International Journal of Analysis and Applications Volume 19, Number 1 (2021), 110-122 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-110 STRONG SOLUTIONS TO 3D-LAGRANGIAN AVERAGED BOUSSINESQ SYSTEM RIDHA SELMI1,3,4,∗, LEILA AZEM2,4 1Department of Mathematics, College of Sciences, Northern Border University, P.O. Box 1321, Arar, 73222, KSA 2Department of Mathematics, College of Sciences and Art (TURAIF), Northern Border University, KSA 3Department of Mathematics, Faculty of Sciences of Gabès, University of Gabès, Gabès, 6072, Tunisia 4Laboratory of partial differential equations and applications (LR03ES04), Faculty of sciences of Tunis, University of Tunis El Manar,Tunis, 1068, Tunisia ∗Corresponding author: Ridha.selmi@nbu.edu.sa Abstract. Under suitable assumptions on the initial data, we prove the existence, uniqueness of the strong solutions to a regularized periodic three-dimensional Lagrangian averaged Boussinesq system, in a Sobolev spaces. Also, we establish the convergence results of this unique strong solution of this regularized Boussinesq system to a strong solution of the three-dimensional Boussinesq system, as the regularizing parameter vanishes. 1. Introduction and statement of main results We consider the following 3D-Lagrangian averaged Boussinesq-α system: Received April 4th, 2020; accepted April 22nd, 2020; published December 17th, 2020. 2010 Mathematics Subject Classification. primary 35A01, 35A02, 35B40; secondary 35B10. Key words and phrases. 3D-Lagrangian averaged Boussinesq-α system; alpha-regularization; existence; uniqueness; convergence. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 110 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-110 Int. J. Anal. Appl. 19 (1) (2021) 111 (1.1)   ∂v ∂t + (u ·∇)v + ∇uT ·v + ∇p = ν∆v + θe3, (t,x) ∈ R+ ×T3 ∂θ ∂t −κ∆θ + (u ·∇)θ = 0, (t,x) ∈ R+ ×T3 v = (1 −α2∆)u, (t,x) ∈ R+ ×T3 div u = div v = 0, (t,x) ∈ R+ ×T3 (u,θ)|t=0 = (u0,θ0), x ∈ T 3, where by T3 we refer to the 3D-torus, ν > 0 represents the viscosity of the fluid and κ > 0 its thermal conductivity. The unknown vector field u, the scalars p and θ stand respectively for the velocity, the pressure and the temperature of the fluid at the point (t,x) ∈ R+ × T3. The superscript MT denotes the transpose of the matrix M. The data (u0,θ0), are respectively the initial free divergence velocity and the initial temperature. The Lagrangian averaged Boussinesq-α model (1.1) is the first to use Lagrangian averaging to address the turbulence closure problem and also in geophysical modeling [4]. The Boussinesq equations were derived in the nineteenth century by Joseph Boussinesq, despite intense study, there still remain many difficulties and open questions concerning them. Namely, the Boussinesq system have an incomplete solution theory. It is not known whether global in time strong solutions exist, and although we know weak solutions exist, we are still unable to prove their uniqueness. Regularization models are a way to come up with a rather well-posed problem theory. The first attempt to regularize the Navier-Stokes equations was made in [9] via smoothing the convective velocity by taking the convolution product against a mollifier. Here, and among many others methods of regularization presented in the literature, we note that α-regularization models are obtained by applying a smoothing via taking the inverse of the Helmholtz operator I − α2∆. The interested reader is referred to [3, 5, 6]. Especially, the LANS-α model acts by modifying the nonlinearity in the Navier-Stokes equations without introducing any extra dissipation [6]. It can be seen as a systematic method for modelling the mean circulatory effects of small-scale turbulence, while preserving the mathematical properties that guarantee existence of a unique, regular solution [3, 5]. The inviscid Lagrangian averaged Euler-α equations were originally derived as Euler–Poincaré equations in the framework of Hamilton’s principle for geometric fluid mechanics [7]. The existence, uniqueness and continuous dependance of solutions to initial date, as well as convergence results of various α-models, as α vanishes can be found in [1, 2, 8, 10, 12] and references therein. Before stating our main results, let us introduce some notations that will be used throughout the paper. Int. J. Anal. Appl. 19 (1) (2021) 112 • For n ∈ N, we denote by Pn the projection into Fourier modes of order up to n, that is Pn( ∑ k∈Z3 ûk e ikx) = ∑ |k|≤n ûk e ikx. • For s > 0, we define the operator Λs acting on Hs(T3) as follows. Let u ∈ Hs(T3) having the Fourier series u(x) = ∑ k∈Z3 ûk e ikx ∈ Hs(T3). Then, we define Λsu(x) = ∑ k∈Z3 |k|sûk eikx ∈ L2(T3). • We denote by ‖ · ‖Ḣs the seminorm ‖Λ s · ‖L2 . This is compatible with the definition of the Sobolev norm. In fact, ‖ · ‖Hs is equivalent to ‖ · ‖L2 + ‖ · ‖Ḣs . Note that in the Fourier setting it is more usual to define an equivalent norm on Hs by ‖u‖ = ( ∑ k∈Z3 (1 + |k|2s)|ûk|2 )1/2 . • We refer to the fractional Laplacian √ −∆ by Λ. • We denote B(u,v) = [(u ·∇)v], u,v ∈ H1(T3), B̃(u,v) = [(∇×v) ×u], u,v ∈ H1(T3). • We define the space H̃s = {u ∈ Hs, div u = 0}. We remark that, for u, v and w ∈ H̃1(T3)( B(u,v),w ) = − ( B(u,w),v ) . • Due to the identity (1.2) (u ·∇)v + 3∑ j=1 vj∇uj = −u× (∇×v) + ∇(v ·u), we obtain ( B̃(u,v),w ) = ( B(u,v),w ) − ( B(w,v),u ) . • By the definitions of the operators B(u,v) and B̃(u,v), we deduce the following properties: Lemma 1.1. i) Let u,v,w ∈ H̃1(T3), then (1.3) ( B(u,v),w ) = − ( B(u,w),v ) which in turn implies that (1.4) ( B(u,v),v ) = 0, u,v ∈ H̃1(T3). Int. J. Anal. Appl. 19 (1) (2021) 113 Also, (1.5) ( B̃(u,v),w ) = ( B(u,v),w ) − ( B(w,v),u ) , u,v,w ∈ H̃1(T3) and hence (1.6) ( B̃(u,v),u ) = 0, u,v ∈ H̃1(T3) ii) Let u ∈ H̃1(T3), v ∈ H̃2(T3), and w ∈ L2(T3), then (1.7) | ( B(u,v),w ) | ≤ c‖∇u‖L2‖∇v‖ 1/2 L2 ‖4v‖1/2 L2 ‖w‖L2. iii) Let u,v,w ∈ H̃1(T3), then (1.8) | 〈 B̃(u,v),w 〉 H̃−1(T3) | ≤ c‖∇u‖L2‖∇v‖L2‖w‖ 1/2 L2 ‖∇w‖L2. Here < .,. > denotes the duality pairing of H̃1(T3) and H̃−1(T3). iv) Let u ∈ H̃1(T3), v ∈ L2(T3) and w ∈ H̃2(T3), then (1.9) | 〈 B̃(u,v),w 〉 H−2 | ≤ c ( ‖u‖1/2 L2 ‖∇u‖1/2 L2 ‖v‖L2‖∆w‖L2 + ‖v‖L2‖∇u‖L2‖∇w‖ 1/2 L2 ‖∆w‖1/2 L2 ) . Proof. See [10] and references therein. � Using the above notations and the identity (1.2), we obtain the following equivalent systems of equations: (1.10)   ∂v ∂t + B̃(u,v) −ν4v = θe3, (t,x) ∈ R+ ×T3 ∂θ ∂t −κ4θ + B(u,θ) = 0, (t,x) ∈ R+ ×T3 v = (1 −α24)u, (t,x) ∈ R+ ×T3 div u = div v = 0, (t,x) ∈ R+ ×T3 (u,θ)|t=0 = (u0,θ0), x ∈ T 3, Our first result is the following existence and uniqueness theorem. Theorem 1.1. Let θ0 ∈ Ḣ1(T3) and u0 ∈ Ḣ2(T3) a divergence free vector field. Then, there exists a unique strong solution (uα,θα) of system (1.10), such that uα ∈ C(R+,Ḣ2(T3)) ∩L2loc(R+,Ḣ 3(T3)) and θα ∈ C(R+,Ḣ1(T3)) ∩L2loc(R+,Ḣ 2(T3)). Int. J. Anal. Appl. 19 (1) (2021) 114 Moreover, ∀ 0 ≤ t ≤ T , this solution satisfies the following energy estimates (1.11) ‖∇θα‖2L2(T3) + ‖∇uα‖ 2 L2(T3) + α 2‖∆uα‖2L2(T3) + κ ∫ t 0 ‖∆θα‖2L2(T3)dτ + ν ∫ t 0 (‖∆uα‖2L2(T3) + α 2‖∇∆uα‖2L2(T3))dτ ≤Kα(T), where Kα(T) = K(T) 2µ + K3(T) 2µα6 (1 + α2) + ‖∇θ0‖2L2(T3) + ‖∇u 0‖2L2(T3) + α 2‖∆u0‖2L2(T3), and µ = min(ν,κ). Here, the function K(t) stands for K(t) = ‖u0‖2L2 + α 2‖∇u0‖2L2 + ‖θ 0‖2L2 + φα(t) and φα is a positive increasing function of time t, defined by (1.12) φα(t) = ( ‖u0‖2 L2 + α2‖∇u0‖2 L2 + ‖θ0‖2 L2 ) (e2t − 1). The proof is based on a Galerkin approximation scheme. While trying to close the energy estimates, the buoyancy force presents some difficulties that we overcome by a Gronwall’s type technique. After that, we run a compactness method based on Aubin-Lions lemma [11]. Our second result is a convergence theorem, as α → 0: Theorem 1.2. Let T > 0, u0 ∈ Ḣ2(T3) a divergence free vector field, θ0 ∈ Ḣ1(T3) and (uα,θα), the solution in [0,T] of system (1.10) and vα = uα −α2∆uα subject of Theorem 1.1. Then, there exists a time T∗ such that 0 < T∗ ≤ T and subsequences uαk , vαk , θαk , a scalar function θ and a divergence free vector field u belonging both of them to L∞([0,T∗],Ḣ1(T3)) ∩L2([0,T∗],Ḣ2(T3)) such that as αk → 0+, one has (1) uαk converges to u and θαk converges to θ weakly in L 2([0,T∗],Ḣ2(T3)) and strongly in L2([0,T∗],Ḣ1(T3)). (2) vαk converges to u weakly in L 2([0,T∗],Ḣ1(T3)) and converges strongly in L2([0,T∗],L2(T3)). (3) uαk (t) converges to u(t) and θαk (t) converges to θ(t) weakly in Ḣ 1(T3) and uniformly over [0,T∗]. Furthermore, (u,θ) is the unique strong solution of the Boussinesq system (Bq) on [0,T∗] associated to the initial data (u0,θ0) and satisfies, for all t ∈ [0,T∗], the energy inequality ‖u(t)‖2 L2(T3) + ‖θ(t)‖ 2 L2(T3) + 2 ∫ t 0 ν‖∇u(τ)‖2L2(T3) + κ‖∇θ(τ)‖ 2 L2(T3)dτ ≤‖u0‖2 L2(T3) + ‖θ 0‖2 L2(T3) + 2φ(T ∗). Here, (Bq) and φ denote respectively (1.10) and φα, for α = 0. Note that the solution of the regularized Lagrangian averaged Boussinesq-α system satisfies an energy inequality that depends on the parameter alpha which provide a singularity, as alpha goes to zero. This is a serious impediment that has to be dealt with when taking the limit as αk → 0. We shall use a compactness Int. J. Anal. Appl. 19 (1) (2021) 115 method to obtain strong convergence. The remainder of this paper is divided into two sections; the first is assigned to prove the existence and uniqueness result. The second is concerned by the proofs of convergence results. 2. Well-posedness result To study the existence and the regularity of strong solutions. We approximate (1.10) by the following system of ordinary differential equations: (2.1)   ∂vn ∂t + PnB̃(un,vn) −ν4vn −θne3 = 0, (t,x) ∈ R+ ×T3 ∂θn ∂t −κ4θn + PnB(un,θn) = 0, (t,x) ∈ R+ ×T3 vn = (1 −α24)un, (t,x) ∈ R+ ×T3 div un = div vn = 0, (t,x) ∈ R+ ×T3 (un,θn)|t=0 = (Pnu0,Pnθ0), x ∈ T 3. Both the bilinear operators on the left are continuous on L2×L2. Then, the above system appears as a system of ordinary differential equations on L2. Thus, the usual Cauchy–Lipschitz theorem yields the existence of a strictly positive maximal time Tn such that a unique solution exists which is continuous in time with value in L2. Next, we obtain uniform estimates, with respect to the approximating parameter n, on the approximate solutions. To do so, we use conservation laws and product lemmas. Taking the L2(T3)-inner product of the equation satisfied by θn in (2.1) against −∆θn and the one satisfied by un against −∆un, to obtain 1 2 d dt ‖∇θn‖2L2(T3) + κ‖δθn‖ 2 L2(T3) = 〈B(un, ,θn), ∆θn〉(2.2) (2.3) 1 2 d dt ( ‖∇un‖2L2(T3) + α 2‖∆un‖2L2(T3) ) + ν ( ‖∆un‖2L2(T3) + α 2‖∇∆un‖2L2(T3) ) = 〈B̃(un,vn), ∆un〉 + 〈θne3,−∆un〉. Now, let us first estimate the right hand sides of (2.2) and (2.3). To do so, we recall the following Sobolev inequalities [11]: for every ϑ ∈ Ḣ1(T3), we have ‖ϑ‖L3 ≤‖ϑ‖ 1/2 L2 ‖ϑ‖1/2 Ḣ1 (2.4) and ‖ϑ‖L6 ≤ c‖ϑ‖Ḣ1.(2.5) Using Hölder’s inequality, (2.4) and (2.5), it holds that |〈B(un,θn), ∆θn〉| ≤ ‖un‖L6(T3)‖∇θn‖L3(T3)‖∆θn‖L2(T3) ≤ c‖un‖Ḣ1‖∇θn‖ 1/2 L2 ‖∇θn‖ 1/2 Ḣ1 ‖∆θn‖L2. Int. J. Anal. Appl. 19 (1) (2021) 116 Hence, one obtains |〈B(un,θn), ∆θn〉| ≤ c‖un‖Ḣ1‖∇θn‖ 1/2 L2 ‖∆θn‖ 3/2 L2 .(2.6) For every ϑ ∈ Ḣ2(T3), the Agmon’s inequality [11] reads ‖ϑ‖L∞ ≤‖ϑ‖ 1/2 Ḣ1 ‖ϑ‖1/2 Ḣ2 .(2.7) The fact that vn = un −α2∆un yields∣∣∣〈B̃(un,vn), ∆un〉∣∣∣ = |〈B(un,vn), ∆un〉−〈B(∆un,vn),un〉| ≤ |〈B(un,un), ∆un〉| + α2 |〈B(un, ∆un), ∆un〉| + |〈B(∆un,un),un〉| + α2 |〈B(∆un, ∆un),un〉| . The first nonlinear term is to be dealt with as follows: |〈B(un,un), ∆un〉| ≤ ‖un‖L∞(T3)‖∇un‖L2(T3)‖∆un‖L2(T3) ≤ c‖un‖ 1/2 Ḣ1 ‖un‖ 1/2 Ḣ2 ‖∇un‖L2(T3)‖∆un‖L2(T3) ≤ c‖un‖ 1/2 Ḣ1 ‖∇un‖L2(T3)‖∆un‖ 3/2 L2(T3) ≤ c‖un‖Ḣ1(T3)‖∆un‖ 1/2 L2(T3)‖∇∆un‖ 3/2 L2(T3), where we used Hölder’s inequality, inequality (2.7) and the facts that ‖∇un‖L2 ≤‖∆un‖L2 and ‖∆un‖L2 ≤ ‖∇∆un‖L2 . Similarly, we have |〈B(∆un,un),un〉| ≤ c‖un‖Ḣ1(T3)‖∆un‖ 1/2 L2(T3)‖∇∆un‖ 3/2 L2(T3). The second nonlinear term is to be dealt with, in the following manner: |〈B(un, ∆un), ∆un〉| ≤ ‖un‖L6(T3)‖∇∆un‖L2(T3)‖∆un‖L3(T3) ≤ c‖un‖Ḣ1‖∇∆un‖L2(T3)‖∆un‖ 1/2 L2(T3)‖∇∆un‖ 1/2 L2(T3) = c‖un‖Ḣ1(T3)‖∆un‖ 1/2 L2(T3)‖∇∆un‖ 3/2 L2(T3). Similarly, we have |〈B(∆un, ∆un),un〉| ≤ ‖∆un‖L3(T3)‖∇∆un‖L2(T3)‖un‖L6(T3) ≤ c‖un‖Ḣ1(T3)‖∆un‖ 1/2 L2(T3)‖∇∆un‖ 3/2 L2(T3). It turns out that ∣∣∣〈B̃(un,vn), ∆un〉∣∣∣ ≤ c(1 + α2)‖un‖Ḣ1(T3)‖∆un‖1/2L2(T3)‖∇∆un‖3/2L2(T3).(2.8) By Cauchy-Schwarz and Young’s inequalities, we get |〈θne3,−∆un〉| ≤ ‖∇θn‖2L2 + ‖∇un‖ 2 L2.(2.9) Int. J. Anal. Appl. 19 (1) (2021) 117 Summing up (2.2) and (2.3) and using (2.6), (2.8) and (2.9), it follows that 1 2 d dt ( ‖∇θn‖2L2(T3) + ‖∇un‖ 2 L2(T3) + α 2‖∆un‖2L2(T3) ) +κ‖∆θn‖2L2(T3) + ν(‖∆un‖ 2 L2(T3) + α 2‖∇∆un‖2L2(T3)) ≤ c‖un‖Ḣ1‖∇θn‖ 1/2 L2 ‖∆θn‖ 3/2 L2 + ‖∇θn‖2L2 + ‖∇un‖ 2 L2 c(1 + α2)‖un‖Ḣ1(T3)‖∆un‖ 1/2 L2(T3)‖∇∆un‖ 3/2 L2(T3). Using Young’s inequality, we absorb the remaining diffusion term in the right hand side and we obtain d dt ( ‖∇θn‖2L2 + ‖∇un‖ 2 L2 + α 2‖∆un‖2L2 ) +κ‖∆θn‖2L2 + ν ( ‖∆un‖2L2 + α 2‖∇∆un‖2L2 ) ≤ c‖un‖4Ḣ1‖∇θn‖ 2 L2 + ‖∇θn‖2L2 + ‖∇un‖ 2 L2 , where c is a generic constant that may change from line to line. Integrating over time, we obtain, for all t ∈ [0,T∗n), ‖∇θn(t)‖2L2(T3) + ‖∇un(t)‖ 2 L2(T3) + α 2‖∆un(t)‖2L2(T3) + κ ∫ t 0 ‖∆θn(τ)‖2L2dτ +ν ∫ t 0 ( ‖∆un(τ)‖2L2(T3) + α 2‖∇∆un(τ)‖2L2(T3) ) dτ ≤ c ∫ t 0 ‖un(τ)‖4Ḣ1 (‖∆un(τ)‖ 2 L2 + ‖∇θn(τ)‖ 2 L2 )dτ + ∫ t 0 (‖∇θn(τ)‖2L2 + ‖∇un(τ)‖ 2 L2 )dτ + ‖∇θ 0‖2L2 + ‖∇u 0‖2L2 + α 2‖∆u0‖2L2. At this point, we give the theorem below that states the existence weak solution. The proof follows exactly the lines of the proof given in [12]: Theorem 2.1. Let θ0 ∈ L2(T3), and u0 ∈ H1(T3) be a divergence-free vector field. Then, for any T > 0 there exists a unique weak solution (uα,θα) to (1.1) in the interval [0,T], where uα ∈ C([0,T],H1(T3)) ∩L2([0,T],H2(T3)), and θα ∈ C([0,T],L2(T3)) ∩L2([0,T],H1(T3)). Moreover, we have for all t ∈ [0,T] : (‖u(t)‖2L2(T3) + α 2‖∇u(t)‖2L2(T3) + ‖θ(t)‖ 2 L2(T3)) +2µ ∫ t 0 (‖∇u(τ)‖2L2(T3) + α 2‖∆u(τ)‖2L2(T3) + ‖∇θ(τ)‖ 2 L2(T3))dτ(2.10) ≤‖u0‖2L2 + α 2‖∇u0‖2L2 + ‖θ0‖ 2 L2 + φα(t)︸ ︷︷ ︸ K(t) where by φα(t) we refer to the function ( ‖u0‖2L2 + α 2‖∇u0‖2L2 +‖θ0‖ 2 L2 ) (e2t−1). Furthermore, this solution is continuously dependent on the initial data (u0,θ0). Moreover, this solution is continuously dependent on the initial data (u0,θ0). In particular, it is unique. Int. J. Anal. Appl. 19 (1) (2021) 118 Using the energy estimate for weak solution (uα,θα) in theorem above, and the expression of the function ρα given by equation (1.12), we infer that∫ t 0 (‖∇θn(τ)‖2L2 + ‖∇un(τ)‖ 2 L2 )dτ ≤ K(T) 2min(ν,κ) and ∫ t 0 ‖un‖4Ḣ1 (‖∆un‖ 2 L2 + ‖∇θn‖ 2 L2 )dτ ≤‖un‖ 4 L∞ T (Ḣ1) (‖θn‖2L2 T (Ḣ1) + ‖un‖2L2 T (Ḣ2) ) ≤ K 2(T) α4 ( K(T) 2κ + K(T) 2να2 ) ≤ K 3(T)(1+α2) 2min(ν,κ)α6 , where K(t) = ‖u0n‖2L2 + α 2‖∇u0n‖2L2 + ‖θ 0 n‖2L2 + φα(T). In conclusion, we obtain the energy estimate (2.11) ‖∇θn(t)‖2L2 + ‖∇un(t)‖ 2 L2 + α2‖∆un(t)‖2L2 + κ ∫ t 0 ‖∆θn(τ)‖2L2dτ +ν ∫ t 0 (‖∆un(τ)‖2L2 + α 2‖∆∇un(τ)‖2L2 )dτ ≤Kα(T), where Kα(T) = K(T) 2µ + K3(T) 2µα6 (1 + α2) + ‖∇θ0‖2L2(T3) + ‖∇u 0‖2L2(T3) + α 2‖∆u0‖2L2(T3), where by µ = min(ν,κ). Notice that the upper bound K(t) in (2.11) is continuous and does not include any singularity with respect to time t. Hence, K(t) rules out the finite time blow-up of the solution near T and the solution can be extended to a global in time solution. The estimation (2.11) provides uniform bounds, with respect to n, of the solution un in L ∞ T (Ḣ 2(T3)), L2T (Ḣ 3(T3)) as well as in L∞T (Ḣ 1(T3)) and L2T (Ḣ 2(T3)) for θn. This allows to use the Aubin-Lions lemma so that we can take the limit as n tends to infinity and then obtain existence. As strong solutions are also weak, uniqueness of strong solution simply follows from uniqueness of weak solution. 3. Convergence result Already, we have proved that an initial data (u0,θ0) gives rise to a global solution (uα,θα). This section is aimed to deal with the convergence result, as the parameter α vanishes. It must be said that the upper bound in the energy estimate (2.11) depends singularly on α and it will fail to control the solution’s norms, as α goes to zero. From (2.10), it is worth mentioning that the dependence of the upper bound in the weak solution’s energy estimate is polynomial, so that such impediment is absent in the weak solution case. To overcome this difficulty, we need to get uniform bounds, with respect to α on ‖∇θα(t)‖2L2(T3) + ‖∇uα(t)‖ 2 L2(T3) + α 2‖∆uα(t)‖2L2(T3), Int. J. Anal. Appl. 19 (1) (2021) 119 and κ ∫ t 0 ‖∆θα(τ)‖2L2dτ + ν ∫ t 0 ( ‖∆uα(τ)‖2L2(T3) + α 2‖∇∆uα(τ)‖2L2(T3) ) dτ. To do so, firstly we mention that ∣∣∣〈B̃(uα,vα), ∆uα〉∣∣∣ ≤ c‖uα‖1/2Ḣ1 ‖uα‖1/2Ḣ2 ‖∇vα‖L2(T3)‖∆uα‖L2(T3) ≤ c‖uα‖ 1/2 Ḣ1 ‖uα‖ 1/2 Ḣ2 (‖uα‖Ḣ1(T3) + α 2‖∇∆uα‖L2(T3))‖∆uα‖L2(T3) ≤ c(‖uα‖6Ḣ1 + α 6‖∆uα‖6L2 ) + ν/2‖∆uα‖ 2 L2(T3) + α 2ν/2‖∇∆uα‖2L2(T3), where we used Agmon’s inequality and Young’s inequality twice. Secondly, using Hölder inequality and Young’s inequality twice, it turns out that |〈(uα ·∇)θα, ∆θα〉| ≤ c‖uα‖4Ḣ1‖∇θα‖ 2 L2 + κ/2‖∆θα‖2L2 ≤ c(‖uα‖6Ḣ1 + ‖∇θα‖ 6 L2 ) + κ/2‖∆θα‖2L2. Thirdly, the bouancy force can be dealt with via Cauchy-Schwarz inequality: |〈θαe3,−∆uα〉| ≤ ‖∇θα‖2L2 + ‖∇uα‖ 2 L2 + α 2‖∆uα‖2L2. Finally, summing up, one obtains (3.1) 1 2 d dt (‖∇θα‖2L2(T3) + ‖∇uα‖ 2 L2(T3) + α 2‖∆uα‖2L2(T3)) +κ/2‖∆θα‖2L2(T3) + ν/2(‖∆uα‖ 2 L2(T3) + α 2‖∇∆uα‖2L2(T3)) ≤ c(‖∇θα‖6L2 + ‖∇uα‖ 6 L2 + α6‖∆uα‖6L2 +‖∇θα‖2L2 + ‖∇uα‖ 2 L2 + α2‖∆uα‖2L2 ), where c is a constant that does not depend on the parameter α. Let g(t) = ‖∇θα(t)‖2L2(T3) +‖∇uα(t)‖ 2 L2(T3) + α2‖∆uα(t)‖2L2(T3). It is clear that g 3 + g ≤ c(g + 1)3. Let h(t) = g(t) + 1. The function g is a non-negative function, so h(0) 6= 0. The estimation (3.1) can be written as dh dt ≤ ch3. We integrate this ordinary differential inequality, to obtain for 0 ≤ t ≤ 1 4Ch2(0) , h(t) ≤ ch(0). Finally, it turns out that for all time t, such that 0 ≤ t ≤ T∗ = min(T, 1 4C(1 + g(0))2 ), we have (3.2) ‖∇θα‖2L2(T3) + ‖∇uα‖ 2 L2(T3) + α 2‖∆uα‖2L2(T3) ≤ c(1 + ‖∇θ0‖2 L2(T3) + ‖∇u 0‖2 L2(T3) + α 2‖∆u0‖2 L2(T3)). Int. J. Anal. Appl. 19 (1) (2021) 120 Integrating (3.1) over (0,T∗) and using (3.2), we obtain (3.3) ∫ T∗ 0 ( κ‖∆θα‖2L2) + ν(‖∆uα‖ 2 L2 + α 2‖∇∆uα‖2L2 ) dt ≤ c(1 + ‖∇θ0‖2 L2(T3) + ‖∇u 0‖2 L2(T3) + α 2‖∆u0‖2 L2(T3)). These are non singular bounds with respect to the parameter α. Since α is intended to vanish, then there exists some fixed value α0, such that 0 < α ≤ α0. We take α = α0 in (3.2) and (3.3), to obtain a uniform bound with respect to α. Namely, the functions θ and u are uniformly bounded in L2([0,T∗],Ḣ2(T3)), as for v, it is uniformly bounded in L2([0,T∗],Ḣ1(T3)). Hence, Banach-Alaoglu theorem [11] allows to extract subsequences (uαk )k, (vαk )k and (θαk )k (that we relabel (uk), (vk) and (θk)) respectively of uα, vα and θα such that (θk,uk) ⇀ (θ,u) in L 2([0,T∗],Ḣ2(T3)) and vk ⇀ u in L2([0,T∗],Ḣ1(T3)), as k → +∞. At this step, we proved the two first results of statements 1 and 2 of Theorem 1.2. To investigate the two second results of these statements, we establish uniform estimates, independent of α, for d dt θαk and d dt uαk . For a fixed positive time, since θαk is uniformly bounded with respect to α, in L2([0,T∗],Ḣ2(T3)), the diffusion ∆θαk belongs to L 2([0,T∗],L2(T3)). Using Sobolev norm definition and product laws we infer that ∫ T∗ 0 ‖div θkuk‖2L2dτ ≤ ∫ T∗ 0 ‖θk‖2Ḣ1‖uk‖ 2 Ḣ2 dτ ≤‖θk‖2L∞ T∗(Ḣ 1) ‖uk‖2L2 T∗(Ḣ 2) . Since uk and θk are, respectively, subsequences of uα and θα, then the energy estimates (3.2) and (3.3) apply also for uk and θk and one can control the advection in L 2([0,T∗],L2(T3)). The above temperature diffusion and convection estimates lead to ‖ d dt θk‖L2 T∗(L 2) ≤ K1, where K1 is a real positive constant. To handle the time derivative of the velocity field uk, we apply the Helmholtz operator (I − α2∆)−1 on the equation satisfied by u = uk in the system (1.10). So, (3.4) d dt uk = ν∆uk − (I −α2∆)−1B̃(uk,vk) − (I −α2∆)−1∇pk + (I −α2∆)−1θke3. For a fixed positif time, since uk is uniformly bounded with respect to α in the space L 2([0,T∗],Ḣ2(T3)), then ∆uk belongs to L 2([0,T∗],L2(T3)). For the remaining terms, we recall that operator (I − α2∆)−1 is bounded from H−2(T3) into L2(T3). Moreover, a direct frequencies computation implies that its norm is uniformly bounded and satisfies (3.5) |||(I −α2∆)−1||| ≤ 1. Int. J. Anal. Appl. 19 (1) (2021) 121 Since θα ∈ L2T∗(Ḣ 2), one infers that ‖(I−α2∆)−1θe3‖L2 T∗(Ḣ 4) ≤ K ′ 2, where K ′ 2 is a real positif constant. To estimate the nonlinear term, one has∫ T∗ 0 ‖(I −α2∆)−1B̃(uk,vk)‖2L2 ≤ ∫ T∗ 0 ‖B̃(uk,vk)‖2H−2 ≤ c‖uk‖ 2 L∞ T∗(Ḣ 1) ‖vk‖2L2 T∗(Ḣ 1) ≤ c‖uk‖2L∞ T∗(Ḣ 1) (∫ T∗ 0 ‖∆uk(τ)‖2L2 + α 2‖∇∆uk(τ)‖2L2dτ ) . Inequalities (3.2) and (3.3) provide uniform bounds to the non-linearity above. Using precedent uniform bounds, thanks to the divergence free-condition, one infers that ‖(I−α2∆)−1∇p‖L2 T∗(Ḣ 4) ≤ K ′′ 2 . So, equation (3.4) implies that ‖ d dt uk‖L2 T∗(L 2) ≤ K2. Using Aubin-Lions lemma, we extract two subsequences relabeled uk and θk, that converge strongly in L2([0,T∗],Ḣ1) and in L2([0,T∗],L2), respectively. We have ‖vk −uk‖2L2([0,T∗],L2) = α 4 ∫ T 0 ( ∑ k∈Z3 |∆̂uk|2) = α4‖uk‖2L2([0,T∗],Ḣ2). As uk belongs to L 2([0,T∗],Ḣ2), vk converges strongly to u in L 2([0,T∗],L2), we have already proved the statements 1 and 2 of Theorem 1.2. Now, we turn to the third statement of Theorem 1.2. For the first result, since (uk,θk) converges strongly to (u,θ) in (L2([0,T∗],Ḣ1))2, then by Cauchy-Schwarz inequality it converges weakly for almost every t ∈ [0,T∗]. In particular, this holds for the supremum. That is (uk(t),θk(t)) converges to (u(t),θ(t)) weakly in Ḣ1(T3) and uniformly over [0,T∗]. To prove the second result, using the precedent bounds of time derivatives, Banach-Alaoglu theorem in Hilbert spaces implies that ( d dt θk, d dt uk) ⇀ ( d dt θ, d dt u) weakly in L2([0,T∗],L2(T3)), ask → +∞, and d dt vk ⇀ d dt u weakly in L2([0,T∗],Ḣ−2(T3)), ask → +∞. Let Λ ∈ Ḣ2 be a vector divergence free and Ξ ∈ L2 a scaler test functions. Taking the inner product and integrating over [0, t], for t ∈ [0,T∗], we obtain 〈θk(t), Ξ〉−〈θk(0), Ξ〉− ∫ t 0 〈θk, ∆Ξ〉dτ + ∫ t 0 〈B(uk,θk), Ξ〉dτ = 0, 〈vk(t), Λ〉−〈vk(0), Λ〉− ∫ t 0 〈vk, ∆Λ〉dτ + ∫ t 0 〈B̃(uk,vk), Λ〉dτ − ∫ t 0 〈θke3, Λ〉dτ = 0. To handle the nonlinear terms, we use a standard compactness argument (thanks to the uniform bounds obtained with respect to αk above) so that B̃(uk,vk) → B(u,u) and B(uk,θk) → B(u,θ). Hence, taking the limit, for every t ∈ [0,T∗]\E, to obtain 〈θ(t), Ξ〉−〈θ(0), Ξ〉− ∫ t 0 〈θ, ∆Ξ〉dτ + ∫ t 0 〈B(u,θ), Ξ〉dτ = 0, Int. J. Anal. 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