The Boussinessq equation is a coupling of the fluid temperature and velocity field. For this paper, we consider the Cauchy problem of 2D nonhomogeneous incompressible Boussinessq equations which read as follows:
where is the density, represents the velocity, stands for the pressure and denotes the temperature of the fluid; is the viscosity coefficient; is the thermal diffusivity.
The initial data is given by
The system (1.1) is a simple model widely used in the modeling of atmospheric motions and oceanic, and it plays an important role in the atmospheric sciences (see  ). The Boussinessq equation is a coupling of the fluid velocity field and the temperature field. In particular, the 2D Boussinesq equations act as a lower-dimensional model of the 3D hydrodynamics equations and can be originated from the conservation laws of mass, energy and momentum (see  ). The 2D Boussinesq equations can be seen as a special case of the 3D incompressible Navier-Stokes and Euler equations, with similar vortex stretching mechanism to 3D incompressible fluids. However, compared with the Navier-Stokes equation and the Euler equation, it has an unknown temperature function and there is a complex nonlinear relationship between the temperature function and the velocity and pressure (see     ). Moreover, the Boussinesq system (1.1) with , there have been a lot of results. When , many scholars have done a lot of research in recent years (Reference    ).
In recent years, the Boussinesq system with has attracted the attention of many mathematicians, and many related research results have emerged. The study of viscous thermal diffusion Boussinesq equations, that is the system (1.1) with and , is popular. Lorca  and Boldrini  gained the existence of global weak solutions for Boussinesq equations with small initial values. And they also studied the existence of local strong solutions under general initial conditions. Recently, much attention has attracted by the density-dependent viscous Boussinesq equations. But, the regularity questions of the case of (1.1) with the initial data can be arbitrarily large, whic is an open problem. Qiu and Yao  showed the local existence and uniqueness of strong solutions of multi-dimensional incompressible density-dependent Boussinesq equations in Besov spaces. The paper  studied regularity criteria for three-dimensional incompressible density-dependent Boussinesq equations. However, there is little research on Boussinesq system when the initial density may include vacuum state or compact support. Because the initial density including the vacuum state will affect the temperature and pressure, the interaction among density, temperature and pressure will increase the non-linear coupling of the system (1.1), thus it may make the problem more complex. Recently, the global existence of strong solutions to the 2D Cauchy problem is given by Lü-Xu-Zhong , the related research refers to  -  . Particularly, the initial density will include vacuum condition and one has compact support and the initial data could be arbitrarily large. Therefore, motivated by , we studied the Boussinesq system with initial density including vacuum and general big data initial. However, the divergence of temperature in the system (1.1) cannot be zero. This enhances the coupling of , and we need to solve some new difficulties.
Now, we make some comments on the analysis of the key ingredients of this paper. If the local solution is extended to the global solution, we need to get global a priori estimates on strong solution to (1.1)-(1.2) in proper higher norms. Because of the strong coupling between temperature and velocity field, the will give rise to some new difficulties. It seems difficult to bound the -norm of u in terms of and . In light of    , we try to estimate on the -norm of and , it can replace the usual with the to multiply by (1.1)2 (see  ). The most important thing is to control the term
According to  , since and , we have the term . And because and combined (2.6) and (2.8), the term in practice can be bounded by (see (3.11)). Next, due to the strong coupled term , we cannot estimate directly the -norm of . Because of multiplying (1.1)3 by , we can't get -norm of and there will be items related to time t. Thus, we can use the instead of the usual to multiply by (1.1)3, and integration by parts that the coupled term can be controlled (see (3.13)). Then, we apply the Stokes system to obtain the -norm of and the -norm of (see (3.16)), and combined with the acts on and multiplied by to get the (3.20) and make further efforts to give the (3.37) (see Lemma 3.3 and 3.5). In addition, it is sufficient to bound the -norm of instead of u. More precisely, using Lemma 2.4 (see (3.35)), reference , and along with the estimate of (see (3.36)), we can find the desired estimates on the -norm of (see (3.40)). Finally, we gain the -norm of and (see (3.56)) and (see (3.57)), which are important to bound the -norm of both and and the -norm of , see Lemma 3.8.
Now, we go back to (1.1). it should be noted here that the notations and conventions employed throughout the paper. For , set
Furthermore, for , , we denote the standard Lebesgue and Sobolve spaces as follows:
Then, we will define precisely what mean by strong solution to (1.1) as follows:
Definition 1.1. (see  ) If all derivatives related to (1.1) for are regular distributions, and system (1.1) satisfy almost everywhere in , then is named a strong solution to (1.1).
In a general way, it can assume that holds
The (1.4) signifies that there is a positive constant such that
Theorem 1.1 In view of (1.4) and (1.5), it assumes that the initial data hold that for any given numbers and ,
In that way, it has a unique global strong solution for the problem (1.1)-(1.2) satisfying that for any ,
It’s about positive constant depending only , , and T. The has the following decay rates, that is for ,
where C depends only on , , , , , and .
Remark 1.1 If the temperature function is zero, i.e., , then (1.1) is the well-known Navier-Stokes equations, and Theorem 1.1 is the same as those results of .
Remark 1.2 Theorem 1.1 goes for arbitrarily large initial data, it can also find the global strong solutions to the 2D incompressible Boussinesq equations with the smallness condition on the initial energy see  .
In next section, we shall first state some basic truths and inequalities. Those things will be employed later in this paper. In the last section is committed to some priori estimates and prove the theorem 1.1.
For the section, we will recall some known truths and elementary inequalities, which will be used frequently later. Then for initial data, it assumes that there is a unique local strong solution. As follows:
Lemma 2.1 see  Assume that satisfies (1.6). Then there exists a small time and a unique strong solution to the problem (1.1)-(1.2) in satisfying (1.8) and (1.9).
Lemma 2.2 (see (  Theorem 1.1)) (Galiardo-Nirenberg). For , , and , there exists some generic constant which may relay on m, q, and r such that for and , we have
The following weighted bounds for elements in can be found in ( , Theorem 1.1).
Lemma 2.3 (see ( , TheoremB.1)) For and , there exists a positive constant C such that for all ,
The Lemma 2.3 combined with the Poincaré inequality gets the following useful results on weighted bounds, we can also refer to (  Lemma 2.4).
Lemma 2.4 (see (  Lemma 2.3)) We can refer to in (1.6), and assume that is a non-negative function such that
for positive constants , , and with . Then for , , there is a positive constant C depending only on , , , , and such that every satisfies
Finally, let and represent BMO and Hardy spaces (see , chapter 4). In the next section, some facts are more important to prove the lemma 3.2.
Lemma 2.5 (see (  TheoremII.1)) (i) There is a positive constant C such that
for all and satisfying
(ii) There is a positive constant C such that
for all .
Proof. (i) Please refer to (  Theorem II.1) for detailed proof.
(ii) The follows together with the Poincaré inequality that for any ball
which directly gives (2.8).
3. Convergence Rate of the Solution
3.1. Lower Order Estimates
Due to , it will estimate the -norm of , as follows:
Lemma 3.1 (see  ) There exists a positive constant C depending only on such that
Then, we will estimate the -norm of and .
Lemma 3.2. There is a positive constant C depending only on , , , , , and such that
Here , and have
Proof. Invoking standard energy estimate, multiplying (1.1)2 by u and integrating the resulting equality over , we get
Multiplying (1.1)3 by and integrating the resulting equality over , we have
The (3.4) combined with (3.5) that gives
Next, multiplying (1.1)2 by and integrating the resulting equality over , we have
Then we can follow form integrating by parts and (2.1) that
Integration by parts together with (1.1)4 gives
For the last inequality, because of the duality of space and BMO (see (  Chapter IV)). And , , and (2.6) yields
The (3.9) combined with (3.10) and (2.8) gives
Next, substituting (3.8) and (3.11) into (3.7) gives
Then, the (1.1)3 multiplied by and integrating the resulting equality by parts over , and together with Hölder’s and (11) that
which combined with (3.12) and (3.6) gives
Due to solves the following Stokes system see 
Using the standard -estimate to (3.15) holds that for any ,
(3.14) combined with and (3.16) gives
where is to be determined. Choosing , it follows from (3.6) and (3.17) that
the (3.18) together with (3.6), (3.17) and (2.1) gives (3.2). Then, (3.17) multiplied by t, we have
the (3.19) combined Gronwall’s inequality with (3.6) gives (3.3). Finally, it finishes the proof of lemma 3.2.
Lemma 3.3 There is some positive constant C depending only on , , , , and such that for ,
Proof: Using to , it follows from a few simple calculations that
Next, (3.22) multiplied by , and integration by parts and (1.1)4, we get
Following the same argument as (  Lemma 3.3) we have the estimates of as
Substituting (3.24) into (3.23) gives
For the left of (3.25), we have
For the right of (3.25), (3.11) together with (3.16), (3.1) and Sobolev's inequality that
Substituting (3.26) and (3.27) into (3.25)
Next, multiplying (3.28) by , it follows from (3.3) and (3.6) that
Then, the (3.29) along with Gronwall's inequality gives
Finally, due to , it deduces from (3.3) to lead to (3.20). The (3.21) is a direct consequence of (3.20) and (3.16). We will finish the proof of Lemma 3.3.
3.2. Higher Order Estimates
It concerns with the estimates on the higher-order derivatives of the strong solution as follow:
Lemma 3.4 For a positive constant C depending only on , , , , , , and T, such that
Proof. For , let satisfy
It combines with (1.1)1 that
in the last inequality of (3.33), it has applied (3.1) and (3.6). Integrating (3.33) and letting , we obtain after using (1.5) that
the (3.34) along with (3.1), (2.2), (3.6) and (3.2) that for any and any ,
(1.1)1 multiplied by and integrating the resulting equality by parts over find that
using the Gronwall’s inequality to (3.36) gives (3.31) and it proves the lemma 3.4.
Lemma 3.5 There is a positive constant C depending on T such that
Proof. We can follow from the (1.1)1 that holds for any ,
Next, employing Lemma 2.2, (3.2) and (3.16), we have ,
It follows from (3.34), (3.1), (2.2) and (3.31) that for any ,
the (3.40) combine with the Gagliardo-Nirenberg inequality shows that
which is deformed and calculated appropriately leads to
Then, the (3.42) and (3.39) implies
Next, using Gronwall’s inequality to (3.38) shows
Then, letting in (3.16) and integrating the resulting equality over , we obtain after using (3.1), (3.2) and (3.3) that
Similarly, setting in (3.16) and integrating the resulting equality over , we deduce from using (3.42), (3.1), (3.2) and (3.3) that
Multiplying (3.16) by t and integrating the resulting equality over , it can obtain after using (3.43), (3.1), (3.2) and (3.3) that
Moreover, it can get from (3.46), (3.47) and (3.48) that
which combined with (3.1) and (3.45) gets (3.37). The Lemma 3.5 is proved.
Lemma 3.6 (see  ) There exists a positive constant C depending on T such that for ,
Proof. First, setting in (1.1)1 that satisfies
Next, we can take the -derivative on both sides of the (3.51) finds
the (3.52) multiplied by and integrating the resulting equality by parts over , and then for any , we obtain that
For the second and the last inequalities of (3.53), it has used (3.35) and (3.31), respectively. Setting in (3.53), and applying Gronwall’s inequality along with (3.37) indicates that
And choosing in (3.53), we will deduce from (3.37) and (3.54) that
Combining (3.54) with (3.31) gives (3.50). The Lemma 3.6 is proved.
Lemma 3.7 There exists a positive constant C such that
Proof. The (1.1)3 multiplied by and integrating the resulting equality by parts over , we have
Substituting (3.59), (3.60) into (3.58), we get
Using Gronwall’s inequality to (3.61), we obtain (3.56).
Next, we will estimate the (3.57). The (1.1)3 Multiplied by and integration by parts over , we find
Submitting , , into (3.62), one has
Multiplying (3.66) by t, and togethering with (3.56) and (3.37), then employing Gronwall’s inequlity, one obtains the (3.57). This completes the Lemma 3.7.
Lemma 3.8 There exists a positive constant C such that
Proof. For any and any , it deduces from (3.40), (3.35) that
Next, it will prove that
With (3.2) at hand, we need only to show
First, it is easy to show that
Then, due to (2.1) and (3.68), we can combine (2.1), (3.2) with (1.1)3 gives
It has used the following facts about (3.72) of the last inequality
According to (3.68) and (2.1), we can give (3.70) by the combination of (3.71), (3.72), (3.37), and (3.56).
Next, differentiating (1.1)2 with respect to t shows
(3.74) multiplied by and integration by parts over , it deduces from (1.1)1 and (1.1)4 that
Submitting , into (3.75) gives
Then, we multiply (3.78) by t, and link to Gronwall’s inequality and (3.37) lead to
Next, differentiating (1.1)3 with respect to t show
Now, the (3.80) multiplied by and integration by parts over , we find
Next, the (3.81) multiplied by t and integration by parts over , and due to (3.70), we have
Finally, it follows from (1.1)3, and (3.73) that
which combine with (3.57), (3.79) and (3.82) gains (3.67). Finally, the proof of Lemma 3.8 is finished.
3.3. Proof of Theorem 1.1
In this section, we will give the proof of Theorem 1.1.
Proof: According to Lemmas 3.1-3.8, using standard theory of local existence, It assumes that there is a such that systems (1.1) and (1.2) have a local and unique strong solution on . Next, we will extend the local solution to all time.
It deduces from (3.67), for any with T finite, and any that,
Then, along with standard embedding
And, due to (3.36), (3.49), and (  Lemma 2.3) we have
we declare that
On the contrary, if , it deduces from (4.2), (4.2), (3.2), (3.6), (3.49), and (3.50) that
conforms to the initial condition (1.6) at . So, we can assume the initial data is the , since the existence and uniqueness of local strong solutions signifies that there is a some , such that Theorem (1.1) holds for . This is contradictory with the hypothesis of in (3.84), so the (3.87) holds. Hence, Lemmas 3.1-3.8 and the local existence and uniqueness of strong solutions indicate that is actually the unique strong solution on for any . This completes the proof of Theorem 1.1.