STRONG SOLUTIONS TO 3D-LAGRANGIAN AVERAGED BOUSSINESQ SYSTEM

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.


Introduction and statement of main results
We consider the following 3D-Lagrangian averaged Boussinesq-α system: where by T 3 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 + × T 3 . The superscript M T denotes the transpose of the matrix M . The data (u 0 , θ 0 ), 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 − α 2 ∆. 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.
• For n ∈ N, we denote by P n the projection into Fourier modes of order up to n, that is • For s > 0, we define the operator Λ s acting on H s (T 3 ) as follows. Let u ∈ H s (T 3 ) having the Fourier series Then, we define • We denote by · Ḣs the seminorm Λ s · L 2 . This is compatible with the definition of the Sobolev norm. In fact, · H s is equivalent to · L 2 + · Ḣs . Note that in the Fourier setting it is more usual to define an equivalent norm on H s by • We refer to the fractional Laplacian √ −∆ by Λ.
• We denote • We define the space H s = {u ∈ H s , div u = 0}. We remark that, for u, v and w ∈ H 1 (T 3 ) • Due to the identity • By the definitions of the operators B(u, v) and B(u, v), we deduce the following properties: which in turn implies that Also, and hence Here < ., . > denotes the duality pairing ofH 1 ( 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) Our first result is the following existence and uniqueness theorem.
Moreover, ∀ 0 ≤ t ≤ T , this solution satisfies the following energy estimates and µ = min(ν, κ). Here, the function K(t) stands for and φ α is a positive increasing function of time t, defined by 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].
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 α k → 0. We shall use a compactness 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.

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: Both the bilinear operators on the left are continuous on L 2 ×L 2 . Then, the above system appears as a system of ordinary differential equations on L 2 . Thus, the usual Cauchy-Lipschitz theorem yields the existence of a strictly positive maximal time T n such that a unique solution exists which is continuous in time with value in L 2 . 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 L 2 (T 3 )-inner product of the equation satisfied by θ n in (2.1) against −∆θ n and the one satisfied by u n against −∆u n , to obtain 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 ∈Ḣ 1 (T 3 ), we have and ϑ L 6 ≤ c ϑ Ḣ1 . ≤ c u n Ḣ1 ∇θ n 1/2 Hence, one obtains | B(u n , θ n ), ∆θ n | ≤ c u n Ḣ1 ∇θ n 1/2 For every ϑ ∈Ḣ 2 (T 3 ), the Agmon's inequality [11] reads The fact that v n = u n − α 2 ∆u n yields The first nonlinear term is to be dealt with as follows: where we used Hölder's inequality, inequality (2.7) and the facts that ∇u n L 2 ≤ ∆u n L 2 and ∆u n L 2 ≤ ∇∆u n L 2 . Similarly, we have | B(∆u n , u n ), u n | ≤ c u n Ḣ1 (T 3 ) ∆u n 1/2 The second nonlinear term is to be dealt with, in the following manner: Similarly, we have It turns out that Using Young's inequality, we absorb the remaining diffusion term in the right hand side and we obtain where c is a generic constant that may change from line to line. Integrating over time, we obtain, for all 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]: Moreover, we have for all t ∈ [0, T ] : where by φ α (t) we refer to the function u 0 2 L 2 + α 2 ∇u 0 2 L 2 + θ 0 2 L 2 (e 2t − 1). Furthermore, this solution is continuously dependent on the initial data (u 0 , θ 0 ). Moreover, this solution is continuously dependent on the initial data (u 0 , θ 0 ). In particular, it is unique.
Using the energy estimate for weak solution (u α , θ α ) in theorem above, and the expression of the function ρ α given by equation (1.12), we infer that where K(t) = u 0 n 2 In conclusion, we obtain the energy estimate 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 u n in L ∞ T (Ḣ 2 (T 3 )), L 2 T (Ḣ 3 (T 3 )) as well as in L ∞ T (Ḣ 1 (T 3 )) and L 2 T (Ḣ 2 (T 3 )) for θ n . 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.

Convergence result
Already, we have proved that an initial data (u 0 , θ 0 ) gives rise to a global solution (u α , θ α ). 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 To do so, firstly we mention that where we used Agmon's inequality and Young's inequality twice.
To investigate the two second results of these statements, we establish uniform estimates, independent of α, for d dt θ α k and d dt u α k . For a fixed positive time, since θ α k is uniformly bounded with respect to α, in ). Using Sobolev norm definition and product laws we infer that Since u k and θ k are, respectively, subsequences of u α and θ α , then the energy estimates (3.2) and (3.3) apply also for u k and θ k and one can control the advection in L 2 ([0, T * ], L 2 (T 3 )). The above temperature diffusion and convection estimates lead to d dt θ k L 2 T * (L 2 ) ≤ K 1 , where K 1 is a real positive constant. To handle the time derivative of the velocity field u k , we apply the Helmholtz operator (I − α 2 ∆) −1 on the equation satisfied by u = u k in the system (1.10). So, For a fixed positif time, since u k is uniformly bounded with respect to α in the space L 2 ([0, T * ],Ḣ 2 (T 3 )), then ∆u k belongs to L 2 ([0, T * ], L 2 (T 3 )). For the remaining terms, we recall that operator (I − α 2 ∆) −1 is bounded from H −2 (T 3 ) into L 2 (T 3 ). Moreover, a direct frequencies computation implies that its norm is uniformly bounded and satisfies (3.5) |||(I − α 2 ∆) −1 ||| ≤ 1.
Since θ α ∈ L 2 T * (Ḣ 2 ), one infers that (I − α 2 ∆) −1 θe 3 L 2 T * (Ḣ 4 ) ≤ K 2 , where K 2 is a real positif constant. To estimate the nonlinear term, one has 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 ( Using Aubin-Lions lemma, we extract two subsequences relabeled u k and θ k , that converge strongly in As u k belongs to L 2 ([0, T * ],Ḣ 2 ), v k converges strongly to u in L 2 ([0, T * ], L 2 ), 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 (u k , θ k ) converges strongly to (u, θ) in (L 2 ([0, T * ],Ḣ 1 )) 2 , then by Cauchy-Schwarz inequality it converges weakly for almost every t ∈ [0, T * ]. In particular, this holds for the supremum. That is (u k (t), θ k (t)) converges to (u(t), θ(t)) weakly To handle the nonlinear terms, we use a standard compactness argument (thanks to the uniform bounds obtained with respect to α k above) so thatB(u k , v k ) → B(u, u) and B(u k , θ k ) → B(u, θ). Hence Especially, every strong solution fulfills the energy estimates (1.11), so one deduces the energy estimates (1.13) by taking the lower limit as α k → 0 + .