A lower bound on the blow-up rate for the Davey–Stewartson system on the torus

Ann. I. H. Poincaré – AN 30 (2013) 691–703

www.elsevier.com/locate/anihpc

A lower bound on the blow-up rate for the Davey–Stewartson system on the torus

Nicolas Godet

University of Cergy-Pontoise, Department of Mathematics, CNRS, UMR 8088, F-95000 Cergy-Pontoise, France Received 5 September 2012; received in revised form 2 December 2012; accepted 3 December 2012

Available online 12 December 2012

Abstract

We consider the hyperbolic–elliptic version of the Davey–Stewartson system with cubic nonlinearity posed on the two- dimensional torus. A natural setting for studying blow-up solutions for this equation takes place in H^s, 1/2< s <1. In this paper, we prove a lower bound on the blow-up rate for these regularities.

©2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

We consider the Davey–Stewartson system defined on the two-dimensional torusT²:=R²/2πZ²: i∂tu−∂_x²u+∂_y²u= −|u|²u+2u∂xφ,

∂_x²+∂_y²

φ=∂_x|u|², (1)

whereu:R×T²→Candφ:R×T²→Rare the unknowns. Rearranging the second equation, we may see this system as a dispersive equation with a hyperbolic linear part and a nonlocal nonlinearity:

i∂_tu+P u= −|u|²u−E

|u|²

u, (t, x)∈R×T², (2)

whereP = −∂_x²+∂_y²andEis the nonlocal operator such that:

E(f )(m, n) = 2m²

m²+n²f (m, n), (m, n)∈Z²\ (0,0)

, E(f )(0, 0)=0.

The Cauchy problem and the blow-up theory for this equation have been studied essentially in the case where the system is posed inR². In this case, the system is locally well posed in the Sobolev spacesL²,H¹(see[5]) and more easily for higher regularitiesH^s,s >1. In[14], T. Ozawa proved that the equation posed onR² enjoys a pseudo-

E-mail address:nicolas.godet@u-cergy.fr.

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.12.001

conformal type symmetry and as for NLS, this allows to construct a blow-up solution by applying this transformation to an explicit stationary (periodic in time for NLS) solution:

u(t, x, y)= 1 1+x²+y². This function is then transformed into

v(t, x, y)= 1 a+btexp

ib 4(a+bt )

−x²+y² 1

1+(_a+^x_bt)²+(_a+^y_bt)², (3) with(a, b)∈R². Note thatv(t )is inH^s(R²)(see[14]) for everys <1 withv(t )L²=√

π but is not inH¹(R²).

The solutionvblows up at timeT = −a/binL²in the sense that theL²blow-up criteria (theL⁴_[0,t]L⁴norm goes to infinity whentgoes toT) is satisfied:

vL⁴([0,t])L⁴(R²)∼ C

(T −t )^1/4 → ∞ astgoes toT , and accumulates all the mass in the origin:

v(t )²→π δ_(0,0) ast→T inD R²

.

The explosion also occurs inH^s,s <1 with the pseudo-conformal bound[14]:

v(t )

H^s ∼ C (T −t )^s.

Note that inR², we have a scaling symmetry; ifusolves (2) then for all λ >0,(t, x, y) →λu(t, λ²t, λx, λy) also solves (2). It is a classical fact that this symmetry automatically implies a lower bound on the blow-up rate: for all blow-up solutionuwith maximal timeT (u) <∞, we have

u(t )

H^s C

(T (u)−t )^s/2. (4)

By analogy with NLS in R², we may ask the question of existence of ground states of the type u(t, x, y)= exp(iωt)Q(x, y)for (1) posed onR²with an exponentially decaying profileQbut the hyperbolicity of the operator

−∂_x²+∂_y²forbids the existence of such solutions at least in the case where the nonlinearity is−|u|²u[6]. Moreover, numerics[11]seems to show that theL²-norm of the solutionu(orv) is the minimal mass for which we may have singularities in finite time. Thus, the functionuplays the role of a ground state but is only polynomially decaying and this requires to work with low regularitiesH^s,s <1.

The aim of this paper is twofold: first give anH^sframework for studying blow-up theory for (2) i.e. show the local well-posedness of (2) for initial data inH^s(T²),s <1 and secondly show that the lower bound on the blow-up rate (4) still holds even if a scaling symmetry does not strictly make sense on the torus. The proof relies on local existence arguments on the dilated torusR²/2π LZ²,L→ ∞and more precisely on bilinear Strichartz estimates. The classical method [2] giving well-posedness from bilinear Strichartz estimates does not work in our setting because of the nonlocal term; we will have to refine the bilinear approach with more general localizations. An interesting question, not solved here, is the localization of the solutionv (3) i.e. construct from va solution of (2). The nonexponential decay ofvis reflected in the estimate

v(t )

H^s(ε|(x,y)|A)C(T −t )^(1−s),

which make perturbation arguments aroundv difficult to apply even on a compact domain. In particular, fors >1, blow-up is not localized and this explains our choice to treat low regularities.

Remark 1.The system we study is called the hyperbolic–elliptic version of the generalized Davey–Stewartson system:

i∂_tu+ε₁∂_x²u+∂_y²u= −|u|²u+2u∂xφ, ε₂∂_x²+∂_y²

φ=∂_x|u|², (5)

whereε_i∈ {−1,+1}. Depending on the values ofε_i, the local well-posedness holds[5,1,13,10,4]but blow-up theory is really well understood only in the elliptic–elliptic caseε₁=ε₂=1[12,17,15]where results are similar to those for NLS.

2. Statement of the result and remarks Let us now give our result.

Theorem 1.Lets >1/2.

1) Eq.(2)is locally well posed in the spaceH^s(T²)in the following sense. There existsb >1/2such that the fol- lowing holds true:for allu₀∈H^s(T²), there exist a timeT >0and a uniqueu∈X_T^s,b(T²)⊂C([0, T], H^s(T²)) satisfyingu(0)=u₀and(2). Here,X^s,b_T denotes the Bourgain space associated to(2)and defined in(17).

2) Letu₀∈H^s(T²)andube the corresponding solution. IfT (u₀)denotes the maximal time of existence ofu, then we have the following possibilities:eitherT (u₀)= +∞orT (u₀) <+∞and in this case, there existsC >0such that for allt∈ [0, T (u0)):

u(t )

H^s(T²) C

(T (u₀)−t )^s/2. (6)

Before giving the proof of Theorem1, let us give some comments. Consider the equation

i∂_tu+P u= −|u|²u, t >0, (x, y)∈T². (7)

Strichartz type estimates hold for the operatorP (see[8,16]) and this with an analysis similar to[2]gives the local well-posedness of (7) inH^s(T²)for alls >1/2. Following[8], it is easy to check that the function defined by

u(t, x, y)=e^it|u0(x+y)|2u₀(x+y) (8)

is a solution of (7) for allu₀∈C^∞(R/2πZ,R). Ifu₀is not a constant function and ifs0, we can check that there existsC >0 such that fort1,

u(t )

H^s Ct^s. (9)

Thus, for s >1/2, we obtain an explicit solution which blows up in infinite time; this contrasts with the usual Schrödinger equation. Using a suitable rescaling of the explicit solution (8), one may also show the local ill-posedness of (7) inH^s,s <1/2 (see the appendix of[3] for a similar discussion). It would be interesting to know if we may construct solutions behaving like (8) for Eq. (2).

3. Proof of the result

Strategy of the proof.To prove Theorem1, the idea is to rescale the torusT²=R²/2πZ²by consideringT_L²= R²/2π LZ²whereL >0 will tend to infinity. In a first step, we perform a Banach fixed point argument in the dilated Bourgain spaceX^s,b_{T ,L}to obtain a local well-posedness result inH^s(T_L²)with the bound on the blow-up time:

T (u₀)F

u_0H^s_(T²

L)

, (10) for some function F independent ofL. This relies on a uniform bilinear Strichartz estimate. In our analysis, it is of importance that dispersive estimates are local in space and time. In particular, if we take L=1, this step will give the first point of Theorem1. In the step 2, we deduce the blow-up lower bound from a scaling argument. The bound (6) which is the same as R² is in accordance with the fact that when L goes to infinityT_L² looks like R² formally. Note that the machinery of the Bourgain spaces is natural to study such questions but we do not exclude the possibility of working on other spaces by adapting harmonic analysis results to the case ofT_L²to treat the opera- torE.

Notations. We denote by e_m,n(x, y)=(2π )⁻¹exp(imx+iny) the usual orthonormal basis of L²(T₁²). When working onT_L², we will keep the same notatione_m,n for the rescaled basis:e_m,n(x, y)=(2π L)⁻¹exp(i(m/L)x+ i(n/L)y). For a functionudefined onT_L², we note

Q(u)=

(m,n)∈Q

c(m, n)e_m,n,

wherec(m, n)are the Fourier coefficients ofu:

c(m, n)= 1 2π L

T_L²

u(x, y)e^i(mLx+Lny)dx dy.

IfQ⊂Z²andRis a dyadic number, we set for a functionu(t, x)defined onR×T_L²:

Q,Ru=

(m,n)∈Q

Rτ−^m_L²₂+_Lⁿ²₂2R

cn,m(τ )e^{2iπ t τ}dτ em,n,

wherecm,n(t)are the Fourier coefficients ofu(t ). WhenQis the cubeQ= {(m, n)∈Z², NMax(|m/L|,|n/L|) 2N}, we will note N= Qand N,R= Q,R.

Step 1.We prove: for alls >1/2,L1 andu0∈H^s(T_L²), there exists a solutionuof i∂_tu+P u= −|u|²u−E

|u|²

u, (x, y)∈T_L², (11)

andα >0,C >0 independent ofLsatisfying the lower bound on the blow-up time:

T (u₀) C u0^α_Hs(T_L²)

. (12)

The point here is that the lower bound depends only on the size of the initial data and not onL. OnT_L², we denote (without changing notations) byP andEthe natural extensions of the operatorsP andEdefined above onT². Hence, symbols are respectively(−m²+n²)/L²and 2m²/(m²+n²).

High regularity.Before looking at low regularities and to convince the reader that (12) holds, let us focus on the easier case of more regular data i.e.s∈N\ {0,1}. Let us prove (12) in this case. LetL1 and consider Eq. (11) and its equivalent formulation

u(t )=e^{it P}u₀+i t

0

e^{i(t−τ )P}u(τ )²u(τ )+Eu(τ )² u(τ )

dτ. (13)

Letu0∈H^s(T_L²). Taking theH^s-norm in (13) and using the triangle inequality, we get for a constantC >0 indepen- dent ofL:

u(t )

H^su0H^s+C T

0

u(τ )²u(τ )

H^s+Eu(τ )² u(τ )

H^s

dτ. (14)

Now we need a Sobolev type inequality with constants independent of the size of the torus.

Lemma 1.Letsbe an integer withs2. There exists a constantC >0such that for allL >0andv, w∈H^s(T_L), vw_Hs(T_L²)Cv_Hs(T_L²)w_Hs(T_L²).

Proof. We first prove that for allσ >1 andv∈H^σ(T_L),vL^∞ CvH^σ for a constantC >0 depending only onσ. Indeed, expandingvin Fourier series, we first get

|v| 1 2π L

(m,n)∈Z²

|v_m,n|.

We make appear theH^σ-norm ofuand use Cauchy–Schwarz inequality to obtain vL^∞ 1

2π L

(m,n)∈Z²

|vm,n|

1+m² L² + n²

L²

σ/2 1+m²

L²+ n² L²

−σ/2

1

2π LvH^σ

(m,n)∈Z²

1+m²

L²+ n² L²

−σ 1/2

. (15)

But we can easily compute the dependence inLof the last sum above by comparing with an integral as follows:

(m,n)∈Z²

1+m²

L² +n² L²

−σ

C

(m,n)∈Z²

1+m²

L²

−σ/2 1+n²

L²

−σ/2

C

dx (1+_L^x22)^σ/2

2

CL².

Thus, there is not more dependence onLin (15) and we obtain the claim. Now we can prove the lemma. Indeed, we first write the Leibniz rule then use the previous claim and an interpolation argument to get

(− )^s/2(vw)

L²CvH^swH^s.

Here, the constant C contains binomial coefficients and therefore is independent of L. Moreover, again with the embeddingH^s→L^∞, we have

vwL²vL^∞wL²CvH^swH^s,

and the last two inequalities end the proof of the lemma. 2

Therefore, coming back to (14), using the boundedness ofEinH^s and Lemma1, we have u(t )

H^s u0H^s +CTu(t )³

H^s,

withC >0 independent of the period. This last estimate allows us to perform a Banach fixed point argument (the Lipschitz property is proved with similar arguments) in a ball of the spaceC([0, T], H^s)of radiusM=2u₀H^s and withT =C/u₀²_H^s, and this proves (12).

Low regularity.This part is the more interesting since, as said above, the explicit blow-up solutionv(t )of the introduction lives only inH^s withs <1. So lets >1/2. We define the Bourgain spaces associated with Eq. (11) as the completion of the space of smooth compactly supported functions onR×T_L²for the norm defined by

u_X^s,b

L =i∂_t+P^b

(− )¹²s

u

L²(R×T_L²),

whereα =(1+α²)^1/2. Note that there exist more convenient equivalent definitions of this space: we may also check that the norm is equivalent to the following

u²_Xs,b L

=

(m,n)∈Z²

1+m²

L² +n² L²

s

R

τ−m²

L² + n² L²

2b

c_m,n(τ )²dτ,

wherec_m,nis the Fourier transform ofc_m,n. A last definition is possible linking the Bourgain norm with Sobolev norm of the free dynamic:

u_X^s,b

L =e^{−it P}u(t )

H^b(R,H^s(T_L²)). (16)

We will work on a finite time interval so that we have to define the localized version of the Bourgain spaces; for u: [0, T] ×T_L²→C:

u_X^s,b

L,T =inf v_X^s,b

L

, v∈X_L^s,bsuch thatv(t )=u(t )for allt∈ [0, T]

. (17)

Let us recall the integral formulation (13):

u(t )=e^{it P}u₀+i t

0

e^{i(t−τ )P}u(τ )²u(τ )+Eu(τ )² u(τ )

dτ.

First, the linear term is easily bounded: ifT 1 andψ (t )denotes a smooth real cut-off function equal to 1 on[0,1]

and with compact support, we get using the definition of the Bourgain spaces (16):

e^{it P}u₀

X^s,b_L,T e^{it P}ψ (t )u₀

X_L^s,bψ (t )u₀

H^b(R,H^s(T_L²))Cu_0Hs(T_L²), (18) whereC= ψ_H^b_(R)is independent of the periodL.

Lemma 2. There exists C >0 such that for all L1, T 1 and all pair (b, b) satisfying 0< b<1/2< b, b+b<1,

t

0

e^{i(t−τ )P}F (τ ) dτ X^s,b_L,T

CT^1−b−bF_X^s,−b L,T

.

Proof. For a fixedL >0, this estimate is classical in the context of the Bourgain spaces. To see that we may chooseC independent ofL, we remark (see[2]) that the proof of such an estimate for a fixedLrelies on the one-dimensional inequality (proved in[7]):

φ

t T

t

0

g(τ ) dτ H^b(R)

CT^1−b−bg_H^−b_(R), (19)

for a cut-off function φ. Then we apply this estimate pointwise withg(τ )=(F (τ, x), e_m,n)e_m,n, take the square, integrate onT_L², multiply by(−m²+n²)/L²and sum for(m, n)∈Z². We then obtain the desired estimate with the same constantCas in (19) thus independent ofL. 2

To treat the nonlinearity in the fixed point argument, we will need the following proposition.

Proposition 1(Trilinear estimate). There exist a pair(b, b)satisfying0< b<1/2< b,b+b<1and a constant C >0such that for everyL1,T >0,u1, u2, u3∈X^s,b_L,T,

u1u2u3_X^s,−b

L,T Cu1_X^s,b

L,Tu2_X^s,b

L,Tu3_X^s,b

L,T

, E(u₁u₂)u₃

X^s,_L,T^−b Cu_1X^s,b

L,Tu_2X^s,b

L,Tu_3X^s,b

L,T

.

Proof. Let us start with a lemma.

Lemma 3(Uniform periodic bilinear Strichartz estimate). There existsC >0such that for everyN₁, N₂1dyadic numbers,(a₁, b₁), (a₂, b₂)∈Z²,L1andu₁, u₂∈L²(T_L²)writing

u₁=

N₁Max(|^m_L−a₁|,|_Lⁿ−b₁|)2N₁

c₁(m, n)e_m,n, u₂=

N₂Max(|^m_L−a₂|,|_Lⁿ−b₂|)2N₂

c₂(m, n)e_m,n

we have the bilinear estimate e^{it P}(u1)e^{it P}(u2)

L²([0,1])L²(T_L²)Cmin(N1, N2)^1/2u1_L²_(T²

L)u2_L²_(T²

L). (20)

Proof. Note that forL=1, linear Strichartz estimates have been proved recently in[16,8]. We first prove the property in the case whereu₁=u₂anda₁=b₁=a₂=b₂=0. So letu=u₁=u₂andN=N₁=N₂. We recall the semiclas- sical Strichartz estimate on the torus of size 1 (see[8]): for allh∈(0,1), for all intervalJ of size hand for allu₀ writing

v0=

h⁻¹Max(|m|,|n|)2h⁻¹

c(m, n)em,n,

for some coefficientc(m, n), we have e^{it P}v₀

L⁴(J )L⁴(T₁²)Cv_0L2(T₁²). (21)

Similarly to the case whereP is the Laplace operator (see[9]), we apply a scaling argument on this estimate to derive a linear Strichartz estimate onT_L²on the time interval[0,1]. Letu0∈L²(T_L²)localized in frequency in[0, N]i.e.

u0=

NMax(|^mL|,|Lⁿ|)2N

c(m, n)em,n, (22)

andv0∈L²(T₁²)defined byv0(x)=u0(Lx). Then computing theL⁴([0,1])L⁴(T_L²)of exp(itP )u0in term ofv0and applying a change of variable, we get

e^{it P}u₀

L⁴([0,1])L⁴(T_L²)=Le^{it P}v₀

L⁴([0,L⁻²])L⁴(T₁²). Remark thatv0writes

v₀=

LNMax(|m|,|n|)2LN

c(m, n)e_m,n,

so that we may apply (21) withh∼LN. We need to consider two cases. IfLN, then[0, L⁻²] ⊂ [0, (LN )⁻¹]and so e^{it P}u₀

L⁴([0,1])L⁴ Le^{it P}v₀

L⁴([0,(LN )⁻¹])L⁴CLv₀L²Cu₀L².

IfL < N, we write[0, L⁻²]as a union of intervals[tk, tk+1]withtk+1−tk∼(LN )⁻¹andk∼N/L. We apply (21) on each[tk, tk+1]and this gives

e^{it P}u₀

L⁴([0,1])L⁴ C N

L

1/4

Lv_0L²C N

L

1/4

u_0L². IfL1, we may in particular summarize the last two inequalities as

e^{it P}u0

L⁴([0,1])L⁴ CN^1/4u0L²,

and this proves (20) ifu1=u2 anda1=b1=a2=b2=0. Now we treat the case u=u1=u2 but without the assumptiona1=b1=a2=b2=0. Let(a, b)∈Z²and write

u=

NMax(|^m_L−a|,|ⁿ_L−b|)2N

e^{Lit2(n2−m2)}c(m, n)e_m,n

=e^iaxe^ibye^{it (a2−b2)}

NMax(|^pL|,|L^q|)2N

c(aL+p, bL+q)e⁻

it

L2(p²−q²+2aLp−2bLq)

ep,q.

Then

u⁴_L4L⁴=

NMax(|^pL|,|L^q|)2N

1

2π Lc(aL+p, bL+q)e⁻

it L2(p²−q²)

e^{ipL(x−2at )}e^{iqL(y+2bt) 4}

L⁴L⁴

=

t

−2atα2π L−2at 2btβ2π L+2bt

NMax(|^p_L|,|^q_L|)2N

e⁻

it

L2(p²−q²) 1

2π Lc(aL+p, bL+q)e^Lipαe^Liqβ

⁴dα dβ dt

=

t

0α2π L 0β2π L

NMax(|^p_L|,|_L^q|)2N

e

it L2(q²−p²)

c(aL+p, bL+q)e_p,q(α, β)

⁴dα dβ dt.

We apply the linear result proved above with(a, b)=(0,0)and this gives u⁴_L4L⁴CN

k,l

c(k, l)^{2 2}

CNu0⁴_L2.

This proves the result whenu1=u2. Note that if we assume another type of localization foru

u=

Max(|^m_L−a|,|_Lⁿ−b|)N

c(m, n)e_m,n,

theL⁴L⁴estimate still holds. It may be seen by remarking that estimate (21) also holds ifu₀is spectrally localized in {(m, n)∈Z², Max(|m|,|n|)2h⁻¹}(see[8]) and using the same analysis as above. Now, we can prove the bilinear estimate in the general case. We assume for instance N₁N₂ and decompose the set A= {(m, n)∈Z², N₂ Max(|a₂−m/L|,|b₂−n/L|)2N2}in small disjoint cubes of the formQ_α=Q_(k,l)= {(m, n)∈A,Max(|k−m/L|,

|l−n/L|)N₁}forα=(k, l)running over a setI. Then for differentαs, the functionse^{it P}(u₀)e^{it P}( _Qαv₀)are almost orthogonal since each function is localized in Fourier in the setD_α:= {(m, n)∈Z², N₁Max(|m/L−a₁|,

|n/L−b₁|)2N1} +Q_α and the setsD_α are almost disjoint in the sense that each point ofZ²belongs to a finite number of setsD_α. Indeed, if(m, n)∈D_α1∩D_α2, then in particular we may write

(m, n)=c+d=e+f,

withd∈Qα₁,f∈Qα₂, andc, e∈ {(m, n)∈Z², N1Max(|m/L−a1|,|n/L−b1|)2N1}. We deduce|c1−e1| =

|f₁−d₁|4N1L. But eachQ_α is of size less than 4N1Land there is a finite number ofQ_αwhose distance toQ_α2 is less than 4N1L. So if we fixα₁, then α₂ runs in a finite number of indexes. Thus, this orthogonality property implies

e^{it P}(u₁)e^{it P}(u₂)²

L²L²C

α∈I

e^{it P}(u₁)e^{it P}( _Qαu₂)²

L²L²

Ce^{it P}(u1)²

L⁴L⁴

α∈I

e^{it P}( Q_αu2)²

L⁴L⁴

CN₁^1/2u₁²_L2N₁^1/2

α∈I

Qαu₂²_L2

CN₁u₁²_L2u₂²_L2. This proves the lemma. 2

Remark 2.Note that ifQ_i denotes the set Qi=

(m, n)∈Z², NiMax m

L−ai

, n

L−bi

2Ni

,

then(N_iL)²|Q_i|and we may rewrite the Strichartz estimate (20) as _Q

1

e^{it P}u

Q₂

e^{it P}v

L²L²C

min(|Q₁|,|Q₂|) L²

1

4uL²vL². (23)

Once we have proved (23), from covering arguments, we may deduce the same estimate for other shapes of Qi

typicallyQi= {(m, n)∈Z², Max(|m/L|,|n/L|)2Ni}or translated sets of the previous one. In the sequel, we will use (23) for these kinds ofQ_i. More precisely, we have the following.

Lemma 4. For allb >1/2, there existC(b) >0,β(b)∈(0,1−b) and ε(b) >0 such that for all dyadic square Q₁, Q₂⊂Z²,R₁,R₂dyadic number,L1andu₀, v₀∈L²(R, L²(T_L²)),

Q1,R1u_{0 Q2,R2}v_0L²_L²C(b)

Min(|Q1|,|Q2|) L²

1/4+ε(b)

(R₁R₂)^β(b)

× Q₁,R₁u0L²L² Q₂,R₂v0L²L², (24) where|Qi|denotes the number of points inQi. Moreover, we may chooseε(b)such thatε(b)goes to0asb goes to1/2.

Proof. As for the proof of (20), we first assumeu=u₀=v₀. Next, from (20), we get for allb >1/2 andf ∈X^0,b_L localized in frequency inQ,

fL⁴L⁴C |Q|

L²

1/8

f_X^0,b

L

.

Again the constantCdoes not depend onLsince the proof (see[2]) relies only on manipulations in time. In particular, for allu,

Q,RuL⁴L⁴C |Q|

L²

1/8

Q,Ru_X^0,b

L

, (25)

for allb >1/2. And this gives using properties of the Bourgain spaces Q,RuL⁴L⁴C

|Q| L²

1/8

R^b Q,RuL²L². (26)

The fact thatb >1/2 in the above estimate will not be enough to conclude so that we need to refine thisL⁴L⁴estimate.

To do so, we compute theL^∞L^∞norm of Q,Ru. From the definition of the projection Q,R, we get using twice Cauchy–Schwarz inequality

Q,RuL^∞L^∞ 1 L

(m,n)∈Q

Rτ−^m2

L2+ⁿ²

L22R

c_m,n(τ )dτ

R^1/2 L

(m,n)∈Q

Rτ−^m2

L2+ⁿ²

L22R

c_m,n(τ )²dτ

1/2

R^1/2 |Q|

L²

1/2

(m,n)∈Q

Rτ−^m_L²2+_Lⁿ²22R

c_m,n(τ )²dτ

1/2

|Q|

L²

1/2

R^1/2 Q,Ru_L²_L². (27)

By interpolation between the trivial inequality Q,Ru_L²_L² Q,Ru_L²_L²and (27), we have Q,RuL⁴L⁴

|Q| L²

1/4

R^1/4 Q,RuL²L². (28)

Letε(b) >0 such thatδ(b):=b(1−8ε(b))+8ε(b)¹₄ ∈(0,1−b)andε(b)→0 asb→1/2. For instance choose δ(b)=3/2−2b. Next, by interpolation between (26) with weight 1−8ε(b)and (28) with weight 8ε(b), we get the expected estimate:

Q,Ru₀L⁴L⁴C |Q|

L²

1/8+ε(b)

R^δ(b) Q,Ru₀L²L². (29)

To deduce (24) from (29), we proceed as in the proof of Strichartz estimate (20): if for instance|Q1|<|Q2|then we decomposeQ2in pieces of size|Q1|and next apply an almost orthogonality argument. We omit this argument and the proof is over. 2

To prove Lemma1, it is enough to prove the trilinear estimate for the global spaceX^s,b_L i.e.T = ∞, then we recover the local in time estimate by taking the infimum on all extensions ofu1, u2, u3∈X_L,T^s,b . Moreover, we only prove the second estimate; the first one is easier. By a duality argument, we have to show the quadrilinear estimate: there exists C >0 such that for allL1,u1, u2, u3, u4∈X_L^s,b:

R×T_L²

E(u₁u₂)u₃u₄

Cu_1X^s,b

L u_2X^s,b

L u_3X^s,b

L u_4X−s,b L

.

In the sequel, we will noteQi= {(m, n)∈Z², NiMax(|m/L|,|n/L|) <2Ni}. Decomposing eachui as u_i=

N_i,R_i

Ni,Ri(u_i),

we have that G=

R×T_L²

E(u1u2)u3u4

becomes G=

R×T_L²

N1,N2,N3,N4

R1,R2,R3,R4

E

N₁,R₁(u1) N₂,R₂(u2)

N₃,R₃(u3) N₄,R₄(u4).

In the summation above, we may restrict indexes toN42(N1+N2+N3). Indeed, the function U=E

N1,R1(u₁) _N2,R2(u₂)

N3,R3(u₃)

is localized in Fourier in the set{(m, n)∈Z², m=m1+m2+m3, n=n1+n2+n3, (m1, n1)∈Q1, (m2, n2)∈Q2, (m3, n3)∈Q3}. Thus, ifN4>2(N1+N2+N3), the integral overT_L²ofU N₄,R₄(u4)is zero. Therefore

G=

N₄2(N₁+N₂+N₃) R₁,R₂,R₃,R₄

α(N₁, N₂, N₃, N₄, R₁, R₂, R₃, R₄), (30)

where

α(N₁, N₂, N₃, N₄, R₁, R₂, R₃, R₄)=

R×T_L²

E

N1,R1(u₁) _N2,R2(u₂)

N3,R3(u₃) _N4,R4(u₄).

Contrary to the case of a typical cubic nonlinearity,αis not symmetric inN1,N2,N3,N4and we need to split the analysis in several cases. The worst situation is when the two lowest frequencies appear in the nonlocal term. Let us first treat this case.

CaseN3=max(N1, N2, N3).Without loss of generality, we may assumeN₁N₂N₃. In this situation, we decompose the setQ₃in small pieces of sizeN₂L. Hence, we may writeQ₃as a disjoint union of sets of the form