Global regularity of the 2D magnetic micropolar fluid flows with mixed partial viscosity

Computers and Mathematics with Applications 70 (2015) 66–72

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Global regularity of the 2D magnetic micropolar fluid flows with mixed partial viscosity✩ Jianfeng Cheng a , Yujun Liu a,b,∗ a

Department of Mathematics, Sichuan University, Chengdu 610064, PR China

b

Department of Mathematics and Computer, Panzhihua University, Panzhihua 617000, PR China

article

info

Article history: Received 3 December 2014 Received in revised form 18 March 2015 Accepted 19 April 2015 Available online 14 May 2015

abstract In this paper, we study the Cauchy problem for the 2D anisotropic magneto-micropolar fluid flows with mixed partial viscosity. We established the global regularity of the 2D anisotropic magneto-micropolar fluid flows with vertical kinematic viscosity, horizontal magnetic diffusion and horizontal vortex viscosity. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Anisotropic magneto-micropolar fluid Global regularity Cauchy problem

1. Introduction and main results The 3D magneto-micropolar fluid flows can be stated as follows

 ∂t u + u · ∇ u + ∇π = (µ + χ )△u + b · ∇ b + 2χ∇ × m,    ∂ b + u · ∇ b = ν△b + b · ∇ u, t  ∂ m + u · ∇ m + 4χ m = κ△m + (α + β)∇∇ · m + 2χ∇ × u, t    ∇ · u = ∇ · b = 0,

(1.1)

where u = (u1 , u2 , u3 ), b = (b1 , b2 , b3 ), m = (m1 , m2 , m3 ) and π denote the velocity field, the magnetic field, the microrotation field and the scalar pressure, respectively. The nonnegative constants µ and ν are the Newtonian kinetic viscosity and the magnetic diffusion coefficient respectively. The parameter χ > 0 is the dynamic micro-rotation viscosity, the nonnegative constants α, β and κ are the angular viscosities. The magnetic micropolar fluid flows describe a micropolar fluid which is moving into a magnetic field (see [1]). Ahmadi and Shahinpoor investigated the stability of the solution to the 3D magnetic micropolar fluid in [1]. Fortunately, the local existence and uniqueness of strong solution were proved by Rojas-Medar in [2]. However, the global regularity of the strong solution with large initial data or finite time singularity to the 3D magneto-micropolar fluid equations is still an open problem. Recently, many mathematicians investigated the blow-up criteria of the strong solutions to the 3D magnetomicropolar fluid equations in [3–6]. As an intermediate case, we want to study the 2D magnetic micropolar fluid equations

✩ This work is supported by Sichuan Youth Science Technology Foundation (2014JQ0003).

∗

Corresponding author at: Department of Mathematics, Sichuan University, Chengdu 610064, PR China. E-mail addresses: [email protected] (J. Cheng), [email protected] (Y. Liu).

http://dx.doi.org/10.1016/j.camwa.2015.04.026 0898-1221/© 2015 Elsevier Ltd. All rights reserved.

J. Cheng, Y. Liu / Computers and Mathematics with Applications 70 (2015) 66–72

67

with partial viscosity. For the 2D case, the velocity field and magnetic field can be understood as u = (u1 (x, y, t ), u2 (x, y, t ), 0) and b = (b1 (x, y, t ), b2 (x, y, t ), 0). We assume that the rotational axis of particles is parallel to the Z -axis, namely, m = (0, 0, m(x, y, t )). Inserting u, b and m of above forms into Eqs. (1.1), one has

 ⊥  ∂t u + u · ∇ u + ∇π = (µ + χ )△u + b · ∇ b + 2χ∇ m, ∂t b + u · ∇ b = ν△b + b · ∇ u,  mt + u · ∇ m + 4χ m = κ△m + 2χ∇ × u, ∇ · u = ∇ · b = 0,

(1.2)

where u = (u1 (x, y, t ), u2 (x, y, t )), b = (b1 (x, y, t ), b2 (x, y, t )) and ∇ ⊥ m = (my , −mx ). The system (1.2) is the 2D micropolar fluid equations when the magnetic b is taken to zero, which is expressed as

 ∂t u + u · ∇ u + ∇π = (µ + χ )△u + 2χ∇ ⊥ m, ∂ m + u · ∇ m + 4χ m = κ△m + 2χ∇ × u, ∇t · u = 0.

(1.3)

For the global regularity of the 2D micropolar fluid equations, Lukaszewicz in [7] obtained the global well-posedness of classical solutions for 2D micropolar fluid with full viscosity. However, whether or not the classical solution of the 2D inviscid micropolar fluid can develop singularity in finite time is a challenging problem. For the 2D micropolar fluid with partial viscosity, the global regularity of the solution to the 2D micropolar fluid with zero angular viscosity has been established by Dong and Zhang in [8]. Xue in [9] established the global well-posedness to 2D micropolar fluid with zero kinetic viscosity or zero angular viscosity. There are more works for the 2D micropolar fluid in [10–13]. If the effect of micro-rotation field is negligible and χ = 0, the system (1.2) reduces to the following famous magnetohydrodynamic equations (MHD), which is written as



∂t u + u · ∇ u + ∇π = µ△u + b · ∇ b, ∂t b + u · ∇ b = ν△b + b · ∇ u, ∇ · u = ∇ · b = 0.

(1.4)

Consider the 2D MHD equations with full viscosity, the existence and uniqueness of classical solution for the large initial data has been obtained in [14]. However, the global regularity issue of inviscid MHD equations is a challenging open problem. As an intermediate case, Cao and Wu established the global regularity of the 2D MHD equations with mixed partial dissipation and magnetic diffusion in [15]. Please see [16–18] to learn more results for 2D MHD equations with partial viscosity. Motivated by their works in [15], our main goal in this paper is to establish the global regularity of 2D magnetic micropolar fluid flows with mixed partial viscosity, which can be expressed as follows

 ⊥  ∂t u + u · ∇ u + ∇π = ∂yy u + b · ∇ b + ∇ m, ∂t b + u · ∇ b = ∂xx b + b · ∇ u,  mt + u · ∇ m = mxx + ∇ × u − 2m, ∇ · u = ∇ · b = 0. Without loss of generality, we take µ = χ = u(x, y, 0) = u0 ,

b(x, y, 0) = b0 ,

1 2

(1.5)

and ν = κ = 1. Here, we impose the initial data as

and m(x, y, 0) = m0 .

(1.6)

Precisely, we have the following results. Theorem 1.1. Suppose that u0 , b0 , m0 ∈ H 2 (R2 ) and div u0 = div b0 = 0, the Cauchy problem (1.5)–(1.6) has a unique global classical solution (u, b, m) with

  u, b, m ∈ L∞ [0, T ); H 2 (R2 ) ,

  ωy , jx , ∇ mx ∈ L2 [0, T ); H 1 (R2 ) ,

for any T > 0 independent of initial data, where ω = ∇ × u and j = ∇ × b represent the vorticity and the current density, respectively. Remark 1.1. Moreover, we consider the 2D anisotropic magneto-micropolar fluid flows with horizontal kinematic viscosity, vertical magnetic diffusion and vertical vortex viscosity,

 ⊥  ∂t u + u · ∇ u + ∇π = ∂xx u + b · ∇ b + ∇ m, ∂t b + u · ∇ b = ∂yy b + b · ∇ u,  mt + u · ∇ m = myy + ∇ × u − 2m, ∇ · u = ∇ · b = 0.

(1.7)

Using the similar arguments in the proof of Theorem 1.1, we can also establish the global regularity for the system (1.7).

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J. Cheng, Y. Liu / Computers and Mathematics with Applications 70 (2015) 66–72

Remark 1.2. Recently, Wang et. al. in [19,20] considered the two-dimensional magneto-micropolar fluid equations as follows,

 1  ∂t u + u · ∇ u + ∇π = △u + b · ∇ b + ∇ ⊥ m,    2   ∂t b + u · ∇ b = b · ∇ u, 1   mt + u · ∇ m = ∇ × u − m,   2   ∇ · u = ∇ · b = 0,

(1.8)

and established a blow-up criterion for the system (1.8). Remark 1.3. As mentioned before, our works are motivated by their works in [15]. They gave a significant approach to deal with the 2D MHD with partial viscosity. Fortunately, we found the approach is valid for the 2D anisotropic magnetomicropolar fluid flows with mixed partial viscosity. We specifically point out that the estimates of I1 and I2 and J1 through J5 in Section 2 were handled in [15]. In order to obtain the estimates for u, b and m in detail, we still handle the estimation of I1 and I2 and J1 through J5 in Section 2. 2. The global regularity of 2D magnetic micropolar fluid flows with mixed partial viscosity Before the proof of our main results, we introduce some simplified notations, which will be used throughout this paper. We denote

∥f ∥2 = ∥f ∥L2 (R2 ) , and

∥f1 , f2 , . . . , fn ∥22 = ∥f1 ∥22 + ∥f2 ∥22 + · · · + ∥fn ∥22 . Next, we recall a very useful lemma throughout this paper, please see [15] for the proof of this lemma. Lemma 2.1. Assume that f , g, h, gx , hy ∈ L2 (R2 ), then



1

1

R2

1

1

|fgh|dxdy ≤ C ∥f ∥2 ∥g ∥22 ∥gx ∥22 ∥h∥22 ∥hy ∥22 ,

(2.1)

where C > 0 is a constant. In order to establish the global regularity of the Cauchy problem (1.5)–(1.6), we first establish the energy estimates of u, b and m. Taking the L2 inner product of the first three equations in (1.5) with u, b and m, respectively, after integrating by parts in R2 and adding the results, we have 1 d

∥u(τ ), b(τ ), m(τ )∥22 + ∥∂y u∥22 + ∥∂x b∥22 + ∥mx ∥22 + 2∥m∥22  = my u1 − mx u2 dxdy + (∂x u2 − ∂y u1 )mdxdy 2 R2   R = −2 ∂y u1 mdxdy − 2 mx u2 dxdy

2 dτ 

R2

R2

 1 ∥∂y u∥22 + ∥mx ∥22 + C ∥u, m∥22 , 2 where we have used the fact ≤



 b · ∇ b · u dxdy + R2

R2

(2.2)

b · ∇ u · b dxdy = 0.

It follows from Gronwall’s inequality that we obtain

∥u(t ), b(t ), m(t )∥22 +

t



∥∂y u(τ ), ∂x b(τ ), mx (τ )∥22 dτ ≤ C ,

(2.3)

0

for any 0 < t ≤ T , where the constant C depends on the initial data u0 , b0 , m0 and T . For notations convenience the same generic constant C denotes various constants that depend on the initial data u0 , b0 , m0 and T . Next, we will establish the lower-order estimates of u, b and m. Taking the operation curl on both sides of the first two equations in (1.5), one has

ωt + u · ∇ω = ωyy + b · ∇ j − △m,

(2.4)

J. Cheng, Y. Liu / Computers and Mathematics with Applications 70 (2015) 66–72

69

and jt + u · ∇ j = jxx + b · ∇ω + 2∂x b1 (∂x u2 + ∂y u1 ) − 2∂x u1 (∂x b2 + ∂y b1 ).

(2.5)

By the micro-rotation equation in (1.5), we have

∇ mt + ∇(u · ∇ m) + 2∇ m = ∇ mxx + ∇ω.

(2.6)

Taking the L inner product of (2.4)–(2.6) with ω, j and ∇ m, respectively, after integrating by parts in R and adding the results, we obtain 2

2

1 d

∥ω(τ ), j(τ ), ∇ m(τ )∥22 + ∥ωy ∥22 + ∥jx ∥22 + ∥∇ mx ∥22 + 2∥∇ m∥22      =2 ∂x u1 ∂x b2 + ∂y b1 j dxdy ∂x b1 ∂x u2 + ∂y u1 j dxdy − 2 2 R2 R  − ∇ m · ∇ u · ∇ m dxdy + 2 ∇ m · ∇ω dxdy

2 dτ 

R2

R2

= I1 + I2 + I3 + I4 ,

(2.7)

where we have used the fact

 R2

(b · ∇ j)ω dxdy +

 R2

(b · ∇ω)j dxdy = 0.

Applying Lemma 2.1, Cauchy–Schwarz inequality and Young’s inequality, we bound the first term I1 on the right hand side of (2.7),

 I1 = 2 R2

∂x b1 (∂x u2 + ∂y u1 )j dxdy 1

1

1

1

≤ C ∥∂x b1 ∥2 ∥j∥22 ∥jx ∥22 ∥∂x u2 + ∂y u1 ∥22 ∥∂xy u2 + ∂yy u1 ∥22   1 1 ≤ ∥ωy ∥22 + ∥jx ∥22 + C ∥∂x b∥22 ∥ω∥22 + ∥j∥22 , 6

(2.8)

4

where we have used the facts

∥∂x u2 + ∂y u1 ∥22 ≤ ∥ω∥22 and ∥∂xy u2 + ∂yy u1 ∥22 ≤ ∥ωy ∥22 . Similarly, one has

 I2 = −2 R2

 =2 R2

∂x u1 (∂x b2 + ∂y b1 )j dxdy

u1 (∂x b2 + ∂y b1 )jx dxdy + 2

 R2

1

1

u1 (∂xx b2 + ∂xy b1 )j dxdy

1

1

1

1

1

1

≤ C ∥jx ∥2 ∥u1 ∥22 ∥∂y u1 ∥22 ∥∂x b2 + ∂y b1 ∥22 ∥∂xx b2 + ∂xy b1 ∥22 + C ∥∂xx b2 + ∂xy b1 ∥2 ∥u1 ∥22 ∥∂y u1 ∥22 ∥j∥22 ∥jx ∥22 ≤

1 4

∥jx ∥22 + C ∥u∥22 ∥∂y u∥22 ∥j∥22 ,

(2.9)

where we have used the facts

∥∂x b2 + ∂y b1 ∥22 ≤ ∥j∥22 and ∥∂xx b2 + ∂xy b1 ∥22 ≤ ∥jx ∥22 . It follows from Lemma 2.1, Cauchy–Schwarz inequality and Young’s inequality for the term I3 on the right hand side of (2.7), which can be bounded by

 I3 = −

R

2

=− R2

∇ m · ∇ u · ∇ m dxdy   2 ∂x u1 mx dxdy − 2 u1 my mxy dxdy − (∂x u2 + ∂y u1 )mx my dxdy R2

3 2

1 2

1 2

R2

1 2

3 2

1

1

1

≤ C ∥mx ∥2 ∥mxy ∥2 ∥∂x u1 ∥2 ∥∂xx u1 ∥2 + C ∥mxy ∥2 ∥my ∥22 ∥u1 ∥22 ∥∂y u1 ∥22 1

1

1

1

1

1

+ C ∥mx ∥2 ∥∂x u2 ∥22 ∥∂xy u2 ∥22 ∥my ∥22 ∥mxy ∥22 + C ∥∂y u1 ∥2 ∥mx ∥22 ∥my ∥22 ∥mxy ∥2   1 1 ≤ ∥ωy ∥22 + ∥∇ mx ∥22 + C ∥u∥22 ∥∂y u∥22 + ∥∂y u∥22 + ∥mx ∥22 ∥ω, ∇ m∥22 . 6

4

(2.10)

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J. Cheng, Y. Liu / Computers and Mathematics with Applications 70 (2015) 66–72

For the last term I4 , one has

 I4 = 2

∇ m · ∇ω dxdy R2

 = −2 R2

mxx ω dxdy + 2



my ωy dxdy

R2

  1 ∥ωy ∥22 + ∥∇ mx ∥22 + C ∥ω∥22 + ∥∇ m∥22 . 6 4 Combining the above estimates of I1 –I4 , we get ≤

1

(2.11)

d

∥ω(τ ), j(τ ), ∇ m(τ )∥22 + ∥ωy ∥22 + ∥jx ∥22 + ∥∇ mx ∥22   ≤ C ∥u∥22 ∥∂y u∥22 + ∥∂x b∥22 + ∥∂y u∥22 + ∥mx ∥22 + 1 ∥ω, j, ∇ m∥22 .

dτ

(2.12)

Consequently, Gronwall’s inequality yields that

∥ω(t ), j(t ), ∇ m(t )∥ + 2 2

t



∥ωy (τ ), jx (τ ), ∇ mx (τ )∥22 dτ ≤ C ,

(2.13)

0

for any t ≤ T . Finally, we will obtain the uniform bounds of ∥∇ω(t )∥22 , ∥∇ j(t )∥22 and ∥△m(t )∥22 for any t ≤ T . Taking the L2 inner product of (2.4) and (2.5) with △ω and △j, respectively, integrating by parts in R2 , we have 1 d 2 dτ



∥∇ω(τ )∥ + ∥∇ω ∥ = − 2 2

2 y 2

 ∇ω · ∇ u · ∇ω dxdy + R2

∇ω · ∇ b · ∇ j dxdy R2

 + R2

b · ∇(∇ j) · ∇ω dxdy +

 R2

△ω△m dxdy,

(2.14)

and 1 d 2 dτ

∥∇ j(τ )∥22 + ∥∇ jx ∥22 = −



 ∇ j · ∇ u · ∇ j dxdy +

R2

 + R2

∇ j · ∇ b · ∇ω dxdy    b · ∇(∇ω) · ∇ j dxdy + 2 ∇ ∂x b1 (∂x u2 + ∂y u1 ) · ∇ j dxdy

 −2 R2

R2

R2

  ∇ ∂x u1 (∂x b2 + ∂y b1 ) · ∇ j dxdy.

(2.15)

Similarly, we take the operator △ on both sides of the micro-rotation equation in (1.5) and take the L2 inner product of this new equation with △m, after integrating by parts in R2 , we obtain 1 d 2 dτ

∥△m(τ )∥ + ∥△ 2 2

mx 22

2 2



∥ + 2∥△m∥ = − R2

△(u · ∇ m)△m dxdy +

 R2

△ω△m dxdy.

(2.16)

Adding the terms (2.14)–(2.16), we find 1 d

∥∇ω(τ ), ∇ j(τ ), △m(τ )∥22 + ∥∇ωy ∥22 + ∥∇ jx ∥22 + ∥△mx ∥22 + 2∥△m∥22   =− ∇ω · ∇ u · ∇ω dxdy + 2 ∇ω · ∇ b · ∇ j dxdy 2 2 R  R   − ∇ j · ∇ u · ∇ j dxdy + 2 ∇ ∂x b1 (∂x u2 + ∂y u1 ) · ∇ j dxdy 2 2 R R      −2 ∇ ∂x u1 (∂x b2 + ∂y b1 ) · ∇ j dxdy − △(u · ∇ m)△m dxdy + 2

2 dτ

R2

R2

= J1 + J2 + J3 + J4 + J5 + J6 + J7 .

△ω△m dxdy R2

(2.17)

Thanks to Lemma 2.1, Cauchy–Schwarz inequality and Young’s inequality, we bound the first term on the right hand side of (2.17),

 J1 = −

∇ω · ∇ u · ∇ω dxdy 2

R =−

R2

∂x u1 ωx2 + ∂y u2 ωy2 dxdy −

 R2

(∂x u2 + ∂y u1 )ωx ωy dxdy

J. Cheng, Y. Liu / Computers and Mathematics with Applications 70 (2015) 66–72 3 2

1 2

1 2

1 2

1 2

1 2

71 1 2

1 2

≤ C ∥ωx ∥2 ∥∂x u1 ∥2 ∥∂xx u1 ∥2 ∥ωxy ∥2 + C ∥∂y u2 ∥2 ∥ωy ∥2 ∥ωxy ∥2 ∥ωyy ∥2 + C ∥∂x u2 + ∂y u1 ∥2 ∥ωx ∥2 ∥ωy ∥2 ∥ωxy ∥2   1 ≤ ∥∇ωy ∥22 + C ∥ω∥22 + ∥ωy ∥22 + 1 ∥∇ω∥22 . (2.18) 12

Similarly, J2 and J3 can be bounded as follows

 J2 = 2

∇ω · ∇ b · ∇ j dxdy R2 1

1

1

1

≤ C ∥∇ b∥2 ∥∇ω∥22 ∥∇ωy ∥22 ∥∇ j∥22 ∥∇ jx ∥22   1 1 ≤ ∥∇ωy ∥22 + ∥∇ jx ∥22 + C ∥∇ b∥22 ∥∇ω∥22 + ∥∇ j∥22 , 12

(2.19)

8

and

 J3 =

∇ j · ∇ u · ∇ j dxdy R2 3

1

1

1

≤ C ∥∇ j∥22 ∥∇ jx ∥22 ∥∇ u∥22 ∥∇ uy ∥22   1 ≤ ∥∇ jx ∥22 + C ∥ω∥22 + ∥ωy ∥22 + 1 ∥∇ j∥22 .

(2.20)

8

For the terms J4 and J5 on the right hand side of (2.17), we have

 J4 = 2

R

2

≤2 R2

  ∇ ∂x b1 (∂x u2 + ∂y u1 ) · ∇ j dxdy (∂x u2 + ∂y u1 )∇∂x b1 · ∇ j dxdy + 2 1

1

 R2

∂x b1 ∇(∂x u2 + ∂y u1 ) · ∇ j dxdy

1

1

≤ C ∥∇∂x b1 ∥2 ∥∇ j∥22 ∥∇ jx ∥22 ∥∂x u2 + ∂y u1 ∥22 ∥∂xy u2 + ∂yy u1 ∥22 1

1

1

1

+ C ∥∂x b1 ∥2 ∥∇ j∥22 ∥∇ jx ∥22 ∥∇(∂x u2 + ∂y u1 )∥22 ∥∇(∂xy u2 + ∂yy u1 )∥22   1 1 ≤ ∥∇ωy ∥22 + ∥∇ jx ∥22 + C ∥ω∥22 + ∥j∥22 + ∥ωy ∥22 + 1 ∥∇ω, ∇ j∥22 , 12

(2.21)

8

and

 J5 = 2 R2

 ≤2 R2

  ∇ ∂x u1 (∂x b2 + ∂y b1 ) · ∇ j dxdy (∂x b2 + ∂y b1 )∇∂x u1 · ∇ j dxdy + 2 1

1

 R2

∂x u1 ∇(∂x b2 + ∂y b1 ) · ∇ j dxdy

1

1

≤ C ∥∂x b2 + ∂y b1 ∥2 ∥∇ j∥22 ∥∇ jx ∥22 ∥∇∂x u1 ∥22 ∥∇∂xy u1 ∥22 1

1

1

1

+ C ∥∇(∂x b2 + ∂y b1 )∥2 ∥∇ j∥22 ∥∇ jx ∥22 ∥∂x u1 ∥22 ∥∂xy u1 ∥22   1 1 ≤ ∥∇ωy ∥22 + ∥∇ jx ∥22 + C ∥ωy ∥22 + ∥ω∥22 + ∥j∥22 + 1 ∥∇ω, ∇ j∥22 , 12

(2.22)

8

where we have used Lemma 2.1, Cauchy–Schwarz inequality, Young’s inequality and the fact 1

∥∇(∂xy u2 + ∂yy u1 )∥22 ≤ ∥∇ωy ∥22 . Similarly, one has

 J6 = −

R

2

≤C

△(u · ∇ m)△m dxdy |∇ u| |D2 m|2 dxdy + C

R2



|D2 u| |∇ m| |D2 m|dxdy

R2 3

1

1

1

1

1

1

1

≤ C ∥△m∥22 ∥△mx ∥22 ∥∇ u∥22 ∥∇ uy ∥22 + C ∥∇ m∥2 ∥△u∥22 ∥△uy ∥22 ∥△m∥22 ∥△mx ∥22   1 1 ≤ ∥∇ωy ∥22 + ∥△mx ∥22 + C ∥ω∥22 + ∥ωy ∥22 + ∥∇ m∥22 + 1 ∥∇ω, △m∥22 . 12

4

(2.23)

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J. Cheng, Y. Liu / Computers and Mathematics with Applications 70 (2015) 66–72

For the last term J7 , we find

 J7 = 2

△ω△m dxdy R2

 = −2 R2

≤

1 12

ωx △mx dxdy + 2

 R2

ωyy △m dxdy

1

∥∇ωy ∥22 + ∥△mx ∥22 + C ∥△m, ∇ω∥22 . 4

(2.24)

Collecting the estimates of J1 –J7 and (2.16), we get d dτ

  ∥∇ω(τ ), ∇ j(τ ), △m(τ )∥22 + ∥∇ωy ∥22 + ∥∇ jx ∥22 + ∥△mx ∥22 ≤ C ∥ω, j, ∇ m, ωy ∥22 + 1 ∥∇ω, ∇ j, △m∥22 .

(2.25)

Thanks to Gronwall’s inequality and (2.13), one deduces

∥∇ω(t ), ∇ j(t ), △m(t )∥ + 2 2

t



∥∇ωy (τ ), ∇ jx (τ ), △mx (τ )∥22 dτ ≤ C ,

(2.26)

0

for any t ≤ T . With above arguments, we obtain the uniform estimates of ∥u(t ), b(t ), m(t )∥2H 2 (R2 ) for any 0 < t ≤ T . Hence, we com-

plete the proof of Theorem 1.1. Acknowledgments

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