In this paper, we consider the system for the nonhomogeneous incompressible Boussinesq equations of Korteweg type as follows
Among them, the free vector field of divergence represents the velocity of the fluid; the scalar function and represent pressure of the fluid and temperature respectively; parameter represents the viscosity coefficient dependent on temperature; and represent density and the capillary coefficient respectively; and the constant represents the thermal diffusivity; f denote the external force.
The initial data is given by
Equation (1.1) governs the motions of the incompressible nonisothermal viscous capillary fluids. Assumed capillarity coefficient , the system (1.1) can simplify to other incompressible equations. Recently, the incompressible equations with density have led scholars to do much research, which gets important results in different academic fields. The study of the system (1.1) with and is more concerned. Cannon and DiBenedetto proved the initial value problem of the Boussinesq equation for incompressible fluid affected by convective heat transfer (see  ). Furthermore, they improved the regularity of the solution when the initial data are smooth. In addition, few scholars further studied the cases of “partial viscosity” (i.e. either the zero diffusivity case, and , or the zero viscosity case, and ). Hou and Li  demonstrated the global well-posedness of the Cauchy problem of viscous Boussinesq equations. Chae  considered the Boussinesq system for incompressible fluid in with either zero diffusion ( ) or zero viscosity ( ). He proved global-in-time regularity in both cases. In addition, the singularity problem of Equation (1.1) is still an unsolved problem in mathematical fluid mechanics. Under the generalized Boussinesq equation approximation, by giving the viscosity and thermal conductivity related to temperature, Lorca  and Boldrini  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  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  considers the stability and zero dissipation limit of the boundary problem of the multidimensional Boussinesq system. But, when the initial conditions include a vacuum situation, there are little relevant researches. For Equation (1.1) we propose the relationship between velocity field, fluid temperature and pressure so as to solve the difficulties caused by vacuum.
If the thermal diffusivity , 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,   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 was obtained by Danchin-Desjardins  and Haspot . For large initial data, Bresch-Desjardins-Lin  analyzed the Korteweg-type compressible fluid model with density-dependent capillary coefficient, and obtained existence results. Recently, Germain-LeFloch  studied the existence, convergence and compactness of the compressible Navier-Stokes-Korteweg model. And both the vacuum and nonvacuum weak solutions were obtained. Moreover, Chen-He-Zhao  discussed the construction of smooth solutions to the Cauchy problem of the fluid models of Korteweg type, and the global solvability results are acquired if the , and satisfy certain conditions (see also   ). Assuming that the influence of temperature is not considered, the Equation (1.1) can be simplified into a general incompressible Korteweg model, then liu-wang-zheng  studied the strong solution of Cauchy problem in this model. Up to now, when the Korteweg term is introduced into Boussinesq equation, this kind of problem is still unknown. For this problem, we have to consider the difficulty of Korteweg term . More importantly, the particularity of vacuum state should also be considered. These are also the core issues of this article.
Our purpose is to study the Cauchy problem for the strong solutions of Equations ((1.1), (1.2)). For convenience, we can set . Since
classifying the term as the pressure term, we can deform the equation as
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 . 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 , and requires us to use some special ideas. We follow the ideas of Lü-Shi-Zhong  studying the incompressible N-S equation and Lü-Xu-Zhong  studying the compressible MHD equation. First, we attempt to estimate on the -norm of and . Using the key technique of   , we multiplied and abandoned the normal (see  ). More importantly, motivated by  , the term was controlled by the basic theories of Hardy and BMO in the second section, which have the term
(see (3.13)). Next, we use the Stokes system (3.18) to get the and (see (3.19)), the key point is to use Gagliardo-Nirenberg
inequality to estimate the value of . Multiplying (1.1)3 by can control the strong coupled term after integration by parts (see (3.16)). And, considered    , we apply to and multiply the resultant equality by to attain the -norm of and the -norm of (see (3.31)), then together with (3.44) to attain . Based on the above treatment of the special term, one can complete the higher order estimates of the solution . Finally, motivated by , our new observation of this paper is to obtain the -norm of and (see (3.76)), which are critical to constraint the -norm of both and and the -norm of , see Lemma 3.8.
Now we will explain the symbols and conventions applied in this article. For , let
Meanwhile, for and , the standard Lebesgue and Sobolve spaces have the following forms:
Next, we show the definition of strong solution to system (1.1) as follows:
Definition 1.1. Assumed the whole derivatives related to system for are regular distributions, Equation (1.1) also hold almost everywhere in , then is considered a strong solution to (1.1).
Moreover, it can be assumed that the initial density satisfies
which implies that exists a positive constant such that
The main conclusions of this paper are given as follows:
Theorem 1.2. Besides (1.5) and (1.6), if the initial data hold that for any constant and ,
and satisfy the compatibility condition
for some and , . Then the problem (1.1)-(1.2) has a unique global strong solution satisfying that for any ,
for positive constant depending only , , and T. In addition, the has the following decay rates, that is for ,
where C depends only on , , , , , and .
Remark 1.3. If there is no influence of fluid temperature, i.e., , then (1.1) reduces to the fluid of Korteweg type, Theorem 1.2 extends the results of Liu and Wang  to the Cauchy problem of global solutions in two-dimensional space. 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.
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:
Lemma 2.1. If that satisfies (1.7). Then there exists a small time and a unique strong solution to the problem (1.1)-(1.2) in that satisfies (1.10) and (1.11).
Lemma 2.2. (Gagliardo-Nirenberg inequality). For , , and , there exists some generic constant which may depend on m, q, and r such that for and , we have
The next weighted bounds can be seen in ( , Theorem 1.1) for elements in .
Lemma 2.3. For and , there exists a positive constant C such that for all ,
Between Lemma 2.3 and the Poincaré inequality, we can get the following key results on weighted bounds, this proof is mentioned in ( , Lemma 2.4).
Lemma 2.4. Let be as in (1.8). 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, set and denote the standard Hardy and BMO spaces (see , chapter IV). Then the next basic fact is very important for proving lemma 3.2 in the section 3.
Lemma 2.5 (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) For the specific proof steps, please see ( , theorem II.1).
(ii) It follows from the Poincaré inequality that for any ball
3. A Priori Estimates of the Solution
Lower Order Estimates
First, because of , we have the following estimate related to the density on the -norm.
Lemma 3.1. There exists a positive constant C depending only on such that
We give the time-independent estimates of and on the -norm.
Lemma 3.2. There exists a positive constant C depending only on , , , , , and such that
Here , and , furthermore, one has
Proof. Applying standard energy estimate, taking the -derivative (i = 1, 2) of (1.4) gives
Multiplying (1.1) by and integrating the resulting equality on , we get
Adding (1.4) × 2u to (1.4) × u2 and integrating the resulting equality on , we have
Multiplying (1.1) by and integrating the resulting equality on , we have
Combining (3.5) with (3.6), (3.7), and then integrating on gives
Next, multiplying (1.1) by and integrating the resulting equality on , we have
Then it follows from integration by parts and Gagliardo-Nirenberg inequality that
Integrating by parts together with (1.1) gives
where one has used the duality of space and BMO one (see , Chapter IV]) in the last inequality. Since , , and (2.6) yields
Equation (3.11) combined with Equation (3.12) and Equation (2.8) gives
Integration by parts together with (1.4), (3.4), (3.8), and Gagliardo-Nirenberg inequality gives
Next, substituting (3.10), (3.13) and (3.14) into (3.9) gives
Then, multiplying (1.1) by and integrating the resulting equality by parts over , it follows from Hölder’s and Gagliardo-Nirenberg inequalities that
which combined with (3.15) and (3.8) gives
Since satisfies the following Stokes system
Applying the standard -estimate to (3.18) (see  ) yields that for any ,
Moreover, since , , are all bounded in , an application of the Gagliardo-Nirenberg inequality results in
substituting (3.20) into (3.19), we get
This combined with (3.17) and (3.19) gives
where is to be determined. Choosing , it follows from (3.22) that
which together with (3.8), (3.22) and Gronwall’s inequality gives (3.2). Then, multiplying (3.22) by t, we have
which combined (3.8) with Gronwall’s inequality gives (3.3). Finally, the above completes the proof of Lemma 3.2.
Lemma 3.3. There exists a positive constant C depending only on , , , , , , and T, such that
Proof. First, for , let satisfy
It follows from (1.1) that
where in the last inequality one has used (3.1) and (3.8). Integrating (3.27) and choosing , we obtain after using (1.6) that
Hence, it follows from (3.28), (3.1), (2.2), (3.8) and (3.2) that for any and any ,
Multiplying (1.1) by and integrating the resulting equality by parts over yield that
this together with Gronwall’s inequality can get (3.25), and the above completes the proof of Lemma 3.3.o
Lemma 3.4. There exists a positive constant C depending only on , , , and such that for ,
Proof. Operating to (1.1)j, one gets by some simple calculations that
Next, multiplying (3.33) by , together with integration by parts and (1.1), we get
Following the same argument as ( , Lemma 3.3) we have the estimates of as
which combined (1.3), (3.4) and the Gagliardo-Nirenberg inequality leads to
Substituting (3.35) and (3.36) into (3.34) and together with (3.2) gives
Due to (3.13) and (3.21), for the right-hand side of (3.37), it follows from (3.13), (3.21), (3.1) and Sobolev’s inequality that
Substituting (3.38) and (3.39) into (3.37)
Next, we estimate . First, the (3.25) combined (3.8) with (3.29) that for any and
The (1.4) combined with (3.8), (3.41) and the Gagliardo-Nirenberg inequality, we derive
which together with (3.41) and the compatibility condition (1.9) yields
Finally, Multiplying (3.40) by and using (3.43), it deduces from Gronwalls inequality and (3.8) to lead to (3.31). The (3.32) is a direct result of (3.31) and (3.21). The proof of Lemma 3.4 is finished.
Lemma 3.5. There exists a positive constant C depending on T such that
Proof. First, it follows from the mass Equation (1.1) that satisfies for any ,
Next, employing Gagliardo-Nirenberg inequality, using (3.2) and (3.19), we have ,
It follows from (3.28), (3.1), (2.2) and (3.25) that for any ,
which together with the Gagliardo-Nirenberg inequality shows that
which is deformed and calculated appropriately leads to
Then, the (3.49) along with (3.46) in particular implies
Next, applying Gronwall’s inequality to (3.45) gives
Setting in (3.19) and integrating the resulting equality over , we obtain after using (3.1), (3.2) and (3.3) that
Similarly, setting in (3.19) and integrating the resulting equality over , we deduce from using (3.49), (3.1), (3.2) and (3.3) that
Multiplying (3.19) by t and integrating the resulting equality over , it can obtain after using (3.50), (3.1), (3.2) and (3.3) that
Furthermore, it is easy to deduce from (3.53), (3.54) and (3.55) that
this together with (3.1) and (3.52) yields (3.44), which completes the proof of Lemma 3.5.o
Lemma 3.6. There exists a positive constant C depending on T such that for ,
Proof. First, setting in (1.1) that satisfies
Taking the -derivative on both side of the (3.58) gives
For any , multiplying (3.59) by and integrating the resulting equality by parts over , we obtain that
where in the last and the second inequalities, we has applied (3.29) and (3.25), respectively. Choosing (3.60), and applying Gronwall’s inequality together with (3.44) indicates that
Setting in (3.60), we deduce from (3.44) and (3.61) that
Next, taking the -derivative again on both side of the (3.59) gives
Similarly, for any , multiplying (3.63) by and integrating the resulting equality by parts over , we can find that
Using (3.64) for Gronwall’s inequality, and according to (3.25), (3.44) and (3.61), we gain the desired estimate (3.57). It completes the proof of the Lemma 3.6. o
Lemma 3.7. There exists a positive constant C such that
Proof. First, multiplying (1.1) by and integrating the resulting equality by parts over , we have
Substituting (3.68), (3.69) into (3.67), we get
Using (3.70) for Gronwall’s inequality, we obtain (3.65).
Next, we estimate the (3.66). Multiplying (1.1)3 by and integrating the resulting equality by parts on , we find
Submitting , , into (3.71), one has
Multiplying (3.75) by t, and together with (3.65) and (3.44), then employing Gronwall’s inequlity, one obtains the (3.66). This completes the Lemma 3.7.o
Lemma 3.8. There exists a positive constant C such that
Proof. First, it is easy to deduce from (3.47), (3.29) that for any and any ,
Next, we prove
With (3.2) at hand, we need only to show
First, it is easy to show that
Then, due to (2.1) and (3.77), combining (2.1), (3.2) with (1.1)
where in the last inequality one has used the follow facts
On the basis of (3.77) and (2.1), (3.79) can be derived by the combination of (3.80), (3.81), (3.44) and (3.65).
Next, differentiating (1.1)2 with respect to t shows
Multiplying (3.83) by and integrating the resulting equality by parts on , it follows (1.1) that
Submitting the above into (3.84) gives
Then, we multiply (3.91) by t, and together with Gronwall’s inequality and (3.44) lead to
Next, differentiating (1.1) with respect to t shows
Now, multiplying (3.93) by and integrating the resulting equality by parts on , we find
Next, multiplying (3.94) by t and integrating the resulting equality by parts on , it follows from (3.79) that
Finally, it follows from (1.1), and (3.82) that
which combined with (3.66), (3.92) and (3.95) indicates (3.76) and the proof of Lemma 3.8 is finished.
4. Proof of Theorem 1.2
In this Section, by the prior estimation in the previous chapter of Lemmas 3.1-3.8, we can complete the proof of Theorem 1.2.
Proof. On the basis of Lemmas 3.1-3.8, through previous result of local existence, there has a such that the Equation (1.1) and (1.2) have unique and local strong solution on . Next, we will extend the local problem to all time.
For any with T finite, one deduces from (3.76) that for any ,
where one has used the standard embedding
Moreover, it follows from (3.30), (3.56), and ( , Lemma 2.3) that
We claim that
Otherwise, if , it follows from (4.2), (4.3), (3.2), (3.8), (3.56), and (3.57) that
which satisfies the initial condition (1.6) at . Thus, taking as the initial data, since the existence and uniqueness of local strong solutions implies that there exists some , such that Theorem 1.2 satisfy . This contradicts the supposition of in (4.1), so the (4.4) holds. Therefore, the existence and uniqueness of local strong solutions and Lemmas 3.1-3.8 show that is in fact the unique strong solution on for any . The above can prove Theorem 1.2.o
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 and . At the same time, we should also consider the difficulties caused by the strong coupling between the velocity and temperature of the fluid. For example, , for such difficult terms, we should carry out ingenious structural analysis and strict calculation and derivation.
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.
Thanks to those who contributed to this article but are not listed in the author list. Thanks to my tutor and classmates for their guidance and help on the model in this paper.
 Cannon, J. and DiBenedetto, E. (1980) The Initial Value Problem for the Boussinesq Equations with Data in Lp. In: Rautmann, R., Ed., Approximation Methods for Navier-Stokes Problems. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg.
 Qiu, H. and Yao, Z. (2017) Well-Posedness for Density-Dependent Boussinesq Equations without Dissipation Terms in Besov Spaces. Computers & Mathematics with Applications, 73, 1920-1931.
 Wang, J. and Xie, F. (2015) Zero Dissipation Limit and Stability of Boundary Layers for the Heat Conductive Boussinesq Equations in a Bounded Domain. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145, 611-637.
 Haspot, B. (2016) Existence of Global Strong Solution for Korteweg System with Large Infinite Energy Initial Data. Journal of Mathematical Analysis and Applications, 438, 395-443.
 Danchin, R. and Desjardins, B. (2001) Existence of Solutions for Compressible Fluid Models of Korteweg Type. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 18, 97-133.
 Bresch, D., Desjardins, B. and Lin, C.K. (2003) On Some Compressible Fluid Models: Korteweg, Lubrication and Shallow Water Systems. Communications in Partial Differential Equations, 28, 843-868.
 Germain, P. and LeFloch, P.G. (2016) Finite Energy Method for Compressible Fluids: The Navie-Stokes-Korteweg Model. Communications on Pure and Applied Mathematics, 69, 3-61.
 Chen, Z., He, L. and Zhao, H. (2017) Global Smooth Solutions to the Nonisothermal Compressible Fluid Models of Korteweg Type with Large Initial Data. Zeitschrift für angewandte Mathematik und Physik, 68, Article No. 79.
 Chen, Z., Chai, X.J., Dong, B.Q. and Zhao, H.J. (2015) Global Classical Solutions to the One-Dimensional Compressible Fluid Models of Korteweg Type with Large Initial Data. Journal of Differential Equations, 259, 4376-4411.
 Liu, Y., Wang, W. and Zheng, S.N. (2018) Strong Solutions to the Cauchy Problem of Two-Dimensional Incompressible Fluid Models of Korteweg Type. Journal of Mathematical Analysis and Applications, 465, 1075-1093.
 Lü, B.Q., Xu, Z.H. and Zhong, X. (2017) Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Promblem of 2D Density-Dependent Magnetohydrodynamic Equations with Vacuum. Journal de Mathématiques Pures et Appliquées, 108, 41-62.
 Lü, B.Q., Shi, X.D. and Zhong, X. (2018) Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Promblem of 2D Density-Dependent Navier-Stokes Equations with Vacuum. Nonlinearity, 31, 2617-2632.
 Li, J. and Xin, Z.P. (2019) Global Well-Posedness and Large Time Asymptotic Behavior of Classical Solutions to the Compressible Navier-Stokes Equations with Vacuum. Annals of PDE, 5, Article No. 37.
 Huang, X.D. and Wang, Y. (2014) Global Strong Solution with Vacuum to the Two-Dimensional Density-Dependent Navier-Stokes System. SIAM Journal on Mathematical Analysis, 46, 1771-1788.
 Hoff, D. (1995) Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data. Journal of Differential Equations, 120, 215-254.
 Lü, B.Q. and Huang, B. (2015) On Strong Solutions to the Cauchy Problem of the Two-Dimensional Compressible Magnetohydrodynamic Equations with Vacuum. Nonlinearity, 28, 509-530.
 Beal, J.T., Kato, T. and Majda, A. (1984) Remarks on the Breakdown of Smooth Solutions for the 3D Euler Equations. Communications in Mathematical Physics, 94, 61-66.
 Liu, S.Q. and Zhang, J.W. (2016) Global Well-Posedeness for the Two-Dimensional Equations of Nonhomogeneous Incompressible Liquid Crystal Flows with Nonnegative Density. Discrete & Continuous Dynamical Systems-B, 21, 2631-2648.
 Liu, Y. and Wang, W. (2018) Strong Solutions to the Cauchy Problem of Two-Dimensional Incompressible Fluid Models of Korteweg Type. Journal of Mathematical Analysis and Applications, 465, 1075-1093.
 Huang, X.D., Li, J. and Xin, Z.P. (2012) Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations. Communications on Pure and Applied Mathematics, 65, 549-585.