Rapha Ël Danchin

Marius Paicu

GLOBAL EXISTENCE RESULTS FOR THE ANISOTROPIC BOUSSINESQ SYSTEM IN DIMENSION TWO

We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions à la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.

Introduction

The Boussinesq system describes the influence of the convection (or convection-diffusion) phenomenon in a viscous or inviscid fluid. It is used as a toy model for geophysical fluids whenever rotation and stratification play an important role (see for example J. Pedlosky's book [START_REF] Pedlosky | Geophysical fluid dynamics[END_REF]). In the two-dimensional case, the Boussinesq system reads:

(B κ,ν )      ∂ t θ + u • ∇θ -κ∆θ = 0 ∂ t u + u • ∇u -ν∆u + ∇Π = θ
e 2 with e 2 = (0, 1), div u = 0.

Above, u = u(t, x) denotes the velocity vector-field and θ = θ(t, x) is a scalar quantity such as the concentration of a chemical substance or the temperature variation in a gravity field, in which case θ e 2 represents the buoyancy force. The nonnegative parameters κ and ν denote respectively the molecular diffusion and the viscosity. In order to simplify the presentation, we restrict ourselves to the whole plan case (that is the space variable x describes the whole R 2 ) and focus on the evolution for positive times (i.e. t ∈ R + ).

In the case where both κ and ν are positive, classical methods allow to establish the global existence of regular solutions (see for example [START_REF] Cannon | The initial value problem for the Boussinesq equations with data in L p[END_REF][START_REF] Guo | Spectral method for solving two-dimensional Newton-Boussineq equation[END_REF]). On the other hand, if κ = ν = 0 then constructing global unique solutions for some nonconstant θ 0 is a challenging open problem (even in the two-dimensional case) which has many similarities with the global existence problem for the three-dimensional incompressible Euler equations.

The intermediate situation where the diffusion acts only on one of the equations has been investigated in a number of recent papers. Under various regularity assumptions on the initial data, it has been shown that for arbitrarily large initial data, systems (B κ,0 ) with κ > 0 and (B 0,ν ) with ν > 0 admit a global unique solution (see for example [START_REF] Abidi | On the global well-posedness for Boussinesq system[END_REF][START_REF] Chae | Global regularity for the 2 -D Boussinesq equations with partial viscous terms[END_REF][START_REF] Danchin | Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux[END_REF][START_REF] Danchin | Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces[END_REF][START_REF] Danchin | Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data[END_REF][START_REF] Hmidi | On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity[END_REF][START_REF] Hmidi | On the global well-posedness of the Boussinesq system with zero viscosity[END_REF]).

In the present paper, we aim at making one more step toward the study of the system with κ = ν = 0 by assuming that the diffusion or the viscosity occurs in the horizontal direction and in one of the equations only. More precisely, we want to consider the following two systems:

(1) 2 1 u + ∇Π = θ e 2 div u = 0 Date: September 29, 2008. 1 Université Paris-Est, Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail: danchin @ univ-paris12.fr.

     ∂ t θ + u • ∇θ = 0 ∂ t u + u • ∇u -ν∂
2 Université Paris 11, Laboratoire de Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France. E-mail: marius.paicu @ math.u-psud.fr.

and

(2)

     ∂ t θ + u • ∇θ -κ∂ 2 1 θ = 0 ∂ t u + u • ∇u + ∇Π = θe 2 div u = 0.
Let us stress that the anisotropic viscosity or diffusion assumptions are consistent with the study of geophysical fluids. It turns out that, in certain regimes and after suitable rescaling, the vertical viscosity (or diffusion) is negligible with respect to the horizontal viscosity (or diffusion) (for more details, one may refer to [START_REF] Chemin | Mathematical Geophysics: An introduction to rotating fluids and to the Navier-Stokes equations[END_REF]). For the standard three-dimensional incompressible Navier-Stokes equations, the first mathematical results concerning anisotropic viscosity have been obtained in [START_REF] Chemin | Fluids with anisotropic viscosity[END_REF][START_REF] Paicu | Équation anisotrope de Navier-Stokes dans des espaces critiques[END_REF].

On the one hand, it is clear that small variations over the classical methods for solving quasilinear hyperbolic systems would give local well-posedness for Systems [START_REF] Abidi | On the global well-posedness for Boussinesq system[END_REF] and [START_REF] Aubin | Un théorème de compacité[END_REF] with initial data in Sobolev spaces with suitably large index. On the other hand, since diffusion occurs in only one direction and one equation, it is not obvious that those solutions are actually global. The present paper is dedicated to the study of global existence for the initial value problem associated to Systems (1) and ( 2) with (possibly) large initial data.

In order to state our main result pertaining to System (1), let us introduce the set √ L of those functions f which belong to every space L p with 2 ≤ p < ∞ and satisfy

(3) f √ L := sup p≥2 f L p √ p-1 < ∞. Theorem 1. Let s ∈]1/2, 1]
. For all function θ 0 ∈ H s ∩ L ∞ and divergence free vector-field

u 0 ∈ H 1 with vorticity ω 0 := ∂ 1 v 2 0 -∂ 2 v 1 0 in √ L, System (1)
with data (θ 0 , u 0 ) admits a unique global solution (θ, u) such that 1 θ ∈ C w (R + ; L ∞ ) ∩ C(R + ; H s-ε ) for all ε > 0 and

u ∈ C w (R + ; H 1 ), ω ∈ L ∞ loc (R + ; √ L) and ∇u ∈ L 2 loc (R + ; √ L).
Remark 1. The assumption that θ 0 ∈ H s (R 2 ) for some s > 1 2 is needed for uniqueness only. It turns out that for less regular initial data one can construct finite energy global weak solutions to System [START_REF] Abidi | On the global well-posedness for Boussinesq system[END_REF], in the spirit of those which have been obtained by J. Leray for the standard Navier-Stokes equation in his pioneering paper [START_REF] Leray | Sur le mouvement d'un liquide visqueux remplissant l'espace[END_REF].

We shall also establish a global well-posedness results for smooth initial data.

Let us now state our main result pertaining to System (2):

Theorem 2. Let 1 < s < 3 2 and θ 0 ∈ H 1 such that |∂ 1 | s θ 0 ∈ L 2 . Let u 0 be a divergence free vector-field with coefficients in H 1 and vorticity ω 0 in L ∞ . Then System (2) with initial data (θ 0 , u 0 ) admits a global unique solution (θ, u) in C w (R + ; H 1 ) such that, in addition,

θ ∈ L ∞ (R + ; H 1 ), |∂ 1 | s θ ∈ L ∞ (R + ; L 2 ), ω ∈ L ∞ loc (R + ; L ∞ ) ∂ 1 θ ∈ L 2 (R + ; H 1 ), |∂ 1 | 1+s θ ∈ L 2 loc (R + ; L 2
). Notation 1. In the above statement, operator |∂ 1 | σ is defined as follows:

|∂ 1 | σ f (x) := 1 4π 2 e ix•ξ |ξ 1 | σ f (ξ) dξ.
Remark 2. Further in the paper, we shall also state the global existence of finite energy weak solutions corresponding to less regular initial data (see Theorem 10).

Remark 3. Like in [START_REF] Danchin | Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data[END_REF], in all the statements pertaining to System (2), one may replace the assumption that u 0 ∈ L 2 (which is slightly restrictive since in the case ω 0 ∈ L 1 it implies that R 2 ω 0 dx = 0) by (u 0 -σ) ∈ L 2 for some fixed smooth stationary radial solution σ to Euler equations. For the sake of simplicity however, we here restrict ourselves to the case where u 0 is in L 2 .

1 In all the paper, we agree that if X is a reflexive Banach space, and I ⊂ R, an interval then Cw(I; X) stands for the set of weakly continuous functions on I with values in X.

Let us emphasize that, in contrast with the previous studies devoted to the Boussinesq system, all the results presented here strongly rely on the fact that the buoyancy force is vertical. As a matter of fact, it is not clear at all that if the direction of the force is changed then having horizontal diffusion allows to improve significantly the results compared to the case κ = ν = 0.

Let us now briefly explain where the direction of the buoyancy force comes into play, and give an insight of the main arguments that we used in the proofs. In the case of System (1), the vorticity ω

:= ∂ 1 u 2 -∂ 2 u 1 satisfies ∂ t ω + u • ∇ω -ν∂ 2 1 ω = ∂ 1 θ
, from which we easily get (at least formally) 1 2

d dt ω 2 L 2 + ν ∂ 1 ω 2 L 2 = - R 2 θ ∂ 1 ω dx.
Taking advantage of the Young inequality and of the fact that θ(t)

L 2 ≤ θ 0 L 2 , it is thus possible to get a bound for ω in L ∞ loc (R + ; L 2 ).
In fact, it turns out that similar arguments enable us to bound stronger norms of the solution so that it will be possible to prove the global existence part of Theorem 1. As the velocity field that we have constructed fails to be Lipschitz, proving uniqueness requires our using losing estimates for transport or transport-diffusion equations in the spirit of [START_REF] Bahouri | Équations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides[END_REF][START_REF] Danchin | Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients[END_REF][START_REF] Danchin | Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux[END_REF][START_REF] Danchin | Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces[END_REF].

If we consider System (2) then the vorticity equation reduces to

∂ t ω + u • ∇ω = ∂ 1 θ,
so that one may write 1 2

d dt ω 2 L 2 = ∂ 1 θ ω dx. Now, it
turns out that the temperature equation in System (2) provides us with the following bound:

θ(t) 2 L 2 + 2κ t 0 ∂ 1 θ(τ ) 2 L 2 (R 2 ) dτ ≤ θ 0 2 L 2 .
Therefore, using Young's inequality, it is still possible to get a global control on ω in L ∞ loc (R + ; L 2 ). In order to get a global result with uniqueness however, we have to consider initial data with much more regularity. Indeed, we have to observe that System (2) contains the Euler system as a particular case (just take θ ≡ 0) so that, according to Yudovich result in [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF] one can barely expect to have uniqueness if the vorticity is not in L ∞ . Now, from the vorticity equation, it is clear that bounding the vorticity in

L ∞ requires that ∂ 1 θ is in L 1 loc (R + ; L ∞ ).
If we assume that θ 0 ∈ H 1 ) then we shall be able to prove that the horizontal smoothing effect ensures that ∇θ is in L 2 loc (R + ; H 1 ). As the space H 1 fails to be embedded in L ∞ however, we will have to assume even more regularity in the horizontal direction, and to check that this additional regularity is preserved for all positive time. These plain considerations explain the assumptions made in the statement of Theorem 2. As for uniqueness, it will follow from adaptations of the Yudovich method in [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF].

The paper unfolds as follows: section 2 is devoted to the study of System (1) whereas Section 3 deals with System (2). A few technical lemmas have been postponed in the final section of the paper (in particular losing a priori estimates for transport equation with anisotropic diffusion).

As usual, we agree that C denotes a harmless positive constant, the meaning of which is clear from the context.

The case of an horizontal viscosity

This part is devoted to the study of the initial value problem for System (1) under various regularity hypotheses. We aim at getting global results for possibly large data.

More precisely, in the first subsection, we prove that for any data (θ 0 , u 0 ) ∈ L2 × H 1 with div u 0 = 0, System (1) admits at least one global solution with finite energy. The next subsection is devoted to a local well-posedness result for smooth data, together with a continuation criterion involving the L ∞ norm of (θ, ∇u). In subsection 2.3, we state a sharper continuation criterion involving a weaker norm of the velocity which is (formally) controlled for all time by System (1). This will enable us to state that the system is globally well-posed in Sobolev spaces with large enough index (see Theorem 5). The last two subsections are devoted to the proof of our global existence and uniqueness result for rough data (namely Theorem 1).

2.1. Global weak solutions. In order to motivate our statement, let us first write out the "natural" energy estimates associated to System [START_REF] Abidi | On the global well-posedness for Boussinesq system[END_REF].

On the one hand, because div u = 0, we have

(4) θ(t) L 2 ≤ θ 0 L 2 .
On the other hand, taking the L 2 inner product of the velocity equation with u, we find that 1 2

d dt u(t) 2 L 2 + 2ν t 0 ∂ 1 u 2 L 2 dτ ≤ θ L 2 u L 2 .
Using Gronwall lemma and (4), we thus get ( 5)

u(t) 2 L 2 + 2ν t 0 ∂ 1 u 2 L 2 dτ ≤ u 0 L 2 + t θ 0 L 2 2 .
Let us stress the fact that the above energy bounds imply that all the components of ∇u except ∂ 2 u 1 are smoothed out for positive time. Indeed, combining the L 2 loc (R + ; L 2 ) bound for ∂ 1 u which is available from [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] with the fact that div u = 0 ensures that ∂ 1 u 1 ∂ 1 u 2 and ∂ 2 u 2 are in L 2 loc (R + ; L 2 ). However, as the last component ∂ 2 u 1 is unlikely to be bounded in L 2 loc (R + ; L 2 ) if no stronger assumption, it is not clear that one may construct global weak solutions for L 2 data (in contrast with the standard Navier-Stokes equations [START_REF] Leray | Sur le mouvement d'un liquide visqueux remplissant l'espace[END_REF] or with the Boussinesq system with isotropic viscosity [START_REF] Danchin | Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux[END_REF]).

This induces us to consider initial velocity fields in H 1 . Now, in order to get a global bound for the H 1 norm of the velocity, one may consider the vorticity equation ( 6)

∂ t ω + u • ∇ω -ν∂ 2 1 ω = ∂ 1 θ.
Combining an energy method with Young's inequality, we get 1 2

d dt ω 2 L 2 + ν ∂ 1 ω 2 L 2 = -θ ∂ 1 ω dx ≤ ν 2 ∂ 1 ω 2 L 2 + 1 2ν θ 2 L 2 .
Therefore,

d dt ω 2 L 2 + ν ∂ 1 ω 2 L 2 ≤ 1 ν θ 2 L 2
, whence, according to (4), ( 7)

ω(t) 2 L 2 + ν t 0 ∂ 1 ω 2 L 2 dτ ≤ ω 0 2 L 2 + t ν θ 0 2 L 2 .
In short, one may formally bound u in L ∞ loc (R + ; H 1 ) and θ in L ∞ (R + ; L 2 ) which leads to the following statement. Theorem 3. Let u 0 ∈ H 1 be a divergence free vector-field and θ 0 ∈ L 2 . System (1) has a unique global solution (θ, u) such that 2

(8) θ ∈ C b (R + ; L 2 ), u ∈ C w (R + ; H 1 ) and u 2 ∈ L 2 loc (R + ; H 2 )
. Proof: It is only a matter of making the above computations rigorous. For that, one may for instance use the Friedrichs method: define the spectral cut-off J n by

J n f (ξ) = 1 [0,n] (|ξ|) f (ξ), n ≥ 1,
and solve the following ODE in the space

L 2 n := f ∈ (L 2 (R 2 )) 3 / supp f ⊂ B(0, n) : (9)      ∂ t θ + J n div (J n θ J n u) = 0, ∂ t u + PJ n div (PJ n u ⊗ PJ n u) -ν∂ 2 1 PJ n u = PJ n (θe 2 ), (θ, u)| t=0 = J n (θ 0 , u 0 ).
From the Cauchy-Lipschitz theorem, we get a unique maximal solution (θ n , u n ) in C 1 ([0, T * n [; L 2 n ). Because J 2 n = J n , P 2 = P and J n P = PJ n , we discover that (θ n , Pu n ) and (J n θ n , J n u n ) are also solutions. By uniqueness, we thus have Pu n = u n (i. e. div u n = 0), J n u n = u n and J n θ n = θ n . Therefore, (10)

     ∂ t θ n + J n div (θ n u n ) = 0, ∂ t u n + PJ n div (u n ⊗ u n ) -ν∂ 2 1 u n = PJ n (θ n e 2 ), div u n = 0.
As Operators J n and PJ n are orthogonal projectors for the L 2 inner product, the above formal calculations remain unchanged. Therefore, we still have as before

θ n (t) 2 L 2 = J n θ 0 2 L 2 ≤ θ 0 2 L 2 , (11)
u n (t) 2 L 2 + 2ν t 0 ∂ 1 u n 2 L 2 dτ ≤ u 0 L 2 + t θ 0 L 2 2 . ( 12
)
This implies that (θ n , u n ) remains bounded in L 2 n for finite time, whence T * n = +∞. Next, applying the curl operator to (10) 2 , we get

∂ t ω n + J n (u n • ∇ω n ) -ν∂ 2 1 ω n = ∂ 1 θ n with ω n = ∂ 1 u 2 n -∂ 2 u 1 n .
Arguing as for proving [START_REF] Chae | Global regularity for the 2 -D Boussinesq equations with partial viscous terms[END_REF], we thus get [START_REF] Danchin | Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux[END_REF] ω

n (t) 2 L 2 + ν t 0 ∂ 1 ω n 2 L 2 dτ ≤ ω 0 2 L 2 + t ν θ 0 2 L 2 .
This implies that (ω n ) n∈N is bounded in L ∞ loc (R + ; L 2 ). Now, it is well known that the divergencefree property entails that

∇u n L 2 = ω n L 2 et ∆u 2 n = ∂ 1 ω n . So one can conclude that • (θ n ) n∈N is bounded in L ∞ (R + ; L 2 ), • (u n ) n∈N is bounded in L ∞ loc (R + ; H 1 ), • (u 2 n ) n∈N is bounded in L 2 loc (R + ; H 2
). This is enough to pass to the limit (up to extraction) in [START_REF] Chemin | Fluids with anisotropic viscosity[END_REF]. Indeed, putting together the continuous embedding H 1 ֒→ L 4 and Hölder inequality, we see that the first two properties imply that (θ n u n ) n∈N is bounded in L 4 loc (R + ; L

3 ) whence, by embedding, in

L 4 loc (R + ; H -1 2 ). Therefore (∂ t θ n ) n∈N is bounded in L 4 loc (R + ; H -3 2 ). Likewise, (∂ t u n ) n∈N is bounded in L 2 loc (R + ; H -1
). Since the embeddings H -3 2 ֒→ L 2 and H -1 ֒→ H 1 are locally compact, the classical Aubin-Lions argument (see e.g. [START_REF] Aubin | Un théorème de compacité[END_REF]) allows to conclude that, up to extraction, sequence (θ n , u n ) n∈N has a limit (θ, u) satisfying System (1) and that

θ ∈ L ∞ (R + ; L 2 ), u ∈ L ∞ loc (R + ; H 1 ) and u 2 ∈ L 2 loc (R + ; H 2 ).
From standard arguments relying on the time continuity of (θ, u) in low norms, it is easy to prove the weak time continuity result. Finally, since θ is transported by the flow of a divergence free vector-field with coefficients in L 2 loc (R + ; H 1 ), we get in addition that θ ∈ C(R + ; L 2 ) (see e.g. [START_REF] Di Perna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]).

Local smooth solutions.

Here we aim at proving the local well-posedness for System [START_REF] Abidi | On the global well-posedness for Boussinesq system[END_REF] with initial data (θ 0 , u 0 ) in H s-1 × H s for some s > 2.

The proof will follow from an energy method once the system has been localized in dyadic frequencies. This localization may be done by means of a nonhomogeneous Littlewood-Paley decomposition. In order to define the dyadic blocks ∆ q used in this decomposition, one may proceed as in [START_REF] Chemin | Fluides parfaits incompressibles[END_REF]: starting from a couple (χ, ϕ) of smooth nonnegative functions such that

Supp χ ⊂ {ξ ∈ R 2 / |ξ| ≤ 4/3}, Supp ϕ ⊂ {ξ ∈ R 2 / 3/4 ≤ |ξ| ≤ 8/3}, χ(ξ) + q≥0 ϕ(2 -q ξ) = 1 for all ξ ∈ R 2 ,
we set ∆ q := 0 if q ≤ -2, ∆ -1 := χ(D) and ∆ q := ϕ(2 -q D) if q ≥ 0.

We also introduce the low frequency cut-off

S q := p≤q-1 ∆ q
and (for technical purposes) the modified low frequency cut-off S q defined by ( 14) S -1 = ∆ -1 = S 0 and S q = S q if q = -1.

It may be easily checked that

u = q≥-1
∆ q u for all tempered distribution u and that the set of tempered distributions u satisfying q≥-1

2 2qs ∆ q u 2 L 2 1 2
< ∞ coincides with the Sobolev space H s , the above left-hand side defining a norm equivalent to the usual one.

Let us now state the main result of this subsection:

Proposition 4. Let (θ 0 , u 0 ) be in H s-1 × H s with s > 2.
Assume that div u 0 = 0. There exists a positive time T depending only (continuously) on ν and on (θ

0 , ω 0 ) H s-1 such that System (1) admits a unique solution (θ, u) in C([0, T ]; H s-1 × H s ). Moreover, u 2 ∈ L 2 ([0, T ]; H s+1 ).
Proof: The uniqueness is a straightforward consequence of a more general result (see Proposition 9) the proof of which is postponed to subsection 2.5. So let us focus on the existence part of the above proposition, which is mostly a consequence of the H s-1 × H s a priori estimates associated to System (1). [START_REF] Abidi | On the global well-posedness for Boussinesq system[END_REF]. We claim that there exists a constant C depending only on s and such that for all t ∈ [0, T ], we have

A priori estimates in H

s-1 × H s . Let (θ, u) ∈ C 1 ([0, T ]; H ∞ ) satisfy
(15) (θ, ω)(t) 2 H s-1 + ν t 0 ∂ 1 ω 2 H s-1 dτ ≤ (θ 0 , ω 0 ) 2 H s-1 e t ν e C t 0 (θ,∇v) L ∞ dτ .
Indeed, applying operator ∆ q to the equation satisfied by θ yields

∂ t ∆ q θ + S q-1 u • ∇∆ q θ = F q (u, θ) with F q (u, θ) := S q-1 u • ∇∆ q θ -∆ q (u • ∇θ).
Taking the L 2 inner product of the above equality with ∆ q θ and using the divergence free condition, we thus get 1 2

d dt ∆ q θ 2 L 2 ≤ F q (u, θ) L 2 ∆ q θ L 2 .
In the appendix (see Inequality (68)), we state that

F q (u, θ) L 2 ≤ C ∇u L ∞ q ′ ≥q-4 2 q-q ′ ∆ q ′ θ L 2 + θ L ∞ |q ′ -q|≤1 ∆ q ′ ω L 2 .
Plugging this inequality in (35) then multiplying both sides by 2 2q(s-1) and summing up over q ≥ -1, we get ( 16)

d dt θ 2 H s-1 ≤ C ∇u L ∞ θ 2 H s-1 + θ L ∞ ω H s-1 θ H s-1 .
In order to get a H s-1 estimate for ω, one may apply ∆ q to the vorticity equation. With the above notation, we get

∂ t ∆ q ω + S q-1 u • ∇∆ q ω -ν∂ 1 ∆ q ω = ∂ 1 ∆ q θ + F q (u, ω).
Taking the L 2 inner product of this inequality with ∆ q ω and using once again the divergence free condition, we get after integration by parts, 1 2

d dt ∆ q ω 2 L 2 + ν ∂ 1 ∆ q ω 2 L 2 = -∆ q θ∂ 1 ∆ q ω dx + F q (u, ω)∆ q ω dx.
Now, we notice that, by virtue of the Young inequality,

-∆ q θ∂ 1 ∆ q ω dx ≤ ∆ q θ 2 L 2 2ν + ν 2 ∂ 1 ∆ q ω 2 L 2 ,
and, according to (70),

F q (u, ω) L 2 ≤ C ∇u L ∞ q ′ ≥q-4 2 q-q ′ ∆ q ′ ω L 2 . Therefore d dt ∆ q ω 2 L 2 + ν ∂ 1 ∆ q ω 2 L 2 ≤ ν -1 ∆ q θ 2 L 2 + C ∆ q ω L 2 ∇u L ∞ q ′ ≥q-4 2 q-q ′ ∆ q ′ ω L 2 .
Multiplying both sides by 2 2q(s-1) then summing up over q ≥ -1, we end up with

d dt ω 2 H s-1 + ν ∂ 1 ω 2 H s-1 ≤ ν -1 θ 2 H s-1 + C ∇u L ∞ ω 2 H s-1 .
It is now clear that adding up this latter inequality to ( 16) then applying Gronwall lemma completes the proof of Inequality (15).

2. The proof of local existence. One can use again the Friedrichs method introduced in the proof of Theorem 3. As Operators J n are orthogonal projectors for all Sobolev spaces, they do not modify the energy estimates leading to Inequality [START_REF] Danchin | Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data[END_REF]. Therefore, the approximate solution (θ n , u n ) to ( 10) satisfies

(θ n , ω n )(t) 2 H s-1 + ν t 0 ∂ 1 ω n 2 H s-1 dτ ≤ J n (θ 0 , ω 0 ) 2 H s-1 e t ν e C t 0 (θn,∇un) L ∞ dτ .
Of course, the L 2 norm of u n is controlled by virtue of [START_REF] Danchin | Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients[END_REF]. As s > 2, the space H s-1 continuously embeds in L ∞ . Since ∇u n H s-1 = ω n H s-1 , the previous inequality thus entails that

X n (t) ≤ (θ 0 , ω 0 ) H s-1 e t 2ν e C t 0 Xn(τ ) dτ with X 2 n (t) := (θ n , ω n )(t) 2 H s-1 + ν t 0 ∂ 1 ω n 2 H s-1 dτ.
This inequality may be easily integrated into

exp -C t 0 X n (τ ) dτ ≥ 1 -2CνX 0 e t 2ν
for all t ≥ 0.

Therefore, there exists a c > 0 such that if we set ( 17)

T := 2ν log c ν (θ 0 , ω 0 ) H s-1 then • (θ n ) n∈N is bounded in L ∞ ([0, T ]; H s-1 ), • (u n ) n∈N is bounded in L ∞ ([0, T ]; H s ), • (u 2 n ) n∈N is bounded in L 2 ([0, T ]; H s+1 ).
Mimicking the compactness argument used for proving Theorem 3, one can now conclude that there exists a solution (θ, u)

satisfying (θ, u) ∈ L ∞ ([0, T ]; H s-1 ×H s ) and u 2 ∈ L 2 ([0, T ]; H s+1 ).
The time continuity follows from the fact that θ and ω satisfy transport equations with H s-1 initial data and a L 2 ([0, T ]; H s-1 ) source term. This completes the proof of Proposition 4.

Global smooth solutions.

Here we aim at proving that the local smooth solutions which have been constructed so far may be extended to all positive time. Exhibiting a polynomial control of ∇u(t) √ L (where the space √ L has been defined in ( 3)) is the cornerstone of this extension. More precisely, we shall first prove that the L 1 ([0, T ]; LL) norm of ∇u with ( 18)

LL := f ∈ S ′ / f LL := sup q≥0 S q f L ∞ q + 1 < ∞
controls the Sobolev regularity of the solutions to System (1). Next, we shall state that, under the hypotheses of Proposition 4, the norm of ∇u in L 2 ([0, T ]; √ L) (which is stronger to the L 1 ([0, T ]; LL) norm) may be bounded for all time by a fixed polynomial the coefficients of which depend only on low norms of the data, and on ν. Combining this with Proposition 4 will lead to the following global existence statement:

Theorem 5. Let (θ 0 , u 0 ) be in H s-1 × H s for some s > 2. Assume that div u 0 = 0. Then system (1) has a unique global solution (θ, u) such that (θ, u) ∈ C(R + ; H s-1 × H s ) and u 2 ∈ L 2 loc (R + ; H s+1
). As a first step for proving Theorem 5, let us show the following lemma:

Lemma 6. Let (θ, u) be a solution to (1) in C([0, T ); H s-1 × H s ) with s > 2. If T 0 ∇u(t) LL dt < ∞
then (θ, u) may be continued beyond T into a smooth solution of (1).

Proof: Putting together the lower bound for the lifespan of (θ, u) given by ( 17) and the uniqueness of smooth solutions, it suffices to state that under the assumptions of the lemma, we have

sup 0≤t<T θ(t) H s-1 + ω(t) H s-1 < ∞.
First, as θ is transported by the vector-field u (which is lipschitz for s > 2 implies H s-1 ֒→ L ∞ ), we get the following control:

θ(t) L ∞ = θ 0 L ∞ for all t ∈ [0, T ).
In consequence, Inequality [START_REF] Danchin | Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data[END_REF] ensures that

(19) (θ, ω)(t) 2 H s-1 + ν t 0 ∂ 1 ω 2 H s-1 dτ ≤ (θ 0 , ω 0 ) 2 H s-1 e (ν -1 +C θ 0 L ∞ )t e C t 0 ∇u L ∞ dτ .
On the other hand, in the appendix, it is shown that

(20) ∇u L ∞ ≤ C 1 + ∇u LL log(e + ω H s-1 )
.

Putting together [START_REF] Hmidi | On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity[END_REF] and ( 20), we deduce that for all t ∈ [0, T ),

log e + (θ, ω)(t) 2 H s-1 ≤ log e + (θ 0 , ω 0 ) 2 H s-1 + C 1 + ν -1 + θ 0 L ∞ t +C t 0 ∇u LL log e + (θ, ω)(t) 2 H s-1 dτ,
whence, according to Gronwall Lemma,

log e + (θ, ω)(t) 2 H s-1 ≤ log e + (θ 0 , ω 0 ) 2 H s-1 + C 1 + ν -1 + θ 0 L ∞ t e C t 0 ∇u LL dτ .
As the argument of exp is, by assumption, bounded for t ∈ [0, T ), we gather that (θ, ω) ∈ L ∞ ([0, T ); H s-1 ), which completes the proof of the lemma.

The next step involves showing that the norm used in the previous lemma is controlled by the system. In fact, we shall state a slightly more accurate result:

Lemma 7. Let (θ, u) be a solution to (1) in C([0, T ); H 2 × H 3 ). There exists a continuous function f : R + → R + depending only (continuously) on ν, u 0 L 2 , θ 0 L 2 ∩L ∞ and ω 0 √ L such that t 0 ∇u 2 √ L dt ≤ f (t) for all t ∈ [0, T ).
Proof: Let us first notice that, because div u = 0 and u is Lipschitz, we have

(21) θ(t) L p = θ 0 L p for all p ∈ [2, ∞] and t ∈ [0, T ).
In order to get a control of ∇u in L 2 ([0, T ); √ L), we are going to state that ( 22)

ω(t) 2 √ L ≤ ω 0 2 √ L + t 2ν θ 0 2 L 2 ∩L ∞ .
For showing that, one may multiply the vorticity equation with |ω| p-2 ω and perform a space integration. As our hypotheses on the solution entail that ω ∈ C 1 ([0, T ); H 1 ) and thus ω ∈ C 1 ([0, T ); L p ) for all p ∈ [2, +∞), we get

1 p d dt |ω| p dx + (p-1)ν |∂ 1 ω| 2 |ω| p-2 dx ≤ (p-1) |θ||∂ 1 ω||ω| p-2 dx, ≤ (p-1)ν |∂ 1 ω| 2 |ω| p-2 dx + p-1 4ν |θ| 2 |ω| p-2 dx ≤ (p-1)ν |∂ 1 ω| 2 |ω| p-2 dx + p-1 4ν θ 2 L p ω p-2 L p . Therefore, d dt ω 2 L p ≤ p -1 2ν θ 2 L p
hence, by virtue of [START_REF] Leray | Sur le mouvement d'un liquide visqueux remplissant l'espace[END_REF],

ω(t) 2 L p ≤ ω 0 2 L p + p -1 2ν θ 0 2 L p t, whence (22).
This does not imply that ∇u ∈ L ∞ ([0, T ); √ L) for the classical result on Calderon-Zygmund operators (see e.g. [START_REF] Chemin | Fluides parfaits incompressibles[END_REF], Chap. 3) gives only that [START_REF] Pedlosky | Geophysical fluid dynamics[END_REF] ∇u(t)

L p ≤ Cp ω(t) L p for all p ∈ [2, ∞).
However, because ∂ 1 ω = ∆u 2 and ∂ 1 u 1 = -∂ 2 u 2 , Inequalities ( 5) and [START_REF] Chae | Global regularity for the 2 -D Boussinesq equations with partial viscous terms[END_REF] entail that

(24) √ ν ∂ i u j L 2 t (H 1 ) ≤ u 0 H 1 + t + t ν θ 0 L 2 for t ∈ [0, T ) and (i, j) = (2, 1).
By virtue of Lemma 13 (see the appendix), we thus get the desired bound for all the components of ∇u except ∂ 2 u 1 . In order to get a suitable bound for ∂ 2 u 1 , one may use the fact that

∂ 2 u 1 = ∂ 1 u 2 -ω.
Putting together Inequalities ( 22) and ( 24), it is now easy to conclude Proof of Theorem 5 : For the sake of simplicity, we restrict ourselves to the case s ≥ 3 so that one may use Lemma 7. The case 2 < s < 3 easily follows from the case s ≥ 3: it is only a matter of smoothing out the initial data then pass to the limit. So let us assume from now on that s ≥ 3 and let us denote by (θ, u) the maximal solution supplied by Proposition 4, and by T * the lifespan of (θ, u). If we assume (by contradiction) that T * is finite then Proposition 7 ensures that

(25) ∇u ∈ L 2 ([0, T * ); √ L).
Remark that the space √ L is continuously embedded in the space LL 1 2 defined by (26) LL

1 2 := f ∈ S ′ / f LL 1 2 := sup q≥0 S q f L ∞ √ q + 1 < ∞ .
Indeed, thanks to Bernstein inequality, there exists a constant C such that for all N ∈ N and p ∈ [2, ∞[, we have

(27) S N ∇u L ∞ ≤ C2 2N p ∇u L p ≤ C2 2N p p -1 ∇u √ L . If we choose p = N + 2 then we get (28) S N ∇u L ∞ ≤ C √ N + 1 ∇u √ L , whence the desired embedding.
It is obvious that LL 1 2 ֒→ LL. Resorting to (25), we thus get ∇u ∈ L 1 ([0, T * ); LL) and Lemma 6 ensures that the solution (θ, u) may be continued beyond T * . This contradicts the definition of T * .

2.4.

Global well-posedness for rough data. In this section, we want to state global existence with uniqueness for a class of data as large as possible. Having in mind the previous subsection, it seems reasonable to require that

θ 0 ∈ L 2 ∩ L ∞ , that u 0 ∈ H 1 and that ω 0 ∈ √ L.
As those regularity assumptions are (formally) conserved by the system during the evolution, we thus expect to get a global solution (θ, u) such that ∇u ∈ L 2 loc (R + ; √ L). This would imply that

u ∈ L 2 loc (R + ; LogLip 1
2 ) where LogLip 1 2 stands for the set of bounded functions f such that sup

x =y |x-y|≤1/2 |f (y) -f (x)| |x -y| log 1 2 (|x -y| -1 ) < ∞.
The above inequality is an obvious corollary of (28) and of Proposition 2.107 in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF].

Even though the vector-field u fails to be lipschitz, it has enough regularity so that we have θ ∈ C(R + ; H s-ε ) for all ε > 0 if we start from θ 0 in H s for some s ∈] -1, 1] (this is in fact a consequence of Theorem 3.12 in [START_REF] Danchin | Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients[END_REF]). These plain observations will lead us to the following statement which obviously contains Theorem 1:

Theorem 8. Let θ 0 ∈ L 2 ∩ L ∞ ,
and u 0 ∈ H 1 with div u 0 = 0 and ω 0 ∈ √ L. Then System (1) admits a global solution (θ, u) such that

(29) θ ∈ C b (R + ; L 2 ) ∩ C w (R + ; L ∞ ) ∩ L ∞ (R + ; L ∞ ), u ∈ C w (R + ; H 1 ), u 2 ∈ L 2 loc (R + ; H 2 ), ω ∈ L ∞ loc (R + ; √ L), ∇u ∈ L 2 loc (R + ; √ L).
If in addition θ 0 ∈ H s for some real number s ∈ (0, 1] then θ ∈ C(R + ; H s-ε ) for all ε > 0. Finally, if s > 1/2 then the solution is unique.

Proof: The uniqueness is a consequence of Proposition 9 below so let us focus on the existence part of the statement. To achieve it, one may smooth out the initial data (θ 0 , u 0 ) n∈N so as to get a sequence (θ 0,n , u 0,n ) n∈N of H ∞ functions which tends to (θ 0 , u 0 ) in (say) L 2 × H 1 . Resorting to Theorem 5, we get a sequence (θ n , u n ) n∈N of smooth global solutions. Moreover, by virtue of Inequalities ( 5), ( 7), ( 21) and ( 22), we have

• (θ n ) n∈N bounded in L ∞ (R + ; L 2 ∩ L ∞ ), • (u n ) n∈N bounded in L ∞ loc (R + ; H 1 ), • (u 2 n ) n∈N bounded in L 2 loc (R + ; H 2 ), • (ω 2 n ) n∈N bounded in L ∞ loc (R + ; √ L).
As explained in the proof of Lemma [START_REF] Chae | Global regularity for the 2 -D Boussinesq equations with partial viscous terms[END_REF], these proprieties imply that (∇u n ) n∈N is bounded in L 2 loc (R + ; √ L). Now, taking advantage of the losing estimates proved in Proposition 16, we deduce that if, in addition, θ 0 ∈ H s for some s ∈ (0, 1) then, for all ε > 0, (θ n ) n∈N is bounded in L ∞ loc (R + ; H s-ε ). In order to conclude to the existence part of the statement, one may use again a compactness argument à la Aubin-Lions as in the proof of Theorem 3.

2.5.

Uniqueness for rough data. The difference (δθ, δu, δΠ) between two solutions (θ 1 , u 1 , Π 1 ) and (θ 2 , u 2 , Π 2 ) satisfies:

(30) ∂ t δθ + div (u 2 δθ) = -div (δu θ 1 ), ∂ t δu + div (u 2 ⊗ δu) + div (δu ⊗ u 1 ) -ν∂ 2 1 δu + ∇δΠ = δθ e 2 .
First of all, let us notice that if θ 1 ∈ L ∞ ([0, T ]; H γ ) then the right-hand side of the first equation is at most in L ∞ ([0, T ]; H γ-1 ). As, under the hypotheses of Theorem 5, u 2 fails to be lipschitz but has gradient in L 2 loc (R + ; √ L), Proposition 16 will enable us to bound δθ in H γ-1-ε provided that div (δu θ 1 ) be bounded in L 1 ([0, T ]; H γ-1 ), a condition which requires a control over the L 1 ([0, T ]; L ∞ ) norm of δu. If the velocity equation had a full Laplace operator then the resulting smoothing effect would be strong enough so as to provide us with a control over this norm. We shall see that in the framework of anisotropic viscosity, one can still get an appropriate bound for δu in L 1 ([0, T ]; L ∞ ) provided γ > 1/2. In short, we expect to be able to control δθ in

L ∞ ([0, T ]; H β-1 ) for some β ∈]1/2, γ[, δu in L ∞ ([0, T ]; H α ) and ∂ 1 δu in L 2 ([0, T [; H α ) for some α ∈]1/2, β[
. This motivates the following statement which implies the uniqueness part of Theorem 8: Proposition 9. Let (θ 1 , u 1 ) and (θ 2 , u 2 ) be two solutions of (1) on [0, T ] × R 2 with the same initial data. Assume that ∇u i ∈ L 1 ([0, T ]; LL 1 2 ) and that there exists some

γ ∈]1/2, 1[ such that θ i ∈ L ∞ ([0, T ]; L ∞ ∩ H γ ) and u i ∈ L ∞ ([0, T ]; H γ ) for i = 1, 2.
Then the two solutions coincide.

Proof: According to the above heuristics, we have to bound

δθ in L ∞ ([0, T ]; H β-1 ), δu in L ∞ ([0, T ]; H α ) and ∂ 1 δu in L 2 ([0, T ]; H α ) for some fixed (α, β) such that 1/2 < α < β < γ.
In order to bound δθ, one may use Proposition 16 with the vector-field u 2 . Because δθ(0) = 0, we have

δθ(t) H β-1 ≤ C t 0 div (δu θ 1 ) H γ-1 for all t ∈ [0, T ].
In order to bound the right-hand side, one may resort to the following Bony's decomposition [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]:

(31) div (δu θ 1 ) = div T δu θ 1 + R(δu, θ 1 ) + 2 i=1 T ∂ i θ 1 δu i ,
where the paraproduct operator T (resp. reminder operator R) is defined by

T f g := q S q-1 f ∆ q g resp. R(f, g) := q ∆ q f ∆ q-1 g+∆ q g+∆ q-1 g .
Let us stress that the condition div δu = 0 has been used in order to have the derivative act on the left for the first two terms of (31).

From standard continuity results for operators T and R (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]) we have

T δu θ 1 + R(δu, θ 1 ) H γ ≤ C δu L ∞ θ 1 H γ .
As for the last term, given that γ -1 < 0, one can write

T ∂ i θ 1 δu i H γ-1 ≤ C ∇θ 1 H γ-1 δu L ∞ . We eventually get (32) δθ L ∞ t (H β-1 ) ≤ C θ 1 L ∞ t (H γ ) δu L 1 t (L ∞ ) .
In order to bound δu, one may use Proposition 17. We get for all t ∈ [0, T ],

δu L ∞ t (H α ) + ∂ 1 δu L 2 t (H α ) ≤ C δθ L 2 t (H β-1 ) + δu • ∇u 1 L 2 t (H β-1 )
for some constant C depending only on α, β, ν and u 2 . Using again the Bony decomposition and arguing exactly as for proving (32), we get

δu • ∇u 1 H β-1 ≤ C δu L ∞ u 1 H β . Therefore, given that u 1 ∈ L ∞ ([0, T ]; H β ), (33) δu L ∞ t (H α ) + ∂ 1 δu L 2 t (H α ) ≤ C δθ L 2 t (H β-1 ) + δu L 2 t (L ∞ ) .
In order to complete the proof of the proposition, it is only a matter of showing that δu L 1 t (L ∞ ) may be bounded in terms of δu L ∞ t (H α ) and of ∂ 1 δu L 2 t (H α ) . This is the only point where the assumption α > 1/2 (and thus γ > 1/2) is going to play a role. First of all, thanks to the trace theorem, one may write (with obvious notation)

H α (R 2 ) ֒→ L ∞ (R x 2 ; H α-1 2 (R x 1 )).
Therefore, one may write (34) δu

L ∞ x 2 (H α-1 2 x 1 ) ≤ C δu H α and ∂ 1 δu L ∞ x 2 (H α-1 2 x 1 ) ≤ C ∂ 1 δu H α .
As for all α ∈]0, 1[, Gagliardo-Nirenberg inequality implies that

δu(•, x 2 ) L ∞ (R) ≤ C δu(•, x 2 ) α H α-1 2 (R) ∂ 1 δu(•, x 2 ) 1-α H α-1 2 (R)
for all x 2 ∈ R,

we have, by combination with (34),

δu L ∞ (R 2 ) ≤ C δu α H α (R 2 ) ∂ 1 δu 1-α H α (R 2 ) .
Coming back to (32) and (33), we deduce that for some constant C depending only on ν, T and on the norms of (θ 1 , u 1 ) and (θ 2 , u 2 ), we have

δθ L ∞ t (H β-1 ) ≤ Ct 1 2 + α 2 δU (t) δU (t) ≤ C t 1 2 δθ L ∞ t (H β-1 ) + t α 2 δU (t) with δU (t) := δu L ∞ t (H α ) + ∂ 1 δu L 2 t (H α ) .
Inserting the first inequality in the second one, one may conclude that δu ≡ 0 (and thus δθ ≡ 0) on a suitably small time interval. Finally, let us notice that our assumptions on the solutions ensure that δθ ∈ C([0, T ]; H β-1 ) and δu ∈ C([0, T ]; H α ). Using a classical connectivity argument, it is now easy to get the uniqueness on the whole interval [0, T ].

The case of an horizontal diffusivity

This section is devoted to the study of System (2). In other words, in contrast with the previous section, we now assume that the velocity satisfies the incompressible Euler equation with buoyancy force whereas the temperature experiences diffusion in the horizontal variable only.

We aim at stating various global existence results for arbitrarily large data. More precisely, we first prove that any data θ 0 ∈ L 2 and u 0 ∈ H 1 with div u 0 = 0 generates a global weak solution with finite energy. The rest of this section is mainly devoted to the proof of Theorem 2. As a first step, in subsection 3.2, we state H 1 a priori estimates for the la temperature. In the next subsection, we prove a uniqueness result for a large class of solutions. As this uniqueness result requires in particular that ∇θ ∈ L 1 loc (R + ; L ∞ ) and that ∇u ∈ L 1 loc ([0, T ]; L), our next task amounts to finding additional regularity conditions on the data which may be propagated globally by the system. It turns out that it is possible to propagate some anisotropic Sobolev regularity over the temperature, and thus to complete the proof of Theorem 2.

3.1. Global weak solutions: the case θ 0 ∈ L 2 and u 0 ∈ H 1 . Let us first derive the formal energy estimates for System (2) in the case θ 0 ∈ L 2 and u 0 ∈ H 1 . First, multiply (2) 1 by θ and integrate over [0, t] × R 2 . We get

(35) θ(t) 2 L 2 + 2κ t 0 ∂ 1 θ(s) 2 L 2 ds ≤ θ 0 2 L 2 .
Combining this with the standard energy estimate for u yields

(36) u(t) L 2 ≤ u 0 L 2 + t θ 0 L 2 .
In order to get a H 1 bound for the velocity, one may consider the vorticity equation:

∂ t ω + u • ∇ω = ∂ 1 θ.
Multiplying by ω then integrating with respect to the space variable, we find that 1 2

d dt ω 2 L 2 ≤ ω L 2 ∂ 1 θ L 2 whence, ω(t) L 2 ≤ ω 0 L 2 + t 0 ∂ 1 θ L 2 ds, ≤ ω 0 L 2 + t 2κ θ 0 L 2 . (37)
Now, using a Friedrichs method quite similar to that of the proof of Theorem 3, we easily get the following statement:

Theorem 10. Let θ 0 ∈ L 2 and u 0 ∈ H 1 with div u 0 = 0. Then System (2) with data (θ 0 , u 0 ) has a global solution (θ, u) such that θ ∈ C w (R + ; L 2 ), ∂ 1 θ ∈ L 2 (R + ; L 2 ) and u ∈ C(R + ; H 1 ).
3.2. H 1 a priori estimates for the temperature. In the present paragraph, we show that one may get (at least formally) a global control over both the H 1 norm of θ and of u.

To start with, let us point out that Inequalities (35), (36) and (37) provide us with a bound for θ in L ∞ (R + ; L 2 ), for ∂ 1 θ in L 2 (R + ; L 2 ) and for u in L ∞ loc (R + ; H 1 ). We claim that if we assume in addition that ∇θ 0 ∈ L 2 then one may bound ∇θ in L ∞ loc (R + ; L 2 ). Indeed, applying operator ∂ i (i = 1, 2) to the equation satisfied by θ yields

∂ t ∂ i θ + u • ∇∂ i θ + ∂ i u • ∇θ -κ∂ 2 1 ∂ i θ = 0.
Let us multiply this equality by ∂ i θ, integrate over R 2 then add up the equalities for i = 1, 2. Integrating by parts where needed and using the fact that div u = 0, we easily find that

(38) 1 2 d dt ∇θ 2 L 2 + κ ∂ 1 ∇θ 2 L 2 + 1≤i,j≤2 ∂ i θ ∂ j u i ∂ j θ dx = 0.
For (i, j) = (2, 2), the terms in the above summation are easy to handle. Indeed, taking advantage of the anisotropic Hölder inequality, one can write

∂ i θ ∂ j u i ∂ j θ dx ≤ ∇u L 2 ∂ 1 θ L 2 x 1 (L ∞ x 2 ) ∇θ L ∞ x 1 (L 2 x 2 ) .
Let us admit the following two inequalities (the proof of which is postponed in the appendix):

(39)

f L 2 x 1 (L ∞ x 2 ) ≤ C f 1 2 L 2 ∂ 2 f 1 2 L 2 and f L ∞ x 1 (L 2 x 2 ) ≤ C f 1 2 L 2 ∂ 1 f 1 2
L 2 . Applying these inequalities to ∂ 1 θ and to ∇θ, and using the fact that ∇u L 2 = ω L 2 , we deduce that

∂ i θ ∂ j u i ∂ j θ dx ≤ C ω L 2 ∇θ L 2 ∂ 1 ∇θ L 2 if (i, j) = (2, 2).
In order to bound the term corresponding to (i, j) = (2, 2), one may use the fact that ∂ 2 u 2 = -∂ 1 u 1 and integrate by parts. We get

∂ 2 u 2 (∂ 2 θ) 2 dx = -∂ 1 u 1 (∂ 2 θ) 2 dx = 2 u 1 ∂ 2 θ ∂ 1 ∂ 2 θ dx.
Therefore, thanks to the anisotropic Hölder inequalities and to (39),

∂ 2 u 2 (∂ 2 θ) 2 dx ≤ 2 ∂ 1 ∂ 2 θ L 2 u 1 L 2 x 1 (L ∞ x 2 ) ∂ 2 θ L ∞ x 1 (L 2 x 2 ) , ≤ C ∂ 1 ∂ 2 θ L 2 u 1 2 L 2 ω 1 2 L 2 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2 L 2 , ≤ C u 1 2 L 2 ω 1 2 L 2 ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 3 2 L 2 .
So finally, Young inequality leads to 1≤i,j≤2

∂ i θ ∂ j u i ∂ j θ dx ≤ κ 2 ∂ 1 ∇θ 2 L 2 + C κ ω 2 L 2 1 + u 2 L 2 κ 2 ∇θ 2 L 2 .
Plugging this inequality in (38) and using Gronwall lemma, we end up with

∇θ(t) 2 L 2 + κ t 0 ∂ 1 ∇θ(s) 2 L 2 ds ≤ ∇θ 0 2 L 2 exp C κ t 0 ω 2 L 2 1 + u 2 L 2 κ 2 dτ .
Putting together ( 35), ( 36) and (37), we conclude that

(40) θ(t) 2 H 1 + κ t 0 ∂ 1 θ(s) 2 H 1 ds ≤ C(t, κ, θ 0 , u 0 ) with C(t, κ, θ 0 , u 0 ) := θ 0 2 H 1 exp Ct κ ω 0 2 L 2 + t κ θ 0 2 L 2 1 + u 0 2 L 2 + t 2 θ 0 2 L 2 κ 2 • 3.3. A uniqueness result.
In this section, we establish a uniqueness result for System (2) under "minimal" assumptions. In order to motivate those assumptions, let us remind that in the isotropic case (that is with a full Laplacian in the temperature equation) which has been investigated in [START_REF] Danchin | Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data[END_REF], uniqueness is true in the class of C 0,1 (R + ; L 2 ) solutions which satisfy in addition (41) ∇θ ∈ L 1 loc (R + ; L ∞ ) and ∇u ∈ L 1 loc (R + ; L). As in the case that we now consider the smoothing effect over the temperature is obviously weaker, we expect the conditions leading to uniqueness to be stronger than (41). We shall prove the following result: Proposition 11. Let (θ 1 , u 1 ) and (θ 2 , u 2 ) be two solutions of (2) with the same data. Assume that both solutions belong to

L ∞ ([0, T ]; H 1 ) ∩ C 0,1 ([0, T ]; L 2 ) and that, in addition, ∂ 1 θ 2 ∈ L 2 ([0, T ]; H 1 ) and ∇u 2 ∈ L 1 ([0, T ]; L). Then (θ 1 , u 1 ) ≡ (θ 2 , u 2 ) on [0, T ] × R 2 .
Proof: With the usual notation, (δθ, δu) satisfies:

∂ t δθ + u 1 • ∇δθ + div (δuθ 2 ) -κ∂ 2 1 δθ = 0 ∂ t δu + u 1 • ∇δu + δu • ∇u 2 = -∇δΠ + δθ e 2 .
From a standard energy method, we get 1 2

d dt δθ 2 L 2 + κ ∂ 1 δθ 2 L 2 ≤ div (θ 2 δu)δθ dx , (42) 1 2 d dt δu 2 L 2 ≤ δu • ∇u 2 δθ dx + δθδu 2 dx . ( 43
)
In order to bound the right-hand side of (42), one may write

(44) div (θ 2 δu)δθ = -θ 2 δu 1 ∂ 1 δθ dx -θ 2 δu 2 ∂ 2 δθ dx.
The first term is easy to deal with: using Cauchy-Schwarz inequality, we get (45)

θ 2 δu 1 ∂ 1 δθ dx ≤ δu L 2 θ 2 L ∞ ∂ 1 δθ L 2 .
Next, applying the following inequality (see the proof in the appendix)

(46) θ 2 L ∞ ≤ C θ 2 1/4 L 2 ∂ 1 θ 2 1/4 L 2 ∂ 2 θ 2 1/4 L 2 ∂ 1 ∂ 2 θ 2 1/4
L 2 , and using Young inequality, we find that (47)

θ 2 δu 1 ∂ 1 δθ dx ≤ C κ θ 2 3/2 H 1 ∂ 1 ∂ 2 θ 2 1/2 L 2 δu 2 L 2 + κ 6 ∂ 1 δθ 2 L 2 .
The second term of ( 44) is more intricate. If we integrate by parts and use the fact that div δu = 0, we get

-θ 2 δu 2 ∂ 2 δθ dx = ∂ 2 θ 2 δu 2 δθ dx + θ 2 ∂ 2 δu 2 δθ dx, = ∂ 2 θ 2 δu 2 δθ dx -θ 2 ∂ 1 δu 1 δθ dx, = A 1 + A 2 + A 3 with A 1 := ∂ 2 θ 2 δu 2 δθ dx, A 2 := θ 2 δu 1 ∂ 1 δθ dx and A 3 := ∂ 1 θ 2 δu 1 δθ dx.
The term A 2 may be bounded according to (47). In order to bound A 3 , we use the anisotropic Hölder inequality and (39). This leads to

|A 3 | ≤ δθ L ∞ x 1 (L 2 x 2 ) ∂ 1 θ 2 L 2 x 1 (L ∞ x 2 ) δu L 2 , ≤ C δθ 1/2 L 2 ∂ 1 δθ 1/2 L 2 ∂ 1 θ 2 1/2 L 2 ∂ 1 ∂ 2 θ 2 1/2
L 2 δu L 2 , whence, resorting again to Young inequality, (48)

|A 3 | ≤ C κ δθ 2 L 2 ∂ 1 θ 2 2 L 2 ∂ 1 ∂ 2 θ 2 2 L 2 + δu 2 L 2 + κ 6 ∂ 1 δθ 2 L 2 .
The term A 1 is the most difficult to deal with. To get an appropriate bound, let us first notice that, as div δu = 0, we may write

δu 2 = (1 -∂ 2 2 ) -1 (δu 2 ) + (1 -∂ 2 2 ) -1 ∂ 2 ∂ 1 δu 1 .

Therefore, integrating by parts, we get

A 1 = A 1 1 + A 2 1 + A 3 1 with A 1 1 := (1 -∂ 2 2 ) -1 (δu 2 ) ∂ 2 θ 2 δθ dx, A 2 1 := -∂ 2 (1 -∂ 2 2 ) -1 (δu 1 ) ∂ 1 ∂ 2 θ 2 δθ dx, A 3 1 := -∂ 2 (1 -∂ 2 2 ) -1 (δu 1 ) ∂ 2 θ 2 ∂ 1 δθ dx.
First of all, we have

|A 3 1 | ≤ ∂ 2 (1 -∂ 2 2 ) -1 δu L 2 x 1 (L ∞ x 2 ) ∂ 2 θ 2 L ∞ x 1 (L 2 x 2 ) ∂ 1 δθ L 2 .
Taking advantage of (39), we get

∂ 2 (1 -∂ 2 2 ) -1 δu 1 L 2 x 1 (L ∞ x 2 ) ≤ C ∂ 2 (1 -∂ 2 2 ) -1 (δu 1 ) 1 2 L 2 ∂ 2 2 (1 -∂ 2 2 ) -1 δu 1 1 2 L 2 ≤ C δu L 2 ∂ 2 θ 2 L ∞ x 1 (L 2 x 2 ) ≤ C ∂ 2 θ 1 2 L 2 ∂ 1 ∂ 2 θ 1 2
L 2 . In consequence, thanks to Young inequality, we have

|A 3 1 | ≤ C κ ∂ 2 θ L 2 ∂ 1 ∂ 2 θ L 2 δu 2 L 2 + κ 6 ∂ 1 δθ 2 L 2 .
To deal with A 2 1 , one may write that, by virtue of (39) and of Young inequality

|A 2 1 | ≤ ∂ 2 (1 -∂ 2 2 ) -1 δu 1 L 2 x 1 (L ∞ x 2 ) ∂ 1 ∂ 2 θ 2 L 2 δθ L ∞ x 1 (L 2 x 2 ) , ≤ C δu L 2 ∂ 1 ∂ 2 θ 2 L 2 δθ 1 2 L 2 ∂ 1 δθ 1 2 L 2 , ≤ C κ δu 2 L 2 ∂ 1 ∂ 2 θ 2 2 L 2 + 3κ 2 δθ 2 L 2 + κ 6 ∂ 1 δθ 2 L 2 .
Finally, for A 1 1 we have

|A 1 1 | ≤ (1 -∂ 2 2 ) -1 (δu 2 ) L 2 x 1 (L ∞ x 2 ) ∂ 2 θ 2 L 2 δθ L ∞ x 1 (L 2 x 2 ) , ≤ C δu L 2 ∂ 2 θ 2 L 2 δθ 1 2 L 2 ∂ 1 δθ 1 2 L 2 , ≤ C κ δu 2 L 2 ∂ 1 θ 2 2 L 2 + 3κ 2 δθ 2 L 2 + κ 6 ∂ 1 δθ 2 L 2 .
Putting together all the previous inequalities, we conclude that

(49) |A 1 | ≤ κ 2 ∂ 1 δθ 2 L 2 + 3κ δθ 2 L 2 + C κ ∂ 1 θ 2 2 H 1 δu 2 L 2 .
Now, inserting Inequalities (47), ( 48) and ( 49) in ( 44), we deduce that there exists an integrable function f 2 over [0, T ] depending only on (θ 2 , u 2 ) and on κ such that (50) 1 2

d dt δθ 2 L 2 ≤ f 2 (t) (δθ, δu) 2 L 2 .
Adapting the well-known Yudovich's argument (see [START_REF] Gérard | Résultats récents sur les fluides parfaits incompressibles bidimensionnels[END_REF] and [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF]), it is now easy to complete the proof of uniqueness. Indeed, from Inequality (43), we get for all

p ∈ [2, ∞[, (51) 1 2
d dt δu 2 L 2 ≤ ∇u 2 L p δu 2 p L ∞ δu 2-2 p L 2 + δθ L 2 δu L 2 .
Setting X ε (t) := (δθ, δu)(t) 2 L 2 + ε 2 for ε > 0, and using (50) and (51), we obtain

d dt X ε ≤ p ∇u 2 L δu 2 p L ∞ X 1-2 p ε + ( 1 2 + f 2 )X ε . Now, if we set Y ε = X ε exp - t 0 ( 1 2 + f 2 (τ )) dτ ) , we have 2 p Y 2 p -1 ε d dt Y ε ≤ 2 ∇u 2 L δu 2 p L ∞ , whence Y ε (t) ≤ ε 2 p + 2 t 0 ∇u 2 L δu 2 p L ∞ dτ p 2
.

Having ε tend to 0, we discover that for all t ∈ R + ,

(δθ, δu)(t) 2 L 2 ≤ δu 2 L ∞ t (L ∞ ) 2 t 0 ∇u 2 L dτ p .
By Sobolev embedding and thanks to [START_REF] Pedlosky | Geophysical fluid dynamics[END_REF] with p = 4, we have

(52) δu L ∞ ≤ C δu L 2 + δω L 4 .
As the assumptions made in the proposition ensure that

ω i ∈ L ∞ ([0, T ]; W 1,4 ) and that u i ∈ L ∞ ([0, T ]; L 2 ), we deduce that δu ∈ L ∞ ([0, T ] × R 2 )
. Therefore, there exists some T 0 > 0 such that the right-hand side of the above inequality tend to 0 when p goes to infinity. This yields uniqueness on [0, T 0 ]. From a standard connectivity argument, it is now easy to conclude to uniqueness on the whole interval [0, T ].

3.4. Anisotropic a priori estimates. If in addition to the H 1 hypothesis on (θ 0 , u 0 ), we assume that ω 0 ∈ L p for some p in [2, ∞[, then the vorticity equation

(53) ∂ t ω + u • ∇ω = ∂ 1 θ implies that (54) ω(t) L p ≤ ω 0 L p + t 0 ∂ 1 θ L p . Now, remind that as θ 0 ∈ H 1 , a bound for ∂ 1 θ in L 2 loc (R + ; H 1
) is available, whence also in L 2 loc (R + ; L p ) by Sobolev embedding. In fact, we even have a more accurate information if ω 0 ∈ √ L. Indeed, Lemma 13 ensures that H 1 is continuously embedded in √ L so that, according to (54), ( 55)

ω(t) √ L ≤ ω 0 √ L + C √ t ∂ 1 θ L 2 t (H 1
) . However, this bound does not imply that ∇u ∈ L 1 loc (R + ; L) so that one cannot get uniqueness by a direct application of Proposition 11. In fact, thanks to [START_REF] Pedlosky | Geophysical fluid dynamics[END_REF], it is obvious that ∇u ∈

L 1 loc (R + ; L) provided ω ∈ L 1 loc (R + ; L 2 ∩ L ∞ ).
According to (54), having ∂ 1 θ in L 1 (R + ; L ∞ ) will entail that the vorticity is bounded.

In order to get this, we shall first show that one may propagate some additional horizontal Sobolev regularity for θ. By virtue of Lemma 14 (see the appendix), this will enable us to estimate ∂ 1 θ in L 1 (R + ; L ∞ ) (and even in L 2 loc (R + ; L ∞ ) actually). More precisely, we assume from now on that (θ 0 , u 0 ) ∈ H 1 (R 2 ) and ω 0 ∈ √ L, and that, in addition, |∂ 1 | 1+s θ 0 ∈ L 2 for some s ∈ (0, 1 2 ]. In order to propagate the additional regularity, one may apply operator |∂ 1 | 1+s to the equation

∂ t θ + u • ∇θ -κ∂ 2
1 θ = 0, and take the L 2 (R 2 ) inner product with |∂ 1 | 1+s θ . After integrating by parts, we find that (56) 1 2

d dt |∂ 1 | 1+s θ 2 L 2 + κ||∂ 1 | 2+s θ 2 L 2 ≤ (|∂ 1 | 1+s u • ∇θ), |∂ 1 | 1+s θ L 2 .
Bounding the right-hand side is the main difficulty. First of all, let us notice that i|∂ 1 | = ∂ 1 R 1 where R 1 stands for the Riesz operator with respect to the first variable. As |∂ 1 | s is a symmetric operator, one may write

(|∂ 1 | 1+s u • ∇θ), |∂ 1 | 1+s θ L 2 = (∂ 1 u • ∇θ), R 1 |∂ 1 | 1+2s θ L 2 ≤ I 1 + I 2 with I 1 := ∂ 1 u • ∇θ, R 1 |∂ 1 | 1+2s θ L 2 and I 2 := u • ∇∂ 1 θ, R 1 |∂ 1 | 1+2s θ L 2 .
The term I 2 is easy to deal with. Indeed, for s ∈ (0, 1/2], we have, according to Hölder and Parseval inequalities,

I 2 ≤ u L ∞ ∇∂ 1 θ L 2 |∂ 1 | 1+2s θ L 2 , ≤ u L ∞ ∂ 1 θ 2 H 1 .
Thanks to (52) and by virtue of Inequalities (36), ( 40) and (55), we thus have (57)

t 0 I 2 (τ ) dτ ≤ C(t, κ, θ 0 , u 0 )
where, from now on, C(t, κ, θ 0 , u 0 ) denotes a positive continuous function depending only on t, κ and on the norm of (θ 0 , u 0 ) in

H 1 × H 1 ∩ W 1,4 .
In order to bound the term I 1 one may write

I 1 ≤ I 1 1 + I 2 1 with I 1 1 := ∂ 1 u 1 ∂ 1 θ, R 1 |∂ 1 | 1+2s θ L 2 , I 2 1 := ∂ 1 u 2 ∂ 2 θ, R 1 |∂ 1 | 1+2s θ L 2 .
For I 1 1 , as ∂ 1 u 1 = -∂ 2 u 2 , integrating by parts yields

I 1 1 ≤ u 2 ∂ 2 ∂ 1 θ, R 1 |∂ 1 | 1+2s θ L 2 + u 2 ∂ 1 θ, R 1 |∂ 1 | 1+2s ∂ 2 θ L 2 , ≤ I 1 1 + I 1 1 with I 1 1 := u 2 ∂ 2 ∂ 1 θ, R 1 |∂ 1 | 1+2s θ L 2 and I 1 1 := |∂ 1 | 2s (u 2 ∂ 1 θ), R 1 |∂ 1 |∂ 2 θ .
In order to bound the term I 1 1 , one may combine Hölder Inequality and (52). As 0 < s ≤ 1/2, we get

I 1 1 ≤ u L ∞ ∂ 1 θ H 1 |∂ 1 | 1+2s θ L 2 , ≤ C u L 2 + ω L 4 ∂ 1 θ 2 H 1 .
In consequence, by virtue of (36), ( 40) and (55), we have (58)

t 0 I 1 1 (τ )dτ ≤ C(t, κ, θ 0 , u 0 ).
As for I 1 1 , we use the fact that (59)

I 1 1 ≤ |∂ 1 | 2s (u 2 ∂ 1 θ) L 2 ∂ 1 ∂ 2 θ L 2 . Because s ∈ (0, 1/2], we have |∂ 1 | 2s (u 2 ∂ 1 θ) L 2 ≤ u 2 ∂ 1 θ H 1 , whence |∂ 1 | 2s (u 2 ∂ 1 θ) L 2 ≤ u 2 ∂ 1 θ L 2 + ∂ 1 θ∇u 2 L 2 + u 2 ∂ 1 ∇θ L 2 , ≤ u L ∞ ∂ 1 θ L 2 + ∇u 2 L 4 ∂ 1 θ L 4 + u L ∞ ∇∂ 1 θ L 2 .
Thanks to the Sobolev embedding H 1 ֒→ L 4 and to ( 23), (52) we get

|∂ 1 | 2s (u 2 ∂ 1 θ) L 2 ≤ C u L ∞ + ω L 4 ∂ 1 θ H 1 .
Coming back to (59) and using (52), one can now conclude that (60)

t 0 I 1 1 (τ )dτ ≤ C(t, κ, θ 0 , u 0 ).
The term I 2 1 is more intricate to deal with. To start with, we integrate by parts to rewrite this term as follows:

I 2 1 ≤ u 2 ∂ 1 ∂ 2 θ R 1 |∂ 1 | 1+2s θ dx + |∂ 1 | s (u 2 ∂ 2 θ)|∂ 1 | 2+s θ dx ,
from which we get the following bound:

I 2 1 ≤ u 2 L ∞ ∂ 1 ∂ 2 θ L 2 |∂ 1 | 1+2s θ L 2 + |∂ 1 | s (u 2 ∂ 2 θ) L 2 |∂ 1 | 2+s θ L 2 .
As s ∈ (0, 1/2], Young inequality enables us to write (61)

I 2 1 ≤ u L ∞ ∂ 1 θ 2 H 1 + κ 2 |∂ 1 | 2+s θ 2 L 2 + 1 2κ |∂ 1 | s (u 2 ∂ 2 θ) 2 L 2 .
Let us admit (see the proof in appendix) that there exists a constant C such that for all s ∈ (0, 1/2] we have

(62) |∂ 1 | s (u 2 ∂ 2 θ) L 2 ≤ C u H 1 ∂ 2 θ L 2 + ∂ 1 ∂ 2 θ L 2 .
Using (52) and plugging (36), ( 40) and ( 55) in (61), we get (63)

t 0 I 2 1 (τ ) dτ ≤ C(t, κ, θ 0 , u 0 ) + κ 2 t 0 |∂ 1 | 2+s θ(τ ) 2 L 2 dτ.
It is now suitable to integrate (56) with respect to time and to plug (57), (58), (60) in (63). We eventually get for all s ∈ (0, 1/2],

(64)

|∂ 1 | 1+s |θ(t) 2 L 2 + κ t 0 |∂ 1 | 2+s θ dτ ≤ |∂ 1 | 1+s |θ 0 2 L 2 + C(t, κ, θ 0 , u 0 ).
Resorting to Lemma [START_REF] Danchin | Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces[END_REF] with s 1 = 1 + s and s 2 = 1, we find that

t 0 ∂ 1 θ 2 L ∞ dτ ≤ C t 0 ∂ 1 θ 2 L 2 + |∂ 1 | 2+s θ 2 L 2 + ∂ 1 ∂ 2 θ 2 L 2 dτ.
Therefore, by virtue of Inequalities ( 40) and (64), we get a bound for ∂ 1 θ in L 2 ([0, t]; L ∞ ) in terms of t and of the norms of the initial data. As explained before, this supplies the desired bound for the vorticity in L ∞ loc (R + ; L ∞ ).

3.5.

A global existence result. This paragraph is devoted to proving the following result (which obviously implies Theorem (2)):

Theorem 12. Let (θ 0 , u 0 ) ∈ H 1 with div u 0 . System (2) has a global solution (θ, u) such that (θ, u) ∈ C w (R + ; H 1 ) and ∂ 1 θ ∈ L 2 loc (R + ; H 1 ). If in addition ω 0 ∈ √ L then one may construct a global solution which also satisfies

ω ∈ L ∞ loc (R + ; √ L).
If in addition ω 0 ∈ L ∞ and there exists s ∈ (0, 1/2] such that |∂ 1 | 1+s θ 0 ∈ L 2 then the above solution is unique, strongly continuous in time with values in H 1 , and satisfies

|∂ 1 | 1+s θ ∈ C(R + ; L 2 ) and |∂ 1 | 2+s θ ∈ L 2 loc (R + ; L 2
). Proof: The result may be obtained by means of the Friedrichs method. With the notation of the previous section, we solve the following ODE in

L 2 n :      ∂ t θ + J n div (J n θ J n u) -κ∂ 2 1 PJ n θ = 0, ∂ t u + PJ n div (PJ n u ⊗ PJ n u) = PJ n (θe 2 ), (θ, u)| t=0 = J n (θ 0 , u 0 ). Cauchy-Lipschitz theorem gives a unique maximal solution (θ n , u n ) in the space C 1 ([0, T * n ); L 2 n ). As J 2 n = J n , P 2 = P et J n P = PJ n , we deduce that Pu n = u n , J n u n = u n and J n θ n = θ n . Therefore (θ n , u n ) satisfies (65) ∂ t θ n + J n div (θ n u n ) -κ∂ 2 1 θ n = 0, ∂ t u n + PJ n div (u n ⊗ u n ) = PJ n (θ n e 2 ).
As usual , because operators J n and PJ n are orthogonal projectors in all the Sobolev spaces, all the previous formal a priori estimates pertaining to Sobolev norms remind true. More precisely, we still have (36), (37) and (40) so that

• (θ n ) n∈N is bounded in L ∞ loc (R + ; H 1 ), • (∂ 1 θ n ) n∈N is bounded in L 2 loc (R+; H 1 ), • (u n ) n∈N is bounded in L ∞ loc (R + ; H 1
). This is fully enough to pass to the limit (up to extraction) in System (65) and to get the first part of the theorem.

In order to construct weak solutions preserving the √ L and the anisotropic regularities, one may smooth out System (2) by means of an artificial viscosity. More precisely, we first solve the following system for ε > 0:

     ∂ t θ + u • ∇θ -κ∂ 2 1 θ -ε∆θ = 0 ∂ t u + u • ∇u + ∇Π -ε∆u = θe 2 div u = 0
supplemented with smoothed out initial data (θ ε 0 , u ε 0 ). Resorting again to the Friedrichs method that has been used in the case ε = 0, and noticing that the cut-off operator J n does not modify the Sobolev estimates, we get a global solution (θ ε , u ε ) in C(R + ; H 1 ) ∩ L 2 loc (R + ; H 2 ) satisfying Inequalities (37) and (40) uniformly with respect to ε.

Actually, using standard methods, one can check that the H 2 regularity controls higher Sobolev norms. As the initial data are in H ∞ , the solution (θ ε , u ε ) thus belongs to all the Sobolev spaces, which will enable us to make the following computations rigorous.

The L p estimate over the vorticity may be proved by multiplying the vorticity equation

∂ t ω ε + u ε • ∇ω ε -ε∆ω ε = ∂ 1 θ ε by |ω ε | p-2 ω ε ,
and performing an integration over R 2 . This gives again

ω ε (t) L p ≤ ω 0 L p + t 0 ∂ 1 θ ε L p ≤ ω 0 L p + C √ pt ∂ 1 θ ε L 2 t (H 1 )
. It is also clear that all the anisotropic Sobolev estimates remain the same, uniformly with respect to ε. Therefore, having ε tend to 0 yields the end of the existence part of Theorem 10.

Finally, the uniqueness result is a mere consequence of Proposition 11. We thus have

∇u L ∞ ≤ S N ∇u L ∞ + q≥N ∆ q ∇u L ∞ ,
whence, using the definition of • LL and Bernstein inequalities,

∇u L ∞ ≤ (N + 1) ∇u LL + C q≥N 2 q ∆ q ∇u L 2 .
Given that ∆ q ∇u L 2 = ∆ q ω L 2 and that 2 -s < 0, we readily get

∇u L ∞ ≤ (N + 1) ∇u LL + C2 N (2-s) ω H s-1 . Now, if C ω H s-1 ≤
v L p ≤ q≥-1 ∆ q v L p , ≤ C q≥-1 2 -2q p 2 q ∆ q v L 2 , ≤ C q≥-1 2 -4q p 1 2 v H 1 , ≤ C √ p -1 v H 1 ,
whence the desired result.

Proof of Inequalities (39): For stating the first inequality, the starting point is the following classical one-dimensional Gagliardo-Nirenberg inequality:

(66) f (x 1 , •) L ∞ x 2 ≤ f (x 1 , •) 1/2 L 2 x 2 ∂ 2 f (x 1 , •) 1/2 L 2 x 2
.

Taking the L 2 x 1 norm of both sides and using Cauchy-Schwarz inequality, we get

f L 2 x 1 (L ∞ x 2 ) ≤ C f 1 2 L 2 (R 2 ) ∂ 2 f 1 2 L 2 (R 2 ) .
For proving the second inequality, it is only a matter of swapping the roles of variables x 1 and x 2 , and using Minkowski's inequality.

Proof of Inequality (46): From (66), we deduce that

f L ∞ ≤ C f 1/2 L ∞ x 1 (L 2 x 2 ) ∂ 2 f 1/2 L ∞ x 1 (L 2 x 2 ) .
Applying the second inequality of (39) to f and ∂ 2 f, it is now easy to complete the proof.

Proof of Inequality (62): Obviously, it suffices to state that

f g L 2 x 2 (H 1/2 x 1 ) ≤ C f H 1 g L 2 + ∂ 1 g L 2 .
For proving the above inequality, we first notice that the standard product laws for onedimensional Sobolev spaces ensure that for all fixed x 2 , we have

(f g)(•, x 2 ) H 1/2 (R) ≤ C f (•, x 2 ) H 1/2 (R) g(•, x 2 ) H 1 (R) . Therefore f g L 2 x 2 (H 1/2 x 1 ) ≤ C f L ∞ x 2 (H 1/2 x 1 ) g L 2 x 2 (H 1 x 1 )
.

Because the trace operator on x 2 = cste is continuous from H 1 (R 2 ) to H 1/2 (R), we get the desired inequality.

In the last part of the paper, anisotropic Sobolev norms have been used several times. Below, we state a sufficient condition under which anisotropic Sobolev spaces are embedded in the set of bounded functions. Lemma 14. For any couple (s 1 , s 2 ) of positive real numbers satisfying 1/s 1 + 1/s 2 < 2 there exists a constant C such that

u L ∞ ≤ C u L 2 + |∂ 1 | s 1 u L 2 + |∂ 2 | s 2 u L 2 .
Proof: Using Fourier variables, we see that

u(ξ) 2 L 1 (R 2 ) ≤ (1 + |ξ 1 | 2s 1 + |ξ 2 | 2s 2 )| u(ξ)| 2 dξ × (1 + |ξ 1 | 2s 1 + |ξ 2 | 2s 2 ) -1 dξ .
Therefore, it suffices to show that

(1 + |ξ 1 | 2s 1 + |ξ 2 | 2s 2 ) -1 dξ < ∞.
If we make the change of variable

ξ 1 = (1 + |ξ 2 | 2s 2 ) 1 2s 1 ζ 1 we get (1 + |ξ 1 | 2s 1 + |ξ 2 | 2s 2 ) -1 dξ = (1 + |ξ 2 | 2s 2 ) -1+ 1 2s 1 (1 + ζ 2s 1 1 ) -1 dζ 1 dξ 2 .
This integral is finite whenever s 1 > 1 2 and s 2 (1 -1 2s 1 ) > 1 2 , a condition which is equivalent to 1/s 1 + 1/s 2 < 2.

4.2.

Losing a priori estimates. The second part of the appendix is mainly devoted to the proof of losing a priori estimates for the following anisotropic Stokes system with convection (67) ∂ t w + v • ∇w -ν∂ 2 1 w + ∇Π = f + g e 2 , div w = 0 in the case where the gradient of the divergence free vector field is only in L 1 ([0, T ]; LL 1 2 ) (where LL 1 2 has been defined in (26)). Remind that those estimates are the key to the proof of uniqueness in Theorem 1. Albeit similar results have been proved before in [START_REF] Danchin | Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients[END_REF], we also prove losing a priori estimates for ordinary transport equations for the reader convenience.

The key to the proof of all those losing a priori estimates is the following commutator estimate (which is also used in the proof of Inequality ( 15)).

Lemma 15. Let v be a divergence free vector-field over R

2 . Let ω := ∂ 1 v 2 -∂ 2 v 1 .
There exists a positive constant C such that for all q ≥ -1, the term F q (v, ρ) := S q-1 v • ∇∆ q ρ -∆ q (v • ∇ρ) (with S q-1 defined in ( 14)) satisfies the following estimates :

F q (v, ρ) L 2 ≤ C ∇v L ∞ q ′ ≥q-4 2 q-q ′ ∆ q ′ ρ L 2 + ρ L ∞ |q-q ′ |≤4 ∆ q ′ ω L 2 , (68)
F q (v, ρ) L 2 ≤ C √ q + 2 ∇v LL 1 2 q ′ 2 -|q-q ′ | ∆ q ′ ρ L 2 . ( 69
)
In the case ρ = ω, we have in addition

(70) F q (v, ω) L 2 ≤ C ∇v L ∞ q ′ ≥q-4 2 q-q ′ ∆ q ′ ω L 2 . Proof: Decompose F q (v, ρ) into F 1 q (v, ρ) + F 2 q (v, ρ) + F 3 q (v, ρ) + F 4 q (v, ρ) with F 1 q (v, ρ) := q ′ ≥-1 [S q ′ -1 v, ∆ q ] • ∇∆ q ′ ρ, F 2 q (v, ρ) := q ′ ≥-1 S q-1 -S q ′ -1 v • ∇∆ q ∆ q ′ ρ, F 3 q (v, ρ) := -∆ q q ′ ≥1 S q ′ -1 ∂ i ρ ∆ q ′ v i , F 4 q (v, ρ) := - q ′ ≥0 ∂ i ∆ q ∆ q ′ v i |α|≤1 ∆ q ′ +α ρ .
Let us emphasize that only the term F 1 q involves low frequencies of v. Taking advantage of the support properties of the function ϕ defined at the beginning of Subsection 2.2, we notice that the summation in the definition of F 1 q may be restricted to those indices q ′ such that |q ′ -q| ≤ 4. Therefore, a standard commutator inequality (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2) ensures that (71)

F 1 q (v, ρ) L 2 ≤ C |q ′ -q|≤4 ∇S q ′ -1 v L ∞ ∆ q ′ ρ L 2 .
For F 2 q (v, ρ), we obtain, according to Hölder and Bernstein inequalities, and to the localization properties of the Littlewood-Paley decomposition, (72)

F 2 q (v, ρ) L 2 ≤ C |q ′ -q|≤1 ∇ ∆q v L ∞ ∆ q ρ L 2 with ∆q := |α|≤4 ∆ q+α .
From the definition of operator S q ′ -1 , the localization properties of operators ∆ q and Bernstein inequalities, we get (73)

F 3 q (v, ρ) L 2 ≤ q ′ ≤q+3 2 q ′′ -q ∆ q ′′ ρ L 2 ∆q ∇v L ∞ .
Notice that one can alternately get the following inequality :

(74)

F 3 q (v, ρ) L 2 ≤ C ρ L ∞ |q ′ -q|≤4 ∆ q ′ ω L 2 .
Indeed, it is only a matter of using that the sum defining F 3 q (v, ρ) may be restricted to q ′ ≥ 1 and thus, according to Bernstein inequalities and to ∇∆ q ′ v L 2 = ∆ q ′ ω L 2 , one may write

S q ′ -1 ∂ i ρ ∆ q ′ v i L 2 ≤ C S q ′ -1 ∂ i ρ L ∞ 2 -q ′ ∇∆ q ′ v i L 2 , ≤ C ρ L ∞ ∆ q ′ ω L 2 .
Finally the term F 4 q (v, ρ) may be bounded as follows:

(75) F 4 q L 2 ≤ q ′ ≥q-3 One can turn to the statement of losing a priori estimates. For technical reasons, we adopt the framework of Besov spaces B σ 2,∞ . As we have H σ ֒→ B σ 2,∞ and B σ 2,∞ ֒→ H σ ′ for all σ > σ ′ , it is of course not difficult to rewrite all those estimates in terms of Sobolev norms.

2 q-q ′ ∆ q ′ ρ L 2 ∇ ∆q ′ v L ∞ . Because ∇ ∆q v L ∞ ≤ C ∇v L ∞ and ∇ ∆q v L ∞ ≤ C q+2 ∇v
For the transport equation, we shall prove the following result (in the spirit of [START_REF] Bahouri | Équations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides[END_REF][START_REF] Danchin | Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients[END_REF]).

Proposition 16. Let ρ satisfy the transport equation

(76) ∂ t ρ + v • ∇ρ = f
with initial data ρ 0 ∈ B s 2,∞ and source term f ∈ L 1 ([0, T ]; B s 2,∞ ). Assume in addition that div v = 0 and that, for some V ∈ L 1 ([0, T ]) we have (77) sup

N ≥0 ∇S N v(t) L ∞ √ 1 + N ≤ V (t).
For all s ∈ (-1, 1), there exists a constant C depending only on s such that for all ε ∈ ]0, (s + 1)/2[ and t ∈ [0, T ], we have

ρ(t) B s-ε 2,∞ ≤ C exp C ε t 0 V (τ ) dτ 2 ρ 0 B s 2,∞ + f L 1 t (B s 2,∞ )
.

Proof: Applying ∆ q to Equation (76), one may write ∂ t ∆ q ρ + S q-1 v • ∇∆ q ρ = ∆ q f + F q (v, ρ) with F q (v, ρ) := S q-1 v • ∇∆ q ρ -∆ q (v • ∇ρ).

Taking the L 2 inner product of this inequality with ∆ q ρ and observing that div S q-1 v = 0, we thus get (78) ∆ q ρ(t)

L 2 ≤ ∆ q ρ 0 L 2 + t 0 ∆ q f L 2 dτ + t 0 F q (v, ρ) L 2 dτ.
From Inequality (69), we readily get for all ε ∈]0, (s + 1)/2[, q ≥ -1 and t ∈ [0, T ],

(79) 2 q(s-ε) F q (v(t), ρ(t))

L 2 ≤ C q + 2 V (t) ρ(t) B s-ε 2,∞
for some constant C depending only on s.

Set η = ε/ T 0 V (τ ) dτ and s t := s-η t 0 V (τ ) dτ for t ∈ [0, T ]. Putting (78) and (79) together yields 2 (2+q)st ∆ q ρ(t) L 2 ≤ 2 (2+q)s ∆ q ρ 0 L 2 2 -η(2+q) t 0 V (τ ′ ) dτ ′ + t 0 2 (2+q)sτ ∆ q f (τ ) L 2 2 -η(2+q) A similar result turns to be true for System 67. In addition, owing to the anisotropic viscosity, we get an extra horizontal smoothing (which was the key to the proof of Proposition 9. More precisely, we have: Proposition 17. Let v and s be as in Proposition 16. Then we have

w(t) B s-ε 2,∞ + ν 1 2 ∂ 1 w L 2 t (B s-ε 2,∞ ) ≤ C(1 + √ νt) exp C ε t 0 V (τ ) dτ 2 ρ 0 B s 2,∞ + f L 1 t (B s 2,∞ ) + ν -1 2 g L 2 t (B s-1 2,∞ ) .
Proof: Let us first apply operator ∆ q to System (67). With the notation introduced in the proof of Proposition 16, we have ∂ t ∆ q w + S q-1 v • ∇∆ q w -ν∂ 2 1 ∆ q w + ∇∆ q Π = ∆ q f + ∆ q g e 2 + F q (v, w) with F q (v, w) satisfying (79).

Taking the L 2 inner product and using the fact that div v = div w = 0, we see that (82) 1 2 d dt ∆ q w 2 L 2 + ν ∂ 1 ∆ q w 2 L 2 = ∆ q f • ∆ q w dx + F q (v, w) • ∆ q w dx + ∆ q g∆ q w 2 dx.

Assume that q ≥ 0. Taking advantage of Parseval equality, one may write ∆ q g∆ q w 2 dx = -(-∆) -1 ∆ q g ∆∆ q w 2 dx, = -(-∆) -1 ∆ q g ∆ q ∂ 2 1 w 2 dx -(-∆) -1 ∆ q g ∆ q ∂ 2 2 w 2 dx.

As div w = 0, integrating by parts yields ∆ q g∆ q w 2 dx = -(-∆) -1 ∆ q g ∆ q ∂ 2 1 w 2 dx + (-∆) -1 ∆ q g ∆ q ∂ 1 ∂ 2 w 1 dx, = ∂ 1 (-∆) -1 ∆ q g ∆ q ∂ 1 w 2 dx -∂ 2 (-∆) -1 ∆ q g ∆ q ∂ 1 w 1 dx.

Next, applying Bernstein and Young inequalities, we deduce that ∆ q g∆ q w 2 dx ≤ C2 -q ∆ q g

L 2 ∂ 1 ∆ q w L 2 ≤ ν 2 ∂ 1 ∆ q w 2 L 2 + C 2ν 2 -2q ∆ q w 2 L 2 .
Then coming back to (82) and integrating, we thus get for all q ≥ 0,

∆ q w 2 L ∞ t (L 2 ) + ν ∂ 1 ∆ q w 2 L 2 t (L 2 ) ≤ ∆ q w 0 2 L 2 +2 ∆ q f 2 L 1 t (L 2 ) + 2 F q (v, w) 2 L 1 t (L 2 ) + C ν 2 -2q ∆ q g 2 L 2
t (L 2 ) .

For q = -1, we merely have

∆ -1 w(t) L 2 ≤ ∆ -1 w 0 L 2 + t 0 ∆ -1 f L 2 + ∆ -1 g L 2 + F -1 (v, w) L 2 dτ. Of course ∂ 1 ∆ -1 w L 2 t (L 2 ) ≤ Ct 1 2 ∆ -1 w L ∞ t (L 2 )
. So finally, for all q ≥ -1, we have ∆ q w L ∞ t (L 2 ) + ν

1 2 ∂ 1 ∆ q w L 2 t (L 2 ) ≤ 2(1 + √ νt) ∆ q w 0 L 2 + ∆ q f L 1 t (L 2 ) + F q (v, w) L 1 t (L 2 ) + C ν 1 2
2 -q ∆ q g L 2 t (L 2 ) . With Inequality (79) at our disposal, it is now easy to conclude the proof of the proposition. It is just a matter of arguing exactly as in Proposition 16.

4 . Appendix 4 . 1 .

 441 A few inequalities. Here we prove a few inequalities which have been used throughout the paper. Proof of Inequality (20) : For proving (20), one may split ∇u into low and high frequencies according to the Littlewood-Paley decomposition. More precisely, for any N ∈ N one may write ∇u = S N ∇u + q≥N ∆ q ∇u.

LL

1 2 ,

 12 Inequalities (71) to (75) enable us to get (68) and (69). Inequality (70) stems from (68).

  ∇u LL then taking N = 0 obviously yields the desired inequality.

				Else,
	one may choose for N the integer part of
	and we still get the desired result.	1 s -2	log 2	C ω H s-1 ∇v LL

Lemma 13.

In dimension two, the Sobolev space H 1 continuously embeds in the space √ L.

Proof: For any p ∈ [2, ∞[ and v ∈ H 1 , using the Littlewood-Paley decomposition and a Bernstein inequality enables us to write

  t τ V (τ ′ ) dτ ′ dτ +C So finally, taking the supremum over q ≥ -1 in (80) and using the above two inequalities, we get which entails the desired inequality given that s ≥ s t ≥ s -ε for all t ∈ [0, T ].

	then the last term may be bounded by
				1 2	sup τ ∈[0,t]	ρ(τ ) B sτ 2,∞
	whereas if q does not satisfy (81) then it may be bounded by
			2C 2 η log 2	0	t	V (τ ) ρ(τ ) B sτ 2,∞ dτ.
	sup τ ∈[0,t] Thanks to Gronwall lemma, we end up with ρ(τ ) B sτ 2,∞ ≤ 2 ρ 0 B s 2,∞ + 2 0	t	f (τ ) B sτ 2,∞ dτ +	C η	0	t	V (τ ) ρ(τ ) B sτ 2,∞ dτ.
	sup t∈[0,T ]	ρ(t) B s t 2,∞	≤ 2e	C η	T 0 V (t) dt	ρ 0 B s 2,∞ +	0	T	f (t) B s t 2,∞	dt ,
	(80)
	Notice that if q satisfies
	(81)			2 + q ≥	4C 2 η 2 log 4

t 0 2+q V (τ )2 -η(2+q) t τ V (τ ′ ) dτ ′ ρ(τ ) B sτ 2,∞ dτ.

We agree that if X is a Banach space, and I ⊂ R, an interval then C b (I; X) stands for the set of continuous bounded functions on I with values in X.