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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 compact 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.

In this paper, we consider the system for the nonhomogeneous incompressible Boussinesq equations of Korteweg type as follows

{ ρ t + div ( ρ u ) = 0, ( ρ u ) t + div ( ρ u ⊗ u ) + ∇ p = μ Δ u + ν ρ ∇ Δ ρ + f , θ t + u ⋅ ∇ θ − κ Δ θ = 0, div u = 0, x = ( x 1 , x 2 ) ∈ Ω , t ≥ 0, (1.1)

Among them, the free vector field of divergence u = ( u 1 , u 2 ) ( x , t ) represents the velocity of the fluid; the scalar function p = p ( x , t ) and θ = θ ( x , t ) represent pressure of the fluid and temperature respectively; parameter μ > 0 represents the viscosity coefficient dependent on temperature; ρ = ρ ( x , t ) and ν represent density and the capillary coefficient respectively; and the constant κ > 0 represents the thermal diffusivity; f denote the external force.

The initial data is given by

ρ ( x , 0 ) = ρ 0 ( x ) , ρ u ( x , 0 ) = ρ 0 u 0 ( x ) , θ ( x , 0 ) = θ 0 ( x ) , x ∈ ℝ 2 . (1.2)

Equation (1.1) governs the motions of the incompressible nonisothermal viscous capillary fluids. Assumed capillarity coefficient ν = 0 , the system (1.1) can simplify to other incompressible equations. Recently, the incompressible equations with density ρ > 0 have led scholars to do much research, which gets important results in different academic fields. The study of the system (1.1) with μ > 0 and κ > 0 is more concerned. Cannon and DiBenedetto proved the initial value problem of the Boussinesq equation for incompressible fluid affected by convective heat transfer (see [

If the thermal diffusivity κ = 0 , 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, [

Our purpose is to study the Cauchy problem for the strong solutions of Equations ((1.1), (1.2)). For convenience, we can set f = 0 . Since

ρ ∇ Δ ρ = ∇ ( ρ Δ ρ ) − ∇ ρ Δ ρ , (1.3)

classifying the term − ν ∇ ( ρ Δ ρ ) as the pressure term, we can deform the equation as

{ ρ t + u ⋅ ∇ ρ = 0, ρ u t + ρ u ⋅ ∇ u + ∇ p = μ Δ u − ν ∇ ρ Δ ρ , θ t + u ⋅ ∇ θ − κ Δ θ = 0, div u = 0, ( x , t ) ∈ ℝ 2 × ℝ + . (1.4)

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 [

‖ p ‖ BMO ‖ ∂ j u i ∂ i u j ‖ H 1 (see (3.13)). Next, we use the Stokes system (3.18) to get the ‖ ∇ 2 u ‖ L r and ‖ ∇ p ‖ L r (see (3.19)), the key point is to use Gagliardo-Nirenberg

inequality to estimate the value of ‖ ∇ ρ Δ ρ ‖ L 2 . Multiplying (1.1)_{3} by Δ θ can control the strong coupled term u ⋅ ∇ θ after integration by parts (see (3.16)). And, considered [

Now we will explain the symbols and conventions applied in this article. For R > 0 , let

B R ≜ { x ∈ 2 | | x | < R } , ∫ ⋅ d x ≜ ∫ 2 ⋅ d x .

Meanwhile, for 1 ≤ r ≤ ∞ and k ≥ 1 , the standard Lebesgue and Sobolve spaces have the following forms:

L r = L r ( ℝ 2 ) , W k , r = W k , r ( ℝ 2 ) , H k = W k , 2 .

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 ( ρ , u , p , θ ) are regular distributions, Equation (1.1) also hold almost everywhere in ℝ 2 × ( 0, T ) , then ( ρ , u , p , θ ) is considered a strong solution to (1.1).

Moreover, it can be assumed that the initial density ρ 0 satisfies

∫ ℝ 2 ρ 0 d x = 1 , (1.5)

which implies that exists a positive constant N 0 such that

∫ B N 0 ρ 0 d x ≥ 1 2 ∫ ρ 0 d x = 1 2 . (1.6)

The main conclusions of this paper are given as follows:

Theorem 1.2. Besides (1.5) and (1.6), if the initial data ( ρ 0 , u 0 , θ 0 ) hold that for any constant a > 1 and q > 2 ,

{ ρ 0 ≥ 0, ρ 0 x ¯ a ∈ L 1 ∩ H 2 ∩ W 2, q , ∇ u 0 ∈ L 2 , ρ 0 u 0 ∈ L 2 , θ 0 ≥ 0, θ 0 x ¯ a 2 ∈ L 2 , ∇ θ 0 ∈ L 2 , div u 0 = 0, (1.7)

where

x ¯ ≜ ( e + | x | 2 ) 1 2 log 2 ( e + | x | 2 ) (1.8)

and satisfy the compatibility condition

− μ Δ u 0 + ∇ p 0 + ν ∇ ρ 0 Δ ρ 0 = ρ 0 g , (1.9)

for some p 0 ∈ H 1 ( B R ) and g ∈ L 2 ( B R ) , div u 0 = 0 . Then the problem (1.1)-(1.2) has a unique global strong solution ( ρ , u , p , θ ) satisfying that for any 0 < T < ∞ ,

{ 0 ≤ ρ ∈ C ( [ 0 , T ] ; L 1 ∩ H 2 ∩ W 2 , q ) , ρ x ¯ a ∈ L ∞ ( 0 , T ; L 2 ∩ H 1 ∩ W 2 , q ) , ρ u , ∇ u , x ¯ − 1 u , t ρ u t , t ∇ p , t ∇ 2 u ∈ L ∞ ( 0 , T ; L 2 ) , θ , θ x ¯ a / 2 , ∇ θ , t θ t , t ∇ 2 θ , t ∇ θ x ¯ a / 2 ∈ L ∞ ( 0 , T ; L 2 ) , ∇ u ∈ L 2 ( 0 , T ; H 1 ) ∩ L ( q + 1 ) / q ( 0 , T ; W 1 , q ) , ∇ p ∈ L 2 ( 0 , T ; L 2 ) ∩ L ( q + 1 ) / q ( 0 , T ; L q ) , ∇ θ ∈ L 2 ( 0 , T ; H 1 ) , θ t , ∇ θ x ¯ a / 2 ∈ L 2 ( 0 , T ; L 2 ) , t ∇ u ∈ L 2 ( 0 , T ; W 1 , q ) , ρ u t , t ∇ u t , t ∇ θ t , t x ¯ − 1 u t ∈ L 2 ( ℝ 2 × ( 0 , T ) ) , (1.10)

and

inf 0 ≤ t ≤ T ∫ B N 1 ρ ( x , t ) d x ≥ 1 4 , (1.11)

for positive constant N 1 depending only ‖ ρ 0 ‖ L 1 , ‖ ρ 0 1 2 u 0 ‖ L 2 , N 0 and T. In addition, the ( ρ , u , p , θ ) has the following decay rates, that is for t ≥ 1 ,

{ ‖ ∇ u ( ⋅ , t ) ‖ L 2 + ‖ ∇ θ ( ⋅ , t ) ‖ L 2 ≤ C t − 1 / 2 , ‖ ∇ 2 u ( ⋅ , t ) ‖ L 2 + ‖ ∇ p ( ⋅ , t ) ‖ L 2 ≤ C t − 1 , (1.12)

where C depends only on μ , κ , ‖ ρ 0 ‖ L 1 ∩ L ∞ , ‖ ρ 0 1 / 2 u 0 ‖ L 2 , ‖ ∇ u 0 ‖ L 2 , and ‖ θ 0 ‖ H 1 .

Remark 1.3. If there is no influence of fluid temperature, i.e., θ = 0 , then (1.1) reduces to the fluid of Korteweg type, Theorem 1.2 extends the results of Liu and Wang [

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 ( ρ 0 , u 0 , θ 0 ) satisfies (1.7). Then there exists a small time T 0 > 0 and a unique strong solution ( ρ , u , p , θ ) to the problem (1.1)-(1.2) in ℝ 2 × ( 0, T 0 ) that satisfies (1.10) and (1.11).

Lemma 2.2. (Gagliardo-Nirenberg inequality). For m ∈ [ 2, ∞ ) , q ∈ ( 1, ∞ ) , and r ∈ ( 2, ∞ ) , there exists some generic constant C > 0 which may depend on m, q, and r such that for f ∈ H 1 ( ℝ 2 ) and g ∈ L q ( ℝ 2 ) ∩ D 1, r ( ℝ 2 ) , we have

‖ f ‖ L m ( ℝ 2 ) m ≤ C ‖ f ‖ L 2 ( ℝ 2 ) 2 ‖ ∇ f ‖ L 2 ( ℝ 2 ) m − 2 , (2.1)

‖ g ‖ C ( ℝ 2 ¯ ) ≤ C ‖ g ‖ L q ( ℝ 2 ) q ( r − 2 ) / ( 2 r + q ( r − 2 ) ) ‖ ∇ g ‖ L r ( ℝ 2 ) 2 r / ( 2 r + q ( r − 2 ) ) . (2.2)

The next weighted L n bounds can be seen in ( [

Lemma 2.3. For h ∈ [ 2, ∞ ) and λ ∈ ( 1 + h / 2 , ∞ ) , there exists a positive constant C such that for all v ∈ D ˜ 1,2 ( ℝ 2 ) ,

( ∫ ℝ 2 | v | h e + | x | 2 ( log ( e + | x | 2 ) ) − λ d x ) 1 / h ≤ C ‖ v ‖ L 2 ( B 1 ) + C ‖ ∇ v ‖ L 2 ( ℝ 2 ) . (2.3)

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. Let x ¯ be as in (1.8). Assume that ρ ∈ L 1 ( ℝ 2 ) ∩ L ∞ ( ℝ 2 ) is a non-negative function such that

‖ ρ ‖ L 1 ( B N 0 ) ≥ M 0 , ‖ ρ ‖ L 1 ( ℝ 2 ) ∩ L ∞ ( ℝ 2 ) ≤ M 1 , (2.4)

for positive constants M 0 , M 1 , and N 0 ≥ 1 with B N 0 ⊂ ℝ 2 . Then for α > 0 , β > 0 , there is a positive constant C depending only on α , β , M 0 , M 1 , and N 0 such that every v ∈ D ˜ 1,2 ( ℝ 2 ) satisfies

‖ v x ¯ − β ‖ L ( 2 + α ) / β ˜ ( ℝ 2 ) ≤ C ‖ ρ 1 / 2 v ‖ L 2 ( ℝ 2 ) + C ‖ ∇ v ‖ L 2 ( ℝ 2 ) , (2.5)

with β ˜ = min { 1, β } .

Finally, set H 1 ( ℝ 2 ) and BMO ( ℝ 2 ) denote the standard Hardy and BMO spaces (see [

Lemma 2.5 (i) There is a positive constant C such that

‖ G ⋅ M ‖ H 1 ( ℝ 2 ) ≤ C ‖ G ‖ L 2 ( ℝ 2 ) ‖ M ‖ L 2 ( ℝ 2 ) , (2.6)

for all G ∈ L 2 ( ℝ 2 ) and M ∈ L 2 ( ℝ 2 ) satisfying

div G = 0, ∇ ⊥ ⋅ M = 0 in D ′ ( ℝ 2 ) . (2.7)

(ii) There is a positive constant C such that

‖ f ‖ BMO ( ℝ 2 ) ≤ C ‖ ∇ f ‖ L 2 ( ℝ 2 ) , (2.8)

for all f ∈ D ˜ 1,2 ( ℝ 2 ) .

Proof. (i) For the specific proof steps, please see ( [

(ii) It follows from the Poincaré inequality that for any ball B ⊂ ( ℝ 2 )

1 | B | ∫ B | v ( x ) − 1 | B | ∫ B v ( y ) d y | d x ≤ C ( ∫ B | ∇ v | 2 d x ) 1 / 2 , (2.9)

o

First, because of div u = 0 , we have the following estimate related to the density on the L ∞ ( 0, T ; L r ) -norm.

Lemma 3.1. There exists a positive constant C depending only on ‖ ρ 0 ‖ L 1 ∩ L ∞ such that

sup t ∈ [ 0, T ] ‖ ρ ‖ L 1 ∩ L ∞ ≤ C . (3.1)

We give the time-independent estimates of ∇ u and ∇ θ on the L ∞ ( 0, T ; L 2 ) -norm.

Lemma 3.2. There exists a positive constant C depending only on μ , κ , ‖ ρ 0 ‖ L ∞ , ‖ ∇ u 0 ‖ L 2 , ‖ ρ 0 u 0 ‖ L 2 , and ‖ θ 0 ‖ H 1 such that

sup t ∈ [ 0 , T ] ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 + ‖ θ ‖ L 2 2 ) + ∫ 0 T ( ‖ ρ u ˙ ‖ L 2 2 + ‖ Δ θ ‖ L 2 2 ) d t ≤ C ∫ 0 T ‖ ∇ u ˙ ‖ L 2 2 d t + C ∫ 0 T ψ d t , (3.2)

Here u ˙ ≜ ∂ t u + u ⋅ ∇ u , and ψ = ‖ ρ x ¯ a ‖ L 1 ∩ H 2 ∩ W 2, q 2 , furthermore, one has

sup t ∈ [ 0 , T ] t ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + ∫ 0 T t ( ‖ ρ 1 2 u ˙ ‖ L 2 2 + ‖ Δ θ ‖ L 2 2 ) d t ≤ C ∫ 0 T ‖ ∇ u ˙ ‖ L 2 2 d t + C ∫ 0 T ψ d t . (3.3)

Proof. Applying standard energy estimate, taking the x i -derivative (i = 1, 2) of (1.4) gives

∂ i ρ t + ∂ i u ⋅ ∇ ρ + u ⋅ ∂ i ( ∇ ρ ) = 0. (3.4)

Multiplying (1.1) by 2 ∇ ρ and integrating the resulting equality on ℝ 2 , we get

d d t ‖ ∇ ρ ‖ L 2 2 + C ‖ ∇ u ‖ L 2 2 + C ‖ ∇ ρ ‖ L 4 4 ≤ C . (3.5)

Adding (1.4) × 2u to (1.4) × u^{2} and integrating the resulting equality on ℝ 2 , we have

d d t ‖ ρ u ‖ L 2 2 + 2 μ ‖ ∇ u ‖ L 2 2 ≤ C ‖ ∇ u ‖ L 2 2 + C ‖ ∇ ρ ‖ L 4 4 . (3.6)

Multiplying (1.1) by θ and integrating the resulting equality on ℝ 2 , we have

d d t ‖ θ ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ≤ C . (3.7)

Combining (3.5) with (3.6), (3.7), and then integrating on [ 0, T ] gives

sup t ∈ [ 0, T ] ( ‖ ρ u ‖ L 2 2 + ‖ ∇ ρ ‖ L 2 2 + ‖ θ ‖ L 2 2 ) + ∫ 0 T ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) d t ≤ C . (3.8)

Next, multiplying (1.1) by u ˙ and integrating the resulting equality on ℝ 2 , we have

∫ ρ | u ˙ | 2 d x = ∫ μ Δ u ⋅ u ˙ d x − ∫ ∇ p ⋅ u ˙ d x − ν ∫ ∇ ρ Δ ρ ⋅ u ˙ d x ≜ F 1 + F 2 + F 3 . (3.9)

Then it follows from integration by parts and Gagliardo-Nirenberg inequality that

F 1 = μ ∫ Δ u ⋅ ( u t + u ⋅ ∇ u ) d x = − μ 2 d d t ‖ ∇ u ‖ L 2 2 − μ ∫ ∂ i u j ∂ i ( u k ∂ k u j ) d x ≤ − μ 2 d d t ‖ ∇ u ‖ L 2 2 + C ‖ ∇ u ‖ L 3 3 ≤ − μ 2 d d t ‖ ∇ u ‖ L 2 2 + C ‖ ∇ u ‖ L 2 2 ‖ ∇ 2 u ‖ L 2 . (3.10)

Integrating by parts together with (1.1) gives

F 2 = − ∫ ∇ p ( u t + u ⋅ ∇ u ) d x = ∫ p ∂ j u i ∂ i u j d x ≤ C ‖ p ‖ BMO ‖ ∂ j u i ∂ i u j ‖ H 1 , (3.11)

where one has used the duality of H 1 space and BMO one (see [

‖ ∂ j u i ∂ i u j ‖ H 1 ≤ C ‖ ∇ u ‖ L 2 ‖ ∇ u ‖ L 2 . (3.12)

Equation (3.11) combined with Equation (3.12) and Equation (2.8) gives

| F 2 | = | ∫ p ∂ j u i ∂ i u j d x | ≤ C ‖ p ‖ BMO ‖ ∇ u ‖ L 2 2 ≤ C ‖ ∇ p ‖ L 2 ‖ ∇ u ‖ L 2 2 . (3.13)

Integration by parts together with (1.4), (3.4), (3.8), and Gagliardo-Nirenberg inequality gives

F 3 = ν 2 ∫ ( ∇ ρ ) 2 ∇ u ˙ d x ≤ C ‖ ∇ ρ ‖ L 4 4 + C ‖ ∇ u ˙ ‖ L 2 2 ≤ C ‖ Δ ( x ¯ − a ⋅ x ¯ a ρ ) ‖ L 2 2 + C ‖ ∇ u ˙ ‖ L 2 2 ≤ C ‖ x ¯ a ρ ‖ H 2 2 + C ‖ ∇ u ˙ ‖ L 2 2 ≤ C ψ + C ‖ ∇ u ˙ ‖ L 2 2 . (3.14)

Next, substituting (3.10), (3.13) and (3.14) into (3.9) gives

μ 2 d d t ‖ ∇ u ‖ L 2 2 + ‖ ρ u ˙ ‖ L 2 2 ≤ C ( ‖ ∇ 2 u ‖ L 2 + ‖ ∇ p ‖ L 2 ) ‖ ∇ u ‖ L 2 2 + C ‖ ∇ u ˙ ‖ L 2 2 + C ψ . (3.15)

Then, multiplying (1.1) by Δ θ and integrating the resulting equality by parts over ℝ 2 , it follows from Hölder’s and Gagliardo-Nirenberg inequalities that

d d t ∫ | ∇ θ | 2 d x + 2 κ ∫ | Δ θ | 2 d x ≤ C ∫ | ∇ u | | ∇ θ | 2 d x ≤ C ‖ ∇ u ‖ L 3 ‖ ∇ θ ‖ L 2 4 3 ‖ ∇ θ ‖ L 2 2 3 ≤ C ‖ ∇ u ‖ L 2 2 ‖ ∇ 2 u ‖ L 2 + C ‖ ∇ θ ‖ L 2 4 + C ‖ Δ θ ‖ L 2 2 , (3.16)

which combined with (3.15) and (3.8) gives

d d t ( μ 2 ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + ‖ ρ u ˙ ‖ L 2 2 + 2 κ ‖ Δ θ ‖ L 2 2 ≤ C ‖ ∇ θ ‖ L 2 4 + C ( ‖ ∇ 2 u ‖ L 2 + ‖ ∇ p ‖ L 2 ) ‖ ∇ u ‖ L 2 2 + C ‖ ∇ u ˙ ‖ L 2 2 + C ψ . (3.17)

Since ( ρ , u , p , θ ) satisfies the following Stokes system

{ − μ Δ u + ∇ p = − ρ u ˙ − ν ∇ ρ Δ ρ , x ∈ ℝ 2 , div u = 0 , x ∈ ℝ 2 , u ( x ) → 0 , | x | → ∞ , (3.18)

Applying the standard L r -estimate to (3.18) (see [

‖ ∇ 2 u ‖ L r + ‖ ∇ p ‖ L r ≤ C ‖ ρ u ˙ ‖ L r + C ‖ ∇ ρ Δ ρ ‖ L r ≤ C ‖ ρ u ˙ ‖ L r + C ‖ ∇ ρ Δ ρ ‖ L r . (3.19)

Moreover, since x ¯ − a , ∇ x ¯ − a , ∇ 2 x ¯ − a are all bounded in ℝ 2 , an application of the Gagliardo-Nirenberg inequality results in

‖ ∇ ρ Δ ρ ‖ L 2 ≤ ‖ ∇ ρ ‖ L ∞ ‖ Δ ρ ‖ L 2 ≤ C ‖ ∇ ρ ‖ W 1 , q q 2 ( q − 1 ) ‖ ∇ ρ ‖ L 2 q − 2 2 ( q − 1 ) ‖ Δ ρ ‖ L 2 = C ‖ ∇ ( x ¯ − a ⋅ x ¯ a ρ ) ‖ W 1 , q q 2 ( q − 1 ) ‖ ∇ ( x ¯ − a ⋅ x ¯ a ρ ) ‖ L 2 q − 2 2 ( q − 1 ) ‖ Δ ( x ¯ − a ⋅ x ¯ a ρ ) ‖ L 2 ≤ C ‖ ρ x ¯ a ‖ W 2 , q q 2 ( q − 1 ) ‖ ρ x ¯ a ‖ H 1 q − 2 2 ( q − 1 ) ‖ ρ x ¯ a ‖ H 2 ≤ C ψ , (3.20)

substituting (3.20) into (3.19), we get

‖ ∇ 2 u ‖ L 2 + ‖ ∇ p ‖ L 2 ≤ ε ‖ ρ u ˙ ‖ L 2 + C ψ . (3.21)

This combined with (3.17) and (3.19) gives

d d t ( μ 2 ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + ‖ ρ u ˙ ‖ L 2 2 + 2 κ ‖ Δ θ ‖ L 2 2 ≤ C ‖ ∇ θ ‖ L 2 4 + C ‖ ∇ u ‖ L 2 4 + ε ‖ ρ u ˙ ‖ L 2 2 + C ‖ ∇ u ˙ ‖ L 2 2 + C ψ , (3.22)

where ε is to be determined. Choosing ε = 1 2 , it follows from (3.22) that

d d t ( μ 2 ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + 1 2 ‖ ρ u ˙ ‖ L 2 2 + 2 κ ‖ Δ θ ‖ L 2 2 ≤ C ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) ( ‖ ∇ θ ‖ L 2 2 + ‖ ∇ u ‖ L 2 2 + C ) + C ‖ ∇ u ˙ ‖ L 2 2 + C ψ , (3.23)

which together with (3.8), (3.22) and Gronwall’s inequality gives (3.2). Then, multiplying (3.22) by t, we have

d d t t ( μ 2 ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) − μ 2 ‖ ∇ u ‖ L 2 2 − ‖ ∇ θ ‖ L 2 2 + 1 2 t ‖ ρ u ˙ ‖ L 2 2 + 2 κ t ‖ Δ θ ‖ L 2 2 ≤ t ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) ( ‖ ∇ θ ‖ L 2 2 + ‖ ∇ u ‖ L 2 2 + C ) + C ‖ ∇ u ˙ ‖ L 2 2 + C ψ , (3.24)

o

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 μ , ‖ ∇ u 0 ‖ L 2 , ‖ ρ 0 ‖ L 1 ∩ L ∞ , ‖ ρ 0 1 2 u 0 ‖ L 2 , ‖ ρ 0 x ¯ a ‖ L 1 , N 0 , and T, such that

sup t ∈ [ 0, T ] ‖ ρ x ¯ a ‖ L 1 ≤ C ( T ) . (3.25)

Proof. First, for N > 1 , let φ N ∈ C 0 ∞ ( B N ) satisfy

0 ≤ φ N ≤ 1, φ N ( x ) = { 1, | x | ≤ N / 2 , 0, | x | ≥ N , | ∇ φ N | ≤ C N − 1 . (3.26)

It follows from (1.1) that

d d t ∫ ρ φ N d x = ∫ ρ u ⋅ ∇ φ N d x ≥ − C N − 1 ( ∫ ρ d x ) 1 / 2 ( ∫ ρ | u | 2 d x ) 1 / 2 ≥ − C ˜ N − 1 , (3.27)

where in the last inequality one has used (3.1) and (3.8). Integrating (3.27) and choosing N = N 1 ≜ 2 N 0 + 4 C ˜ T , we obtain after using (1.6) that

inf t ∈ [ 0, T ] ∫ B N 1 ρ d x ≥ inf t ∈ [ 0, T ] ∫ ρ φ N 1 d x ≥ ∫ ρ 0 φ N 1 d x − C ˜ N 1 − 1 T ≥ ∫ B N 0 ρ 0 d x − C ˜ T 2 N 0 + 4 C ˜ T ≥ 1 4 . (3.28)

Hence, it follows from (3.28), (3.1), (2.2), (3.8) and (3.2) that for any η ∈ ( 0,1 ] and any s > 2 ,

‖ u x ¯ − η ‖ L s / η ≤ C ( ‖ ρ 1 / 2 u ‖ L 2 + ‖ ∇ u ‖ L 2 ) . (3.29)

Multiplying (1.1) by x ¯ a and integrating the resulting equality by parts over ℝ 2 yield that

d d t ∫ ρ x ¯ a d x ≤ C ∫ ρ | u | x ¯ a − 1 log 2 ( e + | x | 2 ) d x ≤ C ‖ ρ x ¯ a − 1 + 8 8 + a ‖ L 8 + a 7 + a ‖ u x ¯ − 4 8 + a ‖ L 8 + a ≤ C ∫ ρ x ¯ a d x + C , (3.30)

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 μ , κ , ‖ ρ 0 ‖ L 1 ∩ L ∞ , ‖ ∇ u 0 ‖ L 2 and ‖ ρ 0 u 0 ‖ L 2 such that for i = 1,2 ,

sup t ∈ [ 0, T ] t i ‖ ρ u ˙ ‖ L 2 2 + ∫ 0 T t i ‖ ∇ u ˙ ‖ L 2 2 d t ≤ C ∫ 0 T ψ d t , (3.31)

and

sup t ∈ [ 0, T ] t i ( ‖ ∇ 2 u ‖ L 2 2 + ‖ ∇ p ‖ L 2 2 ) ≤ C ∫ 0 T ψ d t . (3.32)

Proof. Operating ∂ t + u ⋅ ∇ to (1.1)^{j}, one gets by some simple calculations that

∂ t ( ρ u ˙ j ) + div ( ρ u u ˙ j ) − μ Δ u ˙ j = − μ ∂ i ( ∂ i u ⋅ ∇ u j ) − μ div ( ∂ i u ∂ i u j ) − ∂ j ∂ t p − ( u ⋅ ∇ ) ∂ j p − ν ∂ t ( ∇ ρ Δ ρ ) − ν ( u ⋅ ∇ ) ( ∇ ρ Δ ρ ) . (3.33)

Next, multiplying (3.33) by u ˙ j , together with integration by parts and (1.1), we get

1 2 d d t ∫ ρ | u ˙ | 2 d x + μ ∫ | ∇ u ˙ | 2 d x = − ∫ μ ∂ i ( ∂ i u ⋅ ∇ u j ) u j d x − ∫ μ div ( ∂ i u ∂ i u ˙ j ) u ˙ j d x − ∫ ( u ˙ j ∂ t ∂ j p + u ˙ j ( u ⋅ ∇ ) ∂ j p ) d x − ν ∫ u ˙ j ( ∂ t ( ∂ i ρ Δ ρ ) + ( u ⋅ ∇ ) ( ∂ i ρ Δ ρ ) ) d x ≜ ∑ i = 1 4 H i (3.34)

Following the same argument as ( [

∑ i = 1 3 H i ≤ d d t ∫ p ∂ j u i ∂ i u j d x + C ( ‖ p ‖ L 4 4 + ‖ ∇ u ‖ L 4 4 ) + ε 2 ‖ ∇ u ˙ ‖ L 2 2 . (3.35)

which combined (1.3), (3.4) and the Gagliardo-Nirenberg inequality leads to

H 4 = − ν ∫ ∇ ρ t Δ ρ ⋅ u ˙ j d x − ν ∫ ∇ ρ Δ ρ t ⋅ u ˙ j d x − ν ∫ ( u ⋅ ∇ ) ( ∇ ρ Δ ρ ) ⋅ u ˙ j d x = ν ∫ ∇ ρ ∇ ρ t ∇ u ˙ d x + ν ∫ ( u ⋅ ∇ ) ρ Δ ρ ∇ u ˙ d x ≤ ν ∫ | ∇ ρ | 2 | ∇ u | | ∇ u ˙ | d x ≤ C ‖ Δ ( x ¯ − a ⋅ x ¯ a ρ ) ‖ L 2 6 + C ‖ ∇ u ‖ L 4 4 + C ‖ ∇ u ˙ ‖ L 2 2 ≤ C ‖ ∇ u ‖ L 4 4 + ε 2 ‖ ∇ u ˙ ‖ L 2 2 + C ψ . (3.36)

Substituting (3.35) and (3.36) into (3.34) and together with (3.2) gives

d d t A ( t ) + μ 2 ‖ ∇ u ˙ ‖ L 2 2 ≤ C ( ‖ p ‖ L 4 4 + ‖ ∇ u ‖ L 4 4 ) + C ψ , (3.37)

where

A ( t ) ≜ 1 2 ∫ ρ | u ˙ | 2 d x − ∫ p ∂ j u i ∂ i u j d x ,

satisfies

1 4 ‖ ρ 1 2 u ˙ ‖ L 2 2 − ε ‖ ∇ u ‖ L 2 4 + C ψ ≤ A ( t ) ≤ 1 4 ‖ ρ 1 2 u ˙ ‖ L 2 2 + ε ‖ ∇ u ‖ L 2 4 + C ψ . (3.38)

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

‖ p ‖ L 4 4 + ‖ ∇ u ‖ L 4 4 ≤ C ( ‖ ∇ p ‖ L 4 3 4 + ‖ ∇ 2 u ‖ L 4 3 4 ) ≤ C ‖ ρ u ˙ ‖ L 4 3 4 + C ψ 2 ≤ C ‖ ρ ‖ L 2 2 ‖ ρ u ˙ ‖ L 2 4 + C ψ 2 ≤ C ‖ ρ u ˙ ‖ L 2 4 + C ψ 2 . (3.39)

Substituting (3.38) and (3.39) into (3.37)

d d t A ( t ) + μ 2 ‖ ∇ u ˙ ‖ L 2 2 ≤ C ‖ ρ u ˙ ‖ L 2 4 + C ψ 2 ≤ C ( ‖ ρ u ˙ ‖ L 2 2 + C ψ ) ( A ( t ) + ‖ ∇ u ‖ L 2 4 ) ≤ C ( ‖ ρ u ˙ ‖ L 2 2 + C ψ ) A ( t ) + C ( ‖ ρ u ˙ ‖ L 2 2 + ‖ ∇ u ‖ L 2 4 + C ψ ) . (3.40)

Next, we estimate ‖ ρ u ˙ ‖ L 2 2 . First, the (3.25) combined (3.8) with (3.29) that for any η > 0 and σ > 0

‖ ρ η u ‖ L σ ≤ ‖ ρ η x ¯ 3 a 4 σ ‖ L 4 σ 3 ‖ u x ¯ − 3 a 4 σ ‖ L 4 σ ≤ C ‖ ρ ‖ L ∞ η − 3 4 σ ‖ ρ x ¯ a ‖ L 1 3 4 σ ‖ u x ¯ − 3 a 4 σ ‖ L 4 σ ≤ C ( ‖ ρ u ‖ L 2 + ‖ ∇ u ‖ L 2 ) . (3.41)

The (1.4) combined with (3.8), (3.41) and the Gagliardo-Nirenberg inequality, we derive

∫ ρ | u ˙ | 2 d x ≤ C ∫ ρ | u t | 2 d x + C ∫ ρ [ ( u ⋅ ∇ u ) ] 2 d x ≤ C ∫ ρ | u | 2 | ∇ u | 2 d x + C ∫ ρ − 1 | μ Δ u − ∇ p − κ ∇ ρ Δ ρ | 2 d x + ‖ ρ u ‖ L 4 2 ‖ ∇ u ‖ L 4 2 ≤ C ∫ ρ | u | 2 | ∇ u | 2 d x + C ∫ ρ − 1 | μ Δ u − ∇ p − κ ∇ ρ Δ ρ | 2 d x + C ( ‖ ρ u ‖ L 2 2 + ‖ ∇ u ‖ L 2 2 ) ‖ ∇ u ‖ L 2 ( ‖ ∇ u ‖ L 2 + ‖ ∇ 2 u ‖ L 2 ) ≤ C ∫ ρ | u | 2 | ∇ u | 2 d x + C ∫ ρ − 1 | μ Δ u − ∇ p − κ ∇ ρ Δ ρ | 2 d x + ε ‖ ∇ u ‖ L 2 2 + C ψ , (3.42)

which together with (3.41) and the compatibility condition (1.9) yields

∫ ρ | u ˙ | 2 ( x , 0 ) d x ≤ sup t → 0 + ∫ C ( ρ − 1 | μ Δ u − ∇ p − κ ∇ ρ Δ ρ | 2 + ρ | u | 2 | ∇ u | 2 ) d x + ε ‖ ∇ u ‖ L 2 2 + C ψ ≤ C ( T ) ‖ g ‖ L 2 2 + C ‖ ρ 0 u 0 ‖ L 4 2 ‖ ∇ u 0 ‖ L 4 2 + ε ‖ ∇ u 0 ‖ L 2 2 + C ψ ≤ C ( T ) ‖ g ‖ L 2 2 + C ( ‖ ρ 0 u 0 ‖ L 2 2 + ‖ ∇ u 0 ‖ L 2 2 ) ‖ ∇ u 0 ‖ L 2 ‖ ∇ u 0 ‖ H 1 ≤ C ( T ) . (3.43)

o

Finally, Multiplying (3.40) by t i ( i = 1 , 2 ) 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

sup t ∈ [ 0 , T ] ‖ ρ ‖ H 1 ∩ W 1 , q + ∫ 0 T ( ‖ ∇ 2 u ‖ L 2 2 + ‖ ∇ 2 u ‖ L q q + 1 q + t ‖ ∇ 2 u ‖ L 2 ∩ L q 2 ) d t + ∫ 0 T ( ‖ ∇ p ‖ L 2 2 + ‖ ∇ p ‖ L q q + 1 q + t ‖ ∇ p ‖ L 2 ∩ L q 2 ) d t ≤ C ∫ 0 T ψ 2 d t . (3.44)

Proof. First, it follows from the mass Equation (1.1) that ∇ ρ satisfies for any r ≥ 2 ,

d d t ‖ ∇ ρ ‖ L r ≤ C ( r ) ‖ ∇ u ‖ L ∞ ‖ ∇ ρ ‖ L r . (3.45)

Next, employing Gagliardo-Nirenberg inequality, using (3.2) and (3.19), we have q > 2 ,

‖ ∇ u ‖ L ∞ ≤ C ‖ ∇ u ‖ L 2 q − 2 2 ( q − 1 ) ‖ ∇ 2 u ‖ L q q 2 ( q − 1 ) ≤ C ( ‖ ρ u ˙ ‖ L q q 2 ( q − 1 ) + ψ ) . (3.46)

It follows from (3.28), (3.1), (2.2) and (3.25) that for any s > 2 ,

‖ ρ v ‖ L s ≤ C ‖ ρ x ¯ 3 a 4 s ‖ L 4 s 3 ‖ v x ¯ − 3 a 4 s ‖ L 4 s ≤ C ‖ ρ ‖ L ∞ 4 s − 3 4 s ‖ ρ x ¯ a ‖ L 1 3 4 s ( ‖ ρ 1 / 2 v ‖ L 2 + ‖ ∇ v ‖ L 2 ) ≤ C ( ‖ ρ 1 / 2 v ‖ L 2 + ‖ ∇ v ‖ L 2 ) , (3.47)

which together with the Gagliardo-Nirenberg inequality shows that

‖ ρ u ˙ ‖ L q ≤ C ‖ ρ u ˙ ‖ L 2 2 ( q − 1 ) q 2 − 2 ‖ ρ u ˙ ‖ L q 2 q ( q − 2 ) q 2 − 2 ≤ C ‖ ρ u ˙ ‖ L 2 2 ( q − 1 ) q 2 − 2 ( ‖ ρ u ˙ ‖ L 2 + ‖ ∇ u ˙ ‖ L 2 ) q ( q − 2 ) q 2 − 2 ≤ C ( ‖ ρ u ˙ ‖ L 2 + ‖ ρ u ˙ ‖ L 2 2 ( q − 1 ) q 2 − 2 ‖ ∇ u ˙ ‖ L 2 q ( q − 2 ) q 2 − 2 ) , (3.48)

which is deformed and calculated appropriately leads to

∫ 0 T ‖ ρ u ˙ ‖ L q q + 1 q d t ≤ C ∫ 0 T ‖ ρ u ˙ ‖ L 2 q + 1 q d t + sup t ∈ [ 0, T ] ( t ‖ ρ u ˙ ‖ L 2 2 ) q 2 − 1 2 q ( q 2 − 2 ) ⋅ ∫ 0 T t − q 3 + q 2 − 2 q − 2 2 q ( q 2 − 2 ) ( t ‖ ∇ u ˙ ‖ L 2 2 ) q ( q − 2 ) ( q + 1 ) 2 q ( q 2 − 2 ) d t ≤ C ∫ 0 T ‖ ρ u ˙ ‖ L 2 2 d t + C ∫ 0 T t − q 3 + q 2 − 2 q − 2 q 3 + q 2 − 2 q d t + C ∫ 0 T t ‖ ∇ u ˙ ‖ L 2 2 d t ≤ ∫ 0 T ψ 2 d t , (3.49)

∫ 0 T t ‖ ρ u ˙ ‖ L q 2 d t ≤ ∫ 0 T ( t ‖ ∇ u ˙ ‖ L 2 2 + ψ ) d t ≤ C ∫ 0 T ψ 2 d t . (3.50)

Then, the (3.49) along with (3.46) in particular implies

∫ 0 T ‖ ∇ u ‖ L ∞ d t ≤ C ∫ 0 T ψ 2 d t . (3.51)

Next, applying Gronwall’s inequality to (3.45) gives

sup t ∈ [ 0, T ] ‖ ∇ ρ ‖ L 2 ∩ L q ≤ C ∫ 0 T ψ 2 d t . (3.52)

Setting r = 2 in (3.19) and integrating the resulting equality over [ 0, T ] , we obtain after using (3.1), (3.2) and (3.3) that

∫ 0 T ‖ ∇ 2 u ‖ L 2 2 d t + ∫ 0 T ‖ ∇ p ‖ L 2 2 d t ≤ C ∫ 0 T ψ 2 d t . (3.53)

Similarly, setting r = q in (3.19) and integrating the resulting equality over [ 0, T ] , we deduce from using (3.49), (3.1), (3.2) and (3.3) that

∫ 0 T ‖ ∇ 2 u ‖ L q q + 1 q d t + ∫ 0 T ‖ ∇ p ‖ L q q + 1 q d t ≤ C ∫ 0 T ψ 2 d t . (3.54)

Multiplying (3.19) by t and integrating the resulting equality over [ 0, T ] , it can obtain after using (3.50), (3.1), (3.2) and (3.3) that

∫ 0 T t ‖ ∇ 2 u ‖ L 2 ∩ L q 2 d t + ∫ 0 T t ‖ ∇ p ‖ L 2 ∩ L q 2 d t ≤ C ∫ 0 T ψ 2 d t . (3.55)

Furthermore, it is easy to deduce from (3.53), (3.54) and (3.55) that

∫ 0 T ( ‖ ∇ 2 u ‖ L 2 2 + ‖ ∇ 2 u ‖ L q q + 1 q + t ‖ ∇ 2 u ‖ L 2 ∩ L q 2 ) d t + ∫ 0 T ( ‖ ∇ p ‖ L 2 2 + ‖ ∇ p ‖ L q q + 1 q + t ‖ ∇ p ‖ L 2 ∩ L q 2 ) d t ≤ C ∫ 0 T ψ 2 d t , (3.56)

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 q > 2 ,

sup t ∈ [ 0, T ] ‖ ρ x ¯ a ‖ L 1 ∩ H 2 ∩ W 2, q ≤ C ( T ) . (3.57)

Proof. First, setting ρ = ρ x ¯ a in (1.1) that satisfies

∂ t ( ρ x ¯ a ) + u ⋅ ∇ ( ρ x ¯ a ) − a ρ x ¯ a u ⋅ ∇ log x ¯ = 0. (3.58)

Taking the x i -derivative on both side of the (3.58) gives

0 = ∂ t ∂ i ( ρ x ¯ a ) + u ⋅ ∇ ∂ i ( ρ x ¯ a ) + ∂ i u ⋅ ∇ ( ρ x ¯ a ) − a ∂ i ( ρ x ¯ a ) u ⋅ ∇ log x ¯ − a ρ x ¯ a ∂ i u ⋅ ∇ log x ¯ − a ρ x ¯ a u ⋅ ∂ i ∇ log x ¯ . (3.59)

For any r ∈ [ 2, q ] , multiplying (3.59) by | ∇ ( ρ x ¯ a ) | r − 2 ∂ i ( ρ x ¯ a ) and integrating the resulting equality by parts over ℝ 2 , we obtain that

d d t ‖ ∇ ( ρ x ¯ a ) ‖ L r ≤ C ( 1 + ‖ ∇ u ‖ L ∞ + ‖ u ⋅ ∇ log x ¯ ‖ L ∞ ) ‖ ∇ ( ρ x ¯ a ) ‖ L r + C ‖ ρ x ¯ a ‖ L ∞ ( ‖ | ∇ u | | ∇ log x ¯ | ‖ L r + ‖ | u | | ∇ 2 log x ¯ | ‖ L r ) ≤ C ( 1 + ‖ ∇ u ‖ W 1 , q ) ‖ ∇ ( ρ x ¯ a ) ‖ L r + C ‖ ρ x ¯ a ‖ L ∞ ( ‖ ∇ u ‖ L r + ‖ u x ¯ − 3 5 ‖ L 4 r ‖ x ¯ − 4 3 ‖ L 4 r 3 ) ≤ C ( 1 + ‖ ∇ u ‖ W 1 , q ) ( 1 + ‖ ∇ ( ρ x ¯ a ) ‖ L r + ‖ ∇ ( ρ x ¯ a ) ‖ L q ) , (3.60)

where in the last and the second inequalities, we has applied (3.29) and (3.25), respectively. Choosing r = q (3.60), and applying Gronwall’s inequality together with (3.44) indicates that

sup t ∈ [ 0, T ] ‖ ∇ ( ρ x ¯ a ) ‖ L q ≤ C . (3.61)

Setting r = 2 in (3.60), we deduce from (3.44) and (3.61) that

sup t ∈ [ 0, T ] ‖ ∇ ( ρ x ¯ a ) ‖ L 2 ≤ C . (3.62)

Next, taking the x i -derivative again on both side of the (3.59) gives

0 = ∂ i ∂ i ( ρ x ¯ a ) t + 2 ∂ i u ⋅ ∂ i ∇ ( ρ x ¯ a ) + u ⋅ ∂ i ∂ i ∇ ( ρ x ¯ a ) + ∂ i ∂ i u ⋅ ∇ ( ρ x ¯ a ) − a ∂ i ∂ i ( ρ x ¯ a ) u ⋅ ∇ ln x ¯ − 2 a ∂ i ( ρ x ¯ a ) ∂ i u ⋅ ∇ ln x ¯ − 2 a ∂ i ( ρ x ¯ a ) u ⋅ ∂ i ∇ ln x ¯ − a ρ x ¯ a ∂ i ∂ i u ⋅ ∇ ln x ¯ − 2 a ρ x ¯ a ∂ i u ⋅ ∂ i ∇ ln x ¯ − a ρ x ¯ a u ⋅ ∂ i ∂ i ∇ ln x ¯ . (3.63)

Similarly, for any p ∈ [ 2, q ] , multiplying (3.63) by | ∇ 2 ( ρ x ¯ a ) | p − 2 ∇ 2 ( ρ x ¯ a ) and integrating the resulting equality by parts over ℝ 2 , we can find that

d d t ‖ ∇ 2 ( ρ x ¯ a ) ‖ L p ≤ C ( 1 + ‖ ∇ u ‖ L ∞ + ‖ u ⋅ ∇ ln x ¯ ‖ L ∞ ) ‖ ∇ 2 ( ρ x ¯ a ) ‖ L p + C ‖ ρ x ¯ a ‖ L ∞ ( ‖ ∇ 2 u ‖ L p + ‖ | ∇ u | | ∇ ln x ¯ | ‖ L p + ‖ | u | | ∇ 2 ln x ¯ | ‖ L p ) + C ‖ ρ x ¯ a ‖ L ∞ ( ‖ | ∇ u | | ∇ ln x ¯ | ‖ L p + ‖ | ∇ 2 u | | ∇ ln x ¯ | ‖ L p + ‖ u ⋅ ∇ 3 ln x ¯ ‖ L p )

≤ C ( 1 + ‖ ∇ u ‖ L 2 + ‖ ∇ 2 u ‖ L q + ‖ u x ¯ − 1 4 ‖ L ∞ ) ‖ ∇ 2 ( ρ x ¯ a ) ‖ L p + C ( ‖ ∇ u ‖ L p + ‖ ∇ 2 u ‖ L p + ‖ u x ¯ − 3 5 ‖ L ∞ ‖ x ¯ − 4 3 ‖ L p ) ‖ ∇ 2 ( ρ x ¯ a ) ‖ L 2 ∩ L p + C ( ‖ ∇ u ‖ L p + ‖ ∇ 2 u ‖ L p + ‖ u x ¯ − 1 4 ‖ L ∞ ‖ x ¯ − 3 2 ‖ L p ) ‖ ∇ ( ρ x ¯ a ) ‖ L 2 ∩ L p ≤ C ( 1 + ‖ ∇ 2 u ‖ L 2 ∩ L p ) ( ‖ ∇ 2 ( ρ x ¯ a ) ‖ L p + ‖ ∇ 2 ( ρ x ¯ a ) ‖ L 2 ∩ L p + ‖ ∇ ( ρ x ¯ a ) ‖ L 2 ∩ L p ) ≤ C ( 1 + ψ ) ( ‖ ∇ 2 ( ρ x ¯ a ) ‖ L p + ‖ ∇ 2 ( ρ x ¯ a ) ‖ L 2 ∩ L p + ‖ ∇ ( ρ x ¯ a ) ‖ L 2 ∩ L p ) . (3.64)

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

sup t ∈ [ 0, T ] ‖ θ x ¯ a 2 ‖ L 2 2 + ∫ 0 T ‖ ∇ θ x ¯ a 2 ‖ L 2 2 d t ≤ C ( T ) , (3.65)

sup t ∈ [ 0, T ] ( t ‖ ∇ θ x ¯ a 2 ‖ L 2 2 ) + ∫ 0 T t ‖ Δ θ x ¯ a 2 ‖ L 2 2 d t ≤ C ( T ) . (3.66)

Proof. First, multiplying (1.1) by θ x ¯ a and integrating the resulting equality by parts over ℝ 2 , we have

1 2 d d t ‖ θ x ¯ a 2 ‖ L 2 2 + κ ‖ ∇ θ x ¯ a 2 ‖ L 2 2 = κ 2 ∫ | θ | 2 Δ x ¯ a d x + 1 2 ∫ | θ | 2 u ⋅ ∇ x ¯ a d x ≜ M ^ 1 + M ^ 2 , (3.67)

where

M ^ 1 = κ 2 ∫ | θ | 2 Δ x ¯ a d x ≤ C ∫ | θ | 2 x ¯ a x ¯ − 2 log 4 ( e + | x | 2 ) d x ≤ C ‖ θ x ¯ a 2 ‖ L 2 2 , (3.68)

M ^ 2 = 1 2 ∫ | θ | 2 u ⋅ ∇ x ¯ a d x ≤ C ‖ θ x ¯ a 2 ‖ L 4 ‖ θ x ¯ a 2 ‖ L 2 ‖ u x ¯ − 3 4 ‖ L 4 ≤ C ‖ θ x ¯ a 2 ‖ L 2 2 + κ 2 ‖ ∇ θ x ¯ a 2 ‖ L 2 2 . (3.69)

Substituting (3.68), (3.69) into (3.67), we get

1 2 d d t ‖ θ x ¯ a 2 ‖ L 2 2 + κ 2 ‖ ∇ θ x ¯ a 2 ‖ L 2 2 ≤ C ‖ θ x ¯ a 2 ‖ L 2 2 . (3.70)

Using (3.70) for Gronwall’s inequality, we obtain (3.65).

Next, we estimate the (3.66). Multiplying (1.1)_{3} by Δ θ x ¯ a and integrating the resulting equality by parts on ℝ 2 , we find

1 2 d d t ‖ ∇ θ x ¯ a 2 ‖ L 2 2 + κ ‖ Δ θ x ¯ a 2 ‖ L 2 2 ≤ C ∫ | ∇ u | | ∇ θ | 2 x ¯ a d x + C ∫ | u | | ∇ θ | 2 | ∇ x ¯ a | d x + C ∫ | ∇ θ | | Δ θ | | ∇ x ¯ a | d x ≜ ∑ i = 1 3 M ˜ i , (3.71)

where

M ˜ 1 = C ∫ | ∇ u | | ∇ θ | 2 x ¯ a d x ≤ C ‖ ∇ u ‖ L ∞ ‖ ∇ θ x ¯ a 2 ‖ L 2 2 ≤ C ‖ ∇ u ‖ L 2 q − 2 2 ( q − 1 ) ‖ ∇ 2 u ‖ L q q − 2 2 ( q − 1 ) ‖ ∇ θ x ¯ a 2 ‖ L 2 2 ≤ C ( 1 + ‖ ∇ 2 u ‖ L q q + 1 q ) ‖ ∇ θ x ¯ a 2 ‖ L 2 2 , (3.72)

M ˜ 2 = C ∫ | u | | ∇ θ | 2 | ∇ x ¯ a | d x ≤ C ‖ | ∇ θ | 2 − 2 3 a x ¯ a − 1 3 ‖ L 6 a 6 a − 2 ‖ u x ¯ − 1 3 ‖ L 6 a ‖ | ∇ θ | 2 3 a ‖ L 6 a ≤ C ‖ ∇ θ x ¯ a 2 ‖ L 2 6 a − 2 3 a ‖ ∇ θ ‖ L 4 2 3 a ≤ C ‖ ∇ θ x ¯ a 2 ‖ L 2 2 + C ‖ ∇ θ ‖ L 4 2 ≤ C ‖ ∇ θ x ¯ a 2 ‖ L 2 2 + κ 4 ‖ Δ θ x ¯ a 2 ‖ L 2 2 , (3.73)

M ˜ 3 = C ∫ | ∇ θ | | Δ θ | | ∇ x ¯ a | d x ≤ κ 4 ‖ Δ θ x ¯ a 2 ‖ L 2 2 + C ‖ ∇ θ x ¯ a 2 ‖ L 2 2 . (3.74)

Submitting M ˜ 1 , M ˜ 2 , M ˜ 3 into (3.71), one has

1 2 d d t ‖ ∇ θ x ¯ a 2 ‖ L 2 2 + κ 2 ‖ Δ θ x ¯ a 2 ‖ L 2 2 ≤ C ( 1 + ‖ ∇ 2 u ‖ L q q + 1 q ) ‖ ∇ θ x ¯ a 2 ‖ L 2 2 . (3.75)

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

sup t ∈ [ 0, T ] t ( ‖ ρ 1 2 u t ‖ L 2 2 + ‖ θ t ‖ L 2 2 + ‖ ∇ 2 θ ‖ L 2 2 ) + ∫ 0 T ( t ‖ ∇ u t ‖ L 2 2 + t ‖ ∇ θ t ‖ L 2 2 ) d t ≤ C ( T ) . (3.76)

Proof. First, it is easy to deduce from (3.47), (3.29) that for any η ∈ ( 0,1 ] and any s > 2 ,

‖ ρ η u ‖ L s / η + ‖ u x ¯ − η ‖ L s / η ≤ C . (3.77)

Next, we prove

sup t ∈ [ 0, T ] ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + ∫ 0 T ( ‖ ρ 1 2 u t ‖ L 2 2 + ‖ θ t ‖ L 2 2 + ‖ Δ θ ‖ L 2 2 ) d t ≤ C . (3.78)

With (3.2) at hand, we need only to show

∫ 0 T ( ‖ ρ 1 2 u t ‖ L 2 2 + ‖ θ t ‖ L 2 2 ) d t ≤ C . (3.79)

First, it is easy to show that

‖ ρ 1 2 u t ‖ L 2 2 ≤ ‖ ρ 1 2 u ˙ ‖ L 2 2 + ‖ ρ 1 2 | u | | ∇ u | ‖ L 2 2 ≤ ‖ ρ 1 2 u ˙ ‖ L 2 2 + C ‖ ρ 1 2 u ‖ L 6 2 ‖ ∇ u ‖ L 3 2 ≤ ‖ ρ 1 2 u ˙ ‖ L 2 2 + C ‖ ∇ u ‖ L 2 2 + C ‖ ∇ 2 u ‖ L 2 2 . (3.80)

Then, due to (2.1) and (3.77), combining (2.1), (3.2) with (1.1)

‖ θ t ‖ L 2 2 ≤ C ‖ Δ θ ‖ L 2 2 + ‖ | u | | ∇ θ | ‖ L 2 2 ≤ C ‖ Δ θ ‖ L 2 2 + ‖ ∇ θ x ¯ a 2 ‖ L 2 2 , (3.81)

where in the last inequality one has used the follow facts

‖ | u | | ∇ θ | ‖ L 2 2 ≤ C ‖ u x ¯ − a 4 ‖ L 8 4 ‖ ∇ θ ‖ L 4 2 + C ‖ ∇ θ x ¯ a 2 ‖ L 2 2 ≤ 1 2 ‖ ∇ 2 θ ‖ L 2 2 + C ‖ ∇ θ x ¯ a 2 ‖ L 2 2 . (3.82)

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

ρ u t t + ρ u ⋅ ∇ u t − μ Δ u t + ∇ p t = − ρ t ( u t + u ⋅ ∇ u ) − ρ u t ⋅ ∇ u + ν Δ ρ ∇ ( ∇ ρ ⋅ u ) + ν Δ ( ∇ ρ ⋅ u ) ∇ ρ . (3.83)

Multiplying (3.83) by u t and integrating the resulting equality by parts on ℝ 2 , it follows (1.1) that

1 2 d d t ∫ ρ | u t | 2 d x + μ ∫ | ∇ u t | 2 d x ≤ C ∫ ρ | u | | u t | ( ∇ u t + | ∇ u | 2 + | u | | ∇ 2 u | ) d x + C ∫ ρ | u | 2 | ∇ u | | ∇ u t | d x + C ∫ ρ | u t | 2 | ∇ u | d x + 2 ν ∫ | ∇ 2 ρ | 2 | u | | u t | d x + ν ∫ | ∇ ρ | | ∇ u | | Δ ρ | | u t | d x + ν ∫ | ∇ ρ | | ∇ 2 ρ | | ∇ u | | u t | d x + ν ∫ | ∇ 2 ρ | | ∇ ρ | | u | | ∇ u t | d x + ν ∫ | ∇ ρ | 2 | ∇ u | | ∇ u t | d x ≜ ∑ i = 1 8 I i , (3.84)

where

I 1 ≤ C ‖ ρ u ‖ L 6 ‖ ρ u t ‖ L 2 1 2 ‖ ρ u t ‖ L 6 1 2 ( ‖ ∇ u t ‖ L 2 + ‖ ∇ u ‖ L 4 2 ) + C ‖ ρ 1 4 u ‖ L 12 2 ‖ ρ u t ‖ L 2 1 2 ‖ ρ u t ‖ L 6 1 2 ‖ ∇ 2 u ‖ L 2 ≤ C ‖ ρ u t ‖ L 2 1 2 ( ‖ ρ u t ‖ L 2 + ‖ ∇ u t ‖ L 2 ) 1 2 ( ‖ ∇ u t ‖ L 2 + ‖ ∇ 2 u ‖ L 2 ) ≤ ε ‖ ∇ u t ‖ L 2 2 + C ( 1 + ‖ ρ u t ‖ L 2 2 + ‖ ∇ 2 u ‖ L 2 2 ) , (3.85)

I 2 + I 3 ≤ C ‖ ρ u ‖ L 8 2 ‖ ∇ u ‖ L 4 ‖ ∇ u t ‖ L 2 + ‖ ∇ u ‖ L 2 ‖ ρ u t ‖ L 6 3 2 ‖ ρ u t ‖ L 2 1 2 ≤ ε ‖ ∇ u t ‖ L 2 2 + C ( 1 + ‖ ρ u t ‖ L 2 2 + ‖ ∇ 2 u ‖ L 2 2 ) , (3.86)

I 4 = 2 ν ∫ | x ¯ a ∇ 2 ρ | 2 | x ¯ − a u | | x ¯ − a u t | d x ≤ 2 ν ‖ x ¯ a ∇ 2 ρ ‖ L q 2 ‖ x ¯ − a u ‖ L 2 q q − 2 ‖ x ¯ − a u t ‖ L 2 q q − 2

≤ C ‖ ρ x ¯ a ‖ W 2 , q 2 ( ‖ ρ u ‖ L 2 + ‖ ∇ u ‖ L 2 ) ( ‖ ρ u t ‖ L 2 + ‖ ∇ u t ‖ L 2 ) ≤ ε ‖ ∇ u t ‖ L 2 2 + C ‖ ρ u t ‖ L 2 2 , (3.87)

I 5 + I 6 ≤ 2 ν ∫ | ∇ ρ | | x ¯ a ∇ 2 ρ | | ∇ u | | x ¯ − a u t | d x ≤ 2 ν ‖ ∇ ρ ‖ L 2 ‖ ∇ u ‖ L 4 q q − 2 ‖ x ¯ a ∇ 2 ρ ‖ L q ‖ x ¯ − a u t ‖ L 4 q q − 2 ≤ C ‖ ρ x ¯ a ‖ W 2 , q ( ‖ ∇ 2 u ‖ L 2 q + 2 2 q ‖ ∇ u ‖ L 2 q − 2 2 q + ‖ ∇ u ‖ L 2 ) ( ‖ ρ u t ‖ L 2 + ‖ ∇ u t ‖ L 2 ) ≤ C ( 1 + ‖ ∇ 2 u ‖ L 2 q + 2 2 q ) ( ‖ ρ u t ‖ L 2 + ‖ ∇ u t ‖ L 2 ) ≤ ε ‖ ∇ u t ‖ L 2 2 + C ( 1 + ‖ ρ u t ‖ L 2 2 + ‖ ∇ 2 u ‖ L 2 2 ) , (3.88)

I 7 ≤ ν ‖ x ¯ a ∇ 2 ρ ‖ L q ‖ x ¯ − a u ‖ L 4 q q − 2 ‖ ∇ ρ ‖ L 4 q q − 2 ‖ ∇ u t ‖ L 2 ≤ C ‖ ρ x ¯ a ‖ W 2 , q ‖ ∇ ρ ‖ L 2 q − 2 2 q ‖ ∇ ρ ‖ H 1 q + 2 2 q ‖ ∇ u t ‖ L 2 ≤ ε ‖ ∇ u t ‖ L 2 2 + C , (3.89)

I 8 ≤ ν ‖ ∇ ρ ‖ L 8 2 ‖ ∇ u ‖ L 4 ‖ ∇ u t ‖ L 2 ≤ C ( ‖ ∇ u ‖ L 2 1 2 ‖ ∇ 2 u ‖ L 2 1 2 + ‖ ∇ u ‖ L 2 ) ⋅ ( ‖ ∇ ρ ‖ L 2 1 2 ‖ ∇ 2 ρ ‖ L 2 3 2 + ‖ ∇ ρ ‖ L 2 2 ) ‖ ∇ u t ‖ L 2 ≤ C ( ‖ ∇ u ‖ L 2 1 2 ‖ ∇ 2 u ‖ L 2 1 2 + ‖ ∇ u ‖ L 2 ) ⋅ ( ‖ x ¯ a ρ ‖ H 1 1 2 ‖ x ¯ a ρ ‖ H 2 3 2 + ‖ x ¯ a ρ ‖ H 1 2 ) ‖ ∇ u t ‖ L 2 ≤ ε ‖ ∇ u t ‖ L 2 2 + C ( 1 + ‖ ∇ 2 u ‖ L 2 ) , (3.90)

Submitting the above I i into (3.84) gives

d d t ‖ ρ 1 2 u t ‖ L 2 2 + C ‖ ∇ u t ‖ L 2 2 ≤ C ‖ ρ 1 2 u t ‖ L 2 2 + C ( ‖ ∇ 2 u ‖ L 2 2 + 1 ) . (3.91)

Then, we multiply (3.91) by t, and together with Gronwall’s inequality and (3.44) lead to

sup t ∈ [ 0, T ] t ‖ ρ 1 2 u t ‖ L 2 2 + ∫ 0 T t ‖ ∇ u t ‖ L 2 2 d t ≤ C . (3.92)

Next, differentiating (1.1) with respect to t shows

θ t t + u t ⋅ ∇ θ + u ⋅ ∇ θ t − κ Δ θ t = 0. (3.93)

Now, multiplying (3.93) by θ t and integrating the resulting equality by parts on ℝ 2 , we find

1 2 d d t ∫ | θ t | 2 d x + κ ∫ | ∇ θ t | 2 d x = ∫ u t ⋅ ∇ θ t ⋅ θ d x ≤ C ‖ θ x ¯ a 2 ‖ L 8 a 4 a − 1 ‖ u t x ¯ − 2 a − 1 4 ‖ L 8 a ‖ ∇ θ t ‖ L 2 ≤ μ 2 ( ‖ ∇ u t ‖ L 2 2 + ‖ ρ 1 2 u t ‖ L 2 2 ) + κ 2 ‖ ∇ θ t ‖ L 2 2 . (3.94)

Next, multiplying (3.94) by t and integrating the resulting equality by parts on [ 0, T ] , it follows from (3.79) that

sup t ∈ [ 0, T ] t ‖ θ t ‖ L 2 2 + κ 2 ∫ 0 T t ‖ ∇ θ t ‖ L 2 2 d t ≤ μ 2 ∫ 0 T t ‖ ∇ u t ‖ L 2 2 + C . (3.95)

Finally, it follows from (1.1), and (3.82) that

‖ ∇ 2 θ ‖ L 2 2 ≤ C ‖ θ t ‖ L 2 2 + C ‖ | u | | ∇ θ | ‖ L 2 2 ≤ C ‖ θ t ‖ L 2 2 + 1 2 ‖ ∇ 2 θ ‖ L 2 2 + C ‖ ∇ θ x ¯ a 2 ‖ L 2 2 , (3.96)

o

which combined with (3.66), (3.92) and (3.95) indicates (3.76) and the proof of Lemma 3.8 is finished.

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 T * > 0 such that the Equation (1.1) and (1.2) have unique and local strong solution ( ρ , u , p , θ ) on ℝ 2 × ( 0, T * ] . Next, we will extend the local problem to all time.

Let

T * = sup { T | ( ρ , u , p , θ ) is a strong solution on ℝ 2 × ( 0, T ] } . (4.1)

For any 0 < τ < T < T * with T finite, one deduces from (3.76) that for any q ≥ 2 ,

∇ u , ∇ θ , θ ∈ C ( [ τ , T ] ; L 2 ∩ L q ) , (4.2)

where one has used the standard embedding

Moreover, it follows from (3.30), (3.56), and ( [

ρ ∈ C ( [ 0, T ] ; L 1 ∩ H 1 ∩ W 1, q ) . (4.3)

We claim that

T * = ∞ . (4.4)

Otherwise, if T * < ∞ , it follows from (4.2), (4.3), (3.2), (3.8), (3.56), and (3.57) that

( ρ , u , θ ) ( x , T * ) = lim t → T * ( ρ , u , θ ) ( x , t ) ,

which satisfies the initial condition (1.6) at t = T . Thus, taking ( ρ , u , θ ) ( x , T * ) as the initial data, since the existence and uniqueness of local strong solutions implies that there exists some T * * > T * , such that Theorem 1.2 satisfy T = T * * . 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 3.1-3.8 show that ( ρ , u , p , θ ) is in fact the unique strong solution on ℝ 2 × [ 0, T ] for any 0 < T < T * = ∞ . 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 ‖ ρ u ‖ L 2 and ‖ ∇ u ‖ L 2 . 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, u ⋅ ∇ θ , 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.

The author declares no conflicts of interest regarding the publication of this paper.

Zhang, Q. (2021) Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type. Advances in Pure Mathematics, 11, 346-368. https://doi.org/10.4236/apm.2021.114022