Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space \begin{document}$ \mathbb{R}^{2} $\end{document} with zero density at infinity. We prove that there exists a unique global strong solution provided the initial density and the initial temperature decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Moreover, there is no need to require any Cho-Choe-Kim type compatibility conditions. Our proof relies on the delicate weighted estimates and a lemma due to Coifman-Lions-Meyer-Semmes [J. Math. Pures Appl., 72 (1993), pp. 247-286].

Let Ω = R 2 and we consider the Cauchy problem for (1) with (ρ, u, θ) vanishing at infinity (in some weak sense) and the initial conditions: for given initial data ρ 0 , u 0 and θ 0 .
The Boussinesq system describes the motion of lighter or denser incompressible fluid under the influence of gravitational forces, and has important roles in the atmospheric sciences [20], as well as a model in many geophysical applications [22]. In recent years, the two-dimensional Boussinesq equations (1) with ρ = 1 have attracted significant attention. When Ω = R 2 , Cannon and DiBenedetto [3] studied the Cauchy problem for the Boussinesq equations with full viscosity      u t + u · ∇u − µ∆u + ∇P = θe 2 , θ t + u · ∇θ − κ∆θ = 0, div u = 0, which describe the flow of a viscous incompressible fluid subject to convective heat transfer, where µ > 0, κ > 0 are constants. They found a unique, global in time, weak solution. Furthermore, they improved the regularity of the solution when initial data is smooth. Recently, the result of global existence of smooth solutions to (3) has been generalized to cases of "partial viscosity" (that is, either µ > 0 and κ = 0, or µ = 0 and κ > 0) by Hou and Li [12] and Chae [5] independently. In [12], Hou and Li proved the global well-posedness of the Cauchy problem of viscous Boussinesq equations. They showed that solutions with initial data in H m (m ≥ 3) do not develop finite-time singularities. In [5], Chae considered the Boussinesq system for incompressible fluid in R 2 with either zero diffusion (κ = 0) or zero viscosity (µ = 0). He proved global-in-time regularity in both cases. For more information, we refer the readers to [4,15] for studies in this direction. On the other hand, the two-dimensional initial-boundary value problems of (3) have been analyzed to great extent, please see [13,14] and references therein.
Recently, the density-dependent viscous Boussinesq equations have attracted much attention. The authors [10,26] studied regularity criteria for three-dimensional incompressible density-dependent Boussinesq equations. Qiu and Yao [23] showed the local existence and uniqueness of strong solutions of multi-dimensional incompressible density-dependent Boussinesq equations with κ = 0 in Besov spaces. A blow-up criterion was also shown in [23]. We should point out here the above results always require the initial density is bounded away from zero. By contrast, however, there are few results concerning strong (or classical) solvability of the density-dependent Boussinesq equations with initial data permitting vacuum. Recently, Zhong [27] proved the local existence of strong solutions to the Cauchy problem of the system (1) with κ = 0 and nonnegative density. Naturally, for a partial differential system, one key consideration is to derive the local and global existence of solutions. It should be noticed that, for two-dimensional Cauchy problems, when the far field density is vacuum, it seems difficult to bound the L p -norm of u by √ ρu L 2 and ∇u L 2 for any p ≥ 1, hence the question of the global-in-time existence of solutions to the two-dimensional Cauchy problem of Boussinesq equations, such as (1), is much subtle and remains open. In fact, this is the main aim of this paper. More precisely, we are going to establish the global well-posedness of strong solutions to the Boussinesq system (1) in R 2 , which will generalize the study of [3] to the case of variable density. Now, we wish to define precisely what we mean by strong solutions.
For 1 ≤ r ≤ ∞ and k ≥ 1, we denote the standard Lebesgue and Sobolev spaces as follows: Without loss of generality, we assume that the initial density ρ 0 satisfies which implies that there exists a positive constant N 0 such that where B R := {x ∈ R 2 ||x| < R} for R > 0. Our main result can be stated as follows: Theorem 1.2. For constants q > 2 and a > 1, in addition to (4) and (5), we assume that the initial data (ρ 0 ≥ 0, u 0 , θ 0 ) satisfy wherex := e + |x| 2 1 2 log 2 (e + |x| 2 ).

Remark 1.
It should be noted that the initial density may contain vacuum states and even have compact support. Moreover, the initial data can be arbitrarily large.

Remark 2.
It is worth mentioning that no Cho-Choe-Kim type compatibility condition (see [6,8]) on the initial data is required in Theorem 1.2 for the global existence and uniqueness of strong solutions.
We now make some comments on the analysis of the present paper. Using some key ideas in [16], where Li and Liang dealt with the local well-posedness of classical solutions to the Cauchy problem of two-dimensional compressible Navier-Stokes equations, we first establish that if (ρ 0 , u 0 , θ 0 ) satisfies (4) and (6), then there exists a small T 0 > 0 such that the Cauchy problem (1)-(2) admits a unique strong solution (ρ, u, P, θ) in R 2 × (0, T 0 ] satisfying (8) and (9) (see Theorem 3.1). Thus, to prove Theorem 1.2, we only need to give some global a priori estimates on the strong solutions to system (1)-(2) in suitable higher norms.
It should be pointed out that the crucial techniques of proofs in [10] cannot be adapted to the situation treated here, since it seems difficult to bound the L p (R 2 )norm (p > 2) of u just in terms of √ ρu L 2 (R 2 ) and ∇u L 2 (R 2 ) . To this end, we try to adapt some basic ideas in [19], where the authors investigated the global existence of strong solutions to the 2D Cauchy problem of density-dependent Navier-Stokes equations. However, compared with [19], for the system (1)-(2) treated here, the strong coupling between the velocity field and the temperature, such as u · ∇θ, will bring out some new difficulties.
To overcome these difficulties mentioned above, some new ideas are needed. To deal with the difficulty caused by the lack of Sobolev's inequality, we observe that, in equations (1) 2 , the velocity u is always accompanied by ρ. Motivated by [16], by introducing a weighted function to the density, as well as the Hardy-type inequality in [17] by Lions, the ρ η u L r (Ω) (r > 2, η > 0 and Ω = B R or R 2 ) is controlled in terms of √ ρu L 2 (Ω) and ∇u L 2 (Ω) (see (34) and (152)). After some spatial estimates on ∇θ (i.e., ∇θx a 2 , see (59)), and suitable a priori estimates, we then construct approximate solutions to (1), that is, for density strictly away from vacuum initially, we consider a initial boundary value problem of (1) in any bounded ball B R with radius R > 0. Combining all these ideas stated above, we derive some desired bounds on the gradients of the velocity and the spatial weighted ones on both the density and its gradients where all these bounds are independent of both the radius of the balls B R and the lower bound of the initial density, and then obtain the local existence and uniqueness of solution (see subsection 3.2).
Base on the local existence result (see Theorem 3.1), we attempt to give some appropriate a priori estimates which are needed to obtain the global existence of strong solutions. First, we try to obtain the estimates on the L ∞ (0, T ; L 2 (R 2 ))norm of the gradients of velocity and temperature. On the one hand, motivated by [19], we use material derivativesu := u t + u · ∇u instead of the usual u t , and apply some facts on Hardy and BMO spaces (see Lemma 2.5) to bound the key term R 2 |P ||∇u| 2 dx (see the estimates of I 2 of (118)). Next, after some careful analysis, we derive the desired L 1 (0, T ; L ∞ (R 2 )) bound of the gradient of the velocity ∇u (see (155)), which in particular implies the bound on the L ∞ (0, T ; L q (R 2 ))-norm (q > 2) of the gradient of the density. Moreover, some useful spatial weighted estimates on ρ, θ, ∇θ are derived (see Lemmas 4.6 and 4.7). With the a priori estimates stated above in hand, we can estimate the higher order derivatives of the solution (ρ, u, P, θ) (see Lemma 4.8) to obtain the desired results.
The rest of the paper is organized as follows: In Section 2, we collect some elementary facts and inequalities which will be needed in later analysis. Sections 3 is devoted to establishing the local existence and uniqueness of strong solutions. The main result Theorem 1.2 is proved in Section 4.

2.
Preliminaries. In this section, we will recall some known facts and elementary inequalities which will be used frequently later. First of all, if the initial density is strictly away from vacuum, the following local existence theorem on bounded balls can be shown by similar arguments as in [6,8].
Then there exist a small time T R > 0 such that the equations (1) with the following initial-boundary-value conditions has a unique classical solution (ρ, u, P, θ) on B R × (0, where we denote H k = H k (B R ) for positive integer k.
Next, the following L p -bound for elliptic systems, whose proof is similar to that of [7,Lemma 12], is a direct result of the combination of the well-known elliptic theory [2] and a standard scaling procedure.
Lemma 2.4. For p > 1 and k ≥ 0, there exists a positive constant C depending only on p and k such that Finally, let H 1 (R 2 ) and BM O(R 2 ) stand for the usual Hardy and BM O spaces (see [24,Chapter IV]). Then the following well-known facts play a key role in the proof of Lemma 4.2 in Section 4.
3. Local existence and uniqueness of solutions. In this section, we shall prove the following local existence and uniqueness of strong solutions to the Cauchy problem (1)-(2).

3.1.
A priori estimates. Throughout this subsection, for r ∈ [1, ∞] and k ≥ 0, we denote Moreover, for R > 4N 0 ≥ 4, assume that (ρ 0 , u 0 , θ 0 ) satisfies, in addition to (10), Thus Lemma 2.1 yields that there exists some T R > 0 such that the initial-boundaryvalue problem (1) and (11) has a unique classical solution (ρ, u, P, θ) on B R ×[0, T R ] satisfying (12). Letx, a, and q be as in Theorem 1.2, the main aim of this section is to derive the following key a priori estimate on ψ defined by Proposition 1. Assume that (ρ 0 , u 0 , θ 0 ) satisfies (10) and (18). Let (ρ, u, P, θ) be the solution to the initial-boundary-value problem (1) and (11) on B R × (0, T R ] obtained by Lemma 2.1. Then there exist positive constants T 0 and M both depending only on µ, κ, q, a, N 0 , and E 0 such that where To show Proposition 1, whose proof will be postponed to the end of this subsection, we begin with the following standard energy estimate for (ρ, u, P, θ) and the estimate on the L p -norm of the density.

Lemma 3.2.
Under the conditions of Proposition 1, let (ρ, u, P, θ) be a smooth solution to the initial-boundary-value problem (1) and (11). Then for T 1 such as in Lemma 3.3 and t ∈ (0, where (and in what follows) C denotes a generic positive constant depending only on µ, κ, q, a, N 0 , and E 0 .
Proof. First, since div u = 0, it is easy to deduce from (1) 1 that (see [17, Next, multiplying (1) 2 and (1) 3 by u and θ respectively, then adding the two resulting equations together, and integrating over B R , we have Thus, Gronwall's inequality leads to which together with (22) yields (21) and completes the proof of Lemma 3.2.
Lemma 3.5. Let T 1 be as in Lemma 3.3. Then there exists a positive constant α > 1 such that for all t ∈ (0, Proof. 1. Differentiating (1) 2 with respect to t gives Multiplying (46) by u t and integrating the resulting equality by parts over B R , we obtain after using (1) 1 and (1) 4 that We estimate each term on the right-hand side of (47) as follows.
Hence, by virtue of Lemma 2.1, the initial-boundary-value problem (1) and (11) with the initial data (ρ R 0 , u R 0 , θ R 0 ) has a classical solution (ρ R , u R , P R , θ R ) on B R × [0, T R ]. Moreover, Proposition 1 shows that there exists a T 0 independent of R such that (20) holds for (ρ R , u R , P R , θ R ).
For simplicity, in what follows, we denote Extending (ρ R , u R , P R , θ R ) by zero on R 2 \ B R and denoting it by and sup 0≤t≤T0 ρ Rxa
In what follows, we will use the convention that C denotes a generic positive constant depending on µ, κ, a, q, and the initial data, and use C(α) to emphasize that C depends on α.
First, since div u = 0, we have the following estimate on the L ∞ (0, T ; L r )-norm of the density.
The following lemma concerns the L ∞ (0, T ; L 2 )-norm of the gradients of the velocity and the temperature.

It follows from integration by parts and Gagliardo-Nirenberg inequality that
Integration by parts together with (1) 4 yields where one has used the duality of H 1 and BM O (see [24,Chapter IV]) in the last inequality. Since div(∂ j u) = ∂ j div u = 0 and curl(∇u j ) = 0, we then derive from Lemma 2.5 and (120) that For the term I 3 , by Cauchy-Schwarz inequality, (110), and (117), one has Hence, inserting (119), (121) and (122) into (118) indicates that 3. Multiplying (1) 3 by ∆θ and integrating the resulting equality by parts over R 2 , it follows from Hölder's and Gagliardo-Nirenberg inequalities that which together with (123) gives 4. Since (ρ, u, P, θ) satisfies the following Stokes system applying the standard L r -estimate to (126) yields that for any r ≥ 2, where in the last inequality one has used (110). Then it follows from (125), (127), (110), and (117) that Thus, one has This combined with (117) and Gronwall's inequality yields (111). Finally, applying Gronwall's inequality to (128) multiplied by t, together with (117) gives (112) and finishes the proof of Lemma 4.2.
The following spatial weighted estimate on the density plays an important role in deriving the bounds on the higher order derivatives of the solutions (ρ, u, P, θ).
and sup Proof. 1. Motivated by [11], operating ∂ t + u · ∇ to (1) j 2 , one gets by some simple calculations that which multiplied byu j , together with integration by parts and (1) 4 , leads to We estimate each term on the right-hand side of (138) as follows.
Now, we give the following spatial weighted estimate on the gradient of the density, which has been proved in [19,Lemma 3.6]. We omit the detailed proof here for simplicity.
Next, we shall show the following spatial weighted estimates of θ and ∇θ, which are crucial to derive the estimates on the gradients of both u t and θ t .

4.2.
Proof of Theorem 1.2. With all the a priori estimates in subsection 4.1 at hand, we are ready to prove Theorem 1.2.