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

HAL Id: hal-00793679

https://hal.archives-ouvertes.fr/hal-00793679

Submitted on 25 Feb 2013

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

less regular initial data

Frederic Charve

To cite this version:

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

Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc 2008, XVII (2), pp.221-238. �hal-00793679�

regular initial data

Fr´ed´eric Charve,^∗

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 ε est suffisamment petit, et pour des donn´ees plus r´eguli`eres (partie oscillante initiale dans H˙ ¹² ∩H˙¹ et partie quasig´eostrophique initiale dans H¹).

Classification AMS: 35Q35, 76D05, 76U05, 46M35

Mots cl´es: Equations primitives, syst`eme quasig´eostrophique, dispersion, estimations de Strichartz, interpolation r´eelle.

Abstract: This paper is devoted to the study of the lifespan of the solutions of the primitive equations for less regular initial data. We interpolate 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 parameter εis small enough, and for regular initial data (oscillating part in ˙H¹² ∩H˙¹ and quasigeostrophic part in H¹).

AMS Classification: 35Q35, 76D05, 76U05, 46M35

Key words: Primitive equations, quasigeostrophic system, dispersion, Strichartz esti- mates, real interpolation.

1 Introduction

1.1 The primitive equations The primitive system writes:

(P E_ε)









∂_tU_ε+U_ε.∇U_ε−LU_ε+ 1

εAU_ε= 1

ε(−∇Φ_ε,0) div v_ε = 0

U_ε/t=0 =U_0,ε.

The unknowns are U_ε and Φ_ε. We denote by U_ε a pair (v_ε, θ_ε) where v_ε is a vector field on R³ (three dimensional velocity), θ_ε 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 Φ_ε the pressure, all of them depending on (t, x). The operator L is defined by

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

[email protected]

1

LU_εdef

= (ν∆v_ε, ν^′∆θ_ε), We define :

U_ε.∇U_ε=v_ε.∇U_ε= X3

i=1

v_εⁱ.∂_iU_ε,

and the matrixA by:

Adef

=







0 −1 0 0

1 0 0 0

0 0 0 F⁻¹

0 0 −F⁻¹ 0





.

The small parameter εis called here the Rossby number and F 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 s is a real number, the homogenous (resp. inhomogenous) Sobolev space of order s, which we will denote by H˙^s (resp. H^s), is defined as the space of tempered distributionsu∈ S^′(R³)whose Fourier transformuˆis locally integrable and has the following property:

kuk²H˙^s

def= Z

R³|ξ|^2s|u(ξ)ˆ |²dξ <∞ (resp. kuk²H^s def

= Z

R³

(1 +|ξ|²)^s|u(ξ)ˆ |²dξ <∞).

Although the primitive equations have no scaling anymore, we can easily 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 following results :

Theorem 1.1 (Leray, 1934, [15]) if the initial data U₀ ∈ L²(R³), then there exists for all ε > 0 a Leray solution of the system (P E_ε), U_ε, globally defined in time, belonging to L^∞(R+, L²(R³))∩L²(R+,H˙¹(R³)) and satisfying the following energy inequality (let ν₀ = min(ν, ν^′)>0):

∀t∈R+, kU_ε(t)k²L²(R³)+ 2ν₀ Z t

0 k∇U_ε(t)k²L²(R³)dt≤ kU₀k²L²(R³).

We refer to [5] where we studied the limit of Leray solutions when ε, the Rossby num- ber, 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 ∂₃θ_ε.

Then from this, we define the orthogonal decomposition of U_ε into its quasigeostrophic part, and its oscillating part :

U_ε,QG def

=







−∂₂∆_F⁻¹Ω_ε

∂₁∆_F⁻¹Ω_ε 0

−F ∂₃∆_F⁻¹Ω_ε





, with ∆_F def

= ∂₁²+∂₂²+F²∂₃³, and :

U_ε,oscdef

= U_ε−U_ε,QG =







v_ε¹+∂₂∆_F⁻¹Ω_ε v_ε²−∂₁∆_F⁻¹Ω_ε

v³_ε

θ_ε+F ∂₃∆_F⁻¹Ω_ε





.

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 projectorPon 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, when ε goes to zero, of the weak Leray solutions towards the quasigeostrophic model (see (1) below).

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

Theorem 1 (Fujita and Kato, 1964, [12]) IfU₀∈H˙¹² there exist a unique maximal time T_ε^∗>0, and a unique solution

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

Moreover, ifT_ε^∗ is finite, then we have Z _Tε^∗

0 kU_ε(t)k²_˙

H³²(R³)dt= +∞.

Finally there exists a constantc such that ifkU₀k_H˙¹_2(R3)≤cν₀ thenT_ε^∗ = +∞.

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 6= 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] IfU_0,QG∈H¹(R³)then the quasigeostrophic system (∂_tU_QG−ΓU_QG+Q(U_QG.∇U_QG) = 0

U_QG/t=0 =U_0,QG, (1)

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

∀t∈R+,kUg_QG(t)k²H˙^s + 2cν₀ Z t

0 kUg_QG(t^′)k²H˙^s+1dt^′ ≤C(U_0,QG), (2) whereν₀ = min(ν, ν^′)

The convergence theorem is the following one:

Theorem 3 ([6]) Assume that U₀ ∈ H˙¹(R³)∩H˙¹²(R³) and U_0,QG ∈ L²(R³). Let us define for s ∈ R, E˙^s def

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





∂_tW_ε−LW_ε+1

εPAW_ε=−G W_ε/t=0 =U_0,osc=P(U₀)

(3)

with Gdef

= PP(Ug_QG.∇Ug_QG)−F(ν−ν^′)∆∆⁻²_F







−F ∂₂∂₃² F ∂₁∂₃²

0 (∂₁²+∂₂²)∂₃





Ωg_QG. (4) Then we have the following results:

• W_ε exists globally and is unique in the space E˙^s for everys∈[¹₂,1].

• Moreover kW_εkL²(R+,L^∞)→0as ε→0.

• If we denote by γ_ε def

= U_ε−Ug_QG−W_ε, then if ε is small enough, γ_ε ∈ E˙^s and converges to zero in this space E˙^s for everys∈[¹₂,1].

• Finally ifεis small enoughU_ε is defined for all time inE^s andU_ε−Ug_QG=γ_ε+W_ε goes to zero as εgoes to zero, in the spaceL²(R+, L^∞).

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 paper U_0,QG ∈ H^12+η, U_0,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 H˙ ¹² with small initial data, in order to apply theorem 1. We get the following result : Theorem 4 If the initial data U₀ = U_0,QG+U_0,osc with U_0,QG ∈ H^12+η, U_0,osc ∈ H˙ ¹² then, there exists ε₀ > 0 such that for all ε ≤ ε₀, (P E_ε) has a unique global solution U_ε∈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 argument 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 requiresU_0,QG∈H˙ ¹²,U_0,osc∈H˙¹², and gives local existence of strong so- lutions, with global lifespan and energy if the initial data are small enough (kU₀k_H˙¹₂ ≤ cν₀).

• Theorem 3 requiresU_0,QG∈H¹,U_0,osc∈H˙¹² ∩H˙¹, and gives global strong solutions (and energy and convergence) when the Rossby numberε is 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)≡1 if |x|> ³₂ for example):

U₀=U_0,QG+U_0,osc=

χ(|D|

λ )U_0,QG+U_0,osc

+ (1−χ(|D|

λ ))U_0,QG.

So let us begin with the use of Theorem 3 on the first part (low frequencies for the quasigeostrophic part).

2.2 Study of the low frequencies 2.2.1 Global well-posedness

Let us define U_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 )U_0,QG+U_0,osc.

Theorem 3 gives thenε₀ =ε₀(λ, ...) >0 such that∀ε≤ε₀, the unique solutionU_ε^λglobally exists. Precisely we define U_QG^λ and W_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 )U_0,QG, (5)

and (

∂_tW_ε^λ−LW_ε^λ+¹_εPAW_ε^λ=−G^λ

W_{ε /t=0}^λ =U_0,osc, (6)

whereG^λ =G^λ,b+G^λ,l,

with G^λ,b =PP(U_QG^λ .∇U_QG^λ ), and G^λ,l=−F(ν−ν^′)∆∆⁻²_F







−F ∂₂∂₃² F ∂₁∂₃²

0 (∂₁²+∂₂²)∂3





Ω^λ_QG. (7) Then Theorem 3 givesε₀=ε₀(λ, ...)>0 such that∀ε≤ε₀, the difference of the solutions of the primitive and quasigeostrophic systems, to whom we substract rapid oscillations, γ_ε =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 by V_ε^λ = U_ε −U_ε^λ so 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 estimateU_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 addition, χ(^|D|_λ )U_0,QG ∈ H˙¹ allows us to use the regularity propagation theorem 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_QG by the equation :

∂_tΩ^λ_QG−ΓΩ^λ_QG+U_QG^λ .∇Ω^λ_QG= 0, and for all t < T^∗ (ν₀ = min(ν, ν^′)>0):

kΩ_QG(t)^λk²_L²+ 2cν₀ Z t

0 k∇Ω^λ_QG(τ)k²_L²dτ =kΩ^λ_QG(0)k²_L² ≤C^′kχ(|D|

λ )U_0,QGk²H˙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) :

kU_QG^λ (t)k²L²+ν₀ Z t

0 k∇U_QG^λ (τ)k²L²dτ ≤ kχ(|D|

λ )U_0,QGk²L².

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

∀t∈R+,kU_QG^λ (t)k²H˙^s+ν₀ Z t

0 kU_QG^λ (τ)k²H˙^s+1dτ ≤Ckχ(|D|

λ )U_0,QGk²_H¹ (8)

≤Cλ^12−ηkU_0,QGk_H¹₂+η.

2.2.3 Estimates for W_ε^λ and U_ε^λ

We refer to [6] (section 3.2) for the following estimates : kW_ε^λ(t)k²_˙

H¹² +ν₀ Z t

0 k∇W_ε^λ(t^′)k²_˙

H¹²e

Rt

t′kG^λ,b(τ)k

H˙ 1 2dτ

dt^′ ≤ kU_0,osck²_˙

H¹²e

Rt

0kG^λ,b(τ)k

H˙ 1 2dτ

+ Z t

0

(kG^λ,bk_H˙¹₂ + 1 ν₀kG^lk²_˙

H⁻¹²)e

Rt

t′kG^λ,b(τ)k

H˙ 1 2dτ

dt^′,

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

0 kG^lk²_˙

H⁻¹²dt≤CkU_QG^λ k_L2H˙³²,

contrary to the estimate on G^λ,b, where we could use the argument from [6] : Z ∞

0 kG^bkH˙^sdt≤C





kUg_QGk_L²_(R+,H˙¹)kUg_QGk_L²_(R+,H˙²) if s= ¹₂ kUg_QGk²_L²_(R+,H˙^s+1) if s∈]¹₂,1].

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

Z ∞

0 kG^bk_H˙¹₂dt≤C Z ∞

0 kU_QG^λ .∇U_QG^λ k_H˙¹₂ ≤C Z ∞

0 kU_QG^λ k_H˙³₂⁻η.k∇U_QG^λ k_H˙¹₂+η. Then

kG^bk_L1H˙¹² ≤ kU_QG^λ k_H˙³₂^−η.kU_QG^λ k_H˙³₂^+η So, using the estimate from the previous section, we obtain that :

kW_ε^λk_E˙¹₂ ≤C(kU_0,osck_H˙¹₂ +λ^12−ηkU_0,QGk_H¹₂+η).

Finally, as γ_ε^λ goes to zero, in particular ifε is small enough, its norm in ˙E¹² is less than 1, so we obtain the estimate forU_ε^λ :

kU_ε^λk_E˙¹₂ ≤C(kU_0,osck_H˙¹₂ +λ^12−ηkU_0,QGk_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 U_ε^λ. The Fujita and Kato theorem gives the existence ofU_εinC([0, T_ε^∗[,H˙ ¹²). Then we can define the differenceV_ε^λ =U_ε−U_ε^λ, which satisfies :





∂_tV_ε^λ+V_ε^λ.∇V_ε^λ+V_ε^λ.∇U_ε^λ+U_ε^λ.∇V_ε^λ−LV_ε^λ+1

εAV_ε^λ = 1

ε(−∇Φ^λ_ε,0) V_{ε /t=0}^λ = (1−χ(^|D|_λ ))U_0,QG.

Then the classical schemes of the proofs of the Leray or Fujita-Kato theorems can be adapted on this system to prove the existence of weak solutions, 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 in L², yields:

1 2

d

dtkV_ε^λk²L² +ν₀k∇V_ε^λk²L² ≤ |(V_ε^λ.∇U_ε^λ|V_ε^λ)_L²|.

Then, classical Sobolev injections ( ˙H¹(R³)֒→L⁶(R³) and ˙H¹²(R³))֒→L³(R³), imply : 1

2 d

dtkV_ε^λk²_L² +ν₀k∇V_ε^λk²_L² ≤ ν₀

2|∇V_ε^λk²_L² + C

ν₀kV_ε^λk²_L².k∇U_ε^λk_H˙¹₂. Then a Gronwall estimate implies :

∀t≥0, kV_ε^λ(t)k²L² +ν₀ Z t

0 k∇V_ε^λ(τ)k²L²dτ ≤ k(1−χ(|D|

λ ))U_0,QGk²L²e

2C ν0k∇U_ε^λk²

L2 ˙H 1 2. (9) And if we take the inner product in ˙H¹², we obtain :

1 2

d

dtkV_ε^λk²_˙

H¹² +ν₀k∇V_ε^λk²_˙

H¹² ≤ |(V_ε^λ.∇V_ε^λ|V_ε^λ)_˙

H¹²|+|(V_ε^λ.∇U_ε^λ|V_ε^λ)_˙

H¹²| +|(U_ε^λ.∇U_ε^λ|V_ε^λ)_˙

H¹²|. Using |(f|g)_˙

H¹²| ≤ CkfkL²kgkH˙¹, the fact that L³.L⁶ ֒→ L², and linear interpolation arguments, we get :

1 2

d

dtkV_ε^λk²_˙

H¹² +ν₀k∇V_ε^λk²_˙

H¹² ≤CkV_ε^λk²H˙¹k∇V_ε^λk_H˙¹₂ +CkV_ε^λk_H˙¹₂k∇V_ε^λk_H˙¹₂k∇U_ε^λk_H˙¹₂ +CkU_ε^λk

1 2

H˙¹²k∇U_ε^λk

1 2

H˙¹²k∇V_ε^λk

3 2

H˙¹². The classical inequalityab≤ ^a_p^p + ^b_q^q if ¹_p +¹_q = 1 gives :

1 2

d

dtkV_ε^λk²_˙

H¹² +ν₀k∇V_ε^λk²_˙

H¹² ≤CkV_ε^λk_H˙¹₂k∇V_ε^λk²_˙

H¹² +ν₀

4 k∇V_ε^λk²_˙

H¹²+ C

ν₀kV_ε^λk²_˙

H¹²k∇U_ε^λk²_˙

H¹² +ν₀

4k∇V_ε^λk²_˙

H¹² + C

ν₀³kU_ε^λk²_˙

H¹²k∇U_ε^λk²_˙

H¹²kV_ε^λk²_˙

H¹². And then,

d

dtkV_ε^λk²_˙

H¹² +ν₀k∇V_ε^λk²_˙

H¹² ≤2CkV_ε^λk_H˙¹₂k∇V_ε^λk²_˙

H¹²+ kV_ε^λk²_˙

H¹²

2C

ν₀ k∇U_ε^λk²_˙

H¹² + 2C ν₀³kU_ε^λk²_˙

H¹²k∇U_ε^λk²_˙

H¹²

.

A Gronwall estimate finally gives that for allt∈[0, T_ε^∗[ : kV_ε^λ(t)k²_˙

H¹² + Z t

0

(ν0−2CkV_ε^λ(t^′)k_H˙¹₂)k∇V_ε^λ(t^′)k²_˙

H¹²e^{Rtt′g(τ)dτ}dt^′ ≤ k(1−χ(|D|

λ ))U_0,QGk²_L²e^R0tg(τ)dτ, (10) with g(t) =k∇U_ε^λ(t)k²_˙

H¹²(2C ν₀ +2C

ν₀³kU_ε^λk²_˙

H¹²).

In both cases, we obtain a majoration byk(1−χ(^|D|_λ ))U_0,QGke^kU

ελk

E˙ 1

2, that is k(1−χ(|D|

λ ))U_0,QGke^λ

1−2ηC(kU0,osck

H˙ 1

2,kU0,QGk

H 1 2+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 depends on the estimate of U_QG^λ , we will provide, in the following, an estimate whose right-hand member depends on kχ(^|D|_λ ))U_0,QGk_H¹₂+η instead ofkχ(^|D|_λ ))U_0,QGkH¹.

2.4 Real interpolation

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

Lemma 2.1 Letη >0,U_0,QG∈H¹, andU_QG the unique global solution (we refer to [6]) of the following quasigeostrophic system:

(∂_tU_QG−ΓU_QG+Q(U_QG.∇U_QG) = 0 U_QG/t=0 =U_0,QG.

There exists a constantC such that, for all t≥0, kU_QG(t)k²

H^12+η+ν₀ Z t

0 k∇U_QG(τ)k²

H^12+ηdτ ≤CkU_0,QGk²⁺

1 η

H^12+η.

With this estimate, we will be able to use the results from the previous section onV_ε^λ and prove the global well-posedness asλis 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 2.1 LetE₁andE₂two Banach spaces. The interpolated spaceE= [E₁, E₂]_θ,q withθ∈[0,1] andq ≥1 is defined by:

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

kfk^E =



X

j∈Z

2^jqθK(f, j)^q





1 q

,

and

K(f, j) =^def inf

f1+f2=f(kf₁k^E1+ 2^−jkf₂k^E2) (fi ∈E_i).

The following lemma concerns the case whenE₂ ֒→E ֒→E₁. In the following, we will takeE₁=H¹²,E₂ =H¹ andE =H^12+η.

Lemma 2.2 There exists a constant C(θ, q) such that for any integer j₀ ≥ 1 and any functionf ∈E, the following equivalence holds:



X

j≥j0

2^jqθK(f, j)^q





1 q

≤ kfkE ≤C(θ, q)2^j0



X

j≥j0

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:

• U_0,QG∈L² leads to a global weak solution and a global energy estimate inL².

• U_0,QG∈H˙ ¹² leads to a local unique strong solution, global ifkU_0,QGk_H˙¹₂ ≤cν₀ (see Theorem 1) with global energy estimate in ˙H¹².

• U_0,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 in H¹² (therefore it is small in ˙H¹²), and a modified quasigeostrophic system with initial data inH¹.

For every j∈Z let us decompose

U_0,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, asU_0,QGisH¹, so doesU_0,QG^1,j ), and by definition ofK(U_0,QG, j) as an infimum : kU_0,QG^1,j k_H¹₂ + 2^−jkU_0,QG^2,j kH¹ ≤ 3

2K(U_0,QG, j).

As said earlier, we want to define the corresponding solutions, U_QG^1,j(t) and U_QG(t)^2,j, respectedly given by the Fujita-Kato theorem, or Theorem 2. The problem is that we have no information on the smallness of kU_0,QG^1,j k_H˙¹₂.

But, like in [14], using Lemma 2.2, we can write, that, for every j ≥ 1 (with θ= 2η and q= 2) :

kU_0,QGk_H¹₂^+η ≥2^2jηK(U0,QG, j)≥2^2jη2 3

kU_0,QG^1,j k_H¹₂ + 2^−jkU_0,QG^2,j kH¹

.

In particular,

kU_0,QG^1,j k_H˙¹₂ ≤ kU_0,QG^1,j k_H¹₂ ≤ 3

22^−2jηkU_0,QGk_H¹₂+η.

So if j₀ is defined such that (c being the constant given by the global lifespan for small initial data result) :

3

22^−2j0ηkU_0,QGk_H¹2+η = cν₀

kU_0,QGk_H¹₂^+η, (11)

i.-e.

j₀=E(− 1 2η

log_3kU ^2cν0

0,QGk

H 1 2+η

log 2 ) + 1,

and, for all j ≥ j₀, kU_0,QG^1,j k_H˙¹₂ ≤cν₀. This allows us to apply the Fujita-Kato theorem with small initial data and define U_QG^1,j, solution of :

(∂_tU_QG^1,j −ΓU_QG^1,j +Q(U_QG^1,j.∇U_QG^1,j) = 0

U_QG/t=0^1,j =U_0,QG^1,j , (12)

with the global energy estimate :

∀t∈R+,kU_QG^1,j(t)k²_˙

H¹² +ν₀ Z t

0 k∇U_QG^1,j(τ)k²_˙

H¹²dτ ≤CkU_0,QG^1,j k²_˙

H¹². (13) On the other hand, as U_0,QG^1,j ∈ H¹² ֒→ L² the Leray Theorem says that U_QG^1,j is also a global weak solution, together with the associated energy estimate (in L²), so we finally obtain :

∀t∈R+,kU_QG^1,j(t)k²

H¹² +ν₀ Z t

0 k∇U_QG^1,j(τ)k²

H¹²dτ ≤CkU_0,QG^1,j k²

H¹² ≤Cν₀². (14) Remark 2.1 AsU_0,QG^1,j =U_0,QG−U_0,QG^2,j , it is in fact inH˙¹ so by the regularity propaga- tion theorem, the solution is more regular and the estimate 14 is in fact inH¹.

We now defineU_QG^2,j(t) =U_QG(t)−U_QG^1,j(t),U_QG(t) globally given by Theorem 2, which satisfies the following system :

(∂_tU_QG^2,j −ΓU_QG^2,j +Q(U_QG^2,j.∇U_QG^2,j) +Q(U_QG^2,j.∇U_QG^1,j) +Q(U_QG^1,j.∇U_QG^2,j) = 0

U_QG/t=0^2,j =U_0,QG^2,j . (15)

Its potential vorticity satisfying :

∂_tΩ^2,j_QG−ΓΩ^2,j_QG+U_QG^2,j.∇Ω^2,j_QG+U_QG^2,j.∇Ω^1,j_QG+U_QG^1,j.∇Ω^2,j_QG= 0. (16) We aim to adapt Theorem 1 and get global estimates : if we take the inner product in L² of (15) by U_QG^2,j :

1 2

d

dtkU_QG^2,jk²L² +ν₀k∇U_QG^2,jk²L² ≤ |(U_QG^2,j.∇U_QG^1,j|U_QG^2,j)_L² ≤ kU_QG^2,j.∇U_QG^1,jkL²kU_QG^2,jkL². Using the usual product law L³.L⁶֒→L², we get :

1 2

d

dtkU_QG^2,jk²L² +ν₀k∇U_QG^2,jk²L² ≤ kU_QG^2,jkH˙¹k∇U_QG^1,jk_H˙¹₂kU_QG^2,jkL². The same H¨older estimate as above gives :

1 2

d

dtkU_QG^2,jk²_L² +ν₀k∇U_QG^2,jk²_L² ≤ ν₀

2k∇U_QG^2,jk²_L²+ C

ν₀k∇U_QG^1,jk_H˙¹₂kU_QG^2,jkL²,

and then, classical Gronwall estimates give for all t≥0 : kU_QG^2,j(t)k²L² +ν₀

Z t

0 k∇U_QG^2,j(τ)k²L²dτ ≤CkU_0,QG^2,j k²L²e

2C

ν0k∇U_QG^1,jk²

L2 ˙H 1 2.

On the other hand, as in section 2.2.2, the inner product inL²of (16) by Ω^2,j_QG yields (with the same methods as above) :

kU_QG^2,j(t)k²H˙¹+ν₀ Z t

0 k∇U_QG^2,j(τ)k²H˙¹dτ ≤CkU_0,QG^2,j k²H˙¹e

2C

ν0k∇U_QG^1,jk²

L2 ˙H 1 2.

Collecting these last two estimates, and using (14) we obtain (the new constantCcontains ν₀.) :

kU_QG^2,j(t)k²_H¹+ν₀ Z t

0 k∇U_QG^2,j(τ)k²_H¹dτ ≤CkU_0,QG^2,j k²_H¹. (17) 2.4.3 Application of Lemma 2.2

Since we have decomposed the initial data, we want to use Lemma 2.2 onU_QG^1,j and U_QG^2,j in order to estimateU_QG.

According to section(2.4.1) and to the definition K : kU_0,QGk²

H^12+η ≥ X

j≥j0

2^4jηK(U_0,QG, j, H¹², H¹)²,

After the definition ofU_QG^1,j and U_QG^2,j we can write that (j₀ fixed in the previous section) : kU_0,QGk²

H^12+η ≥ 2 3

X

j≥j0

2^4jη

kU_0,QG^1,j k_H¹₂ + 2^−jkU_0,QG^2,j kH¹

2

. (18)

According to the estimates (14) and (17) we have for allt≥0 : (kU_0,QG^1,j k_H¹₂ ≥CkU_QG^1,j(t)k_H¹₂

kU_0,QG^2,j kH¹ ≥CkU_QG^2,j(t)kH¹, (19)

and 





kU_0,QG^1,j k_H¹₂ ≥C√ν₀kU_QG^1,jk_L2 tH³²

kU_0,QG^2,j kH¹ ≥C√ν₀kU_QG^2,jkL²_tH¹. (20) So if we use (19) into (18), we obtain that :

kU_0,QGk²

H^12+η ≥CX

j≥j0

2^4jη

kU_QG^1,j(t)k_H¹₂ + 2^−jkU_QG^2,j(t)kH¹

2

.

And, usingU_QG(t) =U_QG^1,j(t) +U_QG^2,j(t) and the definition ofK as an infimum : kU_0,QGk²

H^12+η ≥CX

j≥j0

2^4jηK

U_QG(t), j, H¹², H¹2

.

Using Lemma 2.2 allows us to go back toU_QG(t) : kU_0,QGk²

H^12+η ≥C(2^−j0kU_QG(t)k_H¹₂+η)².