Global regularity of 3D magneto-micropolar fluid equations

Applied Mathematics Letters 99 (2020) 105980

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Applied Mathematics Letters www.elsevier.com/locate/aml

Global regularity of 3D magneto-micropolar fluid equations Yinxia Wang, Liuxin Gu School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

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Article history: Received 22 May 2019 Received in revised form 12 July 2019 Accepted 12 July 2019 Available online 19 July 2019

abstract In this paper, we investigate the initial value problem for the three dimensional magneto-micropolar fluid equations for a family of large initial data with finite energy. Global smooth solutions are established by energy method and Serrin’s type non-blow up criterion. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Magneto-micropolar fluid equations Global smooth solutions Serrin’s type non-blow up criterion

1. Introduction We investigate the initial value problem for three dimensional magneto-micropolar fluid equations ⎧ 1 2 ⎪ ⎪ ∂t u − (µ + χ)∆u + u · ∇u − b · ∇b + ∇(p + |b| ) − χ∇ × v = 0, ⎪ ⎪ 2 ⎪ ⎪ ⎨ ∂t v − γ∆v − κ∇∇ · v + 2χv + u · ∇v − χ∇ × u = 0, (1.1) ⎪ ⎪ ⎪ ∂ b − ν∆b + u · ∇b − b · ∇u = 0, t ⎪ ⎪ ⎪ ⎩ ∇ · u = 0, ∇ · b = 0 with initial data t=0:

u = u0 (x), v = v0 (x), b = b0 (x).

(1.2)

Here u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ), b = (b1 , b2 , b3 ) and p are functions of x ∈ R3 and t > 0, which denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. µ is the kinematic viscosity, χ is the vortex viscosity, γ and κ are spin viscosities, and ν1 is the magnetic Reynold. E-mail address: [email protected] (Y. Wang). https://doi.org/10.1016/j.aml.2019.07.011 0893-9659/© 2019 Elsevier Ltd. All rights reserved.

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

2

Due to its important physical background, rich phenomena, mathematical complexity and challenges, the incompressible magneto-micropolar fluid equations have attracted lots of physicists and mathematicians’ attention and many interesting results have been established in [1–18] and [19]. These theory results are mainly concentrated in the following two aspects: on one hand, blow up criterion of smooth solutions or regularity criteria of weak solutions; on the other hand, global regularity. For blow up criteria of smooth solutions or regularity criteria of weak solutions, in the absence of global well-posedness theory, the development of blowup/non-blowup theory is of major importance for both theoretical and practical purposes. Yuan [18] obtained local existence of smooth solutions and blow up criterion of smooth solutions. Yuan [17] established the blow-up criteria for smooth solutions (u, v, b), i.e., u ∈ Lq (0, T ; Lp (R3 )) for 2 q

+

3 p

≤ 1 with 3 < p ≤ ∞, u ∈ C([0, T ); L3 (R3 )) or ∇u ∈ Lq (0, T ; Lp (R3 )) for 2 q

3 2

< p ≤ ∞

3 p

satisfying + ≤ 2. Regularity criterion of weak solution for the magneto-micropolar fluid equations in the 3 Morrey–Campanato space Mp, r and the critical Morrey–Campanato space were established in [2] and [3], respectively. [4] and [19] studied the regularity criterion for the 3D magneto-micropolar fluid equations in Triebel–Lizorkin spaces and Besov spaces, respectively. For global regularity, there are a few results. Cheng and Liu [1] studied the initial value problem for the 2D anisotropic magneto-micropolar fluid flows with mixed partial viscosity and established the global regularity of the 2D anisotropic magneto-micropolar fluid flows with vertical kinematic viscosity, horizontal magnetic diffusion and horizontal vortex viscosity. For more global regularity results for 2D magneto-micropolar fluid equations and its generalized equations, we may refer to [7] and [8]. Inspired by recent work [20] and [21,22] for 3D incompressible Navier–Stokes equations and MHD equations, the main purpose of this paper is to study global existence of smooth solutions to the problem (1.1), (1.2) with the initial data in the following function space

X

=

⏐ { ⏐ (u0 , v0 , b0 )⏐u0 , v0 , b0 ∈ H 1 (R3 ), ∇ · u0 = 0, ∇ · b0 = 0, ζ = min{µ, γ, ν}, 2 ∑

∥u0 ∥L2 ∥∂xi u0 ∥L2 ≤ δ 2 ζ 2 ,

2 ∑

i=1

i=1

2 ∑

2 ∑

2 2

∥b0 ∥L2 ∥∂xi u0 ∥L2 ≤ δ ζ ,

i=1

i=1

2 ∑

2 ∑

i=1

∥v0 ∥L2 ∥∂xi u0 ∥L2 ≤ δ 2 ζ 2 ,

∥u0 ∥L2 ∥∂xi b0 ∥L2 ≤ δ 2 ζ 2 ,

2 ∑

∥u0 ∥L2 ∥∂xi v0 ∥L2 ≤ δ 2 ζ 2 ,

i=1 2 2

∥b0 ∥L2 ∥∂xi b0 ∥L2 ≤ δ ζ ,

2 ∑

∥b0 ∥L2 ∥∂xi v0 ∥L2 ≤ δ 2 ζ 2 ,

i=1

∥v0 ∥L2 ∥∂xi b0 ∥L2 ≤ δ 2 ζ 2 ,

i=1

2 ∑

} ∥v0 ∥L2 ∥∂xi v0 ∥L2 ≤ δ 2 ζ 2 .

i=1

(1.3)

It is well known that the problem (1.1), (1.2) has a unique local smooth solution (u, v, b) for given smooth initial data. If ∥u(t)∥L3 is suitably small, the local solution may be extended to a global one. For the details, we may refer to [17]. If ∥u(t)∥L3 is not small, whether this unique local solution can exist globally is a challenging open problem in mathematical fluid mechanics. To investigate the challenge open problem, we consider the following function space as an example

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

Y

=

3

{ x3 x3 x3 δ δ δ u0 = √ U0 (x1 , x2 , ), v0 = √ V0 (x1 , x2 , ), b0 = √ B0 (x1 , x2 , ) ε ε ε ε ε ε ⏐ ⏐ ⏐U0 , V0 , B0 ∈ H 1 (R3 ), ∇ · U0 = 0, ∇ · B0 = 0, ζ = min{µ, γ, ν}, 2 ∑

2 ∑

∥U0 ∥L2 ∥∂xi U0 ∥L2 ≤ ζ 2 ,

∥U0 ∥L2 ∥∂xi B0 ∥L2 ≤ ζ 2 ,

i=1

i=1

2 ∑

2 ∑

2

∥B0 ∥L2 ∥∂xi U0 ∥L2 ≤ ζ ,

i=1

i=1

2 ∑

2 ∑

∥V0 ∥L2 ∥∂xi U0 ∥L2 ≤ ζ 2 ,

i=1

2 ∑

∥U0 ∥L2 ∥∂xi V0 ∥L2 ≤ ζ 2 ,

i=1 2

∥B0 ∥L2 ∥∂xi B0 ∥L2 ≤ ζ ,

2 ∑

∥B0 ∥L2 ∥∂xi V0 ∥L2 ≤ ζ 2 ,

i=1

∥V0 ∥L2 ∥∂xi B0 ∥L2 ≤ ζ 2 ,

i=1

2 ∑

} ∥V0 ∥L2 ∥∂xi V0 ∥L2 ≤ ζ 2 .

i=1

(1.4) 1

From (1.4), it is easy to find that ∥u0 ∥L3 = δε− 6 ∥U0 ∥L3 , which implies ∥u0 ∥L3 may be as large as possible as ε → 0. Moreover, 2 ∑

∥u0 ∥L2 ∥∂xi u0 ∥L2 = δ 2

i=1

2 ∑

∥U0 ∥L2 ∥∂xi U0 ∥L2 ≤ δ 2 ζ 2 .

(1.5)

i=1

Therefore, it is interesting to study the global existence of smooth solutions to the problem (1.1), (1.2) if the initial data (u0 , v0 , b0 ) ∈ X. The plan of the paper is as follows. Firstly, we give several useful interpolation inequalities and Serrin’s type non-blow up criterion in Section 2, which will play very important roles in the proof of our main result. Section 3 is devoted to prove global existence of smooth solutions to the problem (1.1), (1.2) for a family of large initial data with finite energy by energy method and Serrin’s type non-blow up criterion. Notations. We introduce some notations which are used in this paper. For 1 ≤ p ≤ ∞, Lp = Lp (R3 ) denotes the usual Lebesgue space with the norm ∥ · ∥Lp . The usual Sobolev space of order m is defined by ( ) 12 H m = {u ∈ L2 (R3 )|∇m u ∈ L2 } with the norm ∥u∥H m = ∥u∥2L2 + ∥∇m u∥2L2 . 2. Preliminaries In this section, we state the following interpolation results. The following lemma will play a very important role in proving our main results. Lemma 2.1.

The following interpolation inequalities hold: 3

1

∥f ∥L4 ≤ C∥f ∥L4 2 ∥∇f ∥L4 2 , 1 3

(2.1) 1 3

1 3

∥f ∥L6 ≤ C∥∂x1 f ∥L2 ∥∂x2 f ∥L2 ∥∂x3 f ∥L2

(2.2)

and 1

1

1

1

1

∥f ∥L4 ≤ C∥f ∥L2 2 ∥∂x1 f ∥L8 2 ∥∂x2 f ∥L8 2 ∥∂x1 x3 f ∥L8 2 ∥∂x2 x3 f ∥L8 2 , provided that all norms appearing on the right-hand side of (2.1)–(2.3) are bounded.

(2.3)

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

4

Proof . (2.1), (2.2) are classical, which have been established by Adams and Fourier [23]. (2.3) has been established in [22](see also [24]). For the readers’ convenience, we give the detailed proof. Sobolev embedding theorem and Young’s inequality yield ∥f ∥L4

= ∥∥f ∥L4 (R2x

1 ,x2

)∥ 4 L (Rx3 ) 1

1

≤ C∥∥f ∥L2 2 (R2

x1 ,x2 )

1

∥∂x1 f ∥L4 2 (R2

x1 ,x2 )

1

∥∂x2 f ∥L4 2 (R2

1

1

≤ C∥f ∥L2 2 (R3 ) sup ∥∂x1 f ∥L4 2 (R2

sup ∥∂x2 f ∥L4 2 (R2

x1 ,x2 ) x ∈R 3

x3 ∈R

1

∥

x1 ,x2 ) L4 (R ) x3

1

1

x1 ,x2 )

1

1

≤ C∥f ∥L2 2 ∥∂x1 f ∥L8 2 ∥∂x2 f ∥L8 2 ∥∂x1 x3 f ∥L8 2 ∥∂x2 x3 f ∥L8 2 . Then Lemma 2.1 is proved.

□

To prove global existence of smooth solutions, we also need the following Serrin’s type non-blow up criterion, which has been established by Yuan [17]. Lemma 2.2 ([17]). Assume that u0 , v0 , b0 ∈ H 1 (R3 ) and satisfy ∇ · u0 = ∇ · b0 = 0. If ∫ 0

where [0, T ].

2 q

T

∥u(t)∥qLp dt < ∞, +

3 p

(2.4)

≤ 1 with 3 < p ≤ ∞. Then the solution (u, v, b) to the problem (1.1), (1.2) remains smooth on

3. Global smooth solutions In this section, we investigate global existence of smooth solutions to the problem (1.1), (1.2). By energy method and Serrin’s type non-blow up criterion, we establish global regularity of smooth solutions to the problem (1.1), (1.2). We state our main result as follows. Theorem 3.1. Assume that (u0 , v0 , b0 ) ∈ X for some constant δ > 0. Then there exists a constant δ0 independent of µ, γ, ν, κ, χ such that if δ ≤ δ0 , the problem (1.1), (1.2) has a unique global smooth solution (u, v, b). Proof . Multiplying the first equation of (1.1) by u and integrating with respect to x on R3 , using integration by parts, we obtain ∫ ∫ 1 d 2 2 ∥u(t)∥L2 + (µ + χ)∥∇u(t)∥L2 = b · ∇b · udx + χ (∇ × v) · udx. (3.1) 2 dt R3 R3 Similarly, we get 1 d ∥v(t)∥2L2 + γ∥∇v(t)∥2L2 + κ∥∇ · v∥2L2 + 2χ∥v∥2L2 2 dt

∫ =

(∇ × u) · vdx

χ

(3.2)

R3

and 1 d ∥b(t)∥2L2 + ν∥∇b(t)∥2L2 2 dt

∫ b · ∇u · bdx.

= R3

(3.3)

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

5

Summing up (3.1)–(3.3), we deduce that 1 d (∥u(t)∥2L2 + ∥v(t)∥2L2 + ∥b(t)∥2L2 ) + (µ + χ)∥∇u(t)∥2L2 + 2 dt γ∥∇v(t)∥2L2 + κ∥∇ · v∥2L2 + 2χ∥v∥2L2 + ν∥∇b(t)∥2L2 ∫ ∫ = b · ∇b · udx + χ (∇ × v) · udx+ R3

R3

∫

∫ (∇ × u) · vdx +

χ

(3.4)

R3

b · ∇u · bdx. R3

By integration by parts and Cauchy inequality, we obtain ∫ ∫ χ (∇ × v) · udx + χ (∇ × u) · vdx ≤ χ∥∇u∥2L2 + χ∥v∥2L2 .

(3.5)

Using integration by parts, we obtain ∫ ∫ b · ∇b · udx + b · ∇u · bdx = 0.

(3.6)

R3

R3

R3

R3

Combining (3.4)–(3.6) yields 1 d (∥u(t)∥2L2 + ∥v(t)∥2L2 + ∥b(t)∥2L2 ) + µ∥∇u(t)∥2L2 + 2 dt γ∥∇v(t)∥2L2

+ κ∥∇ ·

v∥2L2

+

χ∥v(t)∥2L2

+

ν∥∇b(t)∥2L2

(3.7)

≤ 0.

Integrating (3.7) with respect to t, we have ∫ t ∥u(t)∥2L2 + ∥v(t)∥2L2 + ∥b(t)∥2L2 + 2 (µ∥∇u(τ )∥2L2 + γ∥∇v(τ )∥2L2 )dτ + 0

∫

t

2 0

(3.8)

(κ∥∇ · v(τ )∥2L2 + χ∥v(τ )∥2L2 + ν∥∇b(τ )∥2L2 )dτ ≤ ∥u0 ∥2L2 + ∥v0 ∥2L2 + ∥b0 ∥2L2 .

Differentiating (1.1) with respect to xi , we arrive at ⎧ 1 ⎪ ⎪ ∂t ∂i u − (µ + χ)∆∂i u + u · ∇∂i u + ∂i ∇(p + |b|2 ) − χ∇ × ∂i v ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ = −∂ u · ∇u + ∂ b · ∇b + b · ∇∂ b, i i i ⎨ ∂t ∂i v − γ∆∂i v − κ∇∇ · ∂i v + 2χ∂i v + u · ∇∂i v − χ∇ × ∂i u = −∂i u · ∇v, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ∂i b − ν∆∂i b + u · ∇∂i b = −∂i u · ∇b + ∂i b · ∇u + b · ∇∂i u, ⎪ ⎪ ⎪ ⎩ ∇ · u = 0, ∇ · b = 0

(3.9)

Taking the inner product with (3.9) and (∂i u, ∂i v, ∂i b), we have

=

d (∥∂i u∥2L2 + ∥∂i v∥2L2 + ∥∂i b∥2L2 ) + (µ + χ)∥∇∂i u∥2L2 + γ∥∇∂i v∥2L2 + dt κ∥∇ · ∂i v∥2L2 + 2χ∥∂i v∥2L2 + ν∥∇∂i b∥2L2 ∫ ∫ ∫ χ (∇ × ∂i v) · ∂i u + χ (∇ × ∂i u) · ∂i v − ∂i u · ∇u · ∂i udx+ R3 R3 R3 ∫ ∫ ∫ ∂i b · ∇b · ∂i udx − ∂i u · ∇v · ∂i vdx − ∂i u · ∇b · ∂i bdx+ R3 R3 R3 ∫ ∂i b · ∇u · ∂i bdx R3

=:

I1 + I2 + I3 + I4 + I5 + I6 .

(3.10)

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

6

By integration by parts and Cauchy inequality, we obtain I1 ≤ χ∥∇∂i u∥2L2 + χ∥∂i v∥2L2 .

(3.11)

It follows from integration by parts, H¨ older inequality, (2.1), (2.3) and Cauchy inequality that I2 ≤ ∥u∥L4 ∥∂i u∥L4 ∥∇∂i u∥L2 1

1

1

1

1

1

7

≤

C∥u∥L2 2 ∥∂x1 u∥L8 2 ∥∂x2 u∥L8 2 ∥∂x1 x3 u∥L8 2 ∥∂x2 x3 u∥L8 2 ∥∂i u∥L4 2 ∥∇∂i u∥L4 2

≤

C∥u∥L2 2 ∥∂i u∥L2 2 ∥∇∂i u∥2L2 .

1

(3.12)

1

Similarly, we obtain I3

= −

3 ∫ ∑ j,k=1

≤

∂i bj bk ∂j ∂i uk dx

R3

∥b∥L4 ∥∂i b∥L4 ∥∇∂i u∥L2 1

1

1

1

1

1

3

≤ C∥b∥L2 2 ∥∂x1 b∥L8 2 ∥∂x2 b∥L8 2 ∥∂x1 x3 b∥L8 2 ∥∂x2 x3 b∥L8 2 ∥∂i b∥L4 2 ∥∇∂i b∥L4 2 ∥∇∂i u∥L2

(3.13)

1

1

≤ C∥b∥L2 2 ∥∂i b∥L2 2 ∥∇∂i b∥L2 ∥∇∂i u∥L2 1

1

≤ C∥b∥L2 2 ∥∂i b∥L2 2 (∥∇∂i u∥2L2 + ∥∇∂i b∥2L2 ),

I4

= −

3 ∫ ∑ j,k=1

∂i uj vk ∂j ∂i vk dx

R3

≤

∥v∥L4 ∥∂i u∥L4 ∥∇∂i v∥L2

≤

C∥v∥L2 2 ∥∂x1 v∥L8 2 ∥∂x2 v∥L8 2 ∥∂x1 x3 v∥L8 2 ∥∂x2 x3 v∥L8 2 ∥∂i u∥L4 2 ∥∇∂i u∥L4 2 ∥∇∂i v∥L2

1

1

1

1

1

1

1

3

1

1

3

(3.14)

5

≤ C∥v∥L2 2 ∥∂i v∥L4 2 ∥∂i u∥L4 2 ∥∇∂i u∥L4 2 ∥∇∂i v∥L4 2 1

1

1

≤ C∥v∥L2 2 ∥∂i v∥L4 2 ∥∂i u∥L4 2 (∥∇∂i u∥2L2 + ∥∇∂i v∥2L2 ) 1

1

1

1

≤ C(∥v∥L2 2 ∥∂i v∥L2 2 + ∥v∥L2 2 ∥∂i u∥L2 2 )(∥∇∂i u∥2L2 + ∥∇∂i v∥2L2 ), 1

I5

1

1

1

≤ C(∥b∥L2 2 ∥∂i b∥L2 2 + ∥b∥L2 2 ∥∂i u∥L2 2 )(∥∇∂i u∥2L2 + ∥∇∂i b∥2L2 )

(3.15)

and I6

= −

3 ∫ ∑ j,k=1

≤

∂i bj uk ∂j ∂i bk dx

R3

∥u∥L4 ∥∂i b∥L4 ∥∇∂i b∥L2 1

1

1

1

1

1

7

≤ C∥u∥L2 2 ∥∂x1 u∥L8 2 ∥∂x2 u∥L8 2 ∥∂x1 x3 u∥L8 2 ∥∂x2 x3 u∥L8 2 ∥∂i b∥L4 2 ∥∇∂i b∥L4 2 1 2

1 4

1 4

1

1

1

1 4

7 4

≤ C∥u∥L2 ∥∂i u∥L2 ∥∂i b∥L2 ∥∇∂i u∥L2 ∥∇∂i b∥L2 ≤ C∥u∥L2 2 ∥∂i u∥L4 2 ∥∂i b∥L4 2 (∥∇∂i u∥2L2 + ∥∇∂i b∥2L2 ) 1

1

1

1

≤ C(∥u∥L2 2 ∥∂i u∥L2 2 + ∥u∥L2 2 ∥∂i b∥L2 2 )(∥∇∂i u∥2L2 + ∥∇∂i b∥2L2 ).

(3.16)

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

7

Inserting (3.11)–(3.16) into (3.10) and summing for i = 1, 2 yields 2 2 ∑ 1 d ∑ (∥∂xi u∥2L2 + ∥∂xi v∥2L2 + ∥∂xi b∥2L2 ) + (µ∥∇∂i u∥2L2 + γ∥∇∂i v∥2L2 + 2 dt i=1 i=1

κ∥∇ · ∂i v∥2L2 + χ∥∂i v∥2L2 + ν∥∇∂i b∥2L2 ) 2 ∑ 1 1 1 1 1 1 1 1 ≤ C (∥u∥L2 2 ∥∂i u∥L2 2 + ∥b∥L2 2 ∥∂i b∥L2 2 + ∥v∥L2 2 ∥∂i v∥L2 2 + ∥v∥L2 2 ∥∂i u∥L2 2 +

(3.17)

i=1 1

1

1

1

∥u∥L2 2 ∥∂i b∥L2 2 + ∥b∥L2 2 ∥∂i u∥L2 2 )(∥∇∂i u∥2L2 + ∥∇∂i v∥2L2 + ∥∇∂i b∥2L2 ). We claim that for any t ∈ [0, T ], it holds 2 ( ∑

∥u∥L2 ∥∂xi u∥L2 + ∥u∥L2 ∥∂xi b∥L2 + ∥u∥L2 ∥∂xi v∥L2 +

i=1

∥b∥L2 ∥∂xi u∥L2 + ∥b∥L2 ∥∂xi b∥L2 + ∥b∥L2 ∥∂xi v∥L2 + ) ∥v∥L2 ∥∂xi u∥L2 + ∥v∥L2 ∥∂xi b∥L2 + ∥v∥L2 ∥∂xi v∥L2 ≤ 9δ 2 ζ 2 .

(3.18)

provided that the initial value (u0 , v0 , b0 ) ∈ X and δ is small enough. Owing to (3.17) and a priori assumption (3.18), we arrive at 2 2 ∑ 1 d ∑ (∥∂xi u∥2L2 + ∥∂xi v∥2L2 + ∥∂xi b∥2L2 ) + (µ∥∇∂i u∥2L2 + γ∥∇∂i v∥2L2 + 2 dt i=1 i=1

κ∥∇ · ∂i v∥2L2 + χ∥∂i v∥2L2 + ν∥∇∂i b∥2L2 )

(3.19)

≤ C1 δζ(∥∇∂i u∥2L2 + ∥∇∂i v∥2L2 + ∥∇∂i b∥2L2 ). Taking δ0 =

1 2C1

suitably small, it follows from (3.19) that 2

2

1 d ∑ 1∑ (∥∂xi u∥2L2 + ∥∂xi v∥2L2 + ∥∂xi b∥2L2 ) + (µ∥∇∂i u∥2L2 + γ∥∇∂i v∥2L2 + 2 dt i=1 2 i=1 κ∥∇ ·

∂i v∥2L2

+

χ∥∂i v∥2L2

+

ν∥∇∂i b∥2L2 )

(3.20)

≤ 0.

Integrating (3.20) with respect to τ ∈ [0, t], we deduce that 2 ∑

(∥∂xi u∥2L2 + ∥∂xi v∥2L2 + ∥∂xi b∥2L2 ) +

i=1

i=1

κ∥∇ · ≤

2 ∑

2 ∫ ∑

∂i v∥2L2

+

χ∥∂i v∥2L2

+

ν∥∇∂i b∥2L2 )dτ

0

t

(µ∥∇∂i u∥2L2 + γ∥∇∂i v∥2L2 + (3.21)

(∥∂xi u0 ∥2L2 + ∥∂xi v0 ∥2L2 + ∥∂xi b0 ∥2L2 ).

i=1

(3.8), (3.21) and (u0 , v0 , b0 ) ∈ X immediately imply (3.18) holds. Thus, we complete the proof of the claim (3.18).

Y. Wang and L. Gu / Applied Mathematics Letters 99 (2020) 105980

8

Finally, we shall prove global regularity by the non-blow up criterion (2.4) in Lemma 2.2. (2.2) and (3.8), (3.21) immediately imply that ∫ t ∥u(τ )∥4L6 dτ 0

∫ ≤

t

C 0

≤

4

4

4

∥∂x1 u(τ )∥L3 2 ∥∂x2 u(τ )∥L3 2 ∥∂x3 u(τ )∥L3 2 dτ 4 3

2 3

C( sup ∥∂x1 u(τ )∥L2 ∥∂x2 u(τ )∥L2 ) 0≤τ ≤t

≤

C

2 ∑ i=1

≤

C

2 ∑

∥∂xi u∥2L2

∫ 0

∫ 0

t

2

4

∥∂x2 u(τ )∥L3 2 ∥∂x3 u(τ )∥L3 2 dτ

(3.22)

t

∥∇u(τ )∥2L2 dτ

(∥∂xi u0 ∥2L2 + ∥∂xi v0 ∥2L2 + ∥∂xi b0 ∥2L2 )(∥u0 ∥2L2 + ∥v0 ∥2L2 + ∥b0 ∥2L2 ).

i=1

Therefore, Lemma 2.2 and (3.22) entail that the problem (1.1), (1.2) has a unique global smooth solution. Theorem 3.1 is proved. □ Acknowledgment The author is partially supported by the NNSF of China (Grant No. 11101144). References [1] J. Cheng, Y. Liu, Global regularity of the 2D magnetic micropolar fluid flows with mixed partial viscosity, Comput. Math. Appl. 70 (2015) 66–72. [2] S. Gala, Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space, Nonlinear Differential Equations Appl. 17 (2010) 181–194. [3] J.B. Geng, X.C. Chen, S. Gala, On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey–Campanato space, Commun. Pure Appl. Anal. 10 (2011) 583–592. [4] C.C. Guo, Z.J. Zhang, J.L. Wang, Regularity criteria for the 3D magneto-micropolar fluid equations in Besov spaces with negative indices, Appl. Math. Comput. 218 (2012) 10755–10758. [5] M. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solutions, Math. Nachr. 188 (1997) 301–319. [6] M. Rojas-Medar, J. Boldrini, Magneto-micropolar fluid motion: existence of weak solutions, Rev. Mat. Complut. 11 (1998) 443–460. [7] D. Regmi, J.H. Wu, Global regularity for the 2D magneto-micropolar equations with partial dissipation, J. Math. Study 49 (2016) 169–194. [8] H.F. Shang, J.F. Zhao, Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion, Nonlinear Anal. 150 (2017) 194–209. [9] Y.X. Wang, Regularity criterion for a weak solution to the three-dimensional magneto-micropolar fluid equations, Bound. Value Probl. 58 (2013) 12. [10] Y.X. Wang, Blow-up criteria of smooth solutions to the three-dimensional magneto-micropolar fluid equations, Bound. Value Probl. 118 (2015) 10. [11] Y.-Z. Wang, Y.X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity, Math. Methods Appl. Sci. 34 (2011) 2125–2135. [12] Y.-Z. Wang, Y.F. Li, Y.X. Wang, Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity, Bound. Value Probl. 11 (2011) 11. [13] Y.X. Wang, K.Y. Wang, Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity, Nonlinear Anal. RWA 33 (2017) 348–362. [14] Z.Y. Xiang, H.Z. Yang, On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative, Bound. Value Probl. 139 (2012) 14. [15] F.Y. Xu, Regularity criterion of weak solution for the 3D magneto-micropolar fluid equations in Besov spaces, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 2426–2433. [16] K. Yamazaki, Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity, Discrete Contin. Dyn. Syst. 35 (2015) 2193–2207.

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[17] B. Yuan, Regularity of weak solutions to magneto-micropolar fluid equations, Acta Math. Sci. 30 (2010) 1469–1480. [18] J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. Methods Appl. Sci. 31 (2008) 1113–1130. [19] Z. Zhang, Z. Yao, X. Wang, A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel–Lizorkin spaces, Nonlinear Anal. 74 (2011) 2220–2225. [20] Z. Lei, F.H. Lin, Y. Zhou, Structure of helicity and global solutions of incompressible Navier–Stokes equation, Arch. Ration. Mech. Anal. 218 (2015) 1417–1430. [21] Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations 259 (2015) 3202–3215. [22] F. Wang, On global regularity of incompressile MHD equations in R3 , J. Math. Anal. Appl. 454 (2017) 936–941. [23] R.A. Adams, J.F. Fournier, Sobolev Spaces, second ed., in: Pure Appl. Math. (Amst.), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. [24] K.Y. Wang, On global regularity of incompressile Navier–Stokes equations in R3 , Commun. Pure Appl. Anal. 8 (2009) 1067–1072.