On the global well-posedness of axisymmetric viscous Boussinesq system in critical Lebesgue spaces

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in \cite{Gallay,Gallay-Sverak}. Roughly speaking, we show essentially that if the initial data $(v_0,\rho_0)$ is axisymmetric and $(\omega_0,\rho_0)$ belongs to the critical space $L^1(\Omega)\times L^1(\mathbb{R}^3)$, with $\omega_0$ is the initial vorticity associated to $v_0$ and $\Omega=\{(r,z)\in\mathbb{R}^2:r>0\}$, then the viscous Boussinesq system has a unique global solution.


INTRODUCTION
The description of the state of a moving stratified fluid in three-dimensional under the Boussinesq approach, taking into account the friction forces is determined by the distribution of the fluid velocity v(t, x) with free-divergence located in position x at a time t, the scalar function ρ(t, x) designates either the temperature in the context of thermal convection, or the mass density in the modeling of geophysical fluids and p(t, x) is the pressure which is relates v and ρ through an elliptic equation. This provides us to the Cauchy problem for the Boussinesq system,  and BMO −1 . It should be noted that these types of solutions are globally well-posed in a time for initial data sufficiently small with respect to the viscosity, except in two-dimensional, see [37,41]. In a similar way, especially in dimension two of spaces, the system (B µ,κ ) was tackled by enormemos authors in various functional spaces and different values for the parameters κ and µ. For the connected subject, we refer to some selected references [25,27,28,30,36,39,44].
Before discussing some theoretical underpinnings results on the well-posedness topic for the viscous Boussinesq system (B µ,κ ) in three-dimensional. First, let us notice that the topic of global existence and uniqueness for (NS µ ) in the general case is a till an open problem in PDEs. It is therefore incumbent upon us to seek a subclass of vector fields which in turns leading to some conservation quantities, and so the global well-posedness result. Such subclass involving to rewriting (NS µ ) under vorticity formulation by applying the "curl" to the momentum equation, which is defined by ω = ∇ × v. Thus, we have: (1.1) Above, the triplet ( e r , e θ , e z ) represents the usual frame of unit vectors in the radial, azimuthal and vertical directions with the notation e r = x 1 r , x 2 r , 0 , e θ = − x 2 r , x 1 r , 0 , e z = (0, 0, 1).
For these flows the vorticity ω takes the form ω ω θ e θ , with Taking advantage to divv = 0, the velocity field can be determined clearly in the half-space Ω = {(r, z) ∈ R 2 : r > 0} by solving the following elliptic system under homogeneous boundary conditions v r = ∂ r v z = 0. The differential system (1.3) is wellknown the axisymmetric Biot-Savart law associated to (NS µ ), see Section 2.1.
A few computations claim that the streatching term ω · ∇v close to v r r ω θ and that ω θ evolves, with the notations v · ∇ = v r ∂ r + v z ∂ z and ∆ = ∂ 2 r + 1 r ∂ r + ∂ 2 z . By setting Π = ω θ r , we discover that Π satisfies Since the dissipative operator (∆ + 2 r ∂ r ) has a good sign, thus the L p −norms are time bounded, that is for t ≥ 0 Π(t) L p ≤ Π 0 L p , p ∈ [1, ∞].
(1.6) Under this pattern, M. Ukhoviskii and V. Yudovich [43], independently O. Ladyzhenskaya [35] succeed to recover (NS µ ) globally in time, whenever v 0 ∈ H 1 and ω 0 , ω 0 r ∈ L 2 ∩ L ∞ . This result was relaxed later by S. Leonardi, J. Màlek, J. Necȃs and M. Pokorný for v 0 ∈ H 2 and weakened recently in [1] by H. Abidi for v 0 ∈ H 1 2 . The majority of aforementioned results are accomplished within the framework of finite energy solutions. For the solutions with infinite energy, in particular in two dimensions many results dealing with the global well-posedness problem have been obtained by numerous authors. Particularly, worth mentioning that Giga, Miyakawa and Osada have been established in [22] that (NS µ ) admits a unique global solutions for initial vorticity is measure. Lately, M. Ben-Artzi [3] has shown that (NS µ ) is globally well-posed whenever the initial vorticity ω 0 belongs to critical Lebesgue space L 1 (R 2 ) who proposed a new formalism based on elementary comparison principles for linear parabolic equations. While, the uniqueness was relaxed later by H. Brezis [6]. Thereafter, this result was improved by I. Gallagher and Th. Gallay [17], where they constructed a solutions globally in time, under the assumption that ω 0 is a finite measure. For more details about this subject we refer the reader to the references [11,18,21].
More recently, the global well-posedness problem for (1.4) was revisited in three-dimensional by Th. Gallay and V.Sverák who established in [20] that (NS µ ) posseses a unique global solutions if the initial velocity is an axisymmetric vector field and its vorticity lying the critical space L 1 (Ω). In addition, they were extending their results in more general case, i.e., ω 0 is a finite measure, where Ω is half-plane Ω = {(r, z) ∈ R 2 : r > 0, z ∈ R} endowed with the product measure drdz.
Actually, their paradigm uses specifically the standard fixed point method to show in particular the local well-posedness for the system (1.4) under the form ∂ t ω θ + div ⋆ (vω θ ) = µ ∆ − 1 r 2 ω θ combined with the special structure of the vorticity, in particular, the axisymmetric Biot-Svart law, where div ⋆ f = ∂ r f r + ∂ z f z in general case. They showed that the local solutions often constructed can be extended to the global one by exploiting some a priori estimates for ω θ in various norms. We point us that the uniqueness topic for initial vorticity is measure was done by the same authors under some smallness condition for the ponctual part. Furthermore, they provide also an asymptotic behavior study for positive vorticity with a finite impulse.
Apropos of (B µ,κ ), the global regularity in dimension three of spaces has received a considerable attention. As shown in [13], for κ = 0 R. Danchin and M. Paicu investigated that (B µ,κ ) is well-posed in time for any dimension in the framework of Leray's and Fujita-Kato's solutions, excpect in three-dimensional and under also smallness condition, the system (B µ,κ ) is also global. Next, in the axisymmetric case, H. Abidi, T. Hmidi and S. Keraani were establishing in [4] that (B µ,κ ) is globally well-posed by rewriting it under vorticity-density formulation: Consequently, the quantity Π = ω θ r solving the equation They assumed that v 0 ∈ H 1 (R 3 ), Π 0 ∈ L 2 (R 3 ), ρ 0 ∈ L 2 ∩L ∞ with supp ρ 0 ∩(Oz) = / 0 and P z (supp ρ 0 ) is a compact set in R 3 , especially to dismiss the violent singularity of ∂ r ρ r , with P z being the orthogonal projector over (Oz). Those results are improved later by T. Hmidi and F. Rousset in [29] for κ > 0 by removing the assumption on the support of the density. Their strategy is deeply based on the coupling between the two equations of the system (1.9) by introducing a new unknown which called coupled function (see, Section 2). In the same way, more recently P. Dreyfuss and second author [14] treated the system in question, where they replaced the full dissipation by an horizontal one in all the equations, and proved the global wellposedness if the axisymmetric initial data In the same direction, in [34] the second and third authors, succeed to solve (B µ,κ ) globally in time for κ = 0 and axisymmetric initial data , by essentially combining the works of [1,13,29].
In the present paper, we are interested to conduct the same results recently obtained in [20] for the viscous Boussinesq system (B µ,κ ) expressed by the following vorticity-density formulation.
  9) for initial data (ω 0 , ρ 0 ) in the critical Lebesgue space L 1 (Ω) ×L 1 (R 3 ), with the following notations Let us denote that the spaces L 1 (Ω) and L 1 (R 3 ) are scale invariant, in the sense for any (ω 0 , ρ 0 ) ∈ L 1 (Ω) × L 1 (R 3 ) and any λ > 0. This is derived from the fact is a symmetry of (1.9). At this stage, we are ready to state the main result of this paper. To be precise, we will prove the following theorem.
be an axisymmetric initial data, then the system (1.9) for κ = 1 admits a unique global mild solution. More precisely, we have: (1.10) (1.11) Furthermore, for every p ∈ [1, ∞], there exists some constantK p (D 0 ) > 0, for which, and for all t > 0 the following statements hold.
A few comments about the previous Theorem are given by the following remarks.
Remark 1.2. By axisymmetric scalar function we mean again a function that depends only on the variable (r, z) but not on the angle variable θ in cylindrical coordinates. We check obviously that the axisymmetric structure is preserved through the time in the way that if (v 0 , ρ 0 ) is axisymmetric without swirl, then the obtained solution is it also. Remark 1.3. The hypothesis ω θ is in L 1 (Ω) doesn't implies generally that the associated velocity v is in L 2 (Ω) space. Consequently, the classical energy estimate is not available to derive a uniform bound for the velocity.
The proof is organized in two parts. The first one cares with the local well-posedness topic for (1.9) in the spirit of Gallay and Svérak [20]. We make use of fixed point-method for an equivalent system (1.9) on product space equipped with an adequate norm with the help of the axisymmetric Biot-Savart law and some norm estimates between the velocity and vorticity. But in our context, we should deal carefully wih the additional term ∂ r ρ r which contributes a singularity over the axe 5 (Oz). The remedy is to hide this term by exploiting the coupling structure of the system (1.9) for κ = 1 and introducing a new unknown functions Γ and Γ in the spirit of [29] by setting Γ = ω θ r − ρ 2 , and Γ = rΓ. A straightforward computation shows that Γ and Γ solve, respectively (1.14) (1. 15) In fact, in the second part, we shall investigate some a priori estimates for the different variables, in order to derive the global regularity for the system in question. Significant properties of the new unknowns, such as the maximum principle, are gained in this transition and Γ evolves a similar equation and keeps the same boundary conditions than Π in the case of the axisymmetric Navier-Stokes without swirl, see (1.5). As consequence, the function Γ (and eventually Γ after some technical computations) satisfies the boundedness estimate as in (1.6) which will be crucial in the process of deriving the global regularity of our solutions. For the reader's convenience, we provide a brief headline of this article. In section 2, we briefly depict the framework that exists regarding the axisymmetric Biot-Savart law. Many results could be spent in explaining this framework in detail, in particular, the relation between the velocity vector field and its vorticity by means of stream function. Along the way, we recall some weighted estimates which will be helpful in the sequel. Afterwards, we focus in the linear equation of (1.9) and some characterization of their associated semigroup, in particular the L p → L q estimate as in two-dimension space. Section 3, mainly treats the local well-posedness topic for the system (1.9). The main tool is the fixed point argument on the product space combined with a few technics about the semigroup estimates. In section 4, we investigate some global a priori estimates by coupling the system (1.9) and introducing the new unknowns Γ and Γ. Considering these latest quantities will be a helpful to derive the global existence for the equivalent system (1.9) and consequently the system (B µ,κ ).

SETUP AND PRELIMINARY RESULTS
In this section we recall some basic tools which will be employed in the subsequent sections. In particular, we devellop the Biot-Savart law in the framework of axisymmetric vector fields, and we study the linear equation associated to the system (1.9), usually specialized to the local existence.

The tool box of Biot-Savart law.
Recalling that in the cylindrical coordinates and in the class of axisymmetric vector fields without swirl the velocity is given by v = (v r , 0, v z ) with v r and v z are independtly of θ −variable, ω θ its vorticity defined from Ω into R by ω θ = ∂ z v r − ∂ r v z and the divergence-free condition divv = 0 turns out to be In this case, it is not difficult to build a scalar function Ω ∋ (r, z) → ψ(r, z) ∈ R which called axisymmetric stream function and satisfying Consequently, one obtains that ψ evolves the following linear elliptic inhomogeneous equation It is evident that L is an elliptic operator of second order, then according to [42], L is invertible with an inverse L −1 . Consequently, the last boundary value problem admits a unique solution given by where the function F : (0, ∞) → R is expressed as follows.
Since, F cannot be expressed as an elementary functions, but it contributes some asymptotic properties near s = 0 and s = ∞ listed in the following proposition. For more details about the proof, see [15,42].
Proposition 2.1. Let F be the function defined in (2.4), then the following assertions are hold.
Thus in view of (2.3), Ψ takes the form 7 with K can be seen as the kernel of the last integral representation. The last formula together with (2.1) claim that there exists a genuine connection between the velocity and its vorticity, namely, axisymmatric Biot-Savart law which reads as follows Here, with the notation and (2.8) A worthwhile property of the kernals K r and K z are given in the following result. For more details about the proof, see [20].
Now, we state the first consequence of the above result, in particular the L p → L q between the velocity and its vorticity, specifically we establish.

Proposition 2.3.
Let v be an axisymmetric velocity vector associated to the vorticity ω θ via the axisymmetric Biot-Savart law (2.6). Then the following assertions are hold. (2.11) Proof. (i) Combining (2.6) and (2.9), we get The last two integrals of the right-hand side are be seen as a singular integral. So, by hypothesis (ii) Let R > 0, then in view of (2.9) we have. 8 where Then by easy computations achieve the estimate.
In the axisymmetric case the weighted estimates practice a decisive role to bound some quantities like r α v in Lebesgue spaces for some α. Now, we state some of them which their proofs can be found in [15,20].
Characterizations of semigroups associated with the linearized equation. We focus on studying the linearized boundary initial value problem associated to the system (1.9) and we state some properties of their semipgroups. Specifically, we consider in the product space Ω × R 3 , with Ω = {(r, z) ∈ R 2 : r > 0} is the half-space by prescribing the homogeneous Dirichlet conditions at the boundary r = 0 for ω θ variable. For (ω 0 , ρ 0 ) ∈ L 1 (Ω) × L 1 (R 3 ), the solution of (2.13) is given explicitely by where (S 1 (t)) t≥0 and (S 2 (t)) t≥0 being respectively the semigroups or evolution operators associated to the dissipative operators (∆ − 1 r 2 ) and ∆. Such are characterized by the following explicit formulae, namely we have. (2.15) Proof. We assume that (ω θ , ρ) solving (2.13), then a straigthforward computations claim that (ω, ρ), with ω = ω θ e θ satisfying the usual heat equation ∂ t ω − ∆ω = 0 and ∂ t ρ − ∆ρ = 0 in R 3 with initial data (ω(0, ·), ρ(0, ·)). Therefore, for every t > 0 we have (2.16) We will develop each term in the cylindrical basis ( e r , e θ , e z ) by writing x = (r cos θ , r sin θ , z) and x = ( r cos θ , r sin θ , z), hence the first equation of (2.16) takes the form , thus we have (2.18) To treat I 1 , we set α = θ − θ 2 then we have Then the last estimate becomes Similarly, Combining the last two estimates and plug them in I 1 we reach the desired estimate. 10 For the second equation in (2.16) we express the density formula in ( e r , e θ , e z ) basis (2.19) Setting The same variable α = θ − θ 2 allows us to write Plug I 2 in (2.19), we get the result. This ends the proof of the Proposition.
The following Proposition provides some asymptotic behavior of the functions N 1 and N 2 near 0 and ∞, which will be fundamental in the sequel. Proposition 2.6. Let N 1 , N 2 : (0, ∞) → R be the functions defined in (2.15). Then the following statements are hold.
Note that lim t↑0 II 2 = 0, so the behavior of N 1 near 0 comes from II 1 . Hence, let us deal with II 1 , we insert the Taylor expansion of the function ζ → 1 √ 1−tζ 2 in the integral of II 1 to obtain .

It is straightforward to show that
Consequently, lim t↑0 II 1 = 1. Combining all the previous quantities, we find the asymptotic behavior of II 1 near 0, that is, By derivation of II 1 , we find the behavior of N ′ 1 . (ii) The Mac Laurin's expansion of the function α → e − sin 2 α t at 0 is given by .
After an easy computations we achieve the estimate.
(iii) To prove this assertion, setting y = sin α √ t in N 2 and we split the integral into two parts, one has We follow the same steps as N 1 . For the second integral in right-hand side, we have Let us observe that the last estimate goes to 0 as t ↑ 0, so the asymptotic behavior of N 2 near 0 comes only from the first integral. To be precise, it is clear that t → 1 √ 1−ty 2 is bounded function whenever 0 < y < 1 Thus, the expansion of the function x → (1 − x) − 1 2 for x = ty 2 enuble us to write (iv) Using the fact sin α ≃ α near 0, then we get We set y = α √ t , clearly that y ↑ 0 as t ↑ ∞ and the power expansion of the function e y near 0 yields the asymptoyic expansion, whereas N ′ 2 is a direct derivative of N 2 expansion.
Some consequences of the previous Proposition are listed in the following remark.
(i) It should be noted that the functions t → N 1 (t) and t → N 2 (t) are decreasing over ]0, ∞[, but the proof seems very hard.
Other nice properties of (S i (t)) t≥0 , with i = 1, 2, in particular the estimate L p → L q are given in the following result.
Here, div ⋆ f = ∂ r f r + ∂ z f z (resp. div f = ∂ r f r + ∂ z f z + f r r ) stands the divergence operator over R 2 (resp. the divergence operator over R 3 in the axisymmetric case).
Proof. (i) We follow the proof of [20] with minor modifications, for this aim let (r, z), ( r, z) ∈ Ω, we will prove the following worth while estimates      1 4πt . (2.23) We distinguish two cases r ≤ 2r and r > 2r.
(ii) By definition for every (r, z) ∈ Ω, we have After an integration by parts, it happens and We proceed by the same manner as above. The fact that the functions N 1 , N ′ 1 and t → t α N 1 (t),t → t α N ′ 1 (t) are bounded, see Remark 2.7, one finds and | f z ( r, z)|d rd z. 14 Together with Young's inequality, we obtain (2.21).
(iii) Let (r, z) ∈ Ω, then we have The two terms III 3 and III 4 ensue by the same argument as in (ii). It remains to treat the term III 5 in the following way The fact that (·/r r) 1/2 N 2 (·/r r) is bounded guided to By pluging the last estimate in (2.25) and combine it with (2.24), it follows Then a new use of Young's inequality leading to the result. To close our claim, it remains to establish that R + ∋ t → S 1 (t) (resp. R + ∋ t → S 2 (t)) is continuous on L p (Ω) (resp. on L p (Ω)). We restrict ourselves only for (S 1 (t)) t≥0 . Let ω 0 ∈ L p (Ω) and define its extension on R 2 by ω 0 which equal to 0 outside of Ω. Thus, in view the change of variables r = r + √ tϑ and z = z + √ tγ, the statement (2.14) takes the form where ϒ(t, r, z, ϑ , γ) = 1 + √ tϑ r Taking the L p −estimate of (2.26), then with the aid of the following Minkowski's integral formula in general case Now, we must establish that ϒ(t, r, z, ϑ , γ) L p (Ω) → 0 as t ↑ 0. To do this, let r > 0 and r + √ tϑ > 0. Writting Therefore On the other hand, it is clear to verify that 1 tϑ ) goes to 1 as t ↑ 0. Thus, Lebesgue's dominated convergence asserts for (ϑ , γ) ∈ R 2 that ϒ(t, r, z, ϑ , γ) L p (Ω) → 0 when t ↑ 0. A new use of Lebesgue's dominated convergence, we finally deduce lim t↑0 S 1 (t)ω 0 (r, z) − ω 0 (r, z) L p (Ω) → 0, (2.27) which accomplished the proof.
In the spirit of Proposition 3.5 in [20], another weighted estimates for the linear semigroup (2.14) are shown in the following proposition, the proof of which can be done by the same reasoning as in the previous proposition, We end this section by recalling the following classical estimate on the heat kernel in dimension three, the proof of which is left to the reader. (2.30)

LOCAL EXISTENCE OF SOLUTIONS
We will explore the aformentioned results and some preparatory topics in the previous sections, we shall scrutinize the local well-posedness issue for the system (1.9). For this reason, we rewrite it in view of the divergence-free condition in the following form The direct treatment of the local well-posedness topic for (3.1) in the spirit of [20] for initial data (ω 0 , ρ 0 ) in the critical space L 1 (Ω) × L 1 (R 3 ) contributes many technical difficulties. This motivates to add the following new unknown ρ rρ which solves We remark that ρ satisfies the same equation as ω θ with additional source term and their variations are in Ω.
To achieve our topic we will handle with the following equivalent integral formulation.
In order to analyze the above system, we will be working in the following Banach spaces.
equipped with the following norms Now, our task is to prove the following result.
admits a unique local solution satisfying Proof. We will proceed by the fixed point theorem in the product space X T = X T × X T × Z T equipped by the norm Notice that by definition, we have For t ≥ 0, define the free part (ω lin (t), ρ lin (t), ρ lin (t)) = S 1 (t)ω 0 , S 1 (t)(rρ 0 ), S 2 (t)ρ 0 , where S 1 (t), S 2 (t) is given in Proposition 2.5. In accordance with the (i)-Proposition 2.8, it is not difficult to check that for (ω 0 , ρ 0 ) ∈ L 1 (Ω) × L 1 (R 3 ), we have for T > 0 and sup 0<t≤T t 1/4 ρ lin (t) On the other hand, the fact that together with (2.28) stated in Proposition 2.9, we further get Combining (3.5), (3.6) and (3.7) to obtain that (ω lin , ρ lin , ρ lin ) ∈ X T . Next, define the following quantity which will be useful in the contraction.
We claim that Λ(ω 0 , ρ 0 , T ) → 0 when T ↑ 0. To do this, we employ the fact ( . Then for every ε > 0 and every On account of (i)-Proposition 2.8 we write Multiply the both sides by t 1/4 and taking the supremum over (0, T ] to get

Thus, by setting
and let T (resp. ε) goes to 0, one deduces Now, we multiply the both sides by t 3/8 and taking the supremum over (0, T ] to deduce Similarly, by putting we shall obtain that lim Now, we are ready to contract the integral formulation (3.3) in X T . Doing so, define for (ω θ , ρ, ρ) ∈ X T the map We aim at estimating Due to the similarity of the first two lines of (3.16), we will restrict ourselves to analyse the first and the third ones. For dτ. 19 Thanks to (2.10), it follows that We show next how to estimate t 0 S 1 (t −τ)∂ r ρ(τ)dτ in L 4 3 (Ω). In view of Proposition 2.9 for α = 0 and β = 3 4 , we get The above estimates combined with (3.8) provide the following inequality As explained above, the estimate of t 0 S 1 (t − τ)div ⋆ v(τ) ρ(τ) dτ can be done along the same lines, so we have we deduce that Let us move to estimate the last line in (3.16). Under the remark div(vρ) = v r r ρ + div ⋆ (vρ), we write dτ (3.20) 20 So, for the first term, we shall apply (2.28) stated in Proposition 2.9 for α = 3 4 and β = 2 to get The second term of the r.h.s. in (3.20), will be done by a similar way as above, but we employ (2.29) in Proposition 2.9 for α = 3 4 and β = 1, one may write Under the conditions 1 The details can be done by following the same approach of Lemma 5.1 from [20] concerning the Navier-Stokes equations, but in our case we have an additional term ∂ r ρ which its integral vanishes over Ω.

22
we shall obtain (3.27) We recall that ρ evolves the same equation as ω θ , so we have (3.28) Finally, to claim similar estimate for J p (ρ, T ), first we write Under the additional condition on p, q 1 , q 2 (3.29) Plugging (3.29) in (3.27) and (3.28) for q = 4 3 , and by denoting we deduce, that Now, to cover all the rang p ∈ ( 4 3 , ∞), we proceed by the following bootstrap algorithm: • For q 1 = q 2 = 4 3 we obviously check that U p (T ) → 0 as T → 0 for all 1 < p < 3 2 . • Next, by taking q 1 = q 2 sufficiently close to 3 2 , we obtain the same result, for all p < 9 5 . • For q 1 = q 2 = 8 5 , the estimate in question holds for all p < 2. • Taking q 1 sufficiently close to 2, the result follows for all p < 3 2 q 2 and for all q 2 < 2. • Finally, we define the sequence p n by p 0 = 4 3 and p n sufficiently close to 3 2 p n−1 , by induction, we find that p n is sufficiently close to ( 3 2 ) n p 0 . Hence, letting n goes to ∞, we can cover all the rage p < ∞, and thus we obtain Our last task of this section is to reach the continuity of the solution stated in (1.10) and (1.11) of the main Theorem 1.1. For this aim, we briefly outline the continuity of ω θ , the rest of quantities can be treated along the same lines. So, we will show that To do so, let 0 < t 0 ≤ t < T ⋆ (t 0 close to 0 for p = 1), so we have (3.31) The first term (free part) is derived by the same manner as in (2.27), that is to say, (3.32) Concerning the second term in the r.h.s of (3.31), (2.21) in Proposition 2.5 provides By virtue of the following interpolation estimate, see, Proposition 2.3 in [20], we have for some which is sufficient to obtain Let us move to the last term of (3.31) which we distinguish two cases for p. For p ∈ (1, ∞), (2.29) stated in Proposition 2.9 for α = 0 and β = 1 p yielding (3.34) and the fact that ρ ∈ L ∞ (0, T * ); L p (R 3 ) ensures that For the case p = 1, we will work with Γ instead of ω θ to avoid the source term ∂ r ρ. The fact rρ L 1 (Ω) = ρ L 1 (R 3 ) leading to so, the continuity of ω θ (·) L p (Ω) relies then on the continuity of Γ(·) L p (Ω) and ρ(·) L 1 (R 3 ) . On the one hand, seen that the equation ofΓ governs the same equation to that of ω θ , but without the source term ∂ r ρ, hence we follow then the same appraoch as above to prove that On the other hand, ρ solve a transport-diffusion equation, for which the continuity property is well-known to hold, thus we skip the details. Therefore Combining the last estimate with (3.32), (3.33) and (3.35), we achieve the result.

GLOBAL EXISTENCE
To reach the global existence for the local solution often formulated in sections 3, we will establish some a priori estimates in Lebesgue spaces. For this target, be a solution of the integral formulation (3.3) and so does (ω θ , ρ) to the differential equation (3.1) associted to initial data (ω 0 , ρ 0 ) ∈ L 1 (Ω) × L 1 (R 3 ), where T * denotes the maximal time of existence. Our basic idea is to couple the system (3.1) by introducing the new unknown Γ = Π − ρ 2 following [29] with Π = ω θ r . Some familiar computations show that Γ obeys (4.1) For our analysis, we need to introduce again the unknown Γ rΓ = ω θ − ρ 2 , which solves ∂ t Γ + div ⋆ (v Γ) − (∆ − 1 r 2 ) Γ = 0 if (t, r, z) ∈ R + × Ω, Γ |t=0 = Γ 0 . (4. 2) The role of the new function Γ (resp. Γ) for the viscous Boussineq system (B µ,κ ) is the same that Π (resp. ω θ ) for the Navier-Stokes equations (NS µ ). For this aim, it is quite natural to treat carefully the properties of Γ and Γ. The starting point of our analysis says that Γ enjoys the strong maximum principle. We will prove the following.
Generally if Γ 0 changes its sign, we procced as follows: we split Γ(t) = Γ + (t) − Γ − (t), where Γ ± solves the following linear equation with the same velocity ∂ t Γ ± + v · ∇Γ ± − (∆ + 2 r ∂ r )Γ ± = 0 if (t, x) ∈ R + × R 3 , Γ ± |t=0 = max(±Γ 0 , 0) ≥ 0. (4.6) Arguiging as above to obtain that Γ ± satisfies (4.4). Thus we have: . If Γ 0 = 0, we distinguish that Γ 0 > 0 or Γ 0 < 0. For this two cases the last inequality is strict and consequently (4.4) is also strict. Therefore, t → Γ(t) L 1 (R 3 ) is strictly decreasing for t = 0, and analogously we deduce that is strictly decreasing over [0, T ]. Now, we state a result which deals with the asymptotic behavior of the coupled function Γ in Lebegue spaces L p (R 3 ). Specifically, we have. Proposition 4.3. Let ρ 0 , ω 0 r ∈ L 1 (R 3 ), then for any smooth solution of (4.1) and 1 ≤ p ≤ ∞, we have Thanks to the well-known Nash's inequality in general case We prove (4.8) for p = 2 n with nonnegative integers n by induction. Assume that (4.8) is true for q = 2 k with k ≥ 0, and let p = 2 k+1 . Combined with (4.11) Thus we have Hence, integrating in time le last inequality yields After a few easy computations, we derive the following By setting C p = 3p 8C 3 2p C q , then (4.8) remains true for p = 2 k+1 . Let us observe that which means that C ∞ is independtly of p. Letting p → ∞, we deduce that For the other values of p, we proceed by complex interpolation to get , combined with (4.12), so the proof is completed.
Next, we recall some a priori estimates for ρ−equation in Lebesgue spaces. To be precise, we have.