A Regularity Criterion in Weak Spaces to Boussinesq Equations

In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in R 3 × ( 0 , T ) . The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces.

We would like to point out that the system (1.1) at θ = 0 reduces to the incompressible Navier-Stokes equations, which has been greatly analyzed. From the viewpoint of the model, therefore, Navier-Stokes flow is viewed as the flow of a simplified Boussinesq equation.
On the other hand, it is desirable to show the regularity of the weak solutions if some partial components of the velocity satisfy certain growth conditions. For the 3D Navier-Stokes equations, there are many results to show such regularity of weak solutions in terms of partial components of the velocity u (see, for example, [4,5,8,11,12,16,19,23,34] and the references cited therein). It is obvious that, for the assumptions of all regularity criteria, it need that every components of the velocity field must satisfies the same assumptions, and it don't make any difference between the components of the velocity field. As pointed out by Neustupa and Penel [21], it is interesting to know how to effect the regularity of the velocity field by the regularity of only one component of the velocity field. In particular, Zhou [35] showed that the solution is regular if one component of the velocity, for example, u 3 satisfies Condition 1.2) can be replaced respectively by the one (see Kukavica and Ziane [20]). Later, Cao and Titi [5] showed the regularity of weak solution to the Navier-Stokes equations under the assumption Motivated by the above work, Zhou and Pokorný [36] showed the following regularity condition while the limiting case u 3 ∈ L ∞ (0, T ; L 10 3 (R 3 )) was covered in [19]. For many other result works, especially the regularity criteria involving only one velocity component, or its gradient, with no intention to be complete, one can consult [32,33] and references therein. However, the conditions (1.2)-(1.5) are quite strong comparing with the condition of Serrin's regularity criterion : 2 Before stating our result, we introduce some notations and function spaces. These spaces can be found in many literatures and papers. For the functional space, L p (R 3 ) denotes the usual Lebesgue space of real-valued functions with norm · L p : On the other hand, the usual Sobolev space of order m is defined by To prove Theorem 2.4, we use the theory of Lorentz spaces and introduce the following notations.
We define the non-increasing rearrangement of f , is defined by Moreover, we define f * * by and Lorentz spaces L p,∞ (R 3 ) by For details, we refer to [2] and [27].
From the definition of the Lorentz space, we can obtain the following continuous embeddings : In order to prove Theorem 2.4, we recall the Hölder inequality in the Lorentz spaces (see, e.g., O'Neil [22]).
holds true for a positive constant C.
The following result plays an important role in the proof of our theorem, the so-called Gagliardo-Nirenberg inequality in Lorentz spaces, its proof can be founded in [17].
holds for a positive constant C and Now we give the definition of weak solution. );

system (1.1) is satisfied in the sense of distributions;
3. the energy inequality, that is, By a strong solution, we mean that a weak solution u of the Navier-Stokes equations (1.1) It is well known that the strong solution is regular and unique.
Our main result is stated as following : θ) is a weak solution to system (1.1). If u 3 and ∇θ satisfy the following conditions 3 Proof of the main result.
In this section, under the assumptions of the Theorem 2.4, we prove our main result. Before proving our result, we recall the following muliplicative Sobolev imbedding inequality in the whole space R 3 (see, for example [5]) : where ∇ h = (∂ x1 , ∂ x2 ) is the horizontal gradient operator. We are now give the proof of our main theorem.
Proof: To prove our result, it suffices to show that for any fixed T > T * , there holds where T * , which denotes the maximal existence time of a strong solution and C T is an absolute constant which only depends on T, u 0 and θ 0 .
The method of our proof is based on two major parts. The first one establishes the bounds L 2 ), while the second gives the bounds of the H 1 −norm of velocity u and temperature θ in terms of the results of part one.
Taking the inner product of (1.1) 1 with −∆ h u, (1.1) 2 with −∆ h θ in L 2 (R 3 ), respectively, then adding the three resulting equations together, we obtain after integrating by parts that is the horizontal Laplacian. For the notational simplicity, we set for t ∈ [Γ, T * ). In view of (2.1), we choose ǫ, η > 0 to be precisely determined subsequently and then select Γ < T * sufficiently close to T * such that for all Γ ≤ t < T * , Integrating by parts and using the divergence-free condition, it is clear that (see e.g. [34]) By appealing to Lemma 2.1, (3.1), and the Young inequality, it follows that where we have used the following Gagliardo-Nirenberg inequality in Lorentz spaces : To estimate the term I 2 of (3.2), first observe that by applying integration by parts and ∇·u = 0, we derive where we have used It follows from Hölder's inequality, (3.1) and Young's inequality that Finally, we we want to estimate I 3 . It follows from integration by parts and Cauchy inequality that Inserting all the estimates into (3.5), Gronwall's type argument using due to (2.1) leads to, for every τ ∈ [Γ, t] (3.6) Next, we analyze the right-hand side of (3.6) one by one. First, due to (3.3) and the definition of J 2 , we have Finally, we deal with the term I 2 (t). Applying Hölder and Young inequalities, one has Hence, choosing η small enough such that Cη < 1 and inserting the above estimates of I 1 (t) and I 2 (t) into (3.6), we derive that for all Γ ≤ t < T * : , which leads to (3.7) Now, we will establish the bounds of H 1 −norm of the velocity magnetic field and microrotationel velocity. In order to do it, taking the inner product of (1.1) 1 with −∆u, (1.1) 2 with −∆b and (1.1) 3 with −∆θ in L 2 (R 3 ), respectively. Then, integration by parts gives the following identity: Integrating by parts and using the divergence-free condition, one can easily deduce that (see e.g. [36]) We treat now the R 3 (u · ∇)θ · ∆θdx-term. By integration by parts, we have Therefore, we have and where the last inequality is obtained by using Cauchy inequality.
Putting all the inequalities above into (3.8) yields Finally, we deal with the term − R 3 (θe 3 )·∆udx. By integration by parts and Cauchy inequality, Combining the above estimates, by Hölder's inequality, Nirenberg-Gagliardo's interpolation inequality and (3.1), we obtain Integrating this last inequality in time, we deduce that for all τ ∈ [Γ, t]   It should be noted that the condition (2.1) is somewhat stronger than in [14], since it is worthy to emphasize that there are no assumptions on the two components velocity field (u 1 , u 2 ). In other word, our result demonstrates that the two components velocity field (u 1 , u 2 ) plays less dominant role than the one compoent velocity field does in the regularity theory of solutions to the Boussinesq equations. In a certain sense, our result is consistent with the numerical simulations of Alzmann et al. in [1].