Global regularity for the 212 D incompressible Hall-MHD system with partial dissipation

J. Math. Anal. Appl. 484 (2020) 123701

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Global regularity for the 2 12 D incompressible Hall-MHD system with partial dissipation Baoying Du Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, PR China Department of Mathematics, Yibin University, Yibin, Sichuan, 644000, PR China

a r t i c l e

i n f o

Article history: Received 3 September 2019 Available online 23 November 2019 Submitted by X. Zhang Keywords: Hall-MHD system 2 12 dimensional Partial dissipation Strong solution Global regularity

a b s t r a c t In this paper, we investigate the Cauchy problem for the 2 12 dimensional incompressible Hall-magnetohydrodynamic equations with partial dissipation. By using energy estimate, we obtain the local existence of strong solution and the global-in-time existence of the strong solution. © 2019 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we are interested in the Cauchy problem for three dimensional incompressible Hallmagnetohydrodynamic (Hall-MHD) equations on the plane of the form ⎧ ⎪ ∂t u + (u · ∇)u + ∇(p + π) = μ1 ux1 x1 + μ2 ux2 x2 + (b · ∇)b, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎨ ∂ b + (u · ∇)b + ∇ × ((∇ × b) × b) = ν b (t, x) ∈ R+ × R2 , t 1 x1 x1 + ν2 bx2 x2 + (b · ∇)u, + 2 ⎪ ∇ · u = ∇ · b = 0, (t, x) ∈ R × R , ⎪ ⎪ ⎩ (u, b)| 2 t=0 = (u0 , b0 ). x ∈ R ,

(1)

where u = (u1 , u2 , u3 ), b = (b1 , b2 , b3 ), here ui = ui (t, x), bi = bi (t, x), i = 1, 2, 3 and p = p(t, x), t ∈ R+ , x = (x1 , x2 ) ∈ R2 . This system is 2 12 D flow for the Hall-MHD system. u(x, t), b(x, t) are the fluid velocity and magnetic field respectively. μ1 , μ2 ≥ 0 are the kinematic viscosity, ν1 , ν2 ≥ 0 are the magnetic diffusion. If μ1 = μ2 = ν1 = ν2 = 0, the above system are ideal MHD equations. In physics, the version of the system (1) in dimension three describes the dynamics of plasma flows with strong shear of magnetic fields. We refer to [1] for the physical backgrounds for the full system, and E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.123701 0022-247X/© 2019 Elsevier Inc. All rights reserved.

2

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

[9,4–8,14,16] for the recent studies of the mathematical problems of the equations. Comparing with the well-known MHD system, the Hall term ∇ × ((∇ × b) × b) is included, which is believed to be a key issue for understanding magnetic reconnection. There are many mathematical results on the 3D usual MHD system, for the existence of global weak solutions [15,24], regularity criterion [17,18], global smooth small solutions [20,25], and so on. But the HallMHD has received little attention from mathematicians. Chae, Degond, and Liu [9] obtained the existence of global weak solutions to 3D incompressible Hall-MHD system, they also obtained local existence of smooth solutions as well as the global existence of smooth solutions for small data. Chae and Lee [4] established the blow-up criterion and obtained the global existence of classical solutions for small data to 3D incompressible Hall-MHD system via blow-up criterion. In [8] it is shown that there exists a weak solution of the time dependent 3D Hall-MHD system on the plane, having the possible set of space-time singularities, whose Hausdroff dimension is at most two. Contrary to the usual MHD system, the global regularity for solutions to the 3D Hall-MHD which depends only on two variables (i.e. 2 21 dimensional Hall-MHD) is still open. Note that 2 21 dimensional Hall-MHD solution has been used in [19] to investigate the influence of the Hall term on width of the magnetic islands of the tearing-mode. In this paper, we investigate the Cauchy problem for 2 21 dimensional incompressible Hall-magnetohydrodynamic equations with partial viscosity. By using energy estimate, we obtain the local existence for strong solution, in addition, we also obtain the global-in-time existence of the strong solution provided that (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 ) is small enough, or the coefficients of partial viscosity and magnetic diffusion are sufficiently large. This work was partially motivated by the well-known works for the Boussinesq system and MHD system with partial dissipation [3,2] in dimension two. For 2D MHD equations, Cao and Wu [2] proved the global regularity for the MHD system with mixed partial dissipation and magnetic diffusion. Du and Zhou [13] established several results for other cases. For the three dimensional MHD equations with partial dissipation and magnetic diffusion, Ye and Zhu [26] obtained the global strong solution under (u0 2L2 + b0 2L2 )(∇u0 2L2 + ∇b0 2L2 ) is small enough or the coefficients of partial viscosity and magnetic diffusion are sufficiently large. For more results of magnetohydrodynamic equations, one can see [10,11,21,22,12]. Our main results in this paper are the following Theorems, the first one is local existence for strong solutions to the system (1) with μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0. Theorem 1.1. Assume that μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0, and (u0 , b0 ) ∈ H 2 (R2 ) with div u0 = div b0 = 0. Then there exists a positive constant T = T (u0 H 2 , b0 H 2 ), such that the system (1) has a strong solution (u, b) on (0, T ) × R2 , satisfying (u, b) ∈ L∞ (0, T ; H 2 (R2 )). Remark 1.1. A similar result can also be obtained for the system (1) with μ1 > 0, μ2 = 0, ν1 > 0, ν2 > 0, the proof procedure is similar to Theorem 1.1, the details are omitted here, and we state it as follows. Theorem 1.2. Assume that μ1 > 0, μ2 = 0, ν1 > 0, ν2 > 0, and (u0 , b0 ) ∈ H 2 (R2 ) with div u0 = div b0 = 0. Then there exists a positive constant T = T (u0 H 2 , b0 H 2 ), such that the system (1) has a strong solution (u, b) on (0, T ) × R2 , satisfying (u, b) ∈ L∞ (0, T ; H 2 (R2 )). Next, we state the global existence for strong solutions to the system (1) with μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0. Theorem 1.3. Assume that μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0, and (u0 , b0 ) ∈ H 2 (R2 ) with div u0 = div b0 = 0. There exists an absolutely positive constant ε0 > 0, independent of u0 , b0 , t and the lower bound of the coefficients of partial viscosity and magnetic diffusion, such that the system (1) has a unique global strong solution (u, b) on (0, ∞) × R2 , satisfying

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

3

(u, b) ∈ L∞ (0, T ; H 2 (R2 )), (∂2 u, j) ∈ L2 (0, T ; H 2 (R2 )),

(2)

(u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 ) ≤ ε0 , M2

(3)

M  min{μ2 , ν1 , ν2 }.

(4)

provided that

where

ω  ∇ × u, j  ∇ × b represent the vorticity and current density, respectively, and ω0  ω(0, x), j0  j(0, x). Remark 1.2. A global regularity result similar to Theorem 1.3 can be obtained for the system (1) with μ1 > 0, μ2 = 0, ν1 > 0, ν2 > 0, the proof procedure is similar to Theorem 1.3, the details are omitted here, and we state it as follows. Theorem 1.4. Assume that μ1 > 0, μ2 = 0, ν1 > 0, ν2 > 0, and (u0 , b0 ) ∈ H 2 (R2 ) with div u0 = div b0 = 0. There exists an absolutely positive constant ε0 > 0, independent of u0 , b0 , t and the lower bound of the coefficients of partial viscosity and magnetic diffusion, such that the system (1) has a unique global strong solution (u, b) on (0, ∞) × R2 , satisfying (u, b) ∈ L∞ (0, T ; H 2 (R2 )), (∂1 u, j) ∈ L2 (0, T ; H 2 (R2 )), provided that (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 ) ≤ ε0 , M2 where M  min{μ1 , ν1 , ν2 }. Remark 1.3. If any one of magnetic diffusion coefficients is zero, the global regularity issue has not been settled because of there exist a main difficult, which arises from the Hall term ∇ × ((∇ × b) × b) due to it is quadratic in the magnetic field and involves second order derivatives. Because of this difficult, the method developed below is challenging to apply the Hall term in this case. Remark 1.4. Compared to [9], on the one hand, the initial data (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 ) instead of (u0 , b0 )H m is sufficiently small (m > 52 ); on the other hand, we also get the existence of strong solution with large initial data. 2. Notations and preliminaries In this section, some simplified notations and useful lemma will be given. We adopt the following simplified notations throughout this paper: ˆ

¨ f dx 

R2

f (t, x)dx1 dx2 , R2

f0  f (0, x).

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

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Notations. Through this paper, C denotes a generic positive constant which will appear hereafter depends only on the initial data and may change from one line to the other. ui and ∂i denote the corresponding ith components of u and ∇, respectively. Next, we will give a known auxiliary lemma, which can be found in [2]. Lemma 2.1. Assume that f, g, ∂2 g, h, ∂1 h are all in L2 (R2 ), then, ˆ

1

1

1

1

|f gh|dx ≤ Cf L2 gL2 2 ∂2 gL2 2 hL2 2 ∂1 hL2 2 . R2

3. Proof of Theorem 1.1 In this section, we will establish the local existence for strong solutions to the system (1). For simplicity, we denote by ∇α = ∂ |α| /∂1α1 ∂2α2 , where α = (α1 , α2 ) ∈ (N ∪ {0})2 with |α| = α1 + α2 ≤ 2. Applying ∇α to the first and the second equations of the system (1), taking the scalar product of the resulting equations with ∇α u and ∇α b, respectively, then summing them over |α| ≤ 2, one gets 1 d (u(t)2H 2 + b(t)2H 2 ) + μ2 ∂2 u2H 2 + ν1 ∂1 b2H 2 + ν2 ∂2 b2H 2 2 dt  ˆ  ˆ α α =− ∇ (u · ∇)u · ∇ udx − ∇α (u · ∇)b · ∇α bdx 0<|α|≤2

+

R2  ˆ ∇α (b · ∇)b · ∇α udx + 0<|α|≤2R2 ˆ

−



0<|α|≤2R2

 ˆ

∇α (b · ∇)u · ∇α bdx

(5)

0<|α|≤2R2

∇α [∇ × ((∇ × b) × b)] · ∇α bdx

0<|α|≤2R2

= K1 + K2 + K3 + K4 + K5 , here, |α| = 0 on the right-hand side is due to (12). The estimate for K1 is subtle. When |α| = 1, we rewrite K1 as ˆ K11 = −

∇(u · ∇)u · ∇udx

R2

ˆ

=− ˆ

(∇u · ∇)u · ∇udx

R2

−∇u1 ∂1 u · ∇u − ∇u2 ∂2 u · ∇udx

= R2

= K111 + K112 . Noticing the fact that ∂1 u1 L2 = ∂2 u2 L2 due to u is divergence free, applying Lemma 2.1 and Young’s inequality, one has 1

1

1

1

|K111 | ≤ C∇uL2 ∇u1 L2 2 ∂1 uL2 2 ∂1 ∇u1 L2 2 ∂12 uL2 2 ≤ C∂2 uH 1 u2H 1 ≤

M ∂2 u2H 1 + Cu4H 1 , 24

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

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and 1

1

1

1

|K112 | ≤ C∇u2 L2 ∂2 uL2 2 ∇uL2 2 ∂12 uL2 2 ∂2 ∇uL2 2 ≤ C∂2 uH 1 u2H 1 ≤

M ∂2 u2H 1 + Cu4H 1 . 24

When |α| = 2, we rewrite it as ˆ K12 = −

∇2 (u · ∇)u · ∇2 udx

R2

ˆ

=−

ˆ (∇2 u · ∇)u · ∇2 udx − 2

R2

(∇u · ∇)∇u · ∇2 udx.

R2

Using Lemma 2.1 and Young’s inequality, one gets 1

1

1

1

K12 ≤ C∇2 uL2 ∇uL2 2 ∇2 uL2 2 ∂1 ∇uL2 2 ∂2 ∇2 uL2 2 1

5

2 2 ≤ C∂2 uH 2 uH 2

≤

10 M ∂2 u2H 2 + CuH3 2 . 24

Thus, summing up the above estimates, one has K1 ≤

10 3M ∂2 u2H 2 + CuH3 2 . 24

(6)

We estimate K2 as follows. When |α| = 1, we rewrite it as ˆ K21 = −

∇(u · ∇)b · ∇bdx

R2

ˆ

=−

(∇u · ∇)b · ∇bdx.

R2

Based on Lemma 2.1 and Young’s inequality, one has 1

1

1

1

K21 ≤ C∇bL2 ∇bL2 2 ∇uL2 2 ∂1 ∇bL2 2 ∂2 ∇uL2 2 ≤ C(∂2 uH 1 + ∂1 bH 1 )bH 1 (uH 1 + bH 1 ) ≤

M (∂2 u2H 1 + ∇b2H 1 ) + Cb2H 1 (u2H 1 + b2H 1 ). 24

When |α| = 2, we rewrite it as ˆ K22 = −

∇2 (u · ∇)b · ∇2 bdx

R2

ˆ

ˆ

=−

(∇2 u · ∇)b · ∇2 bdx − 2

R2

= K221 + K222 .

R2

(∇u · ∇)∇b · ∇2 bdx

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

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By Lemma 2.1 and Young’s inequality, one gets 1

1

1

1

K221 ≤ C∇bL2 ∇2 uL2 2 ∇2 bL2 2 ∂1 ∇2 bL2 2 ∂2 ∇2 uL2 2 ≤ C(∂2 uH 2 + ∇bH 2 )bH 2 (uH 2 + bH 2 ) M (∂2 u2H 2 + ∇b2H 2 ) + Cb2H 2 (u2H 2 + b2H 2 ), 24

≤ and

1

1

1

1

K222 ≤ C∇uL2 ∇2 bL2 2 ∇2 bL2 2 ∂1 ∇2 bL2 2 ∂2 ∇2 bL2 2 ≤ C∇bH 2 bH 2 uH 2 ≤

M ∇b2H 2 + Cb2H 2 u2H 2 . 24

Therefore, one gets |K2 | ≤

3M (∂2 u2H 2 + ∇b2H 2 ) + Cb2H 2 (u2H 2 + b2H 2 ). 24

Similarly, we can estimate K3 + K4 as follows. When |α| = 1, we rewrite it as ˆ ∇(b · ∇)b · ∇u + ∇(b · ∇)u · ∇bdx

K31 = R2

ˆ

ˆ (∇b · ∇)b · ∇udx +

= R2

(∇b · ∇)u · ∇bdx. R2

We can use similar procedure as K21 to deduce that |K31 | ≤

M (∂2 u2H 1 + ∇b2H 1 ) + Cb2H 1 (u2H 1 + b2H 1 ) 24

When |α| = 2, we rewrite it as ˆ

ˆ ∇2 (b · ∇)b · ∇2 udx +

K32 = R2

∇2 (b · ∇)u · ∇2 bdx R2

ˆ

ˆ

(∇2 b · ∇)b · ∇2 udx + 2

= R2

(∇b · ∇)∇b · ∇2 udx

R2

ˆ

ˆ

(∇2 b · ∇)u · ∇2 bdx + 2

+ R2

(∇b · ∇)∇u · ∇2 bdx

R2

= K321 + K322 + K323 + K324 One can use similar procedure as K221 and K222 to obtain |K321 , K322 , K323 , K324 | ≤ By the above estimates, we obtain

M (∂2 u2H 2 + ∇b2H 2 ) + Cb2H 2 (u2H 2 + b2H 2 ) 24

(7)

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

|K3 + K4 | ≤

5M (∂2 u2H 2 + ∇b2H 2 ) + Cb2H 2 (u2H 2 + b2H 2 ). 24

7

(8)

Next, we estimate K5 as follows, by Hölder’s inequality, calculus inequality, interpolation inequality, Sobolev inequality and Young’s inequality, we get K5 = −

 ˆ

∇α [∇ × ((∇ × b) × b)] · ∇α b dx

0<|α|≤2R2 ˆ

=−



∇α (j × b) · ∇α j dx

0<|α|≤2R2 ˆ

=− ≤



[∇α (j × b) − ∇α j × b] · ∇α j dx

0<|α|≤2R2  α

[∇ (j × b) − ∇α j × b]L2 ∇bH 2

0<|α|≤2

≤ ∇bL4 bW 2,4 ∇bH 2 1

3

2 2 ≤ ∇bL4 bH 2 ∇bH 2 1

3

2 2 ≤ bH 2 bH 2 ∇bH 2 M ≤ ∇b2H 2 + C(b2H 2 )3 . 24

Combining (5)–(9), we obtain 1 d (u(t)2H 2 + b(t)2H 2 ) + μ2 ∂2 u2H 2 + ν1 ∂1 b2H 2 + ν2 ∂2 b2H 2 2 dt M ≤ (∂2 u2H 2 + ∇b2H 2 ) + C(u2H 2 + b2H 2 )3 . 2 So we get d (u(t)2H 2 + b(t)2H 2 ) + M (∂2 u2H 2 + ∇b2H 2 ) dt ≤ C(u2H 2 + b2H 2 )3 . Below we set X(t) := u(t)2H 2 + b(t)2H 2 . By the above inequality, we have d X ≤ CX 3 . dt Thanks to nonlinear Gronwall’s inequality, we obtain X(t) ≤  Choose T =

3 4CX 2 (0) ,

X(0) 1 − CX 2 (0)t

.

we have X(t) ≤ 2X(0), ∀t ∈ (0, T ].

(9)

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

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This implies the following priori estimate (u, b)L∞ (0,T ;H 2 (R2 ) ≤ C(u0 2H 2 + b0 2H 2 ). With this estimate, we can conclude the proof of Theorem 1.1.



4. Priori estimates Given a strong solution (u, b) on (0, T ) × R2 , we define A(t)  sup (ω(s)2L2 + j(s)2L2 ), 0 ≤ t ≤ T. 0≤s≤t

In this section, we will establish some priori estimates to the system (1). Proposition 4.1. Under the assumptions of Theorem 1.3, there exists an absolutely positive constant ε0 > 0, independent of u0 , b0 , t, and the lower bound of the coefficients of partial viscosity μ2 and magnetic diffusion ν1 , ν2 , such that if (u0 2L2 +b0 2L2 )(ω0 2L2 +j0 2L2 )/M 2 ≤ ε0 , and (u, b) is a strong solution to the system (1), satisfying: A(T ) ≤ 2(ω0 2L2 + j0 2L2 ),

(10)

A(T ) ≤ (ω0 2L2 + j0 2L2 ),

(11)

then it in fact holds that

here M as in (4). Proof. Proposition 4.1 is an obvious consequence of the following Lemma 4.1 and Lemma 4.2.



Taking the L2 −inner products of the first equation of the system (1) with u, and taking the L2 −inner products of the second equation of system (1) with b, then summing them together, use the integration by parts and the divergence free of u, b to deduce that 1 d (u(t)2L2 + b(t)2L2 ) + M (∂2 u2L2 + ∇b2L2 ) ≤ 0. 2 dt Noticing ∇bL2 ≤ CjL2 due to the fact that b is divergence free, integrating over (0, T ) to above inequality, one can yield the following energy inequality. Lemma 4.1. Let μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0, and (u, b) be a strong solution of the system (1) on [0, T ] × R2 , then ˆT u2L2

+

b2L2

(∂2 u(t)2L2 + j(t)2L2 )dt ≤ u0 2L2 + b0 2L2 ,

+M 0

where M is the positive number determined in (4). Now, we establish the following H 1 -estimates of u, b.

(12)

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

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Lemma 4.2. Assume that the condition (10) of the Proposition 4.1 holds, let (u, b) be a strong solution of the system (1) with μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0 on [0, T ] × R2 , then ˆT sup 0≤t≤T

(ω(t)2L2

+

j(t)2L2 )

(∂2 ω(t)22 + ∇j(t)22 )dt ≤ ω0 2L2 + j0 2L2 ,

+M

(13)

0

provided that (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 )/M 2 ≤ ε0 , and ε0 is sufficiently small. Proof. Applying the operator ∇× to the first and the second equations of system (1). Taking the scalar products of resulting velocity equation with ω, taking the scalar products of resulting magnetic equation with j, then summing them together, we have 1 d (ω(t)2L2 + j(t)2L2 ) + μ2 ∂2 ω2L2 + ν1 ∂1 j2L2 + ν2 ∂2 j2L2 2 dt ˆ ˆ ˆ = − ∇ × (u · ∇)u · ωdx − ∇ × (u · ∇)b · jdx + ∇ × (b · ∇)b · ωdx (14)

R2 ˆ R2 ˆ R2 + ∇ × (b · ∇)u · jdx − ∇ × [∇ × ((∇ × b) × b)] · jdx R2

R2

= I1 + I2 + I3 + I4 + I5 . Noticing u is divergence free, we can rewrite I1 as follows. ˆ I1 = − ∇ × (u · ∇u) · ωdx ˆ

R2

(ω · ∇)u · ω − (u · ∇)ω · ωdx

= R2

ˆ

(ω · ∇)u · ωdx

= R2

ˆ

=

ωi ∂i uk ωk dx R2

ˆ

ˆ

|ω1 ∂1 uk ωk |dx +

≤ R2

|ω2 ∂2 uk ωk |dx R2

= I11 + I12 . Using Lemma 2.1 and Young’s inequality, we get ˆ |I11 | = |ω1 ∂1 uk ωk |dx R2 1

1

1

1

≤ Cωk L2 ω1 L2 2 ∂1 uk L2 2 ∂1 ω1 L2 2 ∂12 uk L2 2 ≤ C∂2 ωL2 ω2L2 ≤

M C ∂2 ω2L2 + ω4L2 , 34 M

here, we use the fact that ∂1 ω1 L2 = ∂2 ω2 L2 due to ∂1 ω1 + ∂2 ω2 = 0.

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

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Similarly, one can estimate I12 as follows. ˆ |I12 | =

|ω2 ∂2 uk ωk |dx R2 1

1

1

1

≤ Cω2 L2 ∂2 u2 L2 2 ωk L2 2 ∂12 uk L2 2 ∂2 ωk L2 2 3

1

≤ C∂2 ωL2 ωL2 2 ∂2 uL2 2 ≤

M C ∂2 ω2L2 + ω3L2 ∂2 uL2 . 34 M

Putting these above estimates together, one gets |I1 | ≤ |I11 | + |I12 | (15)

2M C ∂2 ω2L2 + ω4L2 ∂2 uL2 . 34 M

≤

Using u is divergence free and integration by parts, I2 can be rewritten as ˆ I2 = −

∇ × (u · ∇b) · jdx

R2

ˆ

ˆ

(u · ∇)j · jdx −

=− ˆ

R2

(∂2 u · ∇b3 , −∂1 u · ∇b3 , ∂1 u · ∇b2 − ∂2 u · ∇b1 ) · jdx R2

(−∂2 u · ∇b3 , ∂1 u · ∇b3 , −∂1 u · ∇b2 + ∂2 u · ∇b1 ) · jdx

= R2

ˆ

=−

ˆ ∂2 u · ∇b3 j1 dx +

R2

ˆ ∂1 u · ∇b3 j2 dx −

R2

ˆ ∂1 u · ∇b2 j3 dx +

R2

∂2 u · ∇b1 j3 dx R2

= I21 + I22 + I23 + I24 . The I21 − I24 terms can be estimated as follows. By Lemma 2.1 and Young’s inequality, we have ˆ |I21 | ≤

|∂2 u∇b3 j1 |dx R2 1

1

1

1

≤ C∇b3 L2 ∂2 uL2 2 j1 L2 2 ∂12 uL2 2 ∂2 j1 L2 2 3

1

≤ C(∂2 ωL2 + ∇jL2 )jL2 2 ∂2 uL2 2 ≤ and

M C (∂2 ω2L2 + ∇j2L2 ) + j3L2 ∂2 uL2 , 34 M

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

11

ˆ |I22 | ≤

|∂1 u∇b3 j2 |dx R2 1

1

1

1

≤ C∇b3 L2 ∂1 uL2 2 j2 L2 2 ∂12 uL2 2 ∂1 j2 L2 2 3

1

≤ C(∂2 ωL2 + ∇jL2 )jL2 2 ωL2 2 ≤

M C (∂2 ω2L2 + ∇j2L2 ) + j3L2 ωL2 . 34 M

We can use similar procedure as I21 and I22 to deduce that |I23 , I24 | ≤

M C (∂2 ω2L2 + ∇j2L2 ) + j3L2 ∂2 uL2 . 34 M

Combining these above estimates, one gets: |I2 | ≤ |I21 | + |I22 | + |I23 | + |I24 | 4M C (∂2 ω2L2 + ∇j2L2 ) + (ω4L2  + j4L2 )(∂2 u2L2 + j2L2 ). ≤ 34 M Noticing the fact that b is divergence free, integrating by parts, we have ˆ

ˆ ∇ × (b∇b) · ω +

I3 + I4 = R2

∇ × (b∇u) · jdx R2

ˆ

ˆ (b · ∇)j · ωdx +

= R2

ˆ

(b · ∇)ω · jdx + R2

ˆ

(∂2 b∇b3 , −∂1 b∇b3 , 0) · ωdx R2

(∂2 b∇u3 , −∂1 b∇u3 , ∂1 b∇u2 − ∂2 b∇u1 ) · jdx

+ R2

ˆ

ˆ (∂2 b∇b3 , −∂1 b∇b3 , 0) · ω +

= R2

ˆ

ˆ

∂2 b∇b3 ω1 dx −

= R2

ˆ

−

ˆ

ˆ

∂1 b∇b3 ω2 dx + R2

ˆ

∂1 b∇u2 j3 dx −

∂1 b∇u3 j2 + R2

(∂2 b∇u3 , −∂1 b∇u3 , ∂1 b∇u2 − ∂2 b∇u1 ) · jdx R2

R2

∂2 b∇u3 j1 dx R2

∂2 b∇u1 j3 dx R2

= I31 + I32 + I33 + I34 + I35 + I36 . We estimate I31 − I36 terms as follows. Applying Lemma 2.1 and Young’s inequality, we have ˆ |I31 | ≤

|∂2 b∇b3 ω1 |dx R2 1

1

1

1

≤ C∇b3 L2 ∂2 bL2 2 ω1 L2 2 ∂12 bL2 2 ∂2 ω1 L2 2 3

1

≤ C(∂2 ωL2 + ∇jL2 )jL2 2 ωL2 2 ≤ Similarly, we have

M C (∂2 ω2L2 + ∇j2L2 ) + j3L2 ωL2 . 34 M

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B. Du / J. Math. Anal. Appl. 484 (2020) 123701

12

|I32 , I33 , I34 , I35 , I36 | ≤

M C (∂2 ω2L2 + ∇j2L2 ) + j3L2 ωL2 . 34 M

Collecting these above estimates, one gets |I3 + I4 | ≤ |I31 | + |I32 | + |I33 | + |I34 | + |I35 | + |I36 | 6M C (∂2 ω2L2 + ∇j2L2 ) + j3L2 ωL2 . ≤ 34 M

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To estimate I5 , we need the following notations. J  ∇ × j = (∂2 j3 , −∂1 j3 , ∂1 j2 − ∂2 j1 ) = (J1 , J2 , J3 ). Integrating by parts, one has ˆ

ˆ (b · ∇)jJdx =

R2

bi ∂i jk Jk dx R2

ˆ

=

ˆ

b1 ∂1 j1 J1 dx + R2

R2

ˆ

R2

ˆ

∂1 b1 j2 ∂1 j3 dx − R2

b3 ∂3 j3 J3 dx R2

ˆ

∂1 b1 j1 ∂2 j3 dx −

R2

−

ˆ

b2 ∂2 j2 J2 dx + R2

ˆ

=−

b1 ∂1 j3 J3 dx R2

ˆ

b2 ∂2 j1 J1 dx +

+

ˆ

ˆ b1 ∂1 j2 J2 dx +

R2

ˆ

b1 j2 ∂11 j3 dx + R2

∂2 b2 j1 ∂2 j3 dx − R2

ˆ

ˆ

−

∂1 b2 j2 ∂2 j3 − R2

ˆ

=−

ˆ

ˆ

b2 j2 ∂12 j3 dx +

ˆ

R2

ˆ

∂2 b2 j1 ∂2 j3 dx +

b2 j2 ∂12 j3 dx R2

b2 j1 ∂22 j3 dx R2

ˆ

ˆ

∂2 b2 j2 ∂1 j3 dx −

∂2 b1 j1 ∂1 j3 dx + R2

b1 j1 ∂12 j3 dx

∂2 b2 j2 ∂1 j3 dx +

R2

ˆ

∂1 b1 j1 ∂2 j3 dx +

b1 j2 ∂11 j3 dx R2

R2

ˆ

R2

R2

R2

∂2 b1 j1 ∂1 j3 dx +

b2 j1 ∂22 j3 dx +

R2

ˆ ∂1 b1 j2 ∂1 j3 dx +

R2

ˆ

ˆ

−

ˆ

b1 j1 ∂12 j3 dx +

R2

∂1 b2 j2 ∂2 j3 dx R2

By above conclusion, we can rewrite I5 as ˆ I5 = −

∇ × [∇ × ((∇ × b) × b)] · jdx

R2

ˆ

=− ˆ

∇ × (j × b) · (∇ × j)dx

R2

ˆ

(j · ∇)b · Jdx −

= R2

(b · ∇)j · Jdx R2

ˆ

ˆ

(j · ∇)b · Jdx +

= R2

ˆ ∂1 b1 j1 ∂2 j3 dx −

R2

= I51 + I52 + I53 + I54 + I55 .

ˆ ∂2 b1 j1 ∂1 j3 dx −

R2

ˆ ∂2 b2 j2 ∂1 j3 dx +

R2

∂1 b2 j2 ∂2 j3 dx R2

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

13

By Lemma 2.1 and Young’s inequality, noticing b is divergence free, one has ˆ |I51 | ≤

|j∇bJ|dx R2 1

1

1

1

≤ C∇jL2 ∇bL2 2 jL2 2 ∂1 ∇bL2 2 ∂2 jL2 2 ≤ C∇jL2 ∇jL2 jL2 ≤

M C ∇j2L2 + j2L2 ∇j2L2 , 34 M

and ˆ |I52 | ≤

|∂1 b1 j1 ∂2 j3 |dx R2 1

1

1

1

≤ C∂2 j3 L2 ∂1 b3 L2 2 j1 L2 2 ∂12 b1 L2 2 ∂1 j1 L2 2 ≤ C∇jL2 ∇jL2 jL2 ≤

M C ∇j2L2 + j2L2 ∇j2L2 . 34 M

We can use similar procedure as I52 to get |I53 , I54 , I55 | ≤

M C ∇j2L2 + j2L2 ∇j2L2 . 34 M

By these above estimates, one gets |I5 | ≤ |I51 | + |I52 | + |I53 | + |I54 | + |I55 | ≤

5M C ∇j2L2 + j2L2 ∇j2L2 . 34 M

Combining (14)–(18), we get 1 d (ω(t)2L2 + j(t)2L2 ) + M (∂2 ω22 + ∇j22 ) 2 dt C C M (∂2 ω2L2 + ∇j2L2 ) + j2L2 ∇j2L2 + (ω4L2 + j4L2 )(∂2 u2L2 + j2L2 ). ≤ 2 M M Hence, one has d (ω(t)2L2 + j(t)2L2 ) + M (∂2 ω22 + ∇j22 ) dt C C ≤ j2L2 ∇j2L2 + (ω4L2 + j4L2 )(∂2 u2L2 + j2L2 ). M M Integrating over (0, T ) to above inequality, by (10) and (12), we get

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B. Du / J. Math. Anal. Appl. 484 (2020) 123701

14

ˆT sup 0≤t≤T

(ω(t)2L2

+

j(t)2L2 )

(∂2 ω22 + ∇j22 )dt

+M 0

(u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 ) ≤C sup (ω(t)2L2 + j(t)2L2 ) M2 0≤t≤T ˆT (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 ) ∇j(t)22 dt + ω0 2L2 + j0 2L2 . +C M

(19)

0

Choose ε0 =

1 2C ,

one has (∇b0 2L2 + ∇u0 2L2 )(b0 2L2 + u0 2L2 ) ≤ ε0 M2 ⎧ (∇b0 2L2 + ∇u0 2L2 )(b0 2L2 + u0 2L2 ) 1 ⎪ ⎪ ≤ ⎨C M2 2 =⇒ 2 2 2 2 ⎪ ⎪ ⎩C (∇b0 L2 + ∇u0 L2 )(b0 L2 + u0 L2 ) ≤ M , M 2

which combining (19), yields (13). We complete the proof of Lemma 4.2.  To prove Theorem 1.3, we still need the following priori estimate. Proposition 4.2. Assume that the condition (10) of the Proposition 4.1 holds. Let (u, b) be a strong solution of the system (1) with μ1 = 0, μ2 > 0, ν1 > 0, ν2 > 0 on [0, T ] × R2 , then ˆT sup 0≤t≤T

(∇ω(t)2L2

+

∇j(t)2L2 )

(∂2 ∇ω(t)22 + Δj(t)22 )dt

+M

(20)

0

≤ C(μ2 , ν1 , ν2 , u0 H 2 , b0 H 2 ), provided that (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 )/M 2 ≤ ε0 , and ε0 is sufficiently small. Proof. Operating ∇× to the first equation of system (1) and taking the L2 −inner products with Δω; Operating ∇× to the second equation of system (1) and taking the L2 −inner products with Δj, then summing them together, one gets 1 d (∇ω(t)2L2 + ∇j(t)2L2 ) + μ2 ∂2 ∇ω2L2 + ν1 ∂1 ∇j2L2 + ν2 ∂2 ∇j2L2 2 dt ˆ ˆ ˆ = − ∇ × (u∇u) · Δωdx − ∇ × (u∇b) · Δjdx + ∇ × (b∇b) · Δωdx R2 ˆ R2 ˆ R2 + ∇ × (b∇u) · Δjdx − ∇ × [∇ × ((∇ × b) × b)] · Δjdx R2

R2

= L1 + L2 + L3 + L4 + L5 . Firstly, use integration by parts and the divergence free of u, we can rewrite L1 as

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B. Du / J. Math. Anal. Appl. 484 (2020) 123701

15

ˆ L1 = − ˆ

∇ × (u∇u) · Δωdx

R2

ˆ

(ω · ∇)u · Δωdx −

= R2

(u · ∇)ω · Δωdx R2

ˆ

ˆ ωi ∂i uk ∂jj ωk dx −

= R2

ui ∂i ωk ∂jj ωk dx R2

ˆ

ˆ

∂j ωi ∂i uk ∂j ωk dx −

=− R2

ˆ ωi ∂ij uk ∂j ωk dx +

R2

∂j ui ∂i ωk ∂j ωk dx R2

= L11 + L12 + L13 . For L11 , we have ˆ L11 ≤

ˆ |∂1 ωi ∂i uk ∂1 ωk |dx +

R2

ˆ

≤

|∂2 ωi ∂i uk ∂2 ωk |dx R2

ˆ

|∂1 ω1 ∂1 uk ∂1 ωk |dx + R2

ˆ |∂1 ω2 ∂2 uk ∂1 ωk |dx +

R2

|ω2 ∂i uk ∂2 ωk |dx R2

= L111 + L112 + L113 . Based on Lemma 4.2, we know ωL2 and jL2 is bounded, hence ∇uL2 and ∇b|L2 is bounded due to u and b is divergence free. By Lemma 2.1 and Young’s inequality, we can use the boundedness of ∇uL2 to obtain ˆ |L111 | =

|∂1 ω1 ∂1 uk ∂1 ωk |dx R2 1

1

1

1

≤ C∂1 uk L2 ∂1 ω1 L2 2 ∂1 ωk L2 2 ∂11 ω1 L2 2 ∂12 ω1 L2 2 1

1

≤ C∂2 ∇ωL2 ∂2 ωL2 2 ∇ωL2 2 ≤

M ∂2 ∇ω2L2 + C∂2 ωL2 ∇ωL2 , 76

here, we using the fact that ∂1 ω1 + ∂2 ω2 = 0, ∂1 ω1 L2 = ∂2 ω2 L2 . Similarly, one can use the boundedness of ∇uL2 and ∇bL2 to deduce that ˆ |L112 | =

|∂1 ω2 ∂2 uk ∂1 ωk |dx R2 1

1

1

1

≤ C∂1 ωk L2 ∂1 ω2 L2 2 ∂2 uk L2 2 ∂12 ω2 L2 2 ∂12 uk L2 2 1

1

3

≤ C∂2 ∇ωL2 2 ∂2 ωL2 2 ∇ωL2 2 ≤ and

2 M ∂2 ∇ω2L2 + C∂2 ωL3 2 ∇ω2L2 , 76

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

16

ˆ |L113 | =

|∂2 ωi ∂i uk ∂2 ωk |dx R2 1

1

1

1

≤ C∂i uk L2 ∂2 ωi L2 2 ∂2 ωk L2 2 ∂12 ωi L2 2 ∂22 ωk L2 2 ≤ C∂2 ∇ωL2 ∂2 ωL2 ≤

M ∂2 ∇ω2L2 + C∂2 ω2L2 . 76

For L12 , we have ˆ L12 ≤

ˆ |ω1 ∂11 uk ∂1 ωk |dx +

R2

ˆ |ω2 ∂21 uk ∂1 ωk |dx +

R2

|ωi ∂i2 uk ∂2 ωk |dx R2

= L121 + L122 + L123 . Noticing the fact that ∂1 ω1 L2 = ∂2 ω2 L2 . By Lemma 2.1 and Young’s inequality, we get ˆ |L121 | =

|ω1 ∂11 uk ∂1 ωk |dx R2 1

1

1

1

1

1

≤ C∂11 uk L2 ω1 L2 2 ∂1 ωk L2 2 ∂1 ω1 L2 2 ∂12 ωk L2 2 1

1

3

≤ C∂2 ∇ωL2 2 ∂2 ωL2 2 ∇ωL2 2 ≤

2 M ∂2 ∇ω2L2 + C∂2 ωL3 2 ∇ω|2L2 , 76

and ˆ |L122 | =

|ω2 ∂12 uk ∂1 ωk |dx R2 1

1

≤ C∂12 uk L2 ω2 L2 2 ∂1 ωk L2 2 ∂1 ω2 L2 2 ∂12 ωk L2 2 1

≤ C∂2 ∇ωL2 2 ∂2 ωL2 ∇ωL2 ≤

4 4 M ∂2 ∇ω2L2 + C∂2 ωL3 2 ∇ω|L3 2 . 76

Similarly, we have |L123 | ≤

4 4 M ∂2 ∇ω2L2 + C∂2 ωL3 2 ∇ω|L3 2 . 76

By almost the same argument, one can get ˆ L13 ≤

ˆ |∂1 u1 ∂1 ωk ∂1 ωk |dx +

R2

ˆ |∂1 u2 ∂2 ωk ∂1 ωk |dx +

R2

|∂2 ui ∂i ωk ∂2 ωk |dx R2

= L131 + L132 + L133 . By Lemma 2.1 and Young’s inequality, noticing the boundedness of ∇uL2 , one has

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

17

ˆ |L131 | =

|∂1 u1 ∂1 ωk ∂1 ωk |dx R2 1

1

1

1

≤ C∂1 ωk L2 ∂1 u1 L2 2 ∂1 ωk L2 2 ∂11 u1 L2 2 ∂12 ωk L2 2 1

1

3

≤ C∂2 ∇ωL2 2 ∂2 ωL2 2 ∇ωL2 2 2 M ∂2 ∇ω2L2 + C∂2 ωL3 2 ∇ω|2L2 , 76

≤

here, we use the fact that ∂11 u1 L2 = ∂12 u2 L2 ≤ C∂2 ωL2 due to u is divergence free. Similarly, we have ˆ |L132 | = |∂1 u2 ∂2 ωk ∂1 ωk |dx R2 1

1

1

1

1

1

≤ C∂1 u2 L2 ∂2 ωk L2 2 ∂1 ωk L2 2 ∂12 ωk L2 2 ∂12 ωk L2 2 1

1

≤ C∂2 ∇ωL2 ∂2 ωL2 2 ∇ωL2 2 M ∂2 ∇ω2L2 + C∂2 ωL2 ∇ω|L2 , 76

≤ and

ˆ |L133 | =

|∂2 ui ∂i ωk ∂2 ωk |dx R2 1

1

≤ C∂i ωk L2 ∂2 ωk L2 2 ∂2 uk L2 2 ∂12 ui L2 2 ∂22 ωk L2 2 1

≤ C∂2 ∇ωL2 2 ∂2 ωL2 ∇ωL2 ≤

4 4 M ∂2 ∇ω2L2 + C∂2 ωL3 2 ∇ω|L3 2 . 76

Therefore |L1 | ≤

9M ∂2 ∇ω2L2 + C∂2 ω2L2 ∇ω2L2 . 76

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Using integration by parts, we can rewrite L2 as follows. ˆ L2 = − ∇ × (u∇b) · Δjdx R2

ˆ

=−

ˆ (u · ∇)j · Δjdx −

R2

R2

ˆ

=− ˆ

(∂2 u∇b3 , −∂1 u∇b3 , ∂1 u∇b2 − ∂2 u∇b1 ) · Δjdx ˆ

ui ∂i jk ∂mm jk dx −

R2

(∂2 u∇b3 , −∂1 u∇b3 , ∂1 u∇b2 − ∂2 u∇b1 ) · Δjdx R2

ˆ

∂m ui ∂i jk ∂m jk dx −

= R2

(∂2 u∇b3 , −∂1 u∇b3 , ∂1 u∇b2 − ∂2 u∇b1 ) · Δjdx R2

ˆ

ˆ

∂m ui ∂i jk ∂m jk dx −

= R2

ˆ

R2

= L21 + L22 + L23 + L24 + L25 .

ˆ ∂1 u∇b3 Δj2 dx −

∂2 u∇b3 Δj1 dx + R2

ˆ ∂1 u∇b2 Δj3 dx +

R2

∂2 u∇b1 Δj3 dx R2

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

18

L21 can be estimated as follows. ˆ |L21 | ≤

ˆ |∂1 ui ∂i jk ∂1 jk |dx +

R2

|∂1 ui ∂i jk ∂1 jk |dx R2

= L211 + L212 . Noticing the boundedness of ∇uL2 , one has ˆ |L211 | =

|∂1 ui ∂i jk ∂1 jk |dx R2 1

1

1

1

1

1

≤ C∂i jk L2 ∂1 ui L2 2 ∂1 jk L2 2 ∂12 ui L2 2 ∂11 jk L2 2 1

1

3

≤ CΔjL2 2 ∂2 ωL2 2 ∇jL2 2 2 M Δj2L2 + C∂2 ωL3 2 ∇j|2L2 , 76

≤ and

ˆ |L212 | =

|∂2 ui ∂i jk ∂2 jk |dx R2 1

1

≤ C∂i jk L2 ∂2 ui L2 2 ∂2 jk L2 2 ∂12 ui L2 2 ∂22 jk L2 2 1

1

3

≤ CΔjL2 2 ∂2 ωL2 2 ∇jL2 2 ≤

2 M Δj2L2 + C∂2 ωL3 2 ∇j|2L2 . 76

We can use the boundedness of ∇uL2 and ∇bL2 to obtain ˆ |L22 | ≤

|∂2 u∇b3 Δj1 |dx R2 1

1

1

1

≤ CΔj1 L2 ∂2 uL2 2 ∇b3 L2 2 ∂12 uL2 2 ∂2 ∇bL2 2 1

1

≤ CΔjL2 ∂2 ωL2 2 ∇jL2 2 ≤

M Δj2L2 + C∂2 ωL2 ∇j|L2 . 76

Similarly, we have |L23 , L24 , L25 | ≤

M Δj2L2 + C∂2 ωL2 ∇j|L2 . 76

Hence, one gets |L2 | ≤

6M Δj2L2 + C∂2 ω2L2 ∇j|2L2 . 76

Use integration by parts, we rewrite L3 + L4 as follows.

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B. Du / J. Math. Anal. Appl. 484 (2020) 123701

ˆ

ˆ ∇ × (b∇b) · Δωdx +

L3 + L4 = R2

∇ × (b∇u) · Δjdx R2

ˆ

ˆ (b · ∇)j · Δωdx +

=

19

R2

ˆ (b · ∇)ω · Δjdx −

R2

(j · ∇)b · Δωdx R2

ˆ (∂2 b∇u3 , −∂1 b∇u3 , ∂1 b∇u2 − ∂2 b∇u1 ) · Δjdx

+ R2

= L31 + L32 + L33 + L34 . By virtue of divergence free of b, using integration by parts, one can rewrite L31 + L32 as follows. ˆ

ˆ (b · ∇)j · Δωdx +

L31 + L32 =

(b · ∇)ω · Δjdx

R2

R2

ˆ

ˆ

=

bi ∂i jk ∂mm ωk dx + R2

bi ∂i ωk ∂mm jk dx R2

ˆ =−

ˆ ∂m bi ∂i jk ∂m ωk dx −

R2

bi ∂im jk ∂m ωk dx − R2

ˆ

ˆ ∂m bi ∂i ωk ∂m jk dx −

R2

bi ∂im ωk ∂m jk dx R2

ˆ −∂m bi ∂i jk ∂m ωk −

=

ˆ

R2

∂m bi ∂i ωk ∂m jk dx R2

= L311 + L312 . We estimate L311 as follows. ˆ |L311 | ≤

|∂m bi ∂i jk ∂m ωk |dx R2

ˆ ≤

ˆ |∂1 b1 ∂1 jk ∂1 ωk |dx +

R2

ˆ |∂1 b2 ∂2 jk ∂1 ωk |dx +

R2

|∂2 bi ∂i jk ∂2 ωk |dx R2

= L3111 + L3112 + L3113 . Noticing the boundedness of ∇bL2 , one gets ˆ |L3111 | =

|∂1 b1 ∂1 jk ∂1 ωk |dx R2 1

1

1

1

≤ C∂1 b1 L2 ∂1 jk L2 2 ∂1 ωk L2 2 ∂11 jk L2 2 ∂12 ωk L2 2 ≤ C(∂2 ∇ωL2 + ΔjL2 )(∇ωL2 + ∇jL2 ) ≤

M (∂2 ∇ω2L2 + Δj2L2 ) + C(∇ω2L2 + ∇j2L2 ). 76

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

20

ˆ |L3112 | =

|∂1 b2 ∂2 jk ∂1 ωk |dx R2 1

1

1

1

≤ C∂2 jk L2 ∂1 b2 L2 2 ∂1 ωk L2 2 ∂11 b2 L2 2 ∂12 ωk L2 2 1

1

3

≤ C∂2 ∇ωL2 2 ∇ωL2 2 ∇jL2 2 ≤

2 M ∂2 ∇ω2L2 + C∇ωL3 2 ∇j2L2 , 76

and ˆ |L3113 | =

|∂2 bi ∂i jk ∂2 ωk |dx R2 1

1

1

1

≤ C∂2 bi L2 ∂i jk L2 2 ∂2 ωk L2 2 ∂2i j2 L2 2 ∂12 ωk L2 2 1

1

≤ C(∂2 ∇ωL2 + ΔjL2 )∂2 ωL2 2 ∇jL2 2 ≤

M (∂2 ∇ω2L2 + Δj2L2 ) + C∂2 ωL2 ∇jL2 . 76

Combining these above estimates, one gets

|L311 | ≤

3M (∂2 ∇ω2L2 + Δj2L2 ) + C(∂2 ω2L2 + ∇j2L2 )(∇ω2L2 + ∇j2L2 ). 76

Similarly, one gets

|L312 | ≤

3M (∂2 ∇ω2L2 + Δj2L2 ) + C(∂2 ω2L2 + ∇j2L2 )(∇ω2L2 + ∇j2L2 ). 76

Therefore, we get

|L31 + L32 | ≤

6M (∂2 ∇ω2L2 + Δj2L2 ) + C(∂2 ω2L2 + ∇j2L2 )(∇ω2L2 + ∇j2L2 ). 76

Integrating by parts, we can deduce L33 that ˆ L33 = −

(j · ∇)b · Δωdx

R2

ˆ

=− ˆ =

ji ∂i bk ∂mm ωk dx

R2

∂m ji ∂i bk ∂m ωk dx + R2

= L331 + L332 . L331 can be estimated as

ˆ ji ∂im bk ∂m ωk dx R2

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B. Du / J. Math. Anal. Appl. 484 (2020) 123701

21

ˆ |L331 | =

|∂m ji ∂i bk ∂m ωk dx|dx R2

ˆ

ˆ

|∂1 ji ∂i bk ∂1 ωk |dx +

≤ R2

|∂2 ji ∂i bk ∂2 ωk |dx R2

= L3311 + L3312 . We can use the boundedness of ∇bL2 to obtain ˆ |L3311 | =

|∂1 ji ∂i bk ∂1 ωk |dx R2 1

1

1

1

≤ C∂i bk L2 ∂1 ji L2 2 ∂1 ωk L2 2 ∂11 ji L2 2 ∂12 ωk L2 2 ≤ C(∂2 ∇ωL2 + ΔjL2 )(∇ωL2 + ∇jL2 ) ≤

M (∂2 ∇ω2L2 + Δj2L2 ) + C(∇ω2L2 + ∇j2L2 ), 76

and ˆ |L3312 | =

|∂2 ji ∂i bk ∂2 ωk |dx R2 1

1

1

1

≤ C∂i bk L2 ∂2 ji L2 2 ∂2 ωk L2 2 ∂22 ji L2 2 ∂12 ωk L2 2 1

1

≤ C(∂2 ∇ωL2 + ΔjL2 )∂2 ωL2 2 ∇jL2 2 ≤

M (∂2 ∇ω2L2 + Δj2L2 ) + C∂2 ωL2 ∇jL2 . 76

Similarly, one has ˆ |L332 | ≤

ˆ |ji ∂i1 bk ∂1 ωk |dx +

R2

|ji ∂i2 bk ∂2 ωk |dx R2

= L3321 + L3322 . Applying Lemma 2.1, Young’s inequality and the boundedness of ∇bL2 , we obtain ˆ |L3321 | =

|ji ∂i1 bk ∂1 ωk |dx R2 1

1

1

1

≤ C∂i1 bk L2 ji L2 2 ∂1 ωk L2 2 ∂1 ji L2 2 ∂12 ωk L2 2 1

1

3

≤ C∂2 ∇ωL2 2 ∇ωL2 2 ∇jL2 2 ≤

2 M ∂2 ∇ω2L2 + C∇ωL3 2 ∇j2L2 . 76

We can use similar procedure as L3321 to deduce that |L3322 | ≤ By these above estimates, one gets

2 M |∂2 ∇ω2L2 + C∇ωL3 2 ∇j2L2 . 76

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

22

|L33 | ≤

4M (∂2 ∇ω2L2 + Δj2L2 ) + C(∂2 ω2L2 + ∇j2L2 )(∇ω2L2 + ∇j2L2 ). 76

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One can rewrite L34 as ˆ (∂2 b∇u3 , −∂1 b∇u3 , ∂1 b∇u2 − ∂2 b∇u1 ) · Δjdx

L34 = R2

ˆ

ˆ

ˆ

∂2 b∇u3 Δj1 dx −

= R2

ˆ ∂1 b∇u2 Δj3 dx −

∂1 b∇u3 Δj2 dx + R2

R2

∂2 b∇u1 Δj3 dx R2

= L341 + L342 + L343 + L344 . Using the boundedness of ∇uL2 and ∇bL2 , one has ˆ |L341 | ≤

|∂2 b∇u3 Δj1 |dx R2 1

1

1

1

≤ CΔj1 L2 ∂2 bL2 2 ∇u3 L2 2 ∂12 bL2 2 ∂2 ∇u3 L2 2 1

1

≤ CΔjL2 ∂2 ωL2 2 ∇jL2 2 ≤

M Δj2L2 + C∂2 ωL2 ∇jL2 . 76

One can use similar procedure as L341 to obtain |L342 , L343 , L344 | ≤

M Δj2L2 + C∂2 ωL2 ∇jL2 76

Therefore, we get |L34 | ≤

4M Δj2L2 + C∂2 ωL2 ∇jL2 . 76

(26)

Combining (24)–(26), we have |L3 + L4 | ≤

14M (∂2 ∇ω2L2 + Δj2L2 ) + C(∂2 ω2L2 + ∇j2L2 )(∇ω2L2 + ∇j2L2 ). 76

We rewrite L5 as ˆ L5 = −

∇ × [∇ × ((∇ × b) × b)] · Δjdx

R2

ˆ

=− ˆ

∇ × (j × b) · ΔJdx

R2

ˆ

(j · ∇)b · ΔJdx −

= R2

= L51 + L52 . One can use integration by parts to deduce that

(b · ∇)j · ΔJdx R2

(27)

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

23

ˆ (j · ∇)b · ΔJdx

L51 = R2

ˆ

ji ∂i bk ∂mm Jk dx

= R2

ˆ

ˆ

=−

∂m ji ∂i bk ∂m Jk dx −

R2

ji ∂im bk ∂m Jk dx R2

= L511 + L512 . Using the fact ∂im bk L2 ≤ C∇jL2 , ∂2im bk L2 ≤ CΔjL2 due to b is divergence free, noticing the boundedness of ∇bL2 , one can estimate L511 and L512 as follows. ˆ |L511 | ≤

|∂m ji ∂i bk ∂m Jk |dx R2 1

1

1

1

≤ C∂m Jk L2 ∂m ji L2 2 ∂i bk L2 2 ∂1m ji L2 2 ∂2i bk L2 2 3

≤ CΔjL2 2 ∇jL2 ≤

M Δj2L2 + C∇j2L2 ∇j2L2 , 76

and ˆ |L512 | ≤

|ji ∂im bk ∂m Jk |dx R2 1

1

1

1

≤ C∂m Jk L2 ji L2 2 ∂im bk L2 2 ∂1 ji L2 2 ∂2im bk L2 2 3

≤ CΔjL2 2 ∇jL2 M Δj2L2 + C∇j2L2 ∇j2L2 . 76

≤ Similarly, L52 can be rewritten as

ˆ L52 = −

(b · ∇)j · ΔJdx.

R2

ˆ

=− ˆ =

bi ∂i jk ∂mm Jk dx

R2

ˆ

∂m bi ∂i jk ∂m Jk dx + R2

bi ∂im jk ∂m Jk dx R2

= L521 + L522 . We can use similar procedure as L511 to deduce that |L521 | ≤

M Δj2L2 + C∇j2L2 ∇j2L2 . 76

To estimate L522 , we rewrite L522 as follows

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

24

ˆ L522 =

bi ∂im jk ∂m Jk dx R2

ˆ

ˆ

ˆ

b1 ∂11 j1 ∂1 J1 dx +

= R2

+

b1 ∂11 j2 ∂1 J2 dx + R2

ˆ

b1 ∂12 j2 ∂2 J2 dx +

b2 ∂12 j3 ∂1 J3 dx +

+ R2

=

12 

ˆ

b1 ∂12 j3 ∂2 J3 dx +

b2 ∂12 j1 ∂1 J1 dx + R2

ˆ

b2 ∂12 j2 ∂1 J2 dx R2

ˆ

ˆ

b2 ∂22 j1 ∂2 J1 dx + R2

b1 ∂12 j1 ∂2 J1 dx R2

ˆ

R2

ˆ

b1 ∂11 j3 ∂1 J3 dx + R2

ˆ

R2

ˆ

b2 ∂22 j2 ∂2 J2 dx + R2

b2 ∂22 j3 ∂2 J3 dx R2

Qi .

i=1

Integrating by parts, we can rewrite Q1 − Q12 as follows. ˆ ˆ Q1 = b1 ∂11 j1 ∂1 J1 dx = b1 ∂11 j1 ∂12 j3 dx R2

R2

ˆ

ˆ

∂1 b1 ∂1 j1 ∂12 j3 dx −

=− R2

ˆ

ˆ

b1 ∂11 j2 ∂1 J2 dx = −

Q2 = R2

ˆ ∂1 b1 ∂1 j2 ∂11 j3 dx +

= R2

Q3 =

b1 ∂11 j2 ∂11 j3 dx

R2

ˆ

ˆ

b1 ∂1 j1 ∂112 j3 dx. R2

b1 ∂1 j2 ∂111 j3 dx. R2

b1 ∂11 j3 ∂1 J3 dx R2

ˆ

b1 ∂11 j3 ∂11 j2 − b1 ∂11 j3 ∂12 j1 dx

= R2

ˆ

ˆ ∂1 b1 ∂1 j2 ∂11 j3 dx −

=− R2

ˆ

ˆ

b1 ∂1 j2 ∂111 j3 dx + R2

∂2 b1 ∂1 j1 ∂11 j3 dx + R2

b1 ∂1 j1 ∂112 j3 dx. R2

Similarly, we have ˆ

ˆ

Q4 = −

∂1 b1 ∂2 j1 ∂22 j3 dx −

R2

ˆ Q5 = Q6 = −

ˆ

∂2 b1 ∂1 j2 ∂12 j3 dx + R2

ˆ

b1 ∂2 j1 ∂112 j3 dx. R2

R2

R2

ˆ

∂2 b1 ∂1 j2 ∂12 j3 dx −

b1 ∂1 j2 ∂122 j3 dx. ˆ

ˆ

b1 ∂1 j2 ∂122 j3 dx + R2

R2

ˆ Q7 = −

∂1 b2 ∂2 j1 ∂12 j3 dx −

ˆ

b2 ∂2 j1 ∂112 j3 dx. R2

ˆ

∂2 b2 ∂1 j2 ∂11 j3 dx + R2

b2 ∂1 j2 ∂112 j3 dx. R2

b1 ∂2 j1 ∂122 j3 dx. R2

ˆ

R2

Q8 =

∂2 b1 ∂2 j1 ∂12 j3 dx +

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

ˆ

ˆ

Q9 = −

ˆ

∂1 b2 ∂1 j2 ∂12 j3 dx −

R2

b2 ∂1 j2 ∂112 j3 dx + R2

Q10 = −

∂1 b2 ∂2 j1 ∂12 j3 dx +

∂2 b2 ∂2 j1 ∂22 j3 dx −

ˆ

ˆ

∂2 b2 ∂2 j2 ∂12 j3 dx +

b2 ∂2 j2 ∂122 j3 dx. R2

ˆ

∂1 b2 ∂2 j2 ∂22 j3 dx −

R2

b2 ∂2 j1 ∂222 j3 dx. R2

ˆ

R2

b2 ∂2 j1 ∂112 j3 dx. R2

ˆ

R2

Q12 = −

ˆ

R2

ˆ

Q11 =

25

ˆ

b2 ∂2 j2 ∂122 j3 dx + R2

ˆ ∂2 b2 ∂2 j1 ∂22 j3 dx +

R2

b2 ∂2 j1 ∂222 j3 dx. R2

Combining these above conclusions, one gets L522 =

12 

Qi

i=1

ˆ

ˆ

∂2 b1 ∂1 j1 ∂11 j3 dx −

= R2

R2

∂2 b2 ∂1 j2 ∂11 j3 dx − R2

∂2 b1 ∂2 j1 ∂12 j3 dx R2

ˆ

ˆ

+

ˆ ∂1 b1 ∂2 j1 ∂11 j3 dx + ˆ

∂1 b2 ∂1 j2 ∂12 j3 dx − R2

∂1 b2 ∂2 j2 ∂22 j3 dx R2

= L5221 + L5222 + L5223 + L5224 + L5225 + L5226 . Applying Lemma 2.1 and Young’s inequality, noticing the boundedness of ∇bL2 , we have ˆ |L5221 | ≤

|∂2 b1 ∂1 j1 ∂11 j3 |dx R2 1

1

1

1

≤ C∂11 j3 L2 ∂2 b1 L2 2 ∂1 jk L2 2 ∂22 b1 L2 2 ∂11 j1 L2 2 3

≤ CΔjL2 2 ∇jL2 ≤

M Δj2L2 + C∇j2L2 ∇j2L2 . 76

By almost the same argument, we can get |L5222 , L5223 , L5224 , L5225 , L5226 | ≤

M Δj2L2 + C∇j2L2 ∇j2L2 . 76

Collecting these above estimates, we have |L5 | ≤

9M Δj2L2 + C∇j2L2 ∇j2L2 . 76

Combining (21), (22), (23), (27), (28), we get 1 d (∇ω(t)2L2 + ∇j(t)2L2 ) + M (∂2 ∇ω22 + Δj22 ) 2 dt M ≤ (∂2 ∇ω22 + Δj2L2 ) + C(∇j2L2 + ∂2 ω2L2 )(∇ω2L2 + ∇j2L2 ). 2 Therefore

(28)

26

B. Du / J. Math. Anal. Appl. 484 (2020) 123701

d (∇ω(t)2L2 + ∇j(t)2L2 ) + M (∂2 ∇ω22 + Δj22 ) dt ≤ C(∂2 ω2L2 + ∇j2L2 )(∇ω2L2 + ∇j2L2 ). Applying Gronwall’s inequality to the above inequality, together with (13), yields (20). Thus we complete the proof of Proposition 4.2.  5. Proof of Theorem 1.3 Sketch of proof of Theorem 1.3. Thanks to the local existence result (Theorem 1.1) and the global priori estimates in Proposition 4.1 and Proposition 4.2. By continuity arguments as described in [23] we can obtain the global existence result of Theorem 1.3, provided (u0 2L2 + b0 2L2 )(ω0 2L2 + j0 2L2 )/M 2 ≤ ε0 . The uniqueness of strong solutions can be proved by the standard L2 −method, and the details are omitted here for simplicity. We complete the proof of Theorem 1.3.  Acknowledgments The author would like to thank the anonymous referee for his (her) helpful suggestions and careful reading. References [1] M. Acheritogaray, P. Degond, A. Frouvelle, J.G. Liu, Kinetic formulation and global existence for the Hallmagnetohydrodynamic system, Kinet. Relat. Models 4 (2011) 908–918. [2] C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011) 1803–1822. [3] D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006) 497–513. [4] D. Chae, J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics, J. Differential Equations 256 (2014) 3835–3858. [5] D. Chae, M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations 255 (2013) 3871–3882. [6] D. Chae, S. Weng, Singularity formation for the incompressible Hall-magnetohydrodynamic equations without resistivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1009–1022. [7] D. Chae, J. Wolf, On partial regularity for the steady Hall magnetohydrodynamics system, Comm. Math. Phys. 339 (2015) 1147–1166. [8] D. Chae, J. Wolf, On partial regularity for the 3D non-sationary Hall magnetohydrodynamics equations on the plane, SLAM J. Math. Anal. 48 (2016) 443–469. [9] D. Chae, P. Degond, J.G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014) 555–565. [10] J. Cheng, L. Du, On two-dimensional magnetic Bénard problem with mixed partial viscosity, J. Math. Fluid Mech. 17 (2015) 769–797. [11] J. Cheng, Y. Liu, Global regularity of the 2D magnetic-micropolar fluid flows with mixed partial viscosity, Comput. Math. Appl. 70 (2015) 66–72. [12] L. Du, H. Lin, Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions, Nonlinearity 26 (2013) 219–239. [13] L. Du, D. Zhou, Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal. 47 (2015) 1562–1589. [14] E. Dumas, F. Sueur, On the weak solutions to the Maxwell-Landau-Lifshitz equations and to Hall-magnetohydrodynamic equations, Comm. Math. Phys. 330 (2014) 1179–1225. [15] G. Duvaut, J.L. Lions, Inéquations en thermoéalsticite et magnétohydrodynamique, Arch. Ration. Mech. Anal. 46 (1972) 241–279. [16] J. Fan, S. Huang, G. Nakamura, Well-posedness for the axisymmetric incompressible viscous Hall-magnetohydrodynamic equations, Appl. Math. Lett. 26 (2013) 963–967. [17] C. He, Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations 213 (2005) 235–252. [18] C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152. [19] H. Homann, R. Grauer, Bifurcation analysis of magnetic reconnection in Hall-MHD systems, Phys. D 208 (2005) 59–72. [20] F. Lin, L. Xu, P. Zhang, Global small solutions to 2-D incompressible MHD system, J. Differential Equations 259 (2015) 5440–5485.

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