then there exist a unique solution

  H s by Ḣs , S s (I) by Ṡs (I), and similarly for Proposition 2.12. The proof is the same, replacing Proposition 2.8 by Lemma 2.9. See also e.g.

	[24].

Proposition 2.21 (Conservation laws). Let g such that Assumption A holds and g

  We denote by u L (t) = e it∆ u 0 the solution to the linear Schrödinger equation on R × R d

	2.21
	imply
	(2.46)
	(3.1)

G(|u| 2 ) ≲ |u| p0+2 + |u| p1+2 , so that by Sobolev inequalities, E(u) is well-dened if s ⩾ 1.

3. Profile decomposition

3.1. Proles in homogeneous Sobolev spaces. Let 0 < s < d 2 .

  Arguing as in the proof of the long-time perturbation theory result (Theorem 2.19), and using (3.29), we obtain (3.28).

  Sketch of proof of Lemma 3.16. By Theorem 3.15, (u n (t n )) n is bounded in H s0 (R d ), and

	By (3.53), (3.54), (3.55) and (3.56), we obtain (3.47), which concludes the proof.

□

  Ḣs 0 ⩽ A c (see the similar proof of Theorem 6.5 below). Then φ is global by Property 1.1. Since φ is not the zero solution, the preceding compactness property implies lim inf t→∞ ∥φ(t)∥ Ḣs 0 < m ∈ (0, A 0 ). t n = T + (u) -1/2 n . Extracting subsequences, we can assume by Proposition 3.5 that (u(t n )) n

		1 λ(t)	2 p	φ t,	• -x(t) λ(t)	, t ∈ I max (φ)
	has compact closure in Ḣs0 and
	sup t∈Imax contradiction with the fact that by Property 1.1, φ must be scattering. The proof is complete. □ dp 0 2 > 0, a ∥φ(t)∥ L
	Proof of Theorem 1.2. We argue by contradiction, assuming that there exists a solution u of (2.1)
	such that T + (u) < ∞ and
	lim sup n→∞ ∥u(t)∥ Let ε 0 be such that m + 3ε 0 < A 0 . By conservation of mass, we have indeed,
	(4.3)			lim sup	∥u(t)∥ H s 0 < ∞.
				t→T + (u)
	admits a prole decomposition	(φ j L,n ) n	j⩾1	. By the Pythagorean expansion (3.7) of the Ḣs0
	norm, we have
	(4.4)			∥φ j L (0)∥ 2 Ḣs 0 ⩽ m.
				j⩾1
	By the Pythagorean expansion of the L 2 norm,
	(4.5)

Let

  by (4.8) and Proposition 4.1, we obtain ∀j ∈ 1, J 0 ∩ J c , lim sup Note that since τ ′ n → 0 as n → ∞, we have τ 0 > τ ′ n for large n. Combining with (4.7) and (4.9), we see that the assumptions of Theorem 3.15 are satised on the interval I n = [0, τ ′ n ). By Lemma 3.16, for all sequence

	n→∞	φ j n Ṡs 0 (0,τ ′

n ) < ∞

  2 )|u| 2 + G(|u| 2 ) ≲ |u| p0+2 + |u| p1+2 , one easily checks, using Sobolev inequalities, that Φ is well-dened if u ∈ H s0 ∩ H 1 .

	We prove the following result:
	Proposition 5.1. With the assumptions above, let u be a solution of (2.1) dened on [0, ∞) such
	that there exists x(t), t ∈ [0, ∞) with
	(5.2)

K = {u(t, x + x(t)); t ⩾ 0}

has compact closure in H s0 ∩ H 1 . Then

(5.3)

  |∇u| 2 + |u| 2 + |G ′ (|u| 2 )||u| 2 + |G(|u| 2 )| dx ⩽ ε. |W R (t)| ⩽ CR ∥∇u(t)∥ 2 ∥u(t)∥ 2 ≲ R. ∂ xj u = ∂ j u by u j , u t by ∂ t u andR d by . We have

	such that
	(5.19)
	|x-x(t)|⩾R1(ε)
	By Cauchy-Schwarz inequality, we have
	(5.20)
	For convenience, we denote	d j=1 by	j ,

  This implies by Theorem 1.2 that T + (u) = +∞, a contradiction. Hence (6.7). Next, we see that (6.6), perturbation theory for equation (2.1) and the last assertion in (6.4) implies that for any compact interval I ⊂ I max (u), sup t∈I ∥u(t)∥ H s 0 ⩽ A c .

This implies by Theorem 1.2 that u is global and

sup t∈R ∥u(t)∥ H s 0 ⩽ A c .

By (6.5) and stability theory for equation (2.1), one has

  Thus for all t, there exists x(t) ∈ R d such that |x|<R0 |u(t, x -x(t))| 2 dx ⩾ η/2.this choice of x(t), one can check that K dened by (5.2) is compact. By (6.8), there exists a sequence of times (t n ) n such that

	For By Proposition 5.1, we have
	(6.8)	min t⩾0	Φ(u(t)) -	|P (u)| 2 M (u)	= 0.
	(6.9)	lim tn→∞	Φ(u(t n )) -	|P (u(t n ))| 2 M (u(t n ))	= 0.

  C .We next prove by contradiction that for j ∈ 1, J 0 , I n ⊂ I max ( φj n ) and (6.12) holds. If not, we can assume (inverting time if necessary, and using Theorem 1.2) that for large n, there exists b′ n ∈ (0, b n ] such that [0, b ′ n ] ⊂ 1⩽j⩽J0 I max ( φj

				n ) and
	(6.15)	sup 1⩽j⩽J0	sup 0⩽t⩽b ′ n	φj n (t)	2

and (6.14)

∀j ⩾ J 0 + 1, lim sup n→∞ sup t∈R ∥ φj n (t)∥ H s 0 ⩽ ε.

Dene

(5.4)

where P (u) is momentum. The proof of Proposition 5.1 relies on an asymptotic estimate of x(t)

and a localized virial argument. We start with two lemmas.

Lemma 5.2. With the assumptions above, let u be a solution (2.1) such that there exists x(t), t ⩾ 0 such that K = {u(t, x + x(t)); t ⩾ 0} has compact closure in H 1 . Then (5.5)

Proof. We can assume that x is continuous (see e.g [START_REF] Duyckaerts | Scattering for the non-radial 3D cubic nonlinear Schrödinger equation[END_REF]Proposition 3.2]).

We argue by contradiction, assuming that there exists a sequence t n → +∞, ε 0 > 0 such that

Without loss of generality we may assume x(0) = 0.

For R > 0, we let (5.7)

R n tn ⩾ ε 0 ,

(5.10) where (5.10) follows from (5.6) and t n ⩾ tn . By precompactness of K, for any ε > 0, there exists R 0 (ε) > 0 such that for all t ⩾ 0:

. Thus, φ(x) = x for |x| ⩽ 1 and ∥φ∥ L ∞ < 2d.

We dene (5.12)

By (5.11) we deduce, for t ∈ [0, tn ],

= δ > 0. We x small parameters ε ≪ δ and ε ≪ δ.

From Lemma 5.3, we have

2ε , where R is large (and in particular R > 2εL(ε)). We have

(where we have used ε ≪ δ for the last inequality). This gives a contradiction letting R → ∞, since ε ≪ δ. □

Scattering

This section is dedicated to the proof of Theorems 1.5, 1.6, 1.7 and 1.8. Recall that M (φ) = |φ| 2 . We consider the following property: Property 6.1. There exists m c > 0 such that ∀φ ∈ (

Property 6.1 holds for all m c when g is defocusing. In the case where ι 0 > 0, we have the following: Lemma 6.2. Let g be a nonlinearity that satises Assumption B with ι 0 > 0. Then Property 6.1 is true.

Proof. Using that ι 0 = 1, we have

where G 1 (s) = G(s) -s p0/2 is such that g 1 (u) = G ′ 1 (|u| 2 )u. By Assumption (B), we have g 1 ∈ N (s 0 , p 2 , p 1 ), for some 4 d < p 1 < p 2 < p 0 . Using the denition of N , this implies (6.1)

The claim that there exists m c such that Φ(u) is larger than 1 2 |∇u| 2 when 0 < M (u) < m c follows easily from (6.1) and the generalized Gagliardo-Nirenberg inequalities (See (3.25) in [START_REF] Lewin | The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications[END_REF])

The lower bound for the energy is obtained by the same proof.

□

We have the following result. Lemma 6.3. If Property 6.1 holds, for each φ ∈ H s0 ∩ H 1 such that M (φ) < m c , we have

We next prove Theorems 1.5 and 1.7. The proofs of Theorems 1.6 and 1.8 are similar, but simpler and we omit them. Theorems 1.5 and 1.7 are an immediate consequence of the following result: Theorem 6.4. Let g satisfy Assumption B. Assume that s 0 ⩾ 1, Properties 1.1 and 6.1 hold. Let u be a solution of (2.1) with M (u) < m c and u satises (1.6). Then u scatters forward in time.

We will prove Theorem 6.4 as a consequence of proves that u n scatters for large n, contradicting (6.5). This concludes the proof that there is only one nonzero prole.

Step 4. End of the proof.

We assume that φ 1 is the only nonzero prole. By the same argument as before, we obtain that for large n, I n ⊂ I max ( φ1 n ) and lim

If 1 ∈ J C , we obtain by Proposition 4.1 that lim sup n→∞ ∥φ 1 n ∥ Ṡs 0 (In) < ∞. Thus the assumptions of Theorem 3.15 are satised on I n , a contradiction with (6.5). Thus 1 ∈ J N C . By the same argument, we obtain lim sup n→∞ sup t∈In ∥ w1

Ln (t)∥ H s 0 = 0. Indeed, if not, we would have by the conservation of the H s0 norm for the linear Schrödinger equation (and after extraction of a subsequence) lim n→∞ sup t∈In ∥ w1

Ln (t)∥ H s 0 = ε 0 > 0, and the same strategy as in Steps 2,3 would yield that u n scatters for large n, a contradiction. We have proved 1) in H s0 . By (6.5), t 1,n must be bounded, and we can assume t 1,n = 0 for all n, i.e. (6.18)

which concludes the proof of the claim.

□ Appendix A. Equivalence of Sobolev norms

In this appendix we prove (2.2). We recall Mikhlin multiplier theorem [START_REF] Mikhlin | On the multipliers of Fourier integrals[END_REF], [START_REF] Hörmander | Estimates for translation invariant operators in L p spaces[END_REF]Theorem 2.5