TYPE II BLOW UP FOR THE ENERGY SUPERCRITICAL NLS FRANK MERLE, PIERRE RAPHAËL, AND IGOR RODNIANSKI

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1 TYPE II BLOW UP FOR THE ENERGY SUPERCRITICAL NLS FRANK MERLE, PIERRE RAPHAËL, AND IGOR RODNIANSKI Abstract. We consider the energy super critical nonlinear Schrödinger equation i tu u u u p = in large dimensions d with spherically symmetric data. For all p > pd large enough, in particular in the super critical regime s c = d p >, we construct a family of C finite time blow up solutions which become singular via concentration of a universal profile ut, x λt p r Q e iγt λt with the so called type II quantized blow up rates: λt c ut t l α, l N, l > α = αd, p. The essential feature of these solutions is that all norms below scaling remain bounded lim sup s ut L < for s < s c. t T Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation in 5, 34, which was done using maximum principle techniques following 6. Instead we develop a robust energy method, in continuation of the works in the energy critical case 38, 3, 39, 4 and the L critical case. This shades a new light on the essential role played by the solitary wave and its tail in the type II blow up mechanism, and the universality of the corresponding singularity formation in both energy critical and super critical regimes.. Introduction.. The NLS problem. In this paper we study the focusing nonlinear Schrödinger equation: { i t u u u u NLS p =, t, x R u t= = u R d, ut, x C.. This canonical dissipative model conserves the total energy and mass: Eut = u u p = Eu,. p ut = u..3 The scaling symmetry u λ t, x = λ p uλ t, λx for λ > is an isometry of the homogeneous Sobolev critical space u λ t, Ḣsc = uλ t, Ḣsc for s c = d p.

2 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI We focus on the energy critical and super critical models: s c i.e. p = d d, d 3. These problems are locally well posed in H sc and if the nonlinearity is analytic p = q, q N, then the flow propagates Sobolev regularity and there holds the blow up criterion: T < implies lim ut H s = for s > s c. t T.. Type I and type II blow up for the heat equation. Singularity formation is better understood for the scalar nonlinear heat equation { t u = u u NLH p, t, x R u t= = u R d.4 in dimension d 3, in particular in the radial setting where maximum principle techniques apply. In particular, one can construct time-dependent Lyapunov functionals, based on counting the number of spatial intersections between two solutions. Let us very briefly recall some of the main known facts on singularity formation for.4 in the energy critical and super critical range p >, s c >. The basic object at the heart of the analysis is the self-similar profile. Let us look for solutions to.4 of the explicit form r ut, x = Q b.5 λt λt p where λt is given by the exact self similar-scaling: λt = bt t, b =..6 Q b is then a solution elliptic stationary self-similar equation: Q b bλq b Q p b =, Λ = y, b =..7 p Spherically symmetric solutions of.7 are completely classified. There are two fundamental objects: the regular at the origin constant self-similar solution p Q κ p, κ p =,.8 p and the singular at the origin homogeneous self-similar solution: Rr = c, c = d p p r p p..9 Type I blow up: The regular constant self-similar solution.8 generates a stable blow up dynamics of so called type I with universal blow up rate given by: ut L,. T t p consistent with.5,.6. The existence and stability of this object can be proved using spectral techniques and energy methods,,,, 33, 3. In fact. this blow up regime exists for all p and is not specific to the energy supercritical range. A related analysis has been recently successfully propagated to the case of the wave equation, 7.

3 In the regime < p < p JL there exists another class of regular solutions, decaying at, to the self-similar equation.7 which give rise to the type I unstable blow up, 9, 5. Here, p JL if the Joseph-Lundgren exponent given by { for d, p > p JL = 4 d 4 d for d.. Type II blow up: In the 99 unpublished manuscript by Herrero and Velasquez, announced in 5, proposed a different type of blow up mechanism for p > p JL, based on a threshold structure of the spectrum of the linearized operator, close to.9, H R = Λ pcp r. The spectrum of H R turns out to be completely explicit in suitable weighted spaces. The authors describe a singularity formation in which ut L T t αl p, l N, αl >.3 where α is the phenomenological number.5. The blow up bubble corresponds, in self-similar renormalized variables, ut, x = λt p vs, z, z = r λt, λt = T t, to a profile generated by the singular self-similar solution R: 3 ds dt = λ t,.4 vs, z = Rz e λ js ψ j z lot.5 where λ j is the j-th, j = jl, strictly positive eigenvalue with eigenvector ψ j of the linearized operator H R. The decomposition.5 is singular at the origin and, in particular, does not readily imply the L control.3. It is merely designed to capture the behavior of the solution tail, while the leading order of the solution near the origin is given by a renormalized smooth radial solitary wave Qr solving Q Q p =, Q =. The situation was clarified in the series of works by Matano and Merle 5, 6 through the proof of two fundamental theorems in the radial setting: For < p < p JL, only type I. occurs, with both stable and threshold regimes. For p > p JL, type II occurs as a threshold dynamics between type I and global existence. This requires in particular d, and yields an indirect proof of the existence of type II blow up solutions. We emphasize that an essential tool in the analysis in 5, 6 was a construction of a Lyapunov functional based on the precise counting of intersections of a solution with the singular self-similar solution R. This feature strongly anchors the analysis to the radial setting and to the use of tools reliant on the maximum principle. Following that, using the maximum principle tools developed in 5, 6, Mizoguchi, in 34, 35, has been able to rigorously implement the formal construction of 5 to prove both the existence of solutions with blow up speed.3 and to give a complete classification of radial type II blow up solutions. The difficulty here is that the decomposition.5 is fundamentally singular both at infinity, where all this corresponds to a threshold regime between global solutions and the stable type I blow up dynamics. in a suitable class.

4 4 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI terms have infinite energy, and at the origin, where both R and ψ j are singular 3. The whole analysis consists in deriving.5, first in some weak local L sense, and then propagating this weak control to the L topology in a self-similar window At < z < At, lim At =..6 t T The maximum principle tools developed in 5, 6 are once again essential in this analysis and not at all amenable to an extension of these results to a problem like NLS, or even the non-radial heat equation..3. Critical blow up problems. The past ten years has seen remarkable progress on the question of singularity formation for critical problems, where the scaling symmetry meets a conservation law. For.4, this corresponds to the case p =. Interestingly enough, even maximum principle techniques were not able to address this case, and despite some formal predictions 9, the rigorous derivation of type II blow up solutions has remained open until very recently. A new intuition based on Liouville classification theorem and a new set of energy type techniques have led to the pioneering blow up results on the mass critical gkdv, 7,, to the new classification results of blow up dynamics near the ground state for the mass critical NLS 8, 9, 3, and more recently to a complete classification of the flow near the ground state for the gkdv, 3, 4. Energy critical models have also been a source of important progress in connection with the two dimensional critical geometric equations: the wave maps, the Schrödinger maps and the parabolic harmonic heat flow, 44, 8, 4, 38, 3, 39, 4. New fundamental tools have been developed for the construction of energy critical blow up solutions, in settings where even an existence of singular dynamics had been mostly unknown, and for the analysis of their stability/finite codimensional instability. A continuum of blow up rates were constructed in 8 for the wave map problem, and in for gkdv, while for the parabolic heat flow, a discrete sequence of blow up regimes was rigorously obtained in 4, in agreement with the formal predictions in. In the setting of the nonlinear heat equation.4, these techniques have led to the first construction of type II blow up solutions in the energy critical case p = 3, d = 4, 45. In all these works, the blow up profile is not given by a stationary self-similar solution to.7, but rather by a soliton, i.e. a smooth stationary or time periodic solution to the original PDE, for example for the NLS equation: ut, x = Qxe it, Q Q p =..7 The blow up solution then corresponds to a decomposition ut, x = λt p vs, ye iγt, y = x λt, ds dt = λ t, with vs, y = Qy εs, y, ε..8 The blow up rate λt is never given by the self-similar speed.4, but by its suitable deformations. The ground state which is a smooth stationary solution, as 3 without an obvious cancellation.

5 opposed to the singular self-similar solution.9, turns out, after renormalization, to be the universal attractor of the flow in a suitable topology: lim s εt L = for s > s c..9 t T A robust general strategy for the construction of blow up solutions in the critical regimes emerged from the works 38, 3, 39, 4, 4, and relies on a two step procedure: Construction of a suitable approximate blow up profile through iterated resolutions of elliptic equations. The "tail computation" allows one to derive formally the blow up speed from the behavior of the tail of a profile at infinity. An essential algebraic fact for the analysis is the asymptotic behavior Qr r cd. The parameter cd drives the derivation of the blow up law and the possibility of a blow up with Q profile. Implementation of an energy method to control the full flow via the derivation of "Lyapunov" functionals involving super critical Sobolev norms adapted to the linearized flow near the ground state, which do not rely on spectral estimates and may therefore be easily adapted to various settings Super critical numerology. Let us now come back to the super critical problem s c > and discuss some essential algebraic facts. The problem Q Q p = admits a one parameter family of smooth spherically symmetric solitary waves solutions with the asymptotic behavior Qr Rr = c r p as r,. with c given by.9. A well known characterization of the Joseph-Lundgren exponent. is given through the positivity of the linearized operator closed to Q, see for example 6. Indeed, let = pq p, then: for < p < p JL, has a non positive eigenvalue with well localized eigenvector; for p > p JL, is strictly lower bounded by the Hardy potential d > 4r >.. The proof of. relies on a Sturm-Liouville oscillation argument and is related to the asymptotic expansion Qr = c r p c r γ o r γ, c,.3 5 where { γ = d Discr >, Discr = d 4pc p > p > p JL iff Discr >..4 4 for example, nonlocal non self-adjoint operators as in 4, or quasilinear problems in 3.

6 6 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI We introduce the phenomenological number see Appendix A. α = γ p, α > for p > p JL,.5.5. Statement of the result. Our main claim in this paper is that the asymptotics.3 for p > p JL, replaces the expansion. in the critical case, are perfectly suitable for the implementation of the strategy for a construction of a blow bubble solution with profile Q. The resulting blow up mechanism is type II energy super critical: Theorem. Type II blow up for the super critical NLS equation. Let d. Let α be given by.5 and assume: and α / N, Fix an integer p = q, q N, p > p JL, Discr > 4 d γ / N, and an arbitrary large Sobolev exponent s N,.6 d / N..7 p l N with l > α,.8 s sl as l. Then there exists a radially symmetric initial data u r H s R d, C such that the corresponding solution to. blows up in finite time < T < via concentration of the soliton profile: with: i Blow up speed: ut, r = ii Stabilization of the phase: λt p r Q ε e iγt.9 λt λt = cu o t T T t l α, cu > ;.3 iii Asymptotic stability above scaling: iv Boundedness below scaling: v Behavior of the critical norm: γt γt R as t T ;.3 lim s εt, L = for all s c < s s ;.3 t T lim sup ut H s < for all s < s c ;.33 t T ut Ḣsc = c l α o t T logt t..34

7 7 Comments on Theorem.. On the assumptions on p. The assumption.7 is generic but technical and avoids the appearance of logarithmic losses in the sequence of weighted Hardy inequalities which we use to close our energy estimates. Unlike the situation in the critical case 38, 3, we claim that these logarithms are irrelevant in our setting and, in this sense, the assumption.7 could be removed. Regarding the assumption.6, Discr > 4 is automatic for d 3 and p 3, or for p large enough in dimensions d =,. This assumption is relevant only for the asymptotic development of the solitary wave., and allows for a simple decoupling of the remainder terms. We however claim that it is not essential and we could treat the case Discr > along similar lines. Finally, the assumption p = q makes the nonlinearity analytic, and in particular allows us to estimate the solution in any homogeneous Sobolev norm Ḣs. Given l as in the statement of Theorem., we need to control Ḣ sl norm of the solution with lim sl =. l Hence, a C regularity of the nonlinearity is required for a statement which holds true for all l large enough. However, for a given l a blow up solution satisfying.3 can be constructed for any p pl large enough using the techniques of this paper. Yet, as presented, our analysis does not cover non smooth nonlinearities near the p JL exponent.. On the role of the topology. We stress that the structure of the blow up solution.9,.3 is exactly the same as the one obtained in the energy critical case.9, see in particular 38, 3, 39. This is quite unexpected and reveals the essential role payed by the topology in which the deformation of the ground state is measured. For example, the structure of Q and a theorem from 4 ensures that e ith Q enjoys standard Strichartz estimates, and hence we expect that Q is stable and in fact asymptotically stable as a solution to. with respect to well localized perturbations. This was proved using sup-sub solutions for the nonlinear heat equation in 3. A related phenomenon is the global existence proof by Bejenaru, Tataru for the energy critical Schrödinger map in the vicinity of the ground state harmonic map. However, since Q has infinite energy from.3, if the perturbation is well localized then this kind of flow corresponds to infinite energy solutions. We should also mention here a very recent result of Krieger, Schlag 7 on a global existence of certain solutions to a supercritical septic wave equation in dimension three, arising from the data with an infinite scale invariant norm. On the contrary, the full initial data of Theorem. can be taken to be even compactly supported and, of course, smooth. This means that the initial perturbation ε to Q must possess a far away tail to cancel the slow decay of Q at infinity, and hence ceases to belong to standard spaces in which decay is usually measured. These considerations necessitate the need to work with homogeneous high Sobolev norms for which Q has a finite contribution and for which the decomposition.9 makes complete sense. Let us also note another unexpected feature: the subcritical conservation laws play essentially no role in our analysis. In fact, the whole analysis

8 8 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI takes place in the super critical algebra Ḣσ Ḣs with s c < σ < d s and whether the full solution is or is not of finite energy or mass is irrelevant in the blow up regime under consideration. 3. On the role of the decay of the ground state. The tail computation, initiated in the critical case, allows one to compute explicitly the expected rates of type II blow up directly from the asymptotic expansion of the ground state at spatial infinity, see the strategy of the proof below. It is therefore essential to recall that if Qr r cd,p, p, then the mapping p cd, p is discontinuous at p = For the heat equation this explains why type II blow up holds in the critical case p =, 39, 45, ceases to exist for < p < p JL, 5, and then exists again for p > p JL. 4. On the manifold construction. The statement of Theorem. can be made more precise. Let l N satisfying.8, s, then our initial data is of the form where Q b,a is a deformation of a ground state Q and u = Q b,a ε.35 a = a,..., a L, b = b,..., b L, s L correspond to possible unstable directions of the flow in the suitable neighborhood of Q. Fix a low Sobolev exponent s c < σ < d, H s topology in a we show that for all ε Ḣσ H s small enough in this topology and for all b, b l,..., b L a kl,..., a L small enough, there exists a choice of unstable directions b,..., b l a,..., a kl such that the solution arising from initial data.35 satisfies the conclusions of Theorem.. Here, k l is given by.4. This implies that our blow up solutions are constructed for a codimension l k l > manifold of initial data. Let us insist that our class of initial data includes in particular compactly supported C initial data. As is now standard in the field, this manifold is constructed as a C manifold using a soft Brouwer type fixed point argument. This provides a precise count of the number of directions of instability in this type II blow up regime. Constructing a local Lipschitz manifold would require proving an appropriate local uniqueness statement. The recent analysis 8 clearly suggests that once the existence is shown, by a Brouwer type argument, and with a strong decay on the solution as is the case in the setting of Theorem. then local uniqueness can be obtained by rerunning the machinery on the difference of two solutions, see also 4,. 5. On quantization of blow up rates. The quantization of blow up rates.3 is the same as the one obtained in the case of the heat equation through a complete

9 classification theorem in 35, see also 4. In dispersive settings, a continuum of blow up rates can be constructed, 8, but they correspond to solutions propagating from non-regular data and are therefore never H. See 4 for the study of related phenomena. We expect that the quantized rates.3 are the building blocks to classify type II blow up of smooth solutions near the ground state for.. 6. Comparison with the heat equation. Observe that.9,.3,.3 imply the rate of blow up 9 ut L λ t p T t αl p which, according to.3, is the same as for the nonlinear heat equation. Let us however stress that the decomposition.9 centered on the solitary wave looks very different from the decomposition.5 centered on the singular self-similar solution. In fact, we claim that the sharp description of the blow up behind.9 implies a quantized version of the decomposition.5 in self-similar variables, see the Strategy of the proof below. In other words, our analysis covers, with one set of estimates relying only on energy methods, both the self-similar zone and the zone near the singular point. This is a substantial clarification of the analysis of type II blow up. 7. Other super critical blow up for NLS. In the setting of the energy super critical NLS equation, the sole other example of a blow up phenomenon that we are aware of is the construction of standing ring blow up solutions for the focusing quintic model p = 5 in all dimensions d, 36, 37. These solutions emerge from smooth well localized radial data and concentrate on the sphere r =. The behavior of Sobolev norms is very different, in particular for these ring solutions lim ut Ḣs = for all s >, t T which implies that these blow up solutions are very much connected to the mass conservation law. Theorem. gives the first result of type II blow up for the energy super critical NLS which, following 5, 6, should be understood as a singular regime where according to.33, all norms below scaling remain bounded. Our approach can be extended to the heat and wave equations, and the radial assumption can be removed. The case of the wave equation will be treated in 5. Notations: We collect the main algebraic notations and facts which are used throughout the paper. Super critical numerolgy: Given d, p > p JL, we let: γ = d Discr >, Discr = d 4pc p > and α = γ p >,

10 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI see Appendix A. We define 5 : { k = E d γ, d γ = k δ k, δ k <. k = E d p >, d p = k δ k, δ k <. so that from.7: We let and < δ ± < δ p = max{δ, δ }, < δ p <,.38 k = k k.39 from.5. We will use the relations d γ 4k = 4δ k d 4 p 4k = 4δ k,.4 α k = δ k δ k. We let l α = k l δ l, k l N, < δ l <.4 from.7. Notations for the analysis: Given a large integer, we let: and define the Sobolev exponent: We define the generator Λ of a scaling symmetry Given b >, we define: where We denote: We let the matrix L = k.4 s = k..43 Λu = p u y u. B = b, B = B η.44 η = η, < η..45 B d R = {x = x,..., x d R d, S d R = {x = x,..., x d R d, J = d x i R }, i= d x i = R }. i=, J = Id = where we recall the definition of the integer part: Ex x < Ex, Ex Z.

11 For real vectors: u = u, v = u v, u, v = u v v u v and for complex valued functions: f, g = R fg. R d The nonlinearity fu = u u p. We define the sequence of iterated derivatives D k u = m u for k = m y m u for k = m. We let χ be a smooth radially symmetric cut-off function χx = { for x for x..47 Linearized operator. Given ε C, we identify ε = Rε Iε..48 Near Q the linearization of. generates a linear operator L, given in complex variables by Lε = ε p Qp ε p Qp ε or, equivalently, in terms of.48: where We let the potentials L = and introduce the matrix operator L L = pq p, L = Q p. W = pq p, W = Q p,.49 L = JL = L adapted to the linearized flow of. near Q i s ε = Lε i.e. s ε = Lε.,.5 Observe that L = L L = J LJ, J L = J L..5

12 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI.6. Strategy of the proof. We now give a brief description of the proof of Theorem.. We keep the notations and the strategy close to the ones of the critical case, see in particular 4, with the intent to show the deep unity of the analysis. In what follows, we pick l N, l > α associated with the blow up speed.3, and another integer l, L = k, related to the regularity of the solution and the construction of suitable Lyapunov functionals. i Renormalized flow and iterated resonances. Let us look for a modulated solution ut, r of.4 in the modulated form: ut, r = vs, ye iγ, y = r λt, which leads to the renormalized flow: ds dt = λ t.5 s v i v b Λv ia v iv v p =, b = λ s λ, a = γ s..53 Assuming that the leading par of the solution is given by the ground state profile 6, the remaining linear part of the flow is governed by the matrix Schrödinger operator. L = L The scaling and phase invariances of the problem induce an explicit resonance 7 : L ΛQ Q =. Each component behaves differently at infinity: Q c y p and there holds the fundamental cancellation of the tail at infinity: ΛQ c y γ as y with γ = α p > p..54 We already see here the appearance of the condition p > p JL : for < p < p JL, the asymptotic.54 is false and would instead include oscillations 8, see for example 3. We may now compute the kernel of the powers of L through the iterative scheme LΦ k, = Φ k,, Φ, = ΛQ, LΦk, = Φ k,, Φ, =.55 Q which display a non trivial tail at infinity: J k Φ k, c k,y k γ, J k Φ k, c k, y k p for y this is a theorem for type II blow up in the radial case, 5. 7 This is not an eigenvalue, since neither Q nor ΛQ decay sufficiently fast at infinity. In particular, ΛQ L. 8 a simple way of seeing this is to remark that γ given by.4 is complex valued.

13 3 Note in passing that the positivity of is equivalent to ΛQ > and implies with L Q = the factorization L ± = A ±A ±, A = y y logλq, A = y y logq.57 which simplifies the resolution of Lu = f in the radial sector. ii Tail dynamics. We now implement the approach developed in 38, 3, 4 and claim that Φ k,± k correspond to unstable directions which can be excited in a universal way to produce the type II blow up solutions. To see this, let us look for a slowly modulated solution to.53 of the form vs, y = Q bs,as y with Q b,a = Qy k= b = b,..., b L, a = a,..., a L.58 b k Φ k, y L k= where we expect the a priori bounds and the improved decay estimates L ± a k Φ k, y S k,± y, a, b.59 k= b k b k, a k b k α,.6 S k, y b k y k γ, S k, y b k α y k p, so that S k is in some sense homogeneous of degree k in b but decays better than Φ k. The key point is that this improved decay is possible for a specific regime of the universal dynamical system driving the modes b k i L a k k L : this is the tail computation. In particular, the improved decay.58 for the a k parameters is in agreement with the worst decay.56 of Φ k,, and we bootstrap a regime where the influence of the a terms, i.e., of the phase, is of lower order. Let us now illustrate the tail dynamics. We inject the decomposition.59 into.53 and choose the law, i.e. ODE, for a k s, b k s which cancels the leading order term at infinity: Ob. We cannot adjust the law of b for the first term and obtain from.53 the equation b LΦ, ΛQ =, Φ, as y. c, y γ Oa. We similarly cannot adjust the law of a for the first term and obtain from.53 the equation a LΦ, =, Φ Q, as y. c, y p Ob, b. We consider the imaginary part and obtain b s Φ, b ΛΦ, b LΦ, LS, = b NL Φ,, Q lot where NL T, Q corresponds to nonlinear interaction terms, while the lower order terms come from neglecting some additional contributions which arise after the use

14 4 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI of the a priori bounds.6. When considering the far away tail.56, we have for y large, ΛΦ, γ Φ, = α Φ,, LΦ, = Φ, p and thus b s Φ, b ΛΦ, b LΦ, b s α b b Φ,, and hence the leading order growth for y large is cancelled by the choice b s α b b =. We then solve for LS, = b ΛΦ, αφ, NLΦ,, Q and check that the far away tail is improved: S, b y γ for y. Ob a, a. We now consider the real part and obtain to leading order a s Φ, a b ΛΦ, a LΦ, LS, = a b NL Φ,, Q lot. When considering the far away tail.56, we have for y large, ΛΦ, p p Φ, = Φ,, LΦ, = Φ, and thus a s Φ, b a ΛΦ, a LΦ, a s b a a Φ,, and hence the leading order growth for y large is cancelled by the choice We then solve for a s b a a =. LS, = a b ΛΦ, Φ, NLΦ,, Q and check that the far away tail is improved: S, a b y p for y. Ob k, b k. At the k-th iteration, we obtain an elliptic equation of the form: b k s Φ k, b b k ΛΦ k, b k LΦk, LS k, = b k NL k Φ,,..., Φ k,, Q lot. We have from.56 for tails: and therefore: ΛΦ k, k αφ k, b k s Φ k, b b k ΛΦ k, b k LΦk b k s k αb b k b k Φ k,. The cancellation of the leading order growth occurs for b k s k αb b k b k =. We then solve for the remaining S k, term and check that S k, b k y k γ for y large. Ob a k, a k. We obtain along similar lines: a k s Φ k, b a k ΛΦ k, a k LΦk, LS k, = b k a NL k Φ,,..., Φ k,, Q lot.

15 5 We have from.56 for tails: and therefore: ΛΦ k, kφ k, a k s Φ k, b a k ΛΦ k, a k LΦk a k s kb a k a k Φ k,. The cancellation of the leading order growth occurs for a k s kb a k a k =. We then solve for the remaining S k, term and check that S k, b k p for y large. Note that we neglected here further nonlinear terms in a since a will turn out to be lower order in the regime 9.6. y k iii The universal system of ODE s. The above approach leads to the universal system of ODE s which we stop after the -th iterate: b k s k α b b k b k =, k, b L, a k s kb a k a k =, k L, a L, λs λ = b.6, γ s = a, ds dt =. λ Unlike the critical case, there is no further logarithmic correction to take into account. The system.6 can be solved in a closed form, and a set of explicit solutions is given by { b e j s = c j j L s j s =, j L, s > s >,.6 where a e j l l α, c = c j = αl j l α c j, j l, c j =, j l, l N, l > α. In the original time variable t, this produces λt vanishing in finite blow up time T with: λt T t l α. Moreover, the linearized flow of.6 near this solution is explicit and displays l unstable directions in b and k l unstable directions in a, see Lemma 3.7 and Lemma 3.9. Note that l > α > and hence type II is always unstable. iv. Decomposition of the flow and modulation equations. Let then the approximate solution Q b,a be given by.59, which by construction generates an approximate solution to the renomalized flow.53: Ψ = s Q b,a i Q b,a b ΛQ b,a ia Q b,a Q b,a Q b,a p = Modt Ob where the modulation equation term is roughly of the form: Modt = k= b k s k αb b k b k Φ k, L k= a k s kb a k a k Φ k,. 9 for example ab b α but a b α. On the contrary, the energy critical case treated in 39, 4 formally corresponds to α =, and hence l = is admissible and generates a stable type II regime.

16 6 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI We localize Q b,a in the zone y B to avoid the irrelevant growing tails for y b. We then pick initial data of the form u y = Q b,a y ε y, ε y in some suitable sense and with b, a chosen to be close to the date for the exact solution.6. By a standard modulation argument, we introduce a dynamically modulated decomposition of the flow ut, r = Q bt,at ε = Q bt,at t, t, r λt r λt e iγt wt, r e iγt.63 where the L modulation parameters bt, λt, at, γt are chosen in order to manufacture the orthogonality conditions: εt, L k Φ M, =, k, εt, L k Φ M, =, k L..64 Here Φ M,± y are some fixed directions depending on a large constant M, generating an approximation of the kernel of the powers of L, see section 4.. This orthogonal decomposition, which for each fixed time t directly follows from the implicit function theorem, now allows us to compute the modulation equations governing the parameters bt, λt, at, γt. The Q b,a construction produces the expected modulation equations : L λ s λ b L γ s a b i s i αb b i b i a i s ib a i a i i= ε loc b 3 where ε loc measures a spatially localized norm of the radiation ε. v. The mixed energy/morawetz estimate. According to.65, we need to show now that local norms of ε are under control and do not disturb the dynamical system.6. This is achieved via a high order mixed energy/morawetz type estimates, which in particular provide control of the high order Sobolev norms adapted to the linear flow and based on the powers of the linear operator L. In turn, the orthogonality conditions.64 are sharp enough to ensure the Hardy type coercivity of the iterated matrix operator: E s = J L L k ε, L k ε s ε i= ε y s where s is given by.43. Here the factorization.57 will help simplify the argument. As stated above we can dynamically control this norm thanks to an energy estimate seen on the linearized equation in original variables, i.e., by working with w in.63 and not ε. This strategy was initiated in 44, 38, 3, 4. The outcome is an estimate of the form { } d Es b M ds λ s bδd,p s c λ s, δd, p >.66 s c where the right hand side is controlled by the size of the error in the construction of the approximate blow up profile. Here M corresponds to an additional Morawetz type term needed to control L terms sharply localized on the soliton core. A see Lemma

17 remarkable algebraic fact is that the corresponding virial type quadratic form is coercive thanks to the fact that L > > in Ḣ, see.4. Hence the estimate.66 belongs to the class of mixed energy/morawetz estimates introduced in 38, which have been particularly efficient in blow up settings, see in particular, and which completely avoid the use of spectral tools. We integrate.66 in time using the smallness b M E s to estimate in the regime b b e given by.6: s ε ε y s E s b δd,p, δd, p >,.67 which is good enough to control local norms in ε and close the modulation equations.65. vi. Control of the nonlinear term and low Sobolev norms. The control of high Sobolev norms alone is however not enough to control the nonlinear term and we need a low Sobolev estimate. The bounds following from the conservation laws would be too weak at this point, and we will need the fundamental observation that s c = d p < d s, 7 while Ḣ d almost embeds into L, and hence the space H σ Ḣs, s c < σ < d < s is an algebra. To close the nonlinear term, it therefore suffices to close an estimate for the low Sobolev norm σ ε L for some s c < σ < d. Let us insist that it is essential that this norm is above scaling, any norm of ε below scaling blows up. We then exhibit an energetic Lyapunov functional with the dynamical estimate: { d σ ε } L b ds λ σ sc λ σ sc b δd,p σ ε L b σ scδd,p which upon integration in time yields a bound σ ε L b σ scδd,p, δd, p > which is enough to control of the nonlinear term. vii. Construction of the C manifold. The above scheme designs a bootstrap regime which traps blow up solutions with speed.3. According to Lemma 3.7, Lemma 3.9, such a regime displays k l l > unstable modes and one therefore needs to build the associated stable manifold. We do this in a classical way using a Brouwer fixed point type argument as in 6, and the proof of Theorem. follows. viii. Relation with the decomposition.5. Let us conclude this introduction by making a link between the above construction and the decomposition of previously known type II blow up solutions for the heat equation.5. For this, let us consider the two changes of variables: ut, x = λ p vs, ye iγt = µ p V τ, ze iγt.68

18 8 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI with { y = z = x λt, x µt, ds dt = λ, dτ λt = T t l α µ = T t dt =, µ where the second decomposition corresponds to the self-similar variables.5 in the approach of Herrero-Velasquez: V τ, z = Rz e λ jτ ψ j z lot.69 where λ j is the j-th, j = jl, strictly positive eigenvalue with eigenvector ψ j of the linearized operator H R : H R = iλ pcp r. We now show how our construction and estimates for the renormalized v imply the decomposition.69 in the far field in renormalized variables. We compute b λλ t T t l α and thus z = λ µ y = T t l α z b y. We now estimate the leading order term in the decomposition.59 in the zone z i.e. y B = b by neglecting: the a terms which are lower order, see 4.3, 6.; the S terms which decay better and hence are lower order for z ; the b k terms for k l which are the stable modes and also turn out to be lower order, see 6.9. Using b k b e k s k bk this gives the far away development: l l Q b,a Q b k Φ k, ylot = R c k b k i k y k γ lot = Ryb γ k= k= k= l c k i k z k γ lot, and hence using.68 and the fact that R is homogeneous: µ l p V τ, z = Ry b λ γ c k i k z k lot z = Rzb γ µ l p c k i k z lot. k γ λ We now compute b γ µ λ p T t γ l α T t p l α = e λ lτ, λ l = l α, and obtain the leading order decomposition in the far away zone: with V τ, z = Rz e λ lτ ψ l z lot ψ l z = l c k i k z k γ, λ l = l α. k= k= k=

19 Now a simple computation, see Appendix E, reveals that λ l, ψ l is an eignevalueeigenvector pair for the linearized operator close to the singular self similar solution R. The exact same computation can be done for the heat equation, and the conclusion is the following: the singular decomposition.5 in self similar variables is exactly the long range expansion y b of the regular decomposition.63 in the regime.3. This paper is organized as follows. In section, we collect the main linear properties on the linearized matrix operator L and its iterates. In section 3, we construct the approximate self-similar solutions Q b,a and obtain sharp estimates on the error term Ψ. We also exhibit an explicit solution to the dynamical system.6 and show that it possesses l k l directions of instability. In section 4, we set up the bootstrap argument, Proposition 4.3. In section 5, we construct the main Lyapunov functionals which rely on a mixed energy/morawetz computation. In section 6 we close the bootstrap bounds and build the C manifold of data satisfying the conclusions of Theorem.. Acknowledgments. Part of this work was completed while P.R. was visiting the MIT, Boston, and the Institut du Non Lineaire, Nice, and he would like to thank both institutions for their kind hospitality. P.R and F.M were supported by the senior ERC grant BLOWDISOL. I.R. was supported in part by the NSF grant DMS The linearized Hamiltonian and its iterates We collect in this section the main properties of the linearized Hamiltonian close to Q, which are at the heart of both the construction of the approximate blow up profile and the derivation of coercivity properties required for the high Sobolev energy estimates... The matrix operator. By a standard argument, all smooth radially symmetric solutions to φ φ p =. are dilates of a given normalized ground state profile φr = λ p Qλr, { Q Q p = Q = Let us recall the following Lemma which follows directly from the results in 3, 6: Lemma. Structure of the ground state and positivity of L ±. Let p > p JL, then: i Development of the solitary wave profile for y : there holds k, k y Q = k y R a y γ O where R is given by.9. ii Degeneracy: ΛQ = c y γ O y γg y γgk., a, g >. as y, c..3

20 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI iii Positivity of L ± : for some c p >. iv Positivity of ΛQ: L > > y c p d >.4 4 ΛQ >..5 Proof of Lemma.. The positivity.4 for p > p JL and the associated pointwise lower bound follows from a non trivial Sturm-Liouville oscillation argument, see 6. Now from 3, Thm.5, there holds the asymptotic expansion for p > p JL and y : Qr = c a y p y γ O y γα y γ.6 where γ = d Discr. We recall that α > and from.6: γ γ = Discr > and thus Q = R a y γ O y γg g = min{α, Discr} >..7 The fact that the development.7 propagates to higher derivatives is now a simple consequence of the Q equation., this is left to the reader, and.3 follows. We finally claim that a. Indeed, otherwise from.6: ΛQ = O y γα y γ, and then the bounds d 3 γ = Discr < d 3 γ α = Discr α = 4 d < p imply ΛQ ΛQ y d γ α y d γ dy <..8 By scaling invariance, y ΛQ =. Fix a sufficiently larger and let χ R y be a smooth cut-off function, equal to one for y R. We have ΛQ χ R ΛQ ΛQ y y y R, which, combined with.8, implies χ R ΛQ χ R ΛQ for some strictly positive η. On the other hand, by strict positivity.4 of, χr ΛQ χ R ΛQ χ R ΛQ c C R η y

21 for some positive constant C independent of R, which follows since ΛQ does not vanish identically on any open set. Contradiction... Factorization of L ±. The positivity.4 implies the factorization of L ±. Lemma. Factorization of L ±. Let and the first order operators then V = y logλq, V = y logq.9 A ± u = y u V ± u, A ±u = y d yy d u V ± u, L ± = A ±A ±. Remark.3. The adjoint operators A ± are defined with respect to the Lebesgue measure Auvy d dy = ua vy d dy. y> We collect the following estimate on V ± which follow from.: { V = yλq O as y ΛQ = γ y O as y,. y { 3 V = yq O as y Q = p y O as y,. y { 3 O as Q p y = c p y O. as y. y 4 We also estimate from. with the notations.49: for y, yw j ± = O y j, j Inverting. We rewrite A u = ΛQ y u ΛQ y>, A u = and hence the kernels of A, A are explicit: { A u = on iff u SpanΛQ, A u = on iff u Span Hence with which satisfies the Wronskian relation see 3 for a similar structure. y d ΛQ yy d ΛQ.4 y d ΛQ..5 u = on iff u SpanΛQ, Γ.6 y dx Γ y = ΛQ x d ΛQx.7 Γ ΛQ Γ ΛQ =..8 yd

22 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI We observe the behavior Γ c y d as y, c..9 Moreover, from.3: dx x d ΛQx dx < xd γ where we used from.4 d γ = Discr >. This implies: Γ c y γ as y. The explicit knowledge of the Green s functions allows us to introduce the formal inverse L f = Γ y y fλqx d dx ΛQy in an elementary two step pro- The factorization of allows us to compute L cess 3 : y fγ x d dx.. Lemma.4 Inversion of. Let f be a C radially symmetric function and u = L f be given by., then y y A u = y d fλqx d A u dx, u = ΛQ dx.. ΛQ ΛQ Proof of Lemma.4. We compute from.8 A Γ = Γ ΛQ ΛQ Γ = y d ΛQ. We therefore apply A to. and compute using the cancellation A ΛQ = : A u = y d ΛQ y fλqx d dx.. Hence from.4: y A u u = ΛQ ΛQ dx c uλq. We now estimate at the origin using the formula.,. and the behavior.9: A u y, u y, ΛQ c and thus c u =..4. Inverting L. We rewrite A u = Q y u Q, A u = y d Q yy d Qu.3 and hence the kernels of A, A are explicit: { A u = on iff u SpanQ A u = on iff u Span. y d Q Hence.4 L u = on iff u SpanQ, Γ.5 3 this will avoid tracking cancellations in the formula. induced by the Wronskian relation.8 when estimating the growth of L f.

23 with which satisfies the Wronskian relation We observe the behavior Moreover, from.3: y dx Γ y = Q x d Qx.6 Γ Q Γ Q =..7 yd Γ dx x d Qx 3 c y d as y..8 dx x d 4 p < where we used from.4 d 4 p > d γ >. This implies: c Γ y p as y. The explicit knowledge of the Green s functions allows us to introduce the formal inverse A f = y y d fqx d dx Q and L f = Q A f y Q dx if A f Q dx <, Q.9 y A f Q dx otherwise. Lemma.5 Inversion of L. Let f be a C radially symmetric function and u = L f be given by.9, then L u = f, A u = y y d fqx d dx = A Q f..3 Proof of Lemma.5. From.3,.9: u A u = Q y = A Q f = y y d fqx d dx Q L u = A A u = y d Q y y d QA u = f and. is proved. The definitions.5,.,.9 lead to the formal inverse of L: L L = L Admissible functions. We define a class of admissible functions which display a suitable behavior at infinity: Definition.6 Admissible functions.. Scalar functions: We say a radially symmetric f C R d, R is admissible of degree j, ± R {, } if f and its derivatives admit the bounds: for y, { k, y k y j γ k for j, fy y p j k.3 for j,

24 4 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI. Vector valued functions: We say a radially symmetric C R d, R complex valued function is admissible of degree p, p R R if f and its derivatives admit a bound: for y, k, y k Rfy y p γ k, y k Ify y p p k..33 L naturally acts on the class of admissible functions in the following way: Lemma.7 Action of L, L on admissible functions. Let f be an admissible function of degree p, p N, then: i Λf is admissible of degree p, p. ii J Lf is admissible of degree p, p. iii L Jf is admissible of degree p, p. iv J L f is admissible of degree p, p. Proof of Lemma.7. Proof of i. This is a direct consequence of.33. Proof of ii. Let f be admissible of degree p, p. Then Lf is a smooth radially symmetric function. For y, using.5, the decay.3 and a simple application of the Leibniz rule imply: for y, k y R Lf = k y L If y p p k, k y I Lf = k y Rf y p γ k, and ii follows. Proof of iii. We compute from.3: L L J = L. Let then p, p N, f be admissible of degree p, p and let us show that u = L Jf is admissible of degree p, p. Near the origin, u is bounded from.,.9, and hence from Lu = Jf, u is a smooth radially symmetric function by standard elliptic regularity estimates. Moreover: Ru = Rf, Iu = L If. Inversion of : For y, we use the lower bound from.4 d γ = Discr > to estimate from.: y y A Ru = y d RfΛQx d dx = O ΛQ y d γ x p γd dx = Oy p γ,.34 y A Ru y Ru = ΛQ y ΛQ dx = O γ x p γγ dx = Oy p γ. We conclude from.34,. and then the bound y Ru y p γ, yru y p γ, k y Ru y p γ k, k, y easily follows by induction by taking radial derivatives of the relation Ru = Rf. Inversion of L : Using d 4 > d γ >, p

25 we estimate from.3: A Iu = A f = y d Q y = O y d p We now distinguish cases. If Q Iu = Q A If Q and otherwise from p and.35: y Iu Q A If dx Q This implies from.35,.: y y IfQx d dx x p 4 p d dx A If 5 = Oy p p..35 dx <, then from.9: dx y p y p p, y p y x p dx y p p. y Iu y p p, yiu y p p, and then again a simple induction argument by differentiation of the relation L Iu = If ensures the bound: k y Iu y p p k, k, y. Proof of iv. We compute from.3: J L L =. Let then p, p N, f admissible of degree p, p and let us show that u = J L f is admissible of degree p, p. From.,.9, u is radially symmetric and bounded near the origin, and hence from Lu = Jf, u is a smooth radially symmetric function by standard elliptic regularity estimates. Moreover: Ru = L Rf, Iu = If. Inversion of : For y, we use the lower bound from.4 to estimate from.: A Iu = y d ΛQ d γ > d γ >.36 p y = Oy p p, and then using γ > p again: Iu = ΛQ y A Iu ΛQ dx = O and we easily conclude as above: y IfΛQx d dx = O y d γ x p p γd dx y γ y x p p γ dx = Oy p p k y Iu y p p k, k, y.

26 6 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI Inversion of L : Using.36, we estimate from.3: A Ru = y y d RfQx d dx = O Q y d p y x p γ p d dx = Oy p γ..37 We now distinguish cases. If p γ p <, then from.3: A Rf Q dx = A u Q dx x p γ p dx < and thus from.9: Ru = Q y y p γ. A Rf Q dx y p y x p γ p dx Otherwise, p γ p, but then using α / N from.7: p γ p = p α >..38 Then either A Rf Q dx < in which case: Ru = Q A Rf dx Q y y p y p γ where we used.38 in the last step, or otherwise from.3,.37: y Ru Q A Rf y dx Q y p x p γ p dx y p γ. We then easily conclude as above: k y Ru y p γ k, k, y..6. Generators of the kernel of L i. We now give an explicit example of admissible functions, which will be essential for the analysis. Lemma.8 Generators of the kernel of L i. i Let then J i Φ i is admissible of degree i, i. ii Let the sequence Φ i = L i ΛQ Q, i.39 Ψ i = ΛΦ i J i D i J i Φ i, i, D i = then J i Ψ i is admissible of degree i, i. i α i,.4 Remark.9. Equivalently, let the directions Φ i, = L i Φ,, Φ, = ΛQ, i.4 Φ i, = L i Φ,, Φ, =, i..4 Q

27 A simple computation ensures J i D i J i = and thus with: D i for i = k i i α Ψ i = Ψ i, Ψ i, for i = k, Ψ i, = ΛΦ i, J i D i J i Φ i, = ΛΦ i, i αφ i,.43 Ψ i, = ΛΦ i, J i D i J i Φ i, = ΛΦ i, iφ i,..44 and J i Ψ i, is real valued of degree i,, and J i Ψ i, is imaginary of degree i,. Proof of Lemma.8. Proof of i. Φ is admissible of degree, from.. We now proceed by induction, assume the claim for i and prove for i. By definition, Φ i = L Φ i. For i = k, we have by induction: J i Φ i = J k Φ k = k Φ k is admissible of degree k, k and hence from Lemma.7 iv, J i Φ i = k J L Φ i is admissible of degree i, i. For i = k, we have by induction: J i Φ i = J k Φ k = k JΦ k is admissible of degree k, k and hence from Lemma.7 iii, J i Φ i = k L Φ k = k L JJΦ k is admissible of degree i, i. Proof of ii. We claim a more precise control of J i Φ i for y : k, i, k y J i c,i y i γ Φ i c,i y i p c,i y i γ k c,i y i k..45 p Assume.45, then Ψ i is radially symmetric and satisfies the bound from.4: for y, J i c,i y i γ c Ψ i = Λ D i JΦ i = Λ D i c,i y i O,i y i γ p c,i y i c = O,i y i γ c,i y i. The control of higher derivatives follows similarly, and hence J i Ψ i is admissible of degree i, i. We now prove.45 by induction on i. i = : From., there holds for y : We then invert Φ = ΛQ Q = c, y γ c, y p O y γg, g = min{α, Discr} > O y γ. LΦ = L IΦ = RΦ ΛQ Q 7

28 8 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI From.3: A IΦ = = = y d Q y d Q y y d p ΛQx d Qdx y c x γ O O O c O y γ x γg y c x p cx d γ p O x γ O We now use the lower bounds: d γ p α = d γ = Discr > to conclude: d γ p Discr d γ Discr = > A IΦ = y d p O = cy γ O y g This implies using α > : and hence from.9: IΦ = Q = y c x γ A IΦ dx Q y γ cy d γ A IΦ Q dx = c y p O y γ O x g = c x γ O x γ dx x γ p p O y < x γ p x d dx y g x g dx. O x g dx from our assumption g >. Similarily, using. and since the integral term is the same: y A RΦ = y d Qx d ΛQdx = cy p O ΛQ y g and hence from.: y A RΦ RΦ = ΛQ ΛQ dx = c y γ O y g O = cy p O y g where we used y x p γ O = cy p O y p γ p g = α g > y g dx and g >. The bound.45 for i = now easily follows by differentation.

29 9 i i We invert LΦ i = L IΦ i = RΦ i RΦ i IΦ i case i = k, k. By induction, J i Φ i = k JΦ i satisfies.45. Hence: L IΦ i c =,i y i p O y i p RΦ i c,i y i γ O y i γ From.3 and using d 4 p > d γ > : A IΦ i = = = = y y d Q y d Q y d p y d p = cy i p Since i >, y IΦ i = Q = cy i p Similarily, from.: A RΦ i = = = = RΦ i x d Qdx O y c xi x p c O y γ O O A IΦ Q A IΦ i dy = Q y d ΛQ y d ΛQ O y y p O y γ cy id y. O y 4 p dy = and 4 thus: y p. RΦ i x d ΛQdx O c y d γ O y y d γ O = cy i γ O y c xi γ x γ x O cx id 4 p O O y α O y cy id γ y, O y y x α O x d dx x dx y O cx i O x dx x x d dx cx id γ O O y x dx 4 one easily checks by induction, starting from. with a, that the leading order terms in.45 do not vanish, i.e., c,i, c,i.

30 3 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI and thus: y A RΦ i RΦ i = ΛQ ΛQ dy = = cy i γ O y. y γ O y y O cx i O x dx The bound.45 for i now easily follows by differentiation in y. case i = k, k. By induction, J i Φ i = k Φ i satisfies.45. Hence: L IΦ i = RΦ i c,i y i γ O y i γ c,i y i p O y i p From.3and using d γ p > d γ > : A IΦ i = = = = y y d Q y d Q y d p y d p RΦ i x d Qdx O y c xi γ x p c O y γ O y = cy i γ O O O y γ cy id γ. x y p x d dx cx id γ p O y O x dx If i γ p <, then IΦ i = Q y A IΦ i dx = Q Q = cy i γ p p A IΦ i Q dy < and thus: O y y cx i γ p O = y i γ O y x dx. If i γ Hence p A IΦ i Q, then i γ dy = and: p = i α > from.7. y IΦ i = Q A IΦ i Q = cy i γ p p y dx = Q O y cx i γ p = y i γ O O x dx y.

31 Similarily: A RΦ i = = = = y y d ΛQ y d O ΛQ RΦ i x d ΛQdx c y d γ O y α y d γ O = cy i p O y c xi x γ p O y γ cy id y and thus: y A RΦ i RΦ i = ΛQ ΛQ dy = y γ O y = cy i p O, O y. y O y x x d dx cx id γ p p γ O y 3 O x dx cx iγ p O x dx The bound.45 for i now easily follows by differentiation in y. 3. Construction of the approximate profile This section is devoted to the construction of the approximate blow up profile Q b,a and the study of the associated dynamical system for the parameters b = b,..., b L and a = a,....a L. 3.. Slowly modulated blow up profiles and growing tails. We introduce a simple notion of a homogeneous admissible function. Definition 3. Homogeneous functions. Given parameters b = b m k L, a = a n n L, we say a function Sb, a, y is homogeneous of degree p, p, j, ± N N N if it is a finite linear combination of monomials with m= km k = p, Π k= bm k k L k= ΠL l= an l l f ± kn k = p, m k, n k N with f ± homogeneous of degree j, ± in the sense of Definition.6. We set degs := p, p, j, ±. We are now in position to construct a slowly modulated blow up profile as a deformation of the solitary wave. Proposition 3. Construction of the approximate profile. Let a large integer α = γ, 3. p

32 3 F. MERLE, P. RAPHAËL, AND I. RODNIANSKI and L be given by.4. Let M > be a large enough universal constant, then there exists a small enough universal constant b M, > such that the following holds true. Let two C maps b = b j j L : s, s b, b, with a priori bounds on s, s : { < b < b, b j b j, j a j b jα for j L. Then there exist homogeneous profiles { Sj,± = S j,± b, a, y, j L ± such that with ζ b,a y = j= S,± = a = a j j L : s, s b, b L 3. Q bs,as y = Qy ζ bs,as y 3.3 b j Φ j, y L j= L ± a j Φ j, y S j,± b, a, y, 3.4 with Φ j,± defined in.43,.44, generates an approximate solution to the renormalized flow, see.53: s Q b,a J Q b,a fq b,a b ΛQ b,a Ja Q b,a = Ψ Modt 3.5 with the following properties: i Modulation equations: M odt = 3.6 L L S m, S m, b j s j αb b j b j Φ j, b j b j j= L j= a j s jb a j a j Φ j, where we used the convention { bj = for j a j = for j L and m=j m=j j= S m, a j L m=j m=j S m, a j { S, = S, = S j, = for j L 3. ii Estimate on the profile: S j,± is a finite 5 linear combination of terms S j,±,s j,± with { degs j, = k, k, j,, k k = j, degs j, = k 3.7, k, j,, k k = j, k. { degs j, = k, k, j,, k k = j, k degs j, = k 3.8, k, j,, k k = j, k. and S k j,± b m =, j m L ±, k S k j,± a m =, j m L ±, k 5 the total number of terms is bounded by Cp, L <., 3.9