Global well-posedness and asymptotic behavior for the 2D Boussinesq system with variable viscosity

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J. Math. Anal. Appl. ••• (••••) ••••••

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Global well-posedness and asymptotic behavior for the 2D Boussinesq system with variable viscosity Yanghai Yu ∗ , Mulan Zhou School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui, 241002, PR China

a r t i c l e

i n f o

Article history: Received 16 June 2019 Available online xxxx Submitted by D. Wang Keywords: Boussinesq system Global well-posedness Asymptotic behavior Weak dissipation

a b s t r a c t In this paper, we investigate the global existence and uniqueness of strong solutions to 2D Boussinesq system with variable kinematic viscosity depending on the temperature. Comparing with the previous results given by Abidi and Zhang [4] who considered the critical case α = 1, we weaken the dissipation effect in the temperature equation to the supercritical case α ∈ (0, 1). Furthermore, we obtain the algebraic decay estimate for ||u||L2 . © 2019 Elsevier Inc. All rights reserved.

1. Introduction The purpose of this paper is to consider the Cauchy problem of the two-dimensional Boussinesq system with variable viscosity which reads ⎧ ⎪ ∂t u + u · ∇u − div(2μ(θ)Du) + ∇Π = θe2 , ⎪ ⎪ ⎨ ∂ θ + u · ∇θ + νΛα θ = 0, t ⎪ divu = 0, ⎪ ⎪ ⎩ (u(0, x), θ(0, x)) = (u (x), θ (x)), 0 0

x ∈ R2 , t > 0, x ∈ R2 , t > 0, x ∈ R2 , t ≥ 0, x ∈ R2 ,

(1.1)

where u = (u1 (t, x), u2 (t, x)) denotes the velocity of the fluid, θ and Π stand for the scalar temperature and pressure, respectively. Du represents the 2 × 2 deformation matrix whose (i, j)-th component is given by (Du)i,j = 12 (∂i uj + ∂j ui ) with 1 ≤ i, j ≤ 2. θe2 is buoyancy force with e2 = (0, 1), and the kinematic viscous coefficient μ(θ) is a positive, non-decreasing smooth function on [0, ∞). The parameter ν ≥ 0 represents the thermal diffusivity. The Boussinesq equations arise from a zero order approximation to the coupling between Navier-Stokes (NS) equations and the thermodynamic equations and describe many geophysical phenomena in atmospheric * Corresponding author. E-mail addresses: [email protected] (Y. Yu), [email protected] (M. Zhou). https://doi.org/10.1016/j.jmaa.2019.123668 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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and oceanographic sciences [21,22]. Due to its profound physical background and important mathematical significance, there has been a huge amount of literature on the study of the Boussinesq system by many physicists and mathematicians, for example, see [1,2,5,6,9,13–16,18,20,26–28] and the references therein. Big progresses have been made on the global well-posedness of the system (1.1) especially for the twodimensional case. Here we mainly recall some notable works which are more closely to our problem in terms of Boussinesq system with variable viscosity, namely, ⎧ ∂t u + u · ∇u − div(μ(θ)∇u) + ∇Π = θe2 , ⎪ ⎪ ⎪ ⎪ ⎨ ∂t θ + u · ∇θ − div(ν(θ)∇θ) = 0, ⎪ divu = 0, ⎪ ⎪ ⎪ ⎩ (u(0, x), θ(0, x)) = (u0 (x), θ0 (x)),

x ∈ R2 , t > 0, x ∈ R2 , t > 0, x ∈ R2 , t ≥ 0,

(1.2)

x ∈ R2 ,

Cannon–DiBenedetto [8] showed the global-in-time regularity of the system (1.2) with μ(θ) ≡ μ > 0 and ν(θ) ≡ ν > 0 by classical methods. For this case μ(θ) ≡ μ > 0 and ν(θ) ≡ 0, or μ(θ) ≡ 0 and ν(θ) ≡ ν > 0, Chae [10] and Hou–Li [17] independently proved the global existence of smooth solutions to (1.2) under the framework of Sobolev spaces. When μ(θ) ≡ μ > 0 and ν(θ) ≡ 0, Abidi–Hmidi [3] obtained the global −1 0 well-posedness for (1.2) with initial data satisfying (θ0 , u0 ) ∈ B2,1 × (B∞,1 ∩ L2 ). When ν(θ) ≡ ν > 0 and μ(θ) ≡ 0, Hmidi–Keraani [15] proved the global existence and uniqueness of solutions to (1.2) with 0 0 (u0 , θ0 ) ∈ H s (R2 ) × (Bp,∞ ∩ B2,1 ) where (s, p) ∈ (0, 2] × (2, ∞]. Wang–Zhang [24] established the global existence of smooth solutions to (1.2) under the assumptions that both μ(·) and ν(·) belong to L∞ (R+ ) and have positive lower bounds. Li–Xu [19] generalized the result in [24] to the degenerate case, i.e. μ(θ) ≡ 0 and obtained the global strong solution for arbitrarily large initial data in Sobolev spaces H s with s > 2. To gain a deeper understanding of strength and weaknesses of available mathematics methods and techniques, researches begin to pay attention to the critical Boussinesq system be of the form ⎧ ∂t u + u · ∇u + μΛu + ∇Π = θe2 , ⎪ ⎪ ⎪ ⎪ ⎨ ∂t θ + u · ∇θ + νΛθ = 0, ⎪ divu = 0, ⎪ ⎪ ⎪ ⎩ (u(0, x), θ(0, x)) = (u0 (x), θ0 (x)),

x ∈ R2 , t > 0, x ∈ R2 , t > 0, x ∈ R2 , t ≥ 0,

(1.3)

x ∈ R2 ,

Hmidi–Keraani–Rousset in their series papers [15,16] used a beautiful diagonalization approach and proved the global well-posedness of fractional diffusion Boussenesq system (1.3) with μ = 1 or ν = 1. Recently, Abidi–Zhang [4] studied the Cauchy problem of the system (1.1) with α = 1 and established the global well-posedness provided that the viscosity coefficient μ(θ) is sufficiently close to some positive constant. Subsequently, Yu–Wu–Tang [29] obtained the global well-posedness of the system (1.1) with the weak damping effect to instead of the regularity effect for the thermal conductivity, where the obvious advantage of the damping term lies in that it provides exponential decay of ||θ||Lp . Since it is commonly believed that the dissipation term Λα θ is always good and the power α is more smaller, the issue of global well-posedness of the Boussinesq system (1.1) is more harder. Inspired by the recent work given by Abidi–Zhang [4], we consider the Cauchy problem of the system (1.1) with α ∈ (0, 1) in this paper. Throughout this paper, we always make the following assumptions on μ(θ) and ν in (1.1) μ(θ) ≥ 1, μ(·) ∈ W 2,∞ (R+ ), μ(0) = ν = 1 and for some large enough positive constant C. Now, we state our main result.

1  C||μ(·) − 1||L∞ p∗

(1.4)

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Theorem 1.1. Let α ∈ (0, 1) and p ∈ ( α4 , p∗ ], u0 ∈ H˙ α−1 ∩ H 1 (R2 ) be a solenoidal vector field and θ0 ∈ L1 ∩ L∞ ∩ H˙ 1 (R2 ). There exists a sufficiently small ε0 > 0 such that ||μ(·) − 1||L∞ (R+ ) ≤ ε0 ,

(1.5)

then the system (1.1) has a unique global strong solution (u, θ) satisfying that for any given T > 0, α

θ ∈ C([0, T ]; H 1 ) ∩ L2 ([0, T ]; H 1+ 2 ) and u ∈ C([0, T ]; H 1 ) ∩ L2 ([0, T ]; H 2 ). Furthermore, there holds ||θ||Lp ≤ C t − α (1− p ) ||θ0 ||L1 ∩Lp 2

1

for

2 ≤ p < ∞,

(1.6)

and ||u||L2 ≤ CH0 t α−1 ,

(1.7)

here and in what follows, we denote ·  e + ·

and

H0  1 + ||u0 ||2L2 ∩H˙ α−1 + ||u0 ||4L2 + ||θ0 ||4L1 ∩L2 .

Remark 1.1. Although the proof of Theorem 1.1 largely follows the same manner from [4], we simplify the details and overcome the technical difficulties. On one hand, we establish the decay estimate of ||θ||Lp and give directly the estimate of ||u||L2 . On the other hand, owing to the weak dissipation of the temperature equation, the method of [4] to prove the uniqueness is invalid. We present a simple and different proof than that of [4]. For more details, see Section 4.1. Remark 1.2. Compared with the previous references [4], the decay estimate (1.7) is completely new. Remark 1.3. It is obvious that Theorem 1.1 holds true for μ(θ) = θ + 1 with ||θ0 ||L∞ ≤ ε0 . Remark 1.4. The global well-posedness of system (1.1) with μ(θ) ≡ 0 is an interesting open problem. Next, we introduce some notations and conventions, and recall some standard theories of Besov space which will be used throughout this paper. 2. Notations and Littlewood-Paley theory First, we denote by C strictly positive constants whose values are insignificant and may be different in each occurrence. a  b means that there is a uniform constant C such that a ≤ Cb. We also use the notations ||f1 , f2 ||22  ||f1 ||2L2 + ||f2 ||2L2 and the commutator operator [A, B]  AB − BA, where A and B are any pair of operators on some Banach space. We denote by [b] the integer part of b. For X a Banach space and I an interval of R, we denote by C(I; X) the set of continuous functions on I with values in X, and by Lp (I; X) with p ∈ [1, ∞] stands for the set of measurable functions on I with values in X, such that t → ||f (t)||X ∈ Lp (I). Finally, we denote the Leray projection operator by P = I + ∇(−Δ)−1 div. The following material involving the theories of Littlewood-Paley is standard, we refer the readers to Bahouri–Chemin–Danchin [7]. Let C denote the annulus {ξ ∈ Rd : 3/4 ≤ |ξ| ≤ 8/3} and B denote the ball {ξ ∈ Rd : |ξ| ≤ 4/3}. There exist two radial functions χ ∈ Cc∞ (B(0, 4/3)) and ϕ ∈ Cc∞ (C) both taking values in [0, 1] such that

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ϕ(2−j ξ) = 1 for ξ ∈ Rd \ {0} and χ(ξ) +



ϕ(2−j ξ) = 1 for ξ ∈ Rd .

j≥0

j∈Z

For every u ∈ S  (Rd ), the inhomogeneous dyadic blocks Δj are defined as follows ⎧ ⎪ ⎪ ⎨0, Δj u = χ(D)u, ⎪ ⎪ ⎩ϕ(2−j D)u,

j ≤ −2; j = −1; j ≥ 0.

The inhomogeneous low-frequency cut-off operator Sj is defined by

Sj u =

j−1 

Δq u.

q=−1

˙ j and homogeneous low-frequency cut-off operator For every u ∈ S  (Rd ), the homogeneous dyadic blocks Δ S˙ j are defined as follows ˙ j u = ϕ(2−j D)u and S˙ j u = χ(2−j D)u = ∀j ∈ Z, Δ



˙ q u. Δ

q≤j−1

In the inhomogeneous case, the following Littlewood-Paley decomposition makes sense u=



Δj u, u ∈ S  (Rd ).

j≥−1

Unfortunately, for the homogeneous case, the Littlewood-Paley decomposition is invalid. We need to a new space to modify it, namely,  Sh  u ∈ S  (Rd ) :

 lim ||χ(2−j D)u||L∞ = 0 .

j→−∞

Then we have the formal Littlewood-Paley decomposition in the homogeneous case u=



˙ j u, ∀u ∈ S  . Δ h

j∈Z s We recall now the definition of homogeneous Besov space B˙ p,q . s ˙ (Rd ) is defined by Let s ∈ R and p, q ∈ [1, ∞], the homogeneous Besov space Bp,q

  

1/q s ˙ j uq p = u ∈ Sh (Rd ) : uB˙ p,q 2qjs Δ <∞ . B˙ p,q s (Rd ) = L j∈Z s u ∈ B˙ p,q if and only if there exist a constant C > 0 and a non-negative sequence {ej }j∈Z such that

˙ Lp ≤ Cej 2−js u ˙ s and ej lq = 1. ∀j ∈ Z, Δu Bp,q s Similar definition of inhomogeneous Besov space Bp,q can be found in [7]. We frequently use Bony’s decomposition [7] in the homogeneous context throughout this paper

˙ uv = T˙u v + T˙v u + R(u, v),

(2.1)

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where T˙u v 



˙ jv S˙ j−1 uΔ

˙ and R(u, v) 

j∈Z



˙ v ˙ j uΔ Δ j

˙ v  with Δ j



˙ k v. Δ

|j−k|≤1

j∈Z

Bernstein’s inequalities are fundamental in the analysis involving Besov spaces and these inequalities trade integrability for derivatives. Lemma 2.1. [7] Let B a ball and C be an annulus. There exist constants c, C > 0 such that for all k ∈ N ∪{0}, any positive real number λ and any function f ∈ Lp with 1 ≤ p ≤ q ≤ ∞, we have suppfˆ ⊂ λB ⇒ Dk f Lq = sup ∂ α f Lq ≤ C k+1 λk+d( p − q ) f Lp ,

(2.2)

suppfˆ ⊂ λC ⇒ C −k−1 λk f Lp ≤ Dk f Lp ≤ C k+1 λk f Lp .

(2.3)

suppfˆ ⊂ λC ⇒ etΔ f Lp ≤ Ce−cλt f Lp .

(2.4)

1

1

|α|=k

3. Key a priori estimates First, we establish the following decay estimate in terms of the norm ||θ||Lp with 2 ≤ p < ∞ which is crucial to obtain the basic energy estimate for u. Proposition 3.1. Let (θ, u) be a smooth solution to the following 2D transport-diffusion system with α ∈ (0, 1) ⎧ α ⎪ ⎨ ∂t θ + u · ∇θ + Λ θ = 0, divu = 0, ⎪ ⎩ θ(0, x) = θ (x). 0

(3.1)

Then there exists some constant C which depends only on p and α such that ||θ||Lp ≤ C t − α (1− p ) ||θ0 ||L1 ∩Lp 2

1

2 ≤ p < ∞.

for

(3.2)

Especially, it holds that ||θ||Lp ≤ ||θ0 ||Lp

for

1 ≤ p ≤ ∞.

(3.3)

Proof. The proof follows from that of Remark 2.7 in [12] with minor modifications. Taking the inner product of Equ. (3.1) with |θ|p−2 θ and using the fact divu = 0 yields 1 d ||θ||pLp + p dt

Λα θ · |θ|p−2 θdx = 0. R2

Invoking the following pointwise inequality (see [12])

α

p

Λα θ · |θ|p−2 θdx ≥ C||Λ 2 (θ) 2 ||2L2 R2 4 α and combining the Sobolev embedding H˙ 2 (R2 ) → L 2−α (R2 ), we deduce

1 d ||θ||pLp + C||θ||p 2p ≤ 0. p dt L 2−α

(3.4)

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Note that the interpolation 2(p−1)

α

||θ||Lp  ||θ||L2(p−1)+α ||θ|| 2(p−1)+α 1 2p L 2−α

and ||θ||L1 ≤ ||θ0 ||L1 , we obtain from (3.4) that 2p(p−1)+pα − pα 1 d ||θ||pLp + C||θ0 ||L12(p−1) ||θ||Lp 2(p−1) ≤ 0, p dt

which reduces to

− pα − pα 2(p − 1) d ||θ||Lp2(p−1) ≥ C||θ0 ||L12(p−1) . pα dt Integrating in time over [0, t] gives −

pα

−

pα

||θ||Lp2(p−1) ≥ C||θ0 ||L12(p−1) (e + t), which is nothing but (3.2). This ends the proof of Proposition 3.1.



Proposition 3.2. Let (θ, u) be a smooth solution to (1.1) on [0, T ∗ ). Then under the assumptions (1.4) and (1.5), we have t ||u||2L∞ 2 + t L

||∇u||2L2 ds ≤ CE0 ,

(3.5)

0

where we denote E0  ||u0 ||2L2 + ||θ0 ||2L1 ∩L2 . Proof. Taking the L2 inner product of (1.1)1 with u and using the fact divu = 0, we obtain 1 d ||u||2L2 + ||∇u||2L2 ≤ 2 dt

θu2 dx ≤ ||θ||L2 ||u||L2 ,

(3.6)

R2

which reduces to d ||u||L2 ≤ ||θ||L2 . dt Integrating in time over [0, t] and combining (3.2) yields t ||u||L2 ≤ ||θ0 ||L1 ∩L2

s − α ds ≤ C||θ0 ||L1 ∩L2 . 1

(3.7)

0

Inserting (3.7) into (3.6) and using (3.2) once again implies that the desired result (3.5). This completes the proof of Proposition 3.2. 

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Proposition 3.3. Let (θ, u) be a smooth solution to (1.1) on [0, T ∗ ). Then under the assumptions (1.4) and (1.5), we have   α 2 ||ut ||2L2t L2 + ||∇u||2L∞ ≤ CE exp C||Λ θ|| , 2 2 2 0 Lt L t L

(3.8)

where   E0  (||∇u0 ||2L2 + ||θ0 ||2L1 ∩L2 ) exp C ||θ0 ||L1 ∩L2 2 E0 . Proof. Taking the L2 inner product of Equ. (1.1)1 with ut and using the fact divu = 0, we obtain ||ut ||2L2 −

div(2μ(θ)Du) · ut dx =

R2

(−u · ∇u + θe2 ) · ut dx.

(3.9)

R2

Integrating by parts yields that −

div(2μ(θ)Du) · ut dx =

R2

2μ(θ)Du : (Du)t dx R2





d = dt

μ(θ)Du : Dudx − R2

∂t (μ(θ))Du : Dudx.

(3.10)

R2

Note that the new equation ∂t (μ(θ)) = −u · ∇μ(θ) − μ (θ)Λα θ, we deduce

∂t (μ(θ))Du : Dudx = − R2



(u · ∇(μ(θ)))Du : Dudx +

R2

R2 −1

u · ∇(μ(θ))

= R2



=



−







ui ul ∂i ∂k 2μ(θ)(Du)kl dx

2μ(θ)∂k ui ∂i ul (Du)kl dx −

R2



R2

μ (θ)Λα θDu : Dudx



−

μ (θ)Λα θDu : Dudx

R2

R2

=−



(μ(θ)Du) : (μ(θ)Du)dx −

∂k ui ul ∂i 2μ(θ)(Du)kl dx +

R2

μ (θ)Λα θDu : Dudx

u · ∇udiv(2μ(θ)Du)dx R2

μ (θ)Λα θDu : Dudx.

(3.11)

R2

Here we have used Einstein summation notation for repeated indices. Combining (3.9)–(3.11) and using the fact divu = 0, we obtain ||ut ||2L2 +

d dt



R2



μ(θ)Du : Dudx = (−u · ∇u + θe2 ) · ut dx − μ (θ)Λα θDu : Dudx R2



−

R2

2μ(θ)∂k ui ∂i ul (Du)kl dx R2

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−

u · ∇u(ut + u · ∇u + ∇Π − θe2 )dx,

(3.12)

R2

where we have used the velocity equation in (1.1), namely, div(μ(θ)Du) = ut + u · ∇u + ∇Π − θe2 . Then, applying the Hölder and Young inequalities to (3.12) gives 3 d ||ut ||2L2 + 4 dt





μ(θ)Du : Dudx  ||u · ∇u||2L2 + ||θ||2L2 + ||Λα θ||L2 + ||∇u||L2 ||∇u||2L4

R2



+ Π∂i uk ∂k ui dx .

(3.13)

R2



To deal with the last term R2 Π∂i uk ∂k ui dx , applying the operator “div” to (1.1)1 , one has Π = (−Δ)−1 div ⊗ div(2μ(θ)Du) − (−Δ)−1 div(u · ∇u − θe2 ), from which, div∂i u = ∂i divu = 0 and ∇⊥ · ∇ui = 0, applying the div-curl Lemma [11] yields that





Π∂i uk ∂k ui dx  ||∇u||L2 ||∇u||2L4 + ||(−Δ)−1 div(u · ∇u − θe2 )||BM O ||∂i uk ∂k ui ||H1 R2

 ||∇u||L2 ||∇u||2L4 + ||u · ∇u − θe2 ||L2 ||∇u||2L2 ,

(3.14)

where the fact ||f ||BM O(R2 )  ||f ||H˙ 1 (R2 ) from [7] has been used. Inserting (3.14) into (3.13) and using the Hölder and Young inequalities, one has d 3 ||ut ||2L2 + 4 dt



μ(θ)Du : Dudx

R2

 ||u||2L4 ||∇u||2L4 + (||Λα θ||L2 + ||∇u||L2 )||∇u||2L4 + ||θ||2L2 + ||∇u||4L2  (1 + ||u||L2 )(||Λα θ||L2 + ||∇u||L2 )||∇u||2L4 + ||θ||2L2 + ||∇u||4L2 ,

(3.15)

where we have used the interpolation inequality ||u||2L4 ≤ C||u||L2 ||∇u||L2 . Due to the divergence-free condition divu = 0, we deduce that ∇u = ∇(−Δ)−1 divP (2(μ(θ) − 1)Du) − ∇(−Δ)−1 divP (2μ(θ)Du).

(3.16)

From which and the following 2D Gagliardo-Nirenberg inequality 2 1− 2 √ ||f ||Lp ≤ C p||f ||Lp 2 ||∇f ||L2 p

for p ∈ [2, ∞),

we deduce that for any p ∈ [2, ∞) 2 1− 2 √ ||∇u||Lp ≤ Cp||μ(θ) − 1||L∞ ||∇u||Lp + C p||∇u||Lp 2 ||divP (2μ(θ)Du)||L2 p .

Taking ε0 sufficiently small in (1.5), we obtain for p ∈ [2, p∗ ]

2 1− 2 1− 2 1− 2 1− 2 √ ||∇u||Lp ≤ C p||∇u||Lp 2 ||ut ||L2 p + ||u||L4 p ||∇u||L4 p + ||θ||L2 p .

(3.17)

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In particular, taking p = 4 in (3.17) and using Young’s inequality leads to

||∇u||2L4 ≤ C ||∇u||L2 ||ut ||L2 + ||u||L2 ||∇u||3L2 + ||∇u||L2 ||θ||L2 .

(3.18)

Inserting (3.18) into (3.15) and then using Young’s inequality yields d 1 ||ut ||2L2 + 2 dt



μ(θ)Du : Dudx  (1 + ||u||2L2 )||∇u||4L2 + ||Λα θ||2L2 ||∇u||2L2 + ||θ||2L2 .

(3.19)

R2

By Gronwall’s inequality, (3.2) and (3.5), we obtain (3.8) and complete the proof of Proposition 3.3.



Proposition 3.4. Let (u, θ) be a smooth solution to (1.1) on [0, T ∗ ). Then under the assumptions of (1.4) and (1.5), we have ||θ||L∞ Bp,∞ α/2 + ||θ||  2 α L B t

t

p,∞

 ||θ0 ||Bp,∞ α/2 + ||θ0 ||L∞ ||∇u||L2 Lp t

(3.20)

and ||θ||2L ∞ H˙ α2 + ||θ||2L2 H˙ α t

t



||θ0 ||2H˙ α2

+ ||θ0 ||2L2 ∩L∞ ||∇u||2L2t L2 ln ||θ0 ||Bp,∞ α/2 + ||θ0 ||L∞ ||∇u||L2 Lp . t

(3.21)

˙ j to the θ-equation of system (1.1), we obtain Proof. Applying the operator Δ ˙ j θ = 0. ˙ j θ) + Δ ˙ j (u · ∇θ) + Λα Δ ∂t (Δ ˙ j θ, we get Taking the L2 inner product of the above equation with Δ α 1 d ˙ ˙ j θ||2 2 = − ||Δj θ||2L2 + ||Λ 2 Δ L 2 dt

˙ j (u · ∇θ) · Δ ˙ j θdx. Δ

(3.22)

R2

Next, we need to handle the right-hand side of (3.22). Applying Bony’s decomposition, one has ˙ u · ∇θ = T˙u ∇θ + T˙∇θ u + R(u, ∇θ). Bernstein’s inequality gives rise to ˙ j (T˙∇θ u)||L2  ||Δ



˙ k u||L2 ||S˙ k−1 ∇θ||L∞ ||Δ

|j−k|≤4





˙ k ∇u||L2 ||S˙ k−1 θ||L∞ ||Δ

|j−k|≤4

 cj ||θ||L∞ ||∇u||L2 ˙ ˙ Similarly, due to R(u, ∇θ) = divR(u, θ), we have

with

||cj ||2 (j∈Z)=1 .

(3.23)

JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.10 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

10



˙ j R(u, ˙ ||Δ ∇θ)||L2  2j

˙ k uΔ ˙ k θ||L2 ||Δ

k≥j−3



 2j

˙ k θ||L∞ ˙ k u||L2 ||Δ ||Δ

k≥j−3





2j−k ck ||∇u||L2 ||θ||L∞

k≥j−3

 cj ||∇u||L2 ||θ||L∞ .

(3.24)

Applying the commutator’s argument and divu = 0, one has





˙ j (T˙u ∇θ) · Δ ˙ j θdx = Δ

˙ k ∇θ · Δ ˙ jΔ ˙ j θdx (S˙ k−1 u − S˙ j−1 u)Δ

R2 |j−k|≤4

R2



+



˙ j , S˙ k−1 ]Δ ˙ k ∇θ · Δ ˙ j θdx [Δ

R2 |j−k|≤4



α c2j 2− 2 j ||∇u||L2 (||θ||L∞

+ ||θ||B˙ ∞,2 )||θ||H˙ α . 0

(3.25)

Collecting all the above estimates (3.23)–(3.25) and resuming them into (3.22), we obtain ||θ||2L ∞ H˙ α/2 + ||θ||2L2 H˙ α ≤ ||θ0 ||2H˙ α/2 + C(||θ||2L∞ + ||θ||2L∞ B˙ 0 )||∇u||2L2t L2 . t

t

t

∞,2

(3.26)

On one hand, by the Duhamel principle, we can rewrite the system (1.1)2 as

θ(t, x) = e

−Λα t

t θ0 −

e−Λ

α

(t−s)

(u · ∇θ)ds.

(3.27)

0

Applying the operator Δj with j ≥ −1 to (3.27) yields −ct2jα

||Δj θ||Lp ≤ e

t ||Δj θ0 ||Lp +

e−c(t−s)2 ||[Δj , u · ∇]θ||Lp ds jα

0

≤ e−ct2 ||Δj θ0 ||Lp + jα

t

e−c(t−s)2 ||∇u||Lp ||θ||L∞ ds, jα

(3.28)

0 0 where we have used the embedding L∞ → B∞,∞ and the commutator’s estimate (see [16]) 0 ||[Δj , u · ∇]θ||Lp  ||∇u||Lp ||θ||B∞,∞

with j ≥ −1 and p ∈ [1, ∞].

Invoking the convolution inequality, one has jα/2 p + 2 ||Δj θ||L∞ ||Δj θ||L2t Lp ≤ ||Δj θ0 ||Lp + 2−jα/2 ||θ0 ||L∞ ||∇u||L2t Lp . t L

(3.29)

Multiplying (3.29) by 2jα yields jα p + 2 2jα/2 ||Δj θ||L∞ ||Δj θ||L2t Lp  ||θ0 ||Bp,∞ α/2 + ||θ0 ||L∞ ||∇u||L2 Lp . t L t

Taking the supremum over j ≥ −1 implies that the desired result (3.20).

(3.30)

JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.11 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

11

On the other hand, note that for p > 4/α, using (3.30), we deduce that ||θ||L∞ ≤ ˙0 t B∞,2



˙ k θ||2 ∞ ∞ ||Δ Lt L

12



+

k≤0

≤



˙ k θ||2 ∞ ∞ ||Δ Lt L

12

+



1≤k≤N

˙ k θ||2 ∞ 2 2j ||Δ Lt L

12

+

√

≤ ||θ0 ||L2 +

√

N ||θ0 ||L∞ + 2

2 (p −α 2 )N

||θ0 ||

1

≤ (1 + ||θ0 ||L2 ∩L∞ ) ln 2 (e + ||θ0 ||

12

k≥N

N ||θ0 ||L∞ +

k≤0

˙ k θ||2 ∞ ∞ ||Δ Lt L

α

2 Bp,∞



2 p) 2( p −α)N (2j 2 ||Δj θ||L∞ t L 4

α

k≥N

α 2 Bp,∞

+ ||θ0 ||L∞ ||∇u||L2t Lp

12



+ ||θ0 ||L∞ ||∇u||L2t Lp ),

(3.31)

where we have taken a natural number in the last step  ln(e + ||θ0 ||

α + ||θ0 ||L∞ ||∇u||L2t Lp )  2 Bp,∞ ( α2 − p2 ) ln 2

N=

+1

such that 2 α 2( p − 2 )N ||θ0 ||

α 2 Bp,∞

+ ||θ0 ||L∞ ||∇u||L2t Lp ∼ 1.

Plugging (3.31) into (3.26), we obtain the desired (3.21) and finish the proof of Proposition 3.4.  With Propositions 3.3–3.4 at our hand, we can close the above estimates. Proposition 3.5. Let (u, θ) be a smooth solution to (1.1) on [0, T ∗ ). Then under the assumptions of (1.4) and (1.5), one has ||θ||2L ∞ H˙ α/2 + ln(1 + ||θ||2L∞ (I;B α/2 ) ) + ||θ||2L2 H˙ α t p,∞ t

2 ≤ C 2+C ||θ0 ||L2 ∩L∞ E0 A + B + ln(1 + ||∇u0 ||2L2 + ||θ0 ||2B α/2 ) + ||θ0 ||2H˙ α/2  C1 (3.32) p,∞

and   2 2 2 1 ∩L2 E0 + C1 ||ut ||2L2t L2 + ||∇u||2L∞ C ||θ ||  C2 ; 2 ≤ C(||∇u0 ||L2 + ||θ0 ||L1 ∩L2 ) exp 0 L L t  ||∇u||L2t Lp ≤ C E0 (1 + C2 )  C3 ;  ||θ||L 2 B α ≤ C||θ0 ||Bp,∞ C2 )  C4 , α/2 + C||θ0 ||L∞ E0 (1 + t

p,∞

(3.33) (3.34) (3.35)

where A  C ||θ0 ||L1 ∩L∞ 2 E0 and B  C ||θ0 ||L2 ∩L∞ 2 E0 ln(1 + C||θ0 ||L∞ E0 ) + A. Proof. Considering any interval I = [I − , I + ] ⊂ [0, T ∗ ), we obtain from (3.8) ||ut ||2L2 (I;L2 ) + ||∇u||2L∞ (I;L2 )

    ≤C(||∇u(I − )||2L2 + ||θ0 ||2L1 ∩L2 ) exp C ||θ0 ||L1 ∩L2 2 E0 exp C||Λα θ||2L2 (I;L2 ) .

From (3.21), one gets

(3.36)

JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.12 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

12

||θ||2L ∞ (I;H˙ α/2 ) + ||θ||2L2 (I;H˙ α ) ≤ C||θ(I − )||2H˙ α/2 + C ||θ0 ||L2 ∩L∞ 2 ||∇u||2L2 (I;L2 ) ln ||θ(I − )||Bp,∞ α/2 + ||θ0 ||L∞ ||∇u||L2 (I;Lp ) .

(3.37)

Recalling (3.18), one has ||∇u||2L2 (I;L4 ) ≤C ||∇u||L2 (I;L2 ) ||ut ||L2 (I;L2 ) + ||u||L∞ (I;L2 ) ||∇u||L∞ (I;L2 ) ||∇u||L2 (I;L2 )

+ ||∇u||L∞ (I;L2 ) ||θ||L1 (I;L2 )

≤C E0 ||ut ||L2 (I;L2 ) + ||∇u||L∞ (I;L2 ) ,

(3.38)

from which and (3.17), we get

2 2 2 2 2 1− p 1− p 1− p 1− p √ ||∇u||L2 (I;Lp ) ≤C p||∇u||Lp 2 (I;L2 ) ||ut ||L2 (I;L 2 ) + ||u||L∞ (I;L4 ) ||∇u||L2 (I;L4 ) + ||θ||L2 (I;L2 )

≤C E0 1 + ||ut ||L2 (I;L2 ) + ||∇u||L∞ (I;L2 ) .

(3.39)

Taking |I| = I + − I − sufficiently small such that C ||θ0 ||L2 ∩L∞ 2 ||∇u||2L2 (I;L2 ) ≤ 1, then, gathering (3.39), (3.36) and (3.37) together, we have ||ut ||2L2 (I;L2 ) + ||∇u||2L∞ (I;L2 )

  ≤C(||∇u(I − )||2L2 + 1) exp C ||θ0 ||L1 ∩L2 2 E0 + C||θ(I − )||H˙ α/2

× 1 + ||θ(I − )||Bp,∞ α/2 + C||θ0 ||L∞ E0 (1 + ||ut ||L2 (I;L2 ) + ||∇u||L∞ (I;L2 ) ) ,

which implies that

2 1 + ||ut ||2L2 (I;L2 ) + ||∇u||2L∞ (I;L2 ) ≤C 1 + ||∇u(I − )||2L2 + ||θ(I − )||2B α/2 p,∞   × exp 2C ||θ0 ||L1 ∩L∞ 2 E0 + 2C||θ(I − )||H˙ α/2 .

(3.40)

Taking ln to the above inequality yields

ln 1 + ||ut ||2L2 (I;L2 ) + ||∇u||2L∞ (I;L2 ) ≤



A + C ln(1 + ||∇u(I − )||2L2 + ||θ(I − )||2B α/2 ) + ||θ(I − )||2H˙ α/2 . p,∞ 2

Combining (3.40) and (3.39) leads to

  ||∇u||L2 (I;Lp ) ≤C E0 1 + ||θ(I − )||2B α/2 + ||∇u(I − )||2L2 exp A + C||θ(I − )||2H˙ α/2 . p,∞

From which and (3.37), we infer

||θ||2L ∞ (I;H˙ α/2 ) + ||θ||2L2 (I;H˙ α ) ≤ B + C ln(1 + ||∇u(I − )||2L2 + ||θ(I − )||2B α/2 ) + ||θ(I − )||2H˙ α/2 . p,∞

(3.41)

JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.13 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

13

We deduce from (3.20) and (3.41) that ln(1 + ||θ||2L∞ (I;B α/2 ) ) ≤ p,∞



A + C ln(1 + ||∇u(I − )||2L2 + ||θ(I − )||2B α/2 ) + ||θ(I − )||2H˙ α/2 . p,∞ 2

(3.42)

To close the above estimates, we adopt an argument from [4]. Since the remain process is standard, we omit the details.  Proposition 3.6. Under the assumptions of Theorem 1.1, then one has ||u||L1t B˙ ∞,1 ≤ CC5 , 1

(3.43)

where C5  ||u0 ||B˙ −1 + t||θ0 ||L1 ∩L∞ + E0 (1 + ||θ0 ||L1 ∩L2 ) + C3 (||θ0 ||L1 ∩L2 + C4 ). ∞,1

Proof. By the Duhamel principle, we can write (1.1)1 as the following integral form: t u(t, x) = e

tΔ

e(t−s)Δ P (div(μ(θ) − 1)Du − u · ∇u + θe2 )ds.

u0 +

(3.44)

0

˙ j to (3.44), we obtain Applying the operator Δ t ˙ j u(t) = etΔ Δ ˙ j u0 + Δ

˙ j (div(μ(θ) − 1)Du − u · ∇u + θe2 )ds. e(t−s)Δ P Δ 0

From which and (2.4), we deduce that

˙ j u||L∞ ≤ e ||Δ

−ct22j

t ˙ j u0 ||L∞ + ||Δ

˙ j (div(μ(θ) − 1)Du − u · ∇u + θe2 )||L∞ ds. e−c(t−s)2 ||P Δ 2j

0

Integrating in time over [0, t] yields ˙ j u||L1 L∞  2−2j ||Δ ˙ j u0 ||L∞ + 2−2j ||P Δ ˙ j (div(μ(θ) − 1)Du − u · ∇u + θe2 )||L1 L∞ . ||Δ t t Multiplying the above inequality by 2j and taking 1 (j ∈ Z) norm implies −1 + ||(μ(θ) − 1)Du|| 1 ˙ 0 −1 + ||θ|| 1 ˙ −1 . ||u||L1t B˙ ∞,1  ||u0 ||B˙ ∞,1 1 Lt B∞,1 + ||u · ∇u||L1t B˙ ∞,1 Lt B∞,1         

I1

I2

(3.45)

I3

For the term I2 , by Bony’s decomposition, we obtain that (for more details see [4]) 1

1

||u · ∇u||L1t B˙ −1  ||∇u||2L2t L2 + ||u||L2 ∞ L2 ||∇u||L2t L2 ||u||L2 1 B˙ 1 ∞,1

t

t

∞,1

1 2 2 )||∇u|| 2 2 + ||u||L1t B˙ ∞,1  (1 + ||u||L∞ 1 Lt L t L 4 1  (1 + ||θ0 ||L1 ∩L2 )E0 + ||u||L1t B˙ ∞,1 . 1 4 For the term I3 , we estimate it as follows

(3.46)

JID:YJMAA 14

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.14 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

−1 ≤ ||θ||L1t B˙ ∞,1



˙ j θ||L1 L∞ + 2−j ||Δ t

j≤0







˙ j θ||L1 L∞ 2−j ||Δ t

j≥0

˙ j θ||L1 L1 + 2 ||Δ t j

j≤0



˙ j θ||L1 L∞ 2−j ||Δ t

j≥0

 ||θ||L1t L1 + ||θ||L1t L∞  t||θ0 ||L1 ∩L∞ .

(3.47)

Next, we focus on the estimation of ||(μ(θ) − 1)Du||L1t B˙ ∞,1 . 0 Applying Bony’s decomposition, we write ˙ (μ(θ) − 1)Du = T˙μ(θ)−1 Du + T˙Du (μ(θ) − 1) + R(μ(θ) − 1, Du). By the Hölder inequality, one has ||T˙μ(θ)−1 Du||L1t B˙ ∞,1  0

 

˙ k Du||L1 L∞ ||S˙ k−1 (μ(θ) − 1)Δ t

j∈Z |j−k|≤4

 ||μ(θ) − 1||L∞ ||u||L1t B˙ ∞,1 . 1

(3.48)

To handle with T˙d (μ(θ) − 1), we decompose it as ||T˙d (μ(θ) − 1)||L1t B˙ ∞,1  0



˙ j T˙Du (μ(θ) − 1)||L1 L∞ + ||Δ t

j≤0









˙ j T˙Du (μ(θ) − 1)||L1 L∞ . ||Δ t

j≥0





J1

J2

Using Bernstein and Hölder’s inequalities yields J1 

 

2

(p+2)k p

˙ k (μ(θ) − 1)|| ||S˙ k−1 DuΔ



 

2

2p

L1t L p+2

j≤0 |j−k|≤4 (p+2)k p

˙ k (μ(θ) − 1)||L2 L2 ||∇u||L2t Lp ||Δ t

j≤0 |j−k|≤4

||μ(θ) − 1||L2t L2 ||∇u||L2t Lp ||θ0 ||2L1 ∩L2 C3 , and J2 

 

˙ k (μ(θ) − 1)||L1 L∞ ||S˙ k−1 DuΔ t

j≥0 |j−k|≤4



 

4k ˙ k (μ(θ) − 1)||L2 Lp 2 p ||∇u||L2t Lp ||Δ t

j≥0 |j−k|≤4



 

2( p −α)k ||∇u||L2t Lp ||μ(θ) − 1||L 2 B˙ α 4

t

j≥0 |j−k|≤4

||∇u||L2t Lp ||θ||L 2 B˙ α t

p,∞

C3 C4 . Gathering all the above estimations together, we obtain

p,∞



JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.15 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

15

||T˙Du (μ(θ) − 1)||L1t B˙ ∞,1 C3 (||θ0 ||2L1 ∩L2 + C4 ). 0

(3.49)

˙ To deal with R(μ(θ) − 1, Du), we decompose it as 

˙ ||R(μ(θ) − 1, Du)||L1t B˙ ∞,1  0

˙ j R(μ(θ) ˙ − 1, Du)||L1t L∞ + ||Δ

j≤0









˙ j R(μ(θ) ˙ − 1, Du)||L1t L∞ . ||Δ

j≥0





K1



K2

For the term K1 , we rewrite it as follows: ˙ j R(μ(θ) − 1, Du) = (μ(θ) − 1)Du − T˙μ(θ)−1 Du − T˙Du (μ(θ) − 1). Δ Then we have K1 



2

(p+2)j p

˙ j (μ(θ) − 1)Du − T˙μ(θ)−1 Du − T˙Du (μ(θ) − 1) || ||Δ ˙ j (T˙μ(θ)−1 Du)|| + sup ||Δ

||(μ(θ) − 1)Du||

2p L1t L p+2

j∈Z

||μ(θ) − 1||L2t L2 ||∇u||L2t Lp + sup j∈Z

+ sup j∈Z

2p

L1t L p+2

j≤0



2p L1t L p+2



˙ j (T˙Du (μ(θ) − 1))|| + ||Δ

˙ k ∇u|| ||S˙ k−1 (μ(θ) − 1)Δ

2p

L1t L p+2

|j−k|≤4

˙ k (μ(θ) − 1)|| ||S˙ k−1 ∇uΔ



2p L1t L p+2

2p

L1t L p+2

|j−k|≤4

||μ(θ) − 1||L2t L2 ||∇u||L2t Lp ||θ0 ||2L1 ∩L2 C3 . Along the same line, we have K2 

 

˙ k (μ(θ) − 1)||L2 L∞ ||Δ ˙ k ∇u||L2 L∞ ||Δ t t

j≥0 k≥j−3



 

4 ˙ k (μ(θ) − 1)||L2 Lp ||∇u||L2 Lp 2( p −α)k 2kα ||Δ t t

j≥0 k≥j−3

||θ||L 2 B˙ α ||∇u||L2t Lp t

p,∞

C3 C4 . As a consequence, we get ˙ ||R(μ(θ) − 1, Du)||L1t B˙ ∞,1  C3 (||θ0 ||L1 ∩L2 + C4 ). 0

(3.50)

Combining (3.48)–(3.50), one obtains ||(μ(θ) − 1)∇u||L1t B˙ ∞,1  ||μ(θ) − 1||L∞ ||u||L1t B˙ ∞,1 + C3 (||θ0 ||2L1 ∩L2 + C4 ). 0 1

(3.51)

Resuming (3.46), (3.47), (3.51) into (3.45) and taking ε0 sufficiently small in (1.5) yields 2 −1 + t||θ0 ||L1 ∩L∞ + E0 (1 + ||θ0 ||L1 ∩L2 ) + C3 (||θ0 || 1 ||u||L1t B˙ ∞,1  ||u0 ||B˙ ∞,1 1 L ∩L2 + C4 ).

This ends the proof of Proposition 3.6.



(3.52)

JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.16 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

16

t With the crucial estimation 0 ||∇u||L∞ ds ≤ C at our hand, we next prepare the following global a priori estimates which are essential to prove the Theorem 1.1. Proposition 3.7. Let (u, θ) be a smooth solution to (1.1). Under the assumptions of Theorem 1.1, then there exists a positive constant C which depends on ||μ||W 2,∞ , one has t ||∇θ||2L∞ 2 t L

+

  α ||Λ1+ 2 θ||2L2 ds ≤ C||∇θ0 ||2L2 exp CC5 .

(3.53)

0

Proof. Taking the xi -derivative of the θ equations in (1.1), one has ∂i θt + u · ∇∂i θ + ∂i Λα θ = −∂i u · ∇θ.

(3.54)

Taking the L2 inner product of (3.54) with ∂i θ, using the fact divu = 0 and summing over i, we obtain α 1 d ||∇θ||2L2 + ||Λ1+ 2 θ||2L2 ≤ ||∇u||L∞ ||∇θ||2L2 , 2 dt

which together with (3.43) and the Gronwall inequality yields that (3.53).  4. Proof of Theorem 1.1 4.1. Existence and uniqueness With the global a priori estimations at disposal, by making use of a standard compactness argument of Lions-Aubin Lemma, we can establish the global existence of system (1.1). Since the procedure is rather standard, we omit it here for simplicity. For more details, one may refer similar argument in [1]. It remains to prove the uniqueness of Theorem 1.1. Indeed, let (ui , θi , ∇Πi ), i = 1, 2, be two global smooth solutions to system (1.1) with the same initial data u0 , θ0 . We denote (u, θ, ∇P )  (u2 − u1 , θ2 − θ1 , ∇Π2 − ∇Π1 ). Obviously, from (1.1), the difference pair (u, θ, ∇P ) satisfies ⎧ = θe2 − u · ∇u1 ⎪ ⎪ ∂t u + u2 · ∇u − 2div(μ(θ2 )Du) + ∇P ⎪

⎪ ⎪ ⎪ ⎪ + 2div (μ(θ2 ) − μ(θ1 ))Du1 , ⎪ ⎨ ∂t θ + u2 · ∇θ + Λα θ = −u · ∇θ1 , ⎪ ⎪ ⎪ ⎪ ⎪ divu = 0, ⎪ ⎪ ⎪ ⎩ (u(0, x), θ(0, x)) = (0, 0). Taking the L2 inner product of (4.1)2 with θ yields α 1 d ||θ||2L2 + ||Λ 2 θ||2L2 = 2 dt



Λ− 2 (u · ∇θ1 ) · Λ 2 θdx α

α

R2

≤ ||u · ∇θ1 ||

α

4

L 2+α

||Λ 2 θ||L2

α

≤ ε||Λ 2 θ||2L2 + ||∇θ1 ||2L2 ||u||2

4

Lα

(4.1)

JID:YJMAA

AID:123668 /FLA

Doctopic: Partial Differential Equations

[m3L; v1.279; Prn:21/11/2019; 11:15] P.17 (1-20)

Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

17

4

α

≤ ε||Λ 2 θ, ∇u||2L2 + ||∇θ1 ||Lα2 ||u||2L2 α

≤ ε||Λ 2 θ, ∇u||2L2 + C||u||2L2 ,

(4.2)

where we have used the facts in R2 α

f H˙ −α/2  f L4/(2+α)

1− α

and ||f ||L4/α  ||f ||L22 ||∇f ||L2 2 .

Taking the L2 inner product of (4.1)1 with u yields for p ∈ (4/α, p∗ ] 1 d ||u||2L2 + ||∇u||2L2 ≤ 2 dt



θe2 − u · ∇u1 + 2div((μ(θ2 ) − μ(θ1 ))Du1 ) · udx

R2

 (1 + ||∇u1 ||L∞ )||u, θ||2L2 + ||θ||

2p

 (1 + ||∇u1 ||L∞ )||u, θ||2L2 + ||θ||2

2p

L p−2 L p−2

||∇u1 ||Lp ||∇u||L2 ||∇u1 ||2Lp + ε||∇u||2L2 .

(4.3)

Combining (4.2) and (4.3), we infer that t ||θ, u||2L∞ 2 t L

+

||∇u||2L2t L2



(1 + ||∇u1 ||L∞ )||u, θ||2L2 ds + ||θ||2

2p

||∇u1 ||2L2t Lp

(1 + ||∇u1 ||L∞ )||u, θ||2L2 ds + ||θ||2

2p

.

p−2 L∞ t L

0

t 

p−2 L∞ t L

0

(4.4)

In view of θ-equation in (4.1), by the Hölder inequality, we deduce that t ||θ||

2p

p−2 L∞ t L

≤

||u||

4p

L pα−4

||∇θ1 ||

4

L 2−α

ds

0

≤ ||u||

4p L2t L pα−4

α

||Λ1+ 2 θ1 ||L2t L2

≤ C||u||L2t L2 + ε||∇u||L2t L2 .

(4.5)

Inserting (4.5) into (4.4), due to the zero initial data and Gronwall’s inequality, we get θ = u = 0, which implies the uniqueness part of Theorem 1.1. 4.2. Algebraic decay estimate for ||u||L2 In this subsection, we present the algebraic decay estimate for ||u||L2 . Proposition 4.1. Let θ0 ∈ L1 ∩ L2 (R2 ) and u0 ∈ L2 ∩ H˙ α−1 (R2 ), then under the assumptions (1.4) and (1.5), we have ||u||L2 ≤ CH0 t α−1 .

(4.6)

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Doctopic: Partial Differential Equations

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Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

18

Proof. Note that (3.6) and (3.7), we have 1 d ||u||2L2 + ||∇u||2L2  ||θ0 ||2L1 ∩L2 t − α . dt

(4.7)

Applying Schonbek’s strategy for the classical NS equations in [23], by splitting the phase space R2 into time-dependent domain R2 = S(t) ∪ S c (t), where S(t)  {ξ ∈ R2 : |ξ| ≤ r(t)} for some r(t), which will be chosen later on. Then we deduce from (4.7) that d ||u||2L2 + r2 (t)||u||2L2 ≤r2 (t) dt



|ˆ u(t, ξ)|2 dξ + C||θ0 ||2L1 ∩L2 t − α . 1

(4.8)

S(t)

Taking the Fourier transform with respect to x variables to (3.44), one has

−t|ξ|2

|ˆ u(t, ξ)| ≤e

t |ˆ u0 (ξ)| +

e−(t−s)|ξ|

2



|ξ||Fx ((μ(θ) − 1)Du + u ⊗ u)| + |Fx (θe2 )| ds

e−(t−s)|ξ|

2



ˆ ξ)| ds + | θ(t, |ξ|Fx ((μ(θ) − 1)Du + u ⊗ u)L∞ ξ

0 −t|ξ|2

≤e

t |ˆ u0 (ξ)| + 0

−t|ξ|2

≤e

t |ˆ u0 (ξ)| + |ξ|

t ˆ ξ)|ds. |θ(t,

(μ(θ) − 1)Du + u ⊗ uL1 ds + 0

(4.9)

0

Taking the Fourier transform with respect to x variables to (3.27), one has

ˆ ξ)| ≤e−t|ξ|α |θˆ0 (ξ)| + |θ(t,

t

α

e−(t−s)|ξ| |ξ| Fx (uθ) ds

0 −t|ξ|α

≤e

||θ0 ||L1 + |ξ|e−(·)|ξ| ∗s ||I[0,t] (uθ)(·)||L1 α

Applying Young’s inequality with respect to t variable yields t

2 t

2

2 t α −s|ξ| 2 2(1−α)  ξ)|ds  |θ(s, e ds ||θ0 ||L1 + |ξ| ||uθ||L1 ds

0

0

0

|ξ|−2α + |ξ|2(1−α)

t

t ||u||2L2 ds

0

||θ||2L2 ds. 0

Combining (4.10) and (4.9), one has

|ˆ u(t, ξ)| dξ  2

S(t)

S(t)

e

−2t|ξ|2

t

|ˆ u0 (ξ)| dξ + r (t) 2

4

0

2 ||(μ(θ) − 1)Du + u ⊗ u||L1 ds

(4.10)

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Doctopic: Partial Differential Equations

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Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

t +

 t

2  |θ(ξ)|ds dξ

0

S(t) α−1

19

||u0 ||2L2 ∩H˙ α−1

1)||2L2t L2 ||∇u||2L2t L2

+ r (t)||μ(θ) − 4

4

t

+ r (t)

2 ||u||2L2 ds

0



|ξ|−2α dξ||θ0 ||2L1 + r4−2α (t)

+

t ||u||2L2 ds 0

S(t)

 t

α−1

t

||u0 ||2L2 ∩H˙ α−1

+r

4

||θ||2L2 ds 0

(t)||θ0 ||2L1 ∩L2 E0

t

4

+ r (t)

2 ||u||2L2 ds

0

t + r2(1−α) (t)||θ0 ||2L1 + r4−2α (t)||θ0 ||2L1 ∩L2

||u||2L2 ds 0



H0 t

α−1

+r

2(1−α)

(t) + r

4(1−α)

t

4

(t) + r (t)

2

||u||2L2 ds

(4.11)

0



H0 t α−1 + r2(1−α) (t) + r4(1−α) (t) + r4 (t) t 2 .

(4.12)

Combining (4.8) and (4.12) yields that



t 2 1 d  t r2 (s)ds e0 ||u||2L2 H0 e 0 r (s)ds r2 (t) t α−1 + r4−2α (t) + r6−4α (t) + r6 (t) t 2 + t − α . (4.13) dt Taking r2 (t) =

3 t ln t

in the above inequality and integrating the resulting over [0, t], one has t

ln

3

t ||u||2L2

||u0 ||2L2

+ H0

ln3 s

0

t ||u0 ||2L2

+ H0 0





1 s 2−α ln s

+



2−α 1 1 1 + + s − α ds 3 s ln s

s ln s

ln1+α s

ln3 s  1  ln2 s

+ + 1 + ds s s 1−α s 1−α s (1−α)/α

H0 ln t ,

(4.14)

which implies that 1

||u(t)||L2 ≤CH02 ln−1 t .

(4.15)

Combining (4.11) and (4.8), then taking r2 (t) = β t −1 with β ∈ (1 − α, 1), we have t

β

||u(t)||2L2



t

H0 t

β+α−1

s

β−2

+ 0

ln

−2

s

s



||u(τ )||2L2 dτ ds ,

0

where we have used the following estimate which can be derived from (4.15) t 0

||u(s)||2L2 ds ≤ CH0 t ln−2 t .

(4.16)

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Doctopic: Partial Differential Equations

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Y. Yu, M. Zhou / J. Math. Anal. Appl. ••• (••••) ••••••

20

To obtain the algebraic decay estimate for ||u||L2 , we adopt the standard iteration argument from [25]. Here we omit the details. Then we complete the proof of Proposition 4.1.  Acknowledgments Y. Yu is supported by the Natural Science Foundation of Anhui Province (No. 1908085QA05). References [1] H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Rev. Mat. Iberoam. 23 (2) (2007) 537–586. [2] H. Abidi, Sur l’unicit pour le système de Boussinesq avec diffusion non linéaire, J. Math. Pures Appl. 91 (1) (2009) 80–99. [3] H. Abidi, T. Hmidi, On the global well-posedness for Boussinesq system, J. Differential Equations 233 (1) (2007) 199–220. [4] H. Abidi, P. Zhang, On the global well-posedness of 2D Boussinesq system with variable viscosity, Adv. Math. 305 (2017) 1202–1249. [5] D. Adhikari, C. Cao, J. Wu, The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Differential Equations 249 (5) (2010) 1078–1088. [6] D. Adhikari, C. Cao, J. Wu, Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differential Equations 251 (6) (2011) 1637–1655. [7] H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. [8] J.R. Cannon, E. DiBenedetto, The initial problem for the Boussinesq equations with data in Lp , Lecture Notes in Math. 771 (2) (1980) 129–144. [9] C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal. 208 (3) (2013) 985–1004. [10] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2) (2006) 497–513. [11] R.R. Coifman, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (3) (1993) 247–286. [12] A. Córdoba, D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004) 511–528. [13] T. Hmidi, S. Keraani, Global well-posedness result for two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations 12 (4) (2007) 461–480. [14] T. Hmidi, S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, J. Indiana Univ. Math. 58 (4) (2009) 1591–1618. [15] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations 249 (9) (2010) 2147–2174. [16] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations 36 (3) (2011) 420–445. [17] T. Hou, C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst. 12 (1) (2005) 1–12. [18] A. Larios, E. Lunasin, E.S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations 255 (9) (2013) 2636–2654. [19] D. Li, X. Xu, Global well-posedness of an inviscid 2-D Boussinesq system with nonlinear thermal diffusivity, Dyn. Partial Differ. Equ. 10 (3) (2013) 255–265. [20] J. Li, E.S. Titi, Global well-posedness of the 2D Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal. 220 (3) (2015) 983–1001. [21] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, New York, 2003. [22] J. Pedloski, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [23] M.E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11 (7) (1986) 733–763. [24] C. Wang, Z. Zhang, Global well-posedness for the 2D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math. 228 (1) (2011) 43–62. [25] M. Wiegner, Decay results for weak solutions of the Navier-Stokes equations on Rn , J. Lond. Math. Soc. 35 (2) (1987) 303–313. [26] G. Wu, X. Zheng, Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation, J. Differential Equations 255 (9) (2013) 2891–2926. [27] J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Comm. Math. Phys. 263 (3) (2005) 803–831. [28] X. Wu, Y. Yu, Y. Tang, Global existence and asymptotic behavior for the 3D generalized Hall-MHD system, Nonlinear Anal. 151 (2017) 41–50. [29] Y. Yu, X. Wu, Y. Tang, Global well-posedness for the 2D Boussinesq system with variable viscosity and damping, Math. Methods Appl. Sci. 41 (8) (2018) 3044–3061.