Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation

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Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation Zhuan Ye Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou 221116, Jiangsu, PR China Received 9 December 2018; revised 17 August 2019; accepted 19 August 2019

Abstract This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional α dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α > 0 always possesses a unique global classical solution. © 2019 Elsevier Inc. All rights reserved. MSC: 35Q35; 35B65; 76B03 Keywords: Boussinesq equations; Micropolar equations; Global regularity; Partial dissipation

E-mail address: [email protected]. https://doi.org/10.1016/j.jde.2019.08.037 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction and main results The Boussinesq equations are commonly used to describe natural convection phenomena in geophysics [33,30], as for example in the very important Rayleigh-Bénard problem [8]. The first goal of this paper is to consider the 2D incompressible Boussinesq equations with fractional vertical dissipation in the horizontal velocity, fractional horizontal dissipation in the vertical velocity and zero thermal diffusion, which take the following form ⎧ ∂t u1 + (u · ∇)u1 + x2 u1 + ∂x1 p = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t u2 + (u · ∇)u2 + x1 u2 + ∂x2 p = θ, ⎨ ∇ · u = 0, ⎪ ⎪ ⎪ ⎪ ∂t θ + (u · ∇)θ = 0, ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), θ (x, 0) = θ0 (x),

(1.1)

where u = u(x, t) = (u1 (x, t), u2 (x, t)) denotes the velocity field, p= p(x, t) the pressure, θ = θ (x, t) the temperature. The fractional horizontal operator x1 := −∂x21 and the fractional  vertical operator x2 := −∂x22 are defined through the Fourier transform, namely   x1 f (ξ ) = |ξ1 |fˆ(ξ ),

  x2 f (ξ ) = |ξ2 |fˆ(ξ ),

ξ = (ξ1 , ξ2 ).

Due to the physical background and mathematical relevance, the 2D Boussinesq equations with zero thermal diffusion have been widely studied and considerable attention has been dedicated recently to their well-posedness and regularity. We now summarize some previous works on the 2D Boussinesq equations with zero thermal diffusion. To facilitate the description, we first state the following general form of the system (1.1) ⎧ ∂t u1 + (u · ∇)u1 + ν1 L1 u1 + ∂x1 p = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ∂ u + (u · ∇)u2 + ν2 L2 u2 + ∂x2 p = θ, ⎪ ⎪ ⎨ t 2 ∇ · u = 0, ⎪ ⎪ ⎪ ⎪ ∂t θ + (u · ∇)θ = 0, ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), θ (x, 0) = θ0 (x),

(1.2)

where L1 and L2 are Fourier multiplier operators, namely  L u(ξ ), 1 u(ξ ) = m1 (ξ )

 L u(ξ ). 2 u(ξ ) = m2 (ξ )

The Boussinesq equations can also be used to model large scale atmospheric and oceanic flows, where the viscosity and diffusivity constants are usually different in the horizontal and vertical directions. Based on this direction, much effort has been devoted to the study of the global well-posedness of (1.2) (e.g. [5,23,11,4,20,2,26,28,1,10] and references therein), and here for our purpose we only recall several very related works. To begin with, Chae [5] and Hou-Li [23] independently solved the global regularity problem for the case L1 = L2 = −. Later, Danchin and

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Paicu [11] first examined the 2D Boussinesq equations (1.2) with L1 = L2 = −∂x21 and obtained the global regularity for this case (see also [26] for a sharp uniqueness result). Very recent, when the 2D Boussinesq equations (1.2) admits one of the following cases (1) L1 = −∂x22 , L2 = −∂x21 ;

(2) ν1 = 0, L2 = −,

Adhikari, etc. proved in [2] the global regularity of the corresponding systems. Here it is worthwhile pointing out that by deeply developing the structures of the coupling system, Hmidi, Keraani and Rousset [20] proved the global result for (1.2) in critical case: L1 = L2 = , where 1  := (−) 2 denotes the Zygmund operator. Here and in what follows,  and more general fractional Laplacian operators κ are defined through the Fourier transform, namely κ f (ξ ) = |ξ |κ f(ξ ).  

However, the global well-posedness of the 2D full inviscid Boussinesq equations (i.e., ν1 = ν2 = 0) remains still as an open question, mathematically analogous indeed to the incompressible axi-symmetric swirling three-dimensional Euler equations ([31]). This problem is even discussed in Yudovich’s “eleven great problems of mathematical hydrodynamics” [40]. Very recently, an important progress concerning the 2D inviscid Boussinesq equations has been made by Elgindi and Jeong [16], where they have shown finite-time singularity formation for strong solutions when the fluid domain is a sector of angle less than π . Comparing these two kind results of the two extreme cases, i.e., global-in-time existence for (1.2) with L1 = L2 =  and blowup in finite time for the full inviscid case, it is natural for us to ask whether the solutions exist globally in time or blow up in finite time for these intermediate cases. Therefore, this is the first main objective of this present paper. The first target of this paper is to show that the system (1.1) possesses a unique global classical solutions. More precisely, it is stated as follows. Theorem 1.1. Let (u0 , θ0 ) ∈ H s (R2 ) with s > 2 and ∇ · u0 = 0. Then the system (1.1) admits a unique global solution (u, θ ) such that for any given T > 0, (u, θ ) ∈ L∞ ([0, T ]; H s (R2 )),

1

u ∈ L2 ([0, T ]; H s+ 2 (R2 )).

Remark 1.1. It is obvious to see that the dissipation of the system (1.1) is much weaker compared with [2, Theorem 1.1] and [20, Theorem 1.1]. We also remark that our proof is more or less inspired by Hmidi, Keraani and Rousset [20], whose proof relies heavily on the Besov space techniques. However, we choose not to apply the Littlewood-Paley decomposition to the system itself but to use several refined commutator estimates and some facts involving the Besov spaces. In this sense, we provide an alternative proof of [20, Theorem 1.1]. The second goal of this paper is to consider the 2D micropolar equations. The modern theory of micropolar equations began over forty years ago, when C.A. Eringen [17] published his pioneering works on the micropolar fluid motion equations. The micropolar equations can describe a large number of phenomena that cannot be treated by the classical Navier-Stokes equations for viscous incompressible fluids, such as the suspensions, animal blood, liquid crystals and so on. The 2D micropolar equations take the following form

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⎧ ⎪ ∂t u + (u · ∇)u − (η1 + η2 )u + ∇p = 2η2 ∇ ⊥ w, ⎪ ⎪ ⎪ ⎪ ⎨ ∇ · u = 0,

x ∈ R2 , t > 0,

⎪ ∂t w + (u · ∇)w + 4η2 w − η3 w = 2η2 ∇ × u, ⎪ ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), w(x, 0) = w0 (x),

(1.3)



where u = u(x, t) = u1 (x, t), u2 (x, t) denotes the velocity vector field, w(x, t) describes the micro-rotation velocity, p(x, t) denotes the hydrostatic pressure. The parameter η1 ≥ 0 is the kinematic viscosity, η2 > 0 is the dynamics micro-rotation viscosity, and η3 ≥ 0 is the angular viscosity (see Lukaszewicz [29]). In the 2D case, we use the notation ∇ ⊥ w := (∂x2 w, −∂x1 w),

∇ × u := ∂x1 u2 − ∂x2 u1 .

Due to their mathematical and physical importance, the 2D micropolar equations have attracted considerable attention from the community of mathematical fluids (see [6,15,13,12,14, 18,29,36,37,41]). For 2D micropolar equations (1.3) with full viscosity (i.e., all the viscous coefficients are positive), the global well-posedness of smooth solution has been obtained by Lukaszewicz [29]. Recently, some efforts are devoted to studying the global regularity of the 2D micropolar equations with partial dissipation. Dong and Zhang [15] examined (1.3) with zero micro-rotation viscosity (i.e., η3 = 0), and obtained the global regularity result. Xue [36] proved the global well-posedness in the frame work of Besov spaces for (1.3) with η1 = 0, η2 > 0, η3 > 0 and η2 = η3 . The restriction η2 = η3 is not a key one and the global well-posedness remains true when η2 = η3 . Quite recently, the existence and uniqueness of classical solutions to the 2D micropolar equations with only angular velocity dissipation (the term (η1 + η2 )u in the first equation of (1.3) is absent) were established by [13]. Very recently, the global regularity of 2D micropolar equations with fractional dissipation was obtained when the fractional powers are restricted to some certain regions (see [14]). In certain physical regimes and under suitable scaling, the full Laplacian dissipation can be reduced to a partial dissipation. One remarkable example is the Prandtl boundary layer equation in which only the vertical dissipation is included in the horizontal component, see [34] for details. Actually, the anisotropic fluid equations are widely used in geophysical fluid dynamics as a mathematical model for water flow in lakes and oceans, and also in the study of Ekman boundary layers for rotating fluids (see [19,33] for example). The 2D micropolar equations under consideration assume the form ⎧ ∂t u1 + (u · ∇)u1 − ∂x22 u1 + ∂x1 p = ∂x2 w, ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ∂t u2 + (u · ∇)u2 − ∂x1 u2 + ∂x2 p = −∂x1 w, ⎨ (1.4) ∇ · u = 0, ⎪ ⎪ ⎪ ⎪ ∂t w + (u · ∇)w + w + α w = ∂x1 u2 − ∂x2 u1 , ⎪ ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), w(x, 0) = w0 (x), where u1 and u2 are the horizontal and vertical components of u, respectively. We make the convention that by α = 0 we mean that there is no dissipation in the forth equation of (1.4). The equations of the velocity contain only vertical dissipation in the horizontal velocity equation and horizontal dissipation in the vertical velocity equation. We attempt to prove the global regularity

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of solutions of (1.4) with arbitrarily small fractional powers of the Laplacian (i.e., α > 0) by fully exploiting the special structure of this system. Here we would like to mention that the small α is the main focus of this paper, thus we may assume α ∈ (0, 1). Now let us state the second main result of this paper as follows. Theorem 1.2. Let (u0 , w0 ) ∈ H s (R2 ) with s > 2 and ∇ · u0 = 0. Then the system (1.4) with α > 0 admits a unique global solution (u, w) such that for any given T > 0, (u, w) ∈ L∞ ([0, T ]; H s (R2 )) u ∈ L2 ([0, T ]; H s+1 (R2 )),

α

w ∈ L2 ([0, T ]; H s+ 2 (R2 )).

Remark 1.2. We remark that the key reason we need the positive power α > 0 is to show the boundedness of w(t)L∞ (see Proposition 4.4). As a matter of fact, when α = 0, the w equation of (1.4) reduces to the pure transport equation ∂t w + (u · ∇)w + w = G +

∂x41

2 w, + ∂x42

see (4.4) for the definition of G. It is worthwhile to mention that the following estimate only holds true for p ∈ (1, ∞), but fails for p = ∞,1   

 2  f  ≤ C(p)f Lp . ∂x41 + ∂x42 Lp

As a result, this prevents us from obtaining the desired boundedness of w(t)L∞ . Therefore, it would be interesting and challenging to derive the global regularity result for the system (1.4) with α = 0. This is left for the future. The paper is organized as follows. In Section 2, we collect some useful facts such as Littlewood-Paley analysis, some commutator estimates and so on. Section 3 is devoted to the proof of Theorem 1.1. Finally, in section 4, we shall complete the proof of Theorem 1.2. 2. Preliminaries This section collects the definition of Besov spaces and several inequalities. Let us firstly recall some basic facts about the Littlewood-Paley decomposition. We choose some smooth radial non increasing function χ with values in [0, 1] such that χ ∈ C0∞ (Rn ) is supported in the ball B :=

{ξ ∈ Rn , |ξ | ≤ 43 } and with value 1 on {ξ ∈ Rn , |ξ | ≤ 34 }, then we set ϕ(ξ ) = χ ξ2 − χ(ξ ). One easily verifies that ϕ ∈ C0∞ (Rn ) is supported in the annulus C := {ξ ∈ Rn , 34 ≤ |ξ | ≤ 83 } and satisfy χ(ξ ) +



ϕ(2−j ξ ) = 1,

∀ξ ∈ Rn .

j ≥0

1 Thanks to Prof. Dong Li for pointing this out to me in a private communication.

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Let h = F −1 (ϕ) and h = F −1 (χ), then we introduce the dyadic blocks j of our decomposition by setting  j u = 0, j ≤ −2;

h(y)u(x − y) dy;

−1 u = χ(D)u = Rn



j u = ϕ(2−j D)u = 2j n

h(2j y)u(x − y) dy, ∀j ∈ N.

Rn

We shall also use the following low-frequency cut-off: Sj u = χ(2

−j



D)u =

 k u = 2

−1≤k≤j −1

jn

h(2j y)u(x − y) dy, ∀j ∈ N.

Rn

Meanwhile, we define the homogeneous dyadic blocks as ˙ j u = ϕ(2−j D)u = 2j n 

 h(2j y)u(x − y) dy, ∀j ∈ Z.

Rn

We denote the function spaces of rapidly decreasing functions by S(Rn ), tempered distributions by S  (Rn ), and polynomials by P(Rn ). Let us now recall the definition of inhomogeneous and homogeneous Besov spaces through the dyadic decomposition. s is defined Definition 2.1. Let s ∈ R, (p, r) ∈ [1, +∞]2 . The inhomogeneous Besov space Bp,r  n as a space of f ∈ S (R ) such that s s < ∞}, Bp,r = {f ∈ S  (Rn ); f Bp,r

where

s = f Bp,r

⎧ 1 r ⎪ j rs r ⎪ 2  f  , p ⎪ j L ⎨

for r < ∞,

j ≥−1

⎪ ⎪ js ⎪ ⎩ sup 2 j f Lp , j ≥−1

for r = ∞.

s is defined as Definition 2.2. Let s ∈ R, (p, r) ∈ [1, +∞]2 . The homogeneous Besov space B˙ p,r  n n a space of f ∈ S (R )/P(R ) such that s = {f ∈ S  (Rn )/P(Rn ); f B˙ s < ∞}, B˙ p,r p,r

where

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f B˙ s = p,r

⎧ 1 r ⎪ j rs ˙ r ⎪ 2   f  , p j ⎪ L ⎨

7

for r < ∞,

j ∈Z

⎪ ⎪ js ˙ ⎪ ⎩ sup 2  j f Lp , j ∈Z

for r = ∞.

We now turn to Bernstein’s inequalities, which are fundamental in the analysis involving Besov spaces and these inequalities trade integrability for derivatives (see [3, Lemma 2.1]). Lemma 2.1. Assume 1 ≤ a ≤ b ≤ ∞. Let C be an annulus and B a ball of Rn . Then it holds Suppf⊂ λB Suppf⊂ λC

1

1

k f Lb ≤ C1 λk+n( a − b ) f La , k ≥ 0;

⇒

1

1

C2 λk f Lb ≤ k f Lb ≤ C3 λk+n( a − b ) f La , k ∈ R,

⇒

where C1 , C2 and C3 are constants depending on n, k, a and b only. For the sake of the simplicity, we denote T := −

∂x1 + 3x2

3x1

or

T :=

∂x41

2 . + ∂x42

The operator T is defined by F(T f )(ξ ) = −

|ξ |2 ξ1 Ff (ξ ) |ξ1 |3 + |ξ2 |3

ξ1 Obviously, the functions − |ξ |ξ|3 |+|ξ 2

1

3 2|

and

|ξ |4 ξ14 +ξ24

or

F(T f )(ξ ) =

|ξ |4 Ff (ξ ). ξ14 + ξ24

are homogeneous of degree zero.

We collect some useful properties of this operator T as follows.

Lemma 2.2. Let j ∈ N, then the following statements hold true. (1) For any 1 < q < ∞, there holds T f Lq ≤ Cf Lq .

(2.1)

(2) Let ψ ∈ C0∞ (Rn ). For any 1 ≤ p ≤ ∞, 0 ≤ s < ∞ and f ∈ Lp (R2 ), it holds s ψ(2−j )T f Lp ≤ C2j s f Lp .

(2.2)

(3) Let C is an annulus centered at the origin, then for every function f with spectrum supported on 2j C, there exists a function φ ∈ S(R2 ) whose Fourier transform supported away from the origin, such that T f = 22j φ(2j .)  f.

(2.3)

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Proof of Lemma 2.2. The proof of (2.1) is presented in Lemma 3.1 and Lemma 4.1. The proof of (2.2) can be performed as [21, Proposition 3.1]. Finally, by choosing a suitable bump function, (2.3) follows directly. 2 With the facts of Lemma 2.2 in hand, we will establish the following commutator estimate involving T . Lemma 2.3. Let f be a divergence-free vector field, then it holds for any s ∈ (−1, 1) that s ≤ C ∇f Lp1 gB s [T , f · ∇]gBp,r p where p1 , p2 ∈ [2, ∞] and p ∈ [2, ∞) satisfy p1 = B(AC) denotes the standard commutator notation.

1 p1

+

2 ,r

+ f L2 gL2 ,

1 p2 ,

(2.4)

meanwhile [A, B]C := A(BC) −

Proof of Lemma 2.3. By the Bony decomposition, it gives k [T , f · ∇]g =

    k [T , Sj −1 f · ∇]j g + k [T , j f · ∇]Sj −1 g

|j −k|≤4



+



j g k [T , j f · ∇]



|j −k|≤4

j −k≥−4

:= N1 + N2 + N3 . According to (2.3), there exists a function φ ∈ S(R2 ) whose Fourier transform supported away from the origin such that N1 =

  k φj  (Sj −1 f · ∇j g) − Sj −1 f · (φj  ∇j g) ,

|j −k|≤4

where φj (x) = 22j φ(2j x). Recall the following commutator estimate ([39, Proposition A.3]) h  (fg) − f (h  g)Lp ≤ xhL1 ∇f Lp1 gLp2 , where f, g, h be three functions such that ∇f ∈ Lp1 , g ∈ Lp2 , xh ∈ L1 and p1 , p2 ∈ [2, ∞] and p ∈ [2, ∞) satisfy p1 = p11 + p12 . We thus obtain by using the Berstein inequality that N1 Lp ≤ Cx2kn φ(2k x)L1 ∇Sk−1 f Lp1 k ∇gLp2 ≤ C∇f Lp1 k gLp2 , which implies  ks    2 N1 Lp  r ≤ C∇f Lp1 2ks k gLp2  r ≤ C∇f Lp1 gB s . p ,r l l k

By the same manner, one has

k

2

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N2 Lp ≤ Cx2kn φ(2k x)L1 k ∇f Lp1 Sk−1 ∇gLp2 ≤ C2−k ∇f Lp1 l ∇gLp2 l≤k−2

≤ C∇f Lp1



2−(k−l) l gLp2 ,

l≤k−2

which along with the fact s < 1 leads to    ks   2 N2 Lp  r ≤ C∇f Lp1  2(k−l)(s−1) 2ls l gLp2  r  l k

lk

l≤k−2

≤ C∇f Lp1 gBps ,r . 2

Thank to the incompressibility of f , the final term N3 can be rewritten as N3 =

 

j g) − j f T 

j g k ∇ · T (j f 

j −k≥−4

 

j g) − j f T 

j g k ∇ · T (j f 



=

j −k≥−4, j ≥0

 

−1 g) − −1 f T 

−1 g k ∇ · T (−1 f 



+

−1−k≥−4

:= N31 + N32 ,

j := j −1 + j + j +1 . By the where here and in what follows we have used the notation  Berstein inequality and (2.1)-(2.3), we can check that N31 Lp ≤ C

j −k≥−4, j ≥0

≤C



j −k≥−4, j ≥0

≤C





 

j g)  2k k T (j f     

j g) 2k T (j f  

Lp

Lp



 

j g  + k j f T  

  

j g + j f T  



j gLp2 2k j f Lp1 

j −k≥−4, j ≥0

≤C



2k 2−j j ∇f Lp1 j gLp2

j −k≥−4, j ≥0

≤C



j −k≥−4

 Lp

 

j gLp + j f Lp1 T 

j gLp2 2k j f 

j −k≥−4, j ≥0

≤C

Lp

2k−j ∇f Lp1 j gLp2 ,



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N32 Lp ≤ C





−1 g) Lp + C

−1 g Lp k T (−1 f  k −1 f T 

−1≤k≤3



≤C

−1 gLp + C −1 f 

−1≤k≤3



≤C



−1 gLp −1 f T 

−1≤k≤3

−1 gLp2 + C −1 f Lp1 

−1≤k≤3

≤C

−1≤k≤3





−1 gLp2 −1 f Lp1 T 

−1≤k≤3

−1 gL2p −1 f L2p 

−1≤k≤3

≤ Cχ{−1≤k≤3} f L2 gL2 . According to s > −1, one concludes    ks   (k−j )(s+1) j s 2 N3 Lp  r ≤ C∇f Lp1  p2  + Cf  2 g 2 2 2  g  j L L L l r k

lk

j −k≥−4

≤ C(∇f Lp1 gBps

2 ,r

+ f L2 gL2 ).

2

This completes the proof of Lemma 2.3.

Let us recall the following Hörmander-Mihlin theorem (see [22,32]).

Lemma 2.4. Let (Rf )(x) = F −1 m(ξ ) · Ff (x), and m(ξ ) be a complex-valued bounded function on Rn \ {0} satisfies either (1) Mihlin’s condition  ∂m(ξ )      ≤ C|ξ |−|α| ∂ξ α for all multi-index |α| ≤ 1 + [ n2 ], or (2) Hörmander’s condition sup R R>0

−n+2|α|



R<|ξ |<2R

 ∂m(ξ ) 2     dξ ≤ C ∂ξ α

for all multi-index |α| ≤ 1 + [ n2 ]. Then the following holds true Rf Lp ≤ C(p)f Lp ,

∀p ∈ (1, ∞).

We also need the classical Kato-Ponce commutator estimate (see [24]). Lemma 2.5. Let p, p1 , p3 ∈ (1, ∞) and p2 , p4 ∈ [1, ∞] satisfy 1 1 1 1 1 + = + . = p p1 p2 p3 p4

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Then for s > 0, there exists a positive constant C such that   [s , f ]gLp ≤ C s f Lp1 gLp2 + s−1 gLp3 ∇f Lp4 .

(2.5)

Finally, we recall the following refined commutator estimate in the framework of Besov space (see [38, Lemma 2.6]). Lemma 2.6. Assume that u is a divergence-free vector field, then it holds

s ≤ C(p, r, δ, s) ∇uLp1 g s+δ + u 2 g 2 , [δ , u · ∇]gBp,r L L Bp ,r

(2.6)

2

where (0, 2).

1 p

=

1 p1

+

1 p2

with p ∈ [2, ∞), p1 , p2 ∈ [2, ∞], r ∈ [1, ∞] and s ∈ (−1, 1 − δ) for δ ∈

3. The proof of Theorem 1.1 In this section, we give the proof of Theorem 1.1. Before proving our results, we point out that the local existence in H s for s > 2 can be established through the classical theory of symmetric hyperbolic quasi-linear systems (see, e.g., [35,31]). Therefore, it boils down to establishing global a priori bounds for the solution (u, θ ) in H s . Throughout the paper, C stands for some real positive constants which may be different in each occurrence. We shall write C(λ1 , λ2 , · · ·, λk ) as the constant C depends on the quantities λ1 , λ2 , · · ·, λk . The symbol X ≈ Y stands for C1 Y ≤ X ≤ C2 Y for some constants 0 < C1 ≤ C2 . Let us begin with the basic L2 -energy estimate. Proposition 3.1. Assume (u0 , θ0 ) satisfies the assumptions stated in Theorem 1.1 and let (u, θ ) be the corresponding smooth solution of the system (1.1). Then, for any t > 0, t u(t)2L2

+2

1

1

(x21 u2 (τ )2L2 + x22 u1 (τ )2L2 ) dτ ≤ (u0 L2 + tθ0 L2 )2 ,

(3.1)

0

θ (t)Lp ≤ θ0 Lp ,

∀ p ∈ [2, ∞].

Proof of Proposition 3.1. Testing (1.1)4 by |θ|p−2 θ for 2 ≤ p < ∞ implies 1 d p θ (t)Lp = 0, p dt which yields θ (t)Lp ≤ θ0 Lp . Letting p = ∞, one derives θ (t)L∞ ≤ θ0 L∞ .

(3.2)

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Taking the L2 inner product of (1.1)1 and (1.1)2 with u1 and u2 , respectively, it is easy to verify 1 1 1 d u(t)2L2 + x21 u2 2L2 + x22 u1 2L2 = 2 dt

 u2 θ dx

R2

≤ uL2 θ L2 ≤ uL2 θ0 L2 , which leads to t u(t)2L2

+2

1

1

(x21 u2 (τ )2L2 + x22 u1 (τ )2L2 ) dτ ≤ (u0 L2 + tθ0 L2 )2 . 0

2

This completes the proof of Proposition 3.1.

To obtain the global H 1 -estimate of u, we appeal to the vorticity ( := ∂x1 u2 − ∂x2 u1 ) equation of (1.1) ∂t  + (u · ∇) + (∂x1 x1 u2 − ∂x2 x2 u1 ) = ∂x1 θ. Unfortunately, the external force term ∂x1 θ prevents us from obtaining the global any Lq (q ≥ 2) bound for  directly from the above vorticity equation. In order to overcome this difficulty, we need a new approach and a key new observation to hide the term ∂x1 θ . To this end, taking advantage of the identities u1 = −∂x2 ,

u2 = ∂x1 ,

the vorticity equation can be rewritten as follows ∂t  + (u · ∇) −

3x1 + 3x2 

 = ∂x1 θ.

(3.3)

Denoting

:= − R

∂x1 , + 3x2

3x1

we get from (3.3) that ∂t  + (u · ∇) −

3x1 + 3x2 

) = 0. ( − Rθ

Combining the θ -equation and (3.4), it is obvious to check that

ϒ :=  − Rθ

(3.4)

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satisfies ∂t ϒ + (u · ∇)ϒ −

3x1 + 3x2 

u · ∇]θ. ϒ = [R,

(3.5)

Before further proving, we will establish two lemmas concerning the operator R.

:= − 3 Lemma 3.1. Let R 

∂x1 3 x1 +x2

, then it holds

Lp ≤ C(p)f Lp , Rf

∀p ∈ (1, ∞).

(3.6)

Proof of Lemma 3.1. We can conclude by direct calculations

)(x) = F −1 (Rf



 i|ξ |2 ξ1 · Ff (x) |ξ1 |3 + |ξ2 |3

and √ 2 3 |ξ | ≤ |ξ1 |3 + |ξ2 |3 ≤ 2|ξ |3 . 2 Therefore, for any multi-index α, one has    ∂  |ξ |2 ξ1  −|α|   .  ∂ξ α |ξ |3 + |ξ |3  ≤ C|ξ | 1 2 In view of Lemma 2.4 again, we may conclude the estimate (3.6). 2

The second one concerns the commutator estimate involving the operator R.

:= − 3∂x1 3 and Lemma 3.2. Let R  + x1

x2

1 p

=

1 p1

+

1 p2

with 1 < p, p1 , p2 < ∞, then it holds

u · ∇]θLp ≤ C(p, p1 , p2 )∇uLp1 θ Lp2 . [R,

(3.7)

Proof of Lemma 3.2. The proof is inspired by [27, Corollary 1.4]. More precisely, we first have

j θ − R(∇

u · ∇]θ = R∂

j (uj θ ) − uj R∂

· u θ ). [R,

(3.8)

We remark that the last term of the right hand side of (3.8) vanishes when ∇ · u = 0. On the one hand, it follows from (3.6) that

· u θ)Lp ≤ C(p)∇ · u θ Lp ≤ C(p)∇uLp1 θ Lp2 . R(∇

j for j = 1, 2. Then it yields On the other hand, we set Aj = R∂

(3.9)

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14

j (uj θ ) − uj R∂

j θ = Aj (uj θ ) − uj Aj θ R∂ = Aj (uj θ ) − uj Aj θ − +



γ

∂ γ uj Aj θ − θ Aj uj

|γ |=1



 γ ∂ γ uj Aj θ + θ Aj uj ,

|γ |=1

where γ (ξ ) = i∂ γ A ξ j



 (iξ1 )|ξ |2 (iξj ) . |ξ1 |3 + |ξ2 |3

According to the proof of (3.6), one deduces for |γ | = 1 and 1 < r < ∞ that γ

Aj θ Lr ≤ C(r)θ Lr . Therefore, it implies     γ ∂ γ uj Aj θ + θ Aj uj  

γ

Lp

|γ |=1

≤ C(∇uLp1 Aj θ Lp2 + Aj uj Lp1 θ Lp2 ) ≤ C(p, p1 , p2 )∇uLp1 θ Lp2 .

(3.10)

By letting s = s1 = 1, s2 = 0 in (1.9) of [27, Corollary 1.4], we may obtain     γ ∂ γ uj Aj θ − θ Aj uj  p ≤ C(p, p1 , p2 )∇uLp1 θ Lp2 .(3.11) Aj (uj θ ) − uj Aj θ − L

|γ |=1

Combining (3.10) and (3.11) gives

j θ Lp ≤ C(p, p1 , p2 )∇uLp1 θ Lp2 ,

j (uj θ ) − uj R∂ R∂ which along with (3.8) and (3.9) further implies

u · ∇]θ Lp ≤ C(p, p1 , p2 )∇uLp1 θ Lp2 . [R, We thus complete the proof of Lemma 3.2.

2

We now are able to show the global H 1 -bound of the velocity u. Proposition 3.2. Assume (u0 , θ0 ) satisfies the assumptions stated in Theorem 1.1 and let (u, θ ) be the corresponding smooth solution of the system (1.1). Then, for any t > 0, t ϒ(t)2L2

+ 0

In particular, we have

1

 2 ϒ(τ )2L2 dτ ≤ C0 (t).

(3.12)

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∇u(t)L2 ≤ C0 (t), where C0 (t) depends only on t and the initial data u0 and θ0 . Proof of Proposition 3.2. By multiplying the equation (3.5) by ϒ and using the incompressibility condition, it follows that 1 d ϒ(t)2L2 − 2 dt





3x1 + 3x2

ϒ ϒ dx =



R2

u · ∇]θ ϒ dx. [R,

(3.13)

R2

In view of the Plancherel theorem, we have the estimate 

3x1 + 3x2

−



 ϒ ϒ dx =

R2

R2

|ξ1 |3 + |ξ2 |3 (ξ )|2 dξ |ϒ |ξ |2

√  2 (ξ )| dξ ≥ |ξ | |ϒ 2 R2

√ ≥

1 2  2 ϒ2L2 , 2

(3.14)

where we have applied the following simple fact √ |ξ1 |3 + |ξ2 |3 2 ≥ |ξ |. |ξ |2 2 Keeping in mind (2.4), it can be obtained that 

u · ∇]θ ϒ dx ≤ C[R,

u · ∇]θ  [R,

R2

≤ C(∇uL2 θ 

−1

B2,22

ϒ

1

2 B2,2 1

− 12 B∞,2

+ uL2 θ L2 )(ϒL2 +  2 ϒL2 ) 1

≤ C(ϒL2 θL4 + θ L2 θ L4 + uL2 θ L2 )(ϒL2 +  2 ϒL2 ) ≤

1 1  2 ϒ2L2 + Cϒ2L2 + C. 16

Now we observe, thanks the above estimates (3.13)-(3.15) 1 d ϒ(t)2L2 +  2 ϒ2L2 ≤ C + Cϒ2L2 . dt

After an integration in time and the use of the Gronwall inequality, we get

(3.15)

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t ϒ(t)2L2

+

1

 2 ϒ(τ )2L2 dτ ≤ C0 (t). 0

Moreover, one has ∇u(t)L2 ≤ (t)L2 ≤ ϒ(t)L2 + θ (t)L2 ≤ C0 (t). We thus complete the proof of Proposition 3.2.

2

With Proposition 3.2 at our disposal, we are ready to prove the following key estimate. Proposition 3.3. Assume (u0 , θ0 ) satisfies the assumptions stated in Theorem 1.1 and let (u, θ ) be the corresponding smooth solution of the system (1.1). Then, for any t > 0, 

1 2

t ϒ(t)2L2

+

ϒ(τ )2L2 dτ ≤ C0 (t),

(3.16)

0

where C0 (t) depends only on t and the initial data u0 and θ0 . In particular, for any 0 ≤ T ≤ T , we have T

∇u(τ )B˙ ∞,∞ dτ ≤ C0 (T )(T − T ) 2 + θ0 L∞ (T − T ), 0 1

(3.17)

T

where C0 (T ) depends only on T and the initial data u0 and θ0 . 1

1

Proof of Proposition 3.3. Applying  2 to the equation (3.5) and multiplying it by  2 ϒ, we derive by using ∇ · u = 0 that 1 1 d  2 ϒ(t)2L2 − 2 dt



1

2

3x1 + 3x2 

1

ϒ  2 ϒ dx = I1 + I2 ,

(3.18)

R2

where  I1 := −

1



1

[ 2 , u · ∇]ϒ  2 ϒ dx,

u · ∇]θ  2 ϒ dx.  2 [R, 1

I2 :=

R2

1

R2

According to (3.14), it follows that  − R2

Thanks to (2.6), we obtain



1 2

3x1 + 3x2 

√ 1 2

ϒ  ϒ dx ≥

2 ϒ2L2 . 2

(3.19)

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17

1

I1 ≤ C[ 2 , u · ∇]ϒL2  2 ϒL2 1

1

≈ C[ 2 , u · ∇]ϒB 0  2 ϒL2 ≤ C ∇uL4 ϒ

2,2

1 2 B4,2

1 + uL2 ϒL2  2 ϒL2

1 ≤ C L4 ϒB 1 + uL2 ϒL2  2 ϒL2 2,2

1

1

≤ CL4 (ϒL2 + ϒL2 ) 2 ϒL2 + CuL2 ϒL2  2 ϒL2

L4 )(ϒL2 + ϒL2 ) 12 ϒL2 + CuL2 ϒL2  12 ϒL2 ≤ C(ϒL4 + Rθ 1

1

≤ C(ϒL4 + θ L4 )(ϒL2 + ϒL2 ) 2 ϒL2 + CuL2 ϒL2  2 ϒL2 √ 1 2 ≤ ϒ2L2 + C(ϒL4 + θ L4 )2  2 ϒ2L2 8 1

+ C(ϒL4 + θ L4 + uL2 )ϒL2  2 ϒL2 √ 1 1 2 ≤ ϒ2L2 + C( 2 ϒL2 + θ L4 )2  2 ϒ2L2 8 1

1

+ C( 2 ϒL2 + θL4 + uL2 )ϒL2  2 ϒL2 ,

(3.20)

where we have used the simple fact ϒB 1 ≈ ϒL2 + ∇ϒL2 ≤ ϒL2 + ϒL2 . 2,2

Making use of (3.7), we immediately have

u · ∇]θ L2 ϒL2 I2 ≤ C[R, ≤ C∇uL4 θ L4 ϒL2 ≤ C(ϒL4 + θL4 )θ L4 ϒL2 √ 1 2 ≤ ϒ2L2 + C( 2 ϒL2 + θ L4 )2 θ 2L4 . 8

(3.21)

Putting the above estimates (3.18), (3.19), (3.20) and (3.21) together yields 1 1 1 d  2 ϒ(t)2L2 + ϒ2L2 ≤ C( 2 ϒL2 + θ L4 )2  2 ϒ2L2 + Cθ 4L4 dt 1

1

+ C( 2 ϒL2 + θ L4 + uL2 )ϒL2  2 ϒL2 . Owing to the Gronwall inequality and (3.1), (3.2) as well as (3.12), we arrive at



1 2

t ϒ(t)2L2

+

ϒ(τ )2L2 dτ ≤ C0 (t). 0

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18

and the properties of Besov spaces, it is not hard to verify Recalling the relation ϒ =  − Rθ ∇uB˙ ∞,∞ ≈ B˙ ∞,∞ 0 0

 ˙0 ≤ CϒB˙ ∞,∞ + CRθ 0 B∞,∞ ≤ CϒB˙ 1

2,∞

+ CθB˙ ∞,∞ 0

≤ CϒL2 + Cθ L∞ , which along with (3.16) leads to T

T ∇u(τ )B˙ ∞,∞ dτ ≤ C 0

T

T ϒ(τ )L2 dτ + C

T

θ (τ )L∞ dτ T

⎞1 ⎛ 2 T T 1 ⎜ ⎟ 2

2 ≤ C(T − T ) ⎝ ϒ(τ )L2 dτ ⎠ + C θ0 L∞ dτ T

T

≤ C0 (T )(T − T ) 2 + θ0 L∞ (T − T ). 1

This ends the proof of the proposition. 2 With the estimate (3.16) at our disposal, we are in the position to finish the proof of Theorem 1.1. Proof of Theorem 1.1. Applying operation s with s > 2 to (1.1) and taking the L2 inner product with s u and s θ respectively, adding them up, we can get 1 1 1 d (s u(t)2L2 + s θ (t)2L2 ) + s x22 u1 2L2 + s x21 u2 2L2 2 dt    = s (θ e2 )s u dx − [s , u · ∇]u · s u dx − [s , u · ∇]θ · s θ dx.

R2

R2

R2

On the one hand, recalling the fact u1 = −∂x2 ,

u2 = ∂x1 

and the Plancherel theorem, we observe that 1

1

1

1

s x22 u1 2L2 + s x21 u2 2L2 = s−2 x22 ∂x2 2L2 + s−2 x21 ∂x1 2L2  2 2 + |ξ |s−2 |ξ1 | 2   22 = |ξ |s−2 |ξ2 | 2  L L 3

 22 ≥ c|ξ |s− 2  L 1

3

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19

1

= cs− 2 2L2 1

= cs+ 2 u2L2 , where we have used the simple fact 3

3

3

|ξ1 | 2 + |ξ2 | 2 ≥ c|ξ | 2 for some constant c > 0. In view of the Young inequality, we get         s s   (θ e2 ) u dx  ≤ s θ L2 s uL2 ≤ C(s θ 2L2 + s u2L2 ).    R2 On the other hand, by the Young inequality and (2.5), we find         s s  [ , u · ∇]u ·  u dx  ≤ [s , u · ∇]uL2 s uL2    R2 ≤ C∇uL∞ s u2L2 ,

        s s  [ , u · ∇]θ ·  θ dx  ≤ C[s , u · ∇]θ L2 s θ L2    R2

≤ C(∇uL∞ s θ L2 + ∇θ L∞ s uL2 )s θ L2 1

1

2s

s+ 2 s ≤ C∇uL∞ s θ 2L2 + C∇θ L∞ uL2s+1 uL2s+1 2  2  θL2 1 1 c s 2 ≤ s+ 2 u2L2 + CuLs+1 2 ∇uL∞  θL2 2 2s+1

2s+1

s+1 s + C∇θ Ls+1 ∞  θ  2 L

2s+1 2s+1 1 c s+1 s ≤ s+ 2 u2L2 + C∇uL∞ s θ2L2 + C∇θ Ls+1 ∞  θ 2 . L 2

Summing up the above estimates, we derive 1 d (s u(t)2L2 + s θ (t)2L2 ) + s+ 2 u2L2 dt 2s+1

2s+1

s+1 s ≤ C(1 + ∇uL∞ )(s u2L2 + s θ 2L2 ) + C∇θ Ls+1 ∞  θ  2 . L

(3.22)

Keeping in mind (3.17), the term ∇uL∞ can be controlled by using the logarithmic inequality (see (3.25) below). However, at this moment we have no estimate for ∇θ L∞ . As a matter of fact, to derive any regularity of θ , we need to control ∇uL1 L∞ as θ satisfies only a pure t transport equation. Unfortunately, (3.17) is not enough to help us to achieve this goal. To this

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end, some special techniques are required. Notice that for any given T > 0, according to (3.17), it is not difficult to see that for any small constant  > 0 to be fixed hereafter, there exists T0 = T0 () ∈ (0, T ) such that T ∇u(τ )B˙ ∞, dτ ≤ . 0 ∞

(3.23)

T0

For any T0 ≤ t < T , we denote X (t) := sup (s u(τ )2L2 + s θ (τ )2L2 ), τ ∈[T0 , t]

where s > 2. Consequently, one may verify that X (t) is a nondecreasing function. Then, our main goal is to show lim X (t) < ∞.

t→T −

Sincere θ satisfies ∂t θ + (u · ∇)θ = 0, we have ∂t ∇θ + u · ∇(∇θ ) = −∇u · ∇θ.

(3.24)

Testing (3.24) by |∇θ |p−2 ∇θ for 2 ≤ p < ∞ and using ∇ · u = 0, we obtain 1 d p p ∇θ (t)Lp ≤ ∇uL∞ ∇θLp , p dt which implies d ∇θ (t)Lp ≤ ∇uL∞ ∇θ Lp . dt Letting p = ∞, we get d ∇θ (t)L∞ ≤ ∇uL∞ ∇θ L∞ . dt Thanks to the following logarithmic Sobolev inequality (see [25] for instance) 

 s , ∇uL∞ ≤ C 1 + uL2 + ∇uB˙ ∞, log e +  u 0 2 L ∞ we deduce for any T0 ≤ t < T that

s > 2,

(3.25)

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∇θ (t)L∞

21

 t  ≤ ∇θ (T0 )L∞ exp ∇u(τ )L∞ dτ T0

t

 ≤ C∗ exp C



  s ln(e +  u(τ ) 1 + u(τ )L2 + ∇u(τ )B˙ ∞, 0 2 L dτ ∞

T0

 ≤ C∗ exp

t

  t 

 C(1 + uL2 ) dτ exp C0 ∇u(τ )B˙ ∞, dτ ln e + X (t) 0 ∞ 

T0

T0

  t 

 ∇u(τ )B˙ ∞, dτ ln e + X (t) ≤ C∗ exp C0 0 ∞ T0

≤ C∗ (e + X (t))C0  , where C∗ depends on ∇θ (T0 )L∞ , while C0 > 0 is an absolute constant whose value is independent of , T or T0 . As a result, one has ∇θ (t)L∞ ≤ C∗ (e + X (t))C0 

for any T0 ≤ t < T .

Integrating (3.22) over the interval (T0 , t) and using (3.25) yield s u(t)2L2 + s θ (t)2L2 − s u(T0 )2L2 − s θ (T0 )2L2 t ≤C

s 2 s 2 (1 + ∇u(τ )B˙ ∞, ) ln e +  u(τ ) +  θ (τ ) 0 2 2 L L ∞

T0

× (s u(τ )2L2 + s θ (τ )2L2 ) dτ t +C

2s+1

2s+1

s+1 s ∇θ (τ )Ls+1 dτ, ∞  θ (τ ) 2 L

T0

which together with (3.26) further implies t e + X (t) ≤ e + X (T0 ) + C∗



(1 + ∇u(τ )B˙ ∞, ) ln e + X (τ ) X (τ ) dτ 0 ∞

T0

t + C∗



C0 (2s+1)

2s+1 e + X (τ ) s+1 e + X (τ ) 2s+2 dτ.

T0

Now taking 0 <  ≤

1 2C0 (2s+1) ,

we obtain

(3.26)

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t e + X (t) ≤ e + X (T0 ) + C∗





(1 + ∇u(τ )B˙ ∞, ) ln e + X (τ ) e + X (τ ) dτ. 0 ∞

T0

The combination of this inequality with the Gronwall inequality yields X (t) ≤ (e + X (T0 ))e

A(t)

− e,

T0 ≤ t < T ,

where t A(t) := C∗

(1 + ∇u(τ )B˙ ∞, ) dτ < ∞. 0 ∞ T0

Using (3.23), we observe that for any T0 ≤ t < T A(t) ≤ A(T ) T (1 + ∇u(τ )B˙ ∞, ) dτ 0 ∞

= C∗ T0

≤ C∗ (T − T0 ) + C∗ , which ensures X (t) ≤ (e + X (T0 ))e

C∗ (T −T0 )+C∗ 

− e.

(3.27)

Noting that the right hand side of (3.27) is independent of t for any T0 ≤ t < T , we may conclude that (3.27) is also valid for t = T . Therefore, this further shows T sup ( 0≤t≤T

s

u(t)2L2

+ 

s

θ (t)2L2 ) +



1 s+ 2 u(τ )2L2 dτ ≤ C0 T , X (T0 ) < ∞,

0

which yields the desired global H s -estimate of Theorem 1.1. Moreover, the uniqueness is clear. We thus complete the proof of Theorem 1.1. 2 Finally, we give a remark on the proof of Theorem 1.1. Remark 3.1. Based on the local well-posedness result in H s , there exists a T 0 > 0 such that (u(t), θ (t)) ∈ H s for any t ≤ T 0 . Consequently, we may assume that T > T 0 . With (3.17) in hand, we can prove the global regularity via contradiction. More precisely, we assume that T is the first blow-up time of the local solution. But the estimate (3.27) implies that the solution can be extended past time T . This is a contradiction. Now we point out that, if T  ∈ (T 0 , T ) is the first blow-up time, then we can do it as the case that T is the first blow-up time. Thus, by this approach, we can show (u(t), θ (t)) ∈ H s with t ≤ T for any given T > 0. However, at present,

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23

we are not able to derive any explicit bound for s u(t)L2 + s θ (t)L2 in terms of the initial data, due to the technique used here. 4. The proof of Theorem 1.2 This section is devoted to the proof of Theorem 1.2. Similarly, it is not hard to derive the local existence in H s for s > 2 (see [35,31]). Therefore, the main efforts are devoted to showing the global a priori bound for the solution (u, w) in H s . First, according to the basic energy estimate, we conclude the following estimates. Proposition 4.1. Assume (u0 , w0 ) satisfies the assumptions stated in Theorem 1.2 and let (u, w) be the corresponding smooth solution of the system (1.4). Then, for any t > 0, t u(t)2L2

+ w(t)2L2

+

α

(∇u(τ )2L2 +  2 w(τ )2L2 ) dτ ≤ C0 (t),

(4.1)

0

where C0 (t) depends only on t and the initial data u0 and w0 . Proof of Proposition 4.1. Taking the L2 inner product of (1.4)1 , (1.4)2 and (1.4)4 with u1 , u2 and w, respectively, it is easy to verify α 1 d (u(t)2L2 + w(t)2L2 ) + ∂x2 u1 2L2 + ∂x1 u2 2L2 + 2w2L2 +  2 w2L2 2 dt    = ∂x2 w u1 dx − ∂x1 w u2 dx + (∂x1 u2 − ∂x2 u1 ) w dx

R2



=−

R2

w ∂x2 u1 dx +

R2

R2





w ∂x1 u2 dx +

R2

(∂x1 u2 − ∂x2 u1 ) w dx

R2

≤ C(∂x2 u1 L2 + ∂x1 u2 L2 )wL2 1 1 ≤ ∂x2 u1 2L2 + ∂x1 u2 2L2 + Cw2L2 , 2 2 which yields α d (u(t)2L2 + w(t)2L2 ) + ∂x2 u1 2L2 + ∂x1 u2 2L2 +  2 w2L2 ≤ Cw2L2 . dt

Utilizing the Gronwall inequality, it leads to t u(t)2L2

+ w(t)2L2

+

α

(∂x2 u1 2L2 + ∂x1 u2 2L2 +  2 w2L2 )(τ ) dτ ≤ C0 (t). 0

Noting the fact  = ∂x1 u2 − ∂x2 u1 , one gets

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24

t

t (τ )2L2

dτ ≤

0

(∂x2 u1 (τ )2L2 + ∂x1 u2 (τ )2L2 ) dτ ≤ C0 (t). 0

Notice that the fact ∇uL2 = L2 , we further have t ∇u(τ )2L2 dτ ≤ C0 (t). 0

We therefore complete the proof of the proposition. 2 In order to obtain the global H 1 -estimate of u, it is natural to consider the vorticity ( := ∂x1 u2 − ∂x2 u1 ) equation, which reads as follows ∂t  + (u · ∇) − (∂x31 u2 − ∂x32 u1 ) = −w. Unfortunately, the external force term −w prevents us from obtaining the global any Lq (q ≥ 2) bound for  directly from the above vorticity equation. In order to overcome this difficulty, we need to hide the term −w. To this end, we first make use of the following identities u1 = −∂x2 ,

u2 = ∂x1 ,

to rewrite the vorticity equation as ∂t  + (u · ∇) −

∂x41 + ∂x42 

 = −w.

(4.2)

By denoting R :=

2 , ∂x41 + ∂x42

it follows from (4.2) that ∂t  + (u · ∇) −

∂x41 + ∂x42 

( − Rw) = 0.

(4.3)

By combining the fourth equation of (1.4) and (4.3), it is obvious to see that G :=  − Rw satisfies the equation ∂t G + (u · ∇)G −

∂x41 + ∂x42 

G = [R, u · ∇]w + Rw + Rα w − R.

(4.4)

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Lemma 4.1. Let R :=

2 ∂x4 +∂x4 1

25

, then the following holds true

2

Rf Lp ≤ C(p)f Lp ,

∀p ∈ (1, ∞).

(4.5)

Proof of Lemma 4.1. It is easy to check (Rf )(x) = F −1

  |ξ |4 · Ff (x) ξ14 + ξ24

and 1≤

|ξ |4 ≤ 2, ξ14 + ξ24

∀|ξ | = 0.

Therefore, for any multi-index α, one has    ∂  |ξ |4     ≤ C|ξ |−|α| .  α 4  ∂ξ ξ1 + ξ24  Therefore, Lemma 4.1 is an easy consequence of Lemma 2.4. This ends the proof.

2

Based on the new equation (4.4), we prove that the velocity u admits a global H 1 -bound, as stated in the following proposition. Proposition 4.2. Assume (u0 , w0 ) satisfies the assumptions stated in Theorem 1.2 and let (u, w) be the corresponding smooth solution of the system (1.4). Then, for any t > 0, t G(t)2L2

+

∇G(τ )2L2 dτ ≤ C0 (t), 0

w(t)Lr ≤ C0 (t),

∀ r ∈ (2, ∞).

In particular, we have ∇u(t)L2 ≤ C0 (t), where C0 (t) depends only on t and the initial data u0 and w0 . Proof of Proposition 4.2. Multiplying the equation (1.4)4 by |w|r−2 w with r ∈ (2, ∞), using the divergence-free condition and (4.5) yield 1 d w(t)rLr ≤ r dt

  (|w|r−2 w) dx

R2

≤ CLr wr−1 Lr

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26

≤ C(GLr + RwLr )wr−1 Lr ≤ C(GLr + wLr )wr−1 Lr , where we have used the fact (see [9] for instance)  α w (|w|r−2 w) dx ≥ 0. R2

We thus obtain 1 d w(t)2Lr ≤ C(GLr + wLr )wLr 2 dt ≤ C(G2Lr + w2Lr ) 22

2(1− 2r )

≤ C(GLr2 ∇GL2 ≤

+ w2Lr )

1 ∇G2L2 + C(G2L2 + w2Lr ). 16

(4.6)

In order to close the above inequality, we multiply the equation (4.4) by G, make use of the divergence-free condition and integrate with respect to the space variable to deduce 1 d G(t)2L2 − 2 dt

∂x41 + ∂x42 

R2

 =



G G dx



[R, u · ∇]w G dx +

R2

 Rw G dx +

R2

 Rα w G dx −

R2

R G dx.

(4.7)

R2

By means of the Plancherel theorem, it directly leads to  −

∂x41 + ∂x42 

 G G dx =

R2

R2

≥

1 2

ξ14 + ξ24  )|2 dξ |G(ξ |ξ |2 

 )|2 dξ |ξ |2 |G(ξ

R2

1 ≥ ∇G2L2 , 2 where we have applied the following simple fact ξ14 + ξ24 1 2 ≥ |ξ | . 2 |ξ |2 Keeping in mind (2.4), we can deduce for δ ∈ (0, 1 − 2r ) that

(4.8)

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27

 [R, u · ∇]w G dx ≤ C[R, u · ∇]wB δ−1 GB 1−δ 2,2

2,2

R2

≤ C(∇uL2 wB δ−1 + uL2 wL2 )GδL2 ∇G1−δ L2 ∞,2

≤ C(∇uL2 wLr + uL2 wL2 )GδL2 ∇G1−δ L2 ≤

1 ∇G2L2 + C(1 + ∇u2L2 )(1 + G2L2 + w2Lr ), 16

δ−1 where we have used the simple embedding Lr (R2 ) → B∞,2 (R2 ) for δ ∈ (0, 1 − 2r ). By the Young inequality, it follows

 Rw G dx ≤ wL2 GL2 ≤ C(w2L2 + G2L2 ) ≤ C(1 + G2L2 ), R2

 Rα w G dx ≤ CwL2 α GL2 R2

≤ CwL2 G1−α ∇GαL2 L2 1 ∇G2L2 + C(w2L2 + G2L2 ) 16 1 ≤ ∇G2L2 + C(1 + G2L2 ), 16

≤



R G dx ≤ L2 GL2 R2

≤ C(GL2 + RwL2 )GL2 ≤ C(w2L2 + G2L2 ) ≤ C(1 + G2L2 ). Putting the above estimates into (4.7) and then adding it up with (4.6), we conclude d (G(t)2L2 + w(t)2Lr ) + ∇G2L2 ≤ C(1 + ∇u2L2 )(1 + G2L2 + w2Lr ). dt Thanks to the Gronwall inequality and (4.1), it allows us to conclude t w(t)Lr + G(t)2L2 +

∇G(τ )2L2 dτ ≤ C0 (t),

∀ r ∈ (2, ∞).

0

By the simple fact ∇u(t)L2 ≤ (t)L2 ≤ G(t)L2 + w(t)L2 , it implies ∇u(t)L2 ≤ C0 (t).

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28

2

The proof of Proposition 4.2 is completed.

Proposition 4.3. Assume (u0 , w0 ) satisfies the assumptions stated in Theorem 1.2 and let (u, w) be the corresponding smooth solution of the system (1.4). Then, for any t > 0, t 

G(t)2L2

σ

+

σ ∇G(τ )2L2 dτ ≤ C0 (t) 0

for some 0 < σ ≤ 1 − α2 , where C0 (t) depends only on t and the initial data u0 and w0 . Proof of Proposition 4.3. Applying σ to the equation (4.4) and taking the L2 inner product of the desired equation with σ G, we thus obtain 1 d σ G(t)2L2 − 2 dt



∂x41 + ∂x42 

R2

 =

σ G σ G dx 

σ [R, u · ∇]w σ G dx +

R2





−

σ R σ G dx −

R2

 σ Rw σ G dx +

R2

σ Rα w σ G dx

R2

[σ , u · ∇]G σ G dx.

R2

Similar to (4.8), we obtain  −

∂x41 + ∂x42 

1 σ G σ G dx ≥ σ ∇G2L2 . 2

R2

Using again (2.4), we can deduce for δ ∈ (σ, 1) that  σ [R, u · ∇]w σ G dx ≤ C[R, u · ∇]wB σ −δ σ GB δ 2,2

2,2

R2

≤ C(∇uL2 wB σ −δ + uL2 wL2 )σ G1−δ σ ∇GδL2 L2 ∞,2

≤ C(∇uL2 wLr0 + uL2 wL2 )σ G1−δ σ ∇GδL2 L2 ≤

1 σ ∇G2L2 + C(1 + σ G2L2 ), 16

σ −δ where we have used the estimates of Proposition 4.2 and the embedding Lr0 (R2 ) → B∞,2 (R2 ) 2 with δ−σ < r0 < ∞. For σ ≤ 1 − α2 , we get by using the Young inequality

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29

 σ Rw σ G dx ≤ CwL2 2σ GL2 R2

≤ CwL2 σ G1−σ σ ∇GσL2 L2 

≤

1 σ ∇G2L2 + Cσ G2L2 , 16 α

α

σ Rα w σ G dx ≤ C 2 wL2 2σ + 2 GL2

R2 1−σ − α2

α

≤ C 2 wL2 σ GL2 ≤



σ + α2

σ ∇GL2

α 1 σ ∇G2L2 + Cσ G2L2 + C 2 w2L2 , 16

σ R σ G dx ≤ CL2 2σ GL2 R2

≤ CL2 σ G1−σ σ ∇GσL2 L2 ≤

1 σ ∇G2L2 + Cσ G2L2 . 16

Finally, choosing p

:=

2 ∈ (1, 2), 1+σ

we get by Kato-Ponce inequality (2.5) that  [σ , u · ∇]G σ G dx ≤ C[σ , u · ∇]GLp σ G

p

L p −1

R2

≤ C(∇uL2 σ G ≤ C∇uL2 ∇G

2 p p L 2− 2 p

+ ∇G

σ G

L 2−σ p = C∇uL2 ∇GL2 σ G

2 p

L 2−σ p

σ u

2

Lσ

)σ G

p

L p −1

p

L p −1

2

L 1−σ 1−σ σ ≤ C∇uL2 ∇GL2  GL2 σ ∇GσL2

≤

1 σ ∇G2L2 + C(1 + ∇G2L2 )(1 + σ G2L2 ). 16

Collecting the above estimates together implies α d σ G(t)2L2 + σ ∇G2L2 ≤ C(1 + ∇G2L2 )(1 + σ G2L2 ) + C 2 w2L2 . dt

Thanks to the Gronwall inequality and the estimates of Proposition 4.2, we immediately conclude the proof of Proposition 4.3. 2

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30

At this point it is worth mentioning that Proposition 4.1, Proposition 4.2 and Proposition 4.3 still hold true for the case α = 0. However, in order to establish the boundedness of w(t)L∞ , the condition α > 0 plays an important role. More precisely, we are able to prove the following key bound with the help of α > 0. Proposition 4.4. Assume (u0 , w0 ) satisfies the assumptions stated in Theorem 1.2 and let (u, w) be the corresponding smooth solution of the system (1.4). If the power α > 0, then for any t > 0, t w(t)

L∞

+

∇u(τ )L∞ dτ ≤ C0 (t),

(4.9)

0

where C0 (t) depends only on t and the initial data u0 and w0 . Proof of Proposition 4.4. Applying k operator to the fourth equation of (1.4) yields ∂t k w + (u · ∇)k w + k α w = −[k , u · ∇]w + k .

(4.10)

Multiplying the equation (4.10) by |k w|p−2 k w, using the Hölder inequality and invoking the following estimate (see [7]) 

p

(α k w)|k w|p−2 k w dx ≥ c2αk k wLp , R2

we may deduce that d k wLp + 2αk k wLp ≤ C[k , u · ∇]wLp + Ck Lp . dt Summing up the above inequality from j = −1 to ∞ and using the following fact (see for example [38, Lemma 2.3])     [k , u · ∇]wLp  1 ≤ C∇uLp wB 0 , ∞, 1

lk

it further gives for p ∈ ( α2 , ∞) d α ≤ C∇uLp w 0 w(t)B 0 + wBp,1 0 B∞, 1 + CBp,1 p,1 dt ≤ CLp wB 0 + CB 0 ∞, 1

p,1

≤ C(GLp + wLp )wB 0

∞, 1

+ CGB 0 + CwB 0 p,1

≤ C(GL2 + σ ∇L2 + wLp )w + C(GL2 + σ ∇L2 ) + CwB 0

2 1− αp 0 Bp,1

p,1

p,1

w

2 αp α Bp,1

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31

αp 1 σ α + C(G 2 +  ∇ 2 + wLp ) αp−2 w 0 ≤ wBp,1 L L Bp,1 2

+ C(GL2 + σ ∇L2 ) + CwB 0 , p,1

where we have applied the interpolation inequality wB 0

∞, 1

≤ Cw

2 p Bp, 1

≤ Cw

2 1− αp 0 Bp,1

2

wBαpα , p,1

∀p ∈

2 α

 ,∞ .

This allows us to obtain by further taking p ∈ [ α4 , ∞) αp d σ α ≤ C(G 2 +  ∇ 2 + wLp ) αp−2 w 0 w(t)B 0 + wBp,1 L L Bp,1 p,1 dt

+ C(GL2 + σ ∇L2 ) + CwB 0

p,1

≤ C(1 + G2L2 + σ ∇2L2 + w2Lp )wB 0

p,1

+ C(GL2 +  ∇L2 ). σ

Making use of the estimates of Proposition 4.2 and Proposition 4.3, one easily obtains t (1 + G(τ )2L2 + σ ∇(τ )2L2 + w(τ )2Lp ) dτ ≤ C0 (t),

∀ p < ∞.

0

Thanks to the Gronwall inequality, it guarantees that for any p ∈ [ α4 , ∞) t w(t)B 0 +

α dτ ≤ C0 (t). w(τ )Bp,1

p,1

0

Now it is easy to show ∇uL∞ ≤ ∇uB 0

∞,1

≤ CuL2 + CB 0

∞,1

≤ CuL2 + CGB 0

∞,1

+ CRwB 0

∞,1

≤ CuL2 + CwL2 + CGB 0

∞,1

+ CwB 0

∞,1

α , ≤ CuL2 + CwL2 + CGH σ +1 + CwBp,1

which implies

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32

t ∇u(τ )L∞ dτ ≤ C0 (t). 0

It follows from the equation (1.4)4 that d w(t)Lr ≤ C∇uLr . dt Letting r → ∞, we have d w(t)L∞ ≤ C∇uL∞ . dt As a result, we deduce w(t)L∞ ≤ C0 (t). This completes the proof of the proposition. 2 With the estimate (4.9) at our disposal, we are ready to establish the global H s estimate. Proof of Theorem 1.2. Applying s to the equations (1.4) and taking the L2 inner product of the desired equations with (s u, s w), we obtain α d (s u(t)2L2 + s w(t)2L2 ) + s ∂x2 u1 2L2 + s ∂x1 u2 2L2 + s+ 2 w2L2 dt             s s s s ≤ C     w dx  + C   u  ∇w dx  + C  [s , u · ∇]u s u dx 

R2

R2

    + C  [s , u · ∇]w s w dx .

R2

R2

Recalling the facts  = ∂x1 u2 − ∂x2 u1 and ∇uL2 = L2 , we infer that s ∇uL2 = s L2 ≤ s ∂x2 u1 2L2 + s ∂x1 u2 2L2 . The Young inequality allows us to check         s s C     w dx  + C  s u s ∇w dx  ≤Cs ∇uL2 s wL2 R2

R2

1 ≤ s ∇u2L2 + Cs w2L2 . 8 By the Kato-Ponce inequality (2.5), we immediately conclude

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33

        s s C  [ , u · ∇]u ·  u dx  ≤ [s , u · ∇]uL2 s uL2   R2  ≤ C∇uL∞ s u2L2 . The last term can be bounded by using (4.9) that     C  [s , u · ∇]w s w dx  ≤ C(∇uL∞ s wL2 + ∇wL2s s u

2s

L s−1

)s wL2

R2 s−1

1

1

≤ C∇uL∞ s w2L2 + CwLs∞ s wLs 2 ∇uLs ∞ s−1

× s ∇uLs2 s wL2 2(s−1) 2 1 ≤ s ∇u2L2 + C(∇uL∞ + wLs+1 ∇uLs+1 ∞ ∞) 8

× s w2L2 , where we have applied the following two interpolation inequalities 1

s−1

∇wL2s ≤ CwLs∞ s wLs 2 , s u

1

2s L s−1

s−1

≤ C∇uLs ∞ s ∇uLs2 .

We point out that the above two interpolation inequalities can be deduced from [3, Theorem 2.42]. Combining all the above estimates, we finally get α d (s u(t)2L2 + s w(t)2L2 ) + s ∇u2L2 + s+ 2 w2L2 dt 2(s−1)

2

s 2 s 2 ≤ C(1 + ∇uL∞ + wLs+1 ∇uLs+1 ∞ ∞ )( uL2 +  wL2 ).

By means of the Gronwall inequality and (4.9), it is obvious to see that t 

s

u(t)2L2

+ 

s

w(t)2L2

+

α

(s ∇u(τ )2L2 + s+ 2 w(τ )2L2 ) dτ ≤ C0 (t). 0

This yields the desired global H s -estimate of Theorem 1.2. Moreover, the uniqueness is clear. We therefore complete the proof of Theorem 1.2. 2

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Acknowledgments The author is grateful to the referee and the associated editor for their constructive comments and valuable suggestions which have contributed to the final preparation of our paper. The author was supported by the National Natural Science Foundation of China (No. 11701232) and the Natural Science Foundation of Jiangsu Province (No. BK20170224). References [1] H. Abidi, T. Hmidi, On the global well-posedness for Boussinesq system, J. Differ. Equ. 233 (2007) 199–220. [2] D. Adhikari, C. Cao, H. Shang, J. Wu, X. Xu, Z. Ye, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differ. Equ. 260 (2016) 1893–1917. [3] H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. [4] C. Cao, J. Wu, Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal. 208 (2013) 985–1004. [5] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006) 497–513. [6] Q. Chen, C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differ. Equ. 252 (2012) 2698–2724. [7] Q. Chen, C. Miao, Z. Zhang, A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation, Commun. Math. Phys. 271 (2007) 821–838. [8] P. Constantin, C. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1999) 159–172. [9] A. Cördoba, D. Cördoba, A maximum principle applied to quasi-geostrophic equations, Commun. Math. Phys. 249 (2004) 511–528. [10] R. Danchin, M. Paicu, Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data, Commun. Math. Phys. 290 (2009) 1–14. [11] R. Danchin, M. Paicu, Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci. 21 (2011) 421–457. [12] B. Dong, Z. Chen, Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Contin. Dyn. Syst. 23 (2009) 765–784. [13] B. Dong, J. Li, J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differ. Equ. 262 (2017) 3488–3523. [14] B. Dong, J. Wu, X. Xu, Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst. 38 (2018) 4133–4162. [15] B. Dong, Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differ. Equ. 249 (2010) 200–213. [16] T. Elgindi, I. Jeong, Finite-time singularity formation for strong solutions to the Boussinesq system, arXiv:1802. 09936, 2018. [17] A. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [18] G. Galdi, S. Rionero, A note on the existence and uniqueness of solutions of the micropolar fluid equations, Int. J. Eng. Sci. 15 (1977) 105–108. [19] E. Grenier, N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial date, Commun. Partial Differ. Equ. 22 (1997) 953–975. [20] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differ. Equ. 249 (2010) 2147–2174. [21] T. Hmidi, S. Keraani, F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Commun. Partial Differ. Equ. 36 (2011) 420–445. [22] L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960) 93–139. [23] T.Y. Hou, C. Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst. 12 (2005) 1–12. [24] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math. 41 (1988) 891–907. [25] H. Kozono, T. Ogawa, Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002) 251–278.

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