Regularity criterion for a critical fractional diffusion model of two-dimensional micropolar flows

J. Math. Anal. Appl. 470 (2019) 500–514

Contents lists available at ScienceDirect

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

Regularity criterion for a critical fractional diffusion model of two-dimensional micropolar flows Wen Tan a,b , Bo-Qing Dong a , Zhi-Min Chen a,∗ a

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen 518060, China b

a r t i c l e

i n f o

Article history: Received 16 August 2017 Available online 9 October 2018 Submitted by P. Sacks Keywords: Micropolar equations Critical fractional diffusion Regularity criterion Besov spaces

a b s t r a c t This paper is devoted to the Beale–Kato–Majda (BKM) regularity criterion of the 2D micropolar equations in the critical fractional diffusion situation α + β = 1 for α > 0 and β > 0. With the use of the Littlewood–Paley theory, the global existence of the unique solution (u, w) to the equations is found to be completely determined by the velocity field u. The dynamical behaviour of the scalar angular velocity w is controlled by the velocity u. More precisely, it is shown that the strong solution (u, w) exists globally if T ∇ × u(τ )2L∞ dτ < ∞ 0

for any T < ∞. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The micropolar fluid model, introduced by Eringen [15,16], describes a class of anisotropic fluids consisting of oriented particles suspended in a viscous medium [6,9,14,23,25], for instance, animal blood, liquid crystals, exotic lubricants, etc. Since the micropolar model covers many more phenomena such as anisotropic fluid motions, it can be viewed as a significant generalization of Navier–Stokes flows. In fact, the micropolar equations consist of the incompressible Navier–Stokes equations coupled with the evolution of both micro-rotational effects and micro-rotational inertia of the fluid. In this paper, we consider the fractional diffusion micropolar fluid model governed by the following two-dimensional (2D) fluid motion equations * Corresponding author. E-mail address: [email protected] (Z.-M. Chen). https://doi.org/10.1016/j.jmaa.2018.10.017 0022-247X/© 2018 Elsevier Inc. All rights reserved.

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

⎧ ⎨∂t u + u · ∇u + (ν + κ)(−Δ)α u − 2κ∇ × w + ∇π = 0, ⎩ ∂t w + u · ∇w + γ(−Δ)β w + 4κw − 2κ∇ × u = 0

501

(1)

together with the divergence free property ∇·u=0

(2)

and the initial condition u|t=0 = u0 ,

w|t=0 = w0 .

(3)

Here u is the fluid velocity field, π is the fluid pressure and w is the scalar angular velocity of the rotation of fluid particles. The fractional operator (−Δ)s with s > 0 is defined by the Fourier transform, namely, (−Δ)s = F −1 |ξ|2s F. The consultants α and β are nonnegative. The vorticities are in the classical sense: ∇ × u = ∂x1 u2 − ∂x2 u1 ,

∇ × w = (∂x2 w, −∂x1 w).

The parameter ν > 0 denotes the kinematic viscosity, κ > 0 denotes the micro-rotation viscosity, and γ > 0 is the angular viscosity. The existence and uniqueness problem of the micropolar equations have attracted great attention in recent years [4,20,23,31,32] due to its significance in mathematics and applied fields. When α = β = 1, equations (1)–(3) are reduced to the classical 2D micropolar equations, of which the global existence of regular solutions is well known. In the present paper, however, we are interested in the fractional diffusion situation α, β ≤ 1. The existence of global regular solutions for the 2D micropolar equations was obtained by Dong and Zhang [13] for the absence of the micro-rotation viscosity (i.e., α = 1, β = 0) and by Dong et al. [11] for the absence of the kinematic viscosity (i.e., α = 0, β = 1). Xue [31] considered the vanishing micro-rotation viscosity limit in the case of γ = 0. Yamazaki [32] studied a blowup criterion for the 2D magneto-micropolar equations with zero angular viscosity. Dong et al. [12] recently obtained the existence of global regular solutions for equation (1) in a subcritical diffusion situation (i.e., 0 < α, β < 1 and α+β > 1). In the present paper, we assume that equation (1) is in the critical fractional diffusion α > 0, β > 0, α + β = 1.

(4)

The term “critical fractional diffusion” means that the nonlinear convective parts u · ∇u and u · ∇w are balanced with the diffusion parts (−Δ)α u and (−Δ)β w. The local well-posedness of the micropolar equations described by (1) can be derived by a traditional analysis, which has been widely used in the study of the classical micropolar equations and Boussinesq equations (see [1,7,8,18,19,22,23]). However, the global existence of regular solutions to (1)–(3) in the critical fractional diffusion remains unknown. In view of the global existence problem, it is beneficial to study a blowup mechanism of (1) for mathematical and practical purposes. In recent years, regularity criterions for a variety of fluid models have been widely examined. A fundamental work of this area was given by Beale et al. [3] for a regular solution u of the incompressible Navier–Stokes equations to be extended beyond T < ∞ under the condition T ∇ × u(t)L∞ dt < ∞. 0

(5)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

502

This is known as BKM regularity criterion and is also proved to be true for the classical 3D micropolar equations [10]. It is the purpose of the present paper to extend the BKM regularity criterion to (1)–(3) in the critical fractional diffusion (4). The main result states as follows. Theorem 1.1. Let s > 2, (u0 , w0 ) ∈ H s (R2 )2 × H s (R2 ) with ∇ · u0 = 0, ν > 0, γ > 0 and let κ > 0 be sufficiently small. Then (1)-(4) admit a unique solution (u, w) ∈ C([0, ∞); H s (R2 )2 × H s (R2 )) ∩ L2 (0, ∞; H s+α (R2 )2 × H s+β (R2 )), provided that, for any T < ∞, T ∇ × u(τ )2L∞ dτ < ∞.

(6)

0

Equation (6), without involvement of w, shows that the global existence is essentially determined by the velocity field u rather than the scaler angular velocity w. As in the BKM regularity criterion [3] on the Navier–Stokes equations, the solution of (1)–(4) exists uniquely if both u and w are controlled by the condition T (∇u(t)2L∞ + ∇w(t)2L∞ )dt < ∞ 0

for any T < ∞. Therefore a crucial task herein is to bound ∇wL∞ in terms of ∇uL∞ . To do so, L2 energy estimate is necessary and derived in the first stage. Then, we derive an estimate of ∇wL∞ . For the critical diffusion case (4), the Hilbert space estimate in H s is no longer valid to control the norm ∇wL∞ , as obtained for the partial diffusion case [13]. However, the norm ∇wL∞ can be controlled with the aid 2 +1+ p

of (6). We find that for some  > 0, the bound can be obtained by adopting the Bp,2

(R2 ) Besov space

+1+ 2 Bp,2 p (R2 )

estimate of the micropolar equations and the Sobolev imbedding of the Besov space into the 1 Sobolev space W∞ (R2 ). The Besov space estimate is obtained by using the Littlewood–Paley decomposition together with Bernstein inequalities. Furthermore, we consider the solution in the Hilbert space H s(R2 ) and provide H s estimate involving ∇uL∞ and ∇wL∞ . Finally, with the use of the estimates obtained in the previous steps, we employ the Biot–Savart law and logarithmic Sobolev inequality to control ∇uL∞ and ∇wL∞ by ∇ × uL∞ and uH s + wH s . It follows from the condition (6) that the solution (u, w) is bounded in the Hilbert space H s (R2 ) uniformly with respect to t ∈ [0, T ] by developing a logarithmic estimate technique from [3,33]. Hence the solution can extended uniquely over the time T whenever T < ∞. This paper is organized as follows. Section 2 is for preliminary lemmas and introduction of the Littlewood– Paley theory. Section 3 is devoted to commutator estimates to be used in the derivation of a Besov space estimate and a H s estimate of the micropolar equations. The proof of Theorem 1.1 is given in Section 4. 2. Preliminaries Throughout this paper c represents a generic positive constant independent of the quantities t, x, ξ, f , g, u, w, κ, j and k. For simplicity, A  B denotes the inequality A ≤ cB. The Fourier transform F and the inverse Fourier transform F −1 will be applied to the Schwartz space S(R2 ) and the space of tempered distributions S  (R2 ). For convenience, we will use the fractional operator Λα = F −1 |ξ|α F.

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

503

To define Besov spaces, we use the Littlewood–Paley dyadic decomposition (see, for example, [2,27]) by taking a positive function φ ∈ S(R2 ) such that 4 3

φ(ξ) = 0 for |ξ| >

3 4

and φ(ξ) = 1 for |ξ| <

and the dyadic block symbols φj (ξ) = φ(2−j ξ), ψj (ξ) = φj+1 (ξ) − φj (ξ) for integers j ≥ 0. Thus Littlewood–Paley dyadic blocks are defined as Sj = F −1 φj F, Δj = F −1 ψj F = Sj+1 − Sj for j ≥ 0, Δj = S0 for j = −1. Thus we have Littlewood–Paley dyadic decompositions f=



Δj f for f ∈ S  (R2 ).

(7)

j≥−1 s Definition 2.1. For s ∈ R and p, q ∈ [1, ∞], the Besov space Bp,q (R2 ) is defined as

 s Bp,q (R2 ) = f ∈ S  (R2 );

 s f Bp,q <∞ ,

where s f Bp,q ≡

⎧

1 jsq ⎨ Δj f qLp q + Δ−1 f Lp , j≥0 2

if q < ∞,

⎩ supj≥0 2js Δj f Lp + Δ−1 f Lp ,

if q = ∞.

(8)

s When p = q = 2, the Besov space B2,2 (R2 ) is identical to the Bessel-potential space H s (R2 ) (see 1 [27]). We also adopt the BMO space BMO(R2 ) (see [21]) and the Sobolev space W∞ (R2 ) under the norm 1 = f L f W∞ + ∇f L∞ (see [17]). ∞ The fundamental estimates to the analysis of the present paper are listed in the following.

Lemma 2.2 (See [2, Proposition 2.71]). Let 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ q1 ≤ q2 ≤ ∞. Then, for any real numbers s1 and s2 such that s1 ≥ s2 + p21 − p22 , there holds the following continuous embedding: Bps11 ,q1 (R2 ) → Bps22 ,q2 (R2 ). Lemma 2.3 (See, for example, [2]). For s ≥ 0, 1 ≤ p ≤ q ≤ ∞ and f ∈ S  (R2 ), there hold the following Bernstein inequalities: Λs Sj f Lq  2js+j2( p − q ) Sj f Lp , 1

1

1 js+j2( p − q1 )

2 Δj f Lq  Λ Δj f Lq  2 js

s

∇Δj f Lp  2j Δj f Lp ,

j ≥ 0,

Δj f Lp ,

(9) j ≥ 0,

j ≥ −1.

(10) (11)

Lemma 2.4. ([5]) Let p ∈ [2, ∞), s ∈ [0, 1], f ∈ S  (R2 ) and j ≥ 0. Then the generalized Bernstein inequalities p

22sj Δj f pLp  Λs (|Δj f | 2 )2L2  22sj Δj f pLp .

(12)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

504

Lemma 2.5. ([24]) Let s > 2 and f ∈ H s (R2 ). Then the logarithmic Sobolev inequality ∇f L∞  (1 + ∇f BMO ) log(1 + f H s ) holds true. The logarithmic Sobolev inequality is essential in the derivation of the classical BKM regularity criterion [3,24]. 3. Commutator estimates The symbol [Δj , u] · ∇v represents the commutator operator Δj (u · ∇v) − u · Δj ∇v. In the past decades, the commutator estimates have been studied by many authors [5,22,28–30]. In this section, we adopt two commutator estimates. The first one for H s estimate is known. The second one with respect to a bound in ∇wL∞ and u +1+ p2 −α + w +1+ p2 −β is new and is proved in details. Bp,2

Bp,2

Lemma 3.1 ([13]). Let s > 0, j ≥ 0 and u be a divergence free vector field. Assume that u ∈ H s (R2 )2 ∩ 1 1 W∞ (R2 )2 and v ∈ H s (R2 ) ∩ W∞ (R2 ). Then the commutator estimate [Δj , u] · ∇vL2  cj 2−sj (∇uL∞ vH s + ∇vL∞ uH s )

(13)

holds true for a constant sequence {cj }j≥0 ∈ l2 or {cj }j≥0 l2 < ∞. Lemma 3.2. Let 0 < α < 1, β = 1 − α,  ∈ (0, β), p > 2 +1+ p

∇ · u = 0 and w ∈ Bp,2

2 β−

2 +1+ p

and j ≥ 0. Assume u ∈ Bp,2

(R2 )2 with

(R2 ). Then, it follows

2−αj [P Δj , u] · ∇uLp + 2−βj [Δj , u] · ∇wLp 2 −(+1+ p )j 2 cj ∇uL∞ u +1+ p2 −α + w

(14)

+1+ 2 −β Bp,2 p

Bp,2

for a constant sequence {cj }j≥0 ∈ l2 and the Leray projection operator P = I − ∇Δ−1 ∇·. Proof. It is convenient to use the Bony decomposition (see [2,28]) [P Δj , u] · ∇u =



(P Δj (Sk−1 u · Δk ∇u) − Sk−1 u · P Δj Δk ∇u)

|k−j|≤3

+



P Δj (Δk u · Sk−1 ∇u) +

|k−j|≤3

−

 k≥j

Δk u · Sk−1 P Δj ∇u −



 k ∇u) P Δj (Δk u · Δ

k≥j−3



 k P Δj ∇u ≡ Δk u · Δ

|k−j|≤4

5 

Ji

i=1

 k = Δk−1 + Δk + Δk+1 . The operator P Δj can be understood as a convolution operator: for Δ P Δj f ≡ F −1



⎛ ⎞ 2 2    ξl ξk ξl ξk ⎠ )ψj Ff = F −1 Ψj ∗ f for Ψj = ⎝1 − (1 − ψj . |ξ|2 |ξ|2 l,k=1

l,k=1

(15)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

505

Since Ψj is a Schwartz function supported in the annulus {ξ : 34 2j < |ξ| < 43 2j+1 }, it is readily seen that 

|F −1 Ψj (x)x|dx  2−j .

R2

The summation terms in J1 , J2 and J5 with respect to k are limited by |k − j| ≤ 3. Thus by the Lp estimate of the operator P , the divergence free condition (2) and Lemma 2.3, we have, for a real number τ ∈ (0, 1),      J1 Lp ≤  F −1 Ψj (x − y)(Sj−1 u · Δj ∇u)(y)dy − Sj−1 u(x) · F −1 Ψj (x − y)Δj ∇u(y)dy  R2

R2

Lp

     =  F −1 Ψj (x − y)(x − y) · ∇Sj−1 u x + τ (x − y) · Δj ∇u(y)dy 

Lp

R2

       ∇uL∞  F −1 Ψj (x)(x)Δj ∇u(y − x) dx

Lp

R2

 ∇uL∞ Δj uLp ,

(16)

J2 Lp  Δj (Δj u · Sj−1 ∇u)Lp  ∇uL∞ Δj uLp ,

(17)

J5 Lp  Δj uL∞ Δj ∇uLp  ∇uL∞ Δj uLp .

(18)

and

Similarly, the bounds for the infinite term summations J3 and J4 are obtained as follows J3 Lp 



 k ∇u)L Δj (Δk u · Δ p

k≥j−3





 k uL 2j Δk uLp Δ ∞

(19)

k≥j−3







 k ∇uL  ∇uL 2j Δk uLp 2−k Δ ∞ ∞

k≥j−3

J4 Lp 



2(j−k) Δk uLp ,

k≥j−3

Δk uL∞ Sk−1 Δj ∇uLp

k≥j

 Δj uLp



2(j−k) Δk ∇uL∞

(20)

k≥j

 ∇uL∞ Δj uLp . Combining (15)–(20), we deduce that    [P Δj , u] · ∇uLp  ∇uL∞ Δj uLp + 2(j−k) Δk uLp . k≥j−3

For the term [Δj , u] · ∇wLp , we also use the Bony decomposition [Δj , u] · ∇w =

 |k−j|≤3

(Δj (Sk−1 u · Δk ∇w) − Sk−1 u · Δj Δk ∇w)

(21)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

506

+

 |k−j|≤3

−





Δj (Δk u · Sk−1 ∇w) +

 k ∇w) Δj (Δk u · Δ

k≥j−3



Δk u · Sk−1 Δj ∇w −

 k Δj ∇w. Δk u · Δ

(22)

|k−j|≤4

k≥j

The bound of the second term on the right-hand side of (22) is obtained as         Δj (Δk u · Sk−1 ∇w)   |k−j|≤3 

 Δj (Δj u · Sj−1 ∇w)Lp

Lp

 Δj uL∞ Sj−1 ∇wLp   2−j Δj ∇uL∞ Δk ∇wLp  ∇uL∞



(23)

k≤j−2

2(k−j) Δk wLp .

k≤j−2

Similar to the estimates of the terms J1 , J3 , J4 and J5 , the other terms on the right-hand side of (22) are bounded in the same manner. Hence, in contrast to (21), the corresponding inequality of [Δj , u] · ∇wLp is derived as     [Δj , u] · ∇wLp  ∇uL∞ Δj wLp + 2(k−j) Δk wLp + 2(j−k) Δk wLp . k≤j−2

Since  + 2 +

2 p

− α > 0, p >

2 β−

and  + 2 +

2 p

(24)

k≥j−3

− β > 0, it follows from (21) and (24) that

2−αj [P Δj , u] · ∇uLp + 2−βj [Δj , u] · ∇wLp    2 2 2 2 ≤ 2−(+1+ p )j ∇uL∞ 2(+1+ p −α)j Δj uLp + 2(j−k)(+2+ p −α) 2(+1+ p −α)k Δk uLp +2 +

2 −(+1+ p )j



2

2 (j−k)(+2+ p −β)

2 −(+1+ p )j

for {cj }j≥0 ∈ l2 .

k≥j−3

∇uL∞ 2

k≥j−3

2



2 (+1+ p −β)j

2

Δj wLp +

2 (+1+ p −β)k

Δk wLp





2

k≤j−2

cj ∇uL∞

2(k−j)(−− p +β) 2(+1+ p −β)k Δk wLp 2

(25)

u

+1+ 2 −α p

Bp,2

+ w

+1+ 2 −β p

Bp,2

2

4. Proof of Theorem 1.1 The local existence of the unique solution of (1)–(4) in a small time domain [0, T ] can be obtained by the standard energy method (see [5,13,24]) and hence the proof detail is omitted. By the local time solution extension technique, it remains to show the boundedness lim (u(t)2H s + w(t)2H s ) < ∞

t≤T

(26)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

507

under the validity of the condition (6) for a given constant T > 0. The proof is to be derived in three steps through the combination of L2 estimate, L∞ estimate of ∇w and H s estimate and the derivation of the boundedness (26) due to the Biot–Savart law and logarithmic Sobolev inequality. Firstly, we show the classical energy estimate or L2 estimate of the solution (u, w). Taking the inner product of (1) with (u, w) and then integrating the resultant equation in R2 , we obtain that 1 d (u2L2 + w2L2 ) + (ν + κ)Λα u2L2 + γΛβ w2L2 +4κw2L2 2 dt     ≤ 2κ (∇ × w) · u + (∇ × u)wdx R2

        ≤ 2κ F(∇ × w) · F −1 udξ  + 2κ F(∇ × u)F −1 wdξ  R2



R2

|ξ||Fw(ξ)||F −1 u(ξ)|dξ + 2κ

≤ 2κ R2

≤ 2κΛ



|ξ||Fu(ξ)||F −1 w(ξ)|dξ

R2 α

u2L2

+ 2κΛ

β

w2L2 ,

(27)

where we have used the divergence free condition (2), integration by parts and the Plancherel formula. Thus we have, after the integration with respect to the time in the previous equation, t u(t)2L2

+

w(t)2L2

+2



 (ν − κ)Λα u(τ )2L2 + (γ − 2κ)Λβ w(τ )2L2 + 4κw2L2 dτ

0

≤

u0 2L2

+

w0 2L2 .

This yields, by the assumption of κ being sufficiently small, t u(t)2L2

+

w(t)2L2

+



 νΛα u(τ )2L2 + γΛβ w(τ )2L2 + 4κw2L2 dτ

0

(28)

≤ u0 2L2 + w0 2L2 . Secondly, we show the L∞ estimate of ∇w: ⎛ ∇w(t)L∞  (1 + u0 2H s + w0 2H s )

1+σ 2η

⎛

⎜ ⎝1 + ⎝

t

1 ⎞ ⎞ 2η ⎟ (1 + ∇u(τ )2L∞ )dτ ⎠ ⎠

(29)

0

for some constants η, σ ∈ (0, 1). To do so, we begin with Besov space estimate and then use imbedding properties in the previous section to obtain the desired inequality. Indeed, applying Δj to (1), we have ∂t Δj u + P Δj (u · ∇u) + (ν + κ)(−Δ)α Δj u − 2κP ∇ × Δj w = 0,

(30)

∂t Δj w + Δj (u · ∇w) + γ(−Δ) Δj w + 4κΔj w − 2κ∇ × Δj u = 0,

(31)

β

where the projection property P u = u has been used due to the divergence free condition (2).

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

508

2 Let  ∈ (0, min{β, s − 2}) and p > β− . Taking the inner product of (30) with |Δj u|p−2 Δj u and mulp−2 tiplying (31) by |Δj w| Δj w and then integrating the resultant equations in R2 respectively, we use (2) and integration by parts to obtain

1 d Δj upLp + (ν + κ) p dt

 (Λ2α Δj u) · |Δj u|p−2 Δj udx

R2

    ≤  (P Δj (u · ∇u) − 2κP ∇ × Δj w) · |Δj u|p−2 Δj udx R2

        p−2 ≤  (P Δj (u · ∇u) − u · P Δj ∇u) · |Δj u| Δj udx + 2κ (P ∇ × Δj w) · |Δj u|p−2 Δj udx R2



≤ [P Δj , u] · ∇uLp + 2κP ∇ × Δj wLp



(32)

R2

Δj up−1 Lp ,

and similarly, 1 d Δj wpLp + γ p dt

 Λ2β Δj w|Δj w|p−2 Δj wdx (33)

R2

≤ ([Δj , u] · ∇wLp + 2κ∇ × Δj uLp )Δj wp−1 Lp . Integrating by parts (see [5, (4.3)]), one finds that  (Λ2α Δj u) · |Δj u|p−2 Δj udx ≥ R2

p 2 α Λ |Δj u| 2 2L2 p

(34)

and  (Λ2β Δj w)|Δj w|p−2 Δj wdx ≥ R2

p 2 β Λ |Δj w| 2 2L2 . p

(35)

Hence, substituting (34) into (32) and (35) into (33) respectively, we have p 2(ν + κ) α 1 d Δj upLp + Λ |Δj u| 2 2L2 ≤ ([P Δj , u] · ∇uLp +2κP ∇×Δj wLp )Δj up−1 Lp , p dt p

(36)

and p 2γ β 1 d Δj wpLp + Λ |Δj w| 2 2L2 ≤ ([Δj , u] · ∇wLp + 2κ∇ × Δj uLp )Δj wp−1 Lp . p dt p

(37)

By the definition of Besov spaces and Littlewood–Paley dyadic decomposition (7), the Besov space estimate has to be derived separately for the dyadic blocks Δj with respect to annular supports (j ≥ 0) and Δ−1 = S0 with respect to the disc support. For the case j ≥ 0, by Lemma 2.4, (36) and (37) become respectively  d Δj upLp + 2(ν + κ)22αj Δj upLp  [P Δj , u] · ∇uLp + κP ∇ × Δj wLp Δj up−1 Lp dt

(38)

 d Δj wpLp + 2γ22βj Δj wpLp  [Δj , u] · ∇wLp + κ∇ × Δj uLp Δj wp−1 Lp . dt

(39)

and

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

509

p−2 Dividing (38) by the factor Δj up−2 Lp and (39) by the factor Δj wLp respectively and then summing the two resultant equations, one obtains that

d (Δj u2Lp + Δj w2Lp ) + 2(ν + κ)22αj Δj u2Lp + 2γ22βj Δj w2Lp dt (40)

 κP ∇ × Δj wLp Δj uLp + κ∇ × Δj uLp Δj wLp + [P Δj , u] · ∇uLp Δj wLp + [Δj , u] · ∇wLp Δj uLp , which is bounded (in the sense of ‘’) by   κ2j+1 Δj uLp Δj wLp + κ 22αj Δj u2Lp +22βj Δj w2Lp  1 + 2−2αj [P Δj , u] · ∇u2Lp +2−2βj [Δj , u] · ∇w2Lp κ   1   κ 22αj Δj u2Lp +22βj Δj w2Lp + 2−2αj [P Δj , u] · ∇u2Lp +2−2βj [Δj , u] · ∇w2Lp κ

(41)

after the use of the Lp estimate of the operator P , Lemma 2.3 and the Young inequality. Hence it follows from Lemma 3.2 and the sufficiently small assumption of κ that (40) and (41) can be formulated as d (Δj u2Lp + Δj w2Lp ) + ν22αj Δj u2Lp + γ22βj Δj w2Lp dt   2 ≤ c2−2(+1+ p )j c2j ∇u2L∞ u2 +1+ 2 −α + w2 +1+ 2 −β ,

(42)

  2 d (Δj u2Lp + Δj w2Lp ) ≤ c2−2(+1+ p )j c2j ∇u2L∞ u2 +1+ 2 −α + w2 +1+ 2 −β , dt Bp,2 p Bp,2 p

(43)

Bp,2

p

Bp,2

p

or

where and in what follows the generic constant c may depend on κ. 2 Multiplying (43) by 22(+1+ p )j and summing over j ≥ 0, we deduce that   d  2(+1+ p2 )j 2 (Δj u2Lp + Δj w2Lp )  ∇u2L∞ u2 +1+ 2 −α + w2 +1+ 2 −β . dt Bp,2 p Bp,2 p

(44)

j≥0

Now we consider the remaining case with respect to Δj for j = −1. Dividing (36) by Δ−1 up−2 Lp and (37) by Δ−1 wp−2 Lp and summing the resultant equations, we find that d (Δ−1 u2Lp + Δ−1 w2Lp )  P Δ−1 (u · ∇u)Lp Δ−1 uLp + P ∇ × Δ−1 wLp Δ−1 uLp dt

(45)

+ Δ−1 (u · ∇w)Lp Δ−1 wLp + ∇ × Δ−1 uLp Δ−1 wLp . This implies that, after the use of Lp estimate of P , Lemma 2.3, the divergence free condition (2) and the L2 energy estimate (28), d (Δ−1 u2Lp + Δ−1 w2Lp )  ∇uL∞ u2Lp + uLp wLp + Δ−1 uwL 2p wLp dt p+2  ∇uL∞ u2Lp + uL2 wLp + wL2 u2Lp  (1 + u0 2L2 + w0 2L2 )(1 + ∇u2L∞ )(u2Lp + w2Lp ).

(46)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

510

s By the continuous imbedding Bp,2 (R2 ) → Lp (R2 ) for s > 0, the combination of (44) and (46) implies that

 d u2 +1+ 2 + w2 +1+ 2 dt Bp,2 p Bp,2 p  (1 +

u0 2L2

+

w0 2L2 )(1

+



∇u2L∞ )

u

2

+1+ 2 −α p

Bp,2

 + w +1+ 2 −β .

(47)

2

Bp,2

p

Applying the interpolation inequality (see, for example, [2]) f 

+1+ 2 −ζ p

Bp,2

 f δL2 f 1−δ +1+ 2 Bp,2

p

for ζ ∈ (0, 1) and some δ ∈ (0, 1), we obtain from (28) and (47) that  d 1 + u2 +1+ 2 + w2 +1+ 2 dt Bp,2 p Bp,2 p

 σ  1−η  (1 + u0 2L2 + w0 2L2 )(1 + ∇u2L∞ ) u2L2 + w2L2 1 + u2 +1+ 2 + w2 +1+ 2 p

Bp,2

 1−η  (1 + u0 2L2 + w0 2L2 )1+σ (1 + ∇u2L∞ ) 1 + u2 +1+ 2 + w2 +1+ 2 , Bp,2

p

Bp,2

Bp,2

p

(48)

p

for some η, σ ∈ (0, 1). Dividing by (1 + u2 +1+ 2 + w2 +1+ 2 )1−η and then integrating with respect to t, p

Bp,2

Bp,2

p

the previous inequality gives u(t)2 +1+ 2 + w(t)2 +1+ 2 Bp,2

 1 + u0 

p

Bp,2

2

+ w0 

+1+ 2 Bp,2 p

p

2

+1+ 2 Bp,2 p

+ (1 +

u0 2L2

+

1+σ w0 2L2 ) η

 t

  η1 1 + ∇u(τ )2L∞ dτ .

(49)

0

For the H s bound of the initial (u0 , w0 ) on the previous equation, we use the continuous imbedding (see Lemma 2.2) 2 s−1+ p

s H s (R2 ) = B2,2 (R2 ) → Bp,2

2 +1+ p

(R2 ) → Bp,2

(R2 ).

(50)

For the desired lower bound on the left-hand side of (49), it follows from [26, Theorem 3.3.1] that the lower bound ∇wL∞ is derived from the continuous embedding 2 +1+ p

Bp,2

1 (R2 ) → W∞ (R2 ).

(51)

Hence we obtain (29) by employing the imbedding properties (50) and (51) into (49). Thirdly, we show the validity of the H s estimate 2(s+β) 2(s+β) 2β d (u2H s + w2H s )  (1 + ∇uL∞ )(u2H s + w2H s ) + ∇wLs+2β uHs+2β wLs+2β . s ∞ 2 dt

(52)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

511

Taking the inner product of (30) with Δj u, multiplying (31) by Δj w and then integrating the resultant equations in R2 , we use the L2 projection property of P to obtain 1 d (Δj u2L2 + Δj w2L2 ) + (ν + κ)Λα Δj u2L2 + γΛβ Δj w2L2 + 4κΔj w2L2 2 dt         ≤  Δj (u · ∇u) · Δj u + Δj (u · ∇w)Δj wdx + 2κ (∇ × Δj w) · Δj u + (∇ × Δj u)Δj wdx. R2

(53)

R2

By repeating the manipulation of (27), the second term on the right-hand side of (53) is bounded by 2κΛα Δj u2L2 + 2κΛβ Δj w2L2 . Thus (53) becomes d (Δj u2L2 + Δj w2L2 ) + νΛα Δj u2L2 + γΛβ Δj w2L2 dt     ≤ 2 Δj (u · ∇u) · Δj u + Δj (u · ∇w)Δj wdx

(54)

R2

for small κ. After the use of the divergence free condition (2), Lemma 2.3 and Lemma 3.1, the right-hand side of the previous equation with j ≥ 0 is bounded by     2 [Δj , u] · ∇u · Δj u + [Δj , u] · ∇wΔj wdx R2

(55)

 [Δj , u] · ∇uL2 Δj uL2 + [Δj , u] · ∇wL2 Δj wL2  cj 2−sj (∇uL∞ uH s Δj uL2 + ∇uL∞ wH s Δj wL2 + ∇wL∞ uH s Δj wL2 ). Multiplying (54) or (55) by 22sj and summing the resultant equations over j ≥ 0, we find that   d  2sj 2 (Δj u2L2 + Δj w2L2 ) + ν 22sj Λα Δj u2L2 + γ 22sj Λβ Δj w2L2 dt j≥0



j0

∇uL∞ (u2H s

+

w2H s )

j≥0

(56)

+ ∇wL∞ uH s wH s .

With the use of Lemma 2.3, equation (54) with j = −1 becomes d (Δ−1 u2L2 + Δ−1 w2L2 ) + 2(ν + κ)Λα Δ−1 u2L2 + 2γΛβ Δ−1 w2L2 dt  (Δ−1 (u · ∇u)L2 + ∇ × Δ−1 wL2 )Δ−1 uL2 + (Δ−1 (u · ∇w)L2 + ∇ × Δ−1 uL2 )Δ−1 wL2

(57)

 (1 + ∇uL∞ )(u2L2 + w2L2 ) + ∇wL∞ uL2 wL2  (1 + ∇uL∞ )(u2H s + w2H s ) + ∇wL∞ uH s wH s . Therefore (56) and (57), together with an interpolation inequality and the Young inequality, yield that d (u2H s + w2H s ) + νΛα u2H s + γΛβ w2H s dt  (1 + ∇uL∞ )(u2H s + w2H s ) + ∇wL∞ uH s wH s

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

512

β

s

 (1 + ∇uL∞ )(u2H s + w2H s ) + ∇wL∞ uH s wLs+β Λs+β wLs+β 2 2 2(s+β)

2(s+β)

2β

≤ c(1 + ∇uL∞ )(u2H s + w2H s ) + c∇wLs+2β uHs+2β wLs+2β + γΛs+β w2L2 , s ∞ 2

(58)

which shows (52). Finally, by the classical local strong solution extension property, it remains to show that the u(t)2H s + w(t)2H s is uniformly bounded with respect to t ∈ [0, T ]. Indeed, with the use of (6), we define ⎛ F (t) ≡ max {u(τ )2H s + w(τ )2H s }

and

τ ∈[0,t]

M ≡⎝

T

1 ⎞ 2η

(1 + ∇ × u(τ )2L∞ )dτ ⎠

< ∞.

0

From Lemma 2.5 and the inequality ∇uBMO ≤ ∇ × uL∞

(59)

due to the Biot–Savart law (see, for example, [24]), equation (29) implies that ∇w(t)L∞  (1 +

u0 2H s

+

1+σ w0 2H s ) 2η

 (1 + u0 2H s + w0 2H s )

1+σ 2η

1   t   2η 1 + ∇ × u(τ )2L∞ log2 (1 + u(τ )H s ) dτ 1+



(60)

0

 1 1 + M log η (10 + F (t)) .



By (28), (59), (60) and Lemma 2.5, one finds from (52) that d (u2H s + w2H s ) dt 2(s+β)

2(s+β)

2β

 (1 + ∇uL∞ )(u2H s + w2H s ) + ∇wLs+2β uHs+2β wLs+2β s ∞ 2    1 + (1 + ∇uBMO ) log(1 + uH s ) F (t) + (1 + u0 2H s + w0 2H s )

1+σ s+β η s+2β

 2(s+β)  2(s+β) 2β 1 s+2β uHs+2β w0 Ls+2β 1 + M log η (10 + F (t)) s 2

 (1 + ∇ × uL∞ )(1 + log(1 + F (t)))(1 + F (t))  2(s+β)  s+β  1 s+2β + (1 + u0 2H s + w0 2H s )η 1 + M log η (10 + F (t)) (F (t)) s+2β  (1 + ∇ × u2L∞ )(1 + log(1 + F (t)))(1 + F (t))  2(s+β) 2(s+β) s+β  + (1 + u0 2H s + w0 2H s )η (1 + M ) s+2β log(10 + F (t)) η(s+2β) (F (t)) s+2β β s+β 2 for η  = 1+σ η s+2β + s+2β and after the use of the Cauchy inequality 2∇ × uL∞ ≤ 1 + ∇ × uL∞ . The application of the inequality

into (61) yields

 2(s+β) β log(10 + F (t)) η(s+2β) ≤ (10 + F (t)) s+2β

(61)

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

513

d (u2H s + w2H s ) dt  (1 +

u0 2H s

+

 w0 2H s )η (1

(62) + M)

2(s+β) s+2β

(1 + ∇ ×

u2L∞ )(1

+ log(10 + F (t)))(10 + F (t)),

or, after integration on [0, t], 10 + F (t)  G(t)

(63)

for

G(t) =

 (1+u0 2H s +w0 2H s )η (1+M )

2(s+β) s+2β

t (1+∇ × u(τ )2L∞ )(1+log(10+F (τ )))(10+F (τ ))dτ. 0

Differentiating G(t), we obtain from (63) that 2(s+β)  dG ≤ (1 + u0 2H s + w0 2H s )η (1 + M ) s+2β (1 + ∇ × u(t)2L∞ )(1 + log(10 + F (t)))(10 + F (t)) dt

 (1 +

u0 2H s

+

 w0 2H s )η (1

+ M)

2(s+β) s+2β

(1 + ∇ ×

u(t)2L∞ )(1

(64)

+ log G(t))G(t),

which shows, after dividing the previous equation by (1 + log G(t))G(t) and integrating with respect to t, ⎛ 

log(1 + log G(t))  (1 + u0 2H s + w0 2H s )η ⎝1 +

T

2(s+β) ⎞ η(s+2β) +1

(1 + ∇ × u(τ )2L∞ )dτ ⎠

.

0

This, together with (6), implies that G(t) < ∞ and then F (t) < ∞ uniformly with respect to t ∈ [0, T ]. This gives the validity of the boundedness (26). The proof of Theorem 1.1 is completed. Acknowledgments The research is partially supported by NSFC of China (Grant No. 11571240) and China Postdoctoral Science Foundation (Grant No. 2018M633101). References [1] A. Alghamdi, S. Gala, M. Ragusa, A regularity criterion of weak solutions to the 3D Boussinesq equations, AIMS Math. 2 (2017) 451–457. [2] H. Bahouri, J. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011. [3] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. Math. Phys. 94 (1984) 61–66. [4] Q. Chen, C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Equations 252 (2012) 2698–2724. [5] Q. Chen, C. Miao, Z. Zhang, A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys. 271 (2007) 821–838. [6] Z.M. Chen, W.G. Price, Decay estimates of linearized micropolar fluid flows in R3 space with applications to L3 -strong solutions, Internat. J. Engrg. Sci. 44 (2006) 859–873. [7] Z.M. Chen, Z. Xin, Homogeneity criterion for the Navier–Stokes equations in the whole spaces, J. Math. Fluid Mech. 3 (2001) 152–182. [8] P. Constantin, V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal. 22 (2012) 1289–1321. [9] S.C. Cowin, Polar fluids, Phys. Fluids 11 (1968) 1919–1927.

514

W. Tan et al. / J. Math. Anal. Appl. 470 (2019) 500–514

[10] B.Q. Dong, Z.M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys. 50 (2009) 103525. [11] B.Q. Dong, J. Li, J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations 262 (2017) 3488–3523. [12] B.Q. Dong, J. Wu, X. Xu, Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst. Ser. A 38 (2018) 4133–4162. [13] B.Q. Dong, Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations 249 (2010) 200–213. [14] M.E. Erdogan, Polar effects in the apparent viscosity of suspension, Rheol. Acta 9 (1970) 434–438. [15] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [16] A.C. Eringen, Micropolar fluids with stretch, Internat. J. Engrg. Sci. 7 (1969) 115–127. [17] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. [18] S. Gala, Z. Guo, M. Ragusa, A remark on the regularity criterion of Boussinesq equations with zero heat conductivity, Appl. Math. Lett. 27 (2014) 70–73. [19] S. Gala, M.A. Ragusa, Y. Sawano, H. Tanaka, Uniqueness criterion of weak solutions for the dissipative quasi-geostrophic equations in Orlicz–Morrey spaces, Appl. Anal. 93 (2) (2014) 356–368. [20] G. Galdi, S. Rionero, A note on the existence and uniqueness of solutions of micropolar fluid equations, Internat. J. Engrg. Sci. 14 (1977) 105–108. [21] L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. [22] Q. Jiu, C. Miao, J. Wu, Z. Zhang, The two-dimensional incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal. 46 (2014) 3426–3454. [23] G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, 1999. [24] A. Majda, A.L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. [25] S. Popel, A. Regirer, P. Usick, A continuum model of blood flow, Biorheology 11 (1974) 427–437. s s and Fp,q type, Z. Anal. Anwend. [26] W. Sickel, H. Triebel, Hölder inequalities and sharp embeddings in function spaces of Bp,q 14 (1995) 105–140. [27] H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983. [28] J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier–Stokes equations in Besov spaces, Comm. Math. Phys. 263 (2006) 803–831. [29] J. Wu, Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation, Nonlinear Anal. 67 (2007) 3013–3036. [30] J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations 33 (2008) 285–306. [31] L. Xue, Wellposedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci. 34 (2011) 1760–1777. [32] K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst. 35 (2015) 2193–2207. [33] Z. Ye, Blow-up criterion of smooth solutions for the Boussinesq equations, Nonlinear Anal. 110 (2014) 97–103.