We dedicate to the 2D density-dependent nonhomogeneous incompressible Boussinesq equations with vacuum on
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:
{ ρ t + div ( ρ u ) = 0, ( ρ u ) t + div ( ρ u ⊗ u ) + ∇ p = μ Δ u , θ t + u ⋅ ∇ θ − κ Δ θ = 0, div u = 0, x = ( x 1 , x 2 ) ∈ Ω , t ≥ 0, (1.1)
where ρ = ρ ( x , t ) is the density, u = ( u 1 , u 2 ) ( x , t ) represents the velocity, p = p ( x , t ) stands for the pressure and θ = θ ( x , t ) denotes the temperature of the fluid; μ > 0 is the viscosity coefficient; κ > 0 is the thermal diffusivity.
The initial data is given by
ρ ( x ,0 ) = ρ 0 ( x ) , θ ( x ,0 ) = θ 0 ( x ) , ρ u ( x ,0 ) = ρ 0 u 0 ( x ) , x ∈ ℝ 2 . (1.2)
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 [
In recent years, the Boussinesq system with ρ > 0 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 μ > 0 and κ > 0 , is popular. Lorca [
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 u ⋅ ∇ θ will give rise to some new difficulties. It seems difficult to bound the L p ( ℝ 2 ) -norm of u in terms of ‖ ρ 1 / 2 u ‖ L 2 ( ℝ 2 ) and ‖ ∇ u ‖ L 2 ( ℝ 2 ) . In light of [
H 2 ≜ ∫ p ∂ i u j ∂ j u i d x . (1.3)
According to [
Now, we go back to (1.1). it should be noted here that the notations and conventions employed throughout the paper. For R > 0 , set
B R ≜ { x ∈ ℝ 2 | | x | < R } , ∫ f d x ≜ ∫ ℝ 2 f d x .
Furthermore, for 1 ≤ r ≤ ∞ , k ≥ 1 , we denote the standard Lebesgue and Sobolve spaces as follows:
L r = L r ( ℝ 2 ) , W k , r = W k , r ( ℝ 2 ) , H r = W k , 2 .
Then, we will define precisely what mean by strong solution to (1.1) as follows:
Definition 1.1. (see [
In a general way, it can assume that ρ 0 holds
∫ ℝ 2 ρ 0 d x = 1. (1.4)
The (1.4) signifies that there is a positive constant N 0 such that
∫ B N 0 ρ 0 d x ≥ 1 2 ∫ ρ 0 d x = 1 2 . (1.5)
Theorem 1.1 In view of (1.4) and (1.5), it assumes that the initial data ( ρ 0 , u 0 , θ 0 ) hold that for any given numbers a > 1 and q > 2 ,
{ ρ 0 ≥ 0, ρ 0 x ¯ a ∈ L 1 ∩ H 1 ∩ W 1, 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 , d i v u 0 = 0 , (1.6)
where
x ¯ ≜ ( e + | x | 2 ) 1 2 log 2 ( e + | x | 2 ) . (1.7)
In that way, it has a unique global strong solution ( ρ , u , p , θ ) for the problem (1.1)-(1.2) satisfying that for any 0 < T < ∞ ,
{ 0 ≤ ρ ∈ C ( [ 0, T ] ; L 1 ∩ H 1 ∩ W 1, q ) , ρ x ¯ a ∈ L ∞ ( 0, T ; L 1 ∩ H 1 ∩ W 1, q ) , ρ u , ∇ u , x ¯ − 1 u , t ρ u t , t ∇ p , t ∇ 2 u ∈ L ∞ ( 0, T ; L 2 ) , ∇ θ ∈ L 2 ( 0, T 1 ; H 1 ) , t ∇ u ∈ L 2 ( 0, T ; W 1, q ) , 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 ) ) , (1.8)
and
inf 0 ≤ t ≤ T ∫ B N 1 ρ ( x , t ) d x ≥ 1 4 , (1.9)
It’s about positive constant N 1 depending only ‖ ρ 0 ‖ L 1 , ‖ ρ 0 1 2 u 0 ‖ L 2 , N 0 and T. 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 , (1.10)
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 1.1 If the temperature function is zero, i.e., θ = 0 , 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 [
Lemma 2.2 (see ( [
‖ 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 following weighted L n bounds for elements in D ˜ 1 , 2 ( ℝ 2 ) ≜ { v ∈ H l o c 1 ( ℝ 2 ) | ∇ v ∈ L 2 ( ℝ 2 ) } can be found in ( [
Lemma 2.3 (see ( [
( ∫ 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)
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 (see ( [
‖ 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, let BMO ( ℝ 2 ) and H 1 ( ℝ 2 ) represent BMO and Hardy spaces (see [
Lemma 2.5 (see ( [
‖ 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
d i v 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) Please refer to ( [
(ii) The follows together with the Poincaré inequality that for any ball B ⊂ (ℝ2)
1 | B | ∫ B | v ( x ) − 1 | B | ∫ B v ( y ) d y | d x ≤ C ( ∫ B | ∇ v | 2 d x ) 1 / 2 , (2.9)
which directly gives (2.8).
Due to d i v u = 0 , it will estimate the L ∞ ( 0, T ; L r ) -norm of ρ , as follows:
Lemma 3.1 (see [
sup t ∈ [ 0, T ] ‖ ρ ‖ L 1 ∩ L ∞ ≤ C . (3.1)
Then, we will estimate the L ∞ ( 0, T ; L 2 ) -norm of ∇ θ and ∇ u .
Lemma 3.2. There is 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 , (3.2)
Here u ˙ ≜ ∂ t u + u ⋅ ∇ u , and have
sup t ∈ [ 0, T ] t ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + ∫ 0 T t ( ‖ ρ u ˙ ‖ L 2 2 + ‖ Δ θ ‖ L 2 2 ) d t ≤ C . (3.3)
Proof. Invoking standard energy estimate, multiplying (1.1)2 by u and integrating the resulting equality over ℝ 2 , we get
sup t ∈ [ 0, T ] ‖ ρ u ‖ L 2 2 + ∫ 0 T ‖ ∇ u ‖ L 2 2 d t ≤ C . (3.4)
Multiplying (1.1)3 by θ and integrating the resulting equality over ℝ 2 , we have
sup t ∈ [ 0, T ] ‖ θ ‖ L 2 2 + ∫ 0 T ‖ ∇ θ ‖ L 2 2 d t ≤ C . (3.5)
The (3.4) combined with (3.5) that gives
sup t ∈ [ 0, T ] ( ‖ ρ u ‖ L 2 2 + ‖ θ ‖ L 2 2 ) + ∫ 0 T ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) d t ≤ C . (3.6)
Next, multiplying (1.1)2 by u ˙ and integrating the resulting equality over ℝ 2 , we have
∫ ρ | u ˙ | 2 d x = ∫ μ Δ u ⋅ u ˙ d x − ∫ ∇ p ⋅ u ˙ d x ≜ H 1 + H 2 . (3.7)
Then we can follow form integrating H i ( i = 1 , 2 ) by parts and (2.1) that
H 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 . (3.8)
Integration by parts together with (1.1)4 gives
H 2 = − ∫ ∇ p ( u t + u ⋅ ∇ u ) d x = ∫ p ∂ j u i ∂ i u j d x ≤ C ‖ p ‖ B M O ‖ ∂ j u i ∂ i u j ‖ H 1 , (3.9)
For the last inequality, because of the duality of H 1 space and BMO (see ( [
‖ ∂ j u i ∂ i u j ‖ H 1 ≤ C ‖ ∇ u ‖ L 2 ‖ ∇ u ‖ L 2 . (3.10)
The (3.9) combined with (3.10) and (2.8) gives
| H 2 | = | ∫ p ∂ j u i ∂ i u j d x | ≤ C ‖ p ‖ B M O ‖ ∇ u ‖ L 2 2 ≤ C ‖ ∇ p ‖ L 2 ‖ ∇ u ‖ L 2 2 . (3.11)
Next, substituting (3.8) and (3.11) into (3.7) 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 . (3.12)
Then, the (1.1)3 multiplied by Δ θ and integrating the resulting equality by parts over ℝ 2 , and together with Hölder’s and (11) 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 , (3.13)
which combined with (3.12) and (3.6) 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 . (3.14)
Due to ( ρ , u , p , θ ) solves the following Stokes system see [
{ − μ Δ u + ∇ p = − ρ u ˙ , x ∈ ℝ 2 , d i v u = 0 , x ∈ ℝ 2 , u ( x ) → 0 , | x | → ∞ , (3.15)
Using the standard L r -estimate to (3.15) holds that for any r > 1 ,
‖ ∇ 2 u ‖ L r + ‖ ∇ p ‖ L r ≤ C ‖ ρ u ˙ ‖ L r ≤ C ‖ ρ u ˙ ‖ L r . (3.16)
(3.14) combined with and (3.16) 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 , (3.17)
where ε is to be determined. Choosing ε = 1 2 , it follows from (3.6) and (3.17) 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 ) . (3.18)
the (3.18) together with (3.6), (3.17) and (2.1) gives (3.2). Then, (3.17) multiplied 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 ) , (3.19)
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 μ , κ , ‖ ρ 0 ‖ L 1 ∩ L ∞ , ‖ ∇ u 0 ‖ L 2 , and ‖ ρ 0 1 2 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 , (3.20)
and
sup t ∈ [ 0, T ] t i ( ‖ ∇ 2 u ‖ L 2 2 + ‖ ∇ p ‖ L 2 2 ) ≤ C . (3.21)
Proof: Using ∂ t + u ⋅ ∇ to ( 1.1 ) 2 j , it follows from a few 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 . (3.22)
Next, (3.22) multiplied by u ˙ j , and integration by parts and (1.1)4, we get
1 2 d d t ‖ ρ u ˙ ‖ L 2 2 + μ ‖ ∇ u ˙ ‖ L 2 2 = − ∫ 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 ≜ ∑ i = 1 3 M i . (3.23)
Following the same argument as ( [
∑ i = 1 3 M i ≤ d d t ∫ p ∂ j u i ∂ i u j d x + C ( ‖ p ‖ L 4 4 + ‖ ∇ u ‖ L 4 4 ) + μ 2 ‖ ∇ u ˙ ‖ L 2 2 . (3.24)
Substituting (3.24) into (3.23) gives
d d t ( 1 2 ‖ ρ u ˙ ‖ L 2 2 − ∫ p ∂ j u i ∂ i u j d x ) + μ 2 ‖ ∇ u ˙ ‖ L 2 2 ≤ C ‖ p ‖ L 4 4 + C ‖ ∇ u ‖ L 4 4 . (3.25)
For the left of (3.25), we have
| ∫ p ∂ j u i ∂ i u j d x | ≤ C ‖ ∇ p ‖ L 2 ‖ ∇ u ‖ L 2 2 ≤ C ( ‖ ∇ 2 u ‖ L 2 + ‖ ∇ p ‖ L 2 ) ‖ ∇ u ‖ L 2 2 ≤ C ‖ ρ 1 2 u ˙ ‖ L 2 2 + C ‖ ∇ u ‖ L 2 4 . (3.26)
For the right of (3.25), (3.11) together with (3.16), (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 ‖ ρ ‖ L 2 2 ‖ ρ u ˙ ‖ L 2 4 ≤ C ‖ ρ u ˙ ‖ L 2 4 . (3.27)
Substituting (3.26) and (3.27) into (3.25)
d d t ‖ ρ u ˙ ‖ L 2 2 + μ 2 ‖ ∇ u ˙ ‖ L 2 2 ≤ C ‖ ρ u ˙ ‖ L 2 4 . (3.28)
Next, multiplying (3.28) by t i ( i = 1 , 2 ) , it follows from (3.3) and (3.6) that
d d t t i ‖ ρ u ˙ ‖ L 2 2 − i t i − 1 ‖ ρ u ˙ ‖ L 2 2 + μ 2 ‖ ∇ u ˙ ‖ L 2 2 ≤ C t i ‖ ρ u ˙ ‖ L 2 2 ( ‖ ρ u ˙ ‖ L 2 2 + C ) . (3.29)
Then, the (3.29) along with Gronwall's inequality gives
sup t ∈ [ 0, T ] t i ‖ ρ u ˙ ‖ L 2 2 + ∫ 0 T t i ‖ ∇ u ˙ ‖ L 2 2 d t ≤ ∫ 0 T i t i − 1 ‖ ρ u ˙ ‖ L 2 2 d t . (3.30)
Finally, due to i = 1 , 2 , 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.
It concerns with the estimates on the higher-order derivatives of the strong solution ( ρ , u , p , θ ) as follow:
Lemma 3.4 For a positive constant C depending only on μ , ‖ ρ 0 ‖ L 1 ∩ L ∞ , ‖ ∇ u 0 ‖ L 2 , ‖ 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 ) . (3.31)
Proof. For M > 1 , let φ M ∈ C 0 ∞ ( B M ) satisfy
0 ≤ φ M ≤ 1, φ M ( x ) = { 1, | x | ≤ M / 2 , 0, | x | ≥ M , | ∇ φ M | ≤ C M − 1 . (3.32)
It combines with (1.1)1 that
d d t ∫ ρ φ M d x = ∫ ρ u ⋅ ∇ φ M d x ≥ − C M − 1 ( ∫ ρ d x ) 1 / 2 ( ∫ ρ | u | 2 d x ) 1 / 2 ≥ − C ˜ M − 1 , (3.33)
in the last inequality of (3.33), it has applied (3.1) and (3.6). Integrating (3.33) and letting M = N 1 ≜ 2 N 0 + 4 C ˜ T , we obtain after using (1.5) 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 . (3.34)
the (3.34) along with (3.1), (2.2), (3.6) 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 ) ≤ C . (3.35)
(1.1)1 multiplied by x ¯ a and integrating the resulting equality by parts over ℝ 2 find 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 ¯ − 1 8 + a ‖ L 8 + a ≤ C ∫ ρ x ¯ a d x + C . (3.36)
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
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 ( T ) . (3.37)
Proof. We can follow from the (1.1)1 that ∇ ρ holds for any r ≥ 2 ,
d d t ‖ ∇ ρ ‖ L r ≤ C ( r ) ‖ ∇ u ‖ L ∞ ‖ ∇ ρ ‖ L r . (3.38)
Next, employing Lemma 2.2, (3.2) and (3.16), 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 ) . (3.39)
It follows from (3.34), (3.1), (2.2) and (3.31) 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 ) , (3.40)
the (3.40) combine 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 ) , (3.41)
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 ≤ C , (3.42)
∫ 0 T t ‖ ρ u ˙ ‖ L q 2 d t ≤ C ( ∫ 0 T t ‖ ∇ u ˙ ‖ L 2 2 d t + 1 ) ≤ C . (3.43)
Then, the (3.42) and (3.39) implies
∫ 0 T ‖ ∇ u ‖ L ∞ d t ≤ C . (3.44)
Next, using Gronwall’s inequality to (3.38) shows
sup t ∈ [ 0, T ] ‖ ∇ ρ ‖ L 2 ∩ L q ≤ C . (3.45)
Then, letting r = 2 in (3.16) 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 . (3.46)
Similarly, setting r = q in (3.16) and integrating the resulting equality over [ 0, T ] , we deduce from using (3.42), (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 . (3.47)
Multiplying (3.16) by t and integrating the resulting equality over [ 0, T ] , it can obtain after using (3.43), (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 . (3.48)
Moreover, it can get from (3.46), (3.47) and (3.48) 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 , (3.49)
which combined with (3.1) and (3.45) gets (3.37). The Lemma 3.5 is proved.
Lemma 3.6 (see [
sup t ∈ [ 0, T ] ‖ ρ x ¯ a ‖ L 1 ∩ H 1 ∩ W 1, q ≤ C ( T ) . (3.50)
Proof. First, setting ρ = ρ x ¯ a in (1.1)1 that satisfies
∂ t ( ρ x ¯ a ) + u ⋅ ∇ ( ρ x ¯ a ) − a ρ x ¯ a u ⋅ ∇ log x ¯ = 0. (3.51)
Next, we can take the x i -derivative on both sides of the (3.51) finds
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 ¯ . (3.52)
the (3.52) multiplied by | ∇ ( ρ x ¯ a ) | r − 2 ∂ i ( ρ x ¯ a ) and integrating the resulting equality by parts over ℝ 2 , and then for any r ∈ [ 2, q ] , we obtain that
d d t ‖ ∇ ( ρ x ¯ a ) ‖ L r ≤ C ( 1 + ‖ u ⋅ ∇ log x ¯ ‖ L ∞ + ‖ ∇ u ‖ L ∞ ) ‖ ∇ ( ρ x ¯ a ) ‖ L r + C ‖ ρ x ¯ a ‖ L ∞ ( ‖ | u | | ∇ 2 log x ¯ | ‖ L r + ‖ | ∇ u | | ∇ 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 + ‖ ∇ ( ρ x ¯ a ) ‖ L r + ‖ ∇ ( ρ x ¯ a ) ‖ L q ) ( 1 + ‖ ∇ u ‖ W 1 , q ) . (3.53)
For the second and the last inequalities of (3.53), it has used (3.35) and (3.31), respectively. Setting r = q in (3.53), and applying Gronwall’s inequality along with (3.37) indicates that
sup t ∈ [ 0, T ] ‖ ∇ ( ρ x ¯ a ) ‖ L q ≤ C . (3.54)
And choosing r = 2 in (3.53), we will deduce from (3.37) and (3.54) that
sup t ∈ [ 0, T ] ‖ ∇ ( ρ x ¯ a ) ‖ L 2 ≤ C . (3.55)
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
sup t ∈ [ 0, T ] ‖ θ x ¯ a 2 ‖ L 2 2 + ∫ 0 T ‖ ∇ θ x ¯ a 2 ‖ L 2 2 d t ≤ C ( T ) , (3.56)
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 ) . (3.57)
Proof. The (1.1)3 multiplied 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 , (3.58)
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 , (3.59)
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 . (3.60)
Substituting (3.59), (3.60) into (3.58), 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 . (3.61)
Using Gronwall’s inequality to (3.61), we obtain (3.56).
Next, we will estimate the (3.57). The (1.1)3 Multiplied by Δ θ x ¯ a and integration by parts over 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 , (3.62)
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 , (3.63)
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 , (3.64)
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 . (3.65)
Submitting M ˜ 1 , M ˜ 2 , M ˜ 3 into (3.62), 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 . (3.66)
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
sup t ∈ [ 0, T ] t ( ‖ ρ 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 ) . (3.67)
Proof. For any η ∈ ( 0,1 ] and any s > 2 , it deduces from (3.40), (3.35) that
‖ ρ η u ‖ L s / η + ‖ u x ¯ − η ‖ L s / η ≤ C . (3.68)
Next, it will prove that
sup t ∈ [ 0, T ] ( ‖ ∇ u ‖ L 2 2 + ‖ ∇ θ ‖ L 2 2 ) + ∫ 0 T ( ‖ ρ u t ‖ L 2 2 + ‖ θ t ‖ L 2 2 + ‖ Δ θ ‖ L 2 2 ) d t ≤ C . (3.69)
With (3.2) at hand, we need only to show
∫ 0 T ( ‖ ρ u t ‖ L 2 2 + ‖ θ t ‖ L 2 2 ) d t ≤ C . (3.70)
First, it is easy to show that
‖ ρ u t ‖ L 2 2 ≤ ‖ ρ u ˙ ‖ L 2 2 + ‖ ρ | u | | ∇ u | ‖ L 2 2 ≤ ‖ ρ u ˙ ‖ L 2 2 + C ‖ ρ u ‖ L 6 2 ‖ ∇ u ‖ L 3 2 ≤ ‖ ρ u ˙ ‖ L 2 2 + C ‖ ∇ u ‖ L 2 2 + C ‖ ∇ 2 u ‖ L 2 2 . (3.71)
Then, due to (2.1) and (3.68), we can combine (2.1), (3.2) with (1.1)3 gives
‖ 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 , (3.72)
It has used the following facts about (3.72) of the last inequality
‖ | 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 . (3.73)
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
ρ u t t + ρ u ⋅ ∇ u t − μ Δ u t + ∇ p t = − ρ t ( u t + u ⋅ ∇ u ) − ρ u t ⋅ ∇ u . (3.74)
(3.74) multiplied by u t and integration by parts over ℝ 2 , it deduces from (1.1)1 and (1.1)4 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 | + | ∇ u | 2 ) d x + C ∫ ρ | u | 2 | ∇ u | | ∇ u t | d x + C ∫ ρ | u t | 2 | ∇ u | d x ≜ ∑ i = 1 3 M ¯ i , (3.75)
where
M ¯ 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 ) ≤ μ 4 ‖ ∇ u t ‖ L 2 2 + C ( 1 + ‖ ρ u t ‖ L 2 2 + ‖ ∇ 2 u ‖ L 2 2 ) , (3.76)
M ¯ 2 + M ¯ 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 ≤ μ 4 ‖ ∇ u t ‖ L 2 2 + C ( 1 + ‖ ρ u t ‖ L 2 2 + ‖ ∇ 2 u ‖ L 2 2 ) . (3.77)
Submitting M ¯ 1 , M ¯ 2 + M ¯ 3 into (3.75) gives
d d t ‖ ρ u t ‖ L 2 2 + μ ‖ ∇ u t ‖ L 2 2 ≤ C ‖ ρ u t ‖ L 2 2 + C ( ‖ ∇ 2 u ‖ L 2 2 + 1 ) . (3.78)
Then, we multiply (3.78) by t, and link to Gronwall’s inequality and (3.37) lead to
sup t ∈ [ 0, T ] t ‖ ρ u t ‖ L 2 2 + ∫ 0 T t ‖ ∇ u t ‖ L 2 2 d t ≤ C . (3.79)
Next, differentiating (1.1)3 with respect to t show
θ t t + u t ⋅ ∇ θ + u ⋅ ∇ θ t − κ Δ θ t = 0. (3.80)
Now, the (3.80) multiplied by θ t and integration by parts over 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 u t ‖ L 2 2 ) + κ 2 ‖ ∇ θ t ‖ L 2 2 . (3.81)
Next, the (3.81) multiplied by t and integration by parts over [ 0, T ] , and due to (3.70), we have
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 . (3.82)
Finally, it follows from (1.1)3, and (3.73) 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 , (3.83)
which combine with (3.57), (3.79) and (3.82) gains (3.67). Finally, the proof of Lemma 3.8 is finished.
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 T * > 0 such that systems (1.1) and (1.2) have a local and unique strong solution ( ρ , u , p , θ ) on ℝ 2 × ( 0, T * ] . Next, we will extend the local solution to all time.
Set
T * = sup { T | ( ρ , u , p , θ ) is a strong solution on ℝ 2 × ( 0 , T ] } . (3.84)
It deduces from (3.67), for any 0 < τ < T < T * with T finite, and any q ≥ 2 that,
∇ u , ∇ θ , θ ∈ C ( [ τ , T ] ; L 2 ∩ L q ) . (3.85)
Then, along with standard embedding
And, due to (3.36), (3.49), and ( [
ρ ∈ C ( [ 0, T ] ; L 1 ∩ H 1 ∩ W 1, q ) . (3.86)
we declare that
T * = ∞ . (3.87)
On the contrary, if T * < ∞ , it deduces from (4.2), (4.2), (3.2), (3.6), (3.49), and (3.50) that
( ρ , u , θ ) ( x , T * ) = lim t → T * ( ρ , u , θ ) ( x , t ) .
conforms to the initial condition (1.6) at t = T . So, we can assume the initial data is the ( ρ , u , θ ) ( x , T * ) , since the existence and uniqueness of local strong solutions signifies that there is a some T * * > T * , such that Theorem (1.1) holds for T = T * * . This is contradictory with the hypothesis of T * 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 ( ρ , u , p , θ ) is actually the unique strong solution on ℝ 2 × [ 0, T ] for any 0 < T < T * = ∞ . This completes the proof of Theorem 1.1.
The author declares no conflicts of interest regarding the publication of this paper.
Liu, M. (2019) Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations with Vacuum. Journal of Applied Mathematics and Physics, 7, 2333-2351. https://doi.org/10.4236/jamp.2019.710159