On Universality of Blow-up Profile for L 2 critical nonlinear Schrödinger Equation

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1 On Universality of Blow-up Profile for L critical nonlinear Schrödinger Equation Frank Merle,, Pierre Raphael Université de Cergy Pontoise Institut Universitaire de France Astract We consider finite time low-up solutions to the critical nonlinear Schrödinger equation iu t = u u 4 N u with initial condition u 0 H. Existence of such solutions is known, ut the complete low-up dynamic is not understood so far. For a specific set of initial data, finite time low-up with a universal sharp upper ound on the low-up rate has een proved in [], [3]. We estalish in this paper the existence of a universal low-up profile which attracts low-up solutions in the vicinity of low-up time. Such a property relies on classification results of a new type for solutions to critical NLS. In particular, a new characterization of soliton solutions is given, and a refined study of dispersive effects of NLS in L will remove the possiility of self similar low-up in energy space H. Introduction. Previous results We consider 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 with u 0 H = H R N for N. The prolem we address is the one of formation of singularities in time for solutions to. From a result of Ginire Velo [], is locally well-posed in H. See also [4], [7] for the periodic case and gloal well posedness results. Thus, for u 0 H, there exists 0 < T + such that ut C[0, T, H. Either T = + and the solution is gloal, or T < + and lim t T ut L = +, we say the solution lows up in finite time. Equation is a Hamiltonian system which admits a numer of symmetries in energy space H, explicitly:

2 Space-time translation invariance: if ut, x solves, then so does ut + t 0, x + x 0, t 0, x 0 R R N. Phase invariance: if ut, x solves, then so does ut, xe iγ, γ R. Galilean invariance: if ut, x solves, then so does ut, x βte i β x β t, β R N. Scaling invariance: if ut, x solves, then so does u λ t, x = λ N uλ t, λx, λ > 0, and y direct computation u λ L = u L. Moreover, admits another symmetry which is not in energy space H ut in Σ: Pseudo conformal transformation: if ut, x solves, then so does vt, x = t N u t, x x ei 4t. t These symmetries induce from Ehrenfest law the following conservation laws in the energy space H : L -norm: ut, x = u 0 x ; Energy: Eut, x = ut, x + 4 N ut, x + 4 N = Eu0 ; Momentum: Im uut, x = Im u 0 u 0 x. For u 0 Σ = H {xu L }, the motion of the center of mass is driven y d dt x ut = Im uut, and we moreover have a additional conservation law which is usually expressed as the so called Virial Identity: d dt x ut = 6E 0. 3 Thus, u 0 Σ and E 0 < 0 imply finite time low-up in H.

3 Special solutions play an important role in the description of the dynamic of generic solutions to, at least numerically. They are the so called solitary waves of the form ut, x = e iωt Q ω x, ω > 0, where Q ω solves Q ω + Q ω Q ω 4 N = ωqω, Q ω > 0. 4 There exists a unique solution in H up to translation to 4, see [] and [5]. Q ω is in addition radially symmetric. Letting Q = Q ω=, then Q ω x = ω N 4 Qω x from scaling property. Therefore, one computes Q ω L = Q L. Moreover, Pohozaev identity otained y taking the inner product of 4 y N Q+y Q yields EQ = 0, so that EQ ω = ωeq = 0. In particular, none of the three conservation laws in H of sees the variation of size of the Q ω stationary solutions. Note that for N, 4 admits excited states solutions of larger L norm. For u 0 small in L, u 0 L < Q L, t 0, ut L Cu 0, and the solution is gloal in H. Indeed, this follows from the conservation of the energy, the L -norm and Gagliardo-Nirenerg inequality related to the variational characterization of Q as exhiited y Weinstein in [8]: u H, + 4 N u 4 N + u u N. 5 Q In addition, this condition is sharp: for u 0 L Q L, low-up may occur. Indeed, since EQ = 0 and EQ = Q, there exists u 0ɛ Σ with u 0ɛ L = Q L +ɛ and Eu 0ɛ < 0, and the corresponding solution must low-up from virial identity 3. The case of critical mass u 0 L = Q L has een studied in [0]. The pseudo-conformal transformation applied to the stationary solution e it Qx yields an explicit solution St, x = t N Q x x e i 4t + i t 6 t which lows up at T = 0. Note that St L = Q L. It turns out that St is the unique minimal mass low-up solution in H in the following sense: let u H with u L = Q L, and assume that ut lows up at T = 0, then ut = St up to the symmetries of the equation in H, that is phase, scaling, translation and Galilean invariances. Note that from direct computation ES > 0 and St L C t. 3

4 The case u 0 L > Q L is more complex. The known dynamical results are for initial conditions in the L vicinity of Q, Q u 0 Q + α, α > 0 small. In this situation, from the criticallity of the prolem and variational characterization of the ground state, the solution decomposes as ut, x = e iγt Q + ε t, x xt 7 λ N t λt where εt H δα with δα 0 as α 0, and scaling parameter λt is a priori of order size λt. ut L There still exist in dimension N = solutions of type St y a result of Bourgain Wang, [6], that is solutions with u 0 Q + α, α > 0 small, corresponding to a class of solution ut which write near low-up time ut St + u as t 0, u H, and thus ut L t. On the other hand, numerical simulations, [6], suggest the existence of solutions lowing up like ut L ln lnt t T t in dimension N =. Perelman proves in [5] in dimension N = the existence of a solution of this type and its staility in some space E H. This lnln factor is interpreted as a correction to self similar low-up. Recall indeed that y direct scaling argument, a known lower ound on the low-up rate is ut L Cu 0 T t. 8 In previous papers [], [3], we clarified the situation for initial conditions in H roughly satisfying E 0 < 0 and Q u 0 Q + α with α > 0 and small. From surprising monotonicity properties ased on a refined dispersive property of close to Q under condition E 0 < 0, we proved there the first control on the low-up rate for solutions to which removed in particular the possiility of lowup solutions of type St in this situation. More precisely, let us consider the following property: Spectral Property Let N. Consider the two real Schrödinger operators L = + 4 N N + Q 4 N y Q, L = + N Q 4 N y Q, 9 and the real valued quadratic form for ε = ε + iε H : Hε, ε = L ε, ε + L ε, ε. 0 4

5 Then there exists a universal constant δ > 0 such that ε H, if ε, Q = ε, Q = ε, yq = ε, Q = ε, Q = ε, Q = 0, then: i for N =, Hε, ε δ ε + ε e y for some universal constant < ; ii for N 3, Hε, ε δ ε ; where Q = N Q + y Q, Q = N Q + y Q. Remark Note that this property has een proved in [] for dimension N = and constant = 9 5. We then have: Theorem [], [3] Let N = or N 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 such that Q u 0 < Q + α, E 0 < Im u0 u 0. u 0 L Let ut e the corresponding solution to, then ut lows up in finite time 0 < T < + and there holds for t close to T, ln lnt t ut L C. 3 T t Remark Note that as exhiited in [], assumption may e reduced up to a fixed Galilean transform to E 0 < 0 and Im u 0 u 0 = 0. This is why we call such admissile initial data strictly negative energy initial data throughout the paper. Note that Theorem yields a low-up result in energy space H and not only in Σ. Note moreover that solution St given y 6 does not satisfy the energy condition as ES > 0. Oserve last from lower ound 8 that low-up rate for negative energy solutions to is at most a doule logarithmic correction to the scaling estimate. We propose in this paper to pursue analysis of [], [3]. Our aim is to investigate the question of existence of a universal low-up profile at low-up time, what with notations of decomposition 7 ie after renormalization of the solution corresponds to a limit of Q + εt as t T. Analysis of this kind of prolem for Hamiltonian PDE s requires two kind of informations: rigidity properties of soliton solutions which are in our case the 5

6 natural candidates of asymptotic profiles, and dispersive results in the critical space L. This paper is thus split in two independent parts: -Part A: We estalish various dynamical properties of negative energy solutions to including gloal results in time t, and not only asymptotic estimates near low-up time as in [], [3], among which a characterization of solitons in the set of zero energy solutions to. -Part B: We investigate L dispersive properties of solutions to which classify explicit critical mass low-up solution St as the only non L dispersive low up solution. These results altogether will allow us to prove the universal ehavior of negative energy solutions to at low-up time. The authors deeply thank the referees for their work. Thanks to their careful reading of the manuscript, they have proposed many comments and suggestions which improved the paper and made it clearer.. Part A: Universality of low-up profile Let us note that from explicit computations, one may answer the question of existence of a low-up profile for explicit low-up solution St. Indeed, decomposition 7 holds according to with and St = e iγt λ N t Q + ε t, x λt γt = t, λt = t Q L St L x i εt, y = e 4 t Qy Qy 0 in H as t 0. Note that strong convergence in L is specific to critical mass property St L = Q L. We claim that this result persists for negative energy low-up solutions, or more precisely: Theorem Universality of low-up profile for negative energy solutions Let N = or N assuming Spectral Property holds true. Then there exists α > 0 such that the following is true. Let u 0 H such that Q < u 0 Q + α and Eu 0 Im u0 u 0, 4 u 0 L 6

7 and assume the corresponding solution ut lows up in finite time 0 < T < +. Then there exist parameters λ 0 t = Q L ut L, x 0t R N and γ 0 t R such that e iγ 0t λ N 0 tut, λ 0 tx + x 0 t Q in Ḣ as t T. 5 Remark 3 Ḣ is the space of functions u with u L < +. From the definition of λ 0 t, the function at the left of 5 is uniformly H ounded, and thus 5 is optimal in the sense that it implies y interpolation strong convergence in L p, p, ], and in L loc, whereas strong L convergence is foridden from the conservation of the L norm. Remark 4 In the case of aritrarily large L mass, it is proved in [4] that the solution up to renormalization converges on a susequence t n T to a profile U 0 with the properties that U t has zero energy and is defined for t 0. In our setting, the characterization of U for the full sequence t T as the exact soliton solution is the heart of the matter. The proof of Theorem heavily relies on the negative energy assumption 4 and a priori reaks down for positive energy low-up solutions. Nevertheless, as a Corollary of it, staility of low-up profile still holds on a sequence t n T in the positive energy case. Corollary Weak staility of low-up profile for positive energy solutions Let N = or N assuming Spectral Property holds true. Then there exists α > 0 such that the following is true. Let u 0 H such that Q < u 0 Q + α and assume the corresponding solution ut lows up in finite time 0 < T < +. Then there exist parameters λ 0 t = Q L ut L, x 0t R N, γ 0 t R and a sequence t n T such that e iγ 0t n λ N 0 t n ut n, λ 0 t n x + x 0 t n Q in Ḣ as t n T. 6 Remark 5 In fact, a more precise result is proved as the result holds true assuming ut is defined for t 0 and unounded in H. Let us now give an important Corollary of these results dealing with the speed of low-up collapse and which shows that lower ound 8 is never optimal: Corollary Lower ound on the low-up rate Let N = or N assuming Spectral Property holds true. Then there exists α > 0 such that the following is true. Let u 0 H such that Q < u 0 Q + α and assume the corresponding solution ut lows up in finite time 0 < T < +. Then: 7

8 Negative energy case: assume moreover that u 0 satisfies 4, then T t ut L = +. lim t T Positive energy case: there exists a sequence t n T such that lim T tn ut n L = +. t n T Remark 6 This Corollary removes the possiility of self similar low-up solutions, what from our analysis is very much the heart of the prolem. Let us recall that given 0 > 0, x lnt t i U 0 t, x = Q 0 T t N 0 e T t solves if and only if Q 0 solves the elliptic ODE: Q 0 Q 0 + i 0 N Q 0 + y Q 0 + Q 0 Q 0 4 N = 0. 8 Such solutions are called self similar solutions as they formally satisfy the scaling estimate U 0 t H T t. An important fact, according to [6], is that solutions Q 0 never elong to L from a logarithmic divergence at infinity as Q 0 y y N as y +. Corollary says that even in the time dependent setting, self similar low-up is foridden in energy space H. Remark 7 In [5], J. Bourgain mentions a numer of open prolems regarding low up phenomenon for critical NLS. Results in this paper together with sharp upper ound on low up speed of paper [3] answer this type of questions in the L vicinity of Q. Remark 8 At this stage of the theory, we still do not know whether ut L C ln lnt t T t holds true for all low up solutions. Corollary is a first step in this direction. Lnln upper ound in [], [3] corresponds to a pure local in space low up analysis, whereas from the proof of Corollary, lower ounds are related to gloal dispersive ehavior in space, and in particular in critical space L. Note that on the contrary to the paraolic situation, we are here in a Hamiltonian setting which does not provide us with a Lyapounov type functional to ensure convergence to a simple oject as t T. In our case, linear theory is moreover not enough to otain staility of the low-up profile due to the high degeneracy of the linear structure of around Q, which is indeed of higher order than the numer of symmetries. This difficulty may e ypassed, see [6], 8

9 assuming more decay and regularity control for a finite codimensional set of initial data -as from the proof, one is then in a situation where the focusing part and the radiative parts of the solution do not interact-. This shows the inadequacy of linear theory in a pure H setting. Note again that on the contrary, Theorem holds in the whole energy space H. The only known results of this type for Hamiltonian equations are those y Martel- Merle for the generalized KdV equation { ut + u KdV xx + u 5 x = 0, t, x [0, T R 9 u0, x = u 0 x, u 0 : R R, in papers [7], [8], [9], []. The proof may e summarized in three steps: i Through scaling and compactness arguments, one constructs an asymptotic oject vt solution to 9 which in some sense does not disperse in L. ii From this L non dispersive property and the fact that vt is asymptotic, almost monotonicity properties locally in L allow a gain of regularity and decay on vt. iii Assuming that asymptotic oject vt is not a soliton, a contradiction follows from dispersive properties and decay estimates. Now this approach is not directly successful for the study of, even though we shall argue in the same spirit. To prove Theorem, one needs to otain some dispersive type information for solutions to in critical space L, what corresponds to gloal in space results. Note indeed that so far in [], [3], L gloal information is never used. Our main point is that such gloal L information can e reached not on the solution itself, ut on an asymptotic oject recurrent in time. Let us now precise the main steps of the proof of Theorem which correspond to dynamical properties of negative energy solutions to which have their own interest. Strategy of the proof of Theorem : The proof is also ased on introduction of asymptotic oject vt. Here, we do not know any monotonicity property on the local L norm, ut from negative energy assumption 4, the size of the solution is almost increasing near low-up time. This fact induces non trivial continuity properties which in turn imply a non dispersive structure in L for asymptotic oject vt. More precisely, on the asis of monotonicity properties for the size of ut near low-up time, we exhiit an asymptotic oject vt which satisfies: i Ev = 0 and Im vv = 0; ii vt is defined on the left with t 0, vt L C. Thus, one of the three following cases may hold for vt: Case : vt is gloally defined on the right and ounded: t R, vt L C. 9

10 Case : vt is gloally defined on the right and lows up in infinite time: lim sup vt L = +. t + Case 3: vt lows up in finite time T v < + on the right. Assuming asymptotic oject vt is not a soliton, we have to find a contradiction in each case. Case and Case are removed y the following surprising Theorem which characterizes soliton solutions in the set of zero energy solutions near critical mass: Theorem 3 Finite time low-up for zero energy solutions in H Let N = or N 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 such that Q u 0 Q + α, Eu 0 = Im u0 u 0, u 0 L ut e the corresponding solution to. Then if u 0 is not a soliton up to fixed scaling, phase and translation parameters, ut or u t satisfies the following: u lows up in finite time 0 < T < + and there holds for t close to T ln lnt t ut L C. T t The contradiction in Case 3 is of different nature and relies on dispersive L information on asymptotic vt. This information is otained in a very weak sense using limiting focusing measures at low-up time. Indeed, under hypothesis of Theorem or Theorem 3, one can prove that the limiting measure µ u = lim t T u t, x is well defined. This oject measures in particular the amount of mass which has come into collapse -see Sections 4., 4.4 for precise results of existence and continuity of this oject-. Now in Case 3, from a continuity result in L for limiting focusing measures, the fact that vt is an asymptotic oject implies that it accumulates whole its L mass into low-up in the sense that: µ v = v0 δ x=0. A contradiction which concludes the proof of Theorem then follows from a non existence result of Dirac mass low up solutions to which Part B is devoted. 0

11 .3 Part B: Non existence of Dirac mass low up solutions This part is devoted to the proof of the non existence of Dirac mass low up solutions to which concludes the proof of Theorem. Theorem 4 Non existence of non L -dispersive solutions Let N = or N assuming Spectral Property holds true. Then there exists α > 0 such that the following is true. There is no solution vt H to satisfying: Im vv ; i Q v Q + α and Ev ii v lows up in finite time 0 < T < + and v L for some constant m 0. v t mδ x=0 as t T in the sense of distriutions 0 Remark 9 From the proof, the energy assumption may e replaced y T 0 vt L dt < +. Note moreover that this condition is in turn implied y energy assumption from upper ound 3. Next, the same result holds true with v t, x + xt mδ x=0 for some m 0. We in fact always have m = v0 and xt x T finite as t T. Remark 0 From the structure of in Σ, a noteworthy fact is that this result is in fact equivalent to ruling out directly the existence of self similar solutions in H, that is solutions lowing up like vt L T t, see Remark 6. Oserve that this result is false for positive energy solutions and integraility condition on the gradient of Remark 9 is sharp as for St: ESt > 0, 0 St L = +, St, x Q δ x=0 as t 0. In fact, from assumptions of Theorem 4, vt elongs to Σ and non L dispersive ehavior implies lim t T x vt = 0. Theorem 4 is now equivalent to the following characterization of explicit solution St in Σ: Theorem 5 Characterization of St in Σ Let N = or N assuming Spectral Property holds true. Then there exists α > 0 such that the following is true. Let v 0 Σ with Q v 0 Q + α,

12 vt the corresponding solution to. Assume vt lows up in finite time 0 < T < + and lim x vt = 0, t T then vt = St up to the full set of symmetries of in H. Remark In other words, the only solution in Σ with L norm close to Q L which does not disperse its L mass is the critical mass low-up solution St. This is a classification result in the spirit of the one of minimal L low-up solution of [0]. Recall that the main point there was to prove dispersive estimate without any a priori control on low-up speed, and then conclusion follows from the critical mass assumption and conservation laws in Σ. In our situation of super critical mass, dispersion is used to control the low-up speed and get dispersive relation in Σ, ut conclusion follows from the study of dispersion in L. Note that this result is an open prolem in H for positive energy solutions with vt Ldt = +. T 0 In fact, Theorem 4 and Theorem 5 admit another equivalent formulation related to low-up ehavior of zero energy solutions to. Recall from Theorem 3 that zero energy solutions in H must low-up in finite time on the left or on the right in time. Under additional assumption v 0 Σ, we have surprisingly that low-up occurs on oth sides in time. More precisely: Theorem 6 Blow-up for zero energy solutions in Σ Let N = or N assuming Spectral Property holds true. Then there exists α > 0 and a universal constant C > 0 such that the following is true. Let v 0 Σ with Q < v 0 Q + α and E 0 = Im vv, v L vt the corresponding solution to. Then vt lows up in finite time oth on the right and on the left in time, and estimate 3 holds. Remark Note that the super critical mass assumption removes the possiility for vt to e a soliton. Thus Theorem 6 is a characterization of solitons in Σ as the only zero energy solution to living on an infinite time interval. Note that Theorem 6 is an open prolem in H. Strategy of the proof of Theorems 4, 5, 6: We first note that these Theorems are equivalent formulations of the same result and reduce using conformal invariance and

13 upper ound on the low-up rate for negative energy solutions to prove non existence of a solution vt Σ to satisfying E 0 = Im vv = 0 and x vt 0 as t T. We assume existence of such a solution vt and look for a contradiction. As in [3], first note that vt decomposes as vt, x = λt N ε + Q t t, x xt e iγt λt with εt H small, λt vt and Q H is the non linear approximation of Q introduce in [3]. The proof now follows in two steps: i Using this decomposition, conservation law 3 in Σ seen in the ε variale and a refined analysis on douling time intervals of the norm vt L allow us to prove that vt has a self similar low-up C T t vt L C T t with C C and x εt e C t. Such an estimate relies on the fact that we are in fact dealing with an asymptotic oject, ie a solution without dispersion, as for generic low-up solutions, ehavior x εt + as t T is expected. ii We then exhiit a contradiction to this uniform smallness estimate in Σ. Introducing the tale in a certain regime ζ of self similar solutions related to 8, we exhiit further dispersive relations in variale ε = ε ζ which measure radiation at infinity in space under smallness assumption of ε in Σ, and lead to a contradiction..4 Structure of the paper This paper contains two main independent parts which are each organized as follows: Part A: Section is devoted to recall results exhiited in [], [3] and to adapt them to the zero energy case. In Section 3, we reduce the proof of Theorem to dynamical results for negative energy solutions. In section 4, we prove properties related to existence and continuity of the limiting focusing measure. The last section of Part A is devoted to 3

14 the proofs of Corollary and Corollary. Part B: In section 6, we recall results in [3] in H and slightly sharpen them. In section 7, we reduce the proof of Theorems 4, 5, 6 to the zero energy case in Σ of Theorem 4 to which is devoted the rest of the paper. In section 8, we prove exponential proximity to the self similar regime and exhiit in section 9 a contradiction to the uniform smallness in Σ. Throughout this paper, we will assume in dimension N that Spectral Property holds true recall that it has een proved to hold true for N =. In addition, all results in this paper deal with initial conditions u 0 H in the L vicinity of Q, Q u 0 Q + α for some α > 0 small enough. We also fix some notations. δα > 0 will denote a constant such that δα 0 as α 0. We let the constant given y Spectral Property, = 9 5 for N =. As will e clear from further analysis, we shall not need the precise value of, see Remark. Moreover, given a well-localized function f, we set Note that integration y part yields f = N f + y f and f = N f + y f. f, g = f, g. 4

15 Part A: Universality of low up profile This part is devoted to the proof of Theorem. Decomposition and monotonicity properties for negative energy solutions In this section, we consider a solution ut to for an initial condition u 0 H satisfying Q u 0 Q + α and E 0 Im u0 u 0 u 0 L for some α > 0 small enough. As exhiited in [], first oserve from Galilean invariance that for β R N, u β = ut, x βte i β x β t is a solution to. From, let β = Im u 0 u 0 u0, then u 0 β = u β 0, x satisfies Eu 0 β 0 and Im u 0 β u 0 β = 0. For u 0 satisfying, we prove results on u β and transpose them afterwords -see [3] for more details-. We therefore consider in this section equation for an initial data u 0 H satisfying Q < u 0 Q + α, Eu 0 0 and Im u 0 u 0 = 0, 3 for some α > 0 small enough to e chosen later, and let ut the corresponding solution to with T u, T + u, 0 < T u, T + u +, its maximum time interval existence in H. Our aim in this section is to exhiit a decomposition of the solution ut and first properties of it. This decomposition is indeed the key to otain dispersive estimates which imply gloal in time monotonicity properties and somehow reduce the study of to a finite dimensional prolem. Such an analysis has een proved to hold in the strictly negative energy case in []. In the first susection, we riefly recall key Lemmas of [], and then adapt them in the second susection to the case of zero energy solutions.. Recall of dynamical results for strictly negative energy solutions In this susection, we consider an initial condition u 0 H satisfying Q < u 0 Q + α, Eu 0 < 0 and Im u 0 u 0 = 0, 4 for some α > 0 small enough according to results in []. 5

16 We first recall a classical Lemma of proximity of H functions up to scaling, phase and translation factors to the function Q related to the variational structure of Q under the mass and energy conditions of 3. Recall indeed that for u H, solutions of Eu = 0 and u L = Q L are exactly e iγ 0 λ N 0 Qλ 0 x + x 0 for some fixed parameters λ 0, γ 0, x 0. Lemma Variational characterization of the ground state For all 0 < α α, there exists δα with δα 0 as α 0 such that u H, if Q u Q + α and Eu α u, then there exist parameters λ 0 = Q L u L, γ 0 R and x 0 R N such that Qx e iγ 0 λ N 0 uλ 0 x + x 0 H < δα. Now oserve from the conservation of the energy and the L mass that we may apply Lemma to the solution ut to for all time t T u, T + u and sharpen this decomposition using the implicit function Theorem according to the following Lemma: Lemma Modulation of the solution There exist some continuous functions λ : T u, T + u 0, +, γ : T u, T + u R and x : T u, T + u R N such that t T u, T + u, εt, y = e iγt λ N tut, λty + xt Qy 5 satisfies the following properties: i ε t, Q = 0, ε t, yq = 0 and ε t, Q = 0, 6 where ε = ε + iε in terms of real and imaginary part, and Q, Q given y. ii λt ut L + εt Q H δα, where δα 0 as α 0. 7 L As exhiited in [], decomposition of Lemma is adapted to the study of dispersive effects of in the energy space H. We now introduce a new time scale s = t ε, λ, γ and x are now functions of s. We let 0 dt λ t. s min, s max = s{ T u, T + u }. We summarize in the following Proposition results collected from the study in []. 6

17 Proposition [] There exists a universal constant C > 0 such that λs, γs, xs are C functions of s on s min, s max, and we have the following properties: i Equation of εs : εs satisfies for s s min, s max, y R, s ε L ε = λ s λ Q + x s λ Q + λ s N λ ε + y ε + x s λ ε + γ s ε R ε 8 s ε + L + ε = γ s Q γ s ε + λ s λ N ε + y ε + x s λ ε + R ε 9 where γs = s γs. L = L +, L is the linearized operator close to Q: L + = Q 4 N, L = + Q 4 N, 30 N and the functionals R and R are given y R ε = ε + Q ε + Q 4 N Q 4 N + 4 N + Q 4 N ε, R ε = ε ε + Q 4 N Q 4 N ε. ii A priori estimate on the modulation parameters: s s min, s max, λ s λ + γ s C x s λ δα ε + ε + ε e y, 3 ε e y. iii Invariance induced estimates: s s min, s max, λ se 0 + ε, Q C ε + ε e y, 3 ε, Q s Cδα iv Time mapping estimate: s min, s max =, +. v Virial type estimate: s R, ε, Q s δ 0 ε + vi Evolution of the scaling parameter: s s, s 4 ε, Q + yq L lnλs s λs δα + ε. 33 ε e y δ 0 ε, Q. 34 s s ε, Q. 35 Remark 3 The main input of [] is v and vi altogether. These relations are the keys to somehow reduce the low-up dynamic to a one dimensional prolem for the scaling parameter λs λ ε, Q. 7

18 Sketch of proof of Proposition For the sake of completeness, we riefly recall the proof. See [] for more information on the nature of this result. i is otained y injecting 5 into. ii follows from orthogonality conditions 6 and estimates of interaction terms. iii is a consequence of respectively the conservation of the energy and the momentum. iv is a consequence of ii. Note that the argument is here different from the one in [] and does not rely on the strictly negative energy condition. Assume indeed that s[0, T u + = [0, s 0, 0 < s 0 < +, then from 3, λs C and y integration: s [0, s 0, lnλs Cs 0 from which we conclude that lnλt remains ounded as t T u +. This implies on the one hand T u + = + from 7 and the well posedness of the Cauchy prolem for in H, and on the other hand T u + < + from dt = λ ds. We argue similarly for t Tu. v Virial inequality 34 is the very heart of our analysis. Indeed, the ε equation is from a linear point of view degenerated as exhiited in [9]. Here, using non linear conservation laws, we exhiit a dispersive estimate which avoids this degeneracy. To otain it, we take the inner product of 9 with the well localized function Q, use critical relation L + Q = Q and the conservation of the energy 4 ε Q λ E 0 ε + N + Q 4 N ε + Q 4 N ε δα ε + ε e y. Orthogonality conditions 6 together with non linear degeneracy estimate 33 yield: ε, Q s Hε, ε + λ E 0 δα λ ε + ε e y, where the quadratic form Hε, ε is given y 0. Now conclusion follows from non linear estimates 3, 33 together with positivity property of H, which is proved for N = in [] and conjectured in higher dimension, see Spectral Property stated in Introduction. vi now follows. Indeed, take the inner product of 8 with the well localized function y Q, use L y Q = 4Q and integrate in time with the use of 34, and 35 follows, see Proposition 4 of []. This ends the proof of Proposition. We point out as exhiited in [] that virial estimate 34 is not sharp. Indeed, on the asis of structural and algeraic degeneracies of around Q, we have the following refined virial estimate -see Proposition 7 in []-: Proposition Refined virial estimate There exists a universal constant C > 0 such that the following holds true: set W = y Q + y Q,Q Q and W Q = N W + y W, 8

19 then s R, { + } yq ε, W ε, Q L s + C ε, Q + N Remark 4 In [], aove Proposition is stated for a time s large enough. proof, estimate 36 holds for all times. From the This decomposition of the solution and refined virial estimate 36 are still not sharp enough to otain sharp upper ound 3, and indeed lead according to analysis in [] to: lnt t N ut L C 37 T t where T = T u +. Now sharp upper ound 3 is proved with similar techniques in [3] y introducing a sharp decomposition of the solution closed to modified profiles, what is far more technically involved. In fact, we do not need in Part A sharp estimate 3, the only fact which will e needed to prove some key staility results in the proof of Theorem eing T ut L dt < +, 38 what indeed follows from The last proposition is the maximum principle type result on quantity ε, Q which prevents the norm of the solution from oscillating in time -see 4-, and is the key to results of []. Note that its demonstration heavily relies on the strictly negative energy condition. Proposition 3 Monotonicity for strictly negative energy solutions Under condition 4, there exists a unique s 0 R such that: Moreover, there holds: s s s 0, s < s 0, ε, Q s < 0 ; ε, Q s 0 = 0 ; s > s 0, ε, Q s > s 3 ε, Q δα yq L lnλs s s λs 5 ε, Q + δα 40 s with δα 0 as α 0, and λs < λs. 4 9

20 . Monotonicity properties for zero energy solutions This susection is devoted to the study of zero energy solutions. A first key to the proof of Theorem will e to prove that ground state solution Q has a very specific ehavior in the set of zero energy initial conditions. Let an initial condition u 0 H such that Q < u 0 Q + α, E 0 = 0, Im u 0 u 0 = 0 4 for some α > 0 small enough. Note that the super critical mass condition is equivalent to assuming that u 0 is not a soliton up to fixed scaling, phase and translation parameters. On the one hand, from direct verification, Lemma, Lemma, Proposition and Proposition hold with the same proof for u 0 satisfying 4. On the other hand, we are not ale to prove Proposition 3, only a slightly weaker version of it which will suffice for our analysis. More precisely, we claim: Proposition 4 Partial monotonicity for zero energy solutions Under condition 4, then either ut or u t solutions to satisfies the following: i s 0 R such that ε, Q s 0 > ii Moreover, we then have: s s 0, ε, Q s > 0 and 40 and 4 hold. Proof of Proposition 4 This is a consequence of virial inequality 34. i Assume ε, Q s = 0 for all s R, then from 34, εs = 0 for all s R, so that u 0 L = Q L contradicting 4. Therefore, s 0 R with ε, Q s 0 > 0, and the sign of this scalar product changes from ut to u t which are oth solutions to satisfying 4. ii Assume first that for some s R, there holds ε, Q s = 0 and ε, Q s s 0, then from 34, εs = 0 and u 0 L = Q L what contradicts 4. Thus, if ε, Q is strictly positive for some time, it must remain positive after this time. The almost monotonicity of the scaling parameter now follows from 35 and 39. This ends the proof of Proposition 4. Remark 5 We will prove in section 4. that a zero energy solution satisfying 43 lows up on the right in finite time T u + > 0 and upper ound 3 holds. Now from the aove proof, if for example ε, Q t = 0 = 0, what is ensured y u 0 real valued, then ε, Q must e strictly negative for t < 0 and strictly positive for t > 0, and Proposition 3 holds. One may then prove that the corresponding solution must low-up in finite time T >. There therefore exist zero energy solutions which low-up oth on the right and on the left in time. According to Theorem 6, this is in fact always the case in Σ. This point remains open in H. 0

21 3 Universality of low-up profile for negative energy solutions This section is devoted to the proof of main Theorem assuming key dynamical results which will e proved later. In the first susection, we use Galilean transform to reduce the proof to the study of negative energy and zero momentum solutions to. Moreover, strong convergence 5 in Ḣ will e implied y the corresponding weak H convergence through the conservation of the energy. In the second susection, we then prove it using monotonicity properties in this case. 3. Reduction to the zero momentum case We use Galilean invariance and conservation of the energy to reduce the proof of main Theorem to the one of the following Proposition: Proposition 5 Negative energy and zero momentum case Let u 0 H such that Q < u 0 Q + α, Eu 0 0, Im u 0 u 0 = 0, 44 and assume the corresponding solution lows up in finite time 0 < T < +. Let λt, γt, xt the parameters associated to decomposition of Lemma, then e iγt λ N tut, λtx + xt Q in H as t T. 45 Let us assume Proposition 5 and then conclude the proof of Theorem. Proof of Theorem assuming Proposition 5 Let an initial condition u 0 H satisfying Q < u 0 Q + α and Eu 0 Im u0 u 0, u 0 L and assume the corresponding solution lows up in finite time 0 < T < +. From Galilean invariance, let β = Im u 0 u 0 u0, then u β t, x = ut, x βte i β x β t is a solution to and satisfies 44. From direct computation, u β t L ut L C β u 0 L, and thus u β t lows up at time T. We now may apply Lemma to u β t and exhiit a regular decomposition u β t = ε λ β t N β + Q t, x x βt e iγβt. λ β t

22 This first implies that λ β t u β t L remains ounded from aove and elow as t T, and thus λ β t 0 as t T. Proposition 5 now reads ε β t 0 in H as t T. Weak convergence in H yields strong convergence in L loc : ε β t, y e y dy 0 as t T. We now use the conservation of the energy to get strong convergence in Ḣ. Writing down the conservation of the energy in terms of ε β and estimating interaction terms using the smallness of ε β in H, we let E 0β the energy of u β and get: ε β t λ βt E 0β + ε β t e y 46 so that ε β t L 0 as t T. Now express ut in terms of u β t, ut, x = u β t, x + βte i β x+ β t, and inject regular decomposition of u β, then one gets after some algera: ut, x = ˆε + Q t, x ˆxt e ˆλ N t ˆλt iˆγt, with ˆλt = λ β t, ˆxt = x β t βt, ˆγt = γ β t + β ˆxt β 4 t and We conclude that ˆε goes to zero in Ḣ : ˆεt, y = e iˆλt β y ε β + Qt, y Qy. ˆεt 0 in H and ˆεt L 0 as t T. Set now λ 0 t = Q L ut L, x 0t = ˆxt and γ 0 t = ˆγt, then first from strong convergence in H, ˆλ t ut L Q L ie ˆλt λ 0 t. We then compute λ 0 t N ut, λ0 tx + x 0 te iγ 0t = λ0 t ˆλt N ˆε + Q and 5 follows. This concludes the proof of Theorem assuming Proposition 5. t, λ 0t ˆλt x The rest of this section is devoted to reducing the proof of Proposition 5 to dynamical properties of negative energy solutions to.

23 3. Proof of Proposition 5 Let u 0 H satisfying 44 and ut the corresponding solution which is assumed to lowup at 0 < T u + < +. We denote Tu, 0] the maximum time interval existence on the left. We first apply for all t T, T u + the canonical decomposition given y Lemma to ut with geometrical parameters λ u t, γ u t, x u t and express ut, x = λ N u t ε u + Q We now prove 45 arguing y contradiction. t, x x ut λ u t step : Geometrical control of recurrent oject in time. e iγut. The first step is to define a non trivial recurrent oject in time: from the oundedness of ε u in H, we may assume without loss of generality that there exists a sequence t n T + u such that Let now e iγutn λ N u t n ut n, λt n x + x u t n v0 in H with v0 Q. 47 u n 0 = e iγutn λ N u t n ut n, λt n y + x u t n and u n τ the solution to with initial condition u n 0, explicitly u n τ, x = e iγutn λ u t n N u t n + λ ut n τ, λ u t n x + x u t n, so that u n τ is defined on Tn, T n + with Tn computation, Q u n 0 and u n τ admits a decomposition = T u +t n λ u tn, T + n Q + α, Eu n 0 0, Im u n τ, x = e iγnτ λ N n τ ε n + Q τ, x x nτ λ n τ = T u + t n. From direct λ u tn u n 0u n 0 = 0, 48 with λ n τ = λ ut n + λ ut n τ, 49 λ u t n γ n τ = γ u t n + λ ut n τ γ u t n, 50 x n τ = { } x u t n + λ u t n τ x u t n, λ u t n 5 3

24 ε n τ, y = ε u t n + λ t n τ, y. 5 A asic remark is then that the sequence of initial data u n 0 satisfies: n 0, λ n 0 =, γ n 0 = 0, x n 0 = step : Staility properties of weak convergence. We now exhiit a result of weak convergence of geometrically localized sequences of initial data u n 0 similar to the one exhiited in [8]. Note that we propose a result slightly more general than what we need to prove Proposition 5 as we shall need this generality to prove staility results for positive energy low-up solutions -where existence of the decomposition is indeed ensured only in the vicinity of low-up time according to Lemma -. Lemma 3 Staility of weak convergence Let a sequence of initial data u n 0 H with corresponding solution to u n t, [0, T + n its maximum time interval existence on the right. Assume the following: i Existence of a decomposition: n 0, u n 0 admits a regular decomposition with and u n 0, x = λ N n 0 ε n + Q t, x x n0 λ n 0 e iγn0 n 0, ε n 0 H α 54 ε n 0, Q = 0, ε n 0, yq = 0, ε n 0, Q = 0; 55 ii Geometrical localization: There exists a constant C > 0 such that iii Negative energy limit: Let then v0 H such that n 0, λ n0, x n 0 C; 56 lim sup Eu n n + and vt the corresponding solution to, then there holds: i Geometrical staility: Ev0 0 and u n 0 v0 in H, 58 Q v0 Q + α. 59 4

25 Thus vt admits on its maximum time interval existence [0, T v + a canonical decomposition as in Lemma with parameters λ v t, γ v t, x v t. ii Upper ound on T v + : 0 < T v + lim inf T + n + n. 60 iii Staility of weak convergence: t 0 [0, T v +, there exists nt 0 0 such that n nt 0, u n t admits on [0, t 0 ] a decomposition as in Lemma, and there holds: { t 0 [0, T v + u n t 0, x vt 0, x in C[0, t 0 ], L e 8 y λ n, x n λ v, x v in C[0, t 0 ], R R N 6, where u L e 8 y = u e 8 y. See section 4.3 for the proof. step 3: Monotonicity property and low-up results imply Ev = Im vv = 0 We now study low-up ehavior of asymptotic oject vτ for which we have assumed from 47: v0 is not a soliton. We claim that H invariants of vτ are frozen according to: Ev = 0 and Im vv = 0. 6 We indeed claim as a consequence of monotonicity properties on ut that vτ is gloal on the left: T v = +, 63 so that 6 follows from 59 and low-up Theorem for strictly negative energy solutions for t 0. Indeed, Ev < Im vv v implies low-up in finite time for negative L time, and 59 yields 6. Proof of 63: From finite time low-up assumption on ut, 35 ensures the existence of a time t u > 0 such that ε u, Q t u > 0, and thus 40 and 4 hold for ut, ie Recall now 49: t t t u, λ u t < λ u t. n 0, τ Tn, T n +, λ n τ = λ ut n + λ ut n τ. λ u t n Fix then τ Tv, 0], then from t n T u + and λ u t n 0, there holds for n large enough t u t n + λ ut n τ t n, so that we apply the almost monotonicity of the scaling parameter λ u with t = t n + λ ut n τ and t = t n to conclude: τ T v, 0], Nτ such that n Nτ, λ n τ. 5

26 Now from the staility of weak convergence, we conclude: τ T v, 0], λ v τ ie vτ L C for some universal constant C > 0, and 63 is proved. step 4: Characterization of the ground state in the set of zero energy solutions We are now left with an asymptotic oject vτ which has zero momentum, zero energy and which is gloal on the left in time. Therefore, three different dynamics are possile for vτ on the right: i vτ lows up in finite time 0 < T v + < + on the right; ii vτ lows up in infinite time T v + = + on the right up to a susequence; iii vτ is gloally ounded on the right. A first major input in our analysis is the following: ehavior ii never occurs for negative energy solutions, and ehavior iii characterizes the ground state solution in the set of zero energy solutions. More precisely, let us recall the result stated in the introduction: Theorem 3 Let N = or N 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 such that Q u 0 Q + α, Eu 0 = Im u0 u 0, u 0 L ut e the corresponding solution to. Then if u 0 is not a soliton up to fixed scaling, phase and translation parameters, ut or u t satisfies the following: u lows up in finite time 0 < T < + and there holds for t close to T ln lnt t ut L C. T t See section 4. for the proof. In other words, zero energy solutions which are not a soliton must low-up in finite time either on the left or on the right in time. From Tv = +, 47 ie vτ is not a soliton, and 6, we conclude T v + <, and v lows up in finite time on the right. This result corresponds in the proof to the Liouville Theorem for critical KdV, see [7]. step 5: Limiting focusing measures for negative energy solutions. A step further in our analysis is now to otain some L dispersive information on asymptotic oject vτ. Let us point out that the previous analysis on negative energy solutions to relies on L -loc type dispersive informations, and not gloal in L. L 6

27 dispersive information will e captured here on asymptotic oject vτ in a very weak sense y considering limiting focusing measures at low-up time and continuity properties of this oject for negative energy solutions. Indeed, we first claim as a consequence of upper ound 3, or more precisely 38, the existence of a limiting focusing measure at low-up time. Proposition 6 Existence of a weak focusing measure at low-up time Let u 0 H satisfying Q < u 0 Q + α, E 0 Im u0 u 0, u 0 L and assume the corresponding solution ut lows up in finite time 0 < T < +. Then there exists a continuous function xt : [0, T [ R such that the limiting focusing measure µu = lim t T u t, x + xt is well defined, and the concentration point has a finite limit at low-up time, ie xt xt as t T for some finite xt R N. See section 4. for the proof. The function xt of aove Proposition may e chosen equal to the one of regular decomposition 5, and lim t T u t = µ[u xt ]. Note that a standard concentration result is that any low-up solution to must accumulate at least Q L into low-up in the sense that R > 0, lim inf u t, x Q, t T x x ut R so that µ Q L δ x=0. From the proof, Proposition 6 holds under assumption T 0 ut L dt < +. step 6: Continuity of low-up dynamic for geometrically localized sequences. Recall now that asymptotic oject vτ uild as a weak H limit of a geometrically localized sequence u n τ has een proved to low-up in finite time 0 < T v + < +. The second major input is the following continuity Theorem for oth the low-up time and the limiting focusing measure for negative energy geometrically localized sequences which follows from analysis in []: Theorem 7 Continuity of low-up dynamic under geometrical localization Let u n 0 H a sequence of initial data satisfying n 0, Eu n 0 0 and Im u n 0u n 0 = 0, 7

28 u n t the corresponding solution to with maximum time interval existence on the right [0, T n and decomposition λ n, γ n, x n of Lemma. Assume moreover for some constant C > 0: n 0, λ n0, x n 0 C. 64 Let then v0 H such that u n 0 v0 in H, vt the corresponding solution to, and assume vt lows up in finite time 0 < T + v < +. Then: i Continuity of low-up time: T n T v as n ii Continuity of limiting focusing measure: Let µ n, µ v the limiting focusing measures of respectively u n t, vt, x n T n, x v T v their concentration points, then See section 4.4 for the proof. µ n µ v and x n T n x v T v as n Remark 6 Note that this result heavily relies on the negative energy assumption and is false otherwise. Indeed, the set of low-up solutions is not an open set in H. For example, consider explicit low-up solution St given y 6, and a sequence u 0ε = εs S as ε 0. From u 0ε L < Q L, the corresponding solution u ε t has strictly positive energy and is gloal on R, whereas St lows up in finite time. step 7: Asymptotic oject vτ lows up like a Dirac mass. We now are in position to prove the key dispersive result in L on asymptotic oject v: there exists m 0 such that µ v = mδ x=0. 67 This is in fact an easy consequence of Theorem 7. Indeed, let µ n the limiting focusing measure associated to u n τ, µ u associated to ut, we first claim: R > 0, n 0, µ n x R = µ u x λut nr. 68 Indeed, first recall 5: τ Tn, T n +, x n τ = { λ xu ut n t n + λ u t n τ x u t n }, so that we compute: u n τ, x + x n τ = λ u t n u t n + λ ut n τ, λ u t n x + x n τ + x u t n x R x R 8

29 = = x R [ λ u t n u t n + λ ut n τ, λ u t n x + x u t n + λ ut n τ u t, x + x u t x λ ut nr µ u λut nr as τ T + n ] t = t n + λ ut n τ from T n + = T u + t n λ u tn, and 68 follows. Now oserve that the function fr = µ u x r is a decreasing positive function of r > 0, and let m = lim r 0 fr, then 68, λ u t n 0 as n + and 66 of Theorem 7 yield 67. Remark 7 From estimate 38, one checks that vτ is L compact in the sense that η > 0, A > 0 such that τ [0, T v +, vτ η. x x vt A Thus 67 forces vτ to accumulate its whole L -mass into low-up and one proves: µ v = v0 δ x=0 and m = v0. See Part B for more details. step 8: Non existence of non L dispersive solutions. The last step is now to prove non existence of a zero energy, zero momentum finite time low-up solution which focuses whole its L mass at low-up time according to 67. In fact, for reasons we shall not detail, such a result is equivalent to proving non existence of self similar solutions, that is solutions which would low-up with the scaling rate ut L C T t. Let us recall the result stated in the introduction which will e proved in Part B: Theorem 4 Let N = or N assuming Spectral Property holds true. Then there exists α > 0 such that the following is true. There is no solution vt H to satisfying: Im vv ; i Q v Q + α and Ev ii v lows up in finite time 0 < T < + and v L v t mδ x=0 as t T in the sense of distriutions for some constant m 0. 9

30 See Part B for the proof. Thus from non existence of asymptotic oject vτ, we otain a contradiction to 47, and this concludes the proof of Proposition 5 and Theorem. 4 H dynamic of negative energy solutions This section is devoted to the proofs of results needed to prove Theorem except Theorem 4 which will e proved in Part B. More precisely, the first susection is devoted to the proof of Theorem 3, the second to Proposition 6, the third to Lemma 3 and the last one to Theorem Blow-up in H for zero energy solutions: proof of Theorem 3 We let an initial condition u 0 H such that Q < u 0 Q + α, Eu 0 = 0 and Im u 0 u 0 = 0 for some α > 0 small enough. Note that the super critical mass condition removes the possiility for u 0 to e a soliton. We decompose ut according to Lemma, and recall that Propositions and hold. Moreover, according to Proposition 4, y possily considering u t instead of ut, we may assume: s 0 R such that ε, Q s 0 > 0, and thus s s 0, ε, Q s > 0. The proof of Theorem 3 now follows in four steps similar to those in [], [3], and we riefly recall them in the simplest situation where a ound ut L C otained -see [] for more details-. step : Finite or infinite time low-up: λs 0 as +. lnt t N T t This is a consequence of refined virial inequality 36, which is ased on structural degeneracies of the linear operator L close to Q proved in Proposition 7 of []. Indeed, set now fs = + ε yq, W ε, Q, then L ε, Q fs ε, Q for α > 0 small enough, and view 36 as a differential inequality for f: s R, f s + Cf + N 0. From 39, fs > 0 for s s 0, and integrating this differential inequality yields the existence of a time s s 0 such that s s, ε, Q s fs C 30 s N N+ is. 69