THE INSTABILITY OF BOURGAIN-WANG SOLUTIONS FOR THE L 2 CRITICAL NLS FRANK MERLE, PIERRE RAPHAËL, AND JEREMIE SZEFTEL

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1 THE INSTABILITY OF BOURGAIN-WANG SOLUTIONS FOR THE L CRITICAL NLS FRANK MERLE, PIERRE RAPHAËL, AND JEREMIE SZEFTEL Abstract. We consider the two dimensional L critical nonlinear Schrödinger equation i tu + u + u u = 0. In the pioneering work [], Bourgain and Wang have constructed smooth solutions which blow up in finite time T < + with the pseudo conformal speed u(t L 1 T t, and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical L mass lies on the boundary of both H 1 open sets of global solutions that scatter forward and backwards in time, and solutions that blow up in finite time on the right in the log-log regime ehibited in [8], [31], [41]. We moreover ehibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first ehibited in [6] in the critical mass case. 1. Introduction 1.1. Setting of the problem. We consider in this paper the cubic two dimensional focusing nonlinear Schrödinger equation: (NLS { iut = u u u, (t, R R, u(t 0, = u 0 (, u 0 : R C, t 0 R. (1.1 This is the special case of physical relevance of the N dimensional L critical (NLS: iu t = u u 4 N u, (t, R R N. (1. From a result of Ginibre and Velo [11], (1.1 is locally well-posed in H 1 = H 1 (R and thus, for u 0 H 1, there eists t 0 < T + and a unique solution u(t C([t 0, T, H 1 to (1.1 and either T = +, we say the solution is global, or T < + and then lim t T u(t L = +, we say the solution blows up in finite time. The Cauchy problem can also be solved in the critical L space, [4], in which case from standard Strichartz bound [46], [3], finite time blow up is equivalent to u L 4 ([0,T,L 4 = +. Note also that the equation is time reversible and the flow can similarly be solved backwards in time. (1.1 admits the following conservation laws in the energy space H 1 : L norm : u(t L = u 0 L ; Energy : E(u(t, = 1 u(t, d 1 4 u(t, 4 d = E(u 0 ; Momentum : Im( u(t, u(t, d = Im( u 0 (u 0 (d. 1

2 F. MERLE, P. RAPHAËL, AND J. SZEFTEL A large group of H 1 symmetries leaves the equation invariant: if u(t, solves (1.1, then (λ 0, τ 0, 0, β 0, γ 0 R + R R R R, so does v(t, = λ 0 u(λ 0t + τ 0, λ β 0 te i β 0 ( β 0 t e iγ 0. (1.3 The scaling symmetry v(t, = λ 0 u(λ 0 t, λ 0 leaves the L space invariant and hence the problem is L critical. The additional pseudo conformal symmetry v(τ, = Cu(τ, = 1 ( τ u 1 τ, e i 4τ, (1.4 τ is a continuous transformation for τ 0 in the virial space Σ = {u L } H 1. Following [10], [1], let now Q be the unique H 1 nonzero positive radial solution to Q Q + Q 3 = 0, (1.5 then the variational characterization of Q ensures that initial data u 0 H 1 with u 0 L < Q L yield global and bounded solutions, [48]. Moreover, for u 0 Σ or u 0 H 1 and radial, the solution scatters forward and backward in time, or equivalently from standard Strichartz estimates: u L 4 (R,L 4 < +, (1.6 see [13] and [5], [6] for the defocusing case and references therein. At the critical mass threshold u 0 L = Q L, two new dynamics occur: the solitary wave non dispersive dynamics u(t, = Q(e it, and the critical mass blow up solution generated by the pseudo conformal symmetry S(t, = 1 ( y it (Qe 4 e i t, t R (1.7 t t which blows up with the pseudo conformal speed u(t L 1 t as t 0. From [7], S(t is the unique up to the symmetries critical mass blow up solution. Through the pseudo conformal symmetry, this also classifies the solitary wave as the unique non dispersive critical mass solution in Σ up to the symmetries, see [14] [] for an etension to lower regularity. 1.. Three known dynamics. We focus from now on and for the rest of this paper onto H 1 solutions with mass slightly above the critical one: Q L < u L < Q L + α (1.8 for some small enough universal constant α > 0. A general open problem is to classify the possible dynamics of the flow near the solitary wave Q. Three kind of regimes have been ehibited so far: (i Scattering solutions: Solutions that scatter forward and backward in time in the sense of (1.6 have been ehibited with arbitrary mass, see for eample [39], and from standard Strichartz estimates [46], see also [3], the corresponding set of initial data is open in H 1.

3 (ii Soliton like solutions and pseudo conformal blow up: Solutions that scatter forward to Q with super critical mass u(t Q e it u 0 in H 1 as t + are constructed in the pioneering work by Bourgain and Wang [] in dimensions N = 1,, and in the further etension by Krieger and Schlag [17] in dimension N = 1. Through the pseudo conformal transformation (1.4, they equivalently correspond to super critical mass finite time blow up solutions with eact S(t blow up profile. Theorem 1.1 (Bourgain, Wang []. Let A 0 be a given integer. Let A A 0 a large enough integer. Let z X A = {f H A with (1 + A f L }, and let z C((T, 0], X A be the solution to { i t z + z + z z = 0, z t=0 = z, 3 (1.9 where T < 0 is the maimal time of eistence of z. Assume that z vanishes to high order at the origin: D α z (0 = 0 for α A 1. (1.10 Then for all θ R, there eists t 0 < 0 and a unique solution to u θ BW C([t 0, 0, X A0 to (1.1 with u θ BW (t S(te iθ z(t XA0 t A 0. (1.11 For t < 0, we will denote u θ BW (t the Bourgain Wang solution which blows up at time T = 0 with regular part z and singular part S(te iθ given by Theorem 1.1. Note that S(t scatters to the left as t, and a further simple argument 1 ensures that under the additional smallness assumption z H A < α 1, (1.1 u θ BW scatters to the left in time: u θ BW C((, 0, H A 0, u θ BW (t, L 4 ((, 1],L 4 <. (1.13 Moreover, it blows up at t = 0 with blow up speed given by the pseudo-conformal speed: u θ BW (t L 1 t. (1.14 From the time reversibility of the equation, we let for t > 0 u θ BW (t be the solution which blows up backwards at t = 0 with S(te iθ singular part and z regular part, and u θ BW is global and scatters at + and blows up at the origin with speed (1.14. In general, solutions blowing up with the pseudo conformal blow up speed (1.14 have been conjectured to be unstable and to live on a codimension one manifold, see Krieger, Schlag [17] for further results in this direction. (iii Log-log blow up solutions: Eventually, after the pioneering work by Perelman [38] in dimension N = 1, the eistence of an H 1 open set of initial data leading to finite time blow up solutions in the so called log-log regime is proved in the 1 Indeed, on can show that u θ BW satisfies (4.5, and then Lemma. yields the result.

4 4 F. MERLE, P. RAPHAËL, AND J. SZEFTEL series of works by Merle and Raphaël [8], [9], [41], [30], [31], [3]. These solutions concentrate a universal bubble of mass at blow up u(t, 1 ( (t λ(t Q e iγ(t u in L as t T = T (u < + λ(t for some parameters (t (T R, γ(t + and a blow up speed u(t L = Q L, λ(t = λ(t π(t t (1 + o(1 as t T. (1.15 log log(t t Note that if we normalize the blow up time at T = 0 and consider the pseudo conformal transformation (1.4, then Cu(τ, is global on the right in time and behaves like a spreading bubble with anomalously slow decay: Cu(τ, 4 L log logτ as τ +. ( τ Note that this dynamic is by construction unstable by loglog blow up. Finally, let us observe that the critical mass blow up solution S(t lies on the boundary of both H 1 open sets of global solutions that scatter forward and backwards in time and solutions that blow up in finite time on the right in the log-log regime due to the following eplicit deformations: u η ( 1 = (1 ηs( 1, η > 0, has subcritical mass and thus the corresponding solution is global and scatters on both sides in time, [13]; let u η ( 1 = (1 ηs( 1, η < 0 small, and v η = Cu its pseudo conformal transformation given by (1.4, then v η (1 = (1 ηq has small super critical mass and E(v η < 0 from direct check, hence it blows up in finite time 1 < τ η < + in the log log regime, [8], [9], [30], and thus from (1.4, u η also blows up in finite time T η < 0 in the log log regime Statement of the result. Our aim in this paper is to make some progress towards the understanding of the flow near Q and the proof of the conjectured instability of the pseudo conformal type of blow up (1.14. We prove this instability for the special class of Bourgain Wang solutions. The particularity of these solutions is the decoupling in space between the singular S(t blow up bubble and the residual smooth z part induced by the high degeneracy at blow up point (1.10. More precisely, we claim that Bourgain Wang solutions like the critical mass solution S(t lie on the border of both the H 1 open set of forward and backward scattering solutions, and the H 1 open set of solutions which blow up in finite time in the log-log regime (1.15. The following theorem is the main result of this paper: Theorem 1. (Strong instability of Bourgain-Wang solutions. Let α > 0 be a small enough universal constant and A be a large enough integer. Let z X A radially symmetric satisfying the smallness assumption (1.1 and the degeneracy at blow up point (1.10. Let u 0 BW C((, 0, Σ be the corresponding Bourgain- Wang solution given by Theorem 1.1 with θ = 0. Then there eists a continuous map Γ : [ 1, 1] Σ such that the following holds true. Given η [ 1, 1], let u η (t be the solution to (1.1 with data u η ( 1 = Γ(η, then: Γ(0 = u 0 BW ( 1 ie t < 0, u η=0(t = u 0 BW (t is the Bourgain Wang solution on (, 0 with blow up profile S(t and regular part z ;

5 η (0, 1], u η C(R, Σ is global in time and scatters forward and backwards in the sense of (1.6; η [ 1, 0, u η C((, T η, Σ scatters to the left and blows up in finite time T η < 0 on the right in the log-log regime (1.15 with T η 0 as η 0. (1.17 Theorem 1. will follow trough a complete description of the curve Γ in terms of eplicit modified profiles Q η (t which are deformations of S(t across the pseudo conformal regime, see (.19 for precise definitions. In the log log regime η < 0, the dynamics splits into three parts: the scattering dynamics for t < 1, the pseudo conformal blow up where the solution remains near u BW for t < (1 + δt η, 0 < δ 1, and eventually the log-log blow up for t T η. The control of the solution in each regime requires a specific analysis: Strichartz control at critical L regularity for t < 1, a first monotonicity formula valid in the pseudo conformal regime only, and then the log log machinery based on another monotonicity formula from [8], [9], [30], [31], [41]. In the global scattering regime η > 0, an important outcome of the proof is that we may follow the flow of u η (t for η > 0 past the blow up time t = 0 of the limiting Bourgain Wang solution as η 0. Indeed, a straightforward continuity argument ensures a strong limit before blow up time: t < 0, u η (t u 0 BW (t in Σ as η 0. The important problem is to understand what happens for t > 0, and we recover the chaotic phase behavior first ehibited by Merle [6] in the critical mass case: Theorem 1.3 (Continuation of u η after blow up time. Let 0 < η 1 and u η C(R, Σ be the global scattering solution built in Theorem 1.. Then: (i Identification of the limit points: Let t > 0, then the limit points of the sequence (u η (t η>0 as η 0 are given by u θ BW (t, θ R. (ii Eistence of converging subsequences: θ R, there eists a sequence η n 0 such that t > 0, u ηn (t u θ BW (t in Σ as n +. (1.18 In other words, the regularization after blow up time of the pseudo conformal Bourgain Wang blow up corresponds to a defocalization onto another Bourgain Wang profile with shifted blow up profile S(te iθ and same regular part z, and where the phase shift displays a chaotic behavior as η 0 for t > 0. Eventually, the pseudo conformal symmetry (1.4 yields the following corollary of instability of the manifold of forward scattering to Q. Corollary 1.4 (Instability of the manifold of forward scattering to Q. Let z, u η as in the hypothesis of Theorem 1.. Then there eist continuous maps Γ i : [ 1, 1] Σ, i = 1, such that the solution to (1.1 with data v i η(1 = Γ i (η satisfies the following: v i η=0 C(R+, Σ and scatters to Q to the right: see Lemma 3.5 v i 0(τ Q Cz(τ 0 in L as τ +. 5

6 6 F. MERLE, P. RAPHAËL, AND J. SZEFTEL η (0, 1], vη i C(R, Σ scatters forward as τ + in the sense of (1.6. η [ 1, 0, vη 1 C((0, Tη, Σ blows up in finite time 0 < Tη < + on the right in the log-log regime with Tη + as η 0. η [ 1, 0, vη C((0, +, Σ and spreads to the right with the anomalous decay rate (1.16. Comments on the result 1. Comparison with previous works: The instability result of Theorem 1. and Corollary 1.4 should be compared with the interesting classification results recently obtained by Nakanishi and Schlag [35], [36]. These works lies in the continuation of former and more recent works which address the question of the eistence and uniqueness of critical dynamics [6], [7], [8], [9], [7], [43] and the understanding of the center manifold near the ground state in super critical regimes [45], [17], [15], [16], [1]. In [36], the dynamics of the flow near the ground state solitary wave for the L super critical cubic (NLS in dimension 3 are completely classified through a series of 9 different scenario which in particular identify the stable manifold of forward scattering to Q as a threshold dynamic between the stable dynamics of finite time blow up and scattering. Note in passing that the scenario of anomalous scattering (1.16 does not occur in [36]. In the work [36] however, finite time blow up is obtained through a virial argument without any qualitative information on the attained blow up regime. Moreover, the analysis by Krieger and Schlag [17] suggests that despite some recent important progress on the construction and the understanding of the scattering manifold to Q in super critical settings, it seems substantially more delicate to implement this kind of analysis to the L critical case under consideration which is degenerate and critical with respect to a conservation law.. On the eistence and instability of threshold dynamics: The construction of the deformation Γ in Theorem 1. will not rely on a fied point like argument as in [], [45], [15], [16], [18], [19], [0] but rather on softer compactness arguments as in [5], [3], [17], [43], [1]. Moreover, the threshold Bourgain Wang regime will be achieved as the limit from above and below of a family of solutions which each correspond to a stable dynamics, respectively scattering and log log blow up, and does not need to be constructed a priori. A similar strategy which gives together the eistence and the instability of the threshold dynamics was used by Matano and Merle in their study of the super critical heat equation, see [4] and references therein. The etension of this strategy to the full family of pseudo conformal blow up solutions build in [17] is open. 3. Etensions: We have decided to work for simplicity in dimension N = and for radial data, but similar issues could be addressed for the N dimensional L critical (NLS (1.. The etension to dimension N = 1 and for non radial Bourgain Wang solution in dimension N = 1, is straightforward 3. For higher dimensions, the roughness of the nonlinearity becomes a serious trouble. We will however along the proof of Theorem revisit the Bourgain Wang machinery and our new strategy has two main outcomes. The first one is that we will completely avoid the use of the pseudo conformal symmetry as an eplicit symmetry and use instead 3 The non radial case would follow by introducing modulation on the translation parameter and Galilean deformation of the blow up profile which are lower order deformations, see [33], [40], [43] for the full treatment of these issues in a similar setting.

7 a number of tools developed by Raphaël and Szeftel [43] for the study of minimal mass conformal blow up for an inhomogeneous (NLS equation which yield a direct dynamical investigation of the pseudo conformal blow up without any need for a symmetry. In this sense, we epect the results of Theorem 1. and Theorem 1.3 to etend to a large class of nonlinear critical dispersive problems. The second outcome is that we will seriously diminish the regularity needed for the Bourgain Wang analysis, see in particular Remark 4., which makes a partial etension of the above results in particular to dimension N = 3 more reasonable to attain. This would require a separate analysis which remains to be done. Eventually, the smallness assumption (1.1 on z is used only to ensure global scattering behavior away from blow up. Without smallness, similar results can be proved on the time interval ( T1, T of eistence of the solution to (1.9 with very similar proofs and using in particular the large L mass loglog analysis in [33] Strategy of the proof. Let us give a brief insight into the strategy of the proof of Theorem 1. which follows from two main steps. step 1 Approimate profiles. The first step is to construct approimate profiles Q η (t which are transversal to the pseudo conformal manifold. The instability direction is remarkably enough completely eplicit and corresponds to profiles: Q η (t, = 1 λ η (t [P η (ye ibη(t y 4 for the eplicit choice of modulation parameters: λ η (t = η + t, b η (t = t, γ η (t = 1 + ] ( e iγη(t, λ η (t t 1 7 dτ η + τ, (1.19 and where P η is a suitable small deformation of Q. The profiles P η yield eact well localized scattering solutions for η < 0, and P η=0 = Q so that Q η=0 (t = S(t. For η > 0, the self similar profile P η is unbounded in L and the corresponding solution will blow up in finite time T η < 0 in the log-log regime instead of the self similar regime λ(t Tη t with T η = η formally predicted by (1.19. step Backwards bootstrap of the flow. We now pick an asymptotic smooth profile z satisfying the degeneracy (1.10. We also assume the smallness condition (1.8 which ensures that the solution z to (1.9 is global in time. Our analysis is based on a backwards integration of the flow in the continuation of the works [5], [3], [43]. (i For η > 0, we let an initial data at time t = 0 be u η (t = 0 = Q η (0 + z. We then make a decomposition of the flow u η = Q η + z(t + ũ η (t (1.0

8 8 F. MERLE, P. RAPHAËL, AND J. SZEFTEL and claim a uniform backwards control of ũ η all the way down to some time t = t 1, where t 1 < 0 is independent of η: ũ η (t L ([t 1,0],H 1 1. (1.1 The same argument also yields a similar control on [0, t 1 ], and now from a standard continuity argument, the solution u η is global and scatters at both infinities because so do S(t and z(t. This yields both the scattering part of Theorem 1. and the continuation result of Theorem 1.3 where the chaotic phase is a direct consequence of the eplicit formula for the phase (1.19. The key is therefore the backwards control (1.1. For this, we observe that the decomposition of (1.0 together with the strong decoupling (1.10 induces an almost critical mass dynamics for ũ η. We are thus able to use the Lyapounov machinery ehibited in [43] in the pseudoconformal regime to control the flow backwards in time. (ii For η > 0, an eplicit computation shows that the solution to (1.1 with data u η (T 1 η = Q η (T 1 η + z at T 1 η η blows up in the log log regime forward in time. Here we use the sharp open characterization of the log log set ehibited in [41] which allows one to see directly loglog blow up on the initial data. We now need to run backwards the control of this solution. The key observation is that at the time T 1 η, the solution is still very much in the pseudoconformal regime and the backwards reintegration of the flow is completely similar to the one of (1.1. (iii We now pass to the limit as η 0 and obtain in both cases a Bourgain Wang type solution with regular part z as t 0. It remains to show some uniqueness statement about this type of dynamics to ensure the continuity of the map Γ : η u η ( 1 at η = 0. This is a simple but slightly technical claim which proof relies again on the Lyapounov machinery ehibited in [43], and this is the part which requires a high order degeneracy in (1.10. This paper is organized as follows. In section, we build the deformation Q η of the S(t blow up profile. In section 3, we build the machinery of backwards control of the pseudo conformal flow. The proof of the main theorems is then concluded in section 4. Acknowledgments. F.M. is supported by ANR Projet Blanc OndeNonLin. P.R and J.S are supported by ANR jeunes chercheurs SWAP. Notations: We introduce the differential operator Λf = f + y f (L scaling. Let L = (L +, L the matri linearized operator close to the ground state with: L + = + 1 3Q, L = + 1 Q. (1. We recall that L restricted to radial functions has a generalized nullspace characterized by the following algebraic identities: L Q = 0, L + (ΛQ = Q, L ( y Q = 4ΛQ, L + ρ = y Q, (1.3

9 where ρ is the unique radial H 1 solution to L + ρ = y Q. ( Approimate solutions In this section, we construct the leading order terms in the dynamics we will consider. The regular part will be given by z(t solution to (1.9 for which we recall some well known regularity statements, and the singular dynamics will be approimated by an eplicit deformation of S(t..1. Regular part of the dynamic. From now on and for the rest of the paper, we let m 3 (.1 be an integer and consider z H m+3 radially symmetric with z H m+3 < α (. for some small enough universal constant α > 0. We moreover assume the flatness at the origin: d k z 0 k m, dr k r=0 = 0. (.3 Let z(t be the solution to { i t z + z + z z = 0, z t=0 = z. (.4 then standard Cauchy theory coupled with the smoothness of the nonlinearity ensures: Lemma.1 (Regularity of z. We have z C(R, H m+3. Moreover: (i z scatters ie z L 4 (R,L 4 α ; (.5 (ii there holds the local in time propagation of regularity: z L ([ 1,1],H m+3 α ; (.6 (iii there holds the propagation of degeneracy at the origin: t [ 1, 1], z(t, α ( t + m+1, r z(t, α ( t + m+ 1, r z(t, α ( t + m. (.7 Proof of Lemma.1: The proof is standard and we briefly recall the main facts. (.5 follows from the global in time small data L critical Cauchy theory, [4]. Net, the regularity of the cubic nonlinearity and Strichartz estimates [46] with the dimensional admissible pair (4, 4 imply: z L ([ 1,1],H m+3 z H m+3 + z z L 4 3 ([ 1,1],W m+3, 4 3 α + z L ([ 1,1],H m+3 z L ([ 1,1],L 8 α + z L ([ 1,1],H m+3 z L ([ 1,1],H 1 which ensures (.6 for α small enough. Finally, (.3, (.4, (.6 and a Taylor epansion at (t, = (0, 0 on z yield (.7. This concludes the proof of Lemma.1. Let us also recall the openness of the set of scattering solutions which follows from standard Strichartz estimates again, [4]:

10 10 F. MERLE, P. RAPHAËL, AND J. SZEFTEL Lemma. (The set of scattering solutions is open. Let u C([0 +, H 1 be a global solution to (1.1 which scatters on the right: u L 4 ([0,+,L 4 < +. Then there eists α > 0 such that for all v 0 H 1 with u 0 v 0 L < α, the solution v to (1.1 with initial data v 0 is global v C([0 +, H 1 and scatters to the right v L 4 ([0,+,L 4 < +... Approimate solution for the singular part of the dynamic. Let us now construct the full family of self similar profiles which generalizes the construction in [37], [9]. Let u be a solution to (1.1 and let us pass to the self similar variables which leads to u(t, = 1 λ(t v ( t, λ(t e iγ(t, ds dt = 1 λ, (.8 i s v + v v + ibλv + v v = 0, γ s = 1, λ s = b. (.9 λ Let a parameter η R. We look for specific solutions of the form which thus have to satisfy: b y i v(s, y = P η (ye 4 P η P η + b s + b y P η + P η P η = 0. (.10 4 We now anticipate the law b s + b = η and build the corresponding profiles from standard perturbative and elliptic techniques. Lemma.3 (Eistence of well localized profiles. There eists a universal constant η > 0 and a smooth map η P η H rad defined on ( η, η with the following properties: P η is radially symmetric and real positive, and solves an equation of the form P η P η η 4 y P η + P 3 η = Ψ η (.11 where Ψ η is identically 0 in the case 0 η < η, and some well localized and eponentially small error in the case η < η < 0: for all polynomial q(y, k 0, { } dk q(y dy k Ψ η(y e C k,q η 1 4, SuppΨ η η,. (.1 η L Moreover, for all η < η < η, P η is a smooth function of (η, y with P η η η=0 = 1 ρ, (.13 4 where ρ is the unique H 1 radially symmetric solution to (1.4, and there holds the uniform decay estimates: k 0, η < η < η, d k P η dy k η e ck y, d k dy k (P η Q ηe ck y. (.14 Remark.4. The sign η 0 in (.11 ensures the uniform ellipticity of the operator 1 η y 4 independently of η and hence the uniformity in η of the eponential localization bounds (.14. The profiles P η for η < 0 correspond to the almost self similar solutions build in [9], [31] to which we refer for further details.

11 Remark.5. The invariant of mass and energy can be computed eplicitly in the limit η 0. Indeed, from (1.4, (.13, and standard computations: d { Pη } L dη η=0 = 1 (Q, ρ = 1 4 ( y Q, ΛQ = 1 4 yq L. (.15 For the energy, we multiply (.11 by the Pohozaev multiplier ΛP η and conclude: E(P η = η 8 yq L + O (η + e c η = η 8 yq L + o(η. (.16 Given small parameters (b, η we define: which satisfies: 11 b y i Q η,b = P η e 4, (.17 Q η,b Q η,b + ibλq η,b (b + η y 4 Q η,b + Q η,b Q η,b = Ψ η,b, (.18 b y i where Ψ η,b = 0 if η 0 and Ψ η,b = Ψ η e 4 if η < 0. Following (.8 (.9, we now let Q η (t, = 1 ( λ η (t Q η,b η(t e iγη(t, (.19 λ η (t where the parameters solve the finite dimensional system: for the eplicit solution given by: (λ η s λ η = b η, (γ η s = 1, (b η s + b η = η (.0 λ η (t = η + t, b η (t = t, γ η (t = 1 + t 1 dτ η + τ. (.1 Note that Q η transitions smoothly from an eact global scattering profile for η > 0 to an almost self similar solution for η < 0 which would formally blow up at t = η < 0, and through the S(t blow up solution which blows up in the conformal regime at t = 0 and corresponds to η = Backwards control of the pseudo conformal blow up We develop in this section some analytical tools to control the (NLS flow near Q backwards in time and in the pseudo conformal regime. We will use the decoupling (.7 to reduce ourselves to an almost critical mass setting and use the mied Energy/Morawetz Lyapounov type functional ehibited in [43], and which yields an increasing Lyapounov function hence suitable to integrate the flow backwards from the singularity. As we eplained in the introduction, the backwards integration of the flow after respectively the scattering time for η > 0 or the log-log time for η < 0 is very similar, and we deal with both cases simultaneously. The heart of our analysis is the following: Proposition 3.1 (Backwards control. Let m 3 (3.1 and z H m+3 radially symmetric satisfying (., (.3 for some α > 0 small enough. Let z(t C(R, H m+3 be the solution to (.4. Let η < η (α small enough and a time T η [ 1, 0] such that: η + t λη (t for all t [ 1, T η ]. (3.

12 1 F. MERLE, P. RAPHAËL, AND J. SZEFTEL Let an arbitrary sequence η γ 0 η R and u η be the solution to (1.1 with data at t = T η : u η (T η = Q η (T η e iγ0 η + z(t η. (3.3 Then: (i Local backwards control: there eists a time t 1 < T η independent of η such that u η C([t 1, T η ], H 3 and admits a decomposition of the form ( t, u η (t, = 1 λ(t (Q η,b(t + ε λ(t e i(γ(t+γ0 η + z(t, with the uniform bounds in η: t [t 1, T η ], 1 λ η λ m 1 λ 1 + b η b λ m + γ γ η λ m + ε H 1 λ m+1 + ε 3 H α (3.4 λ m 3 where b η, λ η, γ η are given by (.1. (ii Global backwards control: u η C((, T η, H 3 and scatters backwards: u η L 4 ((,t 1,L 4 1. (3.5 The proof of Proposition 3.1 relies on a bootstrap argument. Indeed, from standard modulation theory 4 and the smallness of λ η (T η, we may consider a time t < T η a priori depending on η such that t [t, T η ]: with w(t, = 1 λ(t (Q η,b(t+ε z u = 1 λ(t (Q η,b(t + ε z + ε ( t, ( t, λ(t e i(γ(t+γ0 η = w + ũ (3.6 e i(γ(t+γ0 η, z(t, = 1 ( λ(t λ(t ε z t, e i(γ(t+γ0 η. λ(t (3.7 The uniqueness of the decomposition (3.6 is ensured through the choice of orthogonality conditions: Re(ε, y Q η,b = 0, Im(ε, ΛQ η,b = 0, Im(ε, ρ = 0 (3.8 where b y i ρ = ρe 4, L + ρ = y Q. Moreover, (λ, b, γ, ε(t η = (λ η, b η, γ η, 0(T η from (.19, (3.3. Thus, from a straightforward continuity argument, we may assume that u satisfies the following a priori bounds: { } ε H 1 sup t [t,t η] λ m+1 K, { 1 sup t [t,t η] λ m 1 λ η λ 1 + b } η b λ m 1, (3.9 where K is a small enough constant. We now claim the following bootstrap Lemma: Lemma 3. (Backwards bootstrap bound on ε. Assume that K in (3.9 has been chosen small enough -independent of η-. Then there eists a small time t 1 < 0 -independent of η- such that u satisfies on [t 1, T η ] the improved bound: sup t [t 1,T η] { 1 λ m 1 4 see [43] for further details λ η λ 1 + b η b λ m + γ γ η λ m + ε H 1 λ m+1 + ε 3 H λ m 3 } α. (3.10

13 13 Remark 3.3. From the proof, the size of t 1 is uniform as α 0. Let us conclude the proof of Proposition 3.1 assuming Lemma 3.. Proof of Proposition 3.1: The local backwards control (3.4 follows from (3.10. We now claim at t = t 1 the uniform control: η η (α small enough, e iγ0 ηu η (t 1 S(t 1 L α (3.11 which together with Remark 3.3, Lemma. and the observation that S(t scatters as t yields the global control (3.5. This concludes the proof of Proposition 3.1 provided (3.11 holds. Now, we turn to the proof of (3.11. Let Then, we have: Q(t 1 = 1 λ(t 1 Q η,b ( t 1, e i(γ(t 1+γ η 0. λ(t 1 S(t 1 e iγ0 η Q(t1 L (3.1 Q P η L + b + t + t λ 1 + γ 1 t Q P η L + b b η + λ η λ 1 + γ γ η + b η + t + t 1 λ η + γ η 1 t η + α, where we used (.14, (.1 and (3.10 in the last inequality. Now, the decomposition (3.6 together with the bounds (., (3.10 and (3.1 yields: u η (t 1 e iγ0 η S(t 1 L z(t 1 L + ũ(t 1 L + S(t 1 e iγ0 η Q(t1 L η + α, which implies (3.11 for η < η (α small enough. This concludes the proof of Proposition 3.1. The rest of this section is devoted to the proof of Lemma 3. which relies on the Lyapounov functional approach developed in [43] Modulation equations. Let us introduced the rescaled time ds dt = 1 λ. Let w be the refined profile given by (3.7 and W its renormalized version: w(t, = Q + z = 1 ( λ(t W t, e i(γ(t+γ0 η, W (s, y = Q λ(t η,b(s (y + ε z (s, y, (3.13 Q = 1 ( λ(t Q η,b t, e i(γ(t+γ0 η, z = 1 ( λ(t λ(t ε z t, e i(γ(t+γ0 η. λ(t From Lemma.3 and (.6, we have the bound: Moreover, w satisfies the equation w L 1, w L 1 λ, w H 3 1 i t w + w + w w = ψ = 1 λ 3 Ψ (t, λ 3. (3.14 e i(γ(t+γ0 η, (3.15 λ(t

14 14 F. MERLE, P. RAPHAËL, AND J. SZEFTEL ( λs Ψ = i λ + b ΛQ η,b (b s + b + η y Q η,b γ s Q b,η Ψ η,b + R, (3.16 R = (Q η,b + ε z Q b,η + ε z Q η,b Q η,b ε z ε z, (3.17 and where γ s = γ s 1. We then decompose u = w + ũ so that ũ satisfies: We rewrite the interaction term as: with i t ũ + ũ + ( u u w w = ψ. (3.18 u u w w = 1 λ 3 [ ε Qη,b + Q η,b ε + R(ε] ( t, e i(γ(t+γ0 η λ(t R(ε = ( W Q η,b ε + ( W Q η,b ε + ( W ε + W ε + ε ε. (3.19 We then decompose Q η,b = Σ + iθ, ε = ε 1 + iε, R(ε = R 1 (ε + ir (ε, R = R1 + i R in terms of real and imaginary part and obtain the equation satisfied by ε: ( λs s ε 1 M (ε + bλε 1 = λ + b (ΛΣ + Λε (b s + b + η y Θ + γ s (Θ + ε + Im(Ψ η,b R R (ε, (3.0 ( λs s ε + M 1 (ε + bλε = λ + b (ΛΘ + Λε (b s + b + η y Σ γ s (Σ + ε 1 Re(Ψ η,b + R 1 + R 1 (ε, (3.1 where (M 1, M are small deformations of the linearized operators (L +, L close to Q: M 1 (ε = ε 1 +ε 1 (3Σ +Θ ε 1 ΣΘε, M (ε = ε +ε (3Θ +Σ ε +ΣΘε 1. We now claim the following preliminary estimates on the decomposition which are a consequence of the orthogonality conditions (3.8: Lemma 3.4 (Preliminary estimates on the decomposition. (i Degeneracy of the unstable direction: there holds (ii Modulation equations: Let then and Proof of Lemma 3.4 (ε 1, Q α λ m+1 + Kλ m+. (3. Mod(t = ( b s + b + η, λ s λ + b, γ s Mod(t α λ m+1 + Kλ m+, (3.3 λ s λ + b α λ m+ + Kλ m+. (3.4 We compute the modulation equations by taking the inner product of (3.0, (3.1 with the well localized in space directions corresponding to the orthogonality conditions (3.8.

15 15 step 1 Inner products. We compute the inner products needed to obtain (3., (3.3, and (3.4. This computation has been made in [43] near P η=0 = Q and the same computation is valid up to O(η ε L terms, and leads to: ( M (ε + bλε 1, Σ + (M 1 (ε + bλε, Θ = O(λ ε L, (3.5 ( M (ε + bε 1, ΛΘ + (M 1 (ε + bλε, ΛΣ = R(ε, Q η,b + O(λ ε L, (3.6 ( M (ε + bλε 1, y Σ + ( M 1 (ε + bλε, y Θ = O(λ ε L, (3.7 ( M (ε + bλε 1, ρ + (M 1 (ε + bλε, ρ 1 = O(λ ε L, (3.8 where ρ = ρ 1 + iρ, and where we used η + b λ from (3. (3.9. step Control of R, R(ε and Ψ η,b. Note first that (.6 yields: Furthermore, (.7 yields: ε z L α, ε z L α λ. (3.9 ε z e y L + ε z e y L α λ m+. (3.30 Thus, the terms involving R given by (3.17 are estimated using (3.9 and (3.30: R L α λ m+. (3.31 The terms involving R(ε are easily estimated using Sobolev, (3.9, (3.30 and the bootstrap bounds (3.9: R(ε e y ε z e c y L ε L + ε L + ε 3 H 1 Kλ m+3. (3.3 Eventually, from (.1 and (3.: for η < η(α small enough. step 3 The law of b. Ψ η,b L e c η α λ m+, (3.33 We take the inner product of the equation (3.0 of ε 1 with ΛΘ and we sum it with the inner product of equation (3.1 of ε with ΛΣ. We obtain after integrating by parts: b s + b + η R(ε, Q η,b + λ ε L + Mod(t ε L + (Kλ + α λ m+ (3.34 where we used the second orthogonality condition in (3.8, the computation of the inner product (3.6, and (3.31-(3.33. step 4 The law of λ. We take the inner product of the equation (3.0 of ε 1 with y Σ and we sum it with the inner product of equation (3.1 of ε with y Θ. We obtain after integrating by parts: λ s λ + b λ ε L + Mod(t ε L + (Kλ + α λ m+, (3.35

16 16 F. MERLE, P. RAPHAËL, AND J. SZEFTEL where we used the first orthogonality condition in (3.8, the computation of the inner product (3.7, and (3.31-(3.33. step 5 The law of γ. We take the inner product of the equation (3.0 of ε 1 with ρ and we sum it with the inner product of equation (3.1 of ε with ρ 1. We obtain after integrating by parts: γ s b s + b + η + λ ε L + Mod(t ε L + (Kλ + α λ m+ (3.36 where we used the third orthogonality condition in (3.8, the computation of the inner product (3.8, and (3.31-(3.33. In view of (3.34-(3.36, we obtain: λ s λ + b λ ε L + (Kλ + α λ m+, (3.37 and b s + b + η + γ s R(ε, Q η,b + λ ε L + (Kλ + α λ m+. (3.38 Furthermore, we have: R(ε, Q η,b (ε 1, Q + ( η + b ε L (3.39 (ε 1, Q + Kλ m+, where we used (.14, (.17, and the fact that η + b λ from (3.9, (.1. The estimates (3.3 and (3.4 then follow from the bootstrap bound (3.9, (3.37, (3.38, (3.39, and the degeneracy (3.. step 6 Proof of the degeneracy (3.. We take the inner product of the equation (3.0 of ε 1 with Σ and we sum it with the inner product of equation (3.1 of ε with Θ. We obtain after integrating by parts: d ( Re(ε, Qη,b λ ε ds L + Mod(t ε L + (Kλ + α λ m+ (3.40 (Kλ + α λ m+ (3.41 where we used the computation of the inner product (3.5, (3.31-(3.33, and the bootstrap bound (3.9. Now from (3.9, (.1: which yields: p 0, Tη t (λ(τ p dτ Tη t t + b λ(t, Thus the time integration from t to T η of (3.40 yields: Re(ε, Q η,b (t Tη t (λ η (τ p dτ t (λ η (t p (λ(t p+1. (3.4 [ K(λ(τ m+1 + α (λ(τ m] dτ Kλ(t m+ + α λ(t m+1, which together with (3.39 concludes the proof of (3. and of Lemma 3.4.

17 3.. Mied energy/morawetz Lyapounov functional. We first rewrite the a priori bound (3.9 as follows: ũ L Kλ m, ũ L Kλ m+1. (3.43 We will also use the a priori bounds from (3.9, (.1 and (3.: η + b λ. (3.44 We let A > 0 be a large enough constant which will be chosen later and let φ : R R be a smooth radially symmetric cut off function with { φ r for r 1, (r = 3 e r (3.45 for r. Let F (u = 1 4 u 4, f(u = u u so that F (u h = Re(f(uh. We claim the following generalized energy estimate on the linearized flow (3.18: Lemma 3.5 (Algebraic generalized energy/morawetz estimate. Let u = w + ũ where w satisfies the bound (3.14, and ũ satisfies (3.18 and the a priori bound Let b, λ satisfying the bounds (3.44 and Let I(ũ = 1 ũ ( b λ I A φ J (ũ = 1 λ I ( ũ L λ, ũ L λ. (3.46 Mod(t λ. (3.47 ũ λ ( ( w ũ R + b { ũ ( λ λ + R + b ( λ R A φ [F (w + ũ F (w F (w ũ ] ũũ ( φ 17, (3.48 w t ( ũ w + ũ w ( ũ, ũ ( ( ũ w + ũ w w, 1 ( 4A { [ L(ũ = I ψ ψ λ + ( w ψ w ψ + i b ( λ A φ then there holds: + i b ( λ φ ψ d I(ũ = J (ũ + L(ũ + O dt ] (3.49 ( } ũ φ λ ψ } ũ, (3.50 ( λ ψ L + ũ L λ + ũ H. ( Remark 3.6. The virtue of (3.51 is to keep track of the quadratic terms I(ũ, J (ũ which will turn out to involve to leading order the coercive quadratic form of the linearized energy near Q given by (Lũ, ũ, and of the leading order linear term L(ũ involving the error ψ.

18 18 F. MERLE, P. RAPHAËL, AND J. SZEFTEL Proof of Lemma 3.5 This Lemma is very similar to Lemma 3.3 in [43] and we briefly recall the proof for the reader s convenience. step 1 Algebraic derivation of the energetic part. We compute from (3.18: { d 1 ũ + 1 ũ [(F dt λ (u F (w F (w ũ ] } (3.5 = R ( t ũ, ũ 1λ ũ + (f(u f(w λ t λ 3 ũ ( R t w, (f(ũ + w f(w f (w ũ = I (ψ, ũ 1λ ũ + (f(u f(w 1 λ I ( (f(u f(w, ũ λ ( t λ 3 ũ R t w, (f(ũ + w f(w f (w ũ ( = I ψ, ũ 1 λ ũ + ( w ũ + ũw 1λ I ũ w λ ( t λ 3 ũ R t w, (wũ + w ũ ( + I ψ 1 λ ũ, (f(w + ũ f(w f (w ũ R ( t w, ũ ũ where we used that f (w ũ = w ũ + w ũ. We first estimate from (3.47: λ t λ 3 ũ = b λ 4 ũ 1 ( λs λ 4 λ + b ũ L = b λ 4 ( ũ ũ + O L λ. (3.53 It remains to estimate the last line in the RHS (3.5. For the quadratic and higher terms, we estimate using the a priori bounds (3.14, (3.46: = (ψ I 1 λ ũ, (f(w + ũ f(w f (w ũ (ψ I 1 λ ũ, (ũ w + ũ w + ũ ũ ψ L (R ũ L 6 ( w L 6 + ũ L 6 + (1 + w L λ ũ 3 L 6 λ ψ L + ũ H 1. (3.54 For the cubic term hitting w t, we replace w t using (3.15, integrate by parts and use (3.14, (3.46 to estimate: w t ũ ũ w H 3/ ũ ũ H 1/ (R + w 3 L ũ 3 6 L + ψ 6 L (R ũ 3 L 6 1 ũ 1/ ũ 5/ λ3/ L H λ ũ H 1 ũ L + ψ L ũ H 1 ũ L λ ψ L + ũ H 1. (3.55

19 Injecting (3.53, (3.54, (3.55 into (3.5 yields the preliminary computation: { d 1 ũ + 1 ũ } dt λ (F (u F (w F (w ũ (3.56 = 1 ( ( λ I w ũ R w t ( ũ w + ũ w + b ũ λ λ ( [ + I ψ ψ ] λ + ( w ψ w ψ ũ ( + O λ ψ L + ũ L λ + ũ H. 1 step Algebraic derivation of the localized virial part. 19 Let then φ(t, = b ( λ A φ, { ( 1 d b ( dt λ I A φ = 1 ( ( I t φ ũũ + R } ũũ [ ] 1 i t ũ φũ + φ ũ. We estimate in brute force using (3.45, (3.47, (3.44: t φ A ( b λ 3 s + b + b λ s λ + b A λ (3.57 from which: ( I t φ ũũ A λ ũ L ũ L ũ L λ + ũ H. ( The second term in (3.57 corresponds to the localized Morawetz multiplier, and we get from (3.18 and the classical Pohozaev integration by parts formula: ( [ ] 1 R i t ũ φũ + φ ũ = (3.59 = b ( ( λ R φ ( ũ, ũ 1 ( ( b α 4 A λ 4 φ ũ b ( ( λ R A φ ( u u w w ũ 1 ( b ( λ R φ ( u u w wũ b ( ( λ R A φ ψ ũ 1 ( b ( λ R φ ψũ. We now epand the nonlinear terms and estimate the cubic and higher terms: b ( ( λ R A φ ( ũ w + ũ w + ũ ũ ũ 1 ( b ( λ R φ ( ũ w + ũ w + ũ ũũ ( ũ 3 L 6 (R + ũ L 6 (R w L 6 (R ũ L (R + 1 λ ( ũ 4 L 4 (R + ũ 3 L 4 (R w L 4 (R ũ L λ + ũ H 1 (3.60

20 0 F. MERLE, P. RAPHAËL, AND J. SZEFTEL where we have used (3.46 and (3.14. The remaining quadratic terms in (3.59 are integrated by parts: b λ R ( = I ( A φ ψ ũ ( [ i b λ A φ ( 1 ( b λ R ψ + i b λ φ ( φ ( α ψ ] ũ ψũ, (3.61 and b ( ( λ R A φ 1 ( b ( λ R φ = b ( ( λ R A φ ( w ũ + w ũ ũ ( w ũ + w ũũ ( ũ w + ũ w w. (3.6 Injecting (3.60, (3.61, (3.6 into (3.59 yields after a further integration by parts: ( R = b λ R [ ] 1 i t ũ φũ + φ ũ ( φ ( ũ, ũ ( + b λ R ( A φ ( ( [ + I i b ( λ A φ 1 ( b 4 A λ 4 ( ũ w + ũ w w ( φ ũ ψ + i b ( ] ( ũ λ φ ψ ũ + O L λ + ũ H. 1 We now inject this together with (3.58 into (3.57 which together with (3.56 concludes the proof of ( Proof of the bootstrap Lemma 3.. We are now in position to close the bootstrap estimates (3.10 as a consequence of the Lyapounov control of Lemma 3.5. step 1 Coercitivity of I. We first claim the coercitivity of I(ũ given by (3.48: I(ũ c 0 ( ũ L + 1 λ ũ L + O ( (K λ + α λ m (3.63 for some universal constant c 0 > 0. Indeed, we first renormalize: I(ũ = 1 { λ ( + bi ε + ( y A φ εε A (F ε (Qη,b + ε z + ε F (Q η,b + ε z F (Q η,b + ε z ε }.

21 We then estimate by homogeneity: [F (Qη,b + ε z + ε F (Q η,b + ε z F (Q η,b + ε z ε ] [ = Re(ε W + ε W ] ( + O W ε 3 + ε 4 = 3 ε 1Q + ε Q + O (( η + b + ε z e c y L + ε z L ε 4 H + ε 3 1 H + ε 4 1 H 1 = 3 ε 1Q + ε Q + O((λ + α ε H 1 where we used the proimity of Q b,η to Q given by (.14 (.17, the bound (3.44 for η and b, the a priori bound (3.9 for ε, and the estimates (3.9 (3.30 for ε z. We thus obtain: I(ũ 1 λ { (L+ ε 1, ε 1 + (L ε, ε + O((λ + α ε H 1 }. (3.64 We now recall the following coercivity property of the linearized energy which is a well known consequence of the variational characterization of Q: Lemma 3.7 (Coercivity of the linearized energy, [49], [8], [31]. There holds for some universal constant c 0 > 0 : ε H 1 radially symmetric, (L + ε 1, ε 1 + (L ε, ε c 0 ε H 1 ( c 0 { (ε1, Q + (ε 1, y Q + (ε, ρ }. The a priori bound (3.9 and the degeneracy (3. imply: (ε 1, Q ( (Kλ + α λ m+1 (K λ + α λ m+. (3.66 Injecting the choice of orthogonality conditions (3.8 and (3.66 into (3.65 and then into (3.64 now yields (3.63 for α small enough. step Coercitivity of J (ũ Let J (ũ be given by (3.49, we claim: J (ũ c b ( λ 4 ε e y A + ε + O ( (K λ + α λ m 1 (3.67 for some universal constant c > 0. Recall the decomposition (3.13, we first claim: ( J (ũ = J 1 (ũ + O A ũ L λ + ũ H ( with J 1 (ũ = 1 λ I ( + b { ũ λ λ ( + λr A φ ( Q ũ R ( + R ( φ Q t ( ũ Q + ũ Q ( ũ, ũ } ( ( ũ Q + ũ Q Q 1 ( 4A. 1 (3.69 ( ũ φ λ

22 F. MERLE, P. RAPHAËL, AND J. SZEFTEL Indeed, we epand w = Q + z and estimate the remaining terms using (.6, (.7, (3.14 and (3.30. We obtain: 1 λ ( Q z + z ũ 1 λ ( ε ze c y L ũ L + z 4 L ũ L ũ L λ + ũ H, 1 and t z( ũ w + ũ w + t Q( ũ z + ũ z t z L 4 ũ L w 4 L λ ε ze c y L ũ L ũ L 4 λ + ũ H, 1 b λ 3 A ũ z z A λ ũ L ũ L λ, ( b λ R ( [ A φ ( ũ Q + ũ Q z + ( ũ z + ũ z w] A ũ L λ, A λ ũ L ( 1 λ [1 + z L + z L ] + 1 λ 3 ε ze c y L and (3.68 follows. We now epand the t Q term from (3.13, (3.47, (3.44: ( i Q t = λ + b Q λ + b ( λ λ Q Mod(t + b + η + O λ ( 3 i = λ + b Q λ + b ( λ λ Q 1 + O, λ which yields: ( R = 1 λ I ( Q t ( ũ Q + ũ Q Q( ũ Q + ũ Q b ( λ R e c λ ( ũ Q + ũ Q Q b ( ( λ R λ ( ũ Q + ũ 1 Q Q + O λ Q L ũ L = 1 ( λ I Q( ũ Q + ũ Q b ( λ R ( ũ Q + ũ Q Q b ( ( λ R λ ( ũ Q + ũ ũ Q Q + O L. We inject this estimate into (3.69 and write the result in renormalized variables: J 1 (ũ = b { ( ( y λ 4 R φ ( ε, ε + ε A (3.70 [(3Σ + Θ ε 1 + 4ΣΘε 1 ε + (Σ + 3Θ ε ] 1 ( y 4A φ ε A ( ( ( y } ( ũ + R A φ y ( ε A Q η,b + ε Q η,b Q η,b + O L. From the proimity of Q η,b to Q and the control of the full L norm, the above quadratic form is for A large enough a small deformation of the localized in A λ λ

23 linearized energy, and hence (3.65, (3.66 and our choice of orthogonality conditions ensure for A large enough: J (ũ c [ ] ( 0 b λ 4 ε e y A + ε + O (K λ + α λ m 1 + A ũ L λ + ũ H 1 O ( (K λ + α λ m 1 where we used (3.9 in the last step. This concludes the proof of (3.67. step 3 Control of L(ũ. We now turn to the control of the leading order linear term L(ũ given by (3.50 and we claim: L(ũ (λk + α Kλ m 1. (3.71 We first derive from (3.16, (3.17, (3.30 and (3.3 the rough bound: Ψ Mod(t e c y + Ψ η,b + R (Kλ + α λ m+1 e c y. (3.7 Together with (3.30 and (3.9, this yields the bound: ( z + Q z ψ ũ 1 λ 4 ( ε z + ε z e c y Ψ ε 3 1 λ 4 ε ze c y L (1 + ε z e c y L λ m+1 ε L α Kλ 3m and thus with L j (ũ = I L(ũ = L 1 (ũ + L (ũ + O ( (K λ + α λ m 1 { [ ψ j ψ j + i b λ φ ( λ + ( Q ψ j Q ψ j + i b ( λ A φ ] } ψ j ũ where according to (3.16: ψ j = 1 ( λ 3 Ψ j e λ, Ψ 1 = Ψ R + Ψ η,b, Ψ = R Ψ η,b. ψ j We start with the second term. We estimate in brute force using (3.31 (3.33: ψ L 1 λ ( Ψ η,b L + R L α λ m. Now, note that (.7 yields the following analog of (3.30: which together with (.1, (3.30 yields: ε z e c y L α λ m+ 5, ψ L 1 λ 3 ( Ψ η,b L + ε z e c y L + ε z e c y L α λ m 1. (3.73 These estimates for ψ and ψ together with (3.44 and the a priori bound (3.43 immediately imply: L (ũ ψ L ũ L + 1 λ ψ L ũ L + A ψ L ũ L + 1 λ ψ L ũ L Kα λ m 1.

24 4 F. MERLE, P. RAPHAËL, AND J. SZEFTEL We now compute L 1 (ũ from the eplicit formula (3.16 and observe that the corresponding terms lie up to O(λ in the general null space of L, and hence a factor λ is gained. In other words, we compute after renormalization: L 1 (ũ Mod(t [ (ε λ 4, L ( y Q + (ε, L Q + O(λ ε L ] + 1 λ s λ 4 λ + b [ (ε 1, L + (ΛQ + O(λ ε L ] K λm+ λ 4 [Kλ + α ] λ m λ 4 (K + α λ m+ (Kλ + α λ m+1 (λk + α Kλ m 1 (3.74 where we used the algebraic identities (1.3, the bounds (3.66, (3.3, (3.4, the orthogonality conditions (3.8, and the a priori bound (3.9. This concludes the proof of (3.71. step 4 Control of ũ. We derive from (3.7 the rough bound: ψ L = 1 λ Ψ L (Kλ + α λ m 1. (3.75 We inject the bounds (3.43, (3.67, (3.71 and (3.75 into (3.51 and obtain the Lyapounov control: d dt I (K λ + α + α Kλ m 1. We integrate this from t to T η and use the coercitivity bound (3.63 and (3.4 to conclude: ũ(t L + ũ(t L λ (K λ(t + α λ(t m + [ ] K + O(α (λ(t m provided t 1 is sufficiently small. Hence Tη t [K λ(τ + α + α K]λ(τ m 1 dτ K K + O(α and thus K α. (3.76 step 5 Integration of the law for the parameters. The control of the parameters now follows by reintegrating the modulation equations. Indeed, from (3.3, (3.76: { } b + η b b = λ λ b + η (b s + b + η ( + η λs λ λ + b s = b η λ η b η + η (b s + b + η + O( α λ m+1 η = O( α λ m η. (3.77 We integrate this bound from t to T η and recall from (.1 that b η + η λ η = 1.

25 Hence: (b(t + η λ(t = 1 + O( ( Tη α λ m + O α (λ η (τ m dτ t = 1 + O( α λ m 1 η. (3.78 (3.78 implies: b + η = λ + O( α λ m+1 η, which together with (3.3 yields: b s + λ = O( α λ m+1 η ie b t + 1 = O( α λ m 1 η. We integrate from t to T η using b(t η + T η = b η (T η + T η = 0 and thus: b(t + t = b(t b η (t = t 5 T η α O(λ m 1 η = O( α λ m η. (3.79 Net, we compare λ with λ η. In view of (3.78, we have: λ = b + η + O( α λ m η = λ η + b + η b η + η + O( α λ m η = λ η + O( b b η + α λ m η, where we used (.1. Together with (3.79, this yields: λ = λ η + O( α λ m η and thus λ 1 λ η α λ m 1. We finally compute the phase using (3.3, (3.9 and (3.76: γ s α λ m+1 and thus d dt (γ γ η 1 λ 1 λ + α λ m 1 α λ m 3, η which after integration in time yields: γ γ η α λ m. Hence, we have obtained the following estimate for the parameters: 1 λ η λ m 1 λ 1 + b η b λ m + γ γ η λ m α. (3.80 step 6 H 3 control. It remains to prove the H 3 bound on [t 1, T η ]: In view of (3.18, ũ satisfies: where ψ is defined by (3.15 (3.16, F 1 is given by: ε H 3 α λ m 3. (3.81 i t ũ + ũ = ψ F 1 F, (3.8 F 1 = Q ũ + Q ũ, F is given by: F = u u w w F 1, and where w has been defined in (3.13. Hence, from standard Strichartz bounds and the smoothing effect of the linear Schrödinger flow, there holds: 3 ũ L [t,tη] L 3 ψ L (1+ F [t,tη] L L [t,tη] H F L 4. ( [t,tη] L 4 3

26 6 F. MERLE, P. RAPHAËL, AND J. SZEFTEL In view of the definition of ψ (3.15 (3.16 and R (3.17, we have: 3 ψ L ( λ 3 Mod(t + Ψ η,b 4 L R L λ 3 ( α λ m+1 + e c y ε z L + e c y ε z L where we used (.1 (3.3 (3.76 in the last inequality. Now, in view of (.7, we have: e c y ε z L + e c y ε z L α λ m+, which in turn yields: 3 ψ L 4 3 α λ m. Thus: 3 ψ L 4 α λ m 5 3 [t,tη] L (3.84 The F 1 term is local in y, hence from the a priori bound (3.43 and (3.76, we obtain: (1 + F 1 H 1 1 λ 3 ũ L + 1 λ ũ H 1 α λ m,, and thus: (1 + F 1 L [t,tη] H 1 α λ m 3. (3.85 The F term is estimated from Sobolev embeddings and standard product estimates in Besov spaces: 3 F L ũ L ( Q L 4 z L + z L 8 + w L 8 ũ L 8 + ũ L 8 + ũ L 8( 3 Q L z L 8 + Q L 8 3 z L + 3 z L z L w L ũ L 8 α λ 1 3 ũ L + α λ m 5 4, where we used in the last inequality the a priori bound given by (3.43 and (3.76 for ũ and the estimate (.6 for z. In turn we obtain: 3 F L 4 3 [t,tη] L 4 3 α λ ũ L [t,tη] L + α λ m 1. Injecting this together with (3.84, (3.85 into (3.83 yields: 3 ũ L [t,tη] L α λ ũ L [t,tη] L + α λ m 3 and (3.81 follows. This concludes the proof (3.10 and of the bootstrap Lemma Instability of Bourgain Wang solutions We are now in position to prove the main results of the paper which proof we split in several propositions. In the whole section, we let and let m 7 (4.1 z H m+3 Σ (4. radially symmetric satisfy (., (.3 with α, η < η (α small enough so that Proposition 3.1 holds. We let z(t C(R, H m+3 Σ be the solution to (.4.

27 We claim that the conclusions of Theorem 1. and Theorem 1.3 hold for this class of profiles Limit as η 0. We first claim that u η constructed by Proposition 3.1 converges strongly as η 0 on (, 0 to a Bourgain Wang solution with profile z. Lemma 4.1 (Limit as η 0. Let a sequence T η 0, T η 0 (4.3 such that (3. holds. Let u η C((, T η ], H 3 Σ be the solution to (3. constructed in Proposition 3.1. Let t 1 < 0 be the corresponding time 5 of local backwards control. Assume that the phase shift satisfies: Then up to a subsequence, t < 0, e iγ0 η e iθ as η 0. (4.4 u η (t u c (t in Σ where u c C((, 0, H 3 Σ is a solution to (1.1 which scatters to the left and satisfies the local control: t [t 1, 0, u c (t S(te iθ z(t L α t m, u c (t S(te iθ z(t H 1 α t m 3. (4.5 Remark 4.. For η > 0 and T η = 0, (3. holds from (.1. Lemma 4.1 thus yields automatically the eistence of a Bourgain solution in the limit η 0 of the solution u η constructed in Proposition 3.1 with γ 0 η = 0, and the obtained solution u c is automatically unstable by scattering. A careful track of constants shows that the whole construction requires m > 1 only and hence a substantial gain on the regularity and more importantly degeneracy in (.3 with respect to the initial Bourgain Wang analysis []. The large m assumption (4.1 is needed first to get uniform bounds on the phase in (3.4 6 and more importantly to prove some uniqueness statement about u c, see Proposition 4.3 below. Proof of Lemma 4.1: step 1 Σ compactness. We first claim that u η (t 1 is compact in Σ as η 0. Using Proposition 3.1 and in particular (3.4, we have the bound: u η (t 1 H 3 1, (4.6 which shows that (u η (t 1 0<η<η is compact in Σ(r < R as η 0 for all R > 0. The Σ compactness of u η (t 1 is now a consequence of a standard localization procedure. Indeed, let a cut off function χ( = 0 for 1 and χ( = 1 for, then ( d χ R u η = dt Im χ R u η u η 1 R, ( d 1 χ R dt u η 1 ( 4 u η 4 = Im χ R u η ( u η + u η u η 1 R 5 independent of η from Proposition where m > is needed 7