Global Well-Posedness and Scattering for the Defocusing <span class="mathjax-formula">${H}^{\frac{1}{2}}$</span>-Subcritical Hartree Equation in <span class="mathjax-formula">${R}^{d}$</span>

www.elsevier.com/locate/anihpc

Global well-posedness and scattering for the defocusing H

-subcritical Hartree equation in R ^d

Changxing Miao

, Guixiang Xu

, Lifeng Zhao

aInstitute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China bDepartment of Mathematics, University of Science and Technology of China, Hefei 230026, China

Received 27 May 2008; received in revised form 8 January 2009; accepted 14 January 2009 Available online 7 February 2009

Abstract

We prove the global well-posedness and scattering for the defocusingH¹²-subcritical (that is, 2< γ <3) Hartree equation with low regularity data inR^d,d3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space H^s(R^d)withs >4(γ−2)/(3γ−4), which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for anys >max(1/2,4(γ−2)/(3γ−4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solutionI u, instead of an interaction Morawetz estimate for the solutionu, and that we make careful analysis of the monotonicity property of the multiplierm(ξ )· ξ^p. As a byproduct of our proof, we obtain that theH^snorm of the solution obeys the uniform-in-time bounds.

©2009 Elsevier Masson SAS. All rights reserved.

MSC:35Q40; 35Q55; 47J35

Keywords:Almost interaction Morawetz estimate; Well-posedness; Hartree equation; I-method; Uniform bound

1. Introduction

In this paper, we study the global well-posedness of the following initial value problem (IVP) for the defocusing H¹²-subcritical (that is, 2< γ <3) Hartree equation.

iut+u=(|x|^−γ∗ |u|²)u, d3,

u(0)=u0(x)∈H^s(R^d), (1.1)

whereH^s denotes the usual inhomogeneous Sobolev space of orders. It is a classical model introduced in [26].

We adopt the following standard notion of local well-posedness, that is, we say that the IVP (1.1) is locally well- posed inH^s if for anyu₀∈H^s, there exists a positive timeT =T (u₀s)depending only on the norm of the initial data, such that a solution to the IVP exists on the time interval[0, T], is unique in a certain Banach space of functional

* Corresponding author.

E-mail addresses:miao_changxing@iapcm.ac.cn (C. Miao), xu_guixiang@iapcm.ac.cn (G. Xu), zhaolifengustc@yahoo.cn (L. Zhao).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.01.003

Fig. 1. The curve “ABC” is described by “s=^4(γ−2)_3γ−4 ”.

X⊂C([0, T], H^s), and the solution map fromH_x^stoC([0, T], H^s)depends continuously. IfT can be taken arbitrarily large, we say that the IVP (1.1) is globally well-posed.

Local well-posedness for the IVP (1.1) inH^sfor anys > ^γ₂−1 was established in [18]. A local solution also exists for H^γ2−1initial data, but the time of existence depends not only on theH^γ2−1norm ofu0, but also on the profile ofu0. For more details on local well-posedness see [18].

L²solutions of (1.1) enjoy mass conservation u(t,·)

L²(R^d)=u₀(·)

L²(R^d).

Moreover,H¹solutions enjoy energy conservation E(u)(t )=1

2∇u(t )²

L²(R^d)+1 4R^d×R^d

1

|x−y|^γu(t, x)²u(t, y)²dx dy=E(u)(0),

which together with mass conservation and the local theory immediately yields global well-posedness for (1.1) with initial data inH¹. A large amount of work has been devoted to global well-posedness and scattering for the Hartree equation, see [7–11,13,15,17–23].

Existence of global solutions inR³to (1.1) corresponding to initial data below the energy threshold was recently obtained in [5] by using the method of “almost conservation laws” or “I-method” (for a detailed description of this method, see [25] or Section 3 below) and the interaction Morawetz estimate for the solution u, where global well- posedness was obtained inH^s(R³)withs >max(1/2,4(γ −2)/(3γ−4))(see Fig. 1). Since authors in [5] used the interaction Morawetz estimate, which involvesH˙^1/2norm of the solution, the restriction conditions¹₂is prerequi- site. In order to resolve IVP (1.1) inH^s,s <¹₂by still using the interaction Morawetz estimate, we need return to the interaction Morawetz estimate for the smoothed out versionI uof the solution, which is used simultaneously in [2]

and [6].

In this paper, we consider the cased3 and we prove the following result:

Theorem 1.1. Let 2< γ <3d, the initial value problem (1.1)is globally well-posedness in H^s(R^d) for any s > ^4(γ_3γ⁻₋²⁾₄ . Moreover the solution satisfies

sup

t∈[0,∞)

u(t )

H^s(R^d)C u0H^s

,

and there is scattering for these solutions, that is, the wave operators exist and there is asymptotic completeness on all ofH^s(R^d).

Remark 1.1.As for the case 3γ <4d, local well-posedness for the IVP (1.1) inH^s holds for anys >^γ₂−1.

Note that in this case, we have γ

2 −11 2,

which satisfies the need of the regularity of the interaction Morawetz estimate. Hence for the case 3γ <4d, we only need to combine “I-method” with the interaction Morawetz estimate for the solution, instead of the interaction Morawetz estimate for the smoothed out versionI uof the solution, to obtain the low regularity global solution of the IVP (1.1), just as in [3] .

Our method follows closely the recent developments in [2,4] and [6], where the main two ingredients are the

“I-method” and an almost Morawetz type estimate following the work of Lin and Strauss [16]. In order to obtain the low regularity global solution of the IVP (1.1), we combine I-method with an interaction Morawetz estimate for the smoothed out versionI uof the solution. By comparison with the interaction Morawetz estimate for the solution in [5], such a Morawetz estimate for an almost solution is the main novelty of this paper, which helps us to lower the need on the regularity of the initial data. In addition, we do not use the monotonicity property of the multiplierm(ξ )· ξ^p in the proof of the almost conservation law.

Last, we organize this paper as following: In Section 2, we introduce some notation and state some important propositions that we will used throughout this paper. In Section 3, we review the I-method, prove the local well- posedness theory forI uand obtain an upper bound on the increment of the modified energy. In Section 4, we prove the “almost interaction Morawetz estimate” for the smoothed out versionI uof the solution. Finally in Section 5, we give the details of the proof of the global well-posedness stated in Theorem 1.1.

2. Notation and preliminaries

2.1. Notation

In what follows, we useAB to denote an estimate of the formACB for some constantC. IfAB and BA, we say thatA≈B. We writeABto denote an estimate of the formAcBfor some small constantc >0.

In additiona :=1+ |a|anda± :=a±with 0< 1. The reader also has to be alert that we sometimes do not explicitly write down constants that depend on theL²norm of the solution. This is justified by the conservation of the L²norm.

2.2. Definition of spaces

We useL^r_x(R^d)to denote the Lebesgue space of functionsf:R^d→Cwhose norm fL^r_x :=

R^d

f (x)^rdx

1 r

is finite, with the usual modification in the caser= ∞. We also use the space–time Lebesgue spacesL^q_tL^r_xwhich are equipped with the norm

u_L^q

tL^r_x:=

J

u(t, x)^q

L^r_xdt

1 q

for any space–time slabJ×R, with the usual modification when eitherqorrare infinity. Whenq=r, we abbreviate L^q_tL^r_xbyL^q_t,x.

As usual, we define the Fourier transform off (x)∈L¹_xby f (ξ )ˆ =(2π )^−d2

R^d

e^−ixξf (x) dx.

We define the fractional differentiation operator|∇x|^αfor any realαby

|∇|^αu(ξ ):= |ξ|^αu(ξ ),ˆ and analogously

∇^αu(ξ ):= ξ^αu(ξ ).ˆ

The (in)homogeneous Sobolev spaceH˙^s(R^d) (H^s(R^d))is given via u_H˙s:=|∇|^su

L²(R^d), uH^s :=∇^su

L²(R^d), and more general (in)homogeneous Sobolev spaces are given via

u_H˙^s,p:=|∇|^su

L^p(R^d), uH^s,p:=∇^su

L^p(R^d).

LetS(t )denote the solution operator to the linear Schrödinger equation iu_t+u=0, x∈R^d.

We denote byX^s,b(R×R^d)the completion ofS(R×R^d)with respect to the following norm uX^s,b=S(−t )u

H_x^sH_t^b=τ+ |ξ|²b

ξ^su(τ, ξ )˜

L²_τL²_ξ(R×R^d), whereu˜is the space–time Fourier transform

˜

u(τ, ξ )=(2π )^−d+21

R×R^d

e^{−i(x·ξ+t τ )}u(t, x) dt dx.

Furthermore for a given time intervalJ, we define uX^s,b(J )=inf

vX^s,b; v=uonJ . 2.3. Some known estimates

Now we recall a few known estimates that we shall need. First we state the following Strichartz estimate [1,14].

Letd3, we recall that a pair of exponents(q, r)is called admissible if 2

q =d 1

2−1

r δ(r), 2q, r∞.

Proposition 2.1.Letd3,(q, r)and(q,˜ r)˜ be any two admissible pairs. Suppose thatuis a solution to iut+u=F (t, x), t∈J, x∈R^d,

u(0)=u₀(x).

Then we have the estimate u_L^q

tL^r_x(J×R^d)u₀L²(R^d)+ F_Lq˜ t Lr˜

x(J×R^d), where the prime exponents denote Hölder dual exponents.

From the above Strichartz estimate and Lemma 2.1 in [12], we have u_L^q_tL^r

x u

X^0,12+

for any admissible(q, r). Then by Sobolev embedding theorem, we have

Proposition 2.2.Letd3. Forr <∞, 0²_qmin(δ(r),1), we have u_L^q_tL^r_xu

X^δ(r)−

2q ,1 2+. While for2q∞, r= ∞, we have

u_L^q_tL∞x u

X

d 2−2

q+,1 2+.

We will also need the Littlewood–Paley projection operators. Specifically, let ϕ(ξ )be a smooth bump function adapted to the ball |ξ|2 which equals 1 on the ball |ξ|1. For each dyadic number N ∈2^Z, we define the Littlewood–Paley operators

P_Nf (ξ ):=ϕ ξ

N f (ξ ),ˆ P_>Nf (ξ ):=

1−ϕ

ξ N

f (ξ ),ˆ

P_Nf (ξ ):=

ϕ

ξ N −ϕ

2ξ N

f (ξ ).ˆ

Similarly we can defineP_<N,P_N, andP_M<·N=P_N−P_M, wheneverMandN are dyadic numbers. We will frequently writef_NforP_Nf and similarly for the other operators.

The Littlewood–Paley operators commute with derivative operators, the free propagator, and the conjugation op- eration. They are self-adjoint and bounded on everyL^p_x andH˙_x^s space for 1p∞ands0. They also obey the following Sobolev and Bernstein estimates

P_NfL^pN^−s|∇|^sP_Nf

L^p, |∇|^sP_Nf

L^q N^s+np−nqP_NfL^p, |∇|^±sP_Nf

L^q N^±s+np−nqP_NfL^p, whenevers0 and 1pq∞.

3. The I-method and the modified local well-posedness

3.1. The I-operator and the hierarchy of energies

Let us define the operator I. For s <1 and a parameterN 1, letm(ξ ) be the following smooth monotone multiplier:

m(ξ ):=

1, if|ξ|< N, (_|^N_ξ|)^1−s, if|ξ|>2N.

We define the multiplier operatorI:H^s→H¹by I u(ξ )=m(ξ )u(ξ ).ˆ

The operatorI is smoothing of order 1−sand we have from Bernstein’s estimates that uH^s0 I u_H^s0+1−s N^1−suH^s0,

u_X^s0,b0 I u_X^s0+1−s,b0 N^1−su_X^s0,b0

for anys0, b0∈R.

We set

E(u) =E(I u), (3.1)

where

E(u)(t )=1

2∇u(t )²

L²+1 4

1

|x−y|^γu(t, x)²u(t, y)²dx dy.

We callE(u) the modified energy. Since we will focus on the analysis of the modified energy, we collect some facts concerning the calculus of multilinear forms used to define the modified energy.

Ifk2 is an even integer, we define a spatial multiplier of orderkto be the functionM_k(ξ₁, ξ₂, . . . , ξ_k)on Γk=

(ξ1, ξ2, . . . , ξk)∈ R^dk

: k j=1

ξj=0

,

which we endow with the standard measureδ(ξ₁+ξ₂+ · · · +ξ_k). IfM_k is a multiplier of orderk, 1j kis an index andl1 is an even integer, the elongationX^l_j(M_k)ofM_kis defined to be the multiplier of orderk+lgiven by

X^l_j(M_k)(ξ₁, ξ₂, . . . , ξ_k+l)=M_k(ξ₁, . . . , ξ_j−1, ξ_j+ · · · +ξ_j+l, ξ_j+l+1, . . . , ξ_k+l).

Also ifM_kis a multiplier of orderkandu₁, u₂, . . . , u_kare functions onR^d, we define thek-linear functional Λ_k(M_k;u₁, u₂, . . . , u_k)=Re

Γk

M_k(ξ₁, ξ₂, . . . , ξ_k) k j=1

ˆ u_j(ξ_j)

and we adopt the notationΛ_k(M_k;u)=Λ_k(M_k;u,u, . . . , u,¯ u). We observe that the quantity¯ Λ_k(M_k;u)is invariant (1) if one permutes the even argumentsξ₂, ξ₄, . . . , ξ_k ofM_k;

(2) if one permutes the odd argumentsξ₁, ξ₃, . . . , ξ_k−1ofM_k; (3) if one makes the change of

Mk(ξ1, ξ2, . . . , ξk−1, ξk)→Mk(−ξ2,−ξ1, . . . ,−ξk,−ξk−1).

Ifuis a solution of (1.1), the following differentiation law holds for the multiplier formsΛk(Mk;u)

∂_tΛ_k(M_k;u)=Λ_k

iM_k k j=1

(−1)^j|ξ_j|²;u

+Λ_k+2

i k j=1

(−1)^j|ξ_j+1,j+2|^{−(d−γ )}X²_j(M_k);u

(3.2) where we used the notational conventionξa,b=ξa+ξb,ξa,b,c=ξa+ξb+ξc, etc. Indeed, note that

i∂_tu_j+u_j =

|x|^−γ∗ |u_j|²

u_j, for 0jk, we take Fourier transformation in spatial variable and deduce

∂tuˆj(ξj)= −i|ξj|²uˆj(ξj)−i

ξ_j=˜ξ_j,j+1,j+2

|˜ξj+1,j+2|^{−(d−γ )}uˆj(˜ξj)uˆ¯j(˜ξj+1)uˆj(˜ξj+2) dξ˜j+1dξ˜j+2,

∂_tuˆ¯_j(ξ_j)= +i|ξ_j|²uˆ¯_j(ξ_j)+i

ξj=˜ξ_j,j+1,j+2

|˜ξ_j+1,j+2|^{−(d−γ )}uˆ¯_j(˜ξ_j)uˆ_j(˜ξ_j+1)uˆ¯_j(˜ξ_j+2) dξ˜_j+1dξ˜_j+2.

From the above identities, we can obtain (3.2).

Using the above notation, the modified energy (3.1) can be written as follows:

E(u) =Λ₂

−1

2ξ₁m₁·ξ₂m₂;u +Λ₄ 1

4|ξ_2,3|^{−(d−γ )}m₁m₂m₃m₄;u where we abbreviatem(ξ_j)asm_j.

Together with the differentiation rules (3.2) and the symmetry properties of k-linear functionalΛ_k(M_k;u), we obtain

∂_tΛ₂

−1

2ξ₁m₁·ξ₂m₂;u

=Λ₂

−i

2ξ₁m₁·ξ₂m₂ 2 j=1

(−1)^j|ξ_j|²;u

+Λ4

−i 2

2 j=1

(−1)^j|ξj+1,j+2|^{−(d−γ )}X_j²(ξ1m1·ξ2m2);u

=Λ₄

i|ξ_2,3|^{−(d−γ )}m²₁|ξ₁|²;u , where we used the fact that₂

j=1(−1)^j|ξ_j|²=0 on the hyperplaneΓ₂. At the same time, we have

∂_tΛ₄ 1

4|ξ_2,3|^{−(d−γ )}m₁m₂m₃m₄;u

=Λ₄

i

4|ξ_2,3|^{−(d−γ )}m₁m₂m₃m₄ 4 j=1

(−1)^j|ξ_j|²;u

+Λ₆

i 4

4 j=1

(−1)^j|ξ_j+1,j+2|^{−(d−γ )}X_j²

|ξ_2,3|^{−(d−γ )}m₁m₂m₃m₄

;u

= −Λ₄

i|ξ_2,3|^{−(d−γ )}|ξ₁|²m₁m₂m₃m₄;u

−Λ₆

i|ξ_2,3|^{−(d−γ )}|ξ_4,5|^{−(d−γ )}m_1,2,3m₄m₅m₆;u

= −Λ4

i|ξ2,3|^{−(d−γ )}|ξ1|²m1m2m3m4;u +Λ6

i|ξ2,3|^{−(d−γ )}|ξ4,5|^{−(d−γ )}m1,2,3(m1,2,3−m4m5m6);u . The fundamental theorem of calculus together with these estimates implies the following proposition, which will be used to prove thatEis almost conserved.

Proposition 3.1.Letube anH¹solution to(1.1). Then for anyT ∈Randδ >0, we have E(u)(T +δ)−E(u)(T ) =

T+δ

T

Λ4(M4;u) dt+

T+δ

T

Λ6(M6;u) dt with

M₄=i|ξ_2,3|^{−(d−γ )}|ξ₁|²m₁(m₁−m₂m₃m₄);

M₆=i|ξ_2,3|^{−(d−γ )}|ξ_4,5|^{−(d−γ )}m_1,2,3(m_1,2,3−m₄m₅m₆).

Furthermore if|ξ_j| N for allj, then the multipliersM₄andM₆vanish onΓ₄andΓ₆, respectively.

3.2. Modified local well-posedness

In this subsection, we shall prove a local well-posedness result for the modified solutionI u and some a priori estimates for it.

LetJ= [t₀, t₁]be an interval of time. We denote byZ_I(J )the following space:

Z_I(J )=S_I(J )∩X^1,

1 2+ I (J ) where

S_I(J )=

u; sup

(q,r)admissible

∇I u

L^q_tL^r_x(J×R^d)<∞ , X_I^1,b(J )=

u; I u

X^1,12+(J×R^d)<∞ .

Proposition 3.2.Let2< γ <3d,s > ^γ₂−1, and consider the IVP iI u_t+I u=I

|x|^−γ∗ |u|²u

, x∈R^d, t∈R, I u(t₀, x)=I u₀(x)∈H¹

R^d

. (3.3)

Then for anyu₀∈H^s, there exists a time intervalJ= [t₀, t₀+δ],δ=δ(I u₀H¹)and there exists a uniqueu∈Z_I(J ) solution to(3.3). Moreover it is continuity with respect to the initial data.

Proof. The proof of this proposition proceeds by the usual fixed point method on the spaceZ_I(J ). Since the estimates are very similar to the ones we provide in the proof of Proposition 3.3 below, in particular (3.9) and (3.10) , we omit the details. 2

Proposition 3.3.Let2< γ <3d and s >^γ₂ −1. Ifuis a solution to the IVP(3.3)on the interval J= [t₀, t₁], which satisfies the following a priori bound

I u⁴

L⁴_tH˙⁻

d−3 4 ,4

x (J×R^d)

< μ,

whereμis a small universal constant, then uZI(J )I u₀H¹.

Proof. We start by obtaining a control of the Strichartz norms. Applying∇to (3.3) and using the Strichartz estimate in Proposition 2.1. For any pair of admissible exponents(q, r), we obtain

∇I u

L^q_tL^r_xI u₀H¹+∇I

|x|^−γ ∗ |u|²u

L

3 t2L

6d 3d+4 x

. (3.4)

Now we notice that the multiplier∇I has symbol which is increasing as a function of|ξ|for anys^γ₂−1. Using this fact one can modify the proof of the Leibnitz rule for fractional derivatives and prove its validity for∇I. See also Principle A.5 in the appendix of [25]. This remark combined with (3.4) implies that

∇I u

L^q_tL^r_xI u₀H¹+∇I

|x|^−γ ∗ |u|²u

L

3 t2L

6d 3d+4 x

I u0H¹+|x|^−γ∗ ∇I

|u|²

L²_tL

2dγ x

u

L⁶_tL

3d+4−3γ6d

x

+|x|^−γ∗ |u|²

L³_tL

3d x4

∇I u

L³_tL

6d 3d−4 x

I u0H¹+∇I u

L³_tL

3d−46d x

u²

L⁶_tL

6d 3d+4−3γ x

(3.5) where we used Hölder’s inequality and Hardy–Littlewood–Sobolev’s inequality.

In order to obtain an upper bound on u

L⁶_tL

6d 3d+4−3γ x

, we perform a Littlewood–Paley decomposition along the following lines. We note that a similar approach was used in [3]. We write

u=u_N0+^∞

j=1

u_Nj, (3.6)

whereu_N0 has spatial frequency support forξN, whileu_Nj is such that its spatial Fourier support transform is supported forξ ≈N_j=2^hj withh_jlogN andj =1,2, . . . .By the triangle inequality and Hölder’s inequality, we have

u

L⁶_tL

6d 3d+4−3γ x

uN₀

L⁶_tL

6d 3d+4−3γ x

+ ∞ j=1

uN_j

L⁶_tL

6d 3d+4−3γ x

u_N0

L⁶_tL

3d+4−3γ6d x

+^∞

j=1

u_Nj^2−γ2

L⁶_tL

3d−6d2 x

u_Nj^γ2−1

L⁶_tL

3d−6d8 x

. (3.7)

On the other hand, by using the definition of the operatorI, the definition of theuN_j’s and the Marcinkiewicz multi- plier theorem, we observe that for some 0< θi<1,i=1, . . . ,4,4

j=1θi=1 uN₀

L⁶_tL

3d+6d4−3γ x

uN₀^θ1

L⁴_tH˙⁻

d−3 4 ,4 x

uN₀^θ2

L⁶_tL

6d 3d−8 x

uN₀^θ3

L⁶_tL

6d 3d−2 x

uN₀^θ_L⁴∞ t L²_x

I u_N0^θ1

L⁴_tH˙⁻

d−34 ,4 x

u_N0¹_Z⁻_{I(J )}^θ1 ,

∇I u_Nj

L⁶_tL

6d 3d−2 x

≈N_j N

N_j

1−s

u_Nj

L⁶_tL

6d 3d−2 x

, j=1,2, . . . , I u_Nj

L⁶_tL

3d−86d x

≈ N

Nj 1−s

u_Nj

L⁶_tL

3d−86d x

, j=1,2, . . . .

Now we use these estimates to obtain the following upper bound on (3.7) u

L⁶_tL

6d 3d+4−3γ x

I uN₀^θ1

L⁴_tH˙⁻

d−3 4 ,4 x

uN₀¹_Z⁻_{I(J )}^θ1

+^∞

j=1

1 Nj

N_j N

1−s

u_NjZI(J )

2−^γ₂ N_j

N

1−s

u_NjZI(J )

γ 2−1

μ^θ41u¹_Z⁻_{I(J )}^θ1 +N^−(2−γ2)uZI(J ), (3.8) which together with (3.5) implies that

∇I u

L^q_tL^r_x I u₀H¹+μ^θ21u³_Z⁻_{I(J )}^2θ1+N^{−(4−γ )}u³_{ZI(J )}. (3.9) Now we shall obtain a control of theX^s,b norm. We use Duhamel’s formula and the theory ofX^s,bspaces [12,25]

to obtain I u

X^1,12+I u₀H¹+∇I

|x|^−γ ∗ |u|²u

X^0,−12+

I u₀H¹+∇I

|x|^−γ ∗ |u|²u

L

3 2+

t L

6d 3d+4+ x

I u0H¹+∇I u

L³_tL

3d−6d4 x

u

L⁶_tL

3d+6d4−3γ x

u

L⁶⁺_t L

3d+6d4−3γ+ x

. (3.10)

An upper bound on u

L⁶_tL

6d 3d+4−3γ x

is given by (3.8). In order to obtain an upper bound on u

L⁶_t⁺L

6d 3d+4−3γ+ x

, we proceed as follows. First we perform a dyadic decomposition and writeuas in (3.6). The triangle inequality applied on (3.6) gives for any 0< δ <^γ₂−1

u

L⁶_t⁺L

6d 3d+4−3γ+ x

uN₀

L⁶_t⁺L

6d 3d+4−3γ+ x

+ ∞ j=1

uN_j

L⁶_t⁺L

6d 3d+4−3γ+ x

= I u_N0

L⁶_t⁺L

3d+4−3γ6d + x

+^∞

j=1

N_j^δ−sN^s−1∇^1−δI u_Nj

L⁶_t⁺L

6d 3d+4−3γ+ x

I u

L⁶_t⁺L

6d 3d+4−3γ+ x

+∇^1−δI u

L⁶_t⁺L

6d 3d+4−3γ+ x

I u

X^1,12+, (3.11)

where we used Proposition 2.2. By applying the inequalities (3.8) and (3.11) to bound the right-hand side of (3.10), we obtain

I u

X^1,12+I u0H¹+μ

θ1

4u³_Z⁻_{I(J )}^θ1 +N^−(2−γ2)u³_{ZI(J )}. (3.12) The desired bound follows from (3.9) and (3.12) by choosingN sufficiently large. 2

3.3. An upper bound on the increment ofE(u)

Decomposition remark. Our approach to prove a decay for the increment of the modified energy is based on obtaining certain multilinear estimates in appropriate functional spaces which areL²-based (more details, see [24]).

Hence, whenever we perform a Littlewood–Paley decomposition of a function we shall assume that the Fourier trans- forms of the Littlewood–Paley pieces are positive. Moreover, we will ignore the presence of conjugates. At the end we will always keep a decay factorC(N₁, N₂, . . .)in order to perform the summations.