Sharp Upper Bound on the Blow up Rate for critical nonlinear Schrödinger Equation

Transcription

1 Sharp Upper Bound on the Blow up Rate for critical nonlinear Schrödinger Equation Frank Merle,, Pierre Raphael Université de Cergy Pontoise Institut Universitaire de France Abstract We consider the critical nonlinear Schrödinger equation iu t = u u 4 N u with initial condition u0, x = u 0. For u 0 H 1, local existence in time of solutions on an interval [0, T is known, and there exist finite time blow up solutions, that is u 0 such that lim t T <+ ut L 2 = +. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to control the blow up rate from above for small in a certain sense blow up solutions with negative energy. In a previous paper [25], we established some blow up properties of NLS in the energy space which implied a control ut L 2 C lnt t N 4 T t solutions which is C T t. and removed the rate of the known explicit blow up In this paper, we prove the sharp upper bound expected from numerics as ln lnt t ut L 2 C T t by exhibiting the exact geometrical structure of dispersion for the problem Introduction 1.1 Setting of the problem We consider in this paper the critical nonlinear Schrödinger equation { iu NLS t = u u 4 N u, t, x [0, T R N u0, x = u 0 x, u 0 : R N C 1 with u 0 H 1 = H 1 R N in dimension N 1. The problem we address is the one of formation of singularities for solutions to 1. 1

2 It is a special case of the following Hamiltonian equation iu t = u u p 1 u 2 where 1 < p < N+2 N 2 and initial condition u 0 H 1. From a result of Ginibre Velo [10], 2 is locally well-posed in H 1. In addition, 1 is locally well-posed in L 2 = L 2 R N from Cazenave Weissler [5]. See also [3], [2] for the periodic case and global well posedness results. Thus, for u 0 H 1, there exists 0 < T + such that ut C[0, T, H 1 and either T = +, we say the solution is global, or T < + and then lim sup t T ut L 2 = +. We first recall the main known facts about equations 1, 2. For u 0 H 1, we have several conservation laws: L 2 -norm: ut, x 2 = u 0 x 2 ; 3 Energy: Eut, x = 1 2 ut, x 2 1 p + 1 ut, x p+1 = Eu 0 ; 4 Momentum: Im uut, x = Im u 0 u 0 x. 5 For p < N, 3, 4 and Gagliardo-Nirenberg inequality imply ut 2 L 2 Cu 0 ut 2α L for some α < 1, so that 2 is globally well posed in H 1 : t [0, T [, ut L 2 Cu 0 and T = +. For p 1+ 4 N, the situation is quite different. Let an initial condition u 0 Σ = H 1 {xu L 2 } and assume Eu 0 < 0, then T < + follows from the so called Viriel Identity. Indeed, the quantity yt = x 2 u 2 t, x is well defined for t [0, T and satisfies y t CpEu 0 where Cp > 0. The positivity of yt yields the conclusion. p = N appears to be the critical power in this problem. From now on, we focus on this critical case p = N. Equation 1 admits a number of symmetries in energy space H 1, explicitly: 2

3 Space-time translation invariance: if ut, x solves 2, then so does ut + t 0, x + x 0, t 0, x 0 R R N. Phase invariance: if ut, x solves 2, then so does ut, xe iγ, γ R. Galilean invariance: if ut, x solves 2, then so does ut, x βte i β 2 x β 2 t, β R N. Scaling invariance: if ut, x solves 1, then so does u λ t, x = λ N 2 uλx, λ 2 t, λ > 0, and by direct computation u λ L 2 = u L 2. Moreover, 1 admits another symmetry which is not in energy space H 1, the so called pseudo conformal transformation: Pseudo conformal transformation: if ut, x solves 1, then so does vt, x = 1 t N 2 u 1 t, x x 2 ei 4t. t This additional symmetry yields the conservation of the pseudo-conformal energy for initial data u 0 Σ, which is most frequently expressed as see [35]: d 2 dt 2 x 2 ut, x 2 = 4 d dt Im x uut, x = 16Eu 0. 6 In addition, special solutions play an important role. They are the so called solitary waves and are of the form ut, x = e iωt W ω x, ω > 0, where W ω solves W ω + W ω W ω 4 N = ωwω. 7 Equation 7 is a standard nonlinear elliptic equation. In dimension N = 1, there exists a unique solution in H 1 up to translation to 7 and infinitely many with growing L 2 -norm for N 2. Nevertheless, from [1], [7] and [13], there is a unique positive solution up to translation Q ω x. Q ω is in addition radially symmetric. Letting Q = Q ω=1, then Q ω x = ω N 4 Qω 1 2 x from scaling property. Therefore, one computes Q ω L 2 = Q L 2. Moreover, multiplying 7 by N 2 Q ω + x Q ω and integrating by parts yields the so called Pohozaev identity: EQ ω = ωeq = 0. In particular, none of the three conservation laws in H 1 3, 4, 5 of 1 sees the variation of size of the Q ω stationary solutions. These two facts are deeply related to the criticality of the problem, that is the value 3

4 p = N. Note that in dimension N = 1, Q writes explicitly Qx = 3 4. ch 2 2x For u 0 small in L 2, u 0 L 2 < Q L 2, t 0, ut L 2 Cu 0, and the solution is global in H 1. Indeed, this follows from the conservation of the energy, the L 2 -norm and Gagliardo-Nirenberg inequality as exhibited by Weinstein in [34]: u H 1, N u 4 N u u 2 2 N. 8 Q 2 In addition, this condition is sharp: for u 0 L 2 Q L 2, blow up may occur. Indeed, since EQ = 0 and EQ = Q, there exists u 0ɛ Σ with u 0ɛ L 2 = Q L 2 +ɛ and Eu 0ɛ < 0, and the corresponding solution must blow up from viriel identity 6. The case of critical mass u 0 L 2 = Q L 2 has been studied in [23]. The pseudo-conformal transformation applied to the stationary solution e it Qx yields an explicit solution St, x = 1 t N 2 Q x x 2 e i 4t + i t 9 t which blows up at T = 0. Note that St L 2 = Q L 2. It turns out that St is the unique minimal mass blow up solution in H 1 in the following sense: let u 1 H 1 with u 1 L 2 = Q L 2, and assume that ut blows up at T = 0, then ut = St up to the symmetries of the equation. Note that from direct computation ES > 0. In previous paper [25], we studied solutions to 1 such that Q 2 < u 0 2 < Q 2 + α 0 and Eu 0 < 0 10 where α 0 is small. Indeed, under assumption 10, from conservation laws and variational characterization of the ground state Q, the solution ut, x remains close to Q in H 1 up to scaling and phase parameters, and also translation in the non radial case. We then were able to define a regular decomposition of the solution of the type ut, x = 1 λt N 2 Q + ɛt, x xt e iγt 11 λt 1 where ɛt H 1 δα 0 with δα 0 0 as α 0 0, λt of order ut, γt R, L 2 xt R N. The question is to understand the blow up phenomenon under a dynamical point of view in H 1 by using this decomposition. This approach was previously successfully applied by Martel-Merle in a different context for the critical generalized KdV equation u t + u xx + u 5 x = 0, t, x [0, T R 4

5 in the sequel of papers [17], [18], [19], [20], [21]. From a surprising monotony property on the basis of decomposition 11, we proved in [25] the first control from above on the blow up rate for negative energy solutions. Explicitly, consider the following property: Spectral Property Let N 2. Consider the two real Schrödinger operators L 1 = N N + 1 Q 4 N 1 y Q, L 2 = + 2 N Q 4 N 1 y Q, 12 and the real valued quadratic form for ε = ε 1 + iε 2 H 1 : Hε, ε = L 1 ε 1, ε 1 + L 2 ε 2, ε 2. Then there exists a universal constant δ 1 > 0 such that ε H 1, if ε 1, Q = ε 1, Q 1 = ε 1, yq = ε 2, Q 1 = ε 2, Q 2 = ε 2, Q = 0, then: i for N = 2, Hε, ε δ 1 ε 2 + ε 2 e 2 y for some universal constant 2 < 2; ii for N 3, Hε, ε δ 1 ε 2 ; where Q 1 = N 2 Q + y Q and Q 2 = N 2 Q 1 + y Q 1. Remark 1 This property has been proved to hold true in dimension N = 1 in [25], and we conjecture from numerics that it indeed holds true at least for low dimension. We then claim: Theorem 1 [25] Let N = 1 or N 2 assuming Spectral Property holds true. Then there exists α > 0 and a universal constant C > 0 such that the following is true. Let u 0 H 1 such that 0 < α 0 = u 0 2 Q 2 < α, E 0 < 1 Im u0 u u 0 L 2 Let ut be the corresponding solution to 1, then ut blows up in finite time 0 < T < + and there holds for t close to T N 1 lnt t 2 ut L 2 C T t In particular, the blow up criterion is in H 1 and not in Σ, and this result removes the possibility of strictly negative energy solutions to 1 blowing up with the rate of the known explicit solution St of critical mass given by 9: St L 2 1 T t. Note that in [4], blow up solutions to 1 which behave locally like St are constructed in the case of super critical mass, and one can check that these solutions have strictly positive energy, at least in the L 2 vicinity of Q. We claim in this paper the following sharp upper bound on the blow up rate: 5

6 Theorem 2 Sharp upper bound on the blow up rate Let N = 1 or N 2 assuming Spectral Property holds true. Then there exists α > 0 and a universal constant C > 0 such that the following is true. Let u 0 H 1 such that 0 < α 0 = u 0 2 Q 2 < α, E 0 < 1 2 Im u0 u u 0 L 2 Let ut be the corresponding solution to 1, then ut blows up in finite time 0 < T < + and there holds for t close to T : Comments on the result 1 ln lnt t ut L 2 C T t 1. Lower bound: By scaling properties, a known lower bound on the blow up rate is ut L 2 Indeed, consider for fixed t [0, T, v t τ, z = ut N 2 L 2 C T t. 16 u t + ut 1 2 L τ, ut 1 2 L z. 2 v t τ, z is a solution to 1 by scaling invariance. We have v t 0 L 2 + v t 0 L 2 C where C is independent of t, and so by the resolution of the Cauchy problem locally in time by fixed point argument in H 1 see [12], there exists τ 0 > 0 independent of t such that v t τ is defined on [0, τ 0 ]. Therefore, for all t, t + ut 1 2 L 2 τ 0 T, which is the desired result. From 15, the rate of blow up of strictly negative energy solutions to 1 is at most a double logarithm correction to the self similar one. 2. Optimality of the result: Numerical simulations in the case N = 2, see Landman, Papanicolaou, Sulem, Sulem [14], and heuristic arguments, see Sulem-Sulem [31], suggest the existence of solutions blowing up as 1 ln lnt t 2 ut L T t In this frame, for N = 1, Perelman in [29] proves the existence of one solution which blows up according to 17 and its stability in some space E H Comments on the result for N 2: In this situation, the proof of Theorem 2 is complete up to the proof of the Spectral Property which is purely related to the variational structure of Q, and is an open problem for the moment for N 2. On the other hand, our analysis is the first result in this setting as all previous results could not bypass the 6

7 lack of differentiability of the non linearity u u 4 N at u = Stability with respect to the initial data: From numerics, rate of blow up 17 should be stable with respect to perturbation of the initial data. Theorem 2 yields a full stability region in the energy space H 1 through condition 14. Note nevertheless that this result is purely perturbative and deals with solutions to 1 which remain in some sense in the vicinity of ground state solution Q. In addition, one can prove from Theorem 2 instability results for the explicit blow up solution St given by 9. Indeed, consider an initial data u 0ε = 1 + εq for ε > 0 and small, and u ε t the corresponding solution to 1. Then u 0ε satisfies the hypothesis of Theorem 2 and therefore blows up at some finite time T ε > 0, T ε + as ε 0, and 15 holds. We now apply the pseudo conformal transformation and set v ε t = 1 1 u t N ε 2 t, x x 2 i e 4t i, t then we obtain a sequence of solutions to 1 such that v ε 1 S 1 in Σ as ε 0, and v ε blows up at T ε = 1 T ε < 0. From T ε strictly negative and the boundedness of xu ε L 2 uniformly in time and ε, v ε satisfies estimate 15 for t close to T ε. This proves that any neighborhood in H 1 of St contains blow up solutions which blow up according to 15 and not to ut L 2 1 T t. A similar result also holds for the solutions constructed in [4]. 5. Structural instability: Note that blow up behavior 17 and therefore the result of Theorem 2 are known to be structurally unstable. Indeed, in dimension N = 2, if we consider the next term in the physical approximation leading to NLS, see [35], we get Zakharov equation { iut = u + nu 1 n c 2 tt = n + u for some large constant c 0. Then in [24], for all c 0 > 0, finite time blow up solutions to 18 are proved to satisfy ut L 2 C T t. 19 Note that this blow up rate is the one of St given by 9. Using a bifurcation argument from 9, one can construct blow up solutions to 18 with the rate of blow up 19, and these appear to be numerically stable, see [11] and [28]. 1.2 Strategy of the proof We briefly sketch in this subsection the proof of Theorem 2. 7

8 Let us first fix some notations. We let 2 the constant given by Spectral Property, 2 = 9 5 for N = 1, and set 1 = 2 2. As will be clear from further analysis, we shall not need the precise value of 2, only the fact that 2 < 2. Moreover, given a well-localized function f, we set Note that integration by part yields f 1 = N 2 f + y f and f 2 = N 2 f 1 + y f 1. f 1, g = f, g 1. As in [25], we reduce to the zero momentum and strictly negative energy case. Observe from Galilean invariance that for β R N, u β = ut, x βte i β 2 x βt is a solution to 1. Now fix β = 2 Im u 0 u 0 u0, then u 2 0 β x = u β 0, x satisfies from 14: 0 < α 0 = u 0 β 2 Q 2 < α, E 0 = Eu 0 β < 0 and Im u 0 β u 0 β = 0, 20 and the proof of Theorem 2 then reduces to proving 15 for u β. We therefore consider equation 1 for an initial data close to Q in L 2, with strictly negative energy and zero momentum. We now focus for the sake of simplicity onto dimension N = 1, see section 5.3 for the higher dimensional case. 1. Non linear decomposition of the solution: Ground state Q admits from [34] the following variational characterization: let v H 1 such that Ev = 0 and v L 2 = Q L 2, then v = λ Qλ 1 x + x 1 e iγ 1 for some fixed parameters λ 1 > 0, x 1, γ 1 R. Then from the assumption of closeness to Q in L 2 and the negative energy condition, this variational characterization allows us to write ux, t = eiγt Q + ε t, x xt λ 1 2 t λt for some functions λt > 0, γt R, xt R, εt a priori small in H 1, such that 21 1 λt u xt L Note that this yields a geometrical interpretation of the norm u x t L 2 which is to be estimated. Moreover, using modulation theory from scaling, phase and translation invariance, 8

9 we may slightly modify λt, γt, xt so that ε = ε 1 + iε 2 satisfies suitable orthogonality conditions, and we chose in [25]: ε 1, Q 2 + yq y = ε 1, yq = ε 2, 1 Q Q yq y + y 2 + yq y = y This strategy which was first applied in [25] is different from previous approaches more based on linear techniques. Consider vt = λt 1 2 ut, λtx + xte iγt which satisfies after a change of time scale ds dt = 1 λ 2 t iv s + v v + v v 4 = i λ s λ 1 2 v + yv y + γ s v + i x s λ v y, 24 and expand v = Q + ε according to 21. This yields the ε equation inherited from 1: i s ε + Lε = i λ s Q y λ 2 + yq + γ s Q + i x s λ Q y + Rε 25 with Rε quadratic in ε = ε 1 + iε 2. In a first approach, from smallness estimate on εt H 1, one may view 25 as a linear equation on the variable ε, and the situation is then as follows -see [33]-: the linear operator L = L +, L is a matrix operator which admits a generalized null space of dimension 6 leading to growing directions in H 1 to the linear equation underlying of these directions are induced by the symmetries of 1, and the last one is induced by an additional degeneracy. The fundamental remark is then as follows: if one considers 25 as a linear equation, one cannot avoid a priori from modulation theory only all growing solutions to the linear equation underlying 25, even when modulating on all symmetries, that is also Galilean invariance and pseudo conformal invariance. 2. Non linear dispersive estimate: We implemented in [25] a more non linear approach. First note that we do not use modulation theory with parameters related to the pseudoconformal transformation or to Galilean invariance, this last symmetry being used only to ensure 20. According to 22, our aim is to obtain a lower estimate on the parameter λt which appears in differential form λs in 25. A first rough estimate allows us to view this λ parameter as a first order term in ε, explicitly: λ s λ + ε 2, Q 1 C ε y 2 + ε 2 e 2 y, 26 in time averaging sense. Now the question is to obtain a dispersive estimate on the norm of ε involved in 26. The key to our analysis is to derive from viriel relation 6 on ut in Σ a viriel type dispersive relation on εt in L 2 loc which writes λ 2 E 0 + ε y 2 + ε 2 e 2 y Cε 2, Q

10 in time averaging sense. A more detailed heuristic explanation of this result is to be found in section 4. A first upper bound on blow up rate u x t L 2 C T t E 0 28 now follows in two steps in [25]. First remark that the same scalar product ε 2, Q 1 is involved in 26 and 27, and one may then prove after some work its positivity: t 0 [0, T such that t 0 t < T, On the other hand, from 27, one can prove a pointwise control Injecting this into 29 formally leads using ds dt = 1 λ 2 λ s λ ε 2t, Q 1 > ε 2, Q 1 C E 0 λ. 30 λ s λ = λ tλ ε 2, Q 1 C E 0 λ ie λ t C E 0 and λt C E 0 T t, and 28 follows from 22. Remark that estimate 26 is fairly general and cannot be improved. On the other hand, viriel estimate 27 is not sharp, and each improvement of this relation yields an improvement of the upper bound on blow up rate one is able to prove. For example, we exhibited in [25] on the basis of linear degeneracies of 1 around Q a refined viriel estimate which roughly writes ε y 2 + ε 2 e 2 y Cε 2, Q 1 4, and leads to a control u x t L 2 C lnt t T t. 4. Formal admissible blow up profiles: We point out two facts from above analysis. First, dynamical study of 1 reduces to a one dimension problem that is the study of geometrical parameter λs λ ε 2, Q 1. Second, the key to the proof of an upper bound on blow up rate is viriel estimate 27. Now observe that to get sharp upper bound 17 is formally equivalent to improving 27 for ε y 2 + ε 2 e 2 y e C ε 2,Q 1, what certainly corresponds to a non linear degeneracy of the structure of 1 around Q which cannot be captured from geometrical decomposition introduced in [25]. Our main point in this paper is to observe that a sharp geometrical decomposition close to a new profile allows us to both capture this non linear degeneracy and preserve monotony properties at the heart of the control of the blow up speed. to: 10

11 Recall 24. As we expect the perturbative study in the vicinity of Q of this equation to somehow reduce to a one dimensional problem on parameter λs λ, we formally introduce a parameter bt which should satisfy λs λ b, and thus the main part of 24 writes: 1 iv s + v v + ib 2 v + yv y + v v 4 0. We now look for formal admissible solutions to this equation in the vicinity of Q, that is parametrized profiles Q b with Q Qb H 1 0 as b 0, expecting to get a sharper decomposition of vt when expanding vt = Q bt + εt. Equivalently, we look for a function bs and a family of functions Q bs parametrized on bs solution to i db ds Qb 1 + Q b bs Q bs + ibs 2 Q bs + yq bs y + Q bs Q bs 4 = 0. It is more convenient to write P bs = e i bs 4 y 2 Q bs to get: i db ds P b db y 2 + P b bs P bs + ds + b2 s 4 P bs + P bs P bs 4 = Equation 31 admits at least two exact solutions and one formal solution under condition Q Q b H 1 small: A first exact solution is: s, bs, P bs = 0, Q, which corresponds to a linearization close to profile Q as in [25]. A second exact solution is: s, bs, P bs = 1 s, Q. This exceptional solution corresponds exactly to solution St to 1 given by 9 and is inherited from the pseudo-conformal symmetry. This solution is unstable. Note that linearization close to this profile has been studied in [4]. A last formal solution is given by: s, bs, P bs = b 0, P b0 where b 0 R is fixed and P b0 solves P b0 P b0 + b2 0 4 y 2 P b0 + P b0 P b0 4 = Solutions to this non linear elliptic PDE are closely related to the so called self similar solutions. Indeed, recall that λs λ b 0 writes from ds dt = 1, λ λ 2 t 2b 2 0 T t, and from direct computation, if P b0 solves 32, then 1 x U b0 t, x = P 2b 0 T t 1 b b 0 T t is a solution to 1 which is called self similar as it formally satisfies the scaling estimate U b0 t H 1 1 T t -recall indeed that lower bound 16 always holds-. 11

12 The situation regarding solutions to 32 is as follows. On the one hand, according to [30], solutions P b0 never belong to L 2 from a logarithmic divergence at infinity as P b0 y 1 as y +. On the other hand, according to expected blow up y behavior 17, we expect solutions to 32 to generate in some sense stable blow up profiles. 5. Sharp geometrical decomposition and proof of sharp upper bound 17: The strategy of the proof is then as follows. We first build regularized self similar profiles, ie for all b constant small enough, solutions P b to 32 on a ball y < 2 b ν 0, 0 < ν 0 << 1, to avoid linear growth of these solutions at infinity. We then set Q b = e i b 4 y 2 Pb which satisfies Q Q b H 1 0 as b 0, and introduce a new sharp geometrical decomposition εt, y = e iγt λ 1/2 tut, λty + xt Q bt y. From variational arguments and modulation theory on the four parameters λt, γt, xt, bt, ε satisfies orthogonality conditions 23 together with an extra condition which gives ε 2, Q 1 = 0 or equivalently bt ε 2, Q 1 at the first order in b. Therefore, one can hope that the new modulation parameter bt now holds for ε 2, Q 1 studied in [25] and governs the whole dynamic. Using non linear degeneracies of the structure of Q b for b close to 0, we then are able to improve dispersive estimate 27 for this decomposition: λ 2 E 0 + ε y 2 + ε 2 e 2 y e C bt 34 in time averaging sense. Moreover, the equation governing the scaling parameter now writes as expected: λ s λ + b C ε y 2 + ε 2 e 2 y 35 in time averaging sense. On the basis of these two facts, we then first derive from the strictly negative energy condition a maximum principle type of property on parameter bt: t 0 [0, T such that t 0 < t < T, bt > 0. This allows us to control oscillations in time in estimate 35 and obtain monotony of the norm near blow up time, or more explicitly: t 1 t 2 < T, λt 2 2λt 1. We then are able to improve uniform estimate 30 for bt C ln lnλt. 12

13 This then leads to the formal differential inequation and to the sharp upper bound λλ t C ln lnλt 1 ln lnt t u x t L 2 C 2. T t This paper is organized as follows. In section 2, we build the regular approximation in H 1 of self similar profiles Q b needed for our analysis and exhibit some degeneracy properties of Q b. In section 3, we build the regular ε decomposition adapted to dispersion with the suitable orthogonality conditions on ε. In section 4, we exhibit the local dispersive inequality in H 1 inherited from the Viriel structure of 1 in Σ. In section 5, we exhibit the sign property of bt and conclude the proof of Theorem 2. Up to section 2.1 and section 5.3, we shall always work with 1 in dimension N = 1. We then focus in section 5.3 onto the higher dimensional case. 2 Approximation in H 1 of self similar profiles Our aim in this section is to construct admissible profiles in H 1 and exhibit some non linear degeneracies of obtained solutions. In this section, we fix A 0 > 2 a large enough constant to be chosen later. Then we set for b R small R b = b A 0 and denote B Rb = {y R N, y R b }, B Rb = {y R N, y = R b }. We introduce a regular even cut-off function φ b x = φ x R b with φz = 0 for z M 0 2 and φz = 1 for z M 0, assuming M 0 > 0 is large enough so that φ b + φ b We also consider the norm on radial functions f C j = max 0 k j f k r L R Nonlinear approximation of the ground state In this subsection, we construct an approximation in H 1 of self-similar solutions to 1 in dimension N 1. Recall from 32 and 33 that these are related to the following elliptic nonlinear partial differential equation N Q b Q b + ib 2 Q b + y Q b + Q b Q b 4 N =

14 y 2 ib for some constant b > 0, or equivalently setting P b = Q b e 4, P b P b + b2 y 2 P b + P b P b 4 N = Global radial regular solutions to 38 never belong to H 1, as they admit from [30] the following asymptotic development as r + P b r = c 1 cos b r N 4 r2 + 1 b lnr + c 2 + Or β, 2 β = 5 2 for N = 1; β = 1 2N N 2 + 4N 4 for N 2, for some constants c 1, c 2. Nevertheless, this equation has a variational structure in H 1 0 B R b, what allows us to derive the following perturbative result: Proposition 1 Regular modified ground states There exist universal constants b > 0 and A 0 > 0 such that the following holds true: for b < b, there exists a unique regular radial solution P b H 1 to { P b P b + b2 y 2 4 P b + P 4 N +1 b = 0, 39 P b r > 0, r < R b ; P b R b = 0. Moreover, let then there holds P b r = P b rφ b r, 40 e A 0 r Pb Q C 3 0 as b 0, 41 and P b is differentiable with respect to b with estimate e A 0 r b P b C 2 0 as b We have the following Corollary as a direct consequence of Proposition 1: Corollary 1 Regular asymptotic profiles There exist universal constants b > 0 and A 0 > 0 such that the following holds true: for b < b b y 2 i, let Q b = P b e 4, then Q b is a radial solution to { Qb Q b + ib N 2 Q b + y Q b + Q b Q b 4 N = 0, 43 Q b R b = 0. Moreover, let Q b r = Q b rφ b r, then e A 0 r Qb Q C 3 0 as b 0, 44 and Q b is differentiable with respect to b with estimate e r 2A 0 b Q b + i y 2 4 Q C 2 0 as b

15 Remark 2 Note that profiles Q b given by Corollary 1 are one possible regularization in H 1 of self similar profiles solutions to 32 -recall again that solutions to 32 never belong to L 2 -. This choice of regularization is by no mean intrinsic for the further study of 1, and other regularizations could be considered. Remark also that the use of C 3 norms is related to the lack of differentiability of non linearity u u 4 N for N large. We shall use in the proof this C 3 regularity to estimate some interaction terms, see for example estimate of 139. Proof of Proposition 1 The proof is divided in several steps. We point out that all arguments are standard and mostly based on the fact that y B Rb, 1 b2 y A This indeed ensures that in the region y R b, operator + 1 b2 y 2 4 is uniformly elliptic with ellipticity constant independent of b, and thus satisfies both the maximum principle and standard regularity results for elliptic equations, see [9]. step 1 Existence of solutions to 39. To build a H 1 0 B R b radial positive solution to 39, we consider the coercive functional on radial functions of H 1 0 B R b F b w = 1 2 w w 2 b2 8 y 2 w 2 and the following minimization problem µ = inf F bw with M = {w Hr0B 1 Rb, w M w 4 N +2 = 1}, 47 where Hr0 1 B R b denotes the set of radial functions of H0 1B R b. Note that 4 N + 2 < 2N N 2 and w H0 1 B Rb, F b w c 0 w 2 + w 2 48 with c 0 independent of b. Let w n a minimizing sequence for 47, then one may assume w n 0, and there is w b non zero such that w n w b in H 1 r0 B R b and w n w b strongly in L 4 N +2 up to a subsequence from compact Sobolev embedding on the compact set B Rb. w b is a minimizer for 47, and from Lagrange multiplier theory, w b satisfies w b w b + b2 y 2 4 w b = ν b w 4 N +1 b in H 1 r0b Rb 15

16 for some multiplier ν b and 0 < 2F b w b = ν b wb 4 N +2 = ν b. From direct calculation, P b = ν N 4 b w b satisfies 39. Standard bootstrap arguments, see [9], ensure P b C 3 B Rb. We conclude moreover from the maximum principle that P b > 0 on the interior of B Rb and P b r R b < 0. Note that from the y dependence of the potential, one cannot directly conclude from [7] that P b is non increasing on B Rb. This is related to the fact that minimization problem 47 on all H0 1B R b and not only on radial functions would provide non radial minimizers accumulating on the boundary. step 2 Uniform estimates on P b. We now view 39 as an ordinary differential equation and use standard tools in this frame. Let P b the positive radial solution to 39 as constructed in the first step. We first claim that there exist universal constants C 1 > 0 and b 1 > 0 such that for all b b 1, one has: 1 C 1 P b L C 1, 49 and sup 0 b b1 P b L r R 0 as R i Proof of 49: To prove the lower bound, we multiply the P b equation by P b and integrate by parts: Pb 2 + P b 2 1 b2 y 2 4 Pb 4 = 0 so that 1 2 P b L for A 0 large enough from 46. The upper bound follows from variational characterization of P b. Indeed, recall from step 1 that P b = ν N 4 b w b where w b is a strictly positive mimimizer to 47, and 2F b w b = ν b. From its variational definition, F b w b is uniformly bounded as b 0, and so is ν b, so that from 48, w b H 1 is uniformly bounded as b 0, and thus P b H 1 C as b 0. This ensures P b L C 1 from standard bootstrap argument. ii Proof of 50: We have from previous argument: P b 2 + P b 2 C uniformly as b 0, so that by interpolation for radial functions for N 2, r N 1 2 P b r C. For N = 1, we equivalently assume by contradiction that there exists ε 0 > 0 small enough, r n + and b n 0 such that w bn r n = ε 0. We may always assume from w bn R bn = 0 that r r n, w bn r ε 0. Extracting a subsequence on w bn r + r n, we obtain a solution v H 1 to { d 2 v k dr 2 0 v + v 5 = 0, vr > 0 for r < 0, v0 = ε 0, vr ε 0 for r 0, where 1 1 A 0 k 0 1. Above system is integrable and vr = k Qk 0 r + r 0 for some r 0 R. We now claim that we are in the dichotomy situation of the concentrationcompactness method, see [15], [16], which is a contradiction for A 0 large enough using the fact that we are considering even functions. Indeed, in this situation, w bn r = h 1 n,mr + 16

17 h 2 n,mr + h 3 n,mr where using also uniform bound on w bn C 2, we have: 1 h 1 n,m 6 + h 2 n,m 6 + h 3 n,m 6 0 as m +, 51 F bn w bn F bn h 1 n,m + F bn h 2 n,m + F bn hn,m 3 0 as m +, 52 and h 1 n,mr n + r vr in H 1, h 2 n,m r n r v r in H 1 as n, m +, and h 3 n,m is even. Note for b small enough and A 0 large enough, we have F 0W F bn w bn 9 10 F 0W where W is the rescaled version of Q with W 6 = 1. There is now a sequence mn + such that letting h i n,mn 1 i 3 = h i n 1 i 3, we have: Moreover, h 1 n 6 v 6 = θ, h 2 n 6 v 6 = θ, lim inf F b n + n h 1 n = lim inf F b n + n h 2 n = 1 2 h 3 n 6 1 2θ. v y 2 + k 0 2 with k 0 k 0 1. From scaling argument on the limit functional, we have and F bn h 3 n 1 2θ 1 3 Fbn w bn lim inf F b n + n h 1 n = lim inf F b n + n h 2 n 9 10 θ 1 3 F0 W for A 0 large enough. Recalling 51 and 52, we get 1 2θ 1 3 v 2 F0 W + 1 2θ δn, δn 0 as n +, F bn w bn and letting n yields: which is a contradiction. 1 4θ + 1 2θ step 3 Uniform exponential decay and C 3 R convergence to Q. Therefore, we get 0 P b r C and up to a subsequence, P b converges as b 0 on every compact set to a positive non zero radial solution P to P P + P 1+ 4 N = 0 with P r 0 as r + from 50. From [13], this characterizes the ground state P = Q. We conclude that P b converges to Q on any compact set as b 0, and from standard boot 17

18 strap arguments, the convergence holds in C 3 loc. We now are in position to prove estimate 41 for b uniformly small enough. Let us first remark that there exists r 0 > 0 such that b small enough, P b r r < 0 for r > r 0. Indeed, remark that r N 1 P b r r is non decreasing for r r 0 see 50 and the P b equation and the fact that P b r R b < 0. Set now g b r = P b d dr P b, then one computes for r r 0 d dr g b g b = d A0 dr P b d2 dr 2 P b P b d A0 dr P b N 1 = + 2 dpb 2 r A 0 dr P b P 4 N A b b2 r for some r 1 > r 0 from 50. We then integrate this differential inequation from r 1 to r [r 1, R b ] to get g b r = P b + d dr P b C r1 e 1 2 r A 0. Using the Pb equation, we moreover conclude d2 P dr 2 b + d r P dr 3 b Ce A now follows from the convergence to Q on compact sets for A 0 large enough. We now turn to the proof of uniqueness and differentiability of P b. Uniqueness follows from the uniqueness of the ground state Q and differentiability from the spectral structure of the linearized operator close to Q. In order to handle the dependence of P b on the domain B Rb, we argue in two steps: given b 0 > 0 fixed, we first prove differentiability at b 0 of a rescaled version of P b which lives on B Rb0, and then the control of the differential of P b = φ b P b at b 0. step 4 Uniqueness and differentiability on a fixed ball. Let b 0 > 0 small enough and b close to b 0, P b given by step 1, and set then T b,b0 H 1 0 B R b0 satisfies b0 T b,b0 b T b,b0 = b0 b N 2 Pb b0 b x, 53 2 T + x 2 2 b0 b,b0 b2 0 T b,b0 4 b + T 1+ 4 N b,b 0 = 0 in B Rb0. Let R = T b,b0 P b0, then first remark from 41 that R L + R H 1 δb 0 with δb 0 0 as b 0 0 uniformly in b for b b 0 b 0 2. We claim the following uniform estimate for b 0 small and b in the vicinity of b 0 : R H 1 C b 0 b b

19 for some universal constant C > 0. Indeed, R is radial and satisfies the following partial differential equation in B Rb0 : where Remark that b0 G 1 R = 1 b L + R = G 1 R + G 2 R + F b, b 0, L + = Q 4 N, N 2 + b 2 x 2 2 b Q 4N 4 P N b 4 b N 0 R, P 4 N N G 2 R = R + P b N P 4 N b 0 2 b0 2 b0 F b, b 0 = 1 P b0 + 1 b b b0 R, b x 2 P b0. G 2 R C R 2 for N 3, G 2 R C R 1+ 4 N for N 4, 55 F b, b 0 L 2 C b 0 b b We now recall the spectral structure of L + as exhibited in [20] -note that the proof there holds in any dimension N 1-. Let µ + < 0 the lowest eigenvalue of L + and φ + a corresponding eigenvector with φ + L 2 = 1, then: Lemma 1 [20] There exists a universal constant δ + > 0 such that for all ω H 1 and radial, there holds L + ω, ω + 1 ω, φ + 2 δ + ω 2 H1. 57 δ + We first compute L + R, R = G 1 R, R + G 2 R, R + F b, b 0, R. We estimate: G 1 R, R = R 1 2 b0 b C A 0 R 2 H 1 + δb 0 R 2 H 1. From 55, 56 and the smallness of R in L, there holds 2 + b 2 x 2 2 b Q 4N 4 P N b 4 b N 0 G 2 R, R δb 0 R 2 H 1, F b, b 0, ω δ + 2 R 2 H 1 + C b 0 2 b b

20 Putting together the three above estimates, we get for b 0 small enough and A 0 large enough such that 1 A 0 δ + 4 the following: Now we estimate from the definition of φ + L + R, R δ + 4 R 2 H 1 + C b 0 2 b b µ + R, φ + = L + R, φ + G 1 R, φ + + G 2 R, φ + + F b, b 0, φ δb 0 R H 1 + C A0 b 0 b b Injecting 58 and 59 into 57 yields 54 for b 0 and 1 A 0 small enough. Note that this implies uniqueness and continuity of P b for b small enough. For the differentiability, we let R = equation: with and from 54: L + R Gb0 R = We have for F b 0 = F b,b 0 b b=b 0 : R b b 0. R H 1 r0 B Rb0 satisfies the following { G 1 R G b0 R } + G 2R + F b, b 0 b b 0 b b 0 G b0 R = b2 0 4 x 2 R Q 4N 4 P N b N 0, R H 1 C b 0. G 1 R G b0 R L 2 0 and F b, b 0 b b 0 F b 0 L 2 0 as b b 0. For N 4, we estimate G 2R 2 L b b 2 C 0 b b 0 2 C R 4 N L R 2+ 8 C N b b 0 2 R 4 N L R 2+ 4 N R 2+ 4 N b b 0 4 N Cb0 b b 0 4 N 0 as b b0. The case N 3 is similar. Therefore, we conclude that L + R Gb0 R F b 0 in L 2 as b b 0. Note that similar calculations as above yield that for b 0 small, L + R G b0 R = 0 implies R = 0. Therefore, L + G b0 1 exists and R L + G b0 1 F b 0 in Hr0 1 B R b0 as b b 0. This concludes the proof of the differentiability of T b,b0 with uniform estimate T b,b0 b b=b 0 H 1 20 C b 0. 60

21 step 5 Uniform exponential decay of the differential on a fixed ball. Tb,b0 Let h = b, then h H b=b r0 1 B R b0 satisfies L + h = G b0 h + F b 0. We have 0 for some β = βn > 0: F b, b 0 C 0,β and F b, b 0 r C + b b=b 0 b b=b 0 b 0 e 1 1 r 2A 0. Therefore, from standard bootstraps arguments, h C 2,β C b 0. We now claim hre A 0 r C 2 B Rb0 C b follows from the maximum principle. Indeed, the operator L = L + + G b0 h = cx with cx = 1 b2 0 y N P 4 N b0 satisfies the maximum principle in any region R 0 y R b0 for R 0 and A 0 large enough from 46 and uniform exponential decay 41. Now h satisfies: r R 0, Lhr C + b 0 e 1 1 r 2A 0. Let gr = e r 2A 0, then one computes for r R0 large enough Lg < C 2 e r 2A 0, and thus: C LKg h b 0 KC 2 e r 2A 0 < 0 for K 2C C 2 b 0. Moreover, Kg hr b = KgR b > 0, and from 60, there exists R 0 such that b 0 small enough, hr 0 C b 0, so that Kg hr 0 Ke R 2A 0 0 C b 0 > 0 for K = 2 C + 1 R b 0 e1 2A 0 0. Therefore there holds on Ω = {R 0 y R b0 }: LKg h < 0, Kg h Ω > 0. From the maximum principle, we conclude hr Kg C + 1 r b 0 e 1 2A 0 on Ω. We argue in the same way to prove h Ke r 2A now follows from standard bootstrap arguments. step 6 Differentiability of P b and uniform estimates on the differential. From the previous steps, Pb is differentiable, the question is to obtain a uniform estimate on the differential. Let b 0 > 0 small enough and b close to b 0, Pb = φ b P b given by step 1. First observe that P b = φ b P b = φ b φ b0 P b + φ b0 P b P b0 + φ b0 P b0. The first 21

22 term satisfies 42 from the continuity of P b from step 4. It thus suffices to prove estimate 42 for the function S = φ b0 P b P b0. Remark again from 41 that S L + S H 1 δb, b 0 with δb, b 0 0 as b, b 0 0, and we claim the following uniform estimate for b 0 small and b in the vicinity of b 0 : S H 1 C b 0 b b 0 62 for some universal constant C > 0. Indeed, S is radial and satisfies in R N : L + S = H 1 S + H 2 S + J 1 b, b 0 J 2 b, b 0, where H 1 S = b 2 4 x Q 4N φb0 P b0 4N S, N H 2 S = S + φ b0 P b N φb0 P b N N φ b0 P b0 4 N S, J 1 b, b 0 = b 2 b 2 0 x 2 4 φ b 0 P b0, J 2 b, b 0 = 2 φ b0 P b P b0 + φ b0 P b P b0 + φ 1+ 4 N b 0 φ b0 P 1+ 4 N b P 1+ 4 N b 0. Note that the structure of this equation is exactly the same like the one of R studied in step 4 and 5. We shall thus apply the same strategy. The only new term we need to control is the one induced by boundary condition on P b, ie J 2 b, b 0, what is now an easy task thanks to steps 4 and 5. Indeed, let T b,b0 given by 53, write P b x = N b 2 b 0 T b b,b0 b 0 x and inject this into the definition of J 2 b, b 0, then from step 4 and uniform estimate 54, we get J 2 b, b 0 L 2 C b 0 b b 0. Moreover, J 2 b, b 0 is differentiable still from step 4 with uniform estimate 61 from step 5. Now observe that each function φ b0, φ b0, φ 1+ 4 N b 0 φ b0 is compactly supported on a ball R b0 M 0 x R b0 M 0 2, so that 61 implies: + 1 r J2 b, b e1 2A 0 0 e C b 0. b r b=b 0 C 2 B Rb0 Once this term is controlled, we argue for S exactly as for R to get estimate 42. This ends the proof of Proposition 1. 22

23 Note that A 0 is now fixed for the rest of this paper such that Proposition 1 holds. Moreover, from now on, we focus for the sake of simplicity onto the one dimensional study N = 1 of 1. See section 5.3 for the higher dimensional case. 2.2 Exponential degeneracy of self similar profiles In this subsection, we exhibit algebraic properties of admissible self-similar profiles Q b constructed from Corollary 1 which will later allow us to capture non linear degeneracies of 1 around Q b. Two kinds of facts were are the heart of the proof in [25] concerning the ground state solution Q: Nonlinear invariants: EQ = 0 and Im Qy Q = 0. Algebraic structure of the linearized operator: The linear operator close to Q for 1 is a matrix operator which writes L = L +, L with L + = + 1 5Q 4 and L = + 1 Q 4, and from [33], algebraic relations hold from the set of symmetries of 1, explicitly: L + Q 1 = 2Q scaling invariance, L + Q y = 0 translation invariance, 63 L Q = 0 phase invariance, L yq = 2Q y Galilean invariance, 64 L y 2 Q = 4Q 1 pseudo conformal invariance. 65 Both facts survive in a certain sense for modified profile Q b from the equation satisfied by Q b and its invariants. We note Q b r = Σ b + iθ b in terms of real and imaginary part. We claim: Proposition 2 Non linear degeneracy of Q b There exists universal constants C > 0 and b > 0 such that for b b, there holds: i Equation of Q b : Qb satisfies Q b Q b + ib Q b 1 + Q b Q b 4 = Ψ b 66 with Ψ b = 2φ b y Q b y + Q b φ b + ibq b yφ b y + φ 5 b φ b Q b Q b 4, 67 23

24 and for any polynomial P y and integer 0 k 1, ii Degeneracy of the energy and the momentum: P yψ k b L e CP,k b. 68 2E Q b = Re Ψ b 1 Qb so that E Q b e C b, 69 Im Q b y Qb = iii Algebraic relations for the linearized operator close to Q b : there holds y Q b y Q b + iby Q b 1 + y Q b Q b 4 = 2 Q b y yψ b ; 71 y 2 Qb y 2 Qb + ib y 2 Q b 1 + y 2 Qb Q b 4 = 4 Q b 1 y 2 Ψ b ; 72 Q b 1 Q b 1 + Q b 1 Q b Q b Q b 2 Σ b Σ b 1 + Θ b Θ b 1 = 2 Q b ib Q b 1 Ψ b Ψ b 1 ib Q b 2 ; 73 Remark 3 Note that letting b = 0, 71, 72 and 73 reduce to 64, 65 and 63 which are related respectively to Galilean, conformal and scaling invariances. Proof of Proposition 2. i follows from direct calculation. Remark indeed that boundary term Ψ b given by 67 is localized on the support of φ b which is itself a neighborhood of fixed size around R b 1 b from 36, and 68 then follows from 44. ii follows from Pohozaev identity, ie multiply 66 by Qb 2 + y Q b y and take the real part to get: 2E Q b = Re Q b Q b + Q b Q b 4, Q b 1 = ReΨ b, Q b 1. This yields 69 from 68. To compute the momentum, write Q b = P y 2 ib b e 4 where P b = φ b P b is radial and real, so that { Im Q b y Qb = Im P b y i b } 2 y P b P b = 0. To prove iii, note first that 66 lies for L Q = 0. Note then that 71 and 72 are derived from 66 and yv yy = yv yy + 2v y, y 2 v yy = y 2 v yy + 4v 1. 24

25 To prove 73, we use scaling invariance as for the proof of L + Q 1 = 2Q. Indeed, write 66 as Q b Q b + Q b Q b 4 = F with F = ib Q b 1 Ψ b. For a given parameter λ > 0, denote w λ y = λ 1 2 Qb λy and F λ y = λ 1 2 F λy, then by rescaling we get w λ λ 2 w λ + w λ w λ 4 = λ 2 F λ. We differentiate this equation with respect to λ at λ = 1. Remark that Q b 1, so that Q b 1 Q b 1 + Q b 1 Q b 4 + Q d b dλ w λ 4 λ=1 = 2F + Q b + F 1. d dλ w λ = λ=1 Now d dλ w λ 4 = 4 Q b 2 Σ b Σ b 1 + Θ b Θ b 1, and 73 follows. This ends the proof of λ=1 Proposition 2. 3 Sharp decomposition of negative energy solutions In this section and the following, we consider ut solution to 1 with initial condition u 0 H 1 such that α 0 = αu 0 = u 0 2 Q 2 < α, E 0 = Eu 0 < 0, Im u 0 x u 0 = 0 for some 0 < α small enough to be chosen later. In this section, we construct a geometrical decomposition of the solution ut related to the fourth dimensional manifold of functions M = {e iγ λ 1 2 Qb λy + x}, which is generated by scaling, phase and translation parameters as in [25], and an extra parameter related to a degeneracy of the problem. This new decomposition will under a precise coordinate frame be able to capture some extra degeneracy in the dispersion at the heart of the proof of sharp upper bound Sharp geometrical decomposition of the solution In this subsection, we construct the sharp geometrical decomposition needed for the proof of Theorem 2 which relies on the variational characterization of the ground state Q and the closeness in H 1 of Q b to Q. Note that the choice of orthogonality conditions will be clear from the next section. The following Lemma is standard in the frame of the so called modulation method. 25

26 Lemma 2 Nonlinear modulation of the solution with respect to M There exists α 2 > 0 such that for α 0 < α 2, there exist some continuous functions λ, γ, x, b : [0, T 0, + R 3 such that t [0, T, εt, y = e iγt λ 1/2 tut, λty + xt Q bt y 74 satisfies the following: i ɛ 1 t, Σ bt 1 + ɛ 2 t, Θ bt 1 ɛ 2 t, yθ bt = 0, 75 ɛ 1 t, yσ bt + = 0, 76 ɛ 1 t, Θ bt 2 + ɛ 2 t, Σ bt 2 = 0, 77 ɛ 1 t, Θ bt 1 + ɛ 2 t, Σ bt 1 = 0, 78 where ε = ɛ 1 + iɛ 2 in terms of real and imaginary part; ii 1 λt u xt L 2 + εt Q x H 1 + bt δα 0, where δα 0 0 as α L 2 Remark 4 Note that letting formally b = 0, orthogonality conditions 75, 76, 77 reduce to 23 introduced in [25]. Now comparing decomposition 74 with 21 introduced in [25], 78 implies at the first order bt ε 2, Q 1 as announced. Proof of Lemma 2 The proof is similar to the one of Lemma 2 in [25]. Let us briefly recall it. Let us start with a classical result of proximity of the solution up to scaling, phase and translation factors to the function Q. From [25], there holds: for α 0 < α 1 and E 0 < 0, t [0, T, there exists γ 0 t R and x 0 t R such that, with λ 0 t = Qx L 2 u xt L 2, Q e iγ0t λ 0 t 1/2 u λ 0 tx + x 0 t < δα H 1 0 where δα 0 0 as α 0 0. Now we sharpen the decomposition using the fact that Q b Q in H 1 as b 0, i.e. we chose λt, γt, xt, bt close to λ 0 t, γ 0 t, x 0 t, 0 such that εt, y = e iγt λ 1/2 tut, λty + xt Q bt y is small in H 1 and satisfies suitable orthogonality conditions 75, 76, 77 and 78. The existence of such a decomposition is a consequence of the implicit function Theorem. 26

27 For δ > 0, let V δ = {v H 1 C; v Q H 1 δ}, and for v H 1 C, λ 1 > 0, γ 1 R, x 1 R, b R small, define ε λ1,γ 1,x 1,by = e iγ 1 λ 1/2 1 vλ 1 y + x 1 Q b. 80 We claim that there exists δ > 0 and a unique C 1 map : V δ 1 λ, 1 + λ γ, γ x, x b, b such that if v V δ, there is a unique λ 1, γ 1, x 1, b such that ε λ1,γ 1,x 1,b = ε λ1,γ 1,x 1,b 1 + iε λ1,γ 1,x 1,b 2 defined as in 80 satisfies ρ 1 v = ε λ1,γ 1,x 1,b 1, Σ b 1 + ε λ1,γ 1,x 1,b 2, Θ b 1 = 0, ρ 2 v = ε λ1,γ 1,x 1,b 1, yσ b + ε λ1,γ 1,x 1,b 2, yθ b = 0, ρ 3 v = ε λ1,γ 1,x 1,b 1, Θ b 2 + ε λ1,γ 1,x 1,b 2, Σ b 2 = 0, ρ 4 v = ε λ1,γ 1,x 1,b 1, Θ b 1 ε λ1,γ 1,x 1,b 2, Σ b 1 = 0. Moreover, there exists a constant C 1 > 0 such that if v V δ, then ε λ1,γ 1,x 1 H 1 + λ γ 1 + x 1 + b C 1 δ. Indeed, we view the above functionals ρ 1, ρ 2, ρ 3, ρ 4 as functions of λ 1, γ 1, x 1, b, v. We first compute at λ 1, γ 1, x 1, b, v = 1, 0, 0, 0, v: ε λ1,γ 1,x 1,b x 1 = v x, ε λ1,γ 1,x 1,b λ 1 = v 2 + xv x, ε λ1,γ 1,x 1,b γ 1 = iv, ε λ1,γ 1,x 1,b b = Qb. b b=0 Now recall from 45 that Q b b=0 = Q and at the point λ 1, γ 1, x 1, b, v = 1, 0, 0, 0, Q, Qb b b=0 y 2 = i 4 Q. Therefore, we obtain ρ 1 = Q 1 2 ρ 1 ρ 1 λ 2, = 0, = 0, ρ1 1 γ 1 x 1 b = 0, ρ 2 ρ 2 ρ 2 = 0, = 0, = 1 λ 1 γ 1 x 1 2 Q 2 2, ρ2 b = 0, ρ 3 ρ 3 = 0, = Q 1 2 ρ 3 λ 1 γ 2, = 0, ρ3 1 x 1 b = 0, ρ 4 ρ 4 ρ 4 = 0, = 0, = 0, ρ4 λ 1 γ 1 x 1 b = 1 4 yq 2 2. The Jacobian of the above functional is non zero, thus the implicit function Theorem applies, and conclusion follows as in [25]. This concludes the proof of Lemma 2. From now on, to simplify notations, we omit the b dependance of Q b and Ψ b given by 67, that is we write Q bt = Σ + iθ and Ψ bt = ReΨ + iimψ. 27

28 3.2 Properties of the decomposition In this subsection, we exhibit properties of the previous decomposition and estimates on the modulation parameters λt, γt, xt and bt. These estimates rely on the equation verified by εt, which is inherited from 1, on smallness estimate 79 and on the conservation of the three H 1 invariants of 1: L 2 norm, energy and momentum. We first introduce a new time scale s = t 0 dt λ 2 t, or equivalently ds dt = 1 λ 2. ɛ, λ, γ, x and b are now functions of s. Let T 1, T 2 0, + 2 respectively the finite negative and positive blow up times of ut from Theorem 1 in [25]. Let us check that when t T 1, T 2, {st} =, +. Indeed, scaling estimate 16 implies λt C T i t 1 2, i = {1, 2}, ie 1 λ 2 t We now claim: C T i t. Lemma 3 Properties of the decomposition There exists universal constants C > 0, α 3 > 0 and a function δα 0 with δα 0 0 as α 0 0 such that for α 0 < α 3, {λs, γs, xs, bs} are C 1 functions of s on R, and we have the following properties: i Equation of εs : εs satisfies for s R, y R, Σ 1 b s b + sɛ 1 M ε + b 2 ε 1 + yε 1 y λs = λ + b Σ 1 + γ s Θ + x s λ Σ y 81 λs 1 + λ + b 2 ε 1 + yε 1 y + γ s ε 2 + x s λ ε 1 y + ImΨ R 2 ɛ, Θ 1 b s b + sɛ 2 + M + ε + b 2 ε 2 + yε 2 y = + λs λ + b Θ 1 γ s Σ + x s λ Θ y 82 λs 1 λ + b 2 ε 2 + yε 2 y γ s ε 1 + x s λ ε 2 y ReΨ + R 1 ɛ, where γs = s γs. M = M +, M is the linear operator close to Q b : M + ε = ε 1 + ε 1 Q b 4 + 4Σ 2 Q b 2 ε 1 4ΣΘ Q b 2 ε 2, 83 M ε = ε 2 + ε 2 Q b 4 + 4Θ 2 Q b 2 ε 2 4ΣΘ Q b 2 ε 1, 84 28

29 and the quadratic functionals R 1 and R 2 are given by R 1 ɛ = ε 1 + Σ ε + Q b 4 Q b 4 + 4Σ 2 Q b 2 ε 1 4ΣΘ Q b 2 ε 2, 85 R 2 ɛ = ε 2 + Θ ε + Q b 4 Q b 4 + 4Θ 2 Q b 2 ε 2 4ΣΘ Q b 2 ε ii Estimates induced by conservation laws: s R, there holds 2λ 2 s E 0 2ε 1, Σ + bθ 1 2ε 2, Θ bσ 1 C ε y 2 + ε 2 e 2 y + e C b, 87 ε 2, Σ y s Cδα 0 ε y 2 + ε 2 e 2 y iii Estimates on the modulation parameters: Moreover, λ s λ + b + γ s + b s C ε y 2 + ε 2 e 2 y e C b, 89 x s λ δα 0 ε y 2 + ε 2 e 2 y e C b. 90 Remark 5 Part i of above proposition is purely algebraic and follows from direct computations. Estimates 87 and 88 correspond to degeneracies of order one in ε scalar products induced by conservation laws, respectively conservation of the energy and conservation of the momentum. Estimate 89 simply means that all modulation parameters may be estimated as order one scalar products in ε, what is a consequence of modulation theory, that is the possibility to fix orthogonality conditions on ε. On the other hand, estimate 90 means that the effect of translation invariance is of smaller order than the one of scaling and phase invariance. This is a consequence of our choice of orthogonality condition 76 and of our use of Galilean invariance to ensure 20. Proof of Lemma 3 See Appendix A. Note that the proof of Lemma 3 is similar to the one of Lemma 5 in [25], and based on the following technical Lemma which allows us to control non linear interaction terms. Let Rε = R 1 ε + ir 2 ε given by 85, 86, F ε given by F ε = ε + Q b 6 Q b 6 6Σ Q b 4 ε 1 6Θ Q b 4 ε { Q b 4 + 4Σ 2 Q b 2 ε Q b 4 + 4Θ 2 Q b 2 ε ΣΘ Q } b 2 ε 1 ε 2, R 1 ε and R 2 ε the formally cubic parts of R 1 and R 2, Rε = R 1 ε + i R 2 ε, R 1 ε = R 1 ε 10Σ 3 + 6ΣΘ 2 ε 2 1 2Σ 3 + 6ΣΘ 2 ε 2 2 4Θ Σ 2 Θε 1 ε 2, 92 R 2 ε = R 2 ε 10Θ 3 + 6ΘΣ 2 ε 2 2 2Θ 3 + 6ΘΣ 2 ε 2 1 4Σ Θ 2 Σε 1 ε 2, 93 and Ψ the boundary term 67, we claim: 29

30 Lemma 4 Control of non linear interactions Let P y a polynomial and integers 0 k 3, 0 l 1, 0 m 2, then for some function δα 0 0 as α 0 0, there holds: ε, P y dk dy Q k b y C P,k ε 2 e 2 y 1 2, ε, P y dk dy Q k b y ε, P y dk dy k Qy δα 0 ε 2 e 2 y 1 2, 94 dm ε P y Q 1 b dy m b ε y 2 + ε 2 e 2 2 y, 95 Rε, P y dk dy Q k b y C ε y 2 + ε 2 e 2 y, F ε + R 1 ε, Σ 1 + R 2 ε, Θ 1 δα 0 ε y 2 + ε 2 e 2 y, dk dy Q k b, P y dl dy l Ψ + dl ε, P y dy l Ψ C e b, Qb dk, P y b dy Q k b y + i y 2 dk Q, P y 4 dy k Qy δα 0. Proof of Lemma 4 Such estimates have been proved to hold in [25] with Q instead of Q b, except for 94 and 95 which are straightforward from respectively 44 and 45. The proof is then completely similar, as we first compute explicitly each term, then replace Q b by Q and remark that 44 may for example be rewritten: y R, 0 k 3, d k dy k Q b Qy 1 δbq1 4A 0 y with δb 0 as b 0. This ends the proof of Lemma 4. 4 The local dispersive estimate in the ε variable Our aim in this section is to exhibit the dispersive structure underlying 1 in the vicinity of the ground state Q. Such a structure has been partially exhibited in [25]. Recall that on the basis of the decomposition introduced there with three parameters only, that is 21, we exhibited a local dispersive relation which roughly wrote ε 2, Q 1 s Hε, ε + λ 2 E 0 30