Soliton Model

On long time dynamic and singularity formation of NLS MASSACHTS ITTUTE by OF TECHNOLOGY Chenjie Fan AUG 0 12017

B.S., Peking University (2012) LIBRARIES Submitted to the Department of Mathematics ARCHIVES in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Signature redacted Author ...... Department of Mathematics May 3rd, 2017

Certified by. Signature redacted ... Gigliola Staffilani LAbby Rockefeller Mauze Professor Thesis Supervisor

Accepted by... Signature redacted ...... William Minicozzi Chairman, Department Committee on Graduate Theses 2 On long time dynamic and singularity formation of NLS by Chenjie Fan

Submitted to the Department of Mathematics on May 3rd, 2017, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract In this thesis, we investigate the long time behavior of focusing mass critical nonlinear Schr6dinger equation (NLS). We will focus on the singularity formation and long time asymptotics. To be specific, there are two parts in the thesis. In the first part, we give a construction of log-log blow up solutions which blow up at m prescribed points simultaneously. In the second part, we show weak convergence to ground state for certain radial blow up solutions to NLS at well chosen time sequence. We also include a lecture note on concentration compactness. Concentration compactness is one of the main tool we use in the second part of the thesis.

Thesis Supervisor: Gigliola Staffilani Title: Abby Rockefeller Mauze Professor

3 I Acknowledgments

I am very fortunate to work with my advisor, Gigliola Staffilani. I am very grateful to her consistent support and encouragement. I want to take this chance to thank her generous sharing of mathematical insights, her patience and her kindness. I have learned a lot from my advisor, not only disperisve PDE, but also how to become a better person.

I also want to thank Larry Guth and David Jerison for being in my thesis committee. I definitely benefit a lot from discussion with them during my five years in MIT.

The material in the thesis benefits a lot from discussion and comments of Patrick

Gerard, Carlos Kenig, Yvan Martel , Pierre RaphaBd, Svetlana Roudenko, Vedran

Sohinger and Boyu Zhang. I want to take this chance to express my thanks.

I have the chance to collaborate with Alice Guionnet, Peter Kleinhenz, Yuqi Song, Andi Wang, Hong Wang, Bobby Wilson, and also with my advisor. That's very nice experience, thank you all.

During my graduate life, I have the chance to discuss mathematics with various mathematicians, and their generous sharing of mathematical ideas and insights are really appreciated. I want to show my thanks here.

I was also very fortunate to spend one semester in MSRI, for the program: New

Challenges in PDE: Deterministic Dynamics and Randomness in High and Infinite

Dimensional Systems, in fall 2015-2016. It is a very precious experience for me. The program not only improved my understanding of my own field, but also helped me see the world in different fields, broadening my view of math.

I am very proud and fortunate to be a graduate student in MIT math department.

The MIT community is known to be nice and supportive, and help me go through graduate life here, which is known to be not so easy. MIT is a wonderful place for me to do research and study.

I want to thank my friends here, though it is impossible to list them all, I want to name a few: Rui Chen, Qiang Guang, Hai-Hao Lu, Xin Sun, Teng Fei, Guo-Zhen

5 Wang, Hong Wang, Wen-Zhe Wei, Wei-Jun Xu, Ben Yang, Yi Zeng, Rui-Xun Zhang, Rui-Xiang Zhang, Xu-Wen Zhu. Thanks for the friendship. And, to all my friends, I want to say thank you.

I want to in particular thank Rui Chen, Hong Wang, Wei-Jun Xu, and Ben Yang.

There were very hard moments in my graduate life, and their support help me walk through that.

During the preparation of the material, I was partially supported by NSF Grant DMS 1069225, DMS 1362509 and DMS 1462401.

Last but not the least, I want to thank my parents, for their unconditional and endless love.

6 Contents

1 Introduction 11 1.1 O verview ...... 11

1.2 Log Log blow up solutions with m blow up points ...... 19

1.3 Weak convergence to ground state ...... 22

1.4 Structure of the Thesis ...... 26

2 On m points log-log blow up solutions 27

2.1 Introduction ...... 27

2.1.1 N otation ...... 30

2.1.2 A quick review of Merle and Rapha81's work and heuristics for

the localization of log-log blow up ...... 31

2.1.3 Strategy and structure of the paper ...... 42

2.2 Description of the initial data and dynamic/modification of system:

one soliton m odel ...... 43

2.2.1 Description of the initial data ...... 43

2.2.2 Modification of the system ...... 44

2.2.3 Description of the dynamic ...... 45

2.3 Description of the initial data and dynamic: multi-soliton model .. . 46

2.3.1 Description of the initial data ...... 46

2.3.2 Modification of the system ...... 48

2.3.3 Description of the dynamic ...... 51

2.3.4 Further remarks on Lemma 2.3.2 ...... 54 2.4 Proof of Lemma 2.2.4: One Soliton Model ...... 55

7 2.4.1 Setting up ...... 55

2.4.2 An overview of the proof ...... 56

2.4.3 Rough control ...... 58

2.4.4 Propagation of regularity ...... 63

2.5 Proof of Lemma 2.3.2: Multi Solitons Model ...... 72

2.5.1 Outline of the Proof ...... 73

2.5.2 Recovering the Lyapounouv functional under bootstrap hypothesis ...... 80

2.5.3 Propagation of regularity under bootstrap hypothesis .... . 84

2.5.4 Proof of bootstrap estimate except (2.3.57), (2.3.54),(2.3.55) . 88

2.6 Proof of Main Theorem ...... 92

2.6.1 Preparation of data ...... 94

2.6.2 Log-log blow up and almost sharp blow up dynamic ...... 96

2.6.3 A quick discussion of blow up at the same time ...... 97

2.6.4 Prescription of blow up points ...... 99

3 On weak convergence to the ground state 105

3.1 Introduction ...... 105

3.2 P relim inary ...... 107

3.2.1 Local well posedness (LWP) and stability ...... 107

3.2.2 Scattering below the mass of the ground state ...... 110

3.2.3 Concentration compactness ...... 110

3.2.4 Variational characterization of the ground state ...... 114

3.3 The dynamic of non-positive energy solution ...... 115

3.4 An overview for the proof for Theorem 1.3.1, Theoreml.3.2 ...... 117

3.4.1 Step 1: First extraction of profile ...... 117

3.4.2 Step 2: Second extraction of profile ...... 117

3.4.3 Step 3: Fast cascade case ...... 118

3.4.4 Step 4: Quasisoliton case ...... 119

3.4.5 Step 5: Approximation argument and conclusion of the proof . 120

8 3.5 Proof of Corollary 3.3.3 ...... 120 3.6 Proof of Lemma 3.4.1 ...... 122

3.7 Proof for Subsection 3.4.4 ...... 125

3.7.1 A quick review of Dodson's work [191 ...... 125 3.7.2 Proof of Lemma 3.4.11 ...... 131 3.7.3 Proof of Lemma 3.4.9 ...... 132 3.7.4 Proof of Lemma 3.4.8 ...... 133 3.7.5 Proof of Lemma 3.7.4 ...... 136 3.8 Proof of Theorem 1.3.1 ...... 137 3.9 Proof of Theorem 1.3.4 ...... 138

A A few technical lemma-ta 141

A.1 The local well posedness of the modified system ...... 141 A.2 Proof of Lemma 2.6.7 ...... 142 A.3 Proof of Lemma 2.6.9 ...... 143

B Lecture notes on concentration compactness 147

B .1 Introduction ...... 147

B .2 N otation ...... 148

B.3 A preliminary model ...... 149

B.4 A working example ...... 154 B.5 One more working example in dispersive PDE ...... 157 B.5.1 Basic setting of profile decomposition in dispersive PDE 157 B.5.2 On defocusing energy critical NLS, (B.5.1) ...... 164

9 10 Chapter 1

Introduction

1.1 Overview

Our thesis is about the long time dynamic of nonlinear Schr6dinger equation, which is a typical example of nonlinear dispersive PDE. Nonlinear dispersive PDEs have been an active research field in recent years. While it is impossible to give a full survey of the field here, we want to highlight certain developments and research directions in recent years, which are closely related to the topic of our thesis. To be more specific ,we mainly restrict ourselves in mass critical nonlinear Schrddinger equation, (1.1.1). The main results of our thesis are also about (focusing) mass critical nonlinear Schr6dinger equation.

Iut+ Au = pu 14/du, u(O) = uo. (1.1.1)

y = -1.

Here d is the dimension, and u is a complex valued function on I x Rd, where I is a time interval. The equation is called focusing when p = -1 and defocusing when p = 1.

We say equation (1.1.1) is a dispersive equation because the linear problem asso-

11 ciated to (1.1.1),

iut+ Au = 0,

U(0) = no.

Is dispersive.

Let etA be the propagator of (1.1.2), one has the well known dispersive estimate,

e uOIILoo -td/2luolILl- (1.1-3)

On the other hand, unlike the heat equation, (1.1.2) is time reversible and it also conserves mass, i.e. it has the following conserved quantity

/ Iu(t, x)1 2dx. (1.1.4)

Based on (1.1.3), one is able to derive the Strichartz estimate

|ie uAOILq([,oo],L'(Rd)) q,r luoII 2, (1.1.5) where 2+ =4,2

One may refer to [711, [401 for more details.

Since the local well posedness (LWP) theory of the nonlinear Schrddinger equation

(1.1.1) is based on the perturbation around linear solutions, Strichartz estimates

(1.1.5) play a fundamental role in the LWP theory for (1.1.1).

The LWP theory for (1.1.1), for t t1, is nowadays standard. It is known that any initial data no in L2 (Rd), gives rise to a unique solution u C C([0, T]; L 2 ), for some time T and the solution depends continuously on the initial data, (9}, see also earlier work [341. Note, when the initial data uo is in Hs, s > 0, one similarly obtain a unique solution u C C([0, T]; H'), for some time T and the solution depends continuously on the initial data uo.

Before we carry on to the discussion about the long time theory of mass critical

Schrbdinger equation (1.1.1), we present the basic conservation laws and symmetries

12 of the equation.

Problem (1.1.1) has three conservation laws:

* Mass:

M(u(t, X)) := Ju(t, x)I 2 dx = M(uo), (1.1.6)

* Energy:

E(u(t, x)) := vu(t, 2 dx +)+ p2 1 u(t, )I2+ dx = E(uo), (1.1.7)

e Momentum:

P(u(t, x)) := Vu(t, x)u(t, x))dx = P(uo), (1.1.8)

and the following symmetry:

1. Space-time translation: If u(t, x) solves (1.1.1), then Vto E R, xo E R, we have

u(t - to, x - xo) solves (1.1.1).

2. Phase transformation: If u solves (1.1.1), then V0O C R, we have ez0ou solves (1.1.1).

3. Galilean transformation: If u(t, x) solves (1.1.1), then V,3 E Rd, we have u(t, x - 3t)ei-(x-2) solves (1.1.1).

4. Scaling: If u(t, x) solves (1.1.1), then VA E R+, we have ux(t, x) := 2u(, A) solves (1.1.1).

5. Pseudo-conformal transformation: If u(t, x) solves (1.1.1), then -(, )e t, solves (1.1.1).

We point out that the sign of p plays a significant role in the study of (1.1.1).

When p = 1, the energy, (1.1.7), of equation (1.1.1) is coercive, and the problem is called defocusing. When p = -1, which would always be the case in our results, the

13 energy is not coercive anymore and can be negative for certain initial data, and the problem is called focusing.

The problem (1.1.1) is called mass critical since the conservation law mass is invariant under scaling symmetry.

As aforementioned, problem (1.1.1) is local wellposed for uo E H', s > 0. We emphasize here, the LWP theory has a fundamental difference between s = 0 and s > 0. When the initial data is in Hs, s > 0, the time T during which one can locally solve (1.1.1) depends only on the H' norm of the initial data, IIuoIIHs. When the initial data is in the critical space L2 , the local existence time T depends on the structure of the initial data, not only on I|uoj|Hs. One then immediately has two conclusions.

1. For the defocusing mass critical Nonlinear Schr6dinger equation, if the initial

data uo is in H1(Rd), then the associated solution u is global, thanks to the

conservation of energy and the H1 LWP theory.

2. Though, the mass is conserved for both focusing and defocusing mass critical

nonlinear Schr6dinger equations, this is not enough to ensure the solution flow be global.

Thus, there are a few natural questions to ask

1. For defocusing mass critical NLS, are solutions of lower regularity global? In

particular, are all L2 solutions to (1.1.1) global?

2. For focusing mass critical NLS, does there exits finite time blow up solution?

3. For both focusing and defocusing mass critical NLS, what is the long time

asymptotic when the solutions are global, and what is the blow up mechanism

when the solutions blows up in finite time.

The first question is totally understood now. Historically, high-low method, [4], and I-method [11] successfully lower the regularity needed to ensure a global flow to defocusing mass critical NLS. Basically, both methods use the frequency localized

14 version of energy, (1.1.7), to illustrate the solution to defocusing NLS is global for initial data uO E Hso, for certain 0 < so < 1. The question is finally answered by

Dodson's work [181, [21], [171, after seminal work [131 regarding defocusing energy critical NLS. It turns out all solution to defocusing mass critical NLS with L 2 initial data are global, and what is more, in long time, those solution behave like linear solution (scattering), and in certain sense, the nonlinear behavior vanishes in long time. Those scattering type results has been proven for a lot of defocusing dispersive equations. In general, one would expect the solution to a defocusing nonlinear dispersive equation would scatter to some linear solution given its critical norm is bounded through the evolution. Mass critical equations are in some sense very specific in the study of this direction, since its critical norm is automatically bounded by the conservation law. In the study of dispersive system, focusing or defocusing, one would be in particular interested in the so-called type-ii solutions, i.e. solutions with an apriori bound for the critical norm. And, in summary, one would expect a type-ii solutions to defocusing nonlinear dispersive equation to be global and scatter to some linear solution. For defocusing mass critical Schr6dinger equations, this is the case.

Now, let us turn to the discussion of the focusing mass critical nonlinear Schrddinger equation, i.e. (1.1.1) with p = -1. In the rest of this overview section, we fix P = -1 in (1.1.1).

Unlike the defocusing case, where the nonlinear dynamic always vanishes asymptotically, it is very easy to see nonlinear objects in the study of the focusing case. If one looks for solitary wave solution of the form Q(x)eit to (1.1.1), then u(t, x) := Q(x)e" solves (1.1.1) if and only if Q(x) solves the elliptic PDE

-AQ(x) + Q = 1Q1 4/dQ. (1.1.9)

Equation (1.1.9) has a unique radial and positive solution, which is usually called ground state and denoted by Q. Q is smooth and decays exponentially '. The solitary wave Q(x)eit is periodic in time, and cannot have linear solution asymptotics, which

'At least for d small.

15 features decay in time. We remark here the ground state Q plays a significant role in the study of focusing NLS.

Applying pesudo conformal symmetry to Q(x), one derive an explicit finite time blow up solution to (1.1.1),

S(t, x) := Q(-)e- . (1.1.10) t2 t

We remark here the mass of S(t, x) is exactly IIQI12.

The explicit blow up solution S(t, x) tells that there does exist blow up solution for focusing NLS. And, indeed, there are a lot of blow up solutions, given by classical virial identity ([35]). Let u be the solution to (1.1.1), direct computation gives

t J Ix1 2 u2dx = 4&9(J xVuU) = 16E(u) = 16E(uo). (1.1.11)

Formula (1.1.11) immediately implies that if xuo C L 2 and E(uo) < 0, then the solution u must blow up in finite time.

On the other hand, one can use pure variational arguments to show, [75], if u solves (1.1.1) with initial data in H1 and mass strictly below the mass of ground state Q, then u is global. One may compare this fact with the virial identity (1.1.11), and we remark here the energy of ground state, E(Q), is zero.

Thus, for focusing NLS there are basically two questions to ask:

1. What is the long time dynamic for solutions with mass below ground state, and

can one lower the regularity of results in [75] to the critical space 2 L2 ?

2. What is the blow up dynamic and no-scattering 3 dynamic?

The first question is now totally understood, it is shown [20] that all L 2 solutions to (1.1.1) with mass strictly below ground state are global in time and will scatter to some linear solution in long time. This result, can be understood as a positive

2We remark here in the critical space L2 , the energy is not defined. 3We would not be too specific here about the meaning of no-scattering, basically one can think about global solutions which are not asymptotic linear.

16 example of the so-called ground state conjecture, proposed in [41], [42], i.e. for a

focusing nonlinear dispersive PDE, there are some minimal nonlinear object playing

certain threshold role, and below the threshold, all solutions are global and scatter

to some linear solutions.

The second question, however, is far from being completely understood. Let us

first focus on the blow up dynamic. Very little can be said in general for blow up

solution. The local theory does suggest for a general blow up solution breaking down

at time T, then necessarily certain mass needs to be concentrated locally in physical space, and if one assume the solution u is in H1, then necessarily IHu(t)IH1 (blow up rate) goes to infinity as t approaches blow up time. The interest lies in a better

characterization of mass concentration phenomena and blow up rate. We will give more specific discussion in Section 1.3.

For blow up dynamics, it seems reasonable to restrict oneself in certain specific regimes and work on examples. As aforementioned, below ground state, [20] indicates no blow up and even the vanishing of nonlinear behavior in long term. Meanwhile, S(t, x) in (1.1.10) gives an example of finite time blow up solution with mass exactly as the mass of the groundstate Q. It is indeed the example and S(t, x) sometimes are called minimal mass blow up solution. It can be shown, [54], all H 1 finite time blow up solution to (1.1.1) with mass exactly as |IQII2, must be S(t,x), modulo natural symmetry of the equation. The next natural regime is when the mass of solution u is just above the ground state

IIQI12 < |Iu(0)11 2 < IIQIF2+ a, here a is some universal small number . (1.1.12)

It is in this regime, (1.1.12), one has the well-studied so-called log-log blow up solutions. Log-log blow up solutions are numerically observed in [481, and first mathematically constructed by [64], and systematically studied in Merle and Rapha8l's series of work [60], [551 , [59], [56], [661, [58]. We will discuss about it in more detail in Section

1.2. But in short, for any non-positive energy solution to (1.1.1) with mass restriction

(1.1.12), Merle and Rapha8l's results imply such solution must blow up in finite time

17 according to log-log law and one should understood the blow up dynamic is almost totally understood in this regime. It is also of interest to construct other examples of blow up solutions, and the first part of our thesis are also in this direction, see more specific introduction in Section 1.2.

Finally, we discuss about long time behavior of general solution to (1.1.1). There is, of course, the possibility that the solution behaves as a linear solution (radiation term) in long time. In some sense, all other solution can be regarded as finite time blow up solution , thanks to the pesudo conformal symmetries '. We already know certain mass must concentrate locally for a solution to blow up. One actually expect more, i.e. the way that mass locally concentrate would exhibit certain universal structure. This is usually called soliton resolution conjecture in the literature. The conjecture suggests for a type-ii solution to a nonlinear dispersive equation, that it should be decoupled into summations of solitary waves with different scales and one single radiation term5 . The current most complete result in this direction is on radial energy critical wave, [24], due to Duyckaerts, Kenig and Merle. Merle and Rapha8l's work, [60], [55] , [59], [56], gives positive answer for the conjecture for focusing mass critical Shcr6dinger equations with mass around ground state and non-positive energy.

We will discuss more details in this aspect in Section 1.3. Our second result can also be understood as an attempt in this direction, though our results are far from soliton resolution at this moment.

Before we conclude this subsection, we point out that the discussion in our overview are only about dispersive equations on Euclidean space. It is also a very active research field the study of dispersive equation on compact manifolds, in particular on tori. A dispersive equation on compact manifolds has a very different dynamic, and the fundamental dispersive estimate (1.1.3) cannot hold because there is not enough space for the solution to disperse. Indeed, for Schrddinger type equation on compact manifolds, Estimate of type, (1.1.3), is in contradiction with mass conservation law.

4This is the case if one works at the regularity of L 2 , pesudo conformal symmetry is a symmetry in L2 , but not in H', Vs > 0. 5 The conjecture is usually about focusing nonlinear dispersive equations, but in some sense, one may also view the scattering type result for type-ii solutions to defocusing equation as positive evidence of the conjecture, in which case, the solutions becomes one single radiation term.

18 We do not address this aspect of dispersive equation in this thesis.

Our thesis includes two parts, the first part gives a construction of log-log blow up solutions which blows up at m prescribed points, the second part proves weak convergence to the ground state for non-scattering radial L2 solution just above the ground state. See following two subsections for more details.

1.2 Log Log blow up solutions with m blow up points

Here we give a more specific introduction for our first part of thesis, and we also state our main result for the first part of thesis.

Let us consider the focusing mass critical nonlinear Schrddinger equation,

ZUt + AU = u14/du,(1.2.1)

u(0) = uo E H'(R d).

We restrict ourselves for H1 initial data. We only consider dimensiond = 1, 2 here.

Recall that the ground state Q is the unique radial positive solution to (1.1.9) and

Q is smooth and decays exponentially. For initial data uo such that IIo1|2 < |1Q112, the associated solution u to (1.2.1) is global and scatter to some linear solution, [201.

Also, all H1 finite time blow up solutions with mass exactly as Q are essentially

S(t, x) as defined in (1.1.10), [541. The next well understood regime is the so-called log-log blow up regime, numerically shown in [64] , first mathematically constructed in [60], and systematically studied by Merle and Raphael in the series of work, [55]

, [59], [56], [66], [58]. Merle and Raphael's work6 indicates that if a solution u to (1.2.1) with mass just above ground state, (1.1.12) and strictly negative energy, then

6 Merle and Rapha8l's work is more general, in particular also includes zero energy solution, we choose to state a (slightly) weaker version to make it more accessible for the readers.

19 the solution must blow up in finite time T with the following asymptotics:

u(t,x)= Q (t,X(t) )e-i-y(t) + E(t, x), SA(t2 A(t) 1 lnfln T- t| R,(t)-y) E R, A(t)- ~ nIIT-t T - t (1.2.2)

lim A(t)IB(t)IHi = 0, t-+T lim x(t) = X". t-+T

We remark that before Merle and Rapha8l's result, it was even unclear whether such solution would blow up in finite time. Virial identity (1.1.11) does not directly work for negative energy solution, since one needs the extra integrability xuO E L2 .

Log-log blow up solutions (1.2.2) and minimal mass blow up solution S(t, x), (1.1.10) are both localized in physical space, and in particular, one can talk about blow up points for those two types of solutions. From the explicit formula (1.1.10), it is natural to say S(t, x) blows up at x = 0 E Rd, whereas from the last formula in

(1.2.2), one may define x., to be the blow up point for log-log blow up solution. This is not always the (clear) case NLS type equation, which, unlike the wave , has the infinite speed of propagation.

Using S(t, x) as a basic building block, Merle, 1521, constructed blow up solutions which blow up simultaneously at k prescribed points, i.e a solution which blows up at finite T, and near blow time has the following asymptotic

k -) ~ x-x( ), A> ,XTX,,-Vi #j. (1.2.3) i=1

Now, we are ready to state our main result for the first part of thesis.

As an analogue of [521, we use log-log blow up solutions as basic building blocks to construct solutions which blow up at (exactly) m points.

To be precise, we show:

Theorem 1.2.1. For d = 1, 2, for each positive integer m, and given any m different points x 1 ,,, ... xm,o in Rd there exists a solution u to (1.2.1) such that u blows up in

20 finite time T, and for t close enough to T,

u(t, x) =Z Q( 'j )ei + B(t. i= A' (t)Ag)

where, for j = 1, ... , m,

1 , and A-n|Tt)|tHI t- T (1.2-5) A 3(t) T -t 3 11-'()IH 0)

i.e 3 can be viewed as an error term. In particular, since the m given points are arbitrary, the solutions do not necessarily have any symmetry restriction.

Remark 1.2.2. The solution we constructed is actually of high regularity, i.e. this is not a low regularity result.

Remark 1.2.3. Essentially the same construction as in the proof of Theorem 1.2.1, works on torus, rd , d -1 2.

Remark 1.2.4. The result of [65] already implies the existence of symmetric solutions which blow up at two points according to log-log law. In fact one first constructs a solution to NLS on the half line/plane H := R+, that blows up at one point according to log-log law and satisfies the Dirichlet condition u = 0 on ORd+, then, by extending this solution symmetrically, one easily derives the solution which blows up at two points in the whole line/plane. Similarly, one can construct solutions that blow up at a even number of points according to log-log law, but they will have a very strong symmetry. See Corollary 1 in [651.

Remark 1.2.5. The work of [52] uses the idea of "integrate from infinity", which means one needs to evolve the data backward. This does not seem to directly work in this setting because of the remainder term E(t, x). In this work, we will evolve the data forward.

Remark 1.2.6. We point out two applications of Theorem 1.2.1. First, it implies the existence of large mass log-log blow up solutions. More general results on this

21 direction have been obtained in [57]. Second, for those who are familiar with standing ring blow up solutions, [67],[68], our construction in the 1D case implies the existence of multiple-standing- ring blow up solutions for quintic NLS on dimension d > 2. To be precise, one can construct a radial solution u to the following Cauchy Problem:

iut + Au = -lul 4 u,

uo E HN2(d) (Rd)

such that u blows up in finite time with log-log blow up rate and near blow up time T, u has the following asymptotic

-I~r - rj(t) (t) +(t, ), (1.2.7) u(t, x) = u(t, r) = A(t)1/2 6 j=1

1 in In T - t( t-+T A ~(t) T t and AjII|(t)IH1 ) 0, (1.2.8) lim rj(t) = rj, > 0. (1.2.9) t-+T

where P is the unique positive L 2 solution which solves (1.1.9) for d = 1.

We will give the proof of Theorem 1.2.1 in Chapter 2, more background and references would also be given there.

1.3 Weak convergence to ground state

While the first part of our thesis is about the construction of certain examples of solutions, the second part of the thesis aims to some classification for solutions to focusing mass critical NLS in certain regime. Unlike our construction in Theorem 1.2.1, which involves actually very smooth solutions, here we want to keep ourselves at the critical regularity, i.e. L2 , for the mass critical equation. Let us consider the Cauchy problem

iut + Au = -Ju| 4 /du)(

u(O, x) = uO E L2 (Rd).

22 We restrict ourselves in dimension d = 1, 2, 3. As aforementioned, all L2 solutions with mass strictly below the mass of the ground state Q are global and will scatter to some linear solution, [20]. Mass concentration is known to happen for finite time blow up solutions, there are a lot of results in this direction, we refer to [611, [7], [631, [38], [12], [73], [74], [4], [45], [3] and references in their works. After Dodson's work, [201, it should not be hard to apply concentration compactness to conclude that for any L2 solution u to (1.3.1) which blows up in finite time T, necessarily a mass of ground state must concentrate as fast as the rate given by scaling. To be more precise, for such u, one would be able to find t, -+ T, and xn C R d, such that

lim JU12 ;> ||Q||1. (1.3.2) n-o I-x.J:!(T-tn) 1/2--

However, as far as we are concerned, it is always of interest to improve the above estimate to at least lim 1u12 > IIQI12. (1.3.3) n-o i-x,,|(T-t,,)1/2

Note (1.3.3) would exclude the possibility of so-called self-similar blow up.

We remark that estimate (1.3.3), for uo E H1 and satisfying (1.1.12), should follow using a rigidity result due to Rapha8l [66], which is highly nontrivial.

Our results describes how mass may concentrate locally. We restrict ourselves to radial solution to (1.3.1) with mass just above the ground state, (1.1.12).

Theorem 1.3.1. Assume u is a radial solution to (1.3.1) which does not scatter forward. Let (T-, T+) be its lifespan, then there exists a time sequence tn - T+(U), and a family of parameters An , Yn such that one has weak convergence

2 Ad/ 2 U (tn, AnX)e-*,n -n Q in L , (1.3.4)

See Definition 3.2.4 for the precise notion of scattering forward.

And, if one further assumes lJull2 = 11Q112, then we can upgrade the above to

Theorem 1.3.2. Assume u is a radial solution to (1.3.1) which does not scatter forward, and ||ull 2 = ||Q12 in Theorem 1.3.1, then there exists a sequence t -+ T*,

23 and a family of parameters A*,n,7 , such that one has strong convergence

An-u(A÷,,x)e-*," -+ Q in L2 . (1.3.5)

Remark 1.3.3. Most of the proof for Theorem 1.3.1 and Theorem 1.3.2, written in this work can be obtainedfor the nonradialcase as well. Indeed, only one step (Lemma

3.4.11) cannot be generalized to the nonradial case. In particular, we do not use

Sobolev embedding or weighted Strichartz estimate for radial solutions. Moreover, our results hold in fact for solutions which are symmetric across any d linearly independent hyperplanes. Nevertheless, the idea in this work is not enough to cover the nonradial case. We will investigate this case in a future work.

We also obtain some partial results for the minimal mass blow up solution to

(1.3.1) at regularity L2 , not necessarily radial.

Theorem 1.3.4. Let u be a general L2 solution to (1.3.1) that blows up at finite time

T and such that ||u(t)11 2 = ||Q112. Then there exist sequences xn,,tn such that

lim 12 > 1Q112. (1.3.6) n+ofx-x,.,<; (T-tn)2/3-

Remark 1.3.5. If one further assumes that the initial data is in H1 and with same mass as Q, then (1.3.6) holds even if one changes the power 2/3 to 1. Indeed, the H1 minimal mass blow up solution is determined in [54] and can be written down explicitly. See also [76].

We do a background exposition to end this subsection. Under assumption (1.1.12), the finite time blow up7 solution to (1.3.1) at regularity H1 has been extensively studied in recent years. Again, we recall the work of [481, [641, [601, [551 , [59], [56], [661, [581 regarding the so-called log-log blow up dynamics. If one assumes the initial data is in H', with nonpostive energy and statisfies assumption (1.1.12), then one can upgrade the sequential convergence in Theorem 1.3.1 to convergence

7 Note that finite time blow up should be understood as one of the long time dynamics rather than a short time dynamics.

24 as t -+ T+, [56]. It is also shown in [561 that for general H' solution to (1.3.1) satisfying (1.1.12), without the sign condition in the energy, Theorem 1.3.1 also holds.

Regarding Theorem 1.3.2, if one assumes the initial data is in H1 and blows up in finite time, then the convergence holds as t - T+ thanks to Merle's complete classification of minimal mass blow up solutions, [541. Moreover, in Theorem 1.3.2, if one assumes u is radial and in H', d = 2,3, and u is global, lJull 2 = lIQl|2, then one can still obtain convergence as t -+ T+, due to [50]. In fact, [50] shows such solution must be a solitary wave. When one assumes d > 4, (our work mainly deals with d = 1, 2,3), if

2 one assumes u is radial and in L , lull 2 = llQII2, global in both sides, then [49] shows such solution must be a solitary wave.

The main purpose of the second part of the thesis is to extend these results,

2 (except the 4 dimensional result [49]), to the lower L regularity. We point out if one only wants to show some sequential weak convergence but does not want to characterize the limit profile, then assumption (1.1.12) or radial assumption may not be necessary, and the method in [25] should be applicable to prove these kinds of results. Indeed, it is pointed out in [25], their methods and strategy should be able to handle general dispersive equations once a suitable profile decomposition is available. However, assumption (1.1.12) and the radial assumption are very important for us to determine that the limit profile is indeed Q.

Results of similar type to Theorem 1.3.1 and Theorem 1.3.2 will also appear naturally when one considers the mass concentration phenomena of finite time blow up solutions to (1.3.1), we refer to [611, [7], [63], [38], [12], [731, [74], [4], [45], [3] and references in their works.

We point out that our work is of course motivated by the recent progress concern- ing the soliton resolution conjecture for the energy critical wave equation, (where more complete and general results are available), see [28], [29], [24], [30], [23], [25], [15],[26], [22], [27] and the references therein.

Roughly speaking, the soliton resolution conjecture suggests that if a generic solution to some dispersive equation stays bounded in a certain critical norm within its lifespan, then the solution should decouple into several solitary waves in different

25 scales and one remainder term. This remainder term should be a single radiation term when the solution is global and a regular term when the solution blows up in finite time. In particular, if one considers a non scattering solution to (1.3.1) which blows up finite time T, with assumption (1.1.12), then one should expect

u(t,x) 1X - x(t)))-/i(t)xeiY(t) + ro(x), as t -+ T. (1.3.7) A(t)-d/2 A(t)

From this perspective, our results indicate that for a radial solution which solves (1.3.1), does not scatter and satisfies (1.1.12), one is able to see the profile Q for a well-chosen time sequence. However, our results are far from the soliton resolution conjecture (1.3.7) (even for radial solutions), since we cannot go to all times t and we cannot exclude the existence of small profiles living at different scales. We will give the proof of Theorem 1.3.1, Theorem 1.3.2, and Theorem 1.3.4 in Chapter 3, more background and reference would also be given there.

1.4 Structure of the Thesis

The material in this thesis are from author's papers, [32], [311. The proof of Theorem 1.2.1 is presented in Chapter 2, and the proof of Theorem 1.3.1, Theorem 1.3.2 and Theorem 1.3.4 is presented in Chapter 3. In the Appendix B, we include a lecture note on concentration compactness.

26 Chapter 2

On m points log-log blow up solutions

This chapter is devoted to the proof of Theorem 1.2.1. Recall the goal is to construct solutions to (1.2.1) which blow up at m prescribed points simultaneously, and behaves like log-log blow up solutions near each blow up points. Log-log blow up solutions are studied in [48], [64], [601, [55] , [59], [561, [66], [58] and a lot of subsequent work.

From now on we fix m E N as the number of blow up points and we fix d = 2, and the case d = 1 would follow similarly.

2.1 Introduction

We first point out, unlike the minimal mass blow up 1 solution S(t, x), the log-log blow up dynamic is actually stable, [66], and indeed it is even stable under H' perturbation, s > 0, [14]. This is actually a favorable feature if one wants to construct multiple blow up points solution. However, unlike minimal mass blow up, which concentrates all the mass near the blow up point, the log-log blow up mechanism is to eject (small but strictly nonzero) mass out of the blow up points, which makes the strategy in

[52] hard to apply.

To construct a solution satisfying Theorem 1.2.1 without relying on certain symmetry property of the initial data, the most intuitive process to follow is that one

'In some sense, all minimal mass blow up cannot be stable.

27 first prepares m log-log blow solutions

u(t,x) = Q( X- (t)eii) + ,j =, M A2 Aj satisfying (1.2.2). Then one shows that the solution to (1.1.1) with initial data

Z7- 1 ui(0) evolves approximately as J:', uj(t), if one assumes that at the initial time all solutions are very close to blowing up, i.e. Aj(0), j = 1, . .. , m is very small, and they are physically separated, i.e. minj 1xI (0) - xj, (0)1 is very large. To achieve this, one needs some mechanism to decouple ul , 2 ... Ur'. Our choice is to require extra smoothness outside the (potential) singular points. Roughly speaking, we find neighborhoods Uj of xjo = xi(0), j = 1, ... , m, and show that the solutions keep very high regularity outside these m neighborhoods. This approach is motivated by the work of Rapha5l and Szeftel, [681. They consider the focusing quintic NLS on Rd and they require their data to be radial and in Hd. By the radial symmetry assumption, the problem can be understood in polar coordinates as a perturbation of the 1D- focusing-quintic NLS. The goal of Rapha8l and Szeftel is to construct solutions that blow up at a sphere (or a ring). The crucial point in their paper is to understand the propagation of singularity/regularity. They show that all the singularities are kept around the sphere where the solution is supposed to blow up, and the solution is kept bounded in H outside the sphere. Note that, thanks to the radial assumption, the authors are using the 1D NLS to model their solutions [68]. See also [67], which indeed is on the same spirit as [681, but in the setting [67], one does not need to work in high regularity.

We show the following theorem.

Theorem 2.1.1 (Propagation of regularity). For any given K1 > 1, K2 > 2, not necessarily integers, if K1 < 2' , then we construct a solution u in HK2 to (1.1.1) that blows up according to the log-log law as (1.2.2) at finite time T and such that

" SUPte(oT)IX(t) - x(0)I < 100'

" u(t, x) is bounded in HK when restricted in Ix - xol > 1/2,

28 Remark 2.1.2. The choice of special numbers !, 1/1000 is (of course) just for con- creteness and simplicity.

Remark 2.1.3. When d=1, and K2 is an integer, and one can prove Theorem 2.1.1 for K1 < K2-1 by sightly modifying the language of [68]. Their method is based on a bootstrap argument and a certain pesudo-energy.. When d=2, Raphael and Szeftel's method does not seem to directly work, and one should be able to use the argument in

[77] to prove Theorem 2.1.1 for K, KK-1 when K2 is an integer. Our proof improves

the previous results in two aspects. We can take K1 < -K and we do not require K 2 to be an integer. Our proof is written more in a harmonic analysis style, relying on the (upside-down) I-method, [111,[691, interpolation and Strichartz estimate.

Remark 2.1.4. When K2 > K = 1, Theorem 2.1.1 is implied by the work of Holmer and Roudenko in [39].

One should understand Theorem 2.1.1 as a proof for the fact that the log-log blow up behavior is local, in the sense that it does not propagate singularity outside the blow up point. This will help us to decouple the m"solitons" , (we sometimes also call them bubbles), in our construction of blow up solution.

Remark 2.1.5. Because of the good localized property of log-log blow up, one can even work on manifolds, see [36] for work in this direction.

Remark 2.1.6. It is not always true in this kind of problems, (if one doesn't put some restriction on the data), that the m "solitons" or m bubbles will be decoupled.

Different bubbles may interact with each other in a strong way. See very recent work

[51] for this direction.

Once we can somehow decouple the m soliton ul, ...u", then we will use some topological argument to construct initial data, and balance those m bubbles and make them blow up at the same time. And, prescription of the blow up points is actually more subtle than making m bubbles blow up at the same time. Fortunately, by taking advantage of the sharp dynamic of log-log blow up, it can still be achieved by certain topological argument. We remark, it is typical that one may rely on

29 soft topological argument rather than pure analysis to prove things like this, see [53),

[651,[161, through one needs to find different topological argument in different settings.

2.1.1 Notation

Throughout this article, a is used to denote a universal small number, 6(a) is a small number depending on a such that limao 6(a) = 0. We use 60, 61,... to denote universal constant (they are usually small, but don't depend on a). We use C to denote a large constant, it usually changes line by line. We also use c, rl and a to denote small constants. For any constant r, we use r+ to mean r 6 where 6 is a small positive constant.

We write A < B when A < CB, for some universal constant C, we write A > B if B < A. We write A ~ B if A < B and B < A. As usual, A <, B means that

A < CUB, where C, is a constant depending on -..

We use A to denote the operator 4 + yV on Hl(Rd). We use the notation

6i := RE, 62 := FjE, i.e. e = E, + ZE2, -- where R is the real part and !a is the imaginary part.

We use usual functional spaces LP, Ci,..., Ck and C , we will also use Sobolev space H', s E R.

If not explicitly pointed out, LP means LP(Rd), so for the other spaces. We also use

L'LP to denote Lq(R; LP(Rd)). When a certain function is only defined on I x Rd, we also use the notation Lq(I; LP(Rd)). Sometimes we use 11f 11to denote 1fJILP.

We use (,) to denote the usual L2 (complex) inner product.

Finally, for a solution u(t, x), we use (T-(u), T+(u)) to denote its lifespan.

30 2.1.2 A quick review of Merle and Rapha*l's work and heuristics for the localization of log-log blow up

Let us quickly review the work of Merle and Rapha6l and highlight the bootstrap structure related to it. At the starting point of their series of work, in [601, they consider a solution u to (1.1.1) with initial data uo E H1 satisfies (1.1.12), with zero momentum and strictly negative energy. They rely on the following variational argument:

Lemma 2.1.7 (Lemma 1 in [60]). For an arbitraryfunction f E H', with energy

E(f) < 0, if also f satisfies (1.1.12), then one can find parameters Ao E R+, xo E Rd, -o E R and e E H', such that

d e A 2of(Aox + xO) = Q + e, (2.1.2) and

EIIH' < 6(a)- (2.1-3)

This lemma implies that for the special solution u(t) to (1.1.1) considered by Merle and Rapha8l, one has the geometric decomposition

A t I A (t)

| e(t)IIH1 4(a). (2.1.5)

Note that one has some freedom in choosing the three parameters A(t), x(t) and -Y(t).

Because of this freedom one can further use the modulation theory to derive the next lemma.

Lemma 2.1.8 (Lemma 2 in [601). Let u(t) be the solution to (3.7.8) with initial data uO, which has zero momentum, strictly negative energy and let uo satisfy (1.1.12).

Then within the lifespan of u(t), there are three unique parameters A(t), x(t), 7(t) and

31 E = El + iE2 E H', such that

1 )(Q+ 6(t)) x - X(t) e_'M, (2.1.6)

Ic(t)IH1 < J(a), (2.1.7) 2 (ei, AQ) = (ei, yQ)= (e2 , A Q) = 0. (2.1.8)

Now the study of (1.1.1) is transferred to the study of the evolution of system {E(t), x(t), A(t), y(t)}. We remark here that the blow up rate is determined by the parameter A(t).

In this setting, Merle and Rapha6l are using the ground state Q to approximate the solution u(t) (up to space translation, scaling and phase transformation). It turns out that sharper results can be obtained by using Qb, a modification of Q, [55], [591. Let's give a brief description of Qb, see Proposition 1 in [59] for details.

Let b E R, i E R+ be small enough, q is fixed. Let us define

2 Rb := - 1-, R- := V/1 -qR, (2.1.9) |b|

and let #b be a smooth cut-off function which equals 1 on lxi < R- and vanishes for jxj > Rb. Then the modified profile Qb := Qbkb, where Qb solves the equation

AQb - Qb + ibAQb + |Qb14/d|Qb 0,

Pb Qbe' 4 > 0 in BRb, Qb(0) E (Q(0) - 6*(r/), Q(0) + E*(r/)), Qb(Rb) = 0. (2.1.10)

Here we also define '1Fb

q/b = -AQb + Qb - ibAQb - QblQb (4/d1 that will be used later.

We now list some useful estimates for Qb:

32 1. Qb is uniformly close to Q in the sense:

2 ______) (, jIyI__n) & 1y1 b-+O |Iee (Qb - Q)c3 + Ie - ( -Qb + i Q)I - 0, (2.1.12)

where f2 0(2) O(r) 10<;r2 1 - dz + 1,>2 2r. (2.1.13)

2. Qb is supported in IyI < .

3. Q has strictly super critical mass:

0 < d Qb112 <

i.e. I|Qb1| 2 _ IIQI| b 2 .

4. Qb is uniformly bound in HS, s E R for all b small enough. (Recall we only consider d=1,2)

IIQbIIHs $ls 1. (2.1.15)

Remark 2.1.9. 0(2) -

Remark 2.1.10. Estimate (2.1.15) is implied by estimate (2.1.12) when s < 3. How-

ever, Merle and RaphaEl consider the C3 rather than the general Ck convergence in (2.1.12) only due to the fact the nonlinearity IQIIQ itself is not smooth enough when d > 3. Thus for d=1,2, since the nonlinearity is algebraic, (2.1.15) holds for all s.

And indeed it is not hard to directly use standard elliptic estimates to prove (2.1.15) for s > 3 once we know this holds for s < 3. This fact is already implicitly used in [68] for d = 1.

Remark 2.1.11. Later in this work many terms will involve Cr but we will be able to fix q, such that Cq is as small as we want.

Remark 2.1.12. Note that since Q decays exponentially and Qb is uniformly close to

Q, it is standard that for given N E N, terms of the form (f, Qi), (VN f, Qb), (yN f, Qb) are controlled by f(2V |2 +|f|2e--I){. This is widely used in [55],[59].

33 With Qb, Merle and Rapha8l modify the lemma 2.1.8 to the following:

Lemma 2.1.13 (Lemma 2 in [551). Let u(t) be the solution to (1.1.1) with initial data uO, which has zero momentum, strictly negative energy and satisfies (1.1.12). Then within the lifespan of the u(t), there are unique parameter {b(t), A(t), x(t), -y(t)} E R x R+ x Rd x R such that

1 (x-(t)\,Yt u(t, x) = 1 (Qi + e) (t)Je- Y, (2.1.16) A(t)2 A (t)) IkIIH1 + IbI 6(c), (2.1.17)

2 2 (ei, IyI Eb) + (e 2 ), IyI eb(t)) = 0, (2.1.18)

(el, yEb) + (62, yeb) = 0, (2.1.19)

2 2 -- (el, A Eh) + (E 2 , A Eb) = 0, (2.1.20)

-(Ei, AEb) + (62, AEb) = 0. (2.1.21)

Here Qb := Eb + iEb.

We note that (2.1.17) a priori assures that the whole analysis is of perturbative nature. Again, the study of (1.1.1) is transferred to the study of the system

{e(t), A(t), -y(t), b(t), x(t)}. To analyze this system it is essential that one considers the slowly varying time variable s rather than the t:

dt _ 1 (2.1.22) ds A2 '

Note that this change of variable changes the lifespan of u (in t variable) to the whole

R, (in s variable), no matter if the original solution u blows up in finite time or not.

Now u satisfying (1.1.1) is equivalent to ei, e2 ,b(s), A(s), x(s),y(s) satisfying the

34 system 2 :

bs8 b + 1 - M_() + bAc =

AAb - R (c) (A + b)AE + i + -VZ + (- + b)(Aci) +'se2 +,(Vi) + ' 2 (2.1.23) ae b Ob) + OsE2 + M+(-E) + bAE2

( + b)A - ~sE + VE+ (- + b)AE2 - yE+ (VE 2) - Rb +Ri(E). (2.1.24)

Here 3 we have i(s) = -s -Y(s), Qb =b + iEb and M = (M+, M_) is the linearized operator near the profile Qb and R 1, R 2 are nonlinear terms. Interested readers may consult (2.31), (2.32) in [59] for more details. Now if one plugs in the four orthogonality condition (2.1.18),(2.1.19),(2.1.20),

(2.1.21), one can obtain a system for the parameters {A, X, ', b}, i.e. four ordinary equations involving {A (s), x(s), j'(s), b(s)}:4

dsd {(E1(t), IyIEb(t) + (E2 (t), Iy|b(t))} 0, (2.1.25)

d d {(El(t), 2 2 A Ob(t))+ (E2(t), A b(t))} = 0, (2.1.27) d {- (ei (t), A2(t) ds + (62(t), AEb(t))} = 0. (2.1.28)

To write down the above ODE system explicitly it requires elementary but involved algebraic computation (see (71), (72),(73), (74) in [66]). A more compact way of writing (2.1.25), (2.1.26), (2.1.27), (2.1.28) is

(bs, ASI s, -Y) = F(bs, As, ,7, Ci, 62), (2.1.29) which justifies the name of ODE system. We call (2.1.25),(2.1.26),(2.1.27), (2.1.28) modulational ODE.

2 Here we slightly abuse notation. For example, x(s) actually means x(t(s)). 3 The evolution of y is of course equivalent to the evolution of ~'. 4Note - = A2 d

35 Now assume that all conclusions in Lemma 2.1.13 hold. Then by applying them into the modulational ODE, one obtains the so-called modulational estimates in the following lemma.

Lemma 2.1.14 (Lemma 5 in [591). Let the assumption of Lemma 2.1.13 hold, and let (2.1.25),(2.1.26),(2.1.27),(2.1.28) hold, then

I + bj + IbsI 5 C IVE1 2 + Je2e-YI) + .1-c" + CA 2IEol, (2.1.30)

2 - IIAQ1121 '1L+A Q)j+j A| o(a)(()Jf Ve2e-2(1) + |eI2n-IY

2 2 +C IVE1 + p-c + CA 1EoI. (2.1.31)

The Fb term will naturally appear in the definition of the linear radiation term (b, which we will discuss later. However, most of the time one only needs to know that

e-(1+Coq)_ _r -(1-Cn)N .b (2.1.32)

We point out that (2.1.32) is (2.17) in [59].

By applying the conservation laws (Energy and Momentum), one can obtain two more crucial estimates in the following lemma.

Lemma 2.1.15 (Lemma 5 in [591). The following two estimates hold:

2 2 2 12(ei, E3) + 2( 2 , 8) C(J IVeI + JeI e-lyl) + p1-Cr + CA IEoI, (2.1.33)

2 I(E2, VE) I < C6o)(J IVe + J Ie2e-IY). (2.1.34)

To derive the blow up rate for the blow up solution u, one performs the following three steps:

1. Based on (2.1.30), explore the fact that b -9, and then transfer the evolution

of A, (which determines the blow up rate) to the evolution of b.

36 2. Obtain a lower bound for b, which gives the upper bound for the blow up rate, [551.

3. Obtain an upper bound for b8, which gives the lower bound for the blow up rate, [591.

The lower bound of b, is given by the following lemma:

Lemma 2.1.16 (Proposition 2 in [591). Let the results of Lemma 2.1.13 hold, let

(2.1.25), (2.1.26), (2.1.27), (2.1.28) hold, let (2.1.30), (2.1.31), (2.1.33),(2.1.34) hold, then one has the estimate

bs > O( IVE1 2 + J E2e-lyl) - CA 2 EO -It . (2.1.35)

The inequality (2.1.35) is called local virial estimate, it is one of the key estimates in the work of Merle and Raphael. The lower bound of b, involves the construction of a certain Lypounouv functional. For this construction one needs to introduce a certain tail term (b or more precisely its cut-off version (b, [56], [59].

Let us first quickly describe (b, one may refer to Lemma 2 in [59] for more details.

Let b, rT, Rb, Rb, #b be as in (2.1.9). Let (b be the unique radial solution to

A(b - (b + ibA(b = XFb, (2.1.36) f IV(b1 2 < .

Here T b is defined in (2.1.11). Note, as mentioned previously, that the Ib term appears naturally when one construct (b, see (2.17) in [59].

What Merle and Rapha8l actually use in [59] is the cut-off version5 of (b, that here we denote by Cb. See their discussion before formula (3.4) in [591. Since later we will use cb in several places, we write here the precise definition. Let A = A(b) = eb, where a is some universal small constant, we let XA be a smooth cut-off function that vanishes outside IxI > A. Then one defines 4b := (bXA. Clearly 4b is supported in

5 The main reason for the introduction of this cut-off is that (b itself is not in L2

37 {ly,< A}. In the rest of this paper, the notation A means A(b) = ETT and note that

A > JI1. The tail term b is introduced to improve the local virial inequality (2.1.35). Es-

sentially, one wants to change the term -171-c' in (2.1.35) to cfb. Let

fi(s) := |lYQbI| + yV ) + (E2, A((,ce)) (EiAim). (2.1.37)

Then (this is highly nontrivial, and is one of the key point in [59])

2 1 2 661 A<|xl<2A {fI(s)}sffiS/I 61( IVEl + / lIl2e- Y1 + C b - CA E0 - 2. (2.1.38)

Here j= E- b. With this, one constructs the Lyapounov functional J, [59] that we write explicitly as

J(s) = (f i [Q12) + 2(Ei, E) + 2(E2, 6) + (I -

where _b 1+ I f1 (b) = |yb + 2QV( j yV). (2.1.40)

One has the following inequality:

2 2 dJ(s) < -Cb (F + fV + I e-' + fA I2 - A2E0 + C Eo. (2.1.41)

Inequality (2.1.41) finally leads to the lower bound of be. In [651 and other related works, one can see that the analysis of the log-log blow up

solutions can be decomposed into two stages:

1. At a certain time to, the initial data evolves into some well prepared data.

2. The well-prepared data admits a suitable bootstrap structure, and analysis can be significantly simplified.

38 One can show for the solution u(t) considered by Merle and Rapha81, that there exists some to such that u(to) satisfies the following:

u(to, X) - (Qbo + c+)( -- O )ei-o. (2.1.42) 0A Also u(to, x) satisfies the following:

1. orthogonality conditions: (2.1.18), (2.1.19), (2.1.20), (2.1.21),

2. the sign condition of b:

bo := b(to) > 0, (2.1.43)

3. closeness to Q

IVO6IIH1 + bo < a, (2.1.44)

4. smallness condition of the error Eo:

IIVEoI 2 + IEo 2 e'l' < I (2.1.45)

5. renormalized energy/momentum control6 :

A0|EoI + Ao|Po| < Firl, (2.1.46)

6. log-log regime 27r 7__ e < Ao < ee . (2.1.47)

Without loss of generality (by translation in time), we can assume to = 0. Now let us focus on the initial data of the form u(to), which from now on we denote with ao.

It turns out that the evolution of the data after to(= 0) is described by the following lemma.

6 1n this case, this condition is actually implied by the log-log regime condition below, we still keep it to make the notation consistent.

39 Lemma 2.1.17. Assume u solves (1.1.1) with initial data uo as (2.1.42). For all

T < T+(u), the following bootstrap argument holds: Let the rescaled time s be defines as s = fo ' + so,so = ebo, if one assumes the bootstrap hypothesis for t E [0, T],

107r ___ b(t) > 0, b(t) + IHe(t)IIH1 < 10al/ 2, e7 t < A(t) < e-eJUM7

lOlns < b(t(s)) , V(t) 2 + Ie(t)I2 (2.1.48)

A(t2 ) 3A(ti), VT > t2 > t1 > 0, (almost monotonicity), Ix(t)I < 1/1000, then one has the bootstrap estimate for t E [0, TI:

b(t) > 0, b(t) + Ie(t)IH < 5 l/2< ee- e K5 t)

2 b(t(s)) - , IIVe(t)11 + 1e(t)1 2 e-ly, < r (.49 51ns Ins b(t(2.1.49)

A(t 2 ) 2A(ti), VT > t2 > ti > 0, (almost monotonicity),

|x(t)I < 1/2000.

Remark 2.1.18. The special numbers 5, 4/5... appearing above, are of course only for technical reason, by sharpening the initial conditions at to, one can push the bootstrap estimates to

|V |2|e 2, ly - i - 6 e(+c)- (1-cO-x 1,E2+ IE2 - b e <_ic ~)< for arbitraryci > 0.

We refer to [67], [681, [651 for a proof. See in particular Proposition 1 in [681. Under the bootstrap regime, the analysis is made easier since one can simplify (2.1.30),(2.1.31), (2.1.33),(2.1.34),(2.1.35),(2.1.41) following the observation below:

For any polynomial P, b >> P(Jb), Fb>> P(A). (2.1.50)

40 Now, the first step of the analysis leading to the log-log blow up solution listed above, that is b ~- is quite clear since by (2.1.30), one has:

AAs -+ bI < IF . (2.1.51)

The local virial inequality (2.1.35), which is used to show the lower bound of b., is simplified further to

bs > -ri-c , (2.1.52) and the Lypounov functional (2.1.41), which is used to show the upper bound of be, is simplified to d -- < -CbFr. (2.1.53) ds

Basically, all those terms involving A 2 EO can be neglected since A is much smaller than b. This is actually one of the key observation in [651, [14].

Recall from the geometric decomposition (2.1.17)

1 - x- X(t -y(t)

At 2 A(t)

Since Qb is supported in yI < - -, and the tail radiation term (b is supported in jyf < A = F-", then the analysis in work of Merle and Rapha8l is taking place in

Ix - x(t)|I A(t)Fga < 1. The only part which connects the local dynamics and the information outside the potential blow up point x(t) is given by the conservation laws

(energy and momentum). Thus, if one considers the local momentum and local energy, one could localize the dynamics, paying the prize that the local energy and momentum are no longer conserved. However, since in the analysis any term appearing with the energy or the momentum is multiplied by a power of A, and A is so small as mentioned above, ultimately we do not really need that the energy and momentum are conserved, but rather that they are bounded. This is the reason we need to investigate whether the log-log blow up regime may keep the solution u(t) very smooth away from the blow up point. This would of course imply that the local momentum and local energy

41 are varying in a bounded (not necessarily small) way, thus finally totally localizing the dynamics and decoupling what happens in the region near the blow up point and away from it.

2.1.3 Strategy and structure of the paper

All arguments and results are valid for both dimensions d = 1 and 2. For simplicity, from now on, we work on d = 2. The main result in Theorem 1.2.1 is proven by describing the dynamics for well-prepared initial data of the form

= 1o(x)Qb, (X - -73' e.+ (2.1.54) uoA )eAj,. +jB. j=1 j, j,o

We call this data a "multi-solitions model". We sometimes call each "soliton" as bubble.

We show that, under certain conditions, m bubbles 1Qbo ( x~xiO) + , j =

1,... ,m evolve as if they do not feel the existence of each other, then we use a topological argument to show the existence of a blow up solution which blows up at m prescribed points according to log-log law.

As mentioned previously, one key element in the proof is to show that solutions generated by the well prepared data in (2.1.54) will keep high regularity outside a small neighborhood of xj,o, j = 1, ... , m.

To make the proof more accessible, we will first discuss an easier model (we call it "one soliton model" ), i.e. we show that yQeb(2)+ E blows up according to the log- log law without propagating singularity outside a small neighborhood of the origin, which is essentially a restatement of Theorem 2.1.1.

We organize the rest of this paper as follows:

In Section 2.2, we introduce the well prepared data for the "one soliton model" and describe its dynamics in Lemma 2.2.4. In Section 2.3, We introduce the well prepared data for the "multi-solitons model" and describe its dynamic in Lemma

2.3.2. In Section 2.4, we prove Lemma 2.2.4 for the "one soliton model". In Section

2.5, we prove Lemma 2.3.2 for the "multi-solitons model". In Section 2.6, we use

42 topological argument to prove the main theorem 1.2.1.

2.2 Description of the initial data and dynamic/modification of system: one soliton model

Let us recall that throughout the paper a will be used to denote a universal small number, though its exact value will be chosen at the very end of work. Also 3(a) is used to denote small constant which depend on a and limO 6(a) = 0.

2.2.1 Description of the initial data

We start with the "one soliton model". We define initial data uO in the following form: uo=+(bo+o)( ), (2.2.1) which satisfies all the property of data described in (2.1.42), i.e. orthogonality condition (2.1.18),(2.1.19),(2.1.20),(2.1.21), and the bounds (2.1.43), (2.1.44),(2.1.45),(2.1.46),(2.1.47).

Moreover we assume that outside the origin the data is smooth in the sense that for some K2 > 1, we have

IIUOIIHK2(IxI 1) $1, (2.2.2) and uo in HK2

Remark 2.2.1. For non-integer values K 2 , formula (2.2.2) means that, there exists a smooth cut-off function x equals 1 in lxi > 1 such that IIXuOIIH K2 < 1.

Now let us restate Theorem 2.1.1.

Lemma 2.2.2. Let u solve (1.1.1), with initial data prepared as in (2.2.1). Then

K2 VK1 < 2 , |U(t)IIHK1(IX,>i < 1 within the lifespan of u.

43 Remark 2.2.3. One should understand Theorem 2.1.1 and Lemma 2.2.2 in the following way: if the initial data is smooth, then partial smoothness will be kept during the log-log blow up process.

2.2.2 Modification of the system

Let us point out here that Lemma 2.1.17 cannot directly be applied to u that solves

(1.1.1) with the prepared data uo. In fact we cannot even say that u satisfies the geometric decomposition in (2.1.17) for t # 0.

Recall that previously a geometric decomposition was obtained via a variational argument, Lemma 2.1.7, (and a further modulation argument), all relying on a negative energy condition and on (1.1.12). In our case, for data described as (2.2.1) we don't even know if the have the negative energy condition.

This does not really matter in the "one soliton model", since one may modify

Lemma 2.1.4 relying on the so-called orbital stability of the soliton and re-establish

Lemma 2.1.13 without negative energy condition.

We do not use this approach here, since we will later deal with "multi-solitons model" and general orbital stability for multi-solitons is a very hard open problem.

Let us consider now a system for {u(t), b(t), A(t), x(t),-y(t)}. We define e(t) = ei + ie2 := u(t) - y(t)(_x(t))ei-Y(t), and we consider the system as

2 iut = -Au -- IuI u

E{(e1 (t),IyI2Eb(t)) + (e2 (t), I|2 e(t))} = 0,

T{(e(t), yF(t) + (e2 (t), yb(t))} = 0, (2.2.3)

2 2 '{ (ei(t), AEb(t)) + (e 2(t), AEb(t))} = 0,

d{-(c(t), AEb(t)) + ( 2 (t), A 2(t)} 0.

{u(t), b(t), A(t), x(t), y(t)}t=o = {uo, bo, Ao, 0, 0}.

The local well posedness of (2.2.3) is straightforward, see Appendix A.1 for more details. Note now the following geometric decomposition automatically hold due to

44 the definition of E.

1 - X - X(t) iyt u(t, X) = (Qb - e) ( (t) e(). (2.2.4) A(t)t At

As pointed out by Merle and RaphaBl, such system should be studied in rescaled time variable rather than in its original time variable. So we define the re-scaled time s as ds = 21, and if one rewrites (2.2.3) in rescaled time variable, E automatically solves (2.1.23), (2.1.24). Since the orthogonality condition (2.1.18), (2.1.19), (2.1.20), (2.1.21) hold at t = 0, by (2.2.3), they hold for all t E [0, T], the life span of (2.2.3).

What's more, the life span of (2.2.3) is exactly the lifespan of the problem (1.1.1) with initial data uo. This is not a trivial fact. Indeed (2.2.3) is an NLS coupled with four ODEs involving {b, A, x, y} and there is the possibility that b may become large, and then this system no longer makes sense since Qb is only defined for small b.

However, as long as (2.2.3) holds, then the bootstrap lemma 2.1.17 works, and this will ensure that b stays bounded and small, and thus the lifespan of (2.2.3) coincides with that of the NLS problem with initial data uo. There is of course another possibility, that the coupled ODE breakdown, i.e. A becomes 0 and this of course, means that the NLS equation blows up.

To summarize, during the study of the NLS (1.1.1) with initial data uo as in

(2.2.1), for any [0, T] in the lifespan of u(t), u(t) satisfies geometric decomposition:

I -X - X(t) eiyt u(t) = A(t) (Qb(t)+E(t)) ( ) eI and orthogonality conditions (2.1.18),(2.1.19), (2.1.20),(2.1.21). In particular the bootstrap lemma 2.1.17 works for these kinds of data.

2.2.3 Description of the dynamic

Now, we can (equivalently) restate Lemma 2.2.2 in the following way.

Lemma 2.2.4. Consider the system (2.2.3) with initial data {uo, bo, A0 , 0, 0} described in subsection 2.2.1, then for any T < T+(uo), we have the following bootstrap argu-

45 ment:

Let the rescaled time s be defined as s = fo' - + so, so = e' , if one assumes boot-

strap hypothesis (2.1.48) for t E [0,T], then one has bootstrap estimate (2.1.49) for

t E [0, T], and following regularity estimate holds for any fixed K1 < 2' ,

IIu(t)IIHK(IxI ) ,< 1,t (E [0,T]. (2.2.5)

2.3 Description of the initial data and dynamic: multi- soliton model

Now we turn to "multi-solitons model".

We introduce some notations. Let XOloc, X1,Ioc be two smooth cut-off functions such that

I 3/4, XO,oc(x) = 1, (2.3.1) 10, 1X| > 1,

LxI 2/3, x1,0CW) = 1, (2.3.2) 0,IxI > 3/4.

Let's define the local version of energy and momentum. For any f E H1 (R2 ), Xo E R2 , let

EOC(f, xo) f1 2 - 1if ,4 (2.3.3) f X0,ioc(x - xo) (IIV

P10c(f, xo) (2.3.4) /R2 (

2.3.1 Description of the initial data

We define an initial data uo of the following form:

m +E0 , uo := - (2.3.5) j=1

46 where

uj,O = Qb,o (x -io _7,, = 1,.... .M (2.3.6) Aj,o \Aj,o / 7o E H1 (R2),7 (2.3.7)

and the properties of E0 are implicitly encoded below. We let

E Aj,oEo(Aj,ox + xj,o)eis'-i, (2.3.8) 3lae ,0 +z 6- ,0 ,j 1 , . . m

We require the orthogonality condition:

2 (2.3.9) (b, IyI ,o) + 2,0, IY Eyb,) = 0, j = 1, ... ,M

(e3, yib 0)+ , yEbj,) 0,j =1,... m (2.3.10)

- (E,o, A2 ej,O) + (i,0, A2 E,,0 ) = 0, j 1, .. .m (2.3.11)

- (ejlo, AEb-,o) + (c,, AEb,) 0,j = 1,... (2.3.12) where Qb Eb + IfE. We further require the following:

1. Non-interaction of any two different solitons:

I1xo- a'o I > 10, Vj f j', (2.3.13)

2. Sign condition and smallness condition of bj,o:

a > bj,o > 0,3j = 1, . . (2.3.14)

3. Smallness condition: m m 1 Aj,o + E|IIH1

47 4. Log-log regime condition:

27r 7__ e5 < A3o < ethe l.ole.o (2.3.16)

5. Smallness of the local error

6 2 01c~ + I~je-'y' < ]F7 0 , 1,. m (2.3.17) JX< jO

6. Tameness of local energy and local momentum

(2.3.18)

AjIEloc(uo, xj,o)I < F1000,j = 1,.. .,m. (2.3.19)

7. Smoothness outside the singularity, here we fix a large N2 , which would be chosen later. (2.3.20) IUoj gN2 (min1

8. uO in HN2.

Remark 2.3.1. Such data can be constructed similarly as those constructed in [68].

2.3.2 Modification of the system

Now let u be the solution to (1.1.1) with initial data uO described as in Subsection 2.3.1. We are expecting that throughout the lifespan of the evolution, we can find

1.-X'(t7)(i'jx parameters {bj(t), Aj(t), xj(t), yj(t)},ml and a function E(t, x) such that the following geometric decomposition holds:

I ( ) u(t, x) Z A(t)Qbi A =1 Ibj(t)l < 6(a), |IllIH1 6 (a), I < J < m, Ej = Aj (Ajx + xgj~e 5I, (2.3.22)

48 as well as the orthogonality condition:

(I, b,) + , yI2 Eb,) = 0, j 1, ... , (2.3.23)

(cjyEb,) + (E2, YEb3 ) = Oj = 1,... ,m, (2.3.24)

-(Ei, A 2 6b,) + (3, A2 Eb,) = 0, j 1, ... , m, (2.3.25)

-(Ec, AEs,) + (2,AEb) = 0,j = 1, ... , m. (2.3.26)

Again Qb Eb + ieb, Ei = + i2*

Since at this point we do not know that the general multi-solitons orbital stability holds, we consider the system for {u(t, x), {bj (t), Aj (t), xj (t), -yj(t)},_. 1} as in Subsection 2.2.2:

rUt+ Au = -uI 2U,

2 ~{(j,Zbyl2Eb)3 + (Ei, IyI (i,)} = Oj = 1,...

f{-(ej, AE,) + (cI,A,)} = , j = 1, ... , m

u(0, x) = uo, Aj (0) = Aj,o, bj (0) = bj,o, x (0) = xj,o, -y(0) = y,, 1, ... , (2.3.27) where d, cj, e is defined by E, as previously noted, and E is defined by

E(t, x) u(t, X) - bj(x ) -- Y(t). (2.3.28) j=1 I

By the orthogonality conditions (2.3.23),(2.3.24),(2.3.25), (2.3.26) of the initial data,

49 one has the orthogonality condition within the lifespan of (2.3.27):

(c, IyI2Eb) + (ey, YI2 Eb)= 0, j 1,.. .M (2.3.29)

(YFb i) + ( ybj)E, = 0, j = 1,... ,m, (2.3.30)

-(, A 2 ) + (Ei, A 2 Eb,) = 0, j = 1,... , m, (2.3.31)

-(Ej, AEsb) + (e2,A~b,) = 0,j = 1,... ,m. (2.3.32)

We discussed previously that the lifespan of (2.2.3) is the same as the lifespan of u. With the same argument one also has that the lifespan of (2.3.27) is the same as lifespan of u. We perform one final simplification. Let us define7:

(x) :=E(x)Xi,oc(x - x), e(t, y) := AEI(Ay + ,33) 23)e

I (t, y) = E3 + iE1, j ,...,M. we point out that our analysis will be performed under the condition A3 < bj, (see Lemma 2.3.2 below). Now, by definition, one has E& = E&Xi,loc( ), and recall Qb is supported in jyj < 1. It is not hard to see the following orthogonality condition for ed .

(_1, Jy12 2-bj)), =E' j =~-1, ... ,m , (2.3.34)

(e, yEIb)+ (4), ysb) =0,j = 1,... ,m (2.3.35)

-(l, A 2 ,) + (Ei, A 2Eb) =0,j = 1,. .. , m, (2.3.36)

-(E, ) + (Ea, AEa) = 0, d e . .,M. (2.3.37)

7 Please note Eand c are different notations.

50 Under the hypothesis Aj < bj, the system (2.3.27) is exactly:

Zit + Au = -IuI 2U,

_{(bIyEb) + (', IyIEb,)} = O,j =1,...

{-(E'j, A2) + (e , A2Es)} =0, j , =1,..., m,

{-t j AE~bj) + (E-, A Ebj)} = 0, j=1,- .. .,M

u(0, x) = uo, Aj (0) = Aj,o, bj (0) = bj,o, x (0) = ,o, -yj(0) = -yj,o, j 1,... m. (2.3.38)

2.3.3 Description of the dynamic

We are now ready to present the main lemma which contains a bootstrap argument used to describe the dynamics of (2.3.38), where the initial data {uo, {bj,O, Aj,o, xj,o, yj,o}m 1} is described as in Subsection 2.3.1.

Lemma 2.3.2. Consider the system (2.3.38) with initial data described as in Sub- section 2.3.1, (with the universal constant a is small enough), then VT < T+(u), the following bootstrap argument holds (and all the estimates are independent of T): 37r Let sj,o := eb, and define the re-scaled time s '= + s8,O, = 1,..., m. If for t C [0, T) the bootstrap hypothesis hold:

" smallness of bj and E:

m m b5I+ 5 If'IIH1 < 10c1/2 (2.3.39) j=1 j=1

" log-log regime, part I:

e- < A,(t) K ee , j = 1, ,m, (2.3.40)

" log-log regime, part II:

51 < b (2.3.41) 1 in s3 - - In sj

* smallness of the local error:

I IVEI 2 + IEj1 2 e- F4 j = 1, ... , (2.3.42)

* almost monotonicity:

VT> t2 > t1 > 0, Aj (t2 ) < 3Aj(t),j= 1,... , (2.3.43)

" control of translationparameters:

1 VO < t < T,1x(t) - X1,oI < 1000, = 1, . m, (2.3.44)

" local control of conserved quantity: for all 0 < t < T, j = 1,...

IE1c(xj (t), u(t)) - Eloc(xj,o, uo)I < 1000, (2.3.45)

IP0oC(xj (t), u(t)) - PrC(xj,o,0 Uo)I < 1000, (2.3.46)

* outside-smoothness:

1 I1uIIHNl(minj{Ix-xj, I>}) ! aX3 A, (2.3.47)

where N1 is some fixed large constant, N1

Then for t C [0, T), the bootstrap estimates hold:

* smallness of bj and e:

jbj+ E I|I'IH1 < 51 (2.3.48) j=1

52 9 log-log regime, part I:

57r 7r__

e-e b A(t) _ e-" , j = 1,.. , (2.3.49)

* log-log regime, part II:

7r b.7 < 57r ,j=1, . .. ,m,7 (2.3.50) 51n sj In sj

* smallness of the local error:

2 2 IVEI + lEc| e- ly' < ],5 1,... , m, (2.3.51)

* almost monoto rnicity:

VT> t 2 > ti > 0, Aj(t 2 ) < 2Aj (ti), j = 1, ... , (2.3.52)

* control of translation parameter:

1 VO < t < T, lXj,t - xj,01 < (2.3.53) 2000,' ,. ,

* local control of conserved quantity: for all 0 < t < T, j = 1, ... ,

IEloc (x(t), u(t)) - Eloc(xj,o, uo)I 500, (2.3.54)

Poc((Xj (t), u (t)) - Poc(xj,o, uo)I < 500, (2.3.55) (2.3.56)

* outside smoothness:

I1UIHNj (minj{Ix-xj,oI <})1 (2.3.57)

53 Remark 2.3.3. According to our notation, (2.3.57) means that

IfUIIHN1(minj{Ix-xj,oI -}) C for some universal constant C. Also by (2.3.15) we have limom-a = oo. Thus, when a is small enough, (2.3.57) is stronger than (2.3.47), i.e. this is a bootstrap

estimate.

Remark 2.3.4. The Lemma 2.3.2 itself implies that T+(u) < 61 (a). Indeed, Lemma 2.3.2 implies that the solution blows up in finite time and according to the log-log law.

Similarly for the solutions in Lemma 2.1.17 and Lemma 2.2.4. This is a standard argument, for details see for example [651. One may also directly look at Subsection

2.6.2, see Remark 2.6.4.

Remark 2.3.5. Note that Lemma 2.3.2 implies that mingjo, jxj(t) - xj'(t)| > 5 for all t < T+(u).

2.3.4 Further remarks on Lemma 2.3.2

Lemma 2.3.2 means that all the bootstrap estimates hold within the lifespan of u generated by the initial data described in Subsection 2.3.1. In particular for initial data of the form

u(t, X) = ( j'U-O)e4' + Eo(t, X) j=1 Aj A with orthogonality condition (2.3.9),(2.3.10),(2.3.11),(2.3.12) and with bounds (2.3.13),(2.3.14)...(2.3.20) the associated solution is smooth in the area min |X - xj,0 > ! (i.e. estimate (2.3.57) holds). 8 Further more, if one looks at Xi,Ioc( - xj(t))u(t, x), j = 1, ... , m, then

X1,10C(x- ux(t))U(t, -) (bj + =1,... ,m (2.3.58)

8We note that by the definition of X1,loc and the bootstrap estimate jx,(t) - Xj,o I < ,then u(t, x) = Xi,loc(x - xj(t))u(t, x) in the region Ix - xjo I<'2

54 and the bootstrap estimates stated in Lemma 2.3.2 simply means that x(x-xj(t))u(t, X) evolves according to the log-log law described in the series of work of Merle and Ra- pahel (until at least one soliton blows up.)

2.4 Proof of Lemma 2.2.4: One Soliton Model

This section is devoted to the proof of Lemma 2.2.2, and we need only to show (2.2.5) since we already have (2.1.17). Fix [0, T], all the estimates below are independent of the choice of T.

2.4.1 Setting up

Recall that u is the solution to (1.1.1) with initial data uo as described in (2.2.1).

Recall as discussed in Subsection 2.2.2, that we consider the system (2.2.3) for {u(t), b(t), A(t), x(t), y(t)}, and the geometric decomposition (2.2.4) holds. Also Lemma

2.1.17 holds. We finally point out that the bootstrap hypothesis (2.1.48) itself implies

ft 1 dT

10 A(T)" In~t 1'2'for01 "Qt)- for p ;> 2, for any t < T. See (51) in [681.

Remark 2.4.1. Note here that we may also use the bootstrap estimate (2.1.49) to

show (2.4.1), since we know that (2.1.48) implies (2.1.49). However, all the argu-

ments in this section only rely on the bootstrap hypothesis (2.1.48), this will become

important when we work on multi-solitons model.

55 2.4.2 An overview of the proof

Recall that for all t E [0, T+(u)), the geometric decomposition holds

1 - - - X(t) -7(t) Ut) =A(t) (Qb + 6)( A(t) and by the bootstrap hypothesis (2.1.48), we know Ix(t)l < for t E [0, T]. Now we fix Xo, such

0,lxi < a0 = xo(x) := (2.4.2) 1, lxi > do .

The key to proving that a log-log blow up solution can propagate some regularity outside its potential blow up point x(t) is the following control:

VT < T+(u), j 2rV(xou(t))I, 1. (2.4.3)

This is pointed out in [681, see formula (63) in [581 for a proof.

We first use the I-method to show the rough control,

Lemma 2.4.2. In the setting of Lemma 2.2.4, the following estimates hold for t E [0, T], for any o- > 0,

||U(t)||HK2 r

Next, we introduce a sequence of cut off functions {xi}'_o, where L is a large but fixed number which will be chosen later:

0, lxi { < a,, (2.4.5) 1, lxi > bi.

Here we require a, < bi < a,-1 , bL 1 The idea is to retreat from Xi-ju to xiu for each 1, showing that xiu has higher regularity than Xl-iu. For convenience of notation we set vi := Xiu. We use the crucial control (2.4.3), Strichartz estimates and interpolation techniques to show:

56 Lemma 2.4.3. For all 0 < v < 1 and for t G [0, T], we have

1 VO- > 0, IIV'v 2 (t)1L2 ,< A )(2.4.6)

For a proof see Subsubsection 2.4.4.

This lemma gives a better L' estimate of v 2 , which of course implies a better L' estimate of vi, i > 2.

We also show:

Lemma 2.4.4. For any 0 < v < 1 and for t c [0, T], we have

1 Va > 0, Iv2IIL- ,NJa A1ta ' (2.4.7)

For a proof see Subsubsection 2.4.4.

Remark 2.4.5. This lemma should be understood as an improvement of the L' estimate. Indeed, from the viewpoint of Sobolev embedding, H' ( 2 ) - L , thus the trivial L' estimate is |Iv2|IL- $ IU(t)IIL- <

Lemma 2.4.4 improves Lemma 2.4.3 immediately:

Lemma 2.4.6. For all 0 < v < 1 and for t E [0, T], we have

|IV'v 3(t)1L2 < 1. (2.4.8)

For a proof see Subsubsection 2.4.4.

Finally, we have the following lemma (which is analogue to Lemma 4 in [68]) to

iterate the gain of regularity for large K 2 .

Lemma 2.4.7. If the following estimate holds for some r > 0 and some i, 1 < i < L-l,

vi(t)|IH ,< 1, t E [0,T]. (2.4.9) then there is a gain of regularity on vi+1, that is

W < (K2 - 1 + r, 1v1+i ,Hf$_ 1, t E [0, T]. (2.4.10) K2

57 For a proof see Subsubsection 2.4.4.

Now we are ready to end the proof of Lemma 2.2.4, i.e. to prove (2.2.5).

Proof of (2.2.5) in Lemma 2.2.4. Lemma 2.4.7 is enough for us to obtain the regularity estimate (2.2.5) for K1 < K2 . To see this consider the end point case in

2 (K -r) = 2. (2.4.10), i.e. if one consider r = K22 - 1+ r, then one obtains r 2 Thus, when K, < -, by choosing the iteration time L large, the desired estimate (2.2.5) follows. l

2.4.3 Rough control

This subsection is devoted to the proof of Lemma 2.4.2, similar type of estimates are derived in [681 by considering some pseudo energy. Here, we rely on I-method.

Without loss of generality, we only show:

I1u(T)|IHK )K2+(.1) 2 a 2 A(T)

LWP interval

Let's introduce the so-called LWP interval as in [141. To make the argument easier, we observe that, under the bootstrap hypothesis (2.1.48), we can show that A(t) is actually strictly decreasing. Indeed, by (2.1.48), b > 0, A < b in the sense of -1 1 (2.1.50). Thus by (2.1.52) and (2.1.50), we obtain A -> 2 b > 0, thus A(t) is strictly decreasing.

We define ko, kT as:

A(t)= . (2.4.13)

Then as in [141, we can perform a bootstrap argument:

58 Lemma 2.4.8. In [0, T], assuming the bootstrap hypothesis

tk+1 - tk < k A 2 (tk), (2.4.14)

we obtain the bootstrap estimate

2 tk+1 - tk 'A (tk) (2.4.15)

This estimate is not sharp, indeed, morally one should have tk+1-tk ~ (ln ln k)A(tk) 2 according to the log-log law. We refer to (2.39) in Lemma 2.4 of [14].

As in [14], we divide all intervals [tk, tk+11 into disjoint intervals U [A- , +11, where

t k =Tk < Tk ... < k -I

Tk ~ T-k T= 4-Atk~) -rjId - Tk= A(tk+1 )2, Vj < J - 2 (2.4.16)

2 TJik - T'k-l K -A(tk+1) ,Vj Jk - 2. 4 Here 61 is a fixed constant which will be chosen later. Now, by Lemma 2.4.8, using the bootstrap estimate (2.4.15) (indeed, we only use the bootstrap hypothesis (2.4.14), which is weaker), we have

Jk 10k 2 ,Z J 10k?3 Iln A(T)1 3. (2.4.17) k

A quick introduction of upside-down I-method

We point out without proof the following classical fact on any LWP interval:

sup,SU 1u(t) - U(-rk)IIHK2 r11. j)u(IIIHK2, (2.4.18)

and directly iterate this over all the LWP intervals (recall we have about Iln A(T)1 3

3 such intervals), to obtain the estimate |1u(T)|HK2 < 0 InA(T)1 for some C > 1, which is clearly weaker than (2.4.11) but actually quite close.

The idea of the original I-method [111 or up-side down I-method [69] are both

59 to improve the estimate (2.4.18) on a LWP interval by working on certain a slowly varying/almost conserved quantity.

We introduce the upside-down I-operator DN:

DNf(() M( N )f((), (2.4.19) where M( ) is a smooth function such that

MJ) ()K2, 1 > 2, (2.4.20) 1%,I(I 1.

It is easy to see that for any f E HK2 we have

|IDNf IL2 $ 1f 11HK2 < NK2 IDNf 1l2. (2.4.21)

Now for some v that solves NLS, the idea of the upside-down I-method is to use

IIDNv 1122 to model IIVI2K, and show that E1 (v) := IIDNvI11 is slow varying, see [691.

Lemma 2.4.9. [Proposition 3.4,Lemma 4.5 in [69] There exists a higher modified energy E2 (or to be more precise, E2), such that for any f in HK2,

E 2 IDNfI2 rj N 1 IDNfI11, (2.4.22)

VM > 0, there exists JO = 60 (M) > 0 such that if v solves (1.1.1) with initial data vo and IIvoIIH1 < M,, then [0, 6o] is in the lifespan of v and E2 (v) is slow varying in the following sense: 1 sup IE2 (v(T)) - E 2(vo) I 5 E2 (vo). (2.4.23) O<-r<5o and

' sup IDNv(t) 2L2 < (1 + C 1 C )IIDNv(t L2(R2), (2.4.24) 0<-r60 N - N- where C is some universal constant

60 Actually in [691 one finds only (2.4.22) and (2.4.23), but (2.4.24) is directly implied.

Formula (2.4.22) can be found in the last formula of the proof of the Proposition 3.4 in [69].

Remark 2.4.10. In [691, the equation is defocusing, while here we are working with a focusing equation. However, when one restricts analysis locally, the two problems actually have no real difference. In [69], the defocusing condition is only used to ensure that any HS, s > 1 solution considered is global. Note [69! deal with both

Euclidean case and Torus case.

Proof of Lemma 2.4.2

Now we are ready to finish the proof of Lemma 2.4.2.

Proof. Recall we need only to show (2.4.11). Let

N = ( )1+O. (2.4.25) A(T)

Here o = o (o-) is a small positive constant chosen later.

We have the following estimate on LWP interval:

1 sup |IDNu(t) L122 (1 + C )IIDNu(iT) 11L2. (2.4.26) tE[Tjk-rk+1] (NA(T))2

The constant C is independent of j, k. We now assume (2.4.26) temporarily and we finish the proof of (2.4.11). By our choice of N, we immediately obtain from (2.4.26) that

sup |IDNu(t)1L2 _ (1 + C 1 DNU(T 2 tE[j -r+1] LAT~ (A(T)-O1)2)LI~uT~I2. (2.4.27)

Then we iterate this estimate and we recall that the total number of LWP intervals is controlled by (2.4.17), thus we obtain

DNu(T) (1 CA(T))C(lnA(T)) HK 2 < 1- (2.4.28)

61 Thus, we arrive to the estimate

IUIIHK2 K2- 2 ANuIK2(1+01 )

The desired estimate (2.4.4) clearly follows if we choose a- < -/K2 . What is now left is to prove (2.4.26). We indeed show that

1 sup IIDNU(t) 11L 2 (1 + C IDN LI2. (2.4.30) 1 2(..0 tE [jk,'rk+1)L (NA(tj))

Since, A(T) < A(tk+1), clearly (2.4.26) follows from (2.4.30). We now prove (2.4.30). Indeed, by scaling or direct computation, let

Uj,k(t, x) := A(tk+l)u(A(tk+l) 2 (-r + t), A(tk+1X)), then Uj,k solves (1.1.1) with initial data A(tk+1)U(Tkj, A(tk+l)x). A direct computation leads to

IIA(tk+l)U( , A(tk+)X)II H1 = IQ EIIH1 < 2 H'-

To apply the upside-down I-method in Lemma 2.4.9, we first choose 60 in Lemma 2.4.9 as Jo(21IQ112.) and then we choose 61 in (2.4.16) as 61= _, and as a consequence for all j, k, Tk - Tk+ A(tk+1 )2.<< Oo-0 By the upside-down I-method (2.4.24), we have:

sup +1 |IDNA(t)Uj,k (t) 112

' \(tk+1l 1 1 1 <(1 + C ., (+ C 1 |DN j)Uj,k 12 (2.4.31)L (NA(ti))- (NA(tj)) -DNA(t)k L2

(1 1 NA(t)Uj,k L2 (NA(tj))i

62 We now observe that for any A > 0 and for any function f we have

A(DNf)(AX) -DNA (Af(Ax)), (2.4.32)

|If|12 = Af(Ax)11 2.

(2.4.30) clearly follows. l

Remark 2.4.11. It is not hard to see that the proof only relies on the fact that one can divide [0, T] into disjoint intervals Uj,klk,j such that

1. I|u(t)||H1 ~ 2-k for t E Ikj(this is equivalent to A(t) ~ 2-k for t E Ik,j.)

2. 1Ik,j ,~ -, Vk, j

3. {Ik,} ,< I In A (T) 13

4. A(T) < A(t), t C [0, T]. and u(O) E HK2

Note that condition 3 follows from the bootstrap Lemma 2.4.8. We finally remark that by further dividing Ik,j, it is very easy to improve condition

2 to IIk,jI < 5 for any fixed 6 > 0.

2.4.4 Propagation of regularity

In this subsection, we give the proof of Lemma 2.4.3, Lemma 2.4.4, Lemma 2.4.6 and

Lemma 2.4.7. Since our proof relies on the Strichartz estimates, for completeness we recall them below.

Strichartz estimates

Consider the equation:

iUt + AU = F, (2.4.33)

U(0, X) = Uo.

63 We write it in the integral form using the Duhamel Formula:

U(t) = etA'Uo - if e(ts-)AF(s)ds, (2.4.34) where eitA is the propagator of linear Schrddinger equation. For notation convenience, let:

FF := -j e(t-s)AF(s)ds.

Then one has the following Strichartz estimates:

V- + - = -, 2< q < oo, q r 2 U Sqr ||Uo 1|2, (2.4.35)

||FFILq({0,t],Lr(R2))

Proof of Lemma 2.4.3

We fix v < 1, and we estimate I|V"v 2II2 . Note that v 2 satisfies:

{itv 2 + Av 2 = AX2VI + 2VX 2Vv1 - V2 lv 12. (2.4.36)

As in previous section, we only need to show:

1 11V~'v 2 (T) 112 3 (T) (2.4.37)

We explain some heuristics for this estimate. We view the system as a perturbation of the linear Schrddinger equation, and the dominating term in the perturbation is the last term on the right side of (2.4.36). We can view (2.4.36) as:

2 ZOtV2 + Av 2 ~ O(Iv 1I v2). (2.4.38)

64 From the viewpoint of persistence of regularity one gets:

I|Vv2(T) 112 o IIV'v 2(0)|| 2e L 4 Ot],L4) (2.4.39)

By estimating

||V1(t)I|L4(R2) JU(t H1, one obtains by (2.4.1)

|| V"v 2 (T) 112 l T)ll. (2.4.40)

This estimate though is too week, of course. On the other hand we didn't even use the key estimate (2.4.3) in the log-log regime. To obtain the stronger estimate (2.4.37) instead of the L norm in (2.4.39) we use a more flexible L'L;. More precisely we replace IIvl(t)1 14o, ) by some IIV1(t)IILq([OT],(R2)) such that (q, r) is an admissible pair. Now, we make the key observation that L 2H 1 and L4 L4 have the same scaling as

L'L; whenever (q, r) is an admissible pair. Thus, by interpolating the two estimates

I T

I T ( 22.4.41 ) 112 < 1, 02/ |VV1(t) we derive fU(t)lqr < ( ln A(t)1) 1/100, (2.4.42) for an admissible (q, r) carefully picked. Then we obtain

IIV'v 2 (T)112 $ IIV'v 2(0)1(2e t)I "i 'Tr) $ el n(A(T)I), (2.4.43) which implies the desired estimate (2.4.37).

We now go to the details. We need the following technical lemmata.

Lemma 2.4.12. Let -+- 1, let q', r' be their dual, let po be defined by 4 =+, q0 ro 0 r 2 P let ho be defined by ho 1 - Ithen we have the following estimate uniform with P0

65 respect to any time interval I

2 IIv1| V v2 ||ILrO II) VvIL2 |LVI ;L2pOI (2.4.44)

IV1|IL2o $ IVllHho, (2.4.46) ho-1/2 1-ho

||V11||L2 (;H ) O (I;H IH) 1 L4(I;H l)

Moreover we can choose qO large enough such that 1 - 1oo < ho < 1, q' < 2.

Lemma 2.4.13. Let -1 = 1, let q', r' be their dual, let g1 , 1 be defined by 1 1 v1 = - V = (1 -v)-1 = letp1 be defned as = . , let h1 be defned

as h1 - 1 = - and let w1 be defined as -= +&, then we have the following estimates uniform with respect to any time interval I

||IV1II'7 V1|V2||,g' , IP7Vv1||L 1||V1|LP1I 2IIL91, (2.4.48) llvIlHl, IV21IL91 2 IIV'V2IIL2, (2.4.49) IVPV1i|LW1 1

||v |q1 1|2||r"(;II V21 L-(I;H') IV11L2(1;H') lV10L-1(I;LPl), (2.4.50)

IlvIILPl $ IIvliHhl, (2.4.51) h 1/2 1

I|vil Lwl(I;Hhl) $ IlviL2(I;H) IvL 4 (I;H ). (2.4.52)

Moreover we can choose q1 large such that 1 - < h1 < 1, w1 ; 4.

The proofs are a direct consequence of Sobolev and H6lder inequalities as well as

standard interpolation

iuIIHs Iu 1 Il2Is1 < s < 2-

We finally remark that all the indices happen to coincide because of scaling reasons.

2 Indeed, we can check that (2q', po), (wi,p1 ) are both admissible pairs.

Now, we pick (qo, ro), (qi, rj) as in Lemma 2.4.12 and Lemma 2.4.13, and we let

all the other associated indices be determined as in these two lemmas. By estimating

66 ||u(t)IIHg T , we have I|v1(t)lI ' 1 . Thus, by (2.4.1) we have Now we have: "U~t"H'1~t) "Vlt)"l <

T (2.4.53)

Also from (2.4.3), we have A(t)1101.IvI(t)II14In IV1i(t)I12 < 1. (2.4.54) I T Using (2.4.53), (2.4.54) and (2.4.46), (2.4.47) in Lemma 2.4.12 we obtain

2 VI 1/<2 I A(T)11/ , L q 0(9, T];L 2po) -I In

and using (2.4.53), (2.4.54) and (2.4.51), (2.4.52) in Lemma 2.4.13 we obtain

1/2 ||V1 II L'1([0 TJ;LP1) < I ln A(T)| .

Thus, we are able to divide [0, T] into Jk disjoint intervals [Tk, Tk+1], k 1, ... , Jk, such

that Jk -n(A(T)) 1,

|v 11 2 .,' < E (2.4.55)

IV1|IL-1([0,T]);LP1 E, where c > 0 is a fixed small constant, to be chosen later.

Now we use the Duhamel formula (2.4.34) for (2.4.36) in [Tk, Tk+1], and we obtain

for any t E [Tk, Tk+11,

||v2(t) - V2(Tk)llftu

T fC ||IV1||H + Cf IV(t)IIHl+v + 11 0 0 e i(t-r)A v 2V2)(T~d1 2

Cf 2 lIu(t) IIH1+v + CI i IVv2 i+([ty,r];L) V(V1 )V211'1Q([tkrJ);Lj)' (2.4.56)

where in the last step we have used Strichartz estimates and Fractional Leibniz rule

(See Theorem A.8, [43}). We remark It is not hard to see that we can choose (qO, r0 ) =

67 (q 1 , rj). Now we plug in the estimates in (2.4.45), (2.4.49) into (2.4.56), to obtain

11v2 (t) - V2(Tk)Ilftzi

+ C( 01|2([-Tk,rk+1];Lp + V IL2q0([-rk,rk+1]-L2po) IV21IHK, 1 IV1IL2([0,T];Hl)) {supE t]] (2.4.57) i.e.

+ C (2.4.58) 11v2(t) - V2(Tk)IIH- C IoTIIu(t) IIH1+v + CESUPtE[tk,rI Iu(t) H-

The term fj' IIu(t)IIH1+v is not hard to control. Interpolating between IJuJIHi and

IIU|IHK2, we obtain K-1-v ) ~ K1 HK2 1, ||U(t)||Hl+v< H (2.4.59)

Now we plug in the estimate |IU(t)IHI ,< 1 and |1U(t)1gK2 A () ' 2+ from Lemma 2.4.2, and we obtain

|Iu(t)|I +v $c+.+Cl 1 (2.4.60) 1" Al+v+Car(t)~

By choosing a small enough such that 1 + v + C- < 2, and using (2.4.1), we have

(2.4.61) I T I~u(t)IHI' +v $1

We plug in (2.4.61) into (2.4.56), and we choose E small enough to obtain

SUPtE[tk,tk+1]IIU(t)IIH < 21U(tk)IIHu + C. (2-4-62)

Iterating this over the Jk -I ln A(T) 12 intervals, we obtain the estimate

A(T) 13 |u(T)IH- < e C1In (2.4.63)

which implies the desired estimate (2.4.37).

68 Proof of Lemma 2.4.4

We now prove Lemma 2.4.4. Indeed, by Lemma 2.4.2 and Lemma 2.4.3, we obtain for any v < 1, & > 0 the estimates

K1 (2.4.64) I|v2 (t)IHK2 $<&

(2.4.65) 11v2(t)||H- '<,& 1

Now, by Sobolev embedding and interpolation, we obtain

1 1 ||V2(t)||Lo(R2) <& JIV2(t)IlHl+& r<- 3 CI I-Ku' (2.4.66)

Since & > 0 is arbitrary and we can choose v as close to 1 as we want, this clearly gives Lemma 2.4.4.

Proof of Lemma 2.4.6

Now we prove Lemma 2.4.6. We fix v and we point out that all the constants in the proof will depend the choice of v. By choosing a- small enough, Lemma 2.4.4 gives for some small c = c, > 0 1 ||V2(t)||L- (2.4.67) r

11v (t)||HV < 1 (2.4.68) 2 AcvlO(t)~

Clearly v3 also satisfies the estimate (2.4.67), (2.4.68), and also it satisfies the equation

2itV3 + Av 3 = AX3v 2 + 2VX 3Vv 2 - v31v 2 12. (2.4.69)

By the Duhamel's formula, we obtain

ei(t-s)A(AX 2 v 3(t) = ei"t'(v3 (0)) + i 3v 2 + 2VX3 Vv 2 - v31v2 1 )(s)ds. (2.4.70)

69 Thus,

(2.4.71) 11v3 (t)I|H- r<11v3()IIHV + 11 v211H" + J v211H1+ + 11V31V2 1211H

As argued previously in (2.4.61), fJ I1v2IIH1+" < 1. Thus, to finish the proof of Lemma

2.4.6, we only need to show

T (2.4.72) fIv 3|v2 12 (t)H 1.-

Indeed,

11v31V2 11H- r IIV3IIH..IV2ILOC + IV31ILooIV2ILoO v2jIH-- (2.4.73)

Now we plug the estimates (2.4.67) and (2.4.68) into (2.4.73) and we recall that v 3 also satisfy estimate (2.4.67) and (2.4.68), we have

(2.4.74) 11v3 1v2 12 (t)IH <

Thus, by (2.4.1), estimate (2.4.72) follows.

Proof of Lemma 2.4.7

Finally we prove Lemma 2.4.7. This is quite straightforward. First, one can directly check (again) that vj+1 satisfies:

2 iztvj+l + Avj = AX+1vj + 2VXj+ 1V7v - vj+11vj1 . (2.4.75)

Since we assume that K2 > 2, then by Lemma 2.4.4, we have (for j > 2):

||vy~)||U (t . (2.4.76)

70 Clearly the same estimates hold for vj+1.

Again by the Duhamel's formula:

vj+1(t) = e"t(Xj+luo) + i ei(t-r)A(AXj+lvj + 2VXj+iVvj -- vj+1Ivj12 )(r)dT, (2.4.77)

We remark here that the main term to control is 2VXj+1 Vvj. Directly from (2.4.77), we obtain:

||j+l(t)I|Hf $ IlXju(O)IKf + Ilv 3(-)IlH + Ilvi(T)IIHr+1 + I j+lVj2 T)II H. (2.4.78)

Clearly, to finish the proof we only need to show

jIt and /t1 (2.4-80)

We first prove (2.4.80). By interpolating between the estimate

|IVjIIHr < 1 and |Iu(T)IIHK2 1o we obtain (2.4.81) A1+CO (T)

Note that the same estimates hold for vj+1 . Thus,

I|vj+1IvjI2(HT) HI oc ;$f (72+C1 (2.4.82)

By choosing - small enough, such that 1 + Ca < 2, (2.4.80) clearly follows from

(2.4.1). Finally, we turn to the proof of (2.4.79). Again by interpolation, we obtain

K 2 (i+l-r) K 2 -if-1 ___r _ )K2 - r ||j|H _1 j HK2 "jT)HK2 r_ (2.4.83) ( 1

71 The key point is our choice of f that ensures K2 (+1-r) < 2, thus by choosing a small K2 -r enough, we have K2 (F+1-r) + Ca < 2. Finally by (2.4.1) estimate (2.4.79) follows . K2 -r

2.5 Proof of Lemma 2.3.2: Multi Solitons Model

This section is devoted to the proof of Lemma 2.3.2.

The proof is involved, hence we first discuss some heuristics . Lemma 2.3.2 is the consequence of the following facts.

1. From previous work, [59],[65], the solution uj to (1.1.1) with initial data Xiloc(x-

x3 ,o)uo , j = 1,.. , m, blows up according to log-log law.

2. Assume some solution v to (1.1.1) blows up according to log-log law, and assume

to is close enough to the blow up time T+(v). Let F = F(t, x) be some smooth

perturbation. Then, the solution i to the following Cauchy problem

ii = -Ai - |If 2 ~)+ F, (2.5.1)

'D(0) = v(to),

still blows up according to the log-log law.

3. By our smoothness condition (2.3.20), we can show that solution to (1.1.1) with

initial data X1,1oc(X - x3,o)uo is smooth in the region Ix - x ;> - j=1,. . ., m.

4. For j = 1, ... , m the function uj X: (X - xj (t)) satisfies the following equation

2 iAtu3 = -- Au - |jj uj + F, (2.5.2)

where

F=- Vxi,1c(x - )dx + Axii0 c(x - xj)u(t, x) + 2Vx, 1 ,ic(x- xj)Vu

2 + (x' 10' - Xiioc)(X - Xj)tu1 u. (2.5.3)

72 The idea is the following loop argument, which is false of course, but it can be made

rigorous by bootstrap argument. If we assume u3 evolves according to the log-log law for each j, we basically know that u is smooth in the region minj{IX- XI} > 1/2. And

since Vxi,1,c(x - xj) and Xi,loc(x - xj) are supported in Ix - xj ;> , this implies that

F above is smooth. Thus fact 4 and fact 2 imply that uj actually evolves according

to the log-log law for each j. Thus, the assumption, that uj evolves according to log-log law, is right.

Let us turn to the details and a rigorous mathematical proof.

2.5.1 Outline of the Proof

Recall that the re-scaled time sj satisfies d =A. for j = 1,... ,m. The system (2.3.38) implies the following:

2 d 3II1 yb) ii I1 2 ~ d6,j IY b 2 E~j~ IY0b 3 =j 1i,... 7M d dsj j, yE) + (E2, YEb,)} = 0, j = 1, . .(. M. d (2.5.4) (Es -A2b) + (,A 2 E,)} = O, j = 1, ... M d d{ (E , Ab ) + (E, AEb)} = Oj = 1,... m. dsj 3

Now, by pure algebraic computation as in [66], one is able to write down almost the same modulation ODE for {bj, Aj, xj, ~'} for j = 1, ... ,m as in formula (71), (72), (73), (74) in [661, where ~'j(sj) := yl - sj. Basically, the only difference is that the

2 A E term in [661 is replaced9 by E(Qbj + E).

Modulation estimates

Now, by exactly the same argument as the proof of Lemma 5 in [591 we have the analogue of Lemma 2.1.14

9 Because in 1661, the solution u has the form u : (Qb + E)(x( )e-', thus all the term 2 E(Qb + E) is equal to A E. In our model , we cannot make this substitution and have to keep the term of form E(Qb + E).

73 Lemma 2.5.1. In the setting of Lemma 2.3.2, the following modulation estimates hold for t G [0,T], and jz=1,...,m:

1 + b C (fV|2 j2e-lI +F C"+C|E (Qs,+di||, (2.5.5) Aj dsj 1+1 dsj | < +

ILj- (Qj, I L+Q21 +11d

- (b2I y 2 1 (2.5.6) <6(a)( IVcjIe ( -q bj + I 126j 1IYI + C I |Vji 2 + rl-C77 + CIE(Qb 3 + F)1.

The proof follows exactly as the proof of (2.36), (2.37) in Lemma 5 of [59].

Estimates by the conservation law

Lemma 2.5.2. In the setting of Lemma 2.3.2, the following estimates hold for t E [0, T], for j = 1,. . .,m

I2(ej, Eb)+F2(Ceb8,)I 0(1 |Ve~I 2 + IJ|2-IYI) +FIt~C7o+CIE(QbJ+gi)|,(2.5.7)

I(ei, VE)I C6(c)(J IVEj2 + Jlk2e-yI + CP(Q" , + d1). (2.5.8)

Proof. The proof of (2.5.7) is exactly the same as the proof of (2.35) in Lemma 5 in

[591. The proof of (2.5.8) is a little different, since in [591 the authors use the zero- momentum condition which is not used here. A direct simple algebraic computation shows that ' P(Q, + E) - -(VEb,, 6-) + (VEb, ) -- 2(Vej, E-).

By a point-wise control IVEb, 1(y) $ e-KIIlyI and Cauchy Schwartz, one has:

|(VEsb, E)I < C( jEjj2e-Kjy )2

74 From here one has that |(VEj,ej)1 < IE|| 2I|Ve||2- Using the general functional analysis fact' 0

2 2 2 J i e-Kjy < K I iVE 1 + J I l e'2- ) the bootstrap hypothesis (2.3.39) and (2.3.42):

3 2 2 II IIL2 < JVEI1I(, + J |jI e-' < 6(a). we have that (2.5.8) follows.

Control of local quantity

We use the bootstrap hypothesis (in particular, the control of local conserved quantity) to show the following

Lemma 2.5.3. In the setting of Lemma 2.3.2, for t c [0,T] andj = 1,...,m the following estimates hold

(2.5.9)

IP(e) + Qbj| < A'. (2.5.10)

Proof. We only prove (2.5.9), and the second inequality follows by a similar argument.

Direct computation shows that

IE(e' + Qb,)(t) - A(t)Eloc (X (t), u(t))I

=A IE(ux1 ,jOc(x - xj)) - Elu(xj(t), u(t))j (2.5.11)

2 4

Note that by a standard Sobolev imbedding and by the bootstrap hypothesis (2.3.47),

'0This estimate holds for all H' functions, see (2.38) in Lemma 5 in [591.

75 we have:

1l,7 2 + jul4 < IlUI4 N(minjj{x-zX~oI}>}) < A ,o}4max J<3I5x-x.I<1

(We will choose N 2 large enough, and thus N1 < can be chosen large enough so that all the desired Sobolev embedding holds.)

On the other hand, by the bootstrap hypothesis (2.3.45), and the assumption on initial data (2.3.18), we obtain

A p Eh t s to c(t), U(t))w

We plug these two estimates into (2.5.11), and we obtain

( 1 )4 JE(di + Obj) I < A (1 + (2.5.12) maxj{Aj,o} )

Finally, by the bootstrap hypothesis (2.3.43), we have Aj(t) < maxj{Aj,o} for j

1,... , m. Thus, (2.5.9) clearly follows from (2.5.12).

Modulation estimates and estimates by conservation law, restated

In this section we summarize what we have found above.

Lemma 2.5.4. In the setting of Lemma 2.3.2, the following estimates hold for t E [0, T], and for j1, .. ., m

1 + b-71+ Idb C (VEJI2 + JleiI2e-yI) + I C, (2.5.13)

1 dx dsj 1 (E , L+Q2| - Aj dsj (2.5.14) 2 1 6()(J IV I2e ,-7)( (j J IEj 2- '), + C pj2+ 1,-C7

12(e-, Eb,) + 2(E2, Eb,)I (J IVejI 2 + lEI2 e-11) + rI-c7 (2.5.15)

76 I(E2, VEsb)I I C( IVE1V)(f2 + J |6Il2e-l). (2.5.16)

Proof. We just need to combine Lemma 2.5.1, Lemma 2.5.2, Lemma 2.5.3. l

Local virial estimate and Lypounov functional control

Below we combine the orthogonality conditions (2.3.34), (2.3.35), (2.3.36), (2.3.37), modulation estimates (2.5.13), (2.5.14) and estimates induced by the (local) conser-

vation laws (2.5.15), (2.5.16), following the work of Merle and Rapha81 to obtain the

analogue of Lemma 7 in [591.

Lemma 2.5.5 (local virial). In the setting of Lemma 2.3.2, the following estimates

hold fort C [0,T], andj= 1,...,m

Ab, 2 2 6 1(] 1VEj1 + Ie4I elyl) _l-c, (2.5.17)

where 61 is a universal constant.

Proof. This lemma is highly nontrivial, and it is actually one of the key elements in

the work of Merle and Rapha8l. However, by applying exactly the same argument, which they used to derive Lemma 7 in [59], one can derive

db > 61( IVEij 2 + lEil2e-lyl) - 7-c -CIE(Qb, ,+ej),j=I 1,...,m. (2.5.18)

Now using estimate (2.5.9) in Lemma 2.5.3, one obtains IE(Qj + ei)I < A, which

is negligible compared to r 1C, since we have (2.3.40). Thus ,(2.5.17) clearly follows from (2.5.18).

Next, we recover the Lyaponouv functional, which is essential to establish the

sharp log-log regime. This is the analogue of Proposition 8 in [591.

Lemma 2.5.6. In the setting of Lemma 2.3.2, the following estimates hold for t G

[0, T], and for j=1,..., m

J.1 2AI i1 21 11 7 I J{2fJ + eil-- +j A} (2.5.19) ds1 3 F~+ V - A3

77 with

F j i ( ) b +2(El,Eb,)2(E,,)(JQI2_Q2-2j 2 )O ~ j) 1 -- (2 .5 .2 0 ) (bfi(bj) - fi(v)dv + b{ 3 , A(Rb) - E, A(2.5)2

where b 12 (2..21 f1 (b) := IyQb + f yVis~b), (2.5.21)

E3 = E - bj , (2.5.22)

and #3j is a non-negative smooth cut-off function, = 1,..., m

OAj (X) = 0, lxi A , 1 JVO3j I

Ajx(X) = 1,l > 3Aj,

where A3 = A(bj) = F -a

Most parts of the proof follows directly the proof of Proposition 8 in [591, however, some extra technical elements need to be treated hence for completeness we will

explain the proof in Subsection 2.5.2.

We have the following control for the scale of the Lyapounouv functional:

Lemma 2.5.7. In the setting of Lemma 2.3.2, the following estimates hold for t E [0, T], and for j = 1 ... , m,

= C*(1 + 0(6(a)), (2.5.24)

where C* is some fixed constant.

Proof. Follow the proof of (5.15) in [591, one can derive 33 - bj. Further refined analysis, will give (2.5.24), see the formula between (5.24) and (5.25) in 159]. 0

78 Bootstrap estimates except (2.3.57)

So far, we already have all the ingredients to prove most of the the bootstrap estimates. In fact for (2.3.48), (2.3.49), (2.3.50),(2.3.51), (2.3.52), (2.3.53) one can follow

the arguments of Planchon and Rapha8l in [651, which we will review for completeness

in Subsection 2.5.4. Here we prove instead (2.3.54), (2.3.55). Actually we only show the details for (2.3.54), since (2.3.55) is similar.

Proof of (2.3.54). A direct computation (using a that solves (1.1.1)) shows that for

j .= m, .. Eloc(x j (t), u (t)) - -E (x j,o)|

<10 EOc (xj (T),u(t))Idr I Td (2.5.25) |o E dt (xj (t), u(t))|dt

where

E = ( 1VU12 _ U14 ) Vxo, 0oc(X - Xj(t)) dI, (2.5.26)

E2 = j XIVxo,loc(x -- x(t)) (2iQ AuVi + 2i+MIU 2 UVit) .1. (2.5.27)

Recall that XO,loc(X - x) vanishes for 1 K -x 1. Using the bootstrap hypothesis (2.3.44), and Sobolev embedding, we obtain

vu2 _ 4) HN(minj (x-xj,oI}!0) 1t 4

J Vxo(x - X) (Auvi - Vuftt + Iu4HN(min -xj,0 (2.5.28) Thus,

E1 + 2 + z ) N(minj{I-xj,oI}.>) (2.5.29)

79 By the bootstrap hypothesis (2.3.47), we have that

HN(minjx -xj,>,) maX3 Ajo}

Note also that by the bootstrap hypothesis (2.3.43), also < for all j.

By the modulation estimate (2.5.14), I- I < 1. Thus, -,'I = I dxiI < -, hence Aj dsjdt A7 dsj A3

.T 1TC

E1 + E2 < dt < AjO- A (2.5.30)

Here we use Aj(t) Aj,o, i.e. (2.3.43).

Finally we have for j 1, ... , m that

C,, Vp < 2. (2.5.31)

This is the analogue of (2.4.1), and it follows exactly the same proof that only relies on the bootstrap hypothesis (2.3.40), (2.3.41). Clearly (2.5.30) and (2.5.31) end the proof.

Propagation of regularity and end of bootstrap estimate

To end the bootstrap estimate, we still have to show (2.3.57), and it will be done in

Subsection 2.5.3.

2.5.2 Recovering the Lyapounouv functional under bootstrap hypothesis

This subsection is devoted to the proof of Lemma 2.5.6. We emphasize again that this proof follows the computation in [59] except for two points:

" We do not use the global energy, and all those A 2E terms in [591 are replaced by E(E + OQb).

" In the definition of the original Lyapounov functional 3 in (2.1.39), there is

80 a term IIQbI2 - |I62 + (i, E) + (e 2 , E) + 2eIIj, that is actually a constant in [59], thanks to the conservation of L 2 mass. In our case, since we analyze the

dynamics locally, the natural substitution is the local mass, which is no longer

a constant. So we need to show that the local mass is slowly varying.

We need the following two lemmas.

Lemma 2.5.8 (analogue of Lemma 6 in [59]). In the setting of Lemma 2.3.2, the following estimates hold for t C [0, T] for some universal constant 61 and j 1, ... , m

d / f f.2+\~ 1ejy 1f2Aj dsh 6y IV~I 2 +]I| 2e-) +CJ'b - W2 (2.5.32) with

f = b(s)|yQabII+ 1 y+1 + (Ej, AR 6) - (Ej, Aa). (2.5.33) f'()=4 2 2 YV~bj~bJ (~A~~

Proof. This lemma is one of the most fundamental points in [59]. We quickly recall its proof. If 1t)(Qb -eX( -))e()solves (1.1.1), then one is able to derive two equations for E := el + ie 2 , i.e. (2.1.23), (2.1.24). Then one takes the inner product of (2.1.24) with A(Eb + AW ) and of (2.1.24) with -A(-Eb + AQs) and sums. The detailed computation is displayed in Appendix B of [59]. Here we follow exactly the same procedure. We pick j = 1, ... , m and we recall that by definition,

X) (Qbj + e)(X 7 )e4iy = X1,oc(X - and Xi,loc(x - xj)u almost solves (1.1.1). One may derive similar equations for E6(y)= ej(y) + ie3 (y) as in (2.1.23), (2.1.24) with some extra terms in right hand side. Since

X1oc(X - xj) =13 in |x - xj| 3 2,3 these extra terms are supported in IyI -> --2Aj (i.e.

Ixx j). Since Qbj, 4bj is supported in jyj < F-, and by the bootstrap hypothesis (2.3.40), p-a < -, then when one pairs these equations with A(Eb, + ARJbj) or

-A(-b, + XA~bj), these extra terms automatically cancel.

As a consequence all the algebraic computation in the appendix B of [59] follow.

81 There is one more difference compared to Appendix B of [591. There the authors

use the energy conservation, (formula(4.17) in [59]), here instead we need to replace

term A 2 Eo in that formula by the term E(Ei + Qb,). Once this is done we follow the argument in 1591 to recover the virial type estimate

I&j 2 -CE(E-7+Qbj), j =1, ... , m. dsjfdf ;>6 51 (V12(fvu\ 2e-y)+ +crbJ - 2 JJAj (2.5.34)

By Lemma 2.5.3 and the bootstrap hypothesis (2.3.40), we obtain

IE(e3 + Qb,)I < A - < F, (2.5.35)

and as a consequence formula (2.5.32).

We have the following Lemma.

Lemma 2.5.9 (analogue of Lemma 7 in [59]). In the setting of Lemma 2.3.2, the following estimates hold for t E [0, T], and for j = 1,...,m

2 2 d{J5AIci} 1l _ +Zo _ f V1 . (2.5.36)

Proof. The proof follows as the proof of Lemma 7 in [59] except, as above, for replacing

2 A E with E(i -+Qb,), which is much smaller than Fi 0 by the bootstrap assumption, and thus negligible. l

The following lemma illustrates the fact that the local mass is slowly varying with

respect to the rescaled time variable sj.

Lemma 2.5.10 (Slow varying of local mass). With the same assumptions as in

Lemma 2.3.2, the following estimate hold

{ dsj~ 2 .b (2.5.37) rlo. d+2(,Fj,fj(2(Ej,jEbj)2)ObjEj|2}1f< 82 Proof. First we observe that

2 (2.5.38) {I IQbI2 + 2(E1, Y) + 2(f2 , 6) + I E1 } lI(b + Ej11.

By (2.3.28), one has the geometric decomposition

M 1 - u(t, x) = Qb, (X )e- Y + E. (2.5.39) j=1 Aj

Also recall that e(y) = x1,ioc(Ajy)cd(y) and that QOb is supported in lxi < 1. Note

that by the bootstrap assumption (2.3.40), - >> 1. We then obtain

IlQb, + IE|| (2.5.40)

=lQbjX1,Ioc(Ay) - eiX, 10c(Ay)2 - IIu(Xi,oc(X -- Xj))1I.

Thus,

2 ds{ Qj 12 + 2(Ej, Eb,) + 2(e2, Eb,) + J E1 }I (2.5.41) 1dxj I|IU112 + 2A NVX1,10c 11U 111U .

Recall now that dsj =A-3 dt~ The first term on left hand side is controlled by the modulation estimate (2.5.14), and the conservation of mass |Iu(t)11 2 = |1u011 2 ,

dsj|6 x - AAj1 ddx- (2.5.42)

The second term on the left hand side is controlled by the bootstrap assumption (2.3.47),

2A J Ivx 11,ocIIuIIvuI < A. (2.5.43)

We finally use the bootstrap assumption (2.3.40) to control Aj and the desired estimate easily follows.

Now we are in good position to finish the proof of Lemma 2.5.19. proof of Lemma 2.5.19. With lemma 2.5.32 and lemma 2.5.9, we follow the proof of

83 Proposition 4 in [591 and we obtain

J AJIE2 800 (bfi(bj) - b fi(v)dv + b{(e', ARibj) - (ej, Al)1

2 < -C ( IVEI + IEil2ey) + FI-C (2.5.44) Now plug in the estimate in Lemma 2.5.10 above and the desired estimate follows. D

2.5.3 Propagation of regularity under bootstrap hypothesis

This subsection mostly follows from the arguments in Section 2.4, indeed, the reason why we write Section 2.4 is to make this subsection more accessible. First, the analogue of (2.4.1) holds,

Lemma 2.5.11. In the setting of Lemma 2.3.2, the following estimates hold for t E [0,T]

dr(),< 2, (2.5.45)

j 1 \j(tp-

Proof. The proof of (2.5.45) only relies on the bootstrap hypothesis (2.3.40), (2.3.41), and it is similar to the proof of (2.4.1). l

We again introduce a sequence of cut-off functionsl fX,}IL

i 0, Ix - xj,ol < a,j =1 or 2, (2.5.46) I, - x3 ,o bl, j = l and 2,

such that al < b, < a,_1, ao = _ < 1, b = < The idea is ( again ) that we want to retreat from xi-iu to xju for each 1, showing that xiu has higher regularity than X1iju. We still use the notation vi = Xiu- The key to gain regularity in Section 2.4 is formula (2.4.3). Similar estimates also holds here: "We still use the notation of x, but the definition of X is different from Section 2.4.

84 Lemma 2.5.12. In the setting of Lemma 2.3.2, the following estimates hold for t E [0,T],

JIV(XoU(t))I 2dt < 1. (2.5.47) J T

Proof. First recall that by the bootstrap hypothesis (2.3.44), if x E suppXo, then minj{IX - x31} > n. Thus we need only to control

- ||(XOU) 11 . IT 0 fminjjj -j1500

Secondly, since u is bounded in L2 and T is bounded (indeed T < 6(a), see Remark

2.3.4, we need only to control

I T II7uII|. do fminj{|x-xj1}00

Indeed

IVU12 f To minj{jx-xjj 1}00 T m 2 (2.5.48) j I 7U12 + 10- l~j< IVu1 fT j= fi 0 n i- -5s

:=E1 + E2 .

The first term is controlled by the bootstrap hypothesis (2.3.47) and (2.3.43). In fact we easily have:

E1 ,

In last step we have applies (2.5.45). The second term E2 is estimated by using the Lyapounov functional Jj in (2.5.19). This is actually one of the key estimates in [581.

85 We recall it here. Note that

2 1 z- X iy Xi,loc(x) 1 for lxi < - and X1,ioc(x - xj) = ++(Qb c(x )e.

We make the observation that since Aj3 < e-- by the bootstrap hypothesis (2.3.40),

when Ix- xj > we have Qb( #) - 0, since SUPPQb C {II < -}. Similarly, in this region the radiation term (bj(xjs( ) also vanishes. Thus, for j 1,... , m

2 2fVU1 I I11 +~lvi- (XXi ei~tjI12 00( (2.5.50) / / =A2 2 A2f 2 Vi12 1V631 Jsoc;AJ jIXI< v~

As a consequence

pSj(t2 ) E2 <- 32*) 23suptE[ O,)fJj - bj < 1. (2.5.51) Jsj (ti)

In the last step, we use (2.5.19) and the fact that the Lyapounov functional Jj ~ bj,

see Lemma 2.5.7. This is enough to end the proof. l

We now again use I-method to recover a rough control:

Lemma 2.5.13. In the setting of Lemma 2.3.2, the following estimates hold for t E [0,T], N2+or |IU(t)IIgN2 (2-.--52)

Proof. Without loss of generality, we only show (2.5.52) for t = T. We remark that

if one defines ~ min{Aj}, then Ilu(t)II ' , and (2.5.52) is equivalent to

HUM- (t)N2+o-_

The proof is almost the same as the proof of (2.4.37) in Lemma 2.4.2. Following

Remark 2.4.11 we only need to show that we can divide [0, T] into disjoint intervals

UIk,h, such that

* IIu(t)IIH- 2k for t C 'k,j

86 * Ik,hJ ~ -1, Vk, h. ln(T) '3. Sl{kh(} I

These clearly follow from the facts below. Fix j, we can divide [0, T] into disjoint intervals UI such that

* Aj(t) 2 --k for t E Ik,j

3 * kh} (I n A3(T))

Now we prove these facts. Then by the bootstrap hypothesis (2.3.43), we can divide

[0,T] into 0 = tko < ... tk < ... tK(T) = T such that Al(t) ~ 2-kfor t E [tk,tk+1, then relying on the bootstrap hypothesis (2.3.40), (2.3.41), similarly as in the proof

of Lemma 2.4.8, one can show that tk+1 - tk vkA2(t) ~/W2-2k. This estimate

1 is enough for us to further divide [tk, tk+1] into disjoint intervals U= 1 [r, -r+ ] such

that our desired Ik,h can be chosen as [k, -rkh+'], see (2.4.16),(2.4.17). l

We now remark that the bootstrap estimate (2.3.57) follows from the lemma below.

Lemma 2.5.14. In the setting of Lemma 2.3.2, the following estimates hold for t E [0,T],

1. Vv < 1, IIV1I|HV $ 1-

2. If there holds estimate for some r > 0, and some 1 < i < L - 1

IIVJIIHr ,< 1, (2.5.53)

then we will have a gain of regularity on vi+1,

V ~ < 2K2 - )- 1 + r, ||VA|H" <1 (2.5.54)

The proof follows as the proof of Lemma 2.4.3, Lemma 2.4.4, Lemma 2.4.6, Lemma 2.4.7, see also Proof of (2.2.5) in Lemma 2.2.4.

87 2.5.4 Proof of bootstrap estimate except (2.3.57), (2.3.54),(2.3.55)

The proof of these estimates can be found (up to a small modification) in [651, [68] and the references therein. See in particular Proposition 1 in 1681. We quickly review those estimates for the convenience of the readers.

Proof of (2.3.48)

We bound the left hand side of (2.3.48) by Ca. The control of bj directly follows from the bootstrap hypothesis (2.3.41), which implies 0 < bj < 107r < b Inlsj,o 3 ,o and we note that by the initial condition (2.3.14), we have bj, o K a. Thus IbjI < a < a. The control of |IVe 112 follows from the bootstrap hypothesis (2.3.42), therefore we have ||Ve~II2 $ 6 < a. The control of Ilg 112 comes from the L2 conservation law. Indeed

2I&|| || IQl2 - lu112 + 0 , IE*,Qbi)) j=1 j=1 (j=1 Note that

(lei, IQb,1) J I V& 2 + J lI 2 e'-M, (2.5.55)

IIQbj,11 = IIQI12 + O(Ibjl2) (2.5.56)

IIUI12= IIuo||2 (2.5.57)

Thus the control of |IVe 112 follows easily from the bootstrap hypothesis (2.3.42) and our choice of initial data.

Proof of (2.3.49),(2.3.50)

We first show (2.3.50). It follows from the local virial estimate (2.5.17) and the control of the Lyapounouv functional (2.5.19). We first show the lower bound of (2.3.50).

88 Note that by (2.5.17), one obtains db> -I p-C, which implies dsj - , ihimle

((1-C'7) de b 3 < 1. dsj )

3tr 8 Recall that sj > j,0 --e *bj,o , thus we have:

(_-C__) 7r(1-Cn) 7r(1-Co) 4 e bi < sj - j,o e b , (2.5.58) s + 3 0 < sj + e 6b, 0 < s'.

Therefore one obtains bj,o > 'rs, which is the lower bound of (2.3.50).

Now we turn to the upper bound of (2.3.50). The control (2.5.19) for the Lypounov functional directly implies dS- (2.5.59) ds -Cb 3 ,.

From Lemma 2.5.7, we know that there exists a C, such that

= C*(1 + 0(j(a)). (2.5.60) 3

Let g3 =J,5, then by combining (2.5.59) and (2.5.60), one obtains:

dg_ 1 dJj 1 1 < -Cg x -C*e gi (2.5.61) ds3 2 sj gj 2.

One obtains d 7,-I-*(I + 0(0(a))) Ie g > 1. (2.5.62) d5 k Thus e(1+0(6(-))) -(1+O(.(-))) e js > Sj - sj,O + e 63,0 > Si, (2.5.63)

3er where we recall that sj,o= e~b,0 This implies the lower bound of (2.3.50).

Now we turn to the proof of (2.3.49). By (2.5.58), (2.5.63), we have

37r b 4-7r (2.5.64) 4 In s3 - -- 3 In sj

89 We only show the upper bound of (2.3.49), the lower bound will be similar.

The computation follows from the proof of Lemma 6 in [671, in fact formula (2.68)

mentioned in that lemma is exactly the upper bound we want.

The modulation estimate (2.5.5) plus the bootstrap hypothesis (2.3.42) imply that

I dA - 1 + (2.5.65) <017.. 3 +bj A3 dsj j dA Thus d= -(1 + O(6(a)))bj. dsj

and from here Aj ((t)) < Aj (sj (t)) = Aj (sj) < Aj,o + IsSO 2lni s'

(note that according to our notation Aj,o = Aj(sjo).) A direct calculation implies that

1 7r s -In A (sj) ; -- In A (sjO) + 2 3lIns ,

37r which further implies that sjA(sj) e . (See (3.106), (3.107) in [67] for more details).

This already implies our desired upper bound.

Proof of (2.3.51)

This is exactly the step 3 of the proof of Proposition 5 in [591, called pointwise control of c by b.

90 Proof of (2.3.52)

The modulation estimate (2.5.5) plus the bootstrap hypothesis (2.3.42) imply

1 dA - 1 +_bd 1 < Cr,% (2.5.66) Aj dsj - bj

Thus dA- dA = -(1 + O(J(a)))bjAj < 0, dsj

(since bj > 0 due to hypothesis (2.3.41)). This clearly implies (2.3.52).

Proof of (2.3.53)

This easily follows from the modulation estimate (2.5.6) and the bootstrap hypothesis

(2.3.40) and (2.3.41), as in [65]. We quickly review it for completeness.

For all t C [0, T],

dx I dt |xi - xjo| < Iot dt (2.5.67) it dxj s,(t) 1 dxI d 1 dt = ds; A = .0 Aj dsA

Note that by (2.5.6), < 6j-|3(a), and by (2.3.40) and (2.3.41), one has Aj(s) <

S 00. Thus one clearly has lxi - xj,ol < 6(a), which easily implies (2.3.53).

Remark 2.5.15. The above computation indeed shows

T+(u) I Idt < o, (2.5.68)

which of course implies limteT+ x3 (t) exists.

91 2.6 Proof of Main Theorem

We will need several parameters throughout the whole section.

1 > a 0 > a1 > a 2 > 0.

Recall our goal is to construct log-log blow up solution u, which blows up at m prescribed points xl,,, ...Xm,,, and has asymptotic near blow up time as log log blow up solutions. We will still focus on dimension d 2. And since we have scaling symmetry, we assume without loss of generality

|xj,, - x3 ,| > 20,j j'. (2.6.1)

It is clear that by Lemma 2.3.2, all initial data uO describe in Subsection 2.3.1, the associated solution u have the following geometric decomposition for t < T+(u).

U(t, X) = IA( bj (x - t ) e-" j(t + E(t, x), (2.6.2) such that for each j,

X1,ioc(X - Xj)u(t, X) A (Qbj + E( - )e-"iYj, and E (1 - E> X1,loc)u in the region {xIIx - xj,ol > , j =1,...,m} and bounded in HNI in this region.

To finish the construction in Theorem 1.2.1, we need to construct initial data uO such that

9 (a) The associated solution u blows up in finite time according to the log-log

law.

* (b) The m points blows up simultaneously, i.e.

Aj (t) )0, j = 1, ... , M. (2.6.3)

92 * (c) The blow up points are as prescribed, i.e.

Xj (t) >-T) X,.'. (2.6.4)

It is easy to check once we get (a), (b), (c), then other requirements in the Theorem

1.2.1 will be automatically satisfied. To see this , just observe

1. Aj -+ 0 -> bj -+ 0 since we have (2.3.49),

2. Thus, Qb, converges strongly to Q in the sense of (2.1.12),

3. E& is bounded in L2 by (2.3.48) and converges to 0 in H 1 as bj -+ 0 by (2.3.51).

Now let us turn to condition (a), (b), (c). Indeed, Lemma 2.3.2 already implies for all data uO, the associated solution u will blow up in finite time with log-log blow up rate nnI nnT - tj IIVu(t)11 2 ~ t (2.6.5)65

Not all data described as in Subsection 2.3.1 will give a solution satisfy condition (b),

(c). Morally speaking, the initial data in Subsection 2.3.1 has two types of parameters, Ai,o and xj,o. Aj,o describes how concentrated the j-th bubble is and the xj,o describes the initial position of jth bubble. Lemma 2.3.2 says the m bubbles evolves with very weak interaction with each other. Thus, we need to choose A3 carefully to make all the bubbles to blow up at the same time, and choose initial position xj,o to make the bubbles blows up at the prescribed points xj, ,. This is achieved by certain topological argument. If one does not want to prescribed the blow up points and only wants to get condition (b), an argument similar to the topological argument in [16] will suffice.

However, if one wants to further prescribe the blow up points, then the problem is actually more tricky, since the choice of Aj,O and xj,o will be coupled.

Before we continue, we point out it is not hard to show limtT+ Xi (t) exists.

Indeed, as remarked, we actually have

IT I| |xIdt < oc. (2.6.6) 0 T

93 The hard part here is to prescribe the limit, i.e. for given xi,o, ...xm,oo, we want to construct solution such that lim xj (t) = Xj,.. (2.6.7) t-+T+

2.6.1 Preparation of data

Let us first fix a smooth-cut off function

Xx = 1, 1(2.6.8) 0, xI > 2.

Let uO = g-Qb( x) + co as described in Subsection 2.2.1. Note Lemma 2.1.17 and

Lemma 2.2.4 hold for those data. We need to choose K 2 large enough for later use. We also require uO be radial. The associated solution will have the geometric decomposition

u(t, x) = At) (Qb(t) + E(t))( At)e-i(t) (2.6.9) such that the conclusion of Lemma 2.1.17 and Lemma 2.2.4. And in the spirit of

Remark 2.1.18, we may further sharpen the initial condition such that u further satisfies -e(1+ 2)

Recall a 2 is the parameter we have fixed at the beginning of this Section.

Note A, b, Y, E(t) depends on t in an continuous way. u(t, x) can be understood as a family of data continuously depending on t. By (2.1.52) and (2.1.50), we obtain - >

2b> 0, since At = A2 AS < 0. Thus, the map from t -+ A(t) is a homeomorphism.

Thus u(t, x) in (2.6.9) can also be understood as a family of data indexed by A. To summarize, we have a family of data index by A small enough,

UA(X) = (Qb(A) + E(A))(-)e- (2.6.10)

94 with

A+b 0, ee b < A

IVE(t)I 2 + IE(t)1 2 < pl-a2

1 I 0000 A2I XO,oc(X - x(t)) (Vt(t) 2 _ - 4 I 1 b(t)

A(t)I JXoioc(X - X(t)) (Vu(t)i(t)) I 00, IIu(t)IIHN2(Ix-1 I, ) < a/10, (2.6.11) and b, 7, c depending on A continuously. Recall XO,1oc is defined in (2.3.1).

We will consider a family of data

UO,A,x UA1,o,A2,0,--Am,,o,x2,,-.XmO (2.6.12) = x(X - Xj,o)u A (x - Xj,0) j=1

Here UA is defined as in (2.6.10). And we also require 1xj,o - 10, j $ j', since we have (2.6.1).

Now let us consider (1.1.1) with initial data uO,A,.. Lemma 2.3.2 will work for uO,A,.. Thus the associated solution ux.,, will satisfy the geometric decomposition in its lifespan

X) 1 (X Xj A,+)e- , (2.6.13) and all the bootstrap estimates in Lemma 2.3.2 holds. For notation convenience, we will write uA,, as u, write Aj,A,x as Aj, write xj,A,x as xj, write 7y,\,x as yj.

Again, in the spirit of Remark 2.1.18, we can further sharpen the condition on

(2.6.11), i.e. make a 2 small enough, such that

(1+al)7r e bj < Aj < e-e j (2.6.14)

95 2.6.2 Log-log blow up and almost sharp blow up dynamic

We first show the solution u with initial data ux,, blows up according to the log-log law, indeed, we show every bubble itself evolves according to the log-log law, and for later use, we need the almost sharp dynamic, and keep in mind condition our data has already been sharpened so that (2.6.14) holds for the associated solution.

Lemma 2.6.1. Let u be the solution with initial data ux,, as in (2.6.12), then for each j = 1, ... m, there is a T such that

AO ln Iln A,o = 27r (1 + O(ai)) Tj, (2.6.15)

Aj (t) 2 In I n Aj (t)| 2 r (I + 0 (a,)) (T - t). (2.6.16)

In particular, since the blow up rate is modeled by min Aj, we have that the solution blow up in finite time T+ and

LnIln(T+ - t) (2.6.17) 11'7~t)12 - T+ _ t

Remark 2.6.2. We implicitly require a to be small enough, as the whole paper.

Remark 2.6.3. Note Lemma 2.3.2 says the m bubbles evolves according to the log- log law without really seeing each other. Tj is the time that the jth bubble which is supposed to blow up. The solution will blow up at T+ = mini T, and the dynamic will be stopped at T+. In particular, if Tj > T, then it means j'th bubble 'blows' up faster than jth bubble, though they may both not blow up.

Remark 2.6.4. The proof of Lemma 2.6.1 needs to use the bootstrap estimate rather than bootstrap hypothesis in Lemma 2.3.2 since we need to get control in term of

0(a1 ). It is not hard to see one can argue as the proof below, and with bootstrap hypothesis rather than bootstrap estimate to show

A ln I ln A,,o I ~ T, (2.6.18)

Aj(t) 2 ln IlnAj(t)j ~ (T - t) (2.6.19)

96 which in particularshows the associated solution u blows up in finite time T+ < 6(a).

Proof of Lemma 2.6.1. We will follow the computation in [591, which is used to show the exact log-log law. 1 dA- 1 - + bl F-I1., -C7 (2.6.20) Aj dsj ba which immediately implies

1 diA- (1 - J(a))bj < -y I d (1 + J(a))bj. (2.6.21) Aj dsj

Note also (2.6.14) implies

b - (1 +0(ai))ir. (2.6.22) InJln Aj| Thus

d 2dA- dA- A InIln Ail =2 2As Iln Aj (1 + J(a)) = 2 (1 + (a)) Ad Iln . (2.6.23)

Now, plug in (2.6.21) and (2.6.22), and choosing a small enough, we get

A In In Ai = -27r (1 + 0(ai)). (2.6.24) which immediately implies (2.6.15), (2.6.16). E

2.6.3 A quick discussion of blow up at the same time

Now, we have a family of data ux,,, and the strategy is to adjust parameters to make the m bubbles to blow up at the same time at the prescribed position. If one only wants to make the m bubble to blow up at same time and does not track the final blow up points, then a topological argument similar to the one in [161 will be enough. We quickly illustrate this. We will fix x1 ,, ...xm,o and A1,0 . And we will adjust A 2 ,0 , ... Am,o to make the m bubbles blow up simultaneously.

Lemma 2.6.5. Fixed x1,o, ... xm,o and A1,0 , (]xj,o - xj',ol > 10,j $ j'), there exist

(32,0, ...-m,o) C [(1 - ao)Al,o, (1 +ao)A1,o] m1 such that, the associated solution u to

97 (1.1.1) with initial data UriOJO,..IOXiO,..xmo as in(2.6.9), will blow up simultaneously at m points, i.e. Aj(t) ~ Aj, (t), t E [0, T+). (2.6.25)

For notation convenience, we write U IO,2 3O,..On,xIO,,...xmO as uo2,... 3m, and we fur-

ther write UJ 2 O,..3m,O as uO. We rewrite (2.6.13) as

1 U'3(t, X) - E Qbt) x-xt) )ei4(t + (t, x). (2.6.26)

Now we prove Lemma 2.6.5 by contradiction. Assume (2.6.25) is not true for any

3 E [(1 - ao)Ai,o, (1 + ao)Ai,o]"m 1 . We consider several maps as following:

* Let F(t, 3) F(t, /32,..., Om) A2,(t) A3,,(t) Am 13(t).

* Let Ta be the first time F(t,/3) hits 0[(1 - ao), (1 + ao)]m' 1 .

* Let G(/3) be F(T,, 0).

Here 3 E [(1 - ao)Aj,o, (1 + ao)A,o]m-1.

Since we assume (2.6.25) is not true for any / E [(1 - ao)Al,o, (1 + ao)A,ojml, then To is always well defined, i.e. Tg < oo. The key point here is that T depends on 3 continuously. Assume this for the moment and let us finish the proof by deriving a contradiction. Note that F is clearly continuous (by standard well posedness theory of NLS) . Since we assume that To depends on 3 continuously, G is also continuous. Make the observation To = 0 for 3 c a[(1 - ao)Al,o, (1 + ao)Aj,o]m'4, and then it is easy to see Gjal(1-ao)eo,(1+ao)Ao]m- is an heomorphsim from 0[(1 - ao)Ai,o, (1 + ao)Aj,o]m 1 to &[(1 - ao), (1 + ao)]"m'. Then we have constructed a continuous map from [(1 - ao)Al,o, (1 + ao)Ai,o"- 1 to 0[(1 - ao), (1 + ao)]m 1 such that its restriction on [(1 - ao)Aj,o, (1 + ao)A,o]"m- 1 is a homeomophism, which is clearly false by classical Homology theory. A contradiction!

We need to check Tp does depend on 3 continously. Note that by LWP of NLS, the map F(t, 3) is continuous and differentiable. Thus, to show that TO depends on 3 continuously, we need only to show that OtF(T, /3) points outward [1-ao, 1+aolm.

98 In order to show this, without loss of generality, we assume 1 T ) = 1 + a0 and check

d(A,') (Tp)>0 dt A > This clearly follows from the fact that

i d 1id 1 d A2,0- -dA > A2,0 dt ' A,, dt 1 3 0, which is equivalent to

1 d 1 d - ( dA1) 3 > -A ,0, (2.6.27) (A1,O)31 ds, ' - (A2,,)31 ds2 2 ' when t = T3 . Note that by (2.6.21) and (2.6.22) , (again, we will choose a small enough), 1 d Ajo = (1 + O(ai)) (2.6.28) 3 2 (A3 ,O) ds ' (A3 ,3) In In Aj, 1 (

Note (T3) = I+ ao, and ao > a,, (2.6.27) follows once one plugs in (2.6.28). This concludes the proof of Lemma 2.6.5.

2.6.4 Prescription of blow up points

To prescribe the blow up points, i.e., to make the solution blows up exactly at the given points x1,o, ...xm,o is more tricky. We will need a topological argument inspired by [53], see also [651. Morally speaking, we are dealing with a family of initial data with parameters A1,0, A 2 ,0 , ...Am,o, xi,o, ... xm,o, and the goal is to adjust those parameters to make the solution blow up at m points and the m points should be the given x 1,2,... xn,. The analysis in Subsection 2.6.3 basically says for any given initial parameter x1,o, ... xm,o, one will be able to find A,o, ... Am,o to make the solution blow up at m points. If one can choose A, 0, ... Am,o according to x1 ,o, ... m,o in a continuous way, then the argument in [53] will be able to help us adjust x1 ,o, ... xn,o to make the m blow up points be exactly the prescribed xi,2,... Xm,O. However, with our arguments in Subsection 2.6.3, the choice of A1,O, ... Am,O is not even uniquely determined by the

99 X1,0, ... Xm,0. Though this can be somehow fixed, it is very unclear whether one can choose A1,0, ... Am, in a continuous way 12 according to x 1,o, ... xm,o. Before we continue, let us first make several important observations

" The sharp dynamic of log-log blow up is known, we should make full use of it.

* The impact of parameters of x1 ,o, ... xm,o is of lower order than A1,, ... Amo.

Our strategy is to choose all the parameters x 1,0, ... xm,o, A1,o, .Am,o simultaneously to make the solutions blow up at exactly m points x 1 ,,,... xm,. Finally, at the technique level, in [53] and [65], they rely on the following topological lemma (they call it index Theorem).

Lemma 2.6.6. Let f be a continuous map from R' to R', let r > 0 and suppose

If(y) - yI < IyI, Vy E &Br (2.6.29)

Then there is yo c Br such f(yo) = 0.

We will need a modified version

Lemma 2.6.7. Let f be a map from Q c Rn to Rn. Let Q be a convex domain and let

&Q be a closed surface which is homeomorhpic to the sphere. We assume the original point is in the Q. Let us further assume for each y in 0Q, we have

0 ( {(1 - t)y + tf(y)It E [0, 1]}. (2.6.30)

Then 0 c f(0).

We will prove Lemma 2.6.7 in Appendix A.2, we point out Lemma A.2 actually implies Lemma 2.6.6.

121f one wants to direct borrow the arguments in [531, one will need an maximal principle type argument, which says the following: Fixed x 1,0 , ..xm,o, let A 1,0, ... Am,o be chosen such that the m bubbles blows up at the same time, then if one further adjusts A1 ,0 to be smaller and keep other parameters unchanged, then the first bubble will blow up first. This is not clear in our setting, and we even think this argument may not hold for log-log blow up solutions.

100 Now, we turn to the prescription of blow up points. As previous mentioned in (2.6.1), we assume xj,, - xj,o I > 20,j # j'. (2.6.31)

We will still consider the data as in (2.6.9) and we will fix A1 ,0 and adjust parameters A2 ,0 , ... Amo, xi,o, ... xm,o to make the m bubbles blows up at the same time in

Xi,00, ... , Xm,oo -

3 Lemma 2.6.8. Fix A 1,o, there exists (/2, ...f m, d1 , ... dm) E [-aoAi,o, aoAi,o]m-l x m (B1 ) , (here B1 C R2 is the unit ball) such that the associated solution u to (1.1.1) with initial data

UA,= uAo,Ao+32 ,..A1,o+,3 0 ,xio+di,...xm,o+dm

will blow up at m given prescribed points x 1 , ,..xm,,,, Z.e.

lim Aj(0) = 0,) 1, ... M. t-+T+(u) (2.6.32) lim xj (t) =- xj'0, = , .. . t-+T+(u)

Here, we use Aj, xj to denote Aj,x,, Ajx,x for notation convenience.

Proof of Lemma 2.6.8. As we previously did in the proof of Lemma 2.6.5, we write

UA1,0,A 1,0+/32,..A1,o+m,o,x,o+d,....xm,o+do as uf 2 ,... 13m,di,...dm. And we further write u82,....3m,di,...dm as uA, where A = (/2, ... 3m, dl, ... dm) E Rm- x R 2" We rewrite (2.6.13) as

1 x - x (t) uA(t, X) = Z QbjA(t)( - 3 ,A )ei TA(t) + xA(t, x). (2.6.33) j=1 Aj,A(t) AjA

Let TA be the blow up time of uA. We now consider the following map:

1 F : [-aoAi,o, aoAio]m x (B1 )m - R"'- x R2,

F(A) := (Y2,A, ---Ym,A, Zl,A, -. Zm,A) (2.6.34)

Yi,A = Ai,A(TA) - A1,A(TA), ZiA = Xj,A(TA) - Xico, i = 2, ... m, j = 1, ... , m.

101 2 Here B1 C R is the unit ball. And Aj,A(TA) and Xj,A(TA) are defined as

Aj,A(TA) = lim AjA(t),j = 1, ... m, t +TA (2.6.35) = lim X,A(t), j = 1, ., m. Xj,A(TA) t-+TA

A),A(TA) is well defined as we have (2.6.21), which implies Aj,A is strictly decreasing.

Xj,A(TA) is well defined as mentioned in Subsection 2.6.3, see also Remark 2.5.15. The point is

Lemma 2.6.9. The map A -+ Aj,A(TA) and the map A - Xj,A(TA) is continuous.

We will prove Lemma 2.6.9 in Appendix A.3.

Note if T(A) = 0 for some A, then UA is the desired solution which blows up according to log-log law at exactly m prescribed points.

Lemma 2.6.9 implies the map F is continuous and we will use Lemma 2.6.7 to show m 0 E F([-aoAl,o, aoA,o]"m x (B1 ) ). (2.6.36)

To achieve this ,we need to show if A in O{[-aoAj,o, aoA,o]"1 x (B1 )'m}, then

0 {tA + (1 - t)F(A), t E [0, 1]}. (2.6.37)

Note if A c &{[-ao0A,o, aoAj,o]"' x (B1 )7"}, then at least one of the following holds

(recall the notation A = (#2, ...1,))

" Case 1:1,3jI= a0A1 0, , for some j=2,..., m.

" Case 2:1d 3= 1, for some j = 1, ... m.

We first show (2.6.37) holds in Case 2. Indeed, by bootstrap estimate (2.3.53) in

Lemma 2.3.2, we have sup

20Il. (Recall our notation 7(A) := (Y2,A, .. Ym,A, Zl,A, ... Zm,A)). Since ldjI = 1, this implies

td + (1 - t)zj,A # 0,V E [0, 11

102 In particular (2.6.37) holds in Case 2.

Next we show (2.6.37) also holds in Case 1. Without loss of generality, we assume

/32 =(ao)Al,o. Using Lemma 2.6.1, we can find T1,A, T2,A such that

j,A(t) In ln A\,A(t)= 27r(1 + O(a,))(Tj,A - t), t < TA,

T1,A = 2ir(1 + O(ai))(A 0 InIlnA,o), (2.6.38)

2 T2,A 27r(1 + 0(ai)) ((1 + ao) A2,0 ln I ln(1 + ao) A,o)I)

(Note this also implies TA < min(T,A, T2,A), and since a, < ao, T1,A < T2,A). Since ao >> a,, it is easy to see

inf (A2(t)In~lA(t)I - Aia(t)IlnA 2 ,A(t)I) <0 (2.6.39) t

(Note it is important here we have < rather than <).

In particular, we have A2,A(TA) > A,A(TA) , ( since TA < T1,A).

Thus, since 02 > 0, and Y2,A = A2,A(TA) - NA(TA) > 0,

0B2 + (1 - t)y2,A # 0, t E [0, 1]. which recovers (2.6.37). This concludes the proof.

103 104 Chapter 3

On weak convergence to the ground state

This chapter is devoted to the proof of Theorem 1.3.1, Theorem 1.3.2, and Theorem

1.3.4. We first briefly illustrate the idea in the proof , see Section 3.4 for a more detailed overview of the proof.

3.1 Introduction

Ever since the work of Kenig and Merle, [41], [421, there is now a road map to obtain results such as Theorem 1.3.1, which includes three ingredients:

1. concentration compactness theorems

2. variational characterization of ground state

3. rigidity theorems

Concentration compactness relies on the study of the linear operator eitA and reme- dies the lack of compactness of classical Strichartz estimate caused by the symmetries of the system.

The concentration compactness will help us reduce the study of the original prob-

105 lem to the study of so-called almost periodic solution, i.e. solution of the form

I X - x(t) u(t, x) = Ad/2 ( Pt( )eix (t), A(t) > 0, x(t), (t) E R 2 ( Ad/

{Pt}t precompact familily in L 2 (Rd).

Such strategy is also called Liouville Theorem in the literature, see for example, [561.

We will give a brief review of the concentration compactness in Subsection 3.2.3.

Variational characterization will help us understand why the ground state Q is special, thus , will help us see the profile Q in the study of (3.1.1). Indeed, the ground state is closely related to the energy, (1.1.7). For any H1 function f such that

E(f) < 0, If 112 11Q112,

f must be essentially ground state Q. Recall that the energy term will naturally appear in the virial identity (1.1.11), which is crucial for the long time dynamic of (1.3.1).

Rigidity theorems will tell us the so called almost periodic solution is special, and one may expect a powerful enough rigidity theorem should fully characterize solutions to (1.3.1) of type (3.1.1), though we cannot prove this here. We will rely on Merle and

Rapha81's results on zero energy solution to (1.3.1) for rigidity argument, see Section

3.3 for a review of Merle and Rapha8l's work. Roughly speaking, an almost periodic solution is some "minimal" element which cannot be split in two piece, whereas zero energy solution, which is not solitary wave Qeit, satisfy log-log dynamic and has a trend to decouple the mass. This would give a strong constraint for almost periodic solution.

It remains to combine those ingredients, and there are two immediate questions.

* Why is nonpositive energy solution relevant in the study?

* We are at the regularity of L2 , how to even make sense of energy?

Virial identity, (1.1.11), indicates the answer of the first question, in particular for the relevance of negative energy solution. To answer the second question, or

106 more precisely, to apply virial identity in the L2 setting 1, one can imagine certain

frequency truncation and space localization are needed, and it is well handled, when

Dodson, [20], proves the scattering result for solutions to (1.3.1) with mass strictly

below ground state. Roughly speaking, let fi be a truncated and localized version of

almost periodic solution u. If the energy is coercive on ft, then one can use Dodson's

argument to conclude ii must be trivial. If the energy is not coercive on ii, then after

certain rescaling, one would be able to see non-positive energy solution, which will

help us to combine the three ingredients in the Kenig-Merle road map.

Now let us go to details in the rest part of the chapter.

3.2 Preliminary

We present the preliminary for this work here, experts may skip this section in the reading.

3.2.1 Local well posedness (LWP) and stability

Classical Strichartz estimates

The local well posedness of (1.3.1) is established using the classical Strichartz Esti-

mates. We recall them below. Consider the linear Schrddinger equation:

Zit + Au = 0, (3.2.1) u(0, x) = uo E L 2 (R d).

We use e" to denote the linear propagator. One has estimates

lie OIIL2(d+2)/d $ lU 0 Lb (3.2.2) tx

4 ei(t-s)Af (s, x)ds n ,

2 'Indeed, in L , not only energy does not directly make sense, the extra integrability xu G L 2 is also missing.

107 We refer to [7] [401,[711 and reference therein for a proof.

Remark 3.2.1. Strichartz estimate holds in more general case, indeed one has

/tfei(t-s)~f (s, x)ds < 11f ILL'.Li;" (3.2.4) 0 IL qLr t t X where (q, r), (, ) is admissible in the sense 1+ = and (q,r, d), (,,d) (2, oo, 2). q r 2 We fix (q, r) (q, f ) = (2(d+2) 2(d+2)) for simplicity. Similarly, the local well posedness and stability holds in more general sense.

In the rest of this section, we quickly recall the classical results in the local well posedness theory without proof. We refer to [81, [101, [71], [71 and the reference there in for a proof.

Local existence

Theorem 3.2.2. Given uO in L2 , there exists T = T(uo) such that there is a unique solution u(t, x) to (1.3.1) with initial data uO in the following sense:

u(t, x) = eit1uo + i ei(tr)A(|u|4/du(r))dT, t C [0, T] (3.2.5)

2 Formula (3.2.5) holds in space C([0, T]; L ) n L (+2)/d( [0 ,T] x Rd).

Blow up criteria

We have the following blow up criteria.

Proposition 3.2.3. Given a solution u with initial data uO, assume (T, T+) is the maximal time interval u can be defined on, if T+ is finite, then

I|IIL2(d+2)/d[,T+)XRd = 00 (3.2.6)

Similarly results holds for T.

Now, we can define the notion of scattering.

108 Definition 3.2.4. We say a solution u to (1.3.1) scatters forward if T+ = oc and

IUI|L2(d+2)/d[O,T+)xRd < oC.. Similarly, we define the notion of scattering backward. If u scatters both backward and forward, then we say u scatters.

Small data theory

Proposition 3.2.5. There exists co > 0 such that if the initial data uo C L 2 and satisfy

I UOIIL2(d+2)/d < E0 (3.2.7) ie t,x then the solution to (1.3.1) with initial data uo is global and one has estimate

|2(td+|| a2)/d < ||U01|2. (3.2.8)

Remark 3.2.6. By Strichartz estimate (3.2.2), it is clear that (3.2.7) holds when

|IuoI 2 is small enough.

Stability

We state a stability result about (1.3.1). This kind of argument is standard nowadays.

One may refer to Lemma 3.9, lemma 3.10 in [13].

Proposition 3.2.7. Let I be a compact interval, and 0 E I. Let ii is a near-solution to (1.3.1) in the sense iIt + Aft + Ifi|4/dil = e, (3.2.9) and the following estimate holds

IIUIIL2(d+2)/d(IxR) M, (3.2.10)

I|f|IL- L2(R) < E, (3.2.11)

I|eI 2(d+2) K e, (3.2.12) L d+4 (Ix R)

109 2 where c < ei e1(M1 , M2). Assume further there exists to E I and uo 6 L such that

IIi(to) - uo|I2 < E, (3.2.13)

Then, there exists a unique solution u(t, x) to the Cauchy Problem

{Ut+ Au + 1u14 /du 0 (3.2.14)

u(to) = u0 . such that

I|u - fil 2(+2> (M ,M E. (3.2.15) LL d (ixR)nL- L2(R)M121 2

In particular

||Ul|L2(d+2>/d(IxR) f

3.2.2 Scattering below the mass of the ground state

The dynamic of (1.3.1) for initial data with mass blow up the ground state is known, due to the following Theorem by Dodson ,[191.

Theorem 3.2.8 (Dodson). Consider the Cauchy Problem (1.3.1), with the initial data uo such that ||UOI|L2 I IL2- (3.2.17)

The solution u to (1.3.1) is global, further more it scatters

Ilu(t, x)I2(+2)/d < o. (3.2.18)

3.2.3 Concentration compactness

Strichartz estimates (3.2.2) lacks compactness due to the symmetry of equation (3.2.1). Note the aforementioned symmetries for the nonlinear equation (1.3.1) also hold for the linear equation. Profile decomposition is the tool to remedy this.

110 Let us start with the following definition,

Definition 3.2.9. Let G := {g = gX 0,t, 0,A0 I xo, to, o (E R, Ao E R+}, where gxo, o,Ao, o is a map L 2(R) -+ L2 (R):

1 - i(- .)a x - X0 gxo, o,Ao,to f(x) := re (e o f)( -). (3.2.19)

Remark 3.2.10. G is a group acting on L 2 and for any g E G, f C L2 , -| f 112 = If 112-

Profile decomposition

One has

2 Proposition 3.2.11 ([621,[6], [3], [72] ). Let {un} _L1 be bounded in L (Rd), then up to extracting subsequence, there exist a family of L 2 functions # , j = 1, 2, --- and group elements gj,n E G , where gj,, = (gn) , such that for all 1 = 1, 2, - -- we have the decomposition

Un= gjn#5 + Wn, (3.2.20) j=1

(here (3.2.20) defines wl .) And the following properties hold

" Asymptotically orthogonality of the group elements: For any j $ j', one has

__ Ay, x -x ___-_ + A ' + 'I '- I+ IAj ,(j,n - + I,n)I+2 I - 4 o (3.2.21) j ,n j,n j,n j,n

" Asymptotically orthogonality of mass:

Vl > 1, lim IM(Un) - ( - M(wn)I = 0. (3.2.22)

* Asymptotically orthogonality of Strichartz norm:

la+2/ = 0. (3.2.23) Vj # j', n--0olim ||ei (gjn#j)e"t(gy,,#j) t,:r

111 * Smallness of remainder term:

lim lim sup 11e6iWLK{I 2(d+2)/d = 0 (3.2.24) 1-+00 n-+o

* Weak limit condition of the remainder term:

Vj < l, gj1W1 0 in L2 (3.2.25)

According to Proposition 3.2.11, we define

Definition 3.2.12. A linear profile is a function f E L2 and a sequence {gn}n C G, or equivalently a function f E L2 with parameters {x, n, An, tn}n c R x R x R+ x R.

Up to extracting subsequence and adjusting the profile, for every profile decomposition as in Proposition 3.2.11, we always assume without loss of generality that limn-+oo - -- = 0 or equal to too. This leads to the following standard definition: j,n

Definition 3.2.13. We call a profile f with parameter{xn, n, , tn},

* Compact profile if t = 0,

* Forward scattering profile if - oo,

* Backward scattering profile if-, = -o.

Nonlinear approximation

To deal with the nonlinear equation (1.3.1), one needs the notion of nonlinear profile.

Definition 3.2.14. Given a linear profile, i.e. a function f E L 2 and a sequence

2 {gn}, c G, or equivalently a function f E L with parameters {xn,1 n~, An, tn}n C RI x R x R+ x R. We say U is the nonlinearprofile associated with this linearprofile if U is a solution to nonlinear Schr6dinger equation

iUt + AU + U14/dU = 0, (3.2.26)

112 and satisfy the estimate

U(- t ) -f = 0. (3.2.27) n- oo Aj,n 2

(3.2.27) is only requiredfor n large enough. For (3.2.27) to make sense, we require U is defined in a neighborhood 2of liman0 - .

Remark 3.2.15. Given a linear profile, the associated nonlinearprofile exists and is unique. The existence and uniqueness of the nonlinear profile basically relies on the local well posedness theory of (1.3.1). This is quite standard, see Notation 2.6 in [28].

Now, we state a nonlinear approximation for NLS. Though we do not give the detailed proof here, we point out it is essentially the consequence of asymptotically orthogonality (3.2.21) and the classical stability theory Proposition 3.2.7, see [3,[62], [72] for more details.

Proposition 3.2.16. Assume that {un} admits a profile decomposition with profiles

(0j;fXj,n, j,n, Aj,n, tj,n}n)j as in Proposition 3.2.11. Let us consider a sequence of

Cauchy problems for a sequence of initial time {tn},

itvn(t, x) + Avn(t, x) + Ivnl4vn(t, x) = 0, (3.2.28)

vn(tn, x ) = un.

For each fixed j, let 473(t, x) be the associated nonlinear profile to Oj. Let

I:ix n e- itI ,n4 12 t - tj,n x - Xjn- 2 jint j'n

If for any {Th}n such that:

Tn - t3, Vj > 0, 1, scatters forward or lim - 2 < T+(Gy), (3.2.29) n-+oo A2

2 1n profile decomposition, one usually needs to extract subsequence many times, we always assume that limnoo - exits or equal to too. We also define the neighborhood of oc as (M, oc) for any M, similarly we define the neighborhood of -oc.

113 (One may extract a subsequence again so that the above limit exists),

then for n large enough, vn, (Di,n are defined in [ta, tn + rn]. Moreover, let

rI := vn - Dj,, - e itn WI (3.2.30) j=1 then one has

2 2 2 lim sup lim sup l rl L ([tn,tn+rn];L )nL (d+ )/d([tn,tn+rn] xR) - (3.2.31) 1-+0o n-+0o

Furthermore, (3.2.21) implies that Vl > 0

2(d+2)

2 2 I (Dj"'i.II L (d+)/d([t.,tn+nl XRd) j=1 (3.2.32)

L2(I+2)/d([IIM ,l m 'xTn-,nXRd) j=1 A 1\j,n 3

In particular, if for any j, 17 scattersforward, then for n large enough, the associated solution to (1.3.1) with initial data u(tn) also scatters forward.

We remark here similar results holds for energy critical wave and energy critical NLS, see [2], [441.

3.2.4 Variational characterization of the ground state

Let us first recall the classical Gagliardo-Nirenberg inequality.

Lemma 3.2.17. Let v C H', then

- 2||| ' E(v) > - ( I 4/d (3.2.33) - IVvI2 1 IIQII22|||

It gives the variational characterization of the ground state Q.

Lemma 3.2.18. Let v C H1 , and

1v12 = E(v) = 0 (3.2.34) I I 2, 114 then

v(x) - A N/ 2Q (Aox + xo)e 70 . (3.2.35)

In [60], Merle and Rapha81 apply concentration compactness type techniques to generalize the above into the following:

Lemma 3.2.19. Let v in H 1, there exists ao > 0, such that for all a < ao, there exists 6(a) > 0, such that if

11v112 < ||Q12 + a, E(v) < all VVII, (3.2.36)

then there exits A 0 = VQ11 2 , xO E R','o cCR such that IIVU11 2'

(3.2.37) d/2U(X A0 -Oe QIIH1 < 6(a). and, lim o 6(a) = 0.

3.3 The dynamic of non-positive energy solution

Throughout this section, we assume the solution u to (1.3.1) has H 1 initial data and satisfies (1.1.12).

The dynamics of non-positive energy solutions are extensively studied in the series work of Merle and Rapha8A, [60], [55] , [59], [56], [66], [58]. We will apply their results in this work. However, we will work from an L 2 based viewpoint rather than H1 viewpoint.

Merle and Rapha81 show all strictly negative energy solutions blows up according to the so-called log-log dynamic, we restate their results as the following,

Theorem 3.3.1. Assume u is a solution to (1.3.1) with H1 initial data, non positive energy, and satisfying assumption (1.1.12), assume further lull2 # 11Q112 if u is of

115 zero energy, then u blows up in finite time according to the so-called log-log law u(t, x) = 12 + (t)Q) )ey(t), x(t) E Rd, y(t) E R, A(t) E R+, IIEIIHI 5(a) Ad/ 2 (t) A (3.3.1) with estimate AT(t) ~ r. (3.3.2) ln |ln T- t

lim (IVE(t, x)12 + E(t, x)l 2e-1xI) = 0. (3.3.3) t-+T

Remark 3.3.2. We do not directly use (3.3.2). Our results rely on the fact such solution will blow up in finite time and the mechanism of blow up is ejecting mass out of the singular point, i.e. the control (3.3.3).

One may refer to [60],[55], [59] for a full proof of Theorem 3.3.1 when the solution is of strictly negative energy. When the solution if of zero energy, one may refer to

Theorem 3 in [59], see also Theorem 4 and Proposition 5 in [56]. For estimate (3.3.3), which is most relevant to our work, one may refer to the formula above (3.7) in page

52 of [591. Strictly speaking, the term appears in [59] is Qb rather than the ground state Q, but Q is just small modification of Q, and converges to Q in a strong way as b -+ 0, see Proposition 1 in [591. And b -+ 0 as t -+ T, see, again, the formula above (3.7) in page 52 of [59].

Now, for the purpose of our work, we write a corollary of Theorem 3.3.1.

Corollary 3.3.3. Assume u is a solution to (1.3.1) with H1 initial data, nonpositive energy, and satisfying assumption (1.1.12), assume further lUll|2 / lQ|112 if U is of zero energy, then there exists 6 = 6(u) > 0, such that VA > 1, there exists T, < T+(u), x1 E Rd, 11 > 0, such that

f Iu(T, x)1 2 > 6, lu(T1, x)1 2 > 6 (3.3.4) i-e poiS in< 3->A.I

See proof in Section 3.5.

116 3.4 An overview for the proof for Theorem 1.3.1, Theoremi.3.2

3.4.1 Step 1: First extraction of profile

First, we will use the profile decomposition and a minimization argument in [25] to show the following:

Lemma 3.4.1. Let u be a solution, not necessarily radial, satisfying the assumption of Theorem 1.3.1, then there exists a sequence t, -+ T+(u), such that u(t") admits a profile decomposition with profiles {q5O, {x,n, Aj,n, ,n, tnjn}j, and there is a unique compact profile, we assume it is $ 1 , such that

* 11#1|12 > |1Q112,

e The associated nonlinear profile (b1, is an almost periodic solution in the sense

of (3.1.1), and it does not scatter forward nor scatter backward.

See Section 3.6 for a proof.

Remark 3.4.2. One may compare this step to the procedure of reduction to the minimal blow up solution in the study of defocusing problem.

Remark 3.4.3. Due to the assumption (1.1.12), there cannot be more than one profile with mass no less than IIQ||1.

3.4.2 Step 2: Second extraction of profile

We need to do some further modification of profile, the following step is very standard when one wants to prove scattering type results. By arguing exactly as Section 4 of [72], we will have

Lemma 3.4.4. Let (D1 be the nonlinear profile as in Lemma 3.4.1, with lifespan

(T, T+). Then, according to Lemma 3.4.1,

1 F ( X (t)ix (t) (t, x) = Pi(- )ed (3.4.1) 4)b ) /2(t) t( A 1(t)

117 Further more, there exists {t,},, t, E (T-, T+), such that

P > Po in L 2 (3.4.2)

And the solution w to (1.3.1) with initial data Po or PO satisfies

w(t, x) = Nd! 2 (t)Lt(N(t)(x - x(t)))ex (t).t > 0, N(t) <; 1 (3.4.3)

And {Lt}t is a precompact L2 family.

Indeed,such a solution w, sometimes also called minimal blow up solution, already partially falls into the framework of Dodson's work [18], [191.

3.4.3 Step 3: Fast cascade case

We exclude the so-called fast cascade, i.e. the case

00jN 3(t) <00C. (3.4.4)

In this regime, for d = 3, Dodson's long time Strichartz estimate, Theorem 1.24 in [18] will indeed imply w is not only a L2 solution, but an H1 solution, see Theorem

3.13 in [18], and furthermore, the energy is zero, see (3.86), (3.87) and Remark 3.14 in [18]. Long time Strichartz estimate also holds for d = 1, 2, with extra technical difficulty, see for [21], [17]. See Theorem 1.9 in [19] for a summary.

Thus, we have

Lemma 3.4.5 (Dodson). Consider w as in Lemma 3.4.4, assume further (3.4.4), 1 then w(0) C H and E(w) = 0.

We then have

Lemma 3.4.6. Consider w as in Lemma 3.4.4, (3.4.4) cannot hold.

1 Proof. The case 11w11 2 = 11Q112 is impossible since by Lemma 3.4.5, w is in H and with zero energy, thus, by Lemma 3.2.18, w = Q(x - xO/AO)e and the solution AO1

118 3 is just a standing wave which implies N(t) - 1 and f' N(t) - o. The case |1Q1 <

lJUll2 < IlQ112 + a is impossible because by Theorem 3.3.1, such solution must blow up in finite time. El

3.4.4 Step 4: Quasisoliton case

It is in this step that we need radial assumption . Since we are considering radial solution, then w in (3.4.3) must also be radial, which imply that x(t),(t) 0.

Remark 3.4.7. Since all we need is x(t), (t) - 0, we may just assume u is symmetric across d linear independent planes. The observation that if u is symmetric

across d linear independent planes then x(t), (t) = 0 has been pointed out by Dodson [18.

Now, we are left with the case fo" N 3 (t) = oc, which is usually called Quasoliton case in the literature. We will show in this case, it must be that llw(0)11 2 - lIQII2-

Lemma 3.4.8. It is impossible that w is of form (3.4.3), x(t), (t) - 0, ||Ql12 <

3 lJw(0) 112 5 |IQ112 + a, and f0 N (t) =0.

And we will further show

Lemma 3.4.9. Assume w is of form (3.4.3), x(t), (t) = 0, Ilw(0)11 2 = 11Q112, and fo7 N 3 (t) = oc, then exist a sequence tn, and parameters A,, 1, such that

lim 11A d/ 2 w(t, A,,x)e-n" - Q112 = 0. (3.4.5) n-+oo

Remark 3.4.10. It is very natural to conjecture that the under the same assumption of Lemma 3.4.9, w is essentially standing wave Qeit. This will be related the classification of finite time blow up solution to (1.3.1) with mass ||uo11 2 = ||Q|12. Such solutions, if one further assume the initial data is in H1 , are completely determined by the result of Merle, [541. At the level of L 2 , it seems to be a very hard problem.

To understand the proof of Lemma 3.4.8, Lemma 3.4.9, one needs to understand how Dodson handles the case ||w(0)11 2 < 1lQ12, 1191. We will give a rather detailed

119 sketch of Dodson's arguments in Section 3.7. Basically, one needs to use Virial identity to explore the decay of the solution and one needs to perform frequency cut-off to explore the coersiveness of energy. We will show the following:

3 Lemma 3.4.11. Assume w is of form (3.4.3), x(t), (t) 0, and fo N(t) = 0o, 3 then there exist sequences t,, < T, Rn > 1, fe" N (t) = Kn, such that

E (X )-)P

We will use Lemma 3.4.11, Lemma 3.2.18 and Corollary 3.3.3 to deduce Lemma 3.4.8, Lemma 3.4.9.

See Section 3.7 for the proof of Lemma 3.4.11, Lemma 3.4.8 and Lemma 3.4.9.

3.4.5 Step 5: Approximation argument and conclusion of the proof

To conclude the proof of Theorem 1.3.1, we use Lemma 3.4.1 to reduce the dynamics of u to the unique compact profile #1, and its associated solution 1ki. Then we use Lemma 3.4.4 to reduce the dynamic of

11w112 = 11QI12, j N 3(t) =0. (3.4.7)

And finally, such solution is characterized by Lemma 3.4.9. We show the detail in Section 3.8.

3.5 Proof of Corollary 3.3.3

We prove Corollary 3.3.3 here. Let u be as in Corollary 3.3.3.

120 First, if u is of strictly negative energy, by Lemma 3.2.18, we have

JIU112 > ||QI12 (3.5.1)

If u is of zero energy, then (3.5.1) is already in the assumption of Corollary 3.3.3.

Thus, we can assume

||uII = IIQ + ||2= IIQI2 + 6j. (3.5.2)

Note mass is a conservation law. By choosing oz in Assumption (1.1.12), 6 0 <

J . 1 IQ2dx We will choose the J(u) in Corollary 3.3.3 as '.

Since Q is of exponential decay, we have that, by (3.3.3), when t is close to T+(u) enough,

< IQI, 16(t, X)I >< J0. (3.5.3)

By (3.5.2), we obtain |2 3 6. (3.5.4)

On the other hand, by (3.5.3)

IQ + E12 > 6o/2. (3.5.5) 1X<1

Now fix any A > 1, using the trivial estimate

E1 21 A fJ 12,-Ixl (3.5.6) JI

By (3.3.3), and T close to T+(u), we have

fE1c2 <6 (3.5.7)

121 Thus combine (3.5.3), (3.5.7) and (3.5.4), we have by triangle inequality that

6 (3.5.8) LA Q + E12 O/2.

Let lo, xO in Lemma 3.3.3 be x(Ti), A(T1 ), then the Corollary follows.

3.6 Proof of Lemma 3.4.1

Lemma 3.4.1 should be compared with the reduction to minimal mass blow up solutions for scattering type problem. Most arguments below are standard in concentration compactness, see for example [41], [42], thus we just sketch it. We will also use a minimization procedure from [25], which makes the whole proof more clear for us.

We remark that we do not use the fact that u is radial here.

First, for any i, -+ T+, up to extracting subsequence, we may assume u(in) admits profile decomposition with profiles {s,{JJ-,, ,n ,, pI},}. If Vj, we have

I||4j||2 < ||Q112, then by Theorem 3.2.8, and the nonlinear approximation argument Proposition 3.2.16, one would derive that u scatters forward, which contradicts our assumption. Thus, there is at least one profile with mass no less than |IQII2. On the other hand, by the asymptotically orthogonality of the mass, (3.2.22), and our assumption (1.1.12), there can only be one profile with mass no less than 11Q1|2. By reordering the profile if necessary, we assume the first profile 01 is the unique profile with

110111 2 11QI12. (3.6.1)

Remark 3.6.1. By Assumption (1.1.12) and the asymptotically orthogonality of mass, all other profiles has mass < 1, which implies there associated nonlinearprofile is global and scattering by the small data theory, Proposition 3.2.5.

Furthermore, $1 must be a compact profile. Indeed, if 01 is a forward scattering profile, using the nonlinear approximation Proposition 3.2.16, we have that a scatter forward, a contradiction. If # 1 is a backward scattering profile, then we use Propositon

3.2.16 for the initial data u(in), but run the (1.3.1) backwards rather than forwards,

122 then we will get a uniform bound for uIIL 2(+2)/d[ot xRd], which again implies u scatters forward, a contradiction. See [411 for similar arguments for energy critical wave, see also [42]. Since 01 is a compact profile, we do not distinguish between 01 and its associated nonlinear profile T1, which is the solution to (1.3.1) with initial data 0 1 .

Finally, we remark V)1 is not uniquely determined by the times sequence {t~ } since one may scale or translate the profile, however, the L 2 norm of V'1 is uniquely determined, since L 2 is invariant under those symmetry .

To find a sequence {tn} such that its associated profile decomposition satisfy

Lemma 3.4.1, we mimic the minimization procedure in Section 4 of [25]. Though that paper deals with energy critical wave and energy critical Schr6dinger, most arguments there are quite general and works whenever there is a satisfying profile decomposition technique.

One will need the so-called double profile decomposition at the technique level, which maybe compared to the diagonal technique which is used in the proof Arzela-

Ascoli Lemma.

Lemma 3.6.2. Assume {fg},,, are uniformly bounded in L 2, assume for Vp, fnP admits a profile decomposition with profiles {g'} such that there exits {7} 3 such that for all p,

S < 00, I eitg II < r. (3.6.2)

And assume for all j, {eitAg(O)}p admits a profile decomposition with profile hj,k, then up to extracting subsequence, there exists n, -+ oo such that { fp,} admits a profile decomposition with profile {hj,k}j,k.

Remark 3.6.3. According to asymptotically orthogonality of mass (3.2.22), we have that for all j, k, |lhj,k 12 1g9 112.

Remark 3.6.4. We will not need to check condition (3.6.2) in our work. because for all the profile decompositions involved in our work, if we reorder the profiles such that

11g3 l jg 3i, Vj _> j', we always have 1191112 2211QJ1 2, and IgI2 ;< ,12Vn > 2 thus (3.6.2) automatically holds.

123 Lemma 3.6.2 is the natural generalization of Lemma 3.16 in [251 for equation

(1.3.1). we refer to [251 for a proof. (Though the proof there is written for energy

critical wave, it also works here.)

Now let us go back to the proof of Lemma 3.4.1.

Let A be the set of the time sequence {fT}, such that limn, in = T+(u) and

{u(n)} admits a profile decomposition. Let #1 be the profile with mass no less than

1IQ112. Recall that 01 may not be uniquely determined by the time sequence, but

1101112 is . We define a map for s = {fn}n E A to R as

E(s) = 1101112 = |II112 (3.6.3)

We have that (3.6.1) implies

inf ;> |1Q112. (3.6.4) sEA

We now claim there exists an so E A such that

.(so) = inf E. (3.6.5) sEA

In fact, by Lemma 3.6.2, there exists E(s,) -+ infsEA S, then apply the double profile decomposition Lemma 3.6.2, one will find {u(in)} admits a profile decomposition

and &({ftn},) = infsEAE.

Let us finish the proof of Lemma 3.4.1. Let so = {tn},, and u(tn) admits a profile decomposition with profiles { j, {Xj,n, ,,, Ajn, tj,n}}j , the associated unique nonlinear profile with mass above ground state be Di, clearly 4) satisfy

IIQ112 l|@112 5 11Q112 + a. (3.6.6)

We further claim 4D must be an almost periodic of form (3.1.1). Indeed, if not, then there exists time sequence {an} within the life span of 4D such that 4D(an)1 admits a profile decomposition and any profile has mass strictly smaller than 1||1112, then us-

124 ing the nonlinear approximation Proposition 3.2.16 and double profile decomposition

Lemma 3.6.2, it is easy to see up to picking a subsequence ,u(t, + A2a,) that admits a profile decomposition and S({t, +2 Aa}n) < C(so), a contradiction. This concludes the proof.

3.7 Proof for Subsection 3.4.4

We prove Lemma 3.4.11, Lemma 3.4.8, Lemma 3.4.9 here.

To prove Lemma 3.4.11, one needs to use the proof in [191. We do a review of the argument in [191 here.

3.7.1 A quick review of Dodson's work [19] warm up and energy tensor

Einstein summation convention.

Let

zut + Au = -ju|P'u. (3.7.1)

(Note in our case, p = 1 + 4/d.)

and let

2 2 T00 =Too(u) := u , T0 =To = uii, (3.7.2) Tk= = 4RtjUk - 6kJA|UI2 - 2 1 +1 p + 1

Then,

&tT00 + O93 TO3 = 0, &9To3 + 0kTJk = 0. (3.7.3)

Let us recall Virial identity as an example. The computation here is just formal,

125 we assume a priori all the quantities in the following computation is finite.

2 2 2 2 Ott |x| tJ T = Ot J x t Too (37.4)

=OtJoix2T1 = o Okx|2Tk = 16E(u).

Almost all results regarding the long time dynamic of (1.3.1) rely on (3.7.4) in some sense. Intuitively, when E(u) is positive, (which is always the case when lIUll2 < l|QIl2

), then f Ix1 2lU1 2 will grow to infinity. On the other hand, since the mass f 1 2 is conserved, this implies that mass are ejected to infinity, which should be understood as a dispersion effect.

We further summarize the last three identities in (3.7.4).

XJ To a d = Ja Tj, = 4E(u). (3.7.5) dtdt 2j.~UOX2 ~ 375

A sketch of Dodson's work

Using (3.7.5) , Dodson shows

Proposition 3.7.1. it is impossible that (3.4.3) holds with f N 3 (t) = oc and

llw112 < llQl12 - q, for some r> 0 (3.7.6) unless w = 0

We will use d = 3 here to present a sketch for the proof of Proposition 3.7.1. We only do the case (t), x(t) = 0, (recall (t), x(t), N(t) in (3.4.3) ). 3. First subcase: (t) = 0, x(t) = 0, N(t) ~ 1 Let TK be the unique time such that

TK 3 (3.7.7) I o N (t)=-K.

All the analysis below is in [0, TK] for w as in (3.4.3). 3 Indeed, the computation is easier for d = 1 or d = 2, however, Dodson's work [19] implicitly use his long time Strichartz estimate, which involves extra technical difficulty for d = 1, 2.

126 We first consider the subcase t) = 0, x(t) = 0, N(t) ~ 1 for w in (3.4.3). We remark that x(t), (t) = 0 when one only considers radial solutions. Now the solution is like a soliton, without dispersion. It is very natural to apply (3.7.5) to get a contradiction. However, w is not in H', so one cannot directly use mass constriction

(3.7.6) to use the coersiveness of energy and one does not have the extra integrability of xu to make sense the left side of (3.7.5). So, very naturally, one needs to do truncation in space and frequency.

We need a cut off version of x, for technical reason, we will need a O(x) such that (x) = 1 for lxi < 1, and O(x) ,< --I, and Ok4(x)xj is semi positive definite 4.

We also define the Fourier truncation, I := IK PKCK, here C is some fixed large constant. Let F(v) := -IVv|4/dv. Now note

(iOt + A)Iw = I(F(w)) = F(I(w)) + {I(F(w) - F(Iw)}. (3.7.8)

Let M(t) be the truncated version of f xjQwjfv-, (M denotes Morawetz action in the literature) :

M(t) = ( )jIKwjIK1- (3.7.9)

Since f JwI2 is bounded, one immediately obtain that

IM(t)I < RK. (3.7.10)

Recall (3.4.3), using the fact that w = N(t)d/ 2 Lt(N(t)x), N(t) ~ 1 and {Lt} is a precompact in L2 , it is easy to upgrade the above to

|M(t)l ;< Ro(K). (3.7.11)

4 This is not hard, indeed, one first constructs some convex f(x) which is like |X1 2 near the origin, slowly grows for lxi > 2 ,and take O(x)xj = Qjf(x). Note we do not need f to be uniformly strictly convex.

127 Now computing the derivative of M(t), one has

-M(t) =E + ak((x/R))xj TkI(I w) dt 1 I 6 {4?RIwjIWk - kjAIII -d 2 d 2 =E1 + f 0jck(0(x/(R)xj)

S1 iW2+4/d =E1 + E2 + E3 + 4 R 2 2( 1Vu1 2 + d (3.7.12)

Here E1, E2, E3 are as the following.

E= -i / )x ({IF(w) - F(Iw)}VIw + IwV{IF(w) - F(Iw)}(n.7.13)

E2 =[Aijk(x/R)x](-6kjIIWi2), (3.7.14)

E3 = k(Vb(x/R)xj) {f4RfwIWk - 2.4/d wI2 (3.7.15)

Here E, is caused by the commutator type error in (3.7.8).

2 One needs to explore the coersiveness of fI VIwl - 1 1Iw I, thus one needs to introduce an extra smooth cut-off function X(x) which is 1 for IxI < 2, and vanishes for lxi > 1. Then

- (3.7.16) 2 2 w 6 E(X(x/R)Iw) fIIxI R 2IVIw1 2+-4

We remark that strictly speaking, (3.7.16) is not completely right, since we neglect the error caused by the commutator (V(XIw) - XVIw), since this is just a sketch, we omit this technical point, one should refer to [191 for more details.

Now, using (3.2.33) and the important condition (3.7.6), one has

E(IIx(x)IwlI) > co(r)(IIX( X)w14/d+2 + IIV(x(x/R)Iw)11 2). (3.7.17)

128 Error E2 , E will be estimated by

E2f| lIwI2, (3.7.18) IE3 - Oxk((xI/R)xj) {4IJwIWk| JvI vIw12+4/d NxotR siv hae R

Note since Bk(OXi) is semipositive definite, we have

E3 > -C1 L R 11w14/d+2 for some C1 > 0. (3.7.19)

E1 is estimated by the commutator type estimate, see Lemma 4.7 in [181.

IIIKF(w) - F(IKw) 112L 2d/d+2[OTK $ OK(1)- (3.7.20)

The proof of the above relies on Dodson's long time Strichartz estimate, Theorem 1.24 in [181 , which is indeed the key ingredient in 118].

Thus,

I TK F 1 $ RoK (1)IVIK wIL 2 L2d/d-2. (3.7.21) 0o E O ()1'-K ILL

And the long time Strichartz estimate will further give, see Lemma 4.5 in [181,

2 I|V JKwIL2 L2d/d- K (3.7.22)

Thus,

IE1I > RoK(1)K. (3.7.23) J TK

We remark that the long time Strichartz estimate is purely analytic, does not relies on the energy structure, i.e. the difference between focusing and defocusing does not matter here.

To summarize,

+M(t) >4E(x(x/R)Iw) + El+ E2 + E3 (3.7.24) > co(TI)IIX(x/R)Iw|L4d+2 + E1 - |E21 - C 1 /JIxR|w|4/d+2

129 In the last step, we plugged in the estimate (3.7.19).

Now, integrate in time on [0, Tk] plug in the estimate for El, (3.7.23), and estimate

for E2, (3.7.18), we recover the estimate (3.26) [19]5.

T j K d M(t) > co()x(x/R)Iw 6 j 11 2+4/d K i KIIIW(t)I2 Iw(t, X)1 -ROK (1)K. fo dt fo Lx f t R2 J>gR (3.7.25)

Estimate (3.7.25) is enough to give a contradiction and conclude u must be zero, one

is refer to [19], in particular (3.26) in [19] for more details. We sketch some standard

arguments below for the convenience of the readers.

The left side of (3.7.25) is controlled by Ro(K), by estimate(3.7.11).

On the other hand, since we assume N(t) ~ 1, IrK K N(t)3 K, thus

TK 11 Ssupt IIu(t)IL2 < 1K (3.7.26)

And, recall again u is of form (3.4.3), N(t) - 1, then for any time interval J of

length ~ 1, local theory of (1.3.1) gives jj iUi12i2 ~ 1. Now using the fact u is of

form (3.7.25), i.e. all the mass of u is uniformly concentrated in physical space and

frequency space, we obtain

IX(x/R)Iw 4/ 2 jj(1). R1, (3.7.27) Ji J fJ J xj>R

Thus, the right side of (3.7.25) is bounded below by

K co(71)K - R2 - OR(K) - Ro(K). (3.7.28)

Here C is some universal constant. Now one obtains that Ro(K) > co(17)K - y -

oR(K) - Ro(K), a contradiction.

The key point to conclude a contradiction by using Ro(K) > co(7)K - R -

5the numerics here are slightly different, because in [191, that part is done for d = 1, and the error caused by El is neglected in this step but treated later.

130 oR(K) - Ro(K) is that here co(r) in (3.7.17) does not depend on R or K, so one can first choose R large enough, then one further choose K large enough to get a contradiction.

General Case: (t), x(t) = 0

Now, for general case with (t), x(t) = 0, (which covers all general radial solution), it is very natural to define the Morawetz action as

M(t) := V(xRt) xN(t)Q VIwIsv. (3.7.29)

However, if one directly relies on the above Morawetz action to argue as previously, one will face the problem that one does not have good control about N'(t). This is handled by Dodson using so called "upcoming algorithm", basically he constructs some slowly oscillating N(t) < N(t) according to the behavior of N(t), and constructs

Morawetz action as

M(t) =v (t))GVIwIth (3.7.30) ( ) R

The proof left follows is in principle as the previous subcase where N(t) - 1, see section 4 in [19] for more details.

As emphasized in the end of the previous case, the key part and the only part Dodson's proof using the fact lu|ll2 < lIQl12 - 77 is that this will gives a universal constant co(17) such that

E(X(N(t)x/R)Iw) > co(,)[IVX(N(t)x/R)IwI2 + IX(N(t)x/R)IwI 4d2]. (3.7.31)

3.7.2 Proof of Lemma 3.4.11

The proof of Lemma 3.4.11 is by contradiction. Indeed, if Lemma 3.4.11 does not hold, then one recovers (3.7.31) even the mass of w is not under the ground state. Then, one argue as the proof of Proposition 3.7.1, which we just reviewed in the previous subsection, to conclude w = 0, which is clearly a contradiction since |Iwl12 llQ1l2-

131 Remark 3.7.2. Proposition 3.7.1 holds for general non radial solution by using a version of interaction Morawetz estimate. Ever since [131, there are a lot of works using interaction version of certain estimates to show results for general solutions rather than radial solution, such as [18], /191 and many others. Howeverwe cannot have a natural useful generalization of Lemma 3.4.11 here. It seems to us if one directly follows the arguments in [19], where a nonradialversion Proposition 3.7.1 is proved, one can only conclude that there exists a sequence tn T, Rn >> N(tn) '1Xfl fj'n N 3 (t) = Kn, such that

x -x . 1 X E(X( x )FGCKfe w(tn)) I -IV<( "fl)PC)()) f 3.7.32) Rn n R'

Formula (3.7.32) is of no use to us, because we do not have control of x, here, thus we are not ensured X(X-n )PGcKfleinX (tn)) contains almost all the mass of w as n -* oc, which will be critical later.

3.7.3 Proof of Lemma 3.4.9

Lemma 3.4.9 is implied by Lemma 3.4.11, with the help of Lemma 3.2.19. We present a short proof here. Recall w is of form (3.4.3), and since we are consider radial solution, we have (t) = 0, x(t) = 0. Since R N>a), K > 1 and {Lt}t is a precompact L 2 family, we easily have

IIw(tn) - x(x/R)P

Now let 5C, := x(x/Rn)P

2 n/Wn(Anx)e - Q in L . (3.7.34)

132 But (3.7.34) follows from Lemma 3.2.19 because

1 IIwnjII: 11W112 ! I1Q112 + 1, (3.7.35) 1 n4

3.7.4 Proof of Lemma 3.4.8

Now we turn to the proof of Lemma 3.4.8. Note we restrict ourselves to radial

solutions. Let tn, Kn, R1 be as in Lemma 3.4.11, let vn = x(x/Rn)P

I1VQ11 2 We apriori have IIVV. 112

a small (3.7.36) | Vn|127 11Q112 + a, enough ,

Thus, when n is large enough, 1/n < a, thus by Lemma 3.2.19, we can find a sequence of 74, such that

I Ad/(f )e n - QIIH1 < 6(a) < 1- (3.7.37)

Note the space translation parameter in Lemma 3.2.19 will not appear because our functions are all radial.

Also recall w(t) = N(t)d/ 2 Lt(N(t)x), Lt is a precompact L2 family, N(t) < 1,thus the condition K >> 1, R > Nimpi

IIw(tn) - Vn|I2 = on(1) (3.7.38)

2 and further it implies that { Nd12 )vN )}n is a precompact L family. Thus, by (3.7.37) we have N(t,) ~ An. Now let f, {Nd/v(n Nt))}In, we have that

1. fn is a precompact L 2 sequence.

2. f, is uniformly bounded.

;< . (This contains the possibility that E(fn) is negative). 3. E(fn) r-

133 Up to extracting a subsequence, we may assume f, strongly converges to fo in L2 , in particular, with (3.7.38), we obtain

||foll = 11w|12, 1Q1 < If 112 < |1Q112 + a (3.7.39)

Now, by Fatou's Lemma, ||follIk < liminf, IfnI.

6 2 By interpolation between L and H', IfnlIL4/d+2 -4 IjfoIlL4/d+2.

Thus, we have the condition

E(fo) < lim inf E(fn) < 0 (3.7.40)

Now, to summarize, with (3.7.38), strong convergence of f,, to fo in L2 , we have

* w is an almost periodic solution,

w(t) = Nd1 2 (t)Lt(N(t),x),t > 0, N(t) < 1.

* Lt, converges strongly in L 2 to f.

* f is in H', and if of nonpositive energy.

These three property is enough to derive a contradiction, we prove something slightly more general for the convenience of future work.

The proof we are left with will not depend on that the radial property, we only need that w is almost periodic in the sense that

1 x -- x(t). w(t) = ItLt( )ex t), t > 0. (3.7.41) Ad (t) AMt)

Let F be the solution to (1.3.1) with initial data f, then by Theorem 3.3.1, F will blow up in finite time T+ > 0, and according to Corollary 3.3.3, there exists

6 Note we do not need the radial Sobolev embedding here

134 Jo = 60 (F) > 0, such that for VA > 0, there exists TA < T+, xo E Rd

2 2 > o, x)1 > 6j. (3.7.42) |ix-x0|$lo /F(TA, x)I JF(TA,|x -X0|kAlo

And note by standard local theory of (1.3.1),

||F(t,x)IiL +2)/d ([,TA] xRd) < CA <00. (3.7.43)

Remark 3.7.3. Note here 60 is fixed once w is fixed, and A can be chosen arbitrary large (by choosing T = TA close to T+ enough.)

On the other hand, since w is an almost periodic solution of form (3.7.41), we claim

Lemma 3.7.4. Assume w is of form (3.7.41), then, for any 6 > 0, there exists

A = A3 such that if for some zo E Rd, to E oo), ho > 0 such that

> -, (3.7.44) W(to)2 2 Jix-zoIsho then we must have

Jw(to, x)12dx < . (3.7.45) fix-z01>Aho2

Lemma (3.7.4) will be proven in Subsection 3.7.5, let us assume it at the moment and finish the proof of Lemma 3.4.8. Let us fix 6o = 6o(w), and picking 61 = 60/2, and let A = A6, as in Lemma 3.7.4 such that (3.7.44) holds , and as mentioned before, we can use Corollary 3.3.3 to find T = TA such that (3.7.42) holds and we emphasize again (3.7.43) holds.

Now since Lt, -+ F(0) in L 2 , then for VE > 0, there exists no = no(e) such that

|ILt.0 - F(0)112

Using the stability argument, Proposition 3.2.7, by choosing e small enough, (according to CA), then (1.3.1) with initial data Lt,, has a solution, we call it v, which exists

135 in [0, TA], such that

Iv(TA) - w(TA)112 <; 60/10. (3.7.47)

By (3.7.42) and triangle inequality, we have

2 6 (3.7.48) /J-xoI v(TA,) 2 > 6o/ 2 , J w(TA, X)1 > o/2.

On the other hand, since v(t) solves (1.3.1) in [0, TA] with initial data L,., and

2 w(t) solves (1.3.1) with w(tn0 ) = Ad/ 1 jLt) / d(x)e'x(no), by the local well posedness theory (uniqueness of the solution), we have w is defined in

[tno, tno + A(tno) 2 TA], and

w(tno + A(tno) 2 TA, X)

xtn (/2 X,z- X(tno) -2tno)A(tno)2A i~Agtn0)igt9)j2TA AA(tno (T/ - ( o)) eAtn)I(t 0 )Tix (tn) (3.7.49)

ei(tno)x 1 V TA - eiYO A(tno)2 ^' A(tn) ) where -o = X(tno) + 2 (tno)A(tno) 2TA A(tno).

2 Let t1 0 := tn + A(tn 0) TA, zo = Zo + xo, and plug in (3.7.48), we obtain

2 6 2 w(ino, X) 2 > 6/2 (3.7.50) I A( 0)IW(ino, X)I > o/ , J

This contradicts Lemma 3.7.4. To finish the proof of Lemma 3.4.6, we are left with the proof of Lemma 3.7.4, which will be done in the following subsection.

3.7.5 Proof of Lemma 3.7.4

Indeed, we only need to prove Lemma 3.7.4 for w(t) = Lt, t > 0. Since the statement of Lemma already takes space translation into account, and the lo takes care of the scaling, and the phase eixW) plays no role in this argument.

Thus, we reduce the proof to following Lemma

Lemma 3.7.5. For a precomact L2 family {Lt}t>0, (not necessarily radial), V6 > 0

136 there exists A > 0 such that if for some lo > 0,

/ |LtO12 > 6, (3.7.51) then

ILto1 2 < 6. (3.7.52) J IXI>Alo

Proof. Since {Lt} is precompact in L 2 , by standard approximation argument, one may without loss of generality may assume L' is uniformly bounded and their support are uniformly compact. Thus(3.7.51) implies 1 < 1it, since Lt is uniformly bounded.

Thus, when A is large enough, clearly (3.7.52) holds. EJ

3.8 Proof of Theorem 1.3.1

Let

11w112 = IQ112, J N(t) 3 = 00 (3.8.1)

Thus, by Lemma 3.4.9, there exists a time sequence sn and parameter {An, 7Y}l such that lim ||Ad/ 2 w(sn, AnX) - Qe--n||2 = 0. (3.8.2)

We point out there is a slight abuse of notation in Lemma 3.4.1, Lemma 3.4.4, and

Lemma 3.4.9. The time sequences {t}, in these lemmas are not necessarily same.

From now on, we change the time sequence in Lemma 3.4.9 to {sn},, and change the time sequence in Lemma 3.4.4 to {ln}. Note by a extracting subsequence, we may without loss of generality assume the 7, in (3.8.2) satisfy -n = Yo. Note phase symmetry is a compact symmetry in L2 We claim (3.8.2) implies

137 Lemma 3.8.1. There exists a time sequence {hn} such that 'I1(hn) admits a profile decomposition with profiles {vj }j. (We will not track the associated parameters here.)

Moreover, there is a profile, we call it v 1 such that v1 = Qe-i?0

Remark 3.8.2. Since 111112 = ||w112 = 1|Q112, this indeed implies v1 is the only profile and one can conclude similar strong convergence results as in (3.8.2).

Proof of Lemma 3.8.1. Recall P., Po, w,

D(l) admits a profile decomposition with only one profile PO, and with stability argument, Proposition 3.2.7 or Proposition 3.2.16, it further implies {1(ln+A1 (l)s,)} admits profile decomposition w(sp) for each p, also note {w(sp)} admits a profile decomposition with only one profile Qet "O by (3.8.2), thus by Lemma 3.6.2, there

2 exists a sequence {np} , such that . 1 (ln, + A(ln,) s ) admits a profile decomposition, one of the profile is Qe.l

To finish the proof of Theorem 1.3.1, let hp be as in Lemma 3.8.1, one simply argues as in the proof of Lemma 3.8.1 we just did, and use Lemma 3.6.2 to do double profile decomposition for {u(tn + A hp)

3.9 Proof of Theorem 1.3.4

As shown in the proof of Lemma 3.4.1, such solution u is of form

I x - x(t) i t u(t, x) = A/2(t)( - )e. (3.9.1)

{ } is a compact L 2 family andl|Vt 12 = IQ112. We assume u blows up in finite time T. To prove Theorem 1.3.4, we need only exclude the possibility

A(t) > (T-- t)2/ 3 -. (3.9.2)

Assume (3.9.2) holds, do a (inverse) pesudo conformal transformation of u, we

138 will get a solution v such that

v(r, y) = 1u(T - Nd 2 (r)Wr(N(T)y + y(T))e'()OY (3.9.3) Td/2 4 r

Where 1 1 N(T) = -- 1 TA(T -) 2 W(z) =VT_ (z)e_ N)i Iz2 ( )) ( (3.9.4)

1 -(T - ()T 1 y(T) = TX(T - -), () =- + 2x(T - -) r T 7

Note the exact value of y(T), (T) actually do not matter. We obtain by (3.9.2)

N(T) < $< 2 < (1)1/3+ (3.9.5) 7 T T

On the other hand, one has the scaling lower bound

A(t) ,< (T - t)1/2 (3.9.6)

Thus,N(T) > - 1 / 2 This further implies W, is also precompact in L2 , since multiplying e2(r75IZ2 s compact perturbation. This contradicts Lemma 3.4.6, since now we have f N 3 (T) <

00.

139 140 Appendix A

A few technical lemma-ta

A.1 The local well posedness of the modified system

We explain briefly why one can always locally solve the systems (2.2.3), (2.3.27).

We only explain the case (2.2.3) here, (2.3.27) is similar.

Indeed, the system is made of NLS coupled with 4 ODE. Since NLS is locally well

posed and ODE systems are always locally well posed, it is no surprise that (2.2.3)

is locally well posed. To construct a solution, one first solve NLS zut = -Au - Iu1 2U

in a time interval [0, T1], and plugs this u(t, x) (which is not unknown in [0, T1 ]) into the last 4 equations, we will obtain 4 ODE on {A(t), b(t), x(t), y(t)}. We do some

computation here to illustrate this. For example, the equation

d 2 {It) lyI Zbw) + (c2 (t), Iy 2b(t))} =0

is now equivalent to

1(yI2QbdX-(X x(t))ei(t)0,)e-i'y(t)) ( dt A(t) A(t) A(t) A(t)

which is equivalent to

- 1( xt ), (Iy|2Q )e-iY(t)) 0, ) A(t) At)(t)

141 which is equivalent to

2 1 2 - x(t) + ilu Q(iAub)()e-iY(t)) A(t) A(t)

+R(U, d 1 (12 Q)(X - X(t) )e-iy(t)) dt A (t) A(t) (A. 1.1) d 1 X - x(t)1 X - X(t) -- R( (tQb( )7 -t (JyJ2Qb)( - )e-syt) dt A(t) A(t)

=0.

Though (A.1.1) is complicate , it is an ODE involving {A(t), b(t), x(t), ' (t)}.

{A(t), b(t), x(t), 7(t)}. Thus the local well posedness theory for (2.2.3) is equivalent to the local well posedness theory for NLS.

A.2 Proof of Lemma 2.6.7

Let us turn to the proof of Lemma 2.6.7 now. This is very standard in algebraic topology. Note Q - {0} is the retract of &Q, i.e. there is a map

r :'R" - {O} -- + oQ, such that

r o t = idaQ, (A.2.1) here t : 0 -+ RT - {O} is the natural inclusion map.

Now we prove Lemma 2.6.7 by contradiction. Assume 0 V f(Q). Then g r o f is well defined and continuous. Note g is map from Q to 0Q.

On the other hand g1,9 is homotopic to the idjan. Indeed, we may write down the homotopy explicitly r o (tf + (1 - t)id).

142 We emphasize here this homotopy is well defined since tf(y) + (1 - t)y # 0 for t E [0, 1] and y E &Q.

Now we have constructed a map g from Q to OQ, and glan is homotopic to idja.-

Note Q is convex domain and &Q is homeomorphic to the sphere. This is a clear contradiction from standard homology theory.

A.3 Proof of Lemma 2.6.9

Before we go to the proof, let us point out that Lemma 2.6.9 basically says that the blow up point (model by Xj,A(t)) and the blow up time (modeled by Aj,A(t)) depending on the initial data (modeled by A) in a continuous way. We remark here that, in general, the problem whether blow up poinst and blow up time depend on the initial data in a continuous way is not an easy problem. Indeed, if one has a sequence initial data uo,,, whose associated solution to (1.1.1) blows up according to the log-log law, and one assumes uo,,, converges to uo in H1 , it is not always true that the associated solution u to (1.1.1) will blow up according to the log-log law. And for NLS, if we don't have some information about the dynamic near the blow up time, we cannot even define the blow up point. However, see [56] for results in this direction.

The proof of Lemma 2.6.9 is much easier, because we are only working on data with finite parameters A. And if A, -+ A, clearly UA still blows up according the log-log law with dynamic described by Lemma 2.3.2. Let us turn to the proof. We recall

A E [-aoAi,o, aoM,olm1 x (Bl)m.

Proof. Recall that to understand the evolution of xj,A(t), Aj,A(t), one needs to consider the system (2.3.27) or equivalently (2.3.38). As explained in Appendix A.1, the system is NLS coupled with 4 ODE. Using the standard stability arguments for NLS and stability argument for ODE, we have that for any T < TA, (recall TA is the blow up time of UA. ) the map A -+ (xj,A(T), Aj,A(T)) is continuous. Now, Lemma 2.6.9 easily follows from the the following lemma

143 Lemma A.3.1. Given A, for any e > 0,there is T < TA and 6 = 6(A) such that for any A' with

IAj,A'(T) - Aj,A(T)I < 6, (A.3.1) then

sup IAj,A,(t) - Aj,A(T)I < e,j = 1,..,m. (A.3.2) tE[T,T]

sup Xj,A(t) - Xj,A'(T)I < eJ = 1, ... m. (A.3.3) tE[T,TA,]

We now prove Lemma A.3.1. Note A is given, and we only need to prove (A.3.2) and (A.3.3) for every given j.

We discuss the two cases.

" Case 1: the jth bubble of uA blows up, i.e. Aj,A(TA) = 0.

" Case 2: the jth bubble of UA blows up, i.e. Aj,A(TA) # 0.

We first discuss Case 1. In this case ,we can choose T close to TA enough, and 6 small enough, such that Aj,A' (T) is small enough,

100 (A.3.4)

Then (A.3.2) follows since dAj,A, < 0 and Aj,A,(t) > 0,Vt < TA,. To prove (A.3.3), one needs modify a little bit the analysis in Subsubsection 2.5.4. as argued in in

Subsubsection 2.5.4.

A' Ixj,A'(t) - Xj,AI(T)I < < fT'IT 1 (T )dr (A.3.5) I t dxjds. Aid1 d T Aj,A

(We have use 1 1 dx, ;< 1 by (2.5.6).) And recall (2.5.45), we further have

1 (A.3.6) Ixj,A'(t) - xj,A,(T)I ,< A ,() I/TA'3 < < E50. I ;TAA1/,(T)

This gives (A.3.3).

144 Now we discuss Case 2. Without loss of generality, we assume Aj,A/ (T) > 6o > 6100, otherwise we just argue as Case 1. Note since UA blows up at TA, at least one bubble blows up at TA, let us assume limtTA AjO,A = 0. Thus, we are able to choose T close enough to TA and 6 small enough such that

Aj, A, (T) >>> Ajo (T), (A.3.7) and of course Aj.,A'(T) < e00.

We remark here we actually may need

1 Ajo ,A' (T) < exp(- exp exp exp exp{ ),) (A.3.8) Aj,a (T) and don't worry about the special form because our arguments are kinds of soft.

The idea is UA is going to blow up so fast that the jth bubble do not have much time to change.

Note it is not clear whether the joth bubble will blow up when UA' blows up, however, we can still estimate blow up time by Lemma 2.6.1, i.e. the blow up time

TA, of UA' is controlled by Tjo,A predicted by Lemma 2.6.1, we have 1

TA , - T , A t,,(T) In ln Ajo,A'(T)|. (A.3.9)

And, using Lemma 2.6.1 again, we can ensure

Aj,A'(t) ~ Aj,A'(T), t C [T, TA/i. (A.3. 10)

Now we use estimate, (we need (2.6.21), (2.6.22))

d 1 dA,, 1 s 3 b-,-A ~ (A.3.11) 0 < dtd-Aj'A A3 jA' .7 j,A' InIln AAI AiA/ dsj AlIA, j,A'

'Lemma 2.6.1 is stated for solutions starting at t = 0, clearly we can also set the starting time at t = T.

145 Combine (A.3.11) and (A.3.10), plug in (A.3.9), we have

sup IA,A'(T) - Aj,A'(t)I < A,(T) in in AJA(T)IA ,,(T) in ln AA-0 A(T) te[T,TA,) (A.3.12) The desired estimate (A.3.2) follows since we have (A.3.8). Similar arguments work for (A.3.3). l

146 Appendix B

Lecture notes on concentration compactness

B. 1 Introduction

These notes were used when the author gave three guest lectures in the course 18.158, Topics in Differential equation, at MIT, during the 2016-2017 Spring term. Those notes do not give too much technical details for the proof of the concentration compactness. The main goal is to motivate the notion and illustrate the applications.

To motivate the notion of concentration compactness, we start with the classical Arzela-Ascoli Lemma.

Lemma B.1.1. Let {fn} be a sequence of C' functions on [0, 1], if both fu, and the derivative of fn are uniformly bounded, then one can find a subsequence {fn,}k which converges uniformly.

In the language of functional analysis, the lemma says the embedding

C1([0, 1]) -+ C([O, 1]) (B.1.1) is compact.

The lesson we can take here is the following:

147 If we have an embedding from a function space X to another function space Y, and X has higher regularity, then one may expect the embedding to enjoy certain compactness.

Before we carry on, we ask one more question. What will happen if we use the whole real line R to replace the finite interval [0, 1] in Lemma B.1.1, i.e. is the embedding C'(R) -+ C*(R) (B.1.2) compact?

The answer is no. And the reason is because the two function spaces are both invariant under translation symmetry, which causes the loss of compactness.

To see embedding (B.1.2) is not compact, we simply consider a C' bump function #, and let f,, = O(x - n). Clearly {fn}, is bounded in C1 (R), and clearly no subsequence can converge in C0 (R).

Another classical example is the embedding H' -+ L 2 . It is known that H 1 (T") -+

L2 (T') is a compact embedding (Rellich Kondrachov theorem). However, the embed-

1 2 ding H ((R"n) -+ L (R"n) is not compact.

2 Exercise: Why the embedding H"(R"n) - L (R") is not compact?

With these observations, we can now start talking about concentration compactness. Basically, one has certain embedding from a regular space X to a less regular space Y, and one asks whether such embedding is compact. While the answer is usually no, due to the lack of compactness, it is sometimes expected that one can use concentration compactness to overcome it.

B.2 Notation

We sometimes write If 11p for|lfILP(Rn)-

A lot of subsequence will be taken, for notation convenience, sometimes we use

{an}, itself, rather than {ank} , to denote a subsequence of {an},.

148 B.3 A preliminary model

The material in this section is essentially a review of (part of) Patrick Gerard's work,

|UIIL4(R3) $ IIUIIH1(R3) (B.3.1)

(the critical Sobolev exponent would be 6, note 2 < 4 < 6.)

In the rest of this section, we restrict ourselves in dimension 3, and we are in Euclidean space.

Remark B.3.1. Note , (B.3.1) says

< 1 -- A >-1/2: L2 -+ L4 (B.3.2) is an embedding.

One will later see, at least at the technical level, that 2 < 4 is an important fact here.

This is not a coincidence, indeed, one may do the following exercise. For any bounded multiplier type operator

Tm : LP -* L'q (B .3.3)

One necessarily has q > p.

Let us go back to Sobolev embedding (B.3.1). Argued similarly as in the introduction, this is not a compact embedding, as both space are invariant under space translation.

To be more precise, again fix a smooth bump function #, and let g,(x) = 0(x - nel), (here el = (0, 0, 1)), clearly g, are bounded in H1 and one cannot find subsequence gl, converging in L4 .

149 The next natural question is the following:

Is the the embedding H' -+ L' a compact embedding modulo space time translation?

In other words, let fa, be bounded in H1 , is it true that one can find a sequence 4 x, E R', such that there is a subsequence {fn,(X - Xnk)}k converging in L . Note the previous example showing the non-compactness of Sobolev embedding is not a counter-example any more.

Although, seeming natural, this is not true. One can consider the following fa. Again, we fixed some bumping function 0, and let

f.(x) = O(x - nel) + #(x + nei). (B.3.4)

One may check it is impossible to find Xz such that there exists subsequence such that {fnk(X - Xn)} converges in L4 .

Profile decomposition or concentration compactness in some sense say the above counter example is the only counter example.

With this in mind, one may "naturally" state the following "lemma":

Lemma B.3.2 (An almost correct lemma). Let fn be bounded in H'(R3 ), up to picking subsequence, one has the following asymptotic decomposition

fn = S#(x - X,) + on(HI) (B.3.5)

To be precise, one is able to find a sequence of H' function #j, (which is usually called profile), and for each j, a sequence of X ,nE R', such that for all J > 0

J fn= 5 (X - X,n) + WJn (B.3.6) j=1

with

lim sup lim sup W,n|Hi = 0 (B.3.7) J n

150 and asymptotic orthogonality,

Vj f j', ,, - Xjn|It 00. (B.3.8)

Note (B.3.8) is very natural. Indeed, assume one has decomposition (B.3.6), and for Ji # j2 (B.3.8) does not hold. Then one could indeed emerge the two profiles #j, and #22 by setting

0*1 () = limq531(x) +- # 2 (x - + ,n) (B.3.9)

Here, up to picking a subsequence, one may assume limn X 1 ,,n - -j2,n x o, thus the limit in (B.3.9) exists in H 1 .

Unfortunately Lemma B.3.2 is not correct, (note we do not even use the L4 space involved in Sobolev embedding).

Let us first observe why Lemma B.3.2 is wrong.

Assuming Lemma B.3.2, then for any sequence of {fn} bounded in H1 , one would be able to capture a non-trivial H1 structure #1 in the sense that

fn(x + X 1,n) - 1 (B.3.10) where weak convergence is in H1 .

Now, let us contradict the above corollary of Lemma B.3.2.

Again let 0 be a smooth bumping function, assume q is supported in a ball of radius 1/2. Let el E R 3 be, for example, (0,0,1). We further assume II0IIH' 1-

Let

fn = $= 5(x - kei). (B.3.11) k1

Then we have ||fnIHi = 1 (note O(x - kei) and O(x - k'e2) does not have common support for any k / k').

However, for any sequence x, e R', fn(x - x,,) weakly converges to zero.

Exercise: Show for any sequence x, E R", fn(x - x,) weakly converges to zero.

151 Thus, Lemma B.3.2 is wrong.

Let us go back the sequence fn as in (B.3.11). If one want do the decomposition as in (B.3.6), one gets #j = 0 for all j, and Wj,n = fn.

Thus, the problem is

lim sup lim sup IIWJ,lHIHl = lim sup IfnIH= 1 $ 0. (B.3.12) J n n

However, make the following observation, for fn as in (B.3.11), we have

IjfnIL4 = - je 11I4 4. (B.3.13) j=1

(As aforementioned, here we do need the fact 4 > 2.)

Thus, instead of (B.3.7), (which does not hold), we have (lettingW2, = fn)

lim sup lim sup 11WJn1IL4 = 0. (B.3.14) J n

From viewpoint of this example, one may state the following correct Lemma (recall again we are in dimension 3)

Lemma B.3.3 (Profile decomposition for H1 -* L4, [331, [381). Let fn be bounded in

H 1, up to picking subsequence, one has the following asymptotic decomposition

fn= S (x - xj,n) + on(L4 ). (B.3.15)

To be precise, one is able to find a sequence of H1 function #j, (which are usually called profile), and for each j, a sequence of xj,, E R3 , such that for all J > 0

J fn = O5(x - xj,n) + Wjn, (B.3.16) j=1

with

lim sup lim sup IIWJ,nIIL4 = 0 (B.3.17) j n

152 and asymptotic orthogonality,

Vj # j', jx,n - Xjn I n-+ o. (B.3.18)

and for each J, we have for any j < J

lim WJn (X + j,n) 0 (B.3.19)

where the weak convergence is in H1 .

Comparing with the wrong statement "Lemma" B.3.2, we have two modifications in Lemma B.3.3.

First, and most importantly, in the control of the error term (B.3.17), we have used the weaker norm L4 rather than the stronger norm H1 .

Second, we add an extra condition (B.3.19). This of course needs to be seen from

the proof (which we will not talk about here). However, let us try to convince the reader this is not surprising.

Indeed, assuming this Lemma, then how should one find all profiles #1, 2, ... ? Basically, one tries to find a sequence {Xn}n such that (up to picking subsequence)

f (x - Xn) converges weakly. Thus, if (B.3.19) is not the case, one may extract extra piece from the remainder term, and merge it with the profiles we already have. We end this section with the following remarks.

The consequence of the lemma indicates the following orthogonality condition: For each J fixed, one has

|i +I|Wj,nJ|H1 +on(1) (B.3.20) I 7 ||I 1f= H j=1

and

||fnII14 ZiI||#I|4 + ||W1,nI14 +on(1) (B.3.21) j=1

And, the orthogonality is due to the orthogonality of the parameter, (B.3.18). The

profiles themselves may not be orthogonal in any sense, for example, it may happen

153 Finally, we remark the structure of the remainder term may be very complex (even after one extracts all the profiles), not necessary in the form of (B.3.11).

B.4 A working example

One may see the Clay lecture notes of Killip and Visan (Section 4.2), [47] and reference therein for more details in this section. The sharp sobolev embedding was studied in [1], [70].

We still work on R 3

Let us consider the equation,

-AU uU (B.4.1) and look for H1 solutions.

It turns out this problem is also related to Sobolev embedding. Note we have

||fl|L6,< ||f||lI.t (B.4.2)

One may look for the sharp constant for inequality (B.4.2), which reduces to the following optimization problem:

Maximize If 116, given |If Iki = 1. Note assuming there is f, such that

Ilf*liftl = 1 and I|f*116 = max lf 116 (B.4.3) lfli'[1=1

It is not hard to see, up to multiplying some constant, f* solves (B.4.1). So how to find the optimizer f*?

Firstly, due to (B.4.2), we know

sup If 116 < 00 (B.4.4) 11f I1,i =1

154 Clearly, one may at least find a sequence f, G f 1 such that

!|fn||Ni =1, and liM |Ifn|I6 = sup 11f116 < 00 (B.4.5) n 1f 1Ui =1

If the embedding (B.4.2) is compact, then we are done. Since we know f" strongly converges in L , (up to picking subsequence.) We may simply let the f, be the H1 weak limit of {fn}, and it will do the job as in (B.4.3).

However, (B.4.2) is not compact, due to the symmetry. This is exactly what we have been talking about in the previous section.

The problem here is more subtle, since there is another symmetry, besides the

space translation, there is scaling symmetry. Let UA - u(x/A), then one may check (recall again we are in dimension 3)

1 IUA11L6 = I JIL6, IIUAI1ft1 = JU .fti. (B.4.6)

It turns out scaling and space translation are the only two relevant symmetry in

this problem, thus we may naturally state the analogue of Lemma B.3.3.

Lemma B.4.1 (Profile decomposition for H 1 -+ L6 , [33]). Let {ff} be a bounded

1 1 sequence in H , then up to picking a subsequence, one is able to find $1, 02, -- in H , and parameters { j ,jn, (A>,n > 0, Xj,n E R3 ), such that for each J > 0, one has

= 0(X Xn) +Wji, (B.4.7) fn(x) Z j,n d

where

lim sup lim sup IWJn|L6 = 0 (B.4.8) J n

and one also has asymptotic orthogonality,

Vj j', | X, '| +| 'nI +V 'I| 4 00, (B.4.9) j,n A3'n Aj,n

155 and for each J, we have for any j < J

lim /\1/ W2, (Aj,n-- + -Cj,n) 0, (B. 4. 10) .n where weak convergence is in ft.

Again, as a consequence, one has the following asymptotic orthogonality For each

J fixed, one has J

I|fn|IIp =3 I|#i|| 11 + IlWy,n||Ii + on(1), (B.4.11) j=1 and J|fn|0H1 = 1||:1j + 11Wj,n|i 1 + On(1). J (B.4.12) LjnJ6 -Z q$IL6 + 11 Wj,n L~6 +O o(1). j=1

Let us now see how could we go back to our optimization problem. The exposition we use here follows the clay lecture notes [471.

Recall, we have a sequence {fn}, such that (B.4.5) holds.

Let Co := max 1 ftf= 11 f 116. Note we have

lim llsfn16 = CO, (B.4.13) n1 and

Vf E -fliI, f 11f6 <- C011f 111i. (B.4.14)

Now, let us perform profile decomposition to this sequence via Lemma B.4.1.

We claim that except for one #1 , all #j = 0, j = 1, and indeed we must have

|| 1ki = 1.

We prove this by contradiction, if this were not true, let us assume 2l#1|j| 1 0- let V1,n = fn - #01(X-7'n). (Here, we only need 0 < E0 1).

By (B.4.11), we have

||01,n||ii = 0+ on(i). (B.4.15)

156 By (B.4.12), we have

||fn|I6= 6|q1||0+ II1,n|' + on(1). (B.4.16)

Now, using (B.4.14), we have

I|fn||I < C6(I0 1 |j1 + 1101,nI11) + on(1) = Cg6((1 - E2)3 + E+ o(1), (B.4.17)

and this contradicts (B.4.13).

Thus, we can claim 0q11 1 = 1, (and are are profiles and remainder term are zero or asymptotically zero), thus I1fnI6 = 110116 + on(1), implying 11#1I6 = Co, and we get the optimizer. In particular, we find one solution to (B.4.1).

To end this section, we remark on Lemma B.4.1. Lemma (B.4.1) essentially says the following: If a bounded H 1 function sequence fn does not asymptotically vanish on L6 , it must be both physically concentrated in some sense, (thus giving the existence of Xj,n), and frequently concentrated in some sense, ( thus giving the existence of

B.5 One more working example in dispersive PDE

We use energy crtical NLS (B.5.1)

ut + AU = Hu,(B.5.1)

u = uo E H 1 (R3 ),

as an example to illustrate how to set up profile decomposition for a dispersive

PDE. We restrict our dimension to be 3 in this section.

B.5.1 Basic setting of profile decomposition in dispersive PDE

For dispersive PDE, the analogue of Sobolev embedding is Strichartz estimates. The fundamental Strichartz estimate associated with the energy critical problem , (B.5.1)

157 'is

Ie uQIIL0 Il'0I2. (B.5.2)

See [40], [71. The norm L jo, plays a very important role in the well posedness theory of (B.5.1).

We introduce some notation here. The pair (q, r) is called admissible if 1+ =-, and we define the X' norm as

If lxi = sup IIVfllLqL; - (B.5.3) (q,r) admissibale

Recall the Strichartz estimate ([40], [7]) for (q, r) admissible

le UO1IL Lr UOL. (B.5.4)

We summarize the key properties of the norm L"oX in the study of (B.5.1)

" Small data theory. There exists E > 0, such that for initial data u0 E ft and

ieA uoIILIo < E, the associated solution u to (B.5.1) is global, and there is the estimate

IluIlL10 < IUflXi Si IluoHl. (B.5.5)

" Blow up criteria: Assume a solution u to (B.5.1) is defined on [0, T) and cannot

be extended beyond T, (i.e. the solution blows up at T), then one must have

0 ull L ({Q,T),L1O) = 00. (B.5.6)

" Scattering criteria ( which can be understood as a stronger version of blow up

criteria): For a solution u which solves (B.5.1), assume there is C > 0, such

that for any (0, t) when the solution u is defined, one has a priori estimate

TnhuLt((s,t),LIO) < C. (B.5.7)

Then the solution must be global (due to blow up criteria), i.e. it must be

158 defined for all t > 0, and the solution scatters to a linear solution eitu+ in the sense lim l1u(t) - eitAu+Ifi = 0 (B.5.8)

* When the initial data is small enough in f 1 , the solution scatters. (By small

data theory and scattering criteria)

* Stability Theory. Assume ii almost solves (B.5.1) in the time interval I (we assume 0 E I) in the sense that

i&it + Ai = |iij4ii + e(t, x), t E I (B.5.9)

and assume further

IIUIIL-1 <$. (B.5.10)

and (B.5.11)

Then there exists e = e(M, E), such that if

I|uo - ii(0)||f1 < E, (B.5.12)

and

IIVeIIL2L/5 E (B.5.13)

(note we simply want the error e be controlled, (2, 6/5) is the dual of (2, 6), which is an admissible pair.)

Then the solution u to (B.5.1) with initial data uO is defined on the whole interval I, and with control

I - i-||xj < C(M, E)E. (B.5.14)

159 In particular, we also have

IIUIILO11o < C(M, E). (B.5.15)

e Summary: For any given interval I, IIUIIL'oLio measures how nonlinear the

solutions are. When IIUIIL10Lio IS very small, then the solution behaves like a

linear solution. And if this norm is globally bounded, nonlinear dynamic finally

vanishes (scattering) in long term. And, for any interval such that this norm is

bounded, some (uniform) perturbation is allowed (stability theory).

One may refer to Lemma 3.9, Lemma 3.10 in [131 for stability theory. One may refer to textbook [711 and the reference therein for the local theory of (B.5.1).

To summarize, we have linear Strichartz esimate (B.5.2). And the norm we controlled in (B.5.2) plays a significant role in the well posedness theory for the nonlinear equation (B.5.1). Again , one would hope this the embedding L 2 to Let'X be compact, but this is not the case due to three symmetries:

" scaling

" space translation

" time translation

To see the last symmetry, note

IleitAeitoAf 1ILL0 le IL0 (B.5.16) and

If I| i = Ile- o'lli (B.5.17)

We remark that ultimately profile decomposition is based on linear analysis, in particular linear Schr6dinger equation

iut + Au = 0, u(0) = u0 E HP. (B.5.18)

160 Now we write down a profile decomposition for (B.5.2), this version is due to Keraani, [44].

Lemma B.5.1 (profile decomposition for energy critical NLS). Let {u"} be a bounded 1 sequence in H . Up to picking a subsequence, there exist 1,...,,... and parameters

{tj,n,xj,,, Aj,.} such that, for each J > 0, let1 V be the linear solution to (B.5.18)

with initial data Oj, j < J,

u,n=Z j(_jin X - X '") + WJn, (B.5.19) j-1 ' j,n Jn

(we use e" to denote the linear propagator to (B.5.18), as usual) with asymptotic vanish of dispersive norm,

lim lim sup IIeitAWJnIILo = 0 (B.5.20)

and asymptotic orthogonality of parameters,

tin - tj'n Xp ~ - Xyn An An n-*oo 0. (B.5.21) Vj,n 'j,n j',n Aj,n

and weak convergence of the remainder term, i.e.

let Wj,, = eit ljW,n, and we have for any J < J

1 A 12WJn(tjn, Aj,nX + X3,n) - 0 in H (B.5.22)

We do have the following orthogonality, for any J

J |Iuo,n|IIi = ||0jlI1 + ||Wjn|i1 + on(1). (B.5.23)

and

IIuo,nH| = n|| 6++ On(1). (B.5.24) j=1

'The linear solution V has exactly the same information as its initial data

161 Note (B.5.23) and (B.5.24) have an important consequence. Let E(u) to denote 6 the energy of u, (i.e. E(f) = I f IVf 12 + I f If1 ), we have for each J

E(Uo,n) = E(#5) + E(wJ~n) + on(1). (B.5.25) j=1

Lemma B.5.1 is not directly suitable to study (B.5.1). Because this Lemma is totally linear, and our problem (B.5.1) is nonlinear.

However, the key point is if we view un (for large n of course) as the initial data for (B.5.1), then its associated solution can be write out explicitly to some extent.

Before we carry on, we need the important notion of nonlinear profiles.

First of all, without loss of generality, we may assume in Lemma B.5.1 for each j, one always has one of the following case

" Compact Profile: for all n, tj,a = 0.

" Forward Scattering Profile: J-+ 0. Ai,?

" Backward Scattering Profile: ' 2> 0. .j,n

Now for each linear profile, i.e a function f in f', and a sequence of parameters tn, An, Xn, (note we always assume tn/A = 0 or goes to +oo), again let F be the solution to (B.5.18) with initial data f.

Definition: We say U is a nonlinear profile associated with the profile {f; tnAn, x} if and only if

* U is the k' solution to (B.5.1), and U is at least defined in a neighborhood of

limn 2 . (The neighborhood of oc should be understood as (M, oo) for some large M).

e For n large enough, one has

lim |IU(-tn/An) - Fl(-tn/An)I|ft 2+ 0. (B.5.26) n

We leave it as an exercise that the nonlinear profile always exists and is unique.

162 Now we are ready to state a nonlinear version of Lemma B.5.1.

Lemma B.5.2 ([44]). Assume a bound ft sequence {uo,n} admits a profile decom-

position with profile {k3 ; tin, Aj,n, xn}. Let U be the associated nonlinearprofile with

respect to Oj, {t,n, Aj,, xjn}. Let T > 0 and assume one has for each j

urn sup ||UJ| -n]O < O0. (B.5.27) n

Then let un be solution to (B.5.1) with initial data uO,n, then for n large enough, un

is defined in [0, T], and Vt E [0, r] (and all the estimates below are uniform on t)

St - ti~ X - xi un(t, x) = U ( A)n n +An (B.5.28) j=1

and

lim IAj,n(t, x) - eitAWJ 2 1 =jf0 (B.5.29)

and 00 |1un|L|10[0,T;L10} Wi ||U| 0+ On(1) < O0. (B.5.30) =1 , 7,n , n

Remark B.5.3. We make a very important remark here. It seems at first glance

(B.5.27) is a very strong assumption and involves the checking of infinity many profiles. This is not the case. Indeed, for any e > 0, due to (B.5.23), there are only finitely many j such that |fj#j jt1 > e. And, for all those j, |I|#|i < e, the small data theory applies and their associated nonlinearprofiles Uj must be defined on the

whole real line with bound ||UI|L| ,< e.

Note this also explains why we have the " < oo" in (B.5.30).

Remark B.5.4 (On proof and generalization of Lemma B.5.1, Lemma B.5.2). We

will not talk about the detailed proof of Lemma B.5.1 here. However, Lemma B.5.2

directly follows from Lemma B. 5.1 using stability theory or local well posedness theory

of (B.5.1). In particular, Lemma of those type cannot capture the energy structure of the equation, and exactly the same Lemma holds for focusing energy critical NLS.

163 Remark B.5.5. Morally speaking, linear profile decomposition asymptotically de- compose a sequence of initial data into decoupled profiles #j (up to symmetry), and a remainder term ww,. The remainder term may contain significant amount of energy but vanishes if one measures it via dispersive norm. In particular, from the Sum- mary we previously discussed about L'o, if one evolves w, by (B.5.1), the outcome is like if one evolve w, linearly via (B.5.18).

Finally, the nonlinearprofile decomposition Lemma B.5.2 basically say, that asymptotically, evolving uo,n is like evolve each profile separately and nonlinearly, and evolving the remainder term linearly, and sum them together.

In particular, if all the profile during evolution does not blow up, the solution un would not blow up.

B.5.2 On defocusing energy critical NLS, (B.5.1)

One of the main questions regrading equation (B.5.1) is the following: Given initial data uo E f 1 , is the associated solution u global and if so what is the asymptotic behaviour?

The actual question is indeed more specific, because people are expecting the solution ro be global and scatter.

More mathematically speaking, one wants to prove that u is defined on whole real line and

||UI|L10 < 00. (B.5.31)

This question is totally answered by the seminal work [13]. And in some sense, the question is answered by asking a harder question 2 , i.e. is it true that u is global and one has a priori estimate

IIuIILlo < C(E(u)). (B.5.32)

Here C(E(u)) means a constant only depending on E(u).

2 Though, we should remark, due to the pioneering work of Bourgain, it is natural to ask this harder question.

164 It is impossible for us to give a complete review of [13] here. But we would use concentration compactness to illustrate some partial results. We remark that it is impossible to only use concentration compactness types techniques to prove the result in [131 . This is because concentration compactness are purely linear and perturbative methods, which are not enough to capture the energy structure of (B.5.1). But in some sense, concentration compactness can put one in a favorable position to apply another ingredient of [13], interaction Morawetz estimate.

In this subsection, we illustrate two partial results towards the proof of (B.5.32) and hopefully give readers idea about how to apply concentration compactness.

The existence of a priori estimate

The material in this section is one of the results in [44].

We show here (B.5.32) is indeed not stronger than (B.5.31). This can be achieved by certain compactness argument.

Lemma B.5.6. If for any solution u to (B.5.1) with E(u) < 00, one has that u is global and

||allL10 < 00, (B.5.33)

Then there exists a (increasing) function C, such that for any solution u to (B.5.1) with E(u) < M, one has

I1u||L1o C(M) (B.5.34)

Let us first give a wrong proof to illustrate why compactness may be involved.

Wrong Proof of Lemma B. 5.6. Assume a bounded space in H1 is compact. We prove

Lemma B.5.6 by contradiction. If Lemma B.5.6 does not hold, one can find M > 0 and a sequence of solution un with E(un) K M such that

I|ufhLlo > n, (B.5.35)

On the other hand, since un(0) are bounded in H', and "a bounded space in

H' is compact", then up to picking a subsequence, one may assume un(0) strongly

165 converges to u,(0). Let uthe solution generated by u*(0), then u* is global and

IIUIILO A < o (B.5.36)

Thus, by stability theory, we have for n large enough,

IIUnhILlO < 2A, (B.5.37) a contradiction.

The proof is wrong since we know a bounded space in f 1 is not compact.

Profile decomposition can help us fix this.

Proof of Lemma B.5.6. Again, we prove it by contradiction. Assume Lemma B.5.6 does not hold, one can find M > 0 and a sequence of solution un with E(un) < M such that

|Un||Lo ;> n. (B.5.38)

By Lemma B.4.1, we may assume up to picking a subsequence, un(0) admits a profile decomposition with profiles {j; {Xn, An, tpn}}j. Let the associated nonlinear profiles are V, then by assumption of the Lemma B.5.6, we have V are globally defined and

||VA|L10 < 00 (B.5.39)

Thus we have that un are defined on the whole real line , (here I cheat a little bit, but very easy to make it rigorous 3.) and

IIUntIL IIl + on(1) <0 0 (B.5.41) j==1

3 The point is that, to show a certain solution u is global and scatters forward, we need only to achieve a prior estimate, i.e. for any t within its lifespan,

IUIILj{[0,t],LIO} <; C, (B.5.40) and the constant C does not depend on t.

166 This contradicts the (B.5.38), since the right side of (B.5.41) does not depend on n

(except for on(1) part.) The proof is already complete, but we perform an extra argument here , hopefully making things more clear.

The " < oo" is already contained in the statement of Lemma B.5.2, i.e. in (B.5.30).

But let us derive it here.

As remarked in Remark B.5.3, for each E > 0, there are at most finite j such that #j > E, we assume those are the first Jo profile. Clearly, we need only show

10 <00 (B.5.42) j=Jo+1

On the other hand , since ||j5||y, < E for j > Jo + 1, by small data theory, we have for j > Jo + 1 ||VAL'1o < ||5|p(B.5.43)

Thus (B.5.42) follows from the (B.5.23) and the uniform bound E(un) < M. (As you see, it is very important 10 > 2 here). l

The existence of critical element

The material in this section can be understood as a rewriting of the result in the first half of [131 from the view point of the Kenig-Merle road map, [41], [421. See also the work of Visan and Killip, [461, which revisits [13].

At first glance, Lemma B.5.6 is not very useful, because the assumptions are too strong. Note before [13], it was even not clear that all f 1 solution are global whereas

Lemma B.5.6 assume all k 1 solutions are global and scatter.

However, one may reformulate Lemma B.5.6 in a more applicable way with essentially the same proof.

Note, by small data theory, it is easy to see there exists some Eo > 0, such that

E(u) < Eo -+ u is global and scatter. (B.5.44)

167 Thus one may look at

Ao := max{B : E(u) < B = u is global and scatter} (B.5.45)

Clearly A 0 > 0 due to (B.5.44). The goal is to show A0 = oo, which would imply all k 1 solutions are global and scatter.

Firstly, it is not hard to rewrite the Lemma B.5.6 in the following form

Lemma B.5.7. Let Ao be as in (B.5.45), then there exists a (increasing) function

C, such that for any solution u to (B.5.1) with E(u) < M < Ao, one has that u is global and

IIuIILt C(M) (B.5.46)

Now, the question is how to prove A0 < o. Proofs for results of those type are usually proof by contradiction. We will give a summary in the next Subsubsection.

One key element .is the following Lemma

Lemma B.5.8 (Existence of critical element). Assume Ao < oo, then there exists a solution u, to (B.5.1) with E(u,) = A 0 , and

IIucIIL1o{(T-,T+);L} = 0O (B.5.47) here we use (T-, T+) be its maximal life span.

The lemma B.5.8, though nontrivial, is not directly applicable, but it turns out u. has very nice properties, (so nice that it cannot exist which leads to the desired contradiction.)

Lemma B.5.9 (u, is almost periodic). Let u, be as in Lemma B.5.8, then there exists

A(t), x(t) depending continuously/smoothly on t E (T-, T+), such that

uc(t, x) - 2 W(x x(t), (B.5.48) A(t)'/ A(t) and {Wt}e(T-,T+) is a precompact family in ft, (here we view t as a parameter.)

168 We briefly sketch the proof of Lemma B.5.8, Lemma B.5.9. The proof is nowadays standard, one may refer to [41],[421, [46] and see also the reference therein.

Sketch. Assume AO < oc, then one would is able to find u, to (B.5.1) such that

E(un) _< Ao + 1/n (B.5.49) and un does not scatter.

Then, by a profile decomposition for {un(O)}, one would be able to conclude there is only one nontrivial profile # whose energy is exactly AO, and # would generate a non-scattering solution uc which also satisfy (B.5.48) as in Lemma B.5.9. I

Kenig-Merle road map

After the work of [41], [42], there is a road map to handle critical problems as (B.5.1), such as showing AO in Lemma B.5.7 to be oc. We do an example in this subsection to illustrate how the road map works and how concentration compactness helps.

We want to show : Any radial solution to (B.5.1) with finite energy is global and scatters. This result is due to [51, [37], and of course finally be implied by later work [13].

The road map basically has three pieces in this particular problem

* Small data theory

* Concentration compactness

" Rigidity theory

The small data theory we need is that any radial solution with energy small enough is global and scatters. This is usually expected and equivalent to local theory for critical problems.

Concentration compactness is a version of the profile decomposition, Lemma B.5.1, based on embedding eitAuo IIL1O < |Iuo II (B.5.50)

169 Note the norm appearing in the LHS of above inequality plays a decisive role in the study of long time dynamic and the norm appear in the RHS of above inequality is exactly the critical norm.

As illustrated in the previous subsubsection, the small data theory and concentration compactness techniques implies:

If not all finite energy solution scatters, there exists some minimal counter example u,, described as in Lemma B.5.8, Lemma B.5.9, which is some times called compact solution or minimal blow up solution.

The rigidity theory basically classifies minimal blow up solution, and in our case, it implies such solution u, cannot exist.

The rest of this section is devoted to the proof that, assuming radial symmetry , uc in Lemma B.5.9 cannot exist.

To give the full proof, it is inevitably that one use certain harmonic analysis and truncation techniques which may make the illustration hard to follow. We sketch the proof, and the reader should be able to fill in extra details.

We follow the presentation in Chapter 5 in [711 , (with slight modification to match the concentration compactness setting.), see in particular Section 5.2, Section 5.3 in [71].

Note that due to the radial assumption, x(t) = 0 in Lemma B.5.8 and Lemma B.5.9.

One may assume {Wt} is not only a precompact family but Wt = W for some

W C H' for simplicity, the proof does not rely on this.

(However, we would not assume W smooth or in L2 . Such assumption would change the nature of the proof.)

Let us first introduce the notion of LWP interval. Note for each to, the local theory illustrates that u, would not really change4 in a time interval I with length

~ A2 (to), (this is the scale indicated by local theory). And we may also derive in I,

A(t) ~ A(to). We also expect and assume without a proof that in such an interval

I, u, is physically localized at scale - A(to) and frequency localized at scale ~- .

4 For any quantitative estimate, one may just assume it does not change.

170 At least at the heuristic level, it is totally OK for one to assume u, ~ I when x r~-, A(t). It is not hard to see that in such an interval I

uCIILioLio - 1, rather than only < 1. (B.5.51)

We will call such an interval LWP interval. Now, we may divide the life span of (T-, T+) into a union of disjoint or essentially disjoint LWP intervals Ik, and assume A(t) - A(tk) = Ak. To show a contradiction to the fact

IIucIILl (T-,T+) = 0 (B.5.52)

we want to show {I} < 00. (B.5.53)

Essentially, one wants to exclude the following two possible regime such that

{IO = 00

" (self similar) blow up

" soliton

A typical regime for self similar blow up is for t close to T+, A(t) - v/T+- t. Though this regime seems very specific, a lot of efforts has been made in the study of dispersive equation to exclude this possibility. A typical regime for soliton is (T-, T+) = R and A(t) ~ 1. We have the following two Lemma, we emphasize hereuc is already fixed.

Lemma B.5.10 (no self-similar blow up). Let to E (T~,T+), and fix A > 0, let

IlI,...I3k be intervals such that

Then k O(B.5.54) d(to, I jJ< A|I ,|I

Then k = OA (1).

171 Lemma B.5.11 (no soliton). For any interval I C (T-, T+), one has estimate

|I11/2 < 1111/2 (B.5.55) IkCI

As we shall see, both Lemma B.5.10 and B.5.11 depends on the structure of the equation, but they are of different nature.

Lemma B.5.10 depends on the mass conservation law, which, though fundamental to Schrddinger type equation, does not see the difference between focusing or defocusing system.

Lemma B.5.11 depends on the Morawetz type argument, and really depends on the defocusing nature of the equation. Basically, for (B.5.1), if it stays bounded for too long, it must decay, and thus soliton cannot really exist.

Integration by parts are inevitable in the rest of this subsubsection. See Section 5.2, 5.3 in [71] for more details.

Sketch of the proof of Lemma B.5.10. Let the local mass be defined as

M'(U) := U12X 2 1/2 ,(B.5.56) where r is a parameter and x is a smooth bump function.

Note for any general k 1 function f, we have (note Ht embeds into L')

M (fy)<,JIfI 1. (B.5.57) r2 M r(f)

If one works a little bit harder, one may improve the above to:

For any (finite or infinite) sequence of rk, rk+1 rk,

iM(f) ,1 i

On the other hand, if we go back to our almost periodic solution uc, by doing

172 integral by parts we have

(B.5.59) dM (uc) f jUj 19j(X2 (r/x)). which implies (by using simple H6lder, and note for any t, Iucfl|l is bounded),

d 1 - (B.5.60) dt-Mr(Uc(t)),< r

Now we are ready to prove Lemma B.5.10.

For notation convenience, we write Ij, as Ii. (Though this is a slight abuse of notation.)

Let us fix a to, and pick t, C Ii, with associated scale A , then

1Ij Afi, (B.5.61)

And we have for any C large enough,

MCmj (Uc (ti))-~ Ai ~ |I li/2 (B.5.62)

On the other hand, by (B.5.60) and the fact d(to, Ih) < AIhI, we have

Choose C large enough (thus A/C small), we have by (B.5.62), (B.5.63)

Mc,\(ucjto)) ~ Ai ~ |Ijil/2 (B.5.64)

By (B.5.64) and (B.5.58), we have the desired estimate

{I } ,< OA(1). (B.5.65)

173 Sketch of the proof of Lemma B.5.11. Let a(x) = Ixk<(x/R) be a truncated version of XIx. After integration by parts and certain H6lder inequality, one may derive the Morawetz estimate

ot J QcVucVa ; LR 16 + O(1/R) (B.5.66) IxIR Thus, for any interval I within the life span of uc, we have

1u16/IxI < sup (B.5.67) J JIX

By doing some optimization, let R - 1I1/2, we derive for some large universal constant C,

U/ CII_/2. (B.5.68) I, fx

The key point is for any LWP interval Ik C I, for t C Ik, uc is localized physically at Ak - IIklI/2, thus we can use (B.5.68) to derive

ZIkCI I'k11 < II

Finally, we leave it as an exercise that Lemma B.5.10 and Lemma B.5.11 imply the desired estimate (B.5.53).

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