Chuanwei Gao,Changxing Miaoand Jianwei Yang-Urbain

1 The Graduate School of China Academy of Engineering Physics,P.O.Box 2101,Beijing 100088,China

2 Institute of Applied Physics and Computational Mathematics,Beijing 100088,China

3 Department of Mathematics,Beijing Institute of Technology,Beijing 100081,China;and LAGA(UMR CNRS 7539),Universit´e,Paris 13,Sorbonne Paris Cit´e,Villetaneuse,France

Received 2 November 2017;Accepted(in revised version)20 June 2018

Abstract. In this paper,we consider the defocusing nonlinear Schr¨odinger equation in space dimensions d≥4.We prove that if u is a radial solution which is priori bounded in the critical Sobolev space,that is,u ∈,then u is global and scatters. In practise,we use weighted Strichartz space adapted for our setting which ultimately helps us solve the problems in cases d ≥4 and . The results in this paper extend the work of[27,Commun. PDEs,40(2015),265-308]to higher dimensions.

Key Words: Nonlinear Schr¨odinger equation,scattering,frequency-localized Morawetz estimae,weighted Strichartz space.

1 Introduction

We consider the Cauchy problem for the nonlinear Schr¨odinger equation(NLS)in Rt×Rdxwith d≥4:

In particular,we call the Eq.(1.1)defocusing whenµ=1,and focusing whenµ=-1. In this paper,we are dedicated to dealing with the defocusing case.

The solutions of Eq.(1.1)are left invariant by the scaling transformation

We proceed by make the notion of solution precise.

for each t∈I.We call I the lifespan of u.We say that u is a maximal-lifespan solution if it cannot be extended to any strictly larger interval. We say u is a global solution if I=R.

The above fact promotes us to define the notion of scattering size and blow up as follows:

Definition 1.2(Scattering size and blow up). We define the scattering size of a solution u:I×Rd→C to(1.1)by

If there exists t0∈I so that S[t0,supI)(u)=∞,then we say u blows up forward in time,correspondingly if there exists t0∈I so that S(infI,t0](u)=∞,then we say u blows up backward in time.

The problem which we concern in this paper can be subsumed into the following conjecture.

then u is global and scatters,with

for some function C:[0,∞)→[0,∞).

For sc/∈{0,1},(1.4)can not be deduced from any available conserved quantity and it is a natural artificial assumption as a substitution of conservation law.

Before addressing our main results, we will make a brief review on the Conjecture 1.1. It is well known that in the critical case,the lifespan of solution depends not only on the Sobolev norm but also the profile of the initial data,thus the fact that(1.4)implies the solution u is global and scatters is not at all obvious.

In the energy-critical setting,the breakthrough was made by Bourgain’s monumental work[1]in which he introduced the induction on energy method. Based on this method and the space-localized Morawetz inequality, the spherically symmetric energy-critical case was resolved in d=3,4.Subsequently,by using the same strategy and the modified interaction Morawetz estimate,Colliander et al.[5]resolved the nonradial case in d=3.For further discussion about the defocusing energy-critical NLS, we refer to [13,20,28,34-36]. For focusing case see[10,15,17].

For further discussion about Conjecture 1.1,we refer to[11,18,23,24].

Now we are in a position to state our main results.

for some function C:[0,∞)→[0,∞).

Adapting the argument in[3],one can obtain the local-in-time theory which serves as a basis for the proof of Theorem 1.1.

1. Local existence: I is an open neighborhood of t0.

2. Blow up: If supI is finite,then u blows up forward in time. If infI is finite,then u blows up backward in time.

Remark 1.2. To prove Theorem 1.2,one may first assume the initial data belongs to Hscxso that the techniques in[3]applies and then establish Theorem 1.2 by using the following stability lemma.

Lemma 1.1. Let d≥4, I be a compact interval,and ˜u:I×Rd→C be a solution to the equation

Suppose

for some small 0＜ε＜ε1(E,L),then there exists a solution u to the Eq.(1.1)with the initial data u0and a constant 0＜c(d)such that

where the definition of S(I)and N(I)can be found in the appendix.

We present the details of the proof of Lemma 1.1 in the Appendix.

Now we can sketch the proof of Theorem 1.1.

1.1 Reduction to a critical solution

To prove Theorem 1.1, we argue by contradiction. Due to Theorem 1.2, we know small initial data implies the theory of global existence and scattering. If Theorem 1.1 fails,there exists a counterexample acting as a threshold. As a consequence of its criticality,such counterexample must concentrate in frequency and physical space at the same time.Further analysis shows that such special solution possesses a wealth of weird properties that a solution should not have in general. Finally,we will show that such properties are inconsistent with the structure of the Eq.(1.1).

Definition 1.3. For A＞0,we define B(A)as follows

Definition 1.4. We say SC(A) holds if for each u0∈B(A), then I=R and SI(u)＜∞.Similarly,we say SC(A,u0)holds if u0∈B(A),then I=R and SI(u)＜∞.

In view of (1.4), to prove Theorem 1.1, it suffices to show that SC(A) holds for each A ＞0. Note that Theorem 1.2 implies SC(A) holds whenever A is sufficiently small.Consequently, if Proposition 1.1 fails, there exists a critical value Acsuch that SC(A)holds when A＜Acbut fails when A＞Ac.In particular,using concentration-compactness method,we can obtain the following key proposition.

The derivation of Theorem 1.1 by now is standard. One can refer to[12,14,18,25-27]for more details.

The critical solution ucin Proposition 1.1 enjoys plenty of additional properties,especially among which is its compactness(modulo scaling),see[14,25]. For brevity,in what follows we abbreviate the critical solution ucas u.

Proposition 1.2. Let u:I×Rdbe the critical spherically symmetric maximal-lifespan solution to(1.1),for each η ＞0,there exists functions N:I →R+,C:R+→R+,such that

for all t∈I.We call N(t)the frequency scale function,and C(η)the compactness modulus function.

Remark 1.3. 1. This definition is adapted to the radial setting.In the general case,one should also take into account the translation. If we consider mass-critical case,one more parameter should be added in(1.13)due to Galilean invariance of(1.1).

2. By the Arzel`a-Ascoli theorem,(1.13)can be rephrased as

3. We claim that there is a constant c＞0 such that

which contradicts the fact that u blows up.

We emphasize that(1.16)has its analogue in Section 6 of[5]which says the potential part must have lower bound. Further, from the compactness property, we may choose c(η)sufficiently small such that

Next we will record more properties of the critical solution which will be used in what follows.

Lemma 1.2(Local Constancy[21]). If u:I×Rd→C is the critical maximal-lifespan solution to(1.1),then there exists δ=δ(u)＞0 so that for all t0∈I

Moreover,N(t)～uN(t0)for |t-t0|≤δN(t0)-2.

Due to Lemma 1.2,we can subdivide the lifespan interval I into several characteristic subintervals Jksuch that

The following result can be directly derived from Lemma 1.2.

Finally we relate the frequency function N(t) to spacetime norm by the following lemma.

Lemma 1.3(Spacetime Bound[21]). Let u:I×Rd→C be the critical maximal-lifespan solution to(1.1),for each interval J ⊂I,we have

By rescaling argument,we can also ensure

at least on the interval J which is one direction of maximal lifespan of u,say[0,sup(I)).For the sake of exposition, we may harmlessly identify J as I. For further discussion,see[31].

To prove Theorem 1.1,it suffices to show that the critical solution in Theorem 1.1 does not exist.To this end,the paper is organized as follows:In Section 2 we will present some basic tools. In Section 3, we will introduce the weighted Strichartz norm and the associated Strichartz estimate. In Section 4, we will establish frequency-localized Morawetz estimate, as a result, we will show that the weighted Strichartz norm of high frequency portion of the solution u will stay bounded,the fact which we will apply directly to rule out the critical solution. In Section 5,we will show that the frequency scale function N(t)can’t go to zero. Together with(1.23),ultimately we will preclude the critical solution in Section 6.

2 Notation and some basic tools

We define the Fourier transform on Rdby

and the homogeneous Sobolev norm as

where

Next we will present the Littlewood-Paley decomposition.

with similar definitions for P＜Nand P≥N. Moreover,we define

whenever M＜N. Also there are the following Bernstein inequalities for the Littlewood-Paley operators:

where 1≤p≤q≤∞.

We will also need the following chain rule for fractional order derivatives. One can turn to[3]for more details.

When the function G is no longer C1, but merely H¨older continuous, we have the following chain rule:

The classical H¨ormander-Mikhlin theorem concerns about the sufficient condition required for a function to be an Lp(1＜p＜∞)multiplier. We should adapt the usual one to be suited for our case and present here the extension form with the power weights. One can refer to[29]for further discussion.

Lemma 2.4. Let T be a H¨ormander-Mikhlin multiplier defined on tempered function f i.e.,

with its symbol m(ξ)satisfying the following pointwise estimate

for all f such that right-hide side is finite.

Remark 2.1. In particular,the operator N-s|▽|sP＜Nand Ns|▽|-sP≥Nare all H¨ormander-Mikhlin multiplier,as well as the frequency localized operator PN,P≷N.

At the end of this section,we will record some fundamental tools.One can find details in[31]and the materials therein.

Lemma 2.5 (Hardy-Littlewood-Sobolev Inequality). Let 1 ＜p,q ＜∞, d ≥1,0 ＜s ＜d, and α,β∈R obey the condition

and the scaling condition

Then for any spherically symmetric u:Rd→C,we have

3 Weighted Strichartz inequality

Motivated by the work of[31]which handled the mass-critical case,we adapt the argument to tackle the case without conserved quantities. In practice,we introduce weighted Strichartz norm suited for our case. To be more precise, we define ‖u‖S(I×Rd)and‖u‖N(I×Rd)respectively as follows:

where ε ＞0 is a sufficiently small constant depending on d and sc. By Lemma 2.4, we obtain that corresponding Bernstein inequalities with respect to the norms ‖u‖S(I×Rd)and‖u‖N(I×Rd).

Lemma 3.1. For any s＞0 and dyadic number N ＞0,we have

The association of‖u‖S(I×Rd)and‖u‖N(I×Rd)with Eq.(1.1)is illuminated by the following weighted Strichartz estimate and radial Sobolev embedding.

Proposition 3.1(Weighted Strichartz estimate[33]). Let u,G:I×Rd→C satisfy(i∂t+Δ)u=G in the sense of distributions,then we have

for all t0∈I.

Using(2.5),we will get the following radial Sobolev embedding.

Lemma 3.2(Radial Sobolev embedding). Let u be spherically symmetric and d ≥4,then we have

Lemma 3.3. If u,v:I×Rd→C are spherically symmetric and d≥4,then

By(2.5),we obtain

By the definition of N,(3.5)implies

Continuing from(3.6),by Lemma 2.1,Lemma 2.2 and(3.3)we have

by the defniition of N,(2.5)implies

Continuing from(3.7),by the H¨older inequality and(3.3)we have

Continuing from(3.8),by Lemma 2.1 we have

RHS of(3.8)

To complete the proof,it suffices to show that

By the local well-posed theory,for example see[2],one has

for any compact interval J contained in the maximal lifespan interval I. As a direct application of(3.4),we obtain the following result which,in some sense,can be viewed as an extension of(3.11)in the weighted norm.

Corollary 3.1. Let u:I×Rd→C be a spherically symmetric maximal-lifespan solution to(1.1)then

Proof. Using(1.4),(3.2)(3.4)and(3.11),we obtain

Thus,we complete the proof.

Next,we will give some refined nonlinear estimates which will be used to control the nonlinear interaction.

Proposition 3.2(Refined nonlinear estimate). Let u,v:I×Rd→C be spherically symmetric,then we have

Proof. By the definition of N and the H¨older inequality and Lemma 2.5, we estimate(3.12a)as

Similarly for(3.12b),we have

Thus,we complete the proof.

Remark 3.1. (3.12a)is very useful when u is low frequency and v is high frequency,as it transfers plenty of derivatives from high frequency to low frequency via the appropriate distribution of weight.

4 Frequency-localized Morawetz estimate

In this part we will primarily establish the following frequency-localized Morawetz inequality.

Proposition 4.1 (Frequency-localized Morawetz estimate). Let d ≥4 and u:I×Rd→C be the critical spherically symmetric maximal-lifespan solution to(1.1)which obeys(1.4),(1.23),then we have

To prove Proposition 4.1,we will first exploit some nontrivial facts about the critical solution u.

Lemma 4.1. Let u:I×Rd→C be the critical spherically symmetric maximal-lifespan solution to(1.1)which obeys(1.4),(1.23). Then for each θ ＞0,we have

Proof. By(1.13)and(1.23),we have that

Thus,we complete the proof of the lemma.

In view of this Lemma 4.1,we can reformulate Proposition 4.1 as follows

Theorem 4.1(Frequency-localized Morawetz estimate I). Let d≥4,0＜η＜1,and u:I×Rdbe the critical spherically symmetric maximal-lifespan solution to(1.1)which satisfies(1.4),(1.23).Then there exits δ＞0 with the following property: given any N ＞0 such that

we have

By scaling invariance of the Eq.(1.1),we may choose N=1.By a limiting argument,we may then take I to be compact. Indeed, observe that by Corollary 3.1, the left-hand side of(4.4)varies continuously on I and goes to zero when I shrinks to a point. Thus,by standard continuity argument, it suffices to show the following bootstrap version of Proposition 4.1.

Proposition 4.2 (Frequency-localized Morawetz estimate II). Let d ≥4, 0 ＜η ＜1, and u:I×Rdbe the critical symmetric solution to(1.1)which satisfies(1.4),(1.23). Then there exits δ＞0 with the following property:

where uhi:=u≥1and ulo:=u＜1,such that we also have bootstrap hypothesis: if

then we have

In order to prove Proposition 4.2,we will primarily establish the corresponding estimate for low and high frequency portion of the solution u.

Lemma 4.2 (Low and high frequency bound). Under the conditions of Proposition 4.2, we have the following estimates:

where ε0(d)＞0 is sufficiently small.

Proof. From the definition of S,Lemmas 3.1(4.5)and(4.6)we derive(4.7a)by choosing δ sufficiently small. (4.7b)comes from(4.7a)and(2.5).Indeed,by Lemma 3.1 and choosing ε0sufficiently small,we have

By(4.7a)and(2.5),we get(4.7b).

Now it suffices to prove(4.7c). We denote Phi:=P≥1.Obviously

By Strichartz estimate(3.2),(4.5)and splitting PhiF(u)into

we have

For the fourth term of(4.8),from Proposition 3.4 and(4.5)we have

For the third term of(4.8),by Lemma 3.1,(3.12a)and(4.5),we have

For the remained term of(4.8),by Lemma 3.1,(3.12a),(4.5)and(4.7a)we have

Putting all these together,we obtain

by Corollary 3.1,we know‖uhi‖S＜∞,after reorganizing the term,we finally derive that

Thus,we complete the proof.

With the above preparation,we are now ready to prove Proposition 4.2.First we need the following particular form of Morawetz inequality which can be found in[31].

Lemma 4.3(Morawetz inequality). Let J be an interval,let d≥3 and let φ,G:J×Rd→C solve the equation

Let ε＞0.If ε is sufficiently small depending on d,then we have

Proof of Proposition 4.2. Let Plo:=P＜1,we substitute φ with φ=ulo,then the corresponding G equals

Using Bernstein inequality and(4.5),we conclude that

Note that by Lemma 2.6

it suffices to estimate

where c is a given constant to be chosen later. By the H¨older inequality and (4.7b), we estimate

In dimension d≥4,by the H¨older inequality,(2.5)and(4.7b)we have

Thus it is reduced to show

We split G into

We can show(4.15)via

For(4.16),by(1.4),Sobolev embedding,Lemma 3.1,(4.7c),Bernstein,we estimate as

Hence,it is remained to prove

From(4.6)we have

and by radial Sobolev embedding(2.5)

By the H¨older inequality,we get

Combining the estimate for(4.16)and(4.17)we have

Corollary 4.1. Let d≥4,and u:I×Rd→C be the spherically symmetric maximal-lifespan solution to(1.1)which obeys(1.4),(1.23)then

In particular,for any N ＞0 being a dyadic integer,we have

Proof. (4.22)comes from(4.7a),(4.7c)and the scaling invariance of the equation. Now we use(4.22)to prove(4.23). Since(4.22)implies(4.23)for N is sufficiently large,it suffices to show that(4.23)also holds for N is small. We may assume N0such that N ≥N0

For any N ＜N0,we have

and

Thus we complete the proof.

5 The non-evacuation of energy

In this part, we will prove that the energy can not evacuate from high frequency to low frequency by showing that N(t)has a lower bound.

Proposition 5.1. Let d ≥4, and let u:I×Rd→C be the critical spherically symmetric maximal-lifespan solution to(1.1)which obeys(1.4),(1.23). Then

Assume for contradiction that we have a critical solution u:I×Rd→C obeying(1.4)and the hypothesis(1.23)but such that

we will obtain the following fact:

Lemma 5.1. Under the conditions of Proposition 5.1,we have

Proof. Let η ＞0 be a small number to be chosen later. By(4.22), there exists＞0 such that

By scaling invariance,we may assume=1,thus

We claim that:

Claim 5.1. For any given δ＞0 such that

then

Assuming the claim,by iterating the above procedure,we will conclude that

and

Let N ≥1,applying P≥Nto both sides of(1.1)we have

Hence,by weighted Strichartz estimate(3.2)we have

for any t0∈I.As inft∈IN(t)=0,we have

Thus

We split F(u)as

So that we have

By(3.4),(5.4b),(5.8),we have

The other term in(5.13)is estimated similarly.

For(5.14),by Lemma 3.1,(3.12a)(5.4a)and(5.8)we obtain

The other term of(5.14)is estimated similarly.

For the(5.15),by Lemma 3.1 and(3.12b)

Since by(5.4a)and(5.7)we have

and

Thus

Combining the separated parts contributed to‖u＞N‖S,we have

By choosing η sufficiently small,we complete the proof.

Proof of Proposition 5.1. Now we can illuminate that inft∈IN(t)=0 is incompatible with energy-conservation. In fact,by(5.2),for sufficiently large N,we have

and for each dyadic number N

Thus,by choosing M sufficiently large

as inft∈IN(t)=0, we may choose a time sequence {ti}∈I such that N(ti)→0, and by dominated convergence theorem we conclude that

By interpolation

where 0＜θ ＜1.Thus

By the energy conservation law of(1.1),(5.21)implies that u≡0,which is impossible.

6 Rule out the critical solution

Theorem 6.1. Let d≥4,and let u:I×Rd→C be the critical maximal-lifespan spherically symmetric solution to(1.1)which obeys(1.4),(1.23). Suppose that u is not identically zero,then I is bounded.

Proof. By (1.13) and the fact that N(t) has lower bound, we may choose N sufficiently small such that

then integrating with respect to the time variable over the interval I,we have

Theorem 6.1 means that u blows up in finite time, thus by Corollary 1.1, N(t) does not have upper bound in I,which is inconsistent with(1.23).

Appendix

In this part,we dedicate to proving Lemma 1.1. First we recall the definition of Strichartz norm and Strichartz estimate.

Definition 6.1(Admissible pair). Let d≥4,we call a pair of exponent(q,r)admissible if

For a time interval I,we define Strichartz norm S(I)as

We also define the dual of S(I)by N(I),we note that

Proposition 6.1(Strichartz estimate). Let u:I×Rd→C be a solution to

and let s≥0,then

for any t0∈I.

In the proof of Lemma 1.1,we need the following result. One can carry over the proof of Lemma 3.4 in[35]verbatim.

Lemma 6.1(Persistence of regularity). Let I be a compact time interval,and u be a solution to(1.10)obeying

then we have

In what follows,we denote

Remark 6.1. The reason we choose the particular form of X(I)and Y(I)stems from the following fact: by dispersive estimate and Hardy-Littlewood-Sobolev inequality we can obtain relatively neat nonlinear estimate

Next we will present some nonlinear estimates.

Lemma 6.3. Let d≥4,then with spacetime norms over I×Rd,we have

Proof. (6.11a) comes directly from the definition of X(I) and Y(I). (6.11b) from Lemma 6.2.

In order to prove Lemma 1.1, we primarily establish the short-time perturbation result.

Lemma 6.4 (Short-time perturbation). Let d ≥4, I be a compact interval, ˜u:I×Rd→C be solution to the equation

Suppose

then there exits u:I×Rd→C solving

satisfying

where c＞0 is a given constant.

By(6.12)and triangle inequality we have

Then using Strichartz estimate(6.9)and(6.11a),we have

We let w=u-˜u,thus w satisfies

By Strichartz estimate(6.9)we have

Thus by choosing δ sufficiently small,we have

By Strichartz estimate and(6.12)and(6.11b),we have

By (6.12) and the persistence of regularity results, we have ‖|▽|sc˜u‖S(I)≤C(δ)E. For(6.13b),by(6.13a)and Strichartz estimate we have

Now(6.13c)can be deduced from Lemma 6.2 and(6.13a).

then we can use the short-time perturbation results and bootstrap argument to obtain Lemma 1.1. □

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under grant No. 11671047 and No. 11726005. Yang Jianwei-Urbain is supported by the LabEx MME-DII.