Global well-posedness for the primitive equations with less regular initial data

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

FRÉDÉRICCHARVE

Global well-posedness for the primitive equations with less regular initial data

Tome XVII, n^o2 (2008), p. 221-238.

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Global well-posedness for the primitive equations with less regular initial data

Fr´ed´eric Charve⁽¹⁾

ABSTRACT.— This paperis devoted to the study of the lifespan of the solutions of the primitive equations for less regular initial data. We in- terpolate the globall well-posedness results for small initial data in ˙H¹² given by the Fujita-Kato theorem, and the result from [6] which gives global well-posedness if the Rossby parameterεis small enough, and for regular initial data (oscillating part in ˙H¹²∩H˙¹and quasigeostrophic part inH¹).

R^´ESUM ´E.— Cet article est consacr´e `a l’´etude du temps d’existence des solutions du syst`eme des ´equations primitives pour des donn´ees moins r´eguli`eres. On interpole les r´esultats d’existence globale `a donn´ees ˙H¹² petites fournis par le th´eor`eme de Fujita-Kato, et le r´esultat de [6] qui donne l’existence globale si le param`etre de Rossby εest suffisamment petit, et pourdes donn´ees plus r´eguli`eres (partie oscillante initiale dans H˙¹²∩H˙¹et partie quasig´eostrophique initiale dansH¹).

1. Introduction 1.1. The primitive equations

The primitive system writes :







∂tUε+Uε.∇Uε−LUε+1

εAUε=1

ε(−∇Φε,0) div vε= 0

U_ε/t=0=U0,ε.

(P Eε)

(∗) Re¸cu le 01/03/2006, accept´e le 15/10/2007

(1) Universit´e Paris-Est, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UMR 8050, Universit´e Paris 12, Bˆatiment P3, 61 avenue du G´en´eral de Gaulle 94010 Cr´eteil, France.

[email protected]

The unknowns are Uεand Φε. We denote byUε a pair (vε, θε) where vε is a vector field on R³ (three dimensional velocity),θε a scalar function (the density fluctuation : in the case of the atmosphere it depends on the scalar (potential) temperature and in the case of the ocean it depends on the temperature and the salinity), and Φε the pressure, all of them depending on (t, x). The operatorLis defined by

LUεdef

= (ν∆vε, ν∆θε), We define :

Uε.∇Uε=vε.∇Uε= 3 i=1

v_εⁱ.∂iUε,

and the matrixAb y :

Adef

=







0 −1 0 0

1 0 0 0

0 0 0 F⁻¹

0 0 −F⁻¹ 0





.

The small parameterεis called here the Rossby number andF is called the Froude number. They are related to the physical Rossby and Froude numbers by the following relations :

Ro=ε, F r=εF.

The smaller isε, the more important are the Coriolis force (induced by the rotation of the earth around its axis) and the vertical stratification of the density.

We refer for example to [6] for the physical meaning of these terms and for a list of physical references.

Definition 1.1. — If sis a real number, the homogenous (resp. inho- mogenous) Sobolev space of orders, which we will denote byH˙^s(resp.H^s), is defined as the space of tempered distributions u∈ S(R³) whose Fourier transform uˆ is locally integrable and has the following property :

u²H˙^s

def=

R³|ξ|^2s|ˆu(ξ)|²dξ <∞ (resp. u²H^s

def=

R³(1 +|ξ|²)^s|u(ξ)ˆ |²dξ <∞).

Although the primitive equations have no scaling anymore, we can eas- ily adapt the proofs of the Leray and Fujita-Kato theorems (thanks to the skewsymmetry of matrix A and the fact that both of these theorems are proved using mainly inner products and energy estimates) to get the follow- ing results :

Theorem 1.2. — (Leray, 1934, [15]) if the initial data U0 ∈ L²(R³), then there exists for all ε > 0 a Leray solution of the system (P Eε), Uε, globally defined in time, belonging toL^∞(R+, L²(R³))∩L²(R+,H˙¹(R³))and satisfying the following energy inequality (let ν0= min(ν, ν)>0) :

∀t∈R+, Uε(t)²L²(R³⁾+ 2ν0

t 0

∇Uε(t)²L²(R³⁾dtU0²L²(R³⁾.

We refer to [5] where we studied the limit of Leray solutions when ε, the Rossby number, goes to zero and introduced the following notations and results in the case of weak solutions : the potential vorticity is defined by

Ωεdef

= ∂1v_ε²−∂2v_ε¹−F ∂3θε.

Then from this, we define the orthogonal decomposition ofUεinto its quasi- geostrophic part, and its oscillating part :

Uε,QGdef

=







−∂2∆F−1

Ωε

∂1∆F−1

Ωε

0

−F ∂3∆F−1

Ωε





,

with ∆Fdef

= ∂₁²+∂₂²+F²∂₃³, and :

Uε,oscdef

= Uε−Uε,QG=







v¹_ε+∂2∆F−1

Ωε

v²_ε−∂1∆F−1

Ωε

v³_ε θε+F ∂3∆F−1

Ωε





.

We have seen in [5] that this decomposition is an orthogonal decomposition and we denoted byP the orthogonal projector onto the potential vorticity free vector fields (which is built the same way as the orthogonal projector P on the divergence free vector fields, also called the Leray projector) and Q = Id− P the orthogonal projector on the quasigeostrophic vectorfiels.

Both of them are homogeneous pseudo differential operators of degree zero.

In [5] we studied the convergence, whenεgoes to zero, of the weak Leray solutions towards the quasigeostrophic model (see (1.1) below).

When the initial data is more regular and even if there is no scale in- variance for this system we can easily adapt the Fujita and Kato theorem (1964) :

Theorem 1. — (Fujita and Kato, 1964, [12]) If U0∈H˙ ¹² there exist a unique maximal time T_ε^∗>0, and a unique solution

Uε∈ C([0, Tε^∗[,H˙ ¹²)∩L²_loc([0, T_ε^∗[,H˙ ³²).

Moreover, if T_ε^∗ is finite, then we have T_ε^∗

0

Uε(t)²_˙

H³2(R³⁾dt= +∞.

Finally there exists a constant c such that ifU0_H˙¹_2(R3)cν0 then Tε^∗= +∞.

Contrary to the Leray solutions, the solutions are unique but we do not know whether they are global in general. The Fujita-Kato theorem also works on the quasigeostrophic system, and again, it does not say whether the unique solution is global if we do not have a small initial data.

Both of these results are general results directly adaptated from the Navier-Stokes case, without using the special structure of the primitive equations. As we have seen in [5], [6], and [7], when the Rossby number ε goes to zero, the system is stabilized as its solutions go to the solutions of the quasigeostrophic model (We refer to [5], [6], and [7] for the study, in the case of the whole space, and forF= 1, of the convergence, asεgoes to zero, of the primitive equations solutions to the quasigeostrophic model.), which is closer to the two-dimensionnal Navier-Stokes system than to the three-dimensionnal one :

Theorem 2. — [6] IfU0,QG∈H¹(R³)then the quasigeostrophic system ∂tUQG−ΓUQG+Q(UQG.∇UQG) = 0

UQG_/t=0 =U0,QG, (1.1)

has a unique global solution in L^∞(R+, H¹)∩L²(R+,H˙¹∩H˙²), with the following energy estimate for alls∈[0,1]:

∀t∈R+,UQG(t)²H˙^s+ 2cν0

t 0

UQG(t)²H˙^s+1dtC(U0,QG), (1.2) whereν0= min(ν, ν).

The convergence theorem is the following one :

Theorem 3. — ([6]) Assume thatU0∈H˙¹(R³)∩H˙¹²(R³)andU0,QG∈ L²(R³). Let us define fors∈R,E˙^sdef

= L^∞(R+,H˙^s)∩L²(R+,H˙^s+1)and let Wε be a solution of the following linear system :

∂tWε−LWε+¹_εPAWε=−G

Wε/t=0=U0,osc=P(U0) (1.3)

with Gdef

= PP(UQG.∇UQG)−F(ν−ν)∆∆⁻_F²







−F ∂2∂₃² F ∂1∂₃²

0 (∂²₁+∂₂²)∂3





 ΩQG. (1.4) Then we have the following results :

• Wε exists globally and is unique in the spaceE˙^sfor every s∈[¹₂,1].

• MoreoverWεL²(R+,L^∞)→0 asε→0.

• If we denote by γε

def=Uε−UQG−Wε, then if ε is small enough, γε∈E˙^s and converges to zero in this spaceE˙^sfor every s∈[¹₂,1].

• Ifε is small enoughUε is defined for all time inE^s.

The aim of this paper is to get global existence results on the solutions of the primitive equations but with less initial regularity. The ˙H¹² regularity being the minimal regularity as we want to apply at least the Fujita and Kato theorem, we will require in this paperU0,QG∈H^12+η,U0,osc∈H˙ ¹² and we will interpolate between theorem 1 and theorem 3, using the arguments given by [14]. The key point is that we cut the initial data into two parts : the first part being regular enough to apply theorem 3, and the second one being ˙H¹² with small initial data, in order to apply theorem 1. We get the following result :

Theorem 4. — If the initial data U0 = U0,QG+U0,osc with U0,QG ∈ H^12+η,U0,osc ∈H˙ ¹² then, there existsε0>0such that for allεε0,(P Eε) has a unique global solution Uε∈L^∞(R+,H˙ ¹²)∩L²(R+,H˙ ³²).

This paper is devoted to the proof of theorem 4 and the structure will be the following : first we will use truncation in order to cut the initial data into two parts. At this point we show how important is the interpolation ar- gument from [14] to get adaptated energy estimates on the quasigeostrophic

system. We then prove this interpolation argument and manage to apply theorem 3. Finally we are able to adapt theorem 1 with small initial data, which concludes the proof.

2. Proof of Theorem 4 2.1. Frequency truncation of the initial data

We have seen that the above theorems give two different results concerning the lifespan of the strong solutions of the primitive equations :

• Theorem 1 requiresU0,QG ∈H˙ ¹²,U0,osc ∈H˙ ¹², and gives local exis- tence of strong solutions, with global lifespan and energy if the initial data are small enough (U0_H˙¹₂ cν0).

• Theorem 3 requiresU0,QG∈H¹,U0,osc∈H˙ ¹²∩H˙¹, and gives global strong solutions (and energy and convergence) when the Rossby num- berεis small enough.

So the idea is to cut our initial data into two parts : on the first one, which is regular (H¹) and whose norm is large, we will be able to apply Theorem 3, and on the second one, which is ˙H¹² with a small norm we will use Theorem 1. We then decompose the initial data in the following way (χ is a C^∞ truncation function : χ(x) ≡1 if x∈ [−1,1] and χ(x) ≡ 0 if |x| > ³₂ for example) :

U0=U0,QG+U0,osc=

χ(|D|

λ )U0,QG+U0,osc

+ (1−χ(|D|

λ ))U0,QG. So let us begin with the use of Theorem 3 on the first part (low frequen- cies for the quasigeostrophic part).

2.2. Study of the low frequencies 2.2.1. Global well-posedness

Let us defineU_ε^λ solution of the primitive equations :





∂tU_ε^λ+U_ε^λ.∇U_ε^λ−LU_ε^λ+¹_εAU_ε^λ=¹_ε(−∇Φ^λ_ε,0) divv_ε^λ= 0

U_ε/t=0^λ =χ(^|D_λ^|)U0,QG+U0,osc.

Theorem 3 gives then ε0 = ε0(λ, ...) > 0 such that ∀ε ε0, the unique solutionU_ε^λ globally exists. Precisely we defineU_QG^λ andW_ε^λ, solutions of the following systems (we refer to [6] for details concerning the convergence of the strong solutions and its proof) :







∂tU_QG^λ −ΓU_QG^λ +Q(U_QG^λ .∇U_QG^λ ) = 0 U_QG/t=0^λ =χ(^|D_λ^|)U0,QG,

(2.5)

and 





∂tW_ε^λ−LW_ε^λ+¹_εPAWε^λ=−G^λ W_{ε /t=0}^λ =U0,osc,

(2.6) whereG^λ=G^λ,b+G^λ,l,

with G^λ,b=PP(U_QG^λ .∇UQG^λ ),

and G^λ,l=−F(ν−ν)∆∆⁻_F²







−F ∂2∂3²

F ∂1∂₃² 0 (∂²₁+∂₂²)∂3





Ω^λ_QG.

(2.7)

Then Theorem 3 gives ε0=ε0(λ, ...)>0 such that ∀εε0, the difference of the solutions of the primitive and quasigeostrophic systems, to whom we substract rapid oscillations, γε =U_ε^λ−U_QG^λ −W_ε^λ, globally exists in ˙E¹² and goes to zero in this space.

The aim is to get estimates in ˙E¹² of and then use it in the system satisfied byVε^λ=Uε−Uε^λso we will outline the adaptation of the estimates given in [6] in the proof of Theorem 3.

2.2.2. Estimates for the limit system

First we have to estimate U_QG^λ in ˙H^s : let us recall that in the proof, the fact that the initial data is in ˙H¹² allows us to use the Fujita and Kato theorem, which gives a local lifespan [0, T^∗,λ[. The fact that, in ad- dition, χ(^|D_λ^|)U0,QG ∈H˙¹ allows us to use the regularity propagation the- orem that provides enough regularity to the potential vorticity which is in C([0, T^∗,λ[, L²)∩L²_loc([0, T^∗,λ[,H˙¹), and it is sufficient for us to define the scalar product inL² of Ω^λ_QGby the equation :

∂tΩ^λ_QG−ΓΩ^λ_QG+U_QG^λ .∇Ω^λQG= 0,

and for allt < T^∗ (ν0= min(ν, ν)>0) : ΩQG(t)^λ2L²+2cν0

t 0

∇Ω^λQG(τ)²L²dτ =Ω^λQG(0)²L² Cχ(|D|

λ )U0,QG²H˙¹. Finally, the fact thatχ(^|D_λ^|)U0,QG is inL² allows us to get the fact that it is a global weak solution, and the following Leray estimate ∀t 0, (ν0 = min(ν, ν)>0) :

U_QG^λ (t)²L²+ν0

t 0

∇U_QG^λ (τ)²L²dτ χ(|D|

λ )U0,QG²L².

Then from this, we easily contradict the usual blow-up criterion, and that implies that there is a unique, global solution and for everys∈[0,1] :

∀t∈R+,UQG^λ (t)²H˙^s+ν0

t 0

UQG^λ (τ)²H˙^s+1dτ Cχ(|D|

λ )U0,QG²H¹

(2.8) Cλ^12−ηU0,QG_H¹₂+η.

2.2.3. Estimates for Wε^λ andUε^λ

We refer to [6] (section 3.2) for the following estimates : W_ε^λ(t)²_H˙¹₂ +ν0

t 0

∇W_ε^λ(t)²_H˙¹₂e t

tG^λ,b(τ)

H˙ 1 2

dτdt

U0,osc²_˙

H¹2e t

0G^λ,b(τ)

H˙ 1 2

dτ+ t

0

(G^λ,b_H˙¹₂+1 ν0

G^l2_˙

H⁻¹2)e t

tG^λ,b(τ)

H˙ 1 2

dτdt,

and there is no change, in lemma 3.2 from [6], for the estimate onG^λ,l : _∞

0

G^l2_˙

H⁻¹2dtCUQG^λ _L2H˙³₂,

contrary to the estimate onG^λ,b, where we could use the argument from [6] : _∞

0

G^bH^˙sdtC









UQGL²(R+,H˙¹)UQGL²(R+,H˙²) if s=1 2 UQG²_L2(R+,H˙^s+1) if s∈]1

2,1].

But heres= ¹₂ and as we cannot afford to use much regularity, we prefer to do it differently, using product laws in Sobolev spaces :

_∞

0

G^b_H˙¹₂dtC _∞

0

UQG^λ .∇UQG^λ _H˙¹₂ C _∞

0

UQG^λ _H˙³₂−η.∇UQG^λ _H˙¹₂+η.

Then

G^b_L1H˙¹2 U_QG^λ _H˙³₂−η.U_QG^λ _H˙³₂+η

So, using the estimate from the previous section, we obtain that : Wε^λ_E˙¹₂ C(U0,osc_H˙¹₂ +λ^12−ηU0,QG_H¹₂+η).

Finally, as γ_ε^λ goes to zero, in particular ifε is small enough, its norm in E˙¹² is less than 1, so we obtain the estimate for U_ε^λ:

Uε^λ_E˙¹₂ C(U0,osc_H˙¹₂ +λ^12−ηU0,QG_H¹₂+η).

But in the following we will use the fact that λgoes to infinity in order to use the results of global well-posedness with small initial data. But in this case the previous estimates explode. We refer to the following section for an explaination of this problem.

2.3. Study of the high frequencies

In the previous section we used Theorem 3 to define Uε^λ. The Fujita and Kato theorem gives the existence ofUεinC([0, Tε^∗[,H˙ ¹²). Then we can define the differenceV_ε^λ=Uε−U_ε^λ, which satisfies :





∂tVε^λ+Vε^λ.∇Vε^λ+Vε^λ.∇Uε^λ+Uε^λ.∇Vε^λ−LVε^λ+¹_εAVε^λ=¹_ε(−∇Φ^λε,0) V_{ε /t=0}^λ = (1−χ(^|D_λ^|))U0,QG.

Then the classical schemes of the proofs of the Leray or Fujita-Kato the- orems can be adapted on this system to prove the existence of weak solu- tions, strong solutions, and global lifespan when the initial data are small.

We won’t give details in this section, we will only write energy estimates (in the proof, such estimates are proved for regularized approximated solutions and obtained as limits).

First, the inner product inL², yields : 1

2 d

dtVε^λ2L²+ν0∇Vε^λ2L²|(Vε^λ.∇Uε^λ|Vε^λ)L²|.

Then, classical Sobolev injections ( ˙H¹(R³) '→ L⁶(R³) and ˙H¹²(R³)) '→ L³(R³), imply :

1 2

d

dtVε^λ2L²+ν0∇Vε^λ2L² ν0

2|∇Vε^λ2L²+ C

ν0Vε^λ2L².∇Uε^λ_H˙¹₂.

Then a Gronwall estimate implies :

∀t0, Vε^λ(t)²L²+ν0

t 0

∇Vε^λ(τ)²L²dτ

(1−χ(|D|

λ ))U0,QG²L²e

2C ν0∇U_ε^λ2

L2 ˙H 1

2. (2.9)

And if we take the inner product in ˙H¹², we obtain : 1

2 d dtVε^λ2_˙

H¹2 +ν0∇Vε^λ2_˙

H¹2 |(Vε^λ.∇Vε^λ|Vε^λ)_˙

H¹2|

+|(Vε^λ.∇Uε^λ|Vε^λ)_H˙ ¹₂|+|(Uε^λ.∇Uε^λ|Vε^λ)_H˙ ¹₂|.

Using |(f|g)_H˙¹₂| CfL²gH˙¹, the fact that L³.L⁶ '→ L², and linear interpolation arguments, we get :

1 2

d dtVε^λ2_˙

H¹2 +ν0∇Vε^λ2_˙

H¹2 CVε^λ2H˙¹∇Vε^λ_H˙¹₂ +CVε^λ_H˙ ¹₂∇Vε^λ_H˙ ¹₂∇Uε^λ_H˙¹₂ +CUε^λ12_˙

H¹2∇Uε^λ12_˙

H¹2∇Vε^λ32_˙

H¹2. The classical inequalityab ^a_p^p+^b_q^q if ¹_p+¹_q = 1 gives :

1 2

d dtVε^λ2_˙

H¹2 +ν0∇Vε^λ2_˙

H¹2 CVε^λ_H˙ ¹₂∇Vε^λ2_˙

H¹2 +ν0

4 ∇Vε^λ2_˙

H¹2+ C

ν0V_ε^λ2_˙

H¹2∇U_ε^λ2_˙

H¹2 +ν0

4∇V_ε^λ2_˙

H¹2 + C ν₀³U_ε^λ2_˙

H¹2∇U_ε^λ2_˙

H¹2V_ε^λ2_˙

H¹2. And then,

d dtVε^λ2_˙

H¹2 +ν0∇Vε^λ2_˙

H¹2 2CVε^λ_H˙¹₂∇Vε^λ2_˙

H¹2+ Vε^λ2_˙

H¹2

2C ν0

∇Uε^λ2_˙

H¹2 +2C ν₀³Uε^λ2_˙

H¹2∇Uε^λ2_˙

H¹2

.

A Gronwall estimate finally gives that for allt∈[0, T_ε^∗[ : Vε^λ(t)²_˙

H¹2 + t

0

(ν0−2CVε^λ(t)_H˙¹₂)∇Vε^λ(t)²_˙

H¹2e t

tg(τ)dτ

dt

(1−χ(|D|

λ ))U0,QG²L²e t

0g(τ)dτ

, (2.10)

with g(t) =∇Uε^λ(t)²_˙

H¹2(2C ν0

+2C ν₀³Uε^λ2_˙

H¹2).

In both cases, we obtain a majoration by(1−χ(^|D_λ^|))U0,QGe^UελE˙12, that is

(1−χ(|D|

λ ))U0,QGe^{λ1−2ηC(U0,oscH˙12,U0,QGH12+η)}

which does not go to zero when λ goes to infinity which is annoying as we want to adapt a theorem with small initial data. So as everything de- pends on the estimate of U_QG^λ , we will provide, in the following, an esti- mate whose right-hand member depends onχ(^|D_λ^|))U0,QG_H¹₂+ηinstead of χ(^|D_λ^|))U0,QGH¹.

2.4. Real interpolation

The aim of this section is to prove the following result :

Lemma 2.1. — Let η > 0, U0,QG ∈ H¹, and UQG the unique global solution (we refer to [6]) of the following quasigeostrophic system :





∂tUQG−ΓUQG+Q(UQG.∇UQG) = 0 U_QG/t=0=U0,QG.

There exists a constant C such that, for allt0, UQG(t)²

H¹2+η+ν0

t 0

∇UQG(τ)²

H¹2+ηdτ CU0,QG^2+1η

H¹2+η.

With this estimate, we will be able to use the results from the previous section on V_ε^λ and prove the global well-posedness as λis large enough to ensure small initial data.

To prove this lemma, we will use the same Calderon method (see [4]) as Gallagher and Planchon in [14] : thanks to interpolation arguments we will be able to control the norm of a part of the initial data and make it as small as we want, so that we can use the Fujita-Kato theorem.

2.4.1. General interpolation results

In this section we will recall classical real interpolation definitions (we refer for example to [3] for a presentation) and present it in the same way as in [14] together with a useful lemma from this paper :

Definition 2.2. — LetE1andE2 two Banach spaces. The interpolated spaceE= [E1, E2]θ,q withθ∈[0,1]andq1 is defined by :

E= [E1, E2]θ,q ={f ∈E1+E2 such that fE <∞}, with

fE=





j∈Z

2^jqθK(f, j)^q





1 q

,

and

K(f, j) =^def inf

f₁+f₂=f(f1E1+ 2^−jf2E2) (fi∈Ei).

The following lemma concerns the case when E2 '→ E '→ E1. In the following, we will takeE1=H¹²,E2=H¹ andE=H^12+η.

Lemma 2.3. — There exists a constantC(θ, q)such that for any integer j01 and any functionf ∈E, the following equivalence holds :





jj₀

2^jqθK(f, j)^q





1 q

fEC(θ, q)2^j0





jj₀

2^jqθK(f, j)^q





1 q

.

We refer to [14] for the proof of this result (section 4.4).

2.4.2. Decomposition and small data

As said in section 2.2.2, we have different results on the quasigeostrophic system :

• U0,QG ∈ L² leads to a global weak solution and a global energy estimate inL².

• U0,QG ∈ H˙ ¹² leads to a local unique strong solution, global if U0,QGH˙¹2 cν0 (see Theorem 1) with global energy estimate in H˙ ¹².

• U0,QG∈H¹leads to a global strong solutions together with a global energy estimate inH¹.

The aim is to decompose the initial data inH^12+η = [H¹², H¹]2η,2 and solve separatedly a quasigeostrophic system with small data inH¹² (there- fore it is small in ˙H¹²), and a modified quasigeostrophic system with initial data inH¹.

For everyj ∈Zlet us decompose

U0,QG =U_0,QG^1,j +U_0,QG^2,j , with U_0,QG^1,j ∈E1=H¹², U_0,QG^2,j ∈E2=H¹, (actually, asU0,QG isH¹, so doesU_0,QG^1,j ), and by definition ofK(U0,QG, j) as an infimum :

U_0,QG^1,j _H¹₂ + 2^−jU_0,QG^2,j H¹ 3

2K(U0,QG, j).

As said earlier, we want to define the corresponding solutions,U_QG^1,j(t) and UQG(t)^2,j, respectedly given by the Fujita-Kato theorem, or Theorem 1.2.

The problem is that we have no information on the smallness ofU_0,QG^1,j _H˙ ¹₂. But, like in [14], using Lemma 2.3, we can write, that, for everyj 1 (withθ= 2η andq= 2) :

U0,QG_H¹₂+η 2^2jηK(U0,QG, j)2^2jη2 3

U_0,QG^1,j _H¹₂ + 2^−jU_0,QG^2,j H¹

.

In particular,

U0,QG^1,j _H˙¹₂ U0,QG^1,j _H¹₂ 3

22^−2jηU0,QG_H¹₂+η.

So ifj0is defined such that (cbeing the constant given by the global lifespan for small initial data result) :

3

22^−2j0ηU0,QG_H¹₂+η = cν0

U0,QG_H¹₂+η

, (2.11)

i.-e.

j0=E(− 1 2η

log_3U ^2cν0

0,QG

H1 2+η

log 2 ) + 1,

and, for all j j0, U_0,QG^1,j _H˙¹₂ cν0. This allows us to apply the Fujita- Kato theorem with small initial data and defineU_QG^1,j, solution of :







∂tU_QG^1,j −ΓU_QG^1,j +Q(UQG^1,j.∇UQG^1,j) = 0 U_QG/t=0^1,j =U_0,QG^1,j ,

(2.12)