A Smoothing Property for the <span class="mathjax-formula">${L}^{2}$</span>-Critical NLS Equations and an Application to Blowup Theory

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A smoothing property for the L ² -critical NLS equations and an application to blowup theory

Sahbi Keraani

, Ana Vargas

aIRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

bDepartamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Received 24 September 2007; accepted 6 March 2008 Available online 21 March 2008

Abstract

In this paper we prove a smoothing property for theL²-critical nonlinear Schrödinger equation and we use it to study the blowup dynamics for singular solutions below the energy level.

©2008 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article, on montre un effet régularisant pour l’équation de SchrödingerL²-critique et on utilise ce résulat pour étudier la dynamique d’explosion pour les solutions singulières ayant des données initiales peu régulières.

©2008 Elsevier Masson SAS. All rights reserved.

MSC:35Q55; 35B40; 35B05

Keywords:Time dependent Schrödinger equation; Blowup; Bourgain spaces

1. Introduction

Consider theL²-critical nonlinear Schrödinger equation i∂_tu+u+κ|u|^4du=0; u|t=0=u₀∈H^s

R^d

. (1)

Here,=_d

j=1∂_x²

j is the Laplace operator onR^dandu:Rt×R^dx→Cis a complex-valued function. The parameter κ equal to 1 (resp.−1) corresponds to the focusing (resp. defocusing) NLS. It is well known (see [7] for instance) that the Cauchy problem (1) is locally well-posed inH^sfor everys0.

* Corresponding author.

E-mail addresses:[email protected] (S. Keraani), [email protected] (A. Vargas).

1 The research of the second author was supported in part by the MEC (Spain) projects MTM2004-00678 and MTM2007-60952 and the UAM- CM project CCG07-UAM/ESP-1664.

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.03.001

The unique solution satisfies the following conservation law u(t, x)²dx= u₀(x)²dx.

Also, ifs1, the energy E(t )=1

2 ∇u(t, x)²dx− κd

4+2d u(t, x)^4d+2dx

is conserved astvaries. Here∇ =(∂_x1, ∂_x2, . . . , ∂_xd)denotes the spatial gradient inR^d.

Fors >0 Eq. (1) is subcritical: the lifespan of the solution depends only on theH^snorm of the data. Define[0, T^∗) to be the forward maximal lifespan of the solution of (1). We have the following blowup alternative: eitherT^∗= +∞

orT^∗<+∞and

tlim↑T∗D^su(t )

L²= +∞.

The spaceL²and the equation have the same scaling. More precisely, ifusolves (1), then, for everyλ >0,so it does u_λ(x, t )=λ^d/2u(λ²t, λx),with datau_λ(0, x)=λu₀(λx).Butu_λ(0,·)L²(R²)= u₀L²(R²) and from this point of view (1) isL²-critical. In this case the situation is more subtle and the time of existence depends on shape of the data.

More precisely, the blow up criterion becomes T^∗<+∞ ⇒ lim

T↑T^∗

[0,T]×R^d

u(t, x)^4d+2dx dt= +∞.

The blowup or “wave collapse” corresponds to self-trapping of beams in laser propagation. A lot of theoretical and numerical works are dedicated to this subject when the initial data belongs to H¹(see [7,21–26,31,38] and the references therein). This theory, which is based on energy arguments, is closely connected to the notion of ground state: the unique positive radial solution of the elliptic problem

Q−Q+ |Q|^4dQ=0.

For a revisitedH¹blowup theory the reader is referred to [19].

For the case of initial data belonging to H^s, with 0s <1, the classical energy arguments do not work. Nev- ertheless, the general consensus is that, in this case too, the same phenomena happen (concentration, universality of blowup profile. . . ).

The first result in this direction is due to J. Bourgain [5] ford=2 ands=0. In fact, by using a refined version of the Strichartz inequality proved in [29] and harmonic analysis techniques, this author proved that if a solution of (1), with initial data in L², blows up at finite timeT^∗>0, then there is concentration of some part of its total mass in small balls of size(T^∗−t )^1/2.Using this work by Bourgain, F. Merle and L. Vega [27] proved, among other things, an asymptotic compactness property inL²(R²)up to the invariance of the equation.

In [20] the first author defines the minimal massδ0as theL²norm necessary to ignite a wave collapse and stresses its role in the blow up mechanism foroneandtwo space dimensions (see also [6]). These results were generalized to higher dimensions by P. Bégout and A. Vargas [1].

Fors close² to 1 and in dimension two, Colliander et al. [9] have proved that the blowup solutions, which are radially symmetric, concentrate at least the mass of the ground state. This result was extended by Tzirakis [35] to dimension 1 and by Visan and Zhang [37] to general dimension. Their proof is based on the so called I-method introduced in [8]. This type of concentration result was already known fors=1.In [18] the first author and T. Hmidi have proved a refined compactness lemma adapted to the blowup theory of NLS and used it to improve the results of [9]: they have removed the assumption of radial symmetry of the initial data and proved thatQis a profile for the singular solutions with minimal mass.

The energy method also proves that in the defocusing case (κ = −1), for data inH¹,the solution is global. For data inH^s(R²), s >2/3,Bourgain [5] proved that the solution of (1)κ=−1is global. This was improved by Fang and

2 More precisely, fors >¹⁺

√11 5 .

Grillakis [14]. Related results for other dimensions have recently appeared in the work of Da Silva, Pavlovic, Staffilani and Tzirakis [12,13]. A global well posedness result fors=0,andd3,with the additional assumption of radial symmetry, has been proved by Tao, Visan and Zhang [33] with different methods (namely, using the results of [20]

and [1], see also [34]).

In this paper we prove the following theorem.

Theorem 1.1.Ford=1,sets₁=3/4.For2d4,sets_d=_d+^d₂.Finally, ford5,sets_d=^d_d(d^2+2d₊₂₎⁻⁸.The solution of (1)with initial datau₀∈H^s(R^d),s > s_d, can be written

u(t )=e^it u₀+w(t ), t∈ [0, T^∗[,

withw∈C([0, T^∗[, H¹(R^d)). Furthermore, ifT^∗is finite then there exists a constantC >0such that ∇w(t )

L² C

√T^∗−t, (2)

for everyt∈ [0, T^∗[.

Remark 1.2.This type of result (i.e. the extra-regularity of the Duhamel part of the solution) was firstly discovered, and used, by Bourgain [5] in the context of the defocusingL²-critical Schrödinger equation with initial data inH^s(R²), s >2/3.

Remark 1.3.Theorem 1.1 says that the blowup phenomenon has anH¹mechanism. In fact, any singular solution can be split en two parts: anH^s part which is global (since it is linear) and anH¹part, which blows up. This explains, in particular, why despite the fact thatu(t )∈H^s, withs <1, all the blowup profiles discovered in [18] belong toH¹ (see Remark 1.7 in [18]).

Combined with the well-known smoothing effects for the linear Schrödinger equation ([10,30] and [36]) Theo- rem 1.1 yields

Corollary 1.4.Under the assumptions of Theorem1.1,u(t )∈H_loc¹ (R^d)for almost everyt∈ [0, T^∗[.

For the rest of this section we takeκ=1 (the focusing case) and assume that, in the context of Theorem 1.1, the maximal time of existenceT^∗is finite. The energyE(w(t ))ofw(t )is, of course, not conserved. However, we believe that a weaker result is true: at blowup time, the potential part ofE(w(t ))is asymptotically equal to the kinetic one.

This result can be easily proved when the initial datau₀ lies inH¹ and we may conjecture that it remains true for initial data inH^s whens > s_d. More precisely we have

Conjecture.Under the assumptions and notations of Theorem1.1, assume thatκ=1andT^∗is finite. Then we have E(w(t ))

∇w(t )²_L2

−→0, ast→T^∗. (3)

Remark 1.5.If this conjecture is true, it would imply that global existence occurs for everyu₀∈H^s, s > s_d,such thatu₀L²<QL².(See Appendix A for the proof of this claim and other discussions.)

The rest of this paper is organized as follows. In Section 2, we recall some function spaces and prove some re- sults needed for the proof of our theorem which is given in Section 3. Some proof of auxiliary results are given in Appendix A.

2. Preliminaries

In this preliminary section, we are going to recall some definitions and prove some basic properties of the objects that will be used in our analysis.

In what follows positive constants will be denoted byCand will change from line to line. If necessary, byC_,..., we denote positive constants depending only on the quantities appearing in the indices.

Let us first recall the dyadic decomposition of the full spaceR^d. Letχ∈C₀^∞(R^d)andϕ∈C₀^∞(R^d\{0})two radially symmetric functions such that

• χ (ξ )+

j0ϕ(2^−jξ )=1, ¹₃χ²(ξ )+

j0ϕ²(2^−jξ )1,

• suppϕ(2^−j·)∩suppϕ(2^−k·)=∅,if|j−k|2,

• j1⇒suppχ∩suppϕ(2^−j)=∅.

For everyv∈Sone defines the inhomogeneous Littlewood–Paley operators P₋₁v=χ (D)v; ∀j∈N, P_jv=ϕ

2^−jD

v and S_j=

−1kj−1

P_k.

From the paradifferential calculus introduced by J.-M. Bony [3] the productuvcan be formally divided into three parts as follows:

uv=T_uv+T_vu+R(u, v), where

T_uv=

j

S_j−1uP_jv, and R(u, v)=

j

P_juP_jv,

withP_j= 1 i=−1

P_j+i=P_j−1+P_j+P_j+1.

T_uvis called paraproduct ofvbyuandR(u, v)the remainder term.

Our proof, which relies on ideas introduced in [5], uses the notion of Bourgain spacesX^s,b.

Definition 2.1(Bourgain spaces).LetI be an interval ofR. For every pair of real numbers(s, b), the spaceX^s,b[I]is the space of functionsufromI toCsuch that

uX^s,b[I]=inf

φX^s,b, φ_|I=u

<∞, where

φX^s,b=

1+ |ξ|^2s

1+λ+ |ξ|^22bφ(ξ, λ)˜ ²dξ dλ ¹

2

.

Here,φ˜denotes the Fourier transform ofφin the variables(x, t ).

We collect in the next proposition some properties of these spaces that will be needed in our proof.

Proposition 2.2.LetI be an interval ofRcontaining0.

(A) For everys∈Randb >1/2,X^s,b[I]→C⁰(I, H^s(R^d)).

(B) For everys, b∈R,(X^s,b)^∗=X^−s,−b.

(C) For any Schwartz time cuttofη∈S(R)andu₀∈H^s(R^d), we have η(t )e^it u0

X^s,b(R×R^d)Cη,bu0H^s(R^d).

(D) For everys∈R,−¹₂< b0,0bb+1,|I|1andf∈X^s,b[I]we have FX^s,b[I]C|I|^1−b+bf_X^s,b_[I],

whereF (t, x)=_t

0e^i(t−s)f (s, x) ds.

(E) For everyb >¹₂ and everya >³₄ifd=1and everya >_d^d₊₂ ifd2, there existsC=C(a, b)such that u∇v

L

d+2 d x,t

CuX^a,bvX^a,b, ∀u, v∈X^a,b.

(F) For every2p^2(d_d⁺²⁾ andγ >^d+₂²(¹₂−_p¹),there existsC=C(p, γ )such that u_L^p

x,t CuX^0,γ, ∀u∈X^0,γ.

(G) For every ^2(d_d⁺²⁾< p+∞, a > (d+2)(_2(d^d₊₂₎−_p¹)andb >¹₂,there existsC=C(p, a)such that u_L^p

x,t Cu_X^a,b, ∀u∈X^a,b.

For the proof of (A)–(D), see [5] and [17]. We will give proofs of (E), (F) and (G) in the next subsections.

2.1. Proof of Proposition 2.2(E)

The key estimate is the following improved Strichartz’s inequality:

Proposition 2.3.Forα=¹₄ ifd=1, forα=¹₂ ifd=2, and for everyα <_d+²₂ ifd3,there existsC=C(α), such that the following estimate

e^it f e^it g

L^d+d2(R^d+1)C M

N α

fL²gL²

holds for allL²functionsf andgwithsuppfˆ⊂ {ξ ∈R^d: |ξ|2M}andsuppgˆ⊂ {ξ ∈R^d: N|ξ|2N}for all 0< MN.

Remark 2.4.For the cased=2,this inequality was proved by Bourgain (see Lemma 111 in [5]).

Remark 2.5.The exponents appearing in this proposition are sharp.

Ford=1 takefˆ=1Mξ2M,gˆ=1_1ξ1+M1/2 and Σ=

(t, x)∈R²: |x+2t|M^−1/2

20 , |t|M⁻¹ 20

, withM1.

It is easy to see (sinceMsmall) that for allξ∈ [M,2M]and(t, x)∈Σ,we have|xξ+t|ξ|²|¹₂.Hence, there is a constantc >0,such that(e^{i(xξ+t|ξ|2)})c.Thus,

e^it f (x)=

2M M

e^{i(xξ+t|ξ|2)}dξ

∼M, ∀(x, t )∈Σ.

Similarly, for every(x, t )∈Σ e^it g(x)=

1+M^1/2 1

e^{i(xξ+t|ξ|2)}dξ =

M^1/2

0

e^{i(η(x+2t )+t|η|2)}dη

∼M^1/2. But|Σ| ∼M^−3/2, and so

e^it f e^it g

L³(R²)CM.

Sinceg2=M^1/2andf2=M^1/4we get the result.

The example in higher dimensionsd2,is quite different. Considerfˆ=1A,where, for someM1, A= {ξ= (ξ₁, ξ₂, . . . , ξ_N):M²ξ₁2M²,|ξ_k|M,k=2,3, . . . , d},andgˆ=1B,whereB= {ξ =(ξ₁, ξ₂, . . . , ξ_d): 1ξ₁

1+M²,|ξ_k|M,k=2,3, . . . , d}.Then, we see that, for|t|_20dM¹ 2,|x₁|_20dM¹ 2,|x₂|,|,|x₃|, . . . ,|x_d|_20dM¹ ,

|e^it f (x)|M^d+1 and|e^it g(x)|M^d+1. Thereforee^it f e^it g

L^d+d2 CM^d+1+d+22, whilef2= g2= M^d+21.This gives the desired result.

Proof of Proposition 2.3. By rescaling, it suffices to consider the caseN=1.IfM∼1,then, this proposition follows from Hölder and Strichartz and estimates. We will only deal with the caseM1.

•Cased2.Proposition 2.3, ford2,follows from the following estimates and an interpolation argument.

Theorem 2.6.(Cf. [32].) Assume thatf andg are functions belonging toL²(R^d),such that suppf ,ˆ suppgˆ⊂ {ξ:

|ξ|2},anddist(suppf ,ˆ suppg)ˆ ∼1. Then, for allp >^d_d⁺₊³₁, e^it f e^it g

L^p_x,t CfL²gL².

Proposition 2.7.Assume thatf andgare functions belonging toL²(R^d),such thatsuppfˆ⊂ {ξ ∈R^d: |ξ|2M} andsuppgˆ⊂ {ξ∈R^d: 1|ξ|2}for someM1/4.Then,

e^it f e^it g

L²_x,t CM^d−21fL²gL².

Proof. We follow the arguments introduced in [4] (see also [28] and [32]). Letψ∈S(R)such thatψ (0)=1 and ψ(τ )ˆ =1 if|τ|<1, ψ (τ )ˆ =0 if|τ|>2.

By the dominated convergence we have e^it f e^it g²

L²_x,t = lim

ε→0

ψ (εt )e^it f (x)e^it f (x)e^it g(x)e^it g(x) dx dt :=lim

ε→0Iε.

A straightforward calculus, using the Plancherel’s identity inx, yields Iε=C

t

ψ (εt )

R^3d

f (ξ )ˆ f (η)¯ˆ g(ζ )ˆ g(ξ¯ˆ +ζ−η)e^{−2it (ξ−η,ζ−η)}dξ dη dζ dt

=C

R^3d

f (ξ )ˆ f (η)¯ˆ g(ζ )ˆ g(ξ¯ˆ +ζ−η)1 εψˆ

2

εξ−η, ζ−η

dξ dη dζ.

Thus, we infer

|Iε|C ε

Γ_ζ,η^ε

f (ξ )ˆ f (η)ˆ g(ζ )ˆ g(ξˆ +ζ−η)dξ dη dζ,

where

Γ_ζ,η^ε :=

ξ∈R^d: ξ−η, ζ−ηε .

By Cauchy–Schwarz’s inequality inξ we get

|Iε|C

Γ_ζ,η^ε

1

εg(ξˆ +ζ −η)f (ξ )ˆ ²dξ ¹

2 1

√εΓ_ζ,η^ε ∩suppfˆ¹²g(ζ )ˆ f (η)ˆ dζ dη.

By the assumptions on suppfˆ and suppgˆ we get easily that for everyη∈suppfˆ and everyζ ∈suppgˆ we have

1

2|η−ζ|3. Therefore, in this caseΓ_ζ,η^ε is a _|η^2ε_−ζ| thick layer (which is orthogonal to the vectorη−ζ) and then Γ_ζ,η^ε ∩suppfˆCεM^d−1.

This implies

IεCM^d−21 1 ε

Γ_ζ,η^ε

g(ξˆ +ζ−η)f (ξ )ˆ ²dξ ¹₂

g(ζ )ˆ f (η)ˆ dζ dη.

We apply Cauchy–Schwarz’s inequality in the variablesζ, η IεCM^d−12

η∈suppfˆ

1 ε

Γ_ζ,η^ε

g(ξˆ +ζ−η)f (ξ )ˆ ²dξ dζ dη ¹₂

fL²gL².

For fixedξ, η,we change variablesζ →ξ +ζ −η=u.Note that, by the definition above,ξ ∈Γ_ζ,η^ε if and only if η∈Γ_ξ^ε_+ζ−η,ξ=Γ_u,ξ^ε .Therefore,

η∈suppfˆ Γ_ζ,η^ε

g(ξˆ +ζ−η)f (ξ )ˆ ²dξ dζ dη=

η∈suppfˆ

u:η∈Γ_u,ξ^ε

g(u)ˆ f (ξ )ˆ ²du dξ dη.

Finally, changing the order of integration,

η∈suppfˆ Γ_ζ,η^ε

g(ξˆ +ζ−η)f (ξ )ˆ ²dξ dζ dη= g(u)ˆ f (ξ )ˆ ²Γ_u,ξ^ε ∩suppfˆdξ du.

As before this leads to

η∈suppfˆ

1 ε

Γ_ζ,η^ε

g(ξˆ +ζ−η)f (ξ )ˆ ²dξ dζ dηCM^d−1f²_L2g²_L2.

The outcome is e^it f e^it g²

L²_x,t =lim

ε→0Iε

CM^d−1f²_L2g²_L2. 2

•Cased=1.Proposition 2.3, ford=1, can be proved by interpolation between the following estimates.

Theorem 2.8.For everyp2there exists a constantC=C(p)such that the following estimate e^it f e^it g

L^p_x,t(R²)C ˆf

L

p p−1(R) ˆg

L

p

p−1(R) (4)

holds for all tempered distributionsf andgsuch that their Fourier transformsfˆandgˆ belong toL

p−1p (R)and are

supported in{ξ: |ξ|2},withdist(suppf ,suppˆ g)ˆ ∼1.

This theorem follows from a classical argument by Fefferman and Stein (see [15]). For sake of completeness we give the proof in Appendix A.

Proposition 2.9.Assume thatf andgare functions defined onRsuch thatsuppfˆ⊂ {ξ∈R: |ξ|2M}andsuppgˆ⊂ {ξ∈R: 1|ξ|2}for someM1/4.Then,

e^it f e^it g

L⁴_x,t CM^3/8fL²gL². Moreover, the exponent3/8is sharp.

Proof of Proposition 2.9. Without loss of generality, we can assume thatgˆ⊂ {ξ∈R: 1ξ2}.For the integersk, M^−1/2−1k2M^−1/2,setI_k= [kM^1/2, (k+1)M^1/2],and define functionsg_k bygˆ_k= ˆg1Ik.Then,g=

kg_k and,

e^it f e^it g

L⁴_x,t =

k

e^it f e^it g_k L⁴_x,t

.

We will use the following well-known orthogonality lemma Lemma 2.10(Orthogonality lemma).

k

e^it f e^it g_k L⁴_x,t

C

k

e^it f e^it g_k²

L⁴_x,t

1/2

.

A similar orthogonality result was first observed by C. Fefferman [16] (see also A. Córdoba [11]). For the sake of completeness we will give a proof in Appendix A. Let us now finish the proof of Proposition 2.9. Using the orthogonality lemma, we are reduced to show that

k

e^it f e^it g_k²

L⁴_x,t

1/2

CM^3/8f2g2. (5)

But, by (4) forp=4,we have

k

e^it f e^it g_k²

L⁴_x,t

1/2

C

k

ˆf²_4/3 ˆg_k²_4/3 1/2

.

By Hölder’s inequality and using the size of the supports offˆandgˆ_k the last expression is bounded by CM^3/8

k

f²2gk²2

1/2

=CM^3/8f2

k

gk²2

1/2

=CM^3/8f2g2,

which gives (5). For the sharpness of the exponent 3/8 we can use the same example in Remark 2.5. 2 Proposition 2.3 ford=1 follows by interpolating between Theorem 2.8 and Proposition 2.9. 2 Using Proposition 2.3, we now prove the following bilinear estimate.

Proposition 2.11.Setβ=β(d)=_d+²₂ifd2andβ=¹₄ifd=1.For everyb∈ ]0, β[, there is a constantCbsuch that,

e^it ψ₁∇e^it ψ₂

L^d+d2(R^d+1)C_bψ_1H^b_(R^d₎ψ_2H¹−b(R^d), for everyψ₁∈H^b(R^d)andψ₂∈H^1−b(R^d).

Proof. Using Bony’s decomposition and the fact thatP_j commutes with the free propagatore^it we write e^it ψ₁e^it ∇ψ₂

L^d+2d (R^d+1)^+∞

j=1

e^it S_j−1ψ₁e^it ∇P_jψ₂

L^d+2d (R^d+1)

+

+∞

j=1

e^it ∇Sj−1ψ2e^it Pjψ1

L^d+d2(R^d+1)

+ ^+∞

j=−1

e^it P_jψ₁e^it ∇ ˜P_jψ₂

L^d+d2(R^d+1)

=I+II+III.

•About II.By Cauchy–Schwarz and Strichartz’s inequalities,

II +∞

j=1 j−2

k=−1

2^kPkψ2L²Pjψ1L²(R^d)

+∞

j=1 j−2

k=−1

2^{b(k−j )}2^(1−b)kP_kψ₂L²2^{j b}P_jψ₁L²

2^−b·2^(1−b)·P_·ψ₂L²,2^·bP_·ψ₁L²

², wheredenotes the convolution in².

If we apply successively Cauchy–Schwarz and Young’s estimates we get IIψ2H^1−b(R^d)ψ1H^b(R^d).

•About III.By Hölder, Strichartz and Bernstein’s inequalities

III= ^+∞

j=−1

e^it (P_jψ₁)e^it (∇ ˜P_jψ₂)

L^d+2d (R^d+1)

+∞

j=−1

2^jPjψ1L²(R^d) ˜Pjψ2L²(R^d)

+∞

j=−1

2^{j b}P_jψ₁L²(R^d)2^{j (1−b)} ˜P_jψ₂L²(R^d)

Cψ₁H^b(R^d)ψ₂H^1−b(R^d).

In the last line we have used the Cauchy–Schwartz inequality.

•AboutI.This is the nontrivial part of the proof. Letα∈ ]b, β(d)[fixed. By Proposition 2.3 we have

I

+∞

j=1 j−2

k=−1

e^it P_kψ₁e^it ∇P_jψ₂

L^d+d2(R^d+1)

^+∞

j=1 j−2

k=−1

2^{(k−j )α}P_kψ₁L²(R^d)2^jP_jψ₂L²(R^d). We write it in an appropriate way,

=

+∞

j=1 j−2

k=1

2^{(j−k)(b−α)}2^kbPkψ1L²2^{j (1−b)}Pjψ2L²(R^d)

=2^{−(b−α)·}2^b·P_·ψ_1L²,2^·(1−b)P_·ψ₂L²

² ψ₁H^b(R^d)ψ_2H¹−b(R^d),

where we have applied successively Cauchy–Schwarz and Young’s inequalities. 2 Now, we are ready to prove Proposition 2.2(E). Write

u_i(x, t )=

e^{i(xξ+t τ )}u˜_i(ξ, τ ) dξ dτ=

e^{i(xξ+t|ξ|2)}e^{it (τ−|ξ|2)}u˜_i(ξ, τ ) dξ dτ.

Change variables,λ=τ− |ξ|²,

=

e^{i(xξ+t|ξ|2)}e^{it λ}u˜_i

ξ, λ+ |ξ|² dξ dλ.

By Fubini,

=

e^{it λ}

e^{i(xξ+t|ξ|2)}u˜_i

ξ, λ+ |ξ|² dξ dλ.

Defineui,λbyuˆi,λ(ξ )= ˜ui(ξ, λ+ |ξ|²),whereˆ denotes the Fourier transform in thex-variable. Then, u_i(x, t )=

e^{it λ}e^it u_i,λ(x) dλ.

Therefore, u1∇u2

L^d+d² =

e^{it λ}

e^{it r}e^it u1,λ(x)e^it ∇u2,r(x) dλ dr L

d+2 d x,t

.

By Minkowski’s inequality,

e^it u_1,λ(x)e^it ∇u_2,r(x)

L

d+2 x,td

dλ dr.

By Proposition 2.11, this is bounded by C

u1,λH^bu2,rH^1−bdλ dr, (6)

for allb < β(d).Notice that

ui,λH^αdλ= uˆi,λ(ξ )²

1+ |ξ|^2α dξ

¹₂ dλ.

Using Cauchy–Schwarz’s inequality, for allε >0, uˆ_i,λ(ξ )²

1+ |ξ|^2α dξ

1+ |λ|1+ε

dλ

¹₂

1+ |λ|₋1−ε

dλ ¹₂

C_ε u˜_i

ξ, λ+ |ξ|²1+ |ξ|^2α

1+ |λ|1+ε

dλ dξ ¹₂

=Cε

u˜i(ξ, τ )1+ |ξ|^2α

1+τ− |ξ|^21+εdτ dξ ¹₂

=C_εu_i

X^α,12+ε. Thus, we estimate (6) by

C_εu₁

X^b,12+εu₂

X^1−b,12+ε, for allε >0.The outcome is

u₁∇u₂

L^d+d2 C_εu₁

X^a,12+εu₂

X^a,12+ε,

for allε >0 andamax(b,1−b)withb < β(d). This is Proposition 2.2(E).

2.2. Proof of Proposition 2.2(F)

Set q= ^d+_d².Using Cauchy–Schwarz and Strichartz’s inequalities, and using (6) as above, we see that, for all ε >0,

u_L^2q

x,t C_εu

X^0,12+ε.

Renamev= ˜u.Then, the last inequality can be written as

˜v_L^2q

x,t Cε

v(ξ, τ )²

1+τ+ |ξ|^21+εdτ dξ ¹₂

,

for allε >0.On the other hand

˜vL² = v(ξ, τ )²dτ dξ ¹

2

.

We can now use the theorem of interpolation with change of measure (see [2] Theorem 5.4.1) to obtain for 2p2q and forθdefined by_p¹=¹⁻₂^θ +_2q^θ,

˜vL^p(I×R²)C_ε v(ξ, τ )²

1+τ + |ξ|^{2θ (1+ε)}dτ dξ ¹₂

, for allε >0.Forv= ˜u,we obtain

uL^p(I×R²)Cεu

X^{0,θ (12+ε)}, for allε >0.This gives Proposition 2.2(F).

2.3. Proof of Proposition 2.2(G)

As above, by interpolation with change of measure with the estimate (F) forp=2q,it suffices to show that wL^∞_x,t Ca,bwX^a,b,

for alla >^d₂ andb >¹₂.This follows form Proposition 2.2(A) and the Sobolev embedding.

3. Proof of Theorem 1.1

Using Proposition 2.2 we are going to prove Theorem 1.1. We will prove that for everyt₀∈ [0, T^∗[there exists a small intervalI containingt0, such thatw∈X^1,12+ε[I],for someε >0. This implies thatw∈X^1,12+ε[J]for every compact interval J ⊂ [0, T^∗[, which is, in view of Proposition 2.2(A), a stronger result than the first assertion in Theorem 1.1. Note also that, via a time translation, it is sufficient to considert0=0.

According to the integral formulation of the initial value problem (1) we have w(t, x)=iκ

t

0

e^i(t−t)u(t, x)^d4u(t, x) dt.

Letφ∈C^∞₀ (R), such that

φ(t)=1 if|t|<1, φ(t )=0 if|t|>2.

ForT >0 we denoteφ_T(t )=φ(_T^t). On[0, T]the functionwis equal to

w(t, x)=iκφT(t ) t

0

e^i(t−t)φ_T(t)u(t, x)^4dφT(t)u(t, x) dt.

Thus, by replacingubyφ_Tuwe can assume thatuis globally defined (and compactly supported) in time. Meanwhile we keep the same notationu.

Note also that the value ofT is not relevant for us in this part of proof. So, the dependence onT of the constants is not explicited.

Let s > s_d fixed and ε=ε(d, s)a sufficiently small positive number (it satisfies a finite number of smallness conditions). Applying Proposition 2.2(D) (withb=¹₂+εandb= −¹₂+ε), one can estimate

∇w

X^0,12+εC∇

|u|^d4u

X^0,−12+ε. (7)

Here and throughout this section the constantsC=C_ε,s,d. For sake of shortness we drop the indices.

The main ingredient of our proof is the following nonlinear estimate.