A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion

Nonlinear Analysis: Real World Applications 21 (2015) 197–206

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion Lei Zhang a , Shan Li b,∗ a

College of Computer Science, Sichuan University, Chengdu 610064, PR China

b

Business School, Sichuan University, Chengdu 610064, PR China

article

abstract

info

Article history: Received 6 January 2014 Received in revised form 6 July 2014 Accepted 14 July 2014

This paper concerns the conditional global regularity of incompressible MHD equations with horizontal dissipation and horizontal magnetic diffusion in two dimension. When only horizontal dissipation and horizontal magnetic diffusion are present, there is no control on the vertical derivatives of velocity field and magnetic field, which is the main difficulty to establish the global regularity. In this paper, we establish a global regularity criterion in terms of one entry of the velocity gradient tensor or one entry of the magnetic field gradient tensor, which extends the recent work (Fan and Ozawa, 2014). © 2014 Elsevier Ltd. All rights reserved.

Keywords: Regularity criterion 2D MHD system Classical solution Horizontal dissipation and magnetic diffusion

1. Introduction and the main results In this paper, we study the incompressible Magnetohydrodynamic fluids with horizontal dissipation and horizontal magnetic diffusion in two-dimension. The 2D viscous MHD flows can be described by the two-dimensional MHD equations ut + u · ∇ u − ν1 ∂xx u − ν2 ∂yy u + ∇ p = b · ∇ b, bt + u · ∇ b − η1 ∂xx b − η2 ∂yy b = b · ∇ u, ∇ · u = ∇ · b = 0,



(1)

where u = (u1 , u2 ), b = (b1 , b2 ), p are the unknown functions of the space (x, y) ∈ R2 and time t ≥ 0 representing the velocity field, the magnetic field and the scalar pressure. The nonnegative parameters ν1 , ν2 are dissipation coefficients and η1 , η2 are the magnetic diffusion coefficients. The system (1) is solved subjected to some given initial data u(x, y, 0) = u0 (x, y) and

b(x, y, 0) = b0 (x, y),

(2)

with divergence free conditions divu0 = 0 and divb0 = 0. Over past many years, the study for the MHD equations has been attracting many mathematicians, and a lot of important results have been achieved (see [1–11] and references therein). For the 2D incompressible MHD equations (1) with full dissipation and magnetic diffusion (see [5,12]) the global existence and the uniqueness of the classical solution have been obtained for any initial data (u0 , b0 ) ∈ H m with m ≥ 2. However, whether or not classical solutions of the 2D incompressible MHD flows without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue in this field.

∗

Corresponding author. E-mail address: [email protected] (S. Li).

http://dx.doi.org/10.1016/j.nonrwa.2014.07.005 1468-1218/© 2014 Elsevier Ltd. All rights reserved.

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L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

Cao and Wu established the global well-posedness of the Cauchy problem (1)–(2) with ν1 = η2 = 0, ν2 > 0 and η1 > 0 in the pioneer work [13]. As a direct consequence of this result, the global regularity result has been obtained for (1)–(2) with ν2 = η1 = 0, ν1 > 0 and η2 > 0 in [13]. For the 2D MHD equations (1) with horizontal dissipation and horizontal magnetic diffusion, some regularity criteria have been established in [14,15]. Namely, when ν1 > 0, η1 > 0, ν2 = η2 = 0, Cao et al. showed that the Cauchy problem (1)–(2) exists a unique global classical solution, provided that

 ∥u1 (t )∥L2r (R2 ) + ∥b1 (t )∥L2r (R2 ) ≤ B0 (t ) r log r + B1 , for 1 ≤ r < ∞, where B0 (t ) is a smooth function of t and B1 depends only on ∥u0 ∥L2r (R2 ) and ∥b0 ∥L2r (R2 ) . Very recently, in [15] Fan and Ozawa established the regularity criterion as

  ∂y u1 , ∂y b1 ∈ L2 0, T ; L2 (R2 ) .

(3)

On another side, in the significant work [16], Cao and Titi provided a sufficient condition in terms of only one of the entries of the gradient tensor to 3D Navier–Stokes equations. Furthermore, the similar results have been extended to 3D MHD system in [4,17]. Motivated by those works, the main goal of this paper is to establish some regularity criteria in terms of only one of the four entries of the gradient velocity tensor or gradient magnetic field tensor and improve the criterion (3). Theorem 1.1. Consider the Cauchy problem (1)–(2) with ν1 > 0, η1 > 0, ν2 = η2 = 0. Assume that (u0 , b0 ) ∈ H s (R2 ) (s > 2) with ∇ · u0 = 0 and ∇ · b0 = 0. Let (u, b) be a unique local smooth solution to the Cauchy problem (1)–(2) in (0, T ) for some finite T > 0, then the solution (u, b) is a strong solution on the interval [0, T ], provided that

  ∂y u1 ∈ L2 0, T ; L2 (R2 )

(4)

  ∂y b1 ∈ L2 0, T ; L2 (R2 ) .

(5)

or

Remark 1. The results indicate that the bound of the only one of the four entries of the gradient of velocity field or only one of the four entries of the gradient of magnetic field can guarantee the global regularity of the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. As a direct consequence, we can obtain the similar criteria to the MHD system (1) with vertical dissipation and vertical magnetic diffusion as follows. Corollary 1. Consider the Cauchy problem (1)–(2) with ν2 > 0, η2 > 0, ν1 = η1 = 0. Assume that (u0 , b0 ) ∈ H s (R2 ) (s > 2) with ∇ · u0 = 0 and ∇ · b0 = 0. Let (u, b) be a unique local smooth solution to the Cauchy problem (1)–(2) in (0, T ) for some finite T > 0, then the solution (u, b) is a strong solution on the interval [0, T ], provided that

  ∂x u2 ∈ L2 0, T ; L2 (R2 ) or

  ∂x b2 ∈ L2 0, T ; L2 (R2 ) . Remark 2. We would like to mention the recent works on the 2D MHD system with mixed partial dissipation and partial magnetic diffusion. Some regularity criteria have been established in the 2D MHD system without magnetic diffusion (ν1 = ν2 > 0 and η1 = η2 = 0) in [18–20]. For the 2D MHD system without dissipation (ν1 = ν2 = 0 and η1 = η2 > 0), local existence of the classical solutions has been shown by Kozono in [21] and the global existence of classical solutions has been shown in [22] for small datum b0 . Furthermore, Lei and Zhou gave a regularity criterion for 2D MHD equations without dissipation in [23]. 2. The proof of theorem In this section, we will prove the main result. To the end of this paper, we adopt the following simplified notations



 fdxdy = R2

fdxdy,

and

∥f ∥2 = ∥f ∥L2 (R2 ) . Before the proof, we introduce an important inequality (see [13] for details),

 R2

1

1

1

1

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

for every f , g , h, gx , hy ∈ L2 (R2 ).

(6)

L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

199

The key point of the proof is to establish the uniform bound of H 2 norm for u and b. Let ω = ∇ × u and j = ∇ × b be the vorticity of velocity field and the current density, respectively. Thanks to the equivalence between ∥(ω, j)∥H 1 (R2 ) and ∥(∇ u, ∇ b)∥H 1 (R2 ) , we will focus on the uniform estimates of the H1 -norm of ω and j throughout the proof. Proof. Step 1. The standard method can be derived by the energy estimate as follows. A direct computation gives that 1 d 

     ∥u(τ )∥22 + ∥b(τ )∥22 + ν1 ∥∂x u1 (τ )∥22 + ∥∂x u2 (τ )∥22 + η1 ∥∂x b1 (τ )∥22 + ∥∂x b2 (τ )∥22 = 0,

2 dτ

(7)

for any τ > 0. By integration in (0, t ) with respect to τ , we get the following energy equality 1 2

 1 ∥u(t )∥22 + ∥b(t )∥22 − ∥u0 , b0 ∥22 + ν1 2

+ η1

t



  ∥∂x u1 (τ )∥22 + ∥∂x u2 (τ )∥22 dτ 0

t





 ∥∂x b1 (τ )∥22 + ∥∂x b2 (τ )∥22 dτ = 0,

(8)

0

for any t ≥ 0, which implies the elementary energy inequality sup 0≤t ≤T

  ∥u(t )∥22 + ∥b(t )∥22 + 2ν1

T





 ∥∂x u1 (t )∥22 + ∥∂x u2 (t )∥22 dt

0

+ 2η1

T



  ∥∂x b1 (t )∥22 + ∥∂x b2 (t )∥22 dt ≤ C ,

(9)

0

for any T ≥ 0, where the constant C which will appear hereafter depends only on the initial data. It is easy to see that the vorticity ω satisfies that

  ωt + u · ∇ω − b · ∇ j = ν1 ∂xxx u2 − ∂xxy u1 ,

(10)

and current density j satisfies jt + u · ∇ j − b · ∇ω = η1 ∂xxx b2 − ∂xxy b1 + 2∂x b1 (∂x u2 + ∂y u1 ) − 2∂x u1 (∂x b2 + ∂y b1 ).





(11)

Then, multiplying Eq. (10) by ω and Eq. (11) by j, adding the resulting equations and integrating in R2 give that

   ∥ω(t )∥22 + ∥j(t )∥22 + ν1 ∂xxy u1 ω − ∂xxx u2 ωdxdy + η1 ∂xxy b1 j − ∂xxx b2 jdxdy 2 dt     = 2 ∂x b1 ∂y u1 jdxdy + 2 ∂x b1 ∂x u2 jdxdy − 2 ∂x u1 ∂y b1 jdxdy − 2 ∂x u1 ∂x b2 jdxdy

1 d 

=I=

4 

Ik .

(12)

k=1

Furthermore, it follows from the divergence free conditions and integration by parts, we have 1 d  2 dt

     ∥ω(t )∥22 + ∥j(t )∥22 + ν1 ∥∇∂x u1 ∥22 + ∥∇∂x u2 ∥22 + η1 ∥∇∂x b1 ∥22 + ∥∇∂x b2 ∥22 = I ,

(13)

here we have used the facts

ωx = ∂xx u2 + ∂yy u2 and ωy = −∂xx u1 − ∂yy u1 ,

(14)

jx = ∂xx b2 + ∂yy b2 ,

(15)

and and

jy = −∂xx b1 − ∂yy b1 .

A direct calculation gives that

∥∇∂x u1 ∥22 = ∥∂xx u1 ∥22 + ∥∂yy u2 ∥22 , ∥∇∂x u2 ∥22 = ∥∂xx u1 ∥22 + ∥∂xx u2 ∥22 , ∥∇∂x b1 ∥22 = ∥∂xx b1 ∥22 + ∥∂yy b2 ∥22 , and

∥∇∂x b2 ∥22 = ∥∂xx b1 ∥22 + ∥∂xx b2 ∥22 ,

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L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

inserting these into (13) yields that 1 d 

   ∥ω(t )∥22 + ∥j(t )∥22 + ν1 2 ∥∂xx u1 ∥22 + ∥∂xx u2 ∥22 + ∥∂yy u2 ∥22   + η1 2 ∥∂xx b1 ∥22 + ∥∂xx b2 ∥22 + ∥∂yy b2 ∥22 = I .

2 dt

(16)

Thanks to the inequality (6), the Cauchy–Schwarz inequality and incompressibility conditions, we have



∂x b1 ∂y u1 jdxdy

I1 = 2

1

1

1

1

≤ C ∥∂x b1 ∥22 ∥∂xy b1 ∥22 ∥∂y u1 ∥22 ∥∂xy u1 ∥22 ∥j∥2 η1 ν1 ≤ ∥∂xy b1 ∥22 + ∥∂xy u1 ∥22 + C ∥∂x b1 ∥2 ∥j∥22 ∥∂y u1 ∥2 8

≤

6

η1

2 yy b2 2

∥∂

8

∥ +

ν1 6

  ∥∂yy u2 ∥22 + C ∥∂x b1 ∥22 + ∥∂y u1 ∥22 ∥j∥22 .

(17)

Similarly,



∂x b1 ∂x u2 jdxdy

I2 = 2

1

1

1

1

≤ C ∥∂x b1 ∥22 ∥∂xx b1 ∥22 ∥∂x u2 ∥22 ∥∂xy u2 ∥22 ∥j∥2 3η1 ν1 ∥∂xx b1 ∥22 + ∥∂xy u2 ∥22 + C ∥∂x b1 ∥2 ∥j∥22 ∥∂x u2 ∥2 ≤ ≤

8 3η1

2

∥∂

8

2 xx b1 2

∥ +

ν1 2

  ∥∂xx u1 ∥22 + C ∥∂x b1 ∥22 + ∥∂x u2 ∥22 ∥j∥22 .

(18)

It follows from the incompressible conditions and integration by parts that

 I3 = −2

∂x u1 ∂y b1 jdxdy



= −2 ∂x u1 ∂y b1 (∂x b2 − ∂y b1 )dxdy  = 2 ∂y u2 ∂y b1 ∂x b2 − ∂y u2 (∂y b1 )2 dxdy = I31 + I32 ,

(19)

moreover,



∂y u2 ∂y b1 ∂x b2 dxdy  = −2 ∂yy u2 b1 ∂x b2 + ∂y u2 b1 ∂xy b2 dxdy

I31 = 2

1

1

1

1

1

1

1

1

≤ C ∥∂yy u2 ∥2 ∥b1 ∥22 ∥∂x b1 ∥22 ∥∂x b2 ∥22 ∥∂xy b2 ∥22 + C ∥∂y u2 ∥22 ∥∂yy u2 ∥22 ∥b1 ∥22 ∥∂x b1 ∥22 ∥∂xy b2 ∥2  ν1  ∂yy u2 2 + 3η1 ∥∂xy b2 ∥2 + C ∥b1 ∥2 ∥∂x b1 ∥2 ∥∂x b2 ∥2 ≤ 2 2 2 2 2 6

8

2 3η1 ν1  + ∂yy u2 2 + ∥∂xy b2 ∥22 + C ∥b1 ∥22 ∥∂x b1 ∥22 ∥∂y u2 ∥22 6

8

   ν1  ∂yy u2 2 + 3η1 ∥∂xx b1 ∥2 + C ∥b∥2 ∥∂x b1 ∥2 ∥ω∥2 + ∥j∥2 , ≤ 2 2 2 2 2 2 3

4

and

 I32 = −2



∂y u2 (∂y b1 )2 dxdy

∂x u1 (∂y b1 )2 dxdy  = −4 u1 ∂y b1 ∂xy b1 dxdy

=2

(20)

L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206 1 2

1 2

1 2

201

1 2

≤ C ∥u1 ∥2 ∥∂y u1 ∥2 ∥∂y b1 ∥2 ∥∂xy b1 ∥2 ∥∂xy b1 ∥2 η1 ≤ ∥∂xy b1 ∥22 + C ∥u1 ∥22 ∥∂y u1 ∥22 ∥∂y b1 ∥22 8

≤

η1 8

∥∂yy b2 ∥22 + C ∥u∥22 ∥∂y u1 ∥22 ∥j∥22 ,

therefore I3 ≤

     ν1  ∂yy u2 2 + 3η1 ∥∂xx b1 ∥2 + η1 ∥∂yy b2 ∥2 + C ∥u∥2 + ∥b∥2 ∥∂x b1 ∥2 + ∥∂y u1 ∥2 ∥ω∥2 + ∥j∥2 . 2 2 2 2 2 2 2 2 2 3

4

8

(21)

Finally,



∂x u1 ∂x b2 jdxdy

I4 = −2



= −2 ∂x u1 ∂x b2 (∂x b2 − ∂y b1 )dxdy  = 2 ∂y u2 (∂x b2 )2 − ∂y u2 ∂x b2 ∂y b1 dxdy  = −2 2u2 ∂x b2 ∂xy b2 − ∂xy u2 b2 ∂y b1 − ∂y u2 b2 ∂xy b1 dxdy = I41 + I42 + I43 .

(22)

Furthermore, using the same technique for the estimate of I1 in (17) yields



u2 ∂x b2 ∂xy b2 dxdy

I41 = −4

1

1

1

1

≤ C ∥u2 ∥22 ∥∂x u2 ∥22 ∥∂x b2 ∥22 ∥∂xy b2 ∥22 ∥∂xy b2 ∥2 3η1 ∥∂xy b2 ∥22 + C ∥u2 ∥22 ∥∂x b2 ∥22 ∥∂x u2 ∥22 ≤ ≤

8 3η1 8

I42 = 2

∥∂xx b1 ∥22 + C ∥u2 ∥22 ∥∂x u2 ∥22 ∥j∥22 ,

(23)

∂xy u2 b2 ∂y b1 dxdy 1

1

1

1

≤ C ∥∂xy u2 ∥2 ∥b2 ∥22 ∥∂y b2 ∥22 ∥∂y b1 ∥22 ∥∂xy b1 ∥22 ν1 η1 ≤ ∥∂xy u2 ∥22 + ∥∂xy b1 ∥22 + C ∥b2 ∥22 ∥∂y b2 ∥22 ∥∂y b1 ∥22 2

≤

ν1 2

8

2 xx u1 2

∥ +

∥∂

η1 8

∥∂yy b2 ∥22 + C ∥b2 ∥22 ∥∂x b1 ∥22 ∥j∥22 ,

(24)

and



∂y u2 b2 ∂xy b1 dxdy

I43 = −2

1

1

1

1

≤ C ∥∂y u2 ∥22 ∥∂xy u2 ∥22 ∥b2 ∥22 ∥∂y b2 ∥22 ∥∂xy b1 ∥2 ≤ ε∥∂xy u2 ∥22 + ε∥∂xy b1 ∥22 + C ∥∂y u2 ∥22 ∥b2 ∥22 ∥∂y b2 ∥22 ν1 η1 ∥∂xx u1 ∥22 + ∥∂yy b2 ∥22 + C ∥b2 ∥22 ∥∂x b1 ∥22 ∥ω∥22 , ≤ 2

(25)

8

therefore I4 ≤ ν1 ∥∂xx u1 ∥22 +

η1

∥∂xx b1 ∥22 +

     η1  ∂yy b2 2 + C ∥u∥2 + ∥b∥2 ∥∂x u2 ∥2 + ∥∂x b1 ∥2 ∥ω∥2 + ∥j∥2 . 2 2 2 2 2 2 2

8 4 Hence, inserting (17), (18), (21) and (26) into (16) yields that

(26)

d 

  2     ∥ω(t )∥22 + ∥j(t )∥22 + ν1 ∥∂xx u∥22 + ∂yy u2 2 + η1 ∥∂xx b∥22 + ∥∂yy b2 ∥22 dt  2     ≤ C ∥u∥2 + ∥b∥2 + 1 ∂y u1  + ∥∂x u2 ∥2 + ∥∂x b1 ∥2 ∥ω∥2 + ∥j∥2 , 2

for any t > 0.

2

2

2

2

2

2

(27)

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L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

Note that the energy inequality (8) gives the boundedness of ∥∂x u2 ∥L2 (0,T ;L2 (R2 )) and ∥∂x b1 ∥L2 (0,T ;L2 (R2 )) at hand. Moreover, if we suppose that the criterion (4) holds, then it follows from (27) and Gronwall’s inequality that



sup 0≤t ≤T

 ∥ω(t )∥22 + ∥j(t )∥22 + ν1

T





 ∥∂xx u(t )∥22 + ∥∂yy u2 (t )∥22 dt + η1

T



  ∥∂xx b(t )∥22 + ∥∂yy b2 (t )∥22 dt ≤ C . (28) 0

0

On another side, by the inequality (6), the Cauchy–Schwarz inequality, Young’s inequality as well as divergence free conditions, we have



∂x b1 ∂y u1 jdxdy

I1 = 2

1

1

1

1

≤ C ∥∂x b1 ∥22 ∥∂xy b1 ∥22 ∥∂y u1 ∥22 ∥∂xy u1 ∥22 ∥∂x b2 − ∂y b1 ∥2 η1 ν1 ≤ ∥∂xy b1 ∥22 + ∥∂xy u1 ∥22 + C ∥∂x b1 ∥2 ∥∂x b2 − ∂y b1 ∥22 ∥∂y u1 ∥2 16

≤

η1

16

4

2 yy b2 2

∥∂

∥ +

ν1 4

   ∥∂yy u2 ∥22 + C ∥∂x b2 ∥22 + ∥∂y b1 ∥22 ∥w∥22 + ∥j∥22 ,

(29)

and

 I3 = −2

∂x u1 ∂y b1 jdxdy 1

1

1

1

≤ C ∥∂x u1 ∥22 ∥∂xy u1 ∥22 ∥∂y b1 ∥22 ∥∂xy b1 ∥22 ∥j∥2   ν1 η1 ≤ ∥∂yy u2 ∥22 + ∥∂yy b2 ∥22 + C ∥∂x u1 ∥22 + ∥∂y b1 ∥22 ∥j∥22 . 4

16

(30)

Hence, combining (18), (29), (30) and (26) yields that I ≤

3ν1

ν1 η1 η1 ∥∂xx u1 ∥22 + ∥∂yy u2 ∥22 + ∥∂xx b1 ∥22 + ∥∂yy b2 ∥22 2 2 2    2 2 2 2 + C ∥∂x u1 ∥2 + ∥∂x b2 ∥2 + ∥∂y b1 ∥2 ∥ω∥2 + ∥j∥22 . 2

(31)

Inserting (31) into (16), one has d 

  2     ∥ω(t )∥22 + ∥j(t )∥22 + ν1 ∥∂xx u∥22 + ∂yy u2 2 + η1 ∥∂xx b∥22 + ∥∂yy b2 ∥22 dt     ≤ C ∥u∥22 + ∥b∥22 + 1 ∥∂x u1 ∥22 + ∥∂x b2 ∥22 + ∥∂y b1 ∥22 ∥ω∥22 + ∥j∥22 . It is easy to see that Gronwall’s inequality and the energy estimates and the criterion (5) give the estimate (28). Step 2. Estimates of ∥∇ω∥2 and ∥∇ j∥2 . To go further, multiplying (10) and (11) by △ω and △j, respectively, then summing them together and integrating in R2 , we get

    ∥∇ω(t )∥22 + ∥∇ j(t )∥22 + ν1 ∂xxx u2 △ω − ∂xxy u1 △ω dxdy 2 dt    + η1 ∂xxx b2 △j − ∂xxy b1 △j dxdy    = 2 ∇ω · ∇ b · ∇ jdxdy − 2 ∂x b1 (∂x u2 + ∂y u1 )△jdxdy − ∇ω · ∇ u · ∇ωdxdy   − ∇ j · ∇ u · ∇ jdxdy + 2 ∂x u1 (∂x b2 + ∂y b1 )△jdxdy

1 d 

=J=

5 

Jk .

(32)

k=1

By using the facts (14) and (15) again, integration by parts a few times, we get 1 d  2 dt

       2 2 ∥∇ω(t )∥22 + ∥∇ j(t )∥22 + ν1 ωxy 2 + ∥ωxx ∥22 + η1 jxy 2 + ∥jxx ∥22 = J .

(33)

L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

203

In the following, we will give the estimates of J, in order to estimate ∥∇ω∥2 and ∥∇ j∥2 . By integration by parts, we have

 J1 = 2

∇ω · ∇ b · ∇ jdxdy 

=2  =2

∂x b1 ωx jx + ∂y b1 ωy jx + ∂x b2 ωx jy + ∂y b2 ωy jy dxdy b2 jx ωxy + b2 ωx jxy + ∂y b1 ωy jx + ∂x b2 ωx jy + b1 ωxy jy + b1 ωy jxy dxdy

= J11 + J12 + J13 + J14 + J15 + J16 ,

(34)

furthermore, in view of the inequality (6) and Young’s inequality, one gets

 J11 = 2

b2 jx ωxy dxdy 1

1

1

1

≤ C ∥b2 ∥22 ∥∂x b2 ∥22 ∥ωxy ∥2 ∥jx ∥22 ∥jxy ∥22  ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b2 ∥2 ∥∂x b2 ∥2 ∥jx ∥2 ≤ 2 2 2 2 2 26

J12

26

 ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b∥2 ∥j∥2 ∥∇ j∥2 , ≤ 2 2 2 2 2 26 26  = 2 b2 ωx jxy dxdy 1

1

1

(35)

1

≤ C ∥b2 ∥22 ∥∂x b2 ∥22 ∥ωx ∥22 ∥ωxy ∥22 ∥jxy ∥2  ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b2 ∥2 ∥ωx ∥2 ∥∂x b2 ∥2 ≤ 2 2 2 2 2 26

J13

26

 ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b∥2 ∥j∥2 ∥∇ j∥2 , ≤ 2 2 2 2 2 26 26  = 2 ∂y b1 ωy jx dxdy 1

1

1

(36)

1

≤ C ∥∂y b1 ∥2 ∥ωy ∥22 ∥ωxy ∥22 ∥jx ∥22 ∥jxy ∥22 ν1 η1 ∥ωxy ∥22 + ∥jxy ∥22 + C ∥ωy ∥2 ∥∂y b1 ∥22 ∥jx ∥2 ≤ 26

J14

26

   ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥j∥2 ∥∇ω∥2 + ∥∇ j∥2 , ≤ 2 2 2 2 2 26 26  = 2 ∂x b2 ωx jy dxdy 1

1

1

(37)

1

≤ C ∥∂x b2 ∥2 ∥ωx ∥22 ∥ωxy ∥22 ∥jy ∥22 ∥jxy ∥22 ν1 η1 ≤ ∥ωxy ∥22 + ∥jxy ∥22 + C ∥ωx ∥2 ∥∂x b2 ∥22 ∥jy ∥2 26

J15

26

   ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥j∥2 ∥∇ω∥2 + ∥∇ j∥2 , ≤ 2 2 2 2 2 26 26  = 2 b1 ωxy jy dxdy 1

1

1

(38)

1

≤ C ∥b1 ∥22 ∥∂y b1 ∥22 ∥ωxy ∥2 ∥jy ∥22 ∥jxy ∥22  ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b1 ∥2 ∥∂y b1 ∥2 ∥jy ∥2 ≤ 2 2 2 2 2 26

26

26

26

 ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b∥2 ∥j∥2 ∥∇ j∥2 , ≤ 2 2 2 2 2 and

 J16 = 2

b1 ωy jxy dxdy 1

1

1

1

≤ C ∥b1 ∥22 ∥∂y b1 ∥22 ∥ωy ∥22 ∥ωxy ∥22 ∥jxy ∥2

(39)

204

L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

 ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b1 ∥2 ∥∂y b1 ∥2 ∥jy ∥2 ≤ 2 2 2 2 2 26

26

26

26

 ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥b∥2 ∥j∥2 ∥∇ω∥2 , ≤ 2 2 2 2 2

(40)

therefore J1 ≤

3ν1 

     ωxy 2 + 3η1 ∥jxy ∥2 + C ∥b∥2 + 1 ∥j∥2 ∥∇ω∥2 + ∥∇ j∥2 . 2

13

2

13

2

2

2

(41)

2

Using the same techniques to estimate J2 , we have



∇(∂x b1 (∂x u2 + ∂y u1 )) · ∇ jdxdy



(∂x u2 + ∂y u1 )(∇∂x b1 ) · ∇ j + ∂x b1 ∇(∂x u2 + ∂y u1 ) · ∇ jdxdy



(∂x u2 + ∂y u1 )(∇∂x b1 ) · ∇ j + ∂x b1 ∇∂x u2 · ∇ j + ∂x b1 ∇∂y u1 · ∇ jdxdy,

J2 = 2

=2 =2

(42)

moreover, 1

1

1

1

1

1

J2 ≤ C ∥∇∂x b1 ∥22 ∥∇∂xy b1 ∥22 ∥∂x u2 + ∂y u1 ∥22 ∥∂x (∂x u2 + ∂y u1 )∥22 ∥∇ j∥2 + C ∥∂x b1 ∥22 ∥∂xx b1 ∥22 ∥∇∂x 1

1

1

1

1

1

× u2 ∥22 ∥∇∂xy u2 ∥22 ∥∇ j∥2 + C ∥∂x b1 ∥22 ∥∂xy b1 ∥22 ∥∇∂y u1 ∥22 ∥∇∂xy u1 ∥22 ∥∇ j∥2   2 2 2 4 η1 ∥∇∂xy b1 ∥22 + C ∥∇∂x b1 ∥23 ∥∂x u2 + ∂y u1 ∥23 ∥∂x (∂x u2 + ∂y u1 )∥23 ∥∇ j∥23 ≤ 26   2 2 2 4 ν1 2 3 3 3 3 + ∥∇∂xy u2 ∥2 + C ∥∇∂x u2 ∥2 ∥∂xx b1 ∥2 ∥∂x b1 ∥2 ∥∇ j∥2 26   2 2 2 4 ν1 2 3 3 3 3 + ∥∇∂xy u1 ∥2 + C ∥∂x b1 ∥2 ∥∂xy b1 ∥2 ∥∇∂y u1 ∥2 ∥∇ j∥2 26   2 2 2 4 η1 ∥jxy ∥22 + C ∥∇∂x b1 ∥23 ∥ω∥23 ∥ωx ∥23 ∥∇ j∥23 ≤ 26     2 2 4 2 2 4 2 2 ν1 ν1 ∥ωxy ∥22 + C ∥∇∂x b1 ∥23 ∥j∥23 ∥ωx ∥23 ∥∇ j∥23 + ∥ωxy ∥22 + C ∥∂yy b2 ∥23 ∥j∥23 ∥∇ω∥23 ∥∇ j∥23 + 26

26

     ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥∂xx b1 ∥2 + ∥∂yy b2 ∥2 + 1 ∥ω∥2 + ∥j∥2 + 1 ∥∇ω∥2 + ∥∇ j∥2 . ≤ 2 2 2 2 2 2 2 2 13

26

(43)

A series of direct calculus gives that

 J3 = −

∇ω · ∇ u · ∇ωdxdy 



=−

 ∂x u1 ωx2 + ∂x u2 ωx ωy + ∂y u1 ωx ωy + ∂y u2 ωy2 dxdy

= J31 + J32 + J33 + J34 .

(44)

Furthermore, with the aid of the inequality (6) and the Cauchy–Schwarz inequality, we obtain

 J31 = −

∂x u1 ωx2 dxdy 1

1

1

1

≤ C ∥∂x u1 ∥22 ∥∂xx u1 ∥22 ∥ωx ∥2 ∥ωx ∥22 ∥ωxy ∥22 ≤

ν1 26

2

2

∥ωxy ∥22 + C ∥ω∥23 ∥∂xx u1 ∥23 ∥∇ω∥22 .

(45)

Similarly, one has

 J32 = −

∂x u2 ωx ωy dxdy 1

1

1

1

≤ C ∥∂x u2 ∥22 ∥∂xy u2 ∥22 ∥ωx ∥2 ∥ωy ∥22 ∥ωxy ∥22 2 2 ν1 ≤ ∥ωxy ∥22 + C ∥ω∥23 ∥∂xx u1 ∥23 ∥∇ω∥22 , 26

(46)

L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

 J33 = −

205

∂y u1 ωx ωy dxdy 1

1

1

1

≤ C ∥∂y u1 ∥22 ∥∂xy u1 ∥22 ∥ωy ∥2 ∥ωx ∥22 ∥wxy ∥22 2 2 ν1 ≤ ∥ωxy ∥22 + C ∥ω∥23 ∥∂yy u2 ∥23 ∥∇ω∥22 ,

(47)

26

and

 J34 = −

∂y u2 ωy2 dxdy 1

1

1

1

≤ C ∥∂y u2 ∥22 ∥∂yy u2 ∥22 ∥ωy ∥2 ∥ωy ∥22 ∥ωxy ∥22 ν1

≤

2

2

26

∥ωxy ∥22 + C ∥ω∥23 ∥∂yy u2 ∥23 ∥∇ω∥22 .

(48)

Combining with (45)–(48), one has J3 ≤

≤

2ν1

2

∥ωxy ∥22 + C ∥ω∥23

13 2ν1



2

2



∥∂xx u1 ∥23 + ∥∂yy u2 ∥23

∥∇ω∥22

  2   ∥ωxy ∥22 + C ∥ω∥22 + 1 ∥∂xx u1 ∥22 + ∂yy u2 2 + 1 ∥∇ω∥22 .

13

(49)

Integrating by parts and using the same techniques above, we obtain

 J4 = −

∇ j · ∇ u · ∇ jdxdy 

= − ∂x u1 j2x + ∂y u1 jx jy + ∂x u2 jx jy + ∂y u2 j2y dxdy     2 = ∂y u2 jx dxdy − ∂y u1 jx jy dxdy − ∂x u2 jx jy dxdy + ∂x u1 j2y dxdy     = −2 u2 jx jxy dxdy − ∂y u1 jx jy dxdy − ∂x u2 jx jy dxdy − 2 u1 jy jxy dxdy 1

1

1

1

1

1

1

1

≤ C ∥u2 ∥22 ∥∂x u2 ∥22 ∥jx ∥22 ∥jxy ∥22 ∥jxy ∥2 + C ∥∂y u1 ∥2 ∥jx ∥22 ∥jxy ∥22 ∥jy ∥22 ∥jxy ∥22 1

1

1

1

1

1

1

1

+ C ∥∂x u2 ∥2 ∥jx ∥22 ∥jxy ∥22 ∥jy ∥22 ∥jxy ∥22 + C ∥u1 ∥22 ∥∂y u1 ∥22 ∥jy ∥22 ∥jxy ∥22 ∥jxy ∥2 η     η   2 2 1  1  jxy 2 + C ∥u2 ∥22 ∥∂x u2 ∥22 ∥jx ∥22 + jxy 2 + C ∥∂y u1 ∥22 ∥jy ∥2 ∥jx ∥2 ≤ 26  26 η   η    2 2 1  1  2  jxy 2 + C ∥∂x u2 ∥2 ∥jx ∥2 ∥jy ∥2 + jxy 2 + C ∥u1 ∥22 ∥∂y u1 ∥22 ∥jy ∥22 + 26

26

  2η1 ∥jxy ∥22 + C ∥u∥22 + 1 ∥ω∥22 ∥∇ j∥22 . ≤

(50)

13

Similarly,

 J5 = −2

  ∇ ∂x u1 (∂x b2 + ∂y b1 ) · ∇ jdxdy



(∂x b2 + ∂y b1 )∇∂x u1 · ∇ jdxdy + 2

=2 =2

∂x u1 (∂x b2 + ∂y b1 )△jdxdy





∂x u1 ∇∂x b2 · ∇ jdxdy +



∂x u1 ∇∂y b1 · ∇ jdxdy,

and then 1

1

1

1

J5 ≤ C ∥∇∂x u1 ∥22 ∥∇∂xx u1 ∥22 ∥∂x b2 + ∂y b1 ∥22 ∥∂y (∂x b2 + ∂y b1 )∥22 ∥∇ j∥2 1

1

1

1

+ C ∥∂x u1 ∥22 ∥∂xx u1 ∥22 ∥∇∂x b2 ∥22 ∥∇∂xy b2 ∥22 ∥∇ j∥2 1

1

1

1

+ C ∥∂x u1 ∥22 ∥∂xy u1 ∥22 ∥∇∂y b1 ∥22 ∥∇∂xy b1 ∥22 ∥∇ j∥2   2 2 2 4 ν1 ∥∇∂xx u1 ∥22 + C ∥∇∂x u1 ∥23 ∥∂x b2 + ∂y b1 ∥23 ∥∂y (∂x b2 + ∂y b1 )∥23 ∥∇ j∥23 ≤ 26

(51)

206

L. Zhang, S. Li / Nonlinear Analysis: Real World Applications 21 (2015) 197–206

 +

η1 26

 + 

η1

2

2

2

4



2 3

2 3

2 3

4 3



∥∇∂xy b2 ∥22 + C ∥∇∂x b2 ∥23 ∥∂xx u1 ∥23 ∥∂x u1 ∥23 ∥∇ j∥23

∥ + C ∥∇∂y b1 ∥2 ∥∂xy u1 ∥2 ∥∂x u1 ∥2 ∥∇ j∥2    2 2 2 2 ν1 η1 ≤ ∥ωxy ∥22 + C ∥∇∂x u1 ∥23 ∥j∥23 ∥∇ j∥22 + ∥jxy ∥22 + C ∥∂yy u2 ∥23 ∥ω∥23 ∥∇ j∥22 26 26   2 2 η1 ∥jxy ∥22 + C ∥ω∥23 ∥∂yy u2 ∥23 ∥∇ j∥22 + 26

∥∇∂

2 xy b1 2

26

      ν1  ωxy 2 + η1 ∥jxy ∥2 + C ∥ω∥2 + ∥j∥2 + 1 ∥∂xx u1 ∥2 + ∂yy u2 2 + 1 ∥∇ j∥2 . ≤ 2 2 2 2 2 2 2 26

13

(52)

Hence, substituting (41), (43), (49), (50) and (51) into (33) yields d 

 ∥∇ω(τ )∥22 + ∥∇ j(τ )∥22 + ν1 ∥ωxy ∥22 + η1 ∥jxy ∥22 dτ   2    ≤ C ∥u∥22 + ∥b∥22 + 1 ∥ω∥22 + ∥j∥22 + 1 ∥∂xx u1 ∥22 + ∂yy u2 2 + ∥∂xx b1 ∥22 + ∥∂yy b2 ∥22 + 1   × ∥∇ω∥22 + ∥∇ j∥22 ,

(53)

for any τ > 0. Thanks to Gronwall’s inequality and the estimates in (9) and (28), we obtain the uniform estimates of ∥∇ω∥L∞ (0,T ;L2 (R2 )) and ∥∇ j∥L∞ (0,T ;L2 (R2 )) for any t ≥ 0. With the aid of the uniform estimates ∥(u, b)∥L∞ (0,T ;L2 (R2 )) and ∥(ω, j)∥L∞ (0,T ;H 1 (R2 )) , we can conclude the uniform bound of ∥(u, b)∥L∞ (0,T ;H 2 (R2 )) , which implies that the solution (u, b) is regular in [0, T ]. This completes the proof of theorem.  The corollary follows directly by using the transformation

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