Global Well-Posedness and Convergence Results to a 3D Regularized Boussinesq System in Sobolev Spaces

We consider a regularized periodic three-dimensional Boussinesq system. For a mean free initial temperature, we use the coupling between the velocity and temperature to close the energy estimates independently of time. Tis allows proving the existence of a global in time unique weak solution. Also, we establish that this solution depends continuously on the initial data. Moreover, we prove that this solution converges to a Leray-Hopf weak solution of the three-dimensional Boussinesq system as the regularizing parameter vanishes.


Introduction
Motivated by ( [1]) and references therein, we consider the regularization to the periodic three-dimensional Boussinesq system (Bq α ) given by where the unknown velocity, the unknown pressure, and the unknown temperature are, respectively, the threedimensional vector u, the scalars p, and the scalar θ.Te parameters ], κ, α > 0 denote, respectively, the viscosity, the thermal conductivity of the fuid, and the regularizing parameter, T 3 � (R/2πZ) 3 is the three-dimensional torus, u 0 is a given divergence-free initial velocity, and θ 0 is a given mean free initial temperature.Te vector e 3 � (0, 0, 1) T .Te periodic three-dimensional Boussinesq system models geophysical fuids such as oceanographic turbulence and atmospheric fronts as well as the Rayleigh-Benard convection [2].More physical application for the Boussinesq system can be found in [3] and related references.It is known that available mathematical methods do not allow proving the global well-posedness of the three-dimensional fuid equations such as the Boussinesq system, especially in Sobolev spaces which are energy spaces frequently used in real word applications.To make practical advances in this feld, researchers took the way of regularisation.In this framework, the idea in [4] was to suggest a particular closure model for the Navier-Stokes equations by approximating the Reynolds stress tensor.Tis model was simplifed in [5] and a mathematical study was performed therein.Existence and uniqueness results in [5] were improved in [6].
Te closest reference to our manuscript is [1], where the author proved that a weak solution exists to (Bq α ), α > 0. Tis solution depends continuously on initial data and it converges to a weak solution of (Bq α�0 ), as the regularizing parameter α ⟶ 0. However, it is clear that in Teorem 1 of [1], the right-hand side of the energy estimate depends on time.Tus, the solution belongs to L ∞ loc (R + , H 1 (T 3 )) as it will blow up when T ⟶ ∞.Tis local aspect is due to the classical arguments based on a brutal application of Cauchy-Schwarz inequality while taking the scalar product of the Buoyancy force θe 3 with the velocity feld u.Tus, this was not a global in-time solution but it should be called a large time solution.Tis insufciency appeared widely in the literature and is still appearing both in the threedimensional case and the two-dimensional case which is supposed to be well understood, see among a wide literature [1,[7][8][9][10][11][12] and references therein.Here, we overcome this insufciency for the range of mean free initial temperature and we make two improvements that are interesting from an applicable point of view.
Te frst is to obtain a global in-time weak solution, under minimal regularity requirements.Tis is the main contribution of this paper.For physicists and engineers, global in-time solutions are closely related to durable in time operating of machines, systems, and networks.Tus, physicists and engineers usually try to start with suitable initial data to avoid blowup in fnite time.For mathematician, global in-time solutions open the way to study the long time behavior [13,14], the existence of the attractors [15], the asymptotic stability [16], and in general, all topics requiring t ⟶ ∞.In numerical analysis of nonlinear system, although the numerical discretization is generally local in time, the existence of a global in-time solution gives the possibility to extend such numerical discretization, by translation in time.Also, based on [1] and using continuity in time, we deduce that our global solution is continuously dependent on the initial data and in particular, it is unique.First, we recall that the uniqueness of weak solution in energy spaces is still an open problem for three-dimensional fuid equations.In the literature, such uniqueness is the main target behind any regularisation.Second, we note that from an applied mathematical point of view, we seek for a nearby solution to arise from nearby initial data.Otherwise, we will never believe in any computer calculations, for example.For physicists and engineers, it is interesting that when starting with an initial state, the system described by a given partial diferential equation should evolve towards an only one future state.
Te second is that our solution converges to a global in-time Leray-Hopf type weak solution of the threedimensional Boussinesq system, as the regularizing parameter α ⟶ 0. Convergence result is one of the main features of the α-regularisation.First, in practical situations, it allows to consider systems with α > 0 as small as required and fully proft from uniqueness and continuous dependence, while keeping nearby a weak solution of the threedimensional Boussinesq system.Second, from a theoretical point of view, it is indeed a diferent mathematical method to prove the existence of a weak solution to the threedimensional Bousssinesq system.Tis solution is the existing limit.Similar results were proved, as the Rossby number vanishes, in [17,18] for example.
Let us mention that starting with a mean free initial temperature, such as sinusoidal initial heating sources, is frequent in natural phenomenons and compulsory in many real word applications; see [19] and the multitude references therein in the case of industrial applications or [20] for applications in medicine and health sciences.In [21,22], authors used the mean free condition to investigate the long time behavior of the solution and to prove an exponential stability result for the periodic 3D Navier-Stokes equations, in critical Sobolev spaces.
Given a Banach space (X, ‖.‖ X ), the Bochner space L p ([0, T], X) is the space of all functions such that If we denote by s a real number, by  u the Fourier transform of u and by S ′ (T 3 ) the Schwartz space, then the homogeneous Sobolev spaces are given by and endowed with the natural norm ‖u‖ _ Te paper is organized as follows.In the following section, we will prove that a continuous global in time weak solution exists and depends continuously on the initial data and in particular, it is unique.In the last section, we will establish that this solution converges to a global in time Leray-Hopf type solution, as the regularizing parameter α ⟶ 0.

Existence Results
In the following, we give formal estimates for a Galerkin approximating scheme to system (Bq α ).We omit the approximating system and the index of the approximating sequence.Interested readers can see [1] for full details.Taking the inner product in L 2 (T 3 ) of ( 1) with u and (2) with θ, we obtain and 1 2 Integrating (2) with respect to x, we infer that the frst Fourier coefcient of θ is conserved during time, that is, Journal of Mathematics Applying the Cauchy-Schwarz inequality and Young inequality to (10), it holds Integrating ( 9) and ( 11) with respect to time and summing, it follows that By Poincaré inequality, one has Above, we have a unitary Poincaré constant.In fact, ≤  k∈Z 3 ,k≠(0,0,0) where we used successively that θ is mean free and that |k| ≥ 1. Te integral with respect to time of (9) gives Finally, we are able to close the estimates independently on time as follows: where A standard compactness argument fnishes the proof of the existence part in Teorem 1.To do so, we take the limit using Aubin compactness lemma [23].Continuity in time of the existing weak solution to (Bq α ) can be proved in a classical manner as in the case of the weak solutions to three-dimensional Navier-Stokes equations Also, details were provided in [7] for the case of the strong solution to (Bq α ).
In [1], the author established the continuous dependence of the weak large time solution with respect to the initial data on [0; T], T > 0. In particular, he deduced that this large time solution was unique.In our case, as the global solution of (Bq α ) is continuous in time, continuous dependence on initial data and uniqueness follow over R + .In conclusion, we have the following theorem.
Theorem 1.Let θ 0 ∈ L 2 (T 3 ) be a mean free scalar function and let u 0 ∈ _ H 1 (T 3 ) be a divergence-free vector feld.Ten, there exists a global in-time weak solution (u α , θ α ) of system (Bq α ) such that u α belongs to Moreover, this solution satisfes the energy estimate (16) and depends continuously on the initial data.In particular, it is unique.

Convergence Results
In this section, we will prove the following theorem.
(1) Te sequence u α k converges to � u and
Taking α � α 0 in the right-hand side of ( 16), we obtain for all t ∈ R + , Above, we added the index α to make precision that the temperature and the velocity depend implicitly on α.By (18), both of θ α and u α are uniformly bounded in L 2 (R + , _ H 1 (T 3 )).
Hence, the Banach-Alaoglu theorem [25] applied in the framework of Hilbert spaces allows to extract subsequences

Proof of Statement (2).
To deal with the strong convergence, we will apply the Aubin-Lions lemma [23].Tis necessitates uniform estimates of the time derivatives of θ α k , of u α k , and of v α k in the appropriate spaces.In the following, K is a real positive constant that may change from line to line.For all positive time, As the Sobolev spaces form a decreasing chain and by defnition of the homogeneous Sobolev norm, Using Sobolev norm properties and Sobolev product laws, it holds that Te above estimates of the difusion and the advection terms lead to where _ H − 3/2 is the dual space of the homogeneous Sobolev ) is the Bochner space as defned in the introduction.Applying the operator (I − α 2 Δ) − 1 to the velocity equation (1), we obtain ∀(x, t) in R + × T 3 , In the following, we will be conformed to the statement of the Aubin lemma [23] and consider a time T > 0. As u α k is bounded independently of α in L 2 ([0, T], _ H 1 (T 3 )), the dissipation Δu α k will be so in the space L 2 ([0, T], _ H − 1 (T 3 )).
For the other terms, we mention that the operator ) and that by frequency calculations, we have where we denote by |‖.‖|, the norm of the operator.Also, as 3 ) ≤ K.As for the convection term, Sobolev norm properties, Sobolev product laws, and classical computations lead to 4

Journal of Mathematics
It is standard to rewrite the pressure in terms of the velocity and the temperature.Also, one applies the divergence operator and the Riesz transform to obtain Using the precedent bounds of the temperature and the velocity, it holds that So, equation (22) implies that Remark 3. It is clear that in ( 21) and (26) as in [1], the constant K � K(α 0 , ], κ, u 0 , θ 0 ).Tus, it is uniform with respect to α.However, the most interesting feature in the present paper is the fact that K is independent of the time T. Tis makes these estimates valid for all time T. Especially, as time goes to infnity.Tis was not the case of convergence result in [1], where estimates for convergence results blow up, as t ⟶ + ∞.
By Aubin-Lions lemma, we extract subsequences relabeled u k and θ k that converge strongly in L we deduce that v k converges strongly to u in L .

Journal of Mathematics
Te solution (u(t), θ(t)) satisfes the energy inequality (17), as we can take the lower limit when α k ⟶ 0.