Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type

In this paper, we study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum. It is proved that there exists a unique strong solution for the two-dimensional Cauchy problem established that the initial density and the initial temperature decay not extremely slow. Particularly, it is allowed to be arbitrarily large for the initial data and vacuum states for the initial density, even including the com-pact support. Moreover, when the density depends on the Korteweg term with the viscosity coefficient and capillary coefficient, we obtain a consistent priority estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, when the pressure and external force are not affected, we deform the fluid models of Korteweg type, we can obtain the large time decay rates of the gradients of velocity, temperature and pressure.


Introduction
In this paper, we consider the system for the nonhomogeneous incompressible the viscosity and thermal conductivity related to temperature, Lorca [4] and Boldrini [5] showed the initial value problem of viscous incompressible systems. In some recent studies, the density-dependent viscous incompressible Boussinesq system caused wide attention. Qiu and Yao [6] get the local well-posedness for the density-dependent Boussinesq Equation (1.1) and consider the regularity problem of the smooth solutions for this equation in Besov spaces. The paper [7] considers the stability and zero dissipation limit of the boundary problem of the propose the relationship between velocity field, fluid temperature and pressure so as to solve the difficulties caused by vacuum.
If the thermal diffusivity 0 κ = , the system (1.1) is referred to as Korteweg model. Research on the compressible Navier-Stokes-Korteweg fluid model has been developed. For small initial data, [8] [9] provided the existence problems of the global strong solutions for Korteweg system in Besov space. And the global existence of weak solutions in the whole space 2  was obtained by Danchin-Desjardins [10] and Haspot [11]. For large initial data, Bresch-Desjardins-Lin [12] analyzed the Korteweg-type compressible fluid model with density-dependent Now, we explain the estimation of the complex term in this model. It is worth noting that when the initial data meets (1.7), the uniqueness and existence result of the strong solution of (1.1)-(1.2) Cauchy problem has been discussed in [17]. In order to extend the local situation to large-time, we need not only lower order estimate on strong solution of (1.1)-(1.2), but also a priori estimates with higher norm.
In this article, the estimate of terms ( )  (3.31)), then together with (3.44) to attain 2,q a W x ρ . Based on the above treatment of the special term, one can complete the higher order estimates of the solution ( ) , , , u p ρ θ . Finally, motivated by [17], our new observation of this paper is to obtain the 2 L -norm of 1 2 x θ and 1 2 x θ ∇ (see (3.76)), which are critical to constraint the ( ) ( ) Now we will explain the symbols and conventions applied in this article. For where ( ) ( )   , , , u p ρ θ has the following decay rates, that is for where C depends only on µ , κ , When the initial data is large, there is no other compatibility conditions are considered for the global existence of the strong solutions.
The following sections of the article are introduced as follows: first, in Section 2, we give some basic facts and important inequalities, which can be applied in the calculations below. Next, in Section 3, we will give the priori estimates. In Section 4, we will attain the important result of this paper, Theorem 1.2, based on the previous.

Preliminaries
In this section, we recall the relevant results obtained by previous mathematicians and state our main results. Then, we begin with the unique and local strong solution. As follows: q r q r r q r r r q r The next weighted n L bounds can be seen in ( [28], Theorem 1.1) for elements in Finally, set ( ) BMO  denote the standard Hardy and BMO spaces (see [29], chapter IV). Then the next basic fact is very important for proving lemma 3.2 in the section 3.
(ii) It follows from the Poincaré inequality that for any ball

A Priori Estimates of the Solution Lower Order Estimates
First, because of div 0 u = , we have the following estimate related to the density on the Lemma 3.1. There exists a positive constant C depending only on We give the time-independent estimates of u ∇ and θ ∇ on the There exists a positive constant C depending only on µ , κ , Proof. Applying standard energy estimate, taking the i x -derivative (i = 1, 2) of (1.4) gives Multiplying (1.1) by 2 ρ ∇ and integrating the resulting equality on 2  , we get Adding (1.4) × 2u to (1.4) × u 2 and integrating the resulting equality on 2  , we have Multiplying (1.1) by θ and integrating the resulting equality on 2  , we have Combining ( .
Then it follows from integration by parts and Gagliardo-Nirenberg inequality Integrating by parts together with (1.1) gives where one has used the duality of 1  space and BMO one (see [28], Chapter IV]) in the last inequality. Since Equation (3.11) combined with Equation (3.12) and Equation (2.8) gives Next, substituting (3.10), (3.13) and (3.14) into (3.9) gives ( ) Then, multiplying (1.1) by θ ∆ and integrating the resulting equality by parts over 2  , it follows from Hölder's and Gagliardo-Nirenberg inequalities that 3 2 2 Since ( ) , , , u p ρ θ satisfies the following Stokes system ( ) , , div 0, , 0, , Applying the standard r L -estimate to (3.18) (see [31]) yields that for any 1 r > , are all bounded in 2  , an application of the Gagliardo-Nirenberg inequality results in where ε is to be determined. Choosing , Proof. First, for It follows from (1.1) that ( ) ( ) where in the last inequality one has used (3.1) and (3.8 Next, multiplying (3.33) by j u  , together with integration by parts and (1.1), we get Due to (3.13) and (3.21), for the right-hand side of (3.37), it follows from  .
Next, we estimate which together with the Gagliardo-Nirenberg inequality shows that which is deformed and calculated appropriately leads to where in the last and the second inequalities, we has applied (3.29) and (          This contradicts the supposition of * T in (4.1), so the (4.4) holds. Therefore, the existence and uniqueness of local strong solutions and Lemmas 3.

Conclusions
For the general incompressible Navier-Stokes flow equation, there is no external force action, we can under the low estimate to a prior estimate of velocity and pressure. In this article, we study the two-dimensional incompressible Boussinesq the equations of Korteweg type model, and fluid temperature contains not only depends on the density of viscous coefficient, and influenced by external forces.
On the one hand, we should overcome the trouble of unbounded region when making the estimation, and carefully consider the special terms On the other hand, the Korteweg fluid model contains high order derivative terms of density, and the system we consider is in the case of large initial values, which makes it difficult to prove the global existence of strong solutions. In order to prove the global existence of the strong solution, we introduce the derivative of the random body and the auxiliary energy estimation of the fundamental inequality.