Regularity criteria for weak solution to the 3D magnetohydrodynamic equations

Acta Mathematica Scientia 2012,32B(3):1063–1072 http://actams.wipm.ac.cn

REGULARITY CRITERIA FOR WEAK SOLUTION TO THE 3D MAGNETOHYDRODYNAMIC EQUATIONS∗

)1

Wang Yuzhu (




Wang Shubin (

)2

)1

Wang Yinxia (

1. School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China 2. Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China

Abstract In this article, regularity criteria for the 3D magnetohydrodynamic equations are investigated. Some sufficient integrability conditions on two components or the gradient of two components of u + B and u − B in Morrey-Campanato spaces are obtained. Key words Magnetohydrodynamic equations; regularity criteria; Morrey-Campanato space 2000 MR Subject Classification

1

35K15; 35K45

Introduction

This article concerns about the regularity of weak solutions to the viscous incompressible magnetohydrodynamics (MHD) equations in R3 ⎧ 1 2 1 ⎪ ⎪ ⎪ ∂t u + u · ∇u − B · ∇B − Re Δu + ∇(p + 2 |B| ) = 0, ⎪ ⎨ 1 (1.1) ΔB = 0, ∂t B + u · ∇B − B · ∇u − ⎪ ⎪ Rm ⎪ ⎪ ⎩ ∇ · u = 0, ∇ · B = 0 with the initial condition t=0:

u = u0 (x), B = B0 (x),

x ∈ R3 .

(1.2)

Here, u = (u1 , u2 , u3 ), B = (B1 , B2 , B3 ), and P = p + 12 |B|2 are non-dimensional quantities corresponding to the flow velocity, the magnetic field, and the total kinetic pressure at the point (x, t), while u0 (x) and B0 (x) are the given initial velocity and initial magnetic field with ∇·u0 = 0 and ∇ · B0 = 0, respectively. The non-dimensional number Re > 0 is the Reynolds number, and Rm > 0 is the magnetic Reynolds number. The 3-dimensional magnetohydrodynamic equations govern the dynamics of electrically conducting fluids. Examples of such fluids include ∗ Received

January 22, 2011; revised March 9, 2011. This work was supported in part by the NNSF of China (11101144, 11171377) and Research Initiation Project for High-level Talents (201031) of North China University of Water Resources and Electric Power.

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plasmas, liquid metals, and salt water. MHD theory has the broad applications to many branches of sciences, for example, geophysics, astrophysics, and engineering problems [8]. The local well-posedness of the Cauchy problem (1.1)–(1.2) in the usual Sobolev spaces H s (R3 ) was established in [15] for any given initial data u0 , B0 ∈ H s (R3 ), s ≥ 3. But whether this unique local solution can exist globally is a challenge open problem in mathematical fluid mechanics. There were numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1)–(1.2) (see [1–3, 5–7, 16–21, 23]). Serrin-type regularity criteria in term of the velocity only were established in [5] and [17]. Zhou [19] also investigated the generalized viscous MHD equations, and Serrin-type regularity criteria were obtained. Logarithmically improved regularity criteria for the 3D viscous MHD equations were obtained in [2] and [23]. A regularity criterion was proved in [1] by adding condition on the velocity in the Besov spaces. Zhou and Gala [20] proved regularity for u and ∇ × u in the multiplier spaces. Wu [16] considered the velocity field in the homogeneous Besov space. Regularity criterion was obtained by imposing a condition on the pressure in [18]. Suggested by the results in [4, 14], one may presume upon that there should have some cancelation between the velocity field and the magnetic field. In particular, some regularity criteria in terms of the combination of u and B were established (see [3, 6]). Gala [3] proved the regularity criterion in terms of the combination of u and B in multiplier spaces X˙ r = M (H˙ r , L2 ). The space X˙ r was characterized by Mazya [13] in terms of Sobolev capacities. X˙ r was used in the study of the uniqueness of weak solutions for the Navier-Stokes equations in [9–11]. The purpose of this article is to improve the regularity criteria of weak solutions via two components or the gradient of two components of u + B and u − B in the Morrey-Campanato space M˙ 2, 3r with 0 < r < 1. Zhou and Gala [22] established a Serrin-type regularity criterion in terms of the pressure for Leray weak solutions to the three-dimensional Navier-Stokes equation in Morrey-Campanato space M˙ 2, r3 with 0 < r < 1. For this purpose, we reformulate (1.1)–(1.2) as follows. Formally, if the first equation of MHD equations (1.1) plus and minus the second one, respectively, then (1.1)–(1.2) can be rewritten as ⎧ ⎪ ⎪ ⎨ ∂t U − μΔU − νΔV + V · ∇U + ∇P = 0, ⎪ ⎪ ⎩

∂t V − μΔV − νΔU + U · ∇V + ∇P = 0,

(1.3)

∇ · U = 0, ∇ · V = 0

with the initial condition t=0:

U = U0 (x), V = V0 (x).

(1.4)

Here, U = u + B, V = u − B, U0 = u0 + B0 , V0 = u0 + B0 and μ=

1 1 1 1 + , ν= − . 2Re 2Rm 2Re 2Rm

Note that μ > |ν|. The article is organized as follows. We first state some preliminaries on function spaces and some important inequalities in Section 2. Then, we give definition of weak solution and state main results in Section 3 and then prove main results in Section 4 and Section 5, respectively.

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Preliminaries

Before stating our main results, we recall the definition and some properties of the spaces that we are going to use. These spaces play an important role in studying the regularity of solutions to nonlinear differential equations. Definition 2.1 For 1 < p ≤ q ≤ +∞, the Morrey-Campanato space M˙ p,q is defined by   3 3 M˙ p,q = f ∈ Lploc (R3 )|f M˙ p,q = sup sup R q − p f Lp(B(x,R)) < ∞ , x∈R3 R>0

where B(x, R) denotes the ball with center at x and of radius R. It is verified that M˙ p,q is a Banach space under the norm  · M˙ p,q . Furthermore, it is checked that 3 f (λ·)M˙ p,q = λ− q f M˙ p,q , λ > 0. Morrey-Campanato spaces can be seen as a complement to Lp spaces. In fact, for p ≤ q, we have Lq = M˙ q,q ⊂ M˙ p,q . We have the following comparison between Lorentz spaces and Morrey-Campanato spaces: for 3 3 p ≥ 2, L r (R3 ) ⊂ L r ,∞ (R3 ) ⊂ M˙ p, r3 (R3 ), where Lp,∞ denotes the usual Lorentz (weak Lp ) space. In the proof of our main results, we need the following lemma given in [21]. Lemma 2.2 For 0 ≤ r < 32 , the space Z˙ r is defined as the space of f (x) ∈ L2loc (R3 ) with f Z˙ r =

sup gB˙ r ≤1

f gL2 < ∞.

(2.1)

2,1

Then, f ∈ M˙ 2, r3 if and only if f ∈ Z˙ r with equivalence of norms. Noting the fact that  r ⊂ H˙ r , 0 < r < 1, L2 H˙ 1 ⊂ B˙ 2,1 we have X˙ r ⊂ M˙ 2, r3 , where X˙ r denotes the point-wise multiplier space from H˙ r to L2 . We need the following lemma basically established in [12]. For completeness, the proof will be also sketched here. Lemma 2.3 For 0 < r < 1 , the inequality r f B˙ r ≤ Cf 1−r L2 ∇f L2

(2.2)

2,1

holds, where C is a positive constant depending on r. Proof It follows from the definition of Besov spaces that f B˙ r = 2ir Δi f L2 2,1

i∈Z

≤



2ir Δi f L2 +

i≤j

≤

i≤j jr

22ir



 12

2i(r−1) 2i Δi f L2

i>j

Δi f 2L2

 12

≤ C(2 f L2 + 2



22i(r−1)

i>j

i≤j j(r−1)

+

 12

22i Δi f 2L2

 12

i>j

f H˙ 1 )

r = C(2jr A−r + 2j(r−1) A1−r )f 1−r ˙ 1, L2 f H

(2.3)

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where A =

f H˙ 1 f L2

. Taking j such that

1 2

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≤ 2jr A−r ≤ 1, from (2.3) we obtain

1 −1 1−r r r r )f  f B˙ r ≤ (1 + 2j(r−1) A1−r )f 1−r ˙ 1 ≤ C(1 + ( ) L2 f H L2 ∇f L2 . 2,1 2 Therefore, we have completed the proof of Lemma 2.3.

3

Main Results Before stating our main results, we introduce some function spaces. Let ∞ (R3 ) = {ϕ ∈ (C ∞ (R3 ))3 : ∇ · ϕ = 0} ⊂ (C ∞ (R3 ))3 . C0,σ

∞ with respect to L2 -norm:  · L2 . Hσr is the The subspace is obtained as the closure of C0,σ ∞ r closure of C0,σ with respect to the H -norm r

uH r = (I − Δ) 2 uL2 , r ≥ 0. Before stating our main results, we give the definition of weak solution to (1.1)–(1.2). Definition 3.1 Given u0 , B0 ∈ L2σ (R3 ), a pair (u, B) on R3 × (0, T ) is called a weak solution to (1.1)–(1.2), provided that (1) u, B ∈ L∞ (0, T ; L2(R3 )) L2 (0, T ; H 1(R3 )), (2) ∇ · u = 0, ∇ · B = 0 in the sense of distribution, (3) for all φ ∈ C0∞ (R3 × (0, T )) with ∇ · φ = 0,

T 1 ∇u · ∇φ + ∇φ : (u ⊗ u − B ⊗ B))dxdt = −(u0 , φ(0)) (u · ∂t φ − Re 3 R 0 and

T

R3

0

(B · ∂t φ −

1 ∇B · ∇φ + ∇φ : (u ⊗ B − B ⊗ u))dxdt = −(B0 , φ(0)). Rm

The weak solution (U, V ) to (1.3)–(1.4) can be defined in a similar way as follows. Definition 3.2 Given U0 , V0 ∈ L2σ (R3 ), a pair (U, V ) on R3 × (0, T ) is called a weak solution to (1.3)–(1.4), provided that (1) U, V ∈ L∞ (0, T ; L2 (R3 )) L2 (0, T ; H 1 (R3 )), (2) ∇ · U = 0, ∇ · V = 0 in the sense of distribution, (3) for all φ ∈ C0∞ (R3 × (0, T )) with ∇ · φ = 0,

T (U · ∂t φ − μ∇U · ∇φ − ν∇V · ∇φ + ∇φ : (V ⊗ U ))dxdt = −(U0 , φ(0)) R3

0

and

0

T

R3

(V · ∂t φ − μ∇V · ∇φ − ν∇U · ∇φ + ∇φ : (U ⊗ V ))dxdt = −(V0 , φ(0)).

˜ = (U1 , U2 , 0) = (u1 + B1 , u2 + B2 , 0), V˜ = In what follows, we use the notations: U ˜ = (∂1 , ∂2 , 0). (V1 , V2 , 0) = (u1 − B1 , u2 − B2 , 0), and ∇ 1 Theorem 3.1 Let (u0 , B0 ) ∈ Hσ (R3 ). Assume that (u, B) is a weak solution to (1.1)– (1.2) on some interval [0, T ] with 0 < T ≤ ∞. If 2 2 ˙ 3 (R3 )), V˜ ∈ L 1−r ˙ 3 (R3 )), ˜ ∈ L 1−r (0, T ; M (0, T ; M U 2, r 2, r

0 < r < 1,

(3.1)

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then, U, V ∈ L∞ (0, T ; H 1(R3 ))



L2 (0, T ; H 2 (R3 )).

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(3.2)

Hence, (u, B) is smooth in R3 × [0, T ]. Theorem 3.2 Let (u0 , B0 ) ∈ Hσ1 (R3 ). Assume that (u, B) is a weak solution to (1.1)– (1.2) on some interval [0, T ] with 0 < T ≤ ∞. If 2 2 ˙ 3 (R3 )) 0 < r < 1, ˜ ∈ L 2−r (0, T ; M˙ 2, 3r (R3 )), ∇V˜ ∈ L 2−r (0, T ; M ∇U 2, r

then, U, V ∈ L∞ (0, T ; H 1(R3 ))



L2 (0, T ; H 2 (R3 )).

(3.3)

(3.4)

Hence, (u, B) is smooth in R3 × [0, T ].

4

Proof of Theorem 3.1 Multiplying the first equation of (1.3) by −ΔU and using integration by parts, we obtain

1 d ΔV · ΔU dx + (V · ∇)U · ΔU dx. (4.1) ∇U (t)2L2 + μΔU (t)2L2 = −ν 2 dt R3 R3

Similarly, we obtain 1 d ∇V (t)2L2 + μΔV (t)2L2 = −ν 2 dt

R3

ΔV · ΔU dx +

R3

(U · ∇)V · ΔV dx.

(4.2)

Summing up (4.1) and (4.2), we deduce that d (∇U (t)2L2 + ∇V (t)2L2 ) + 2μ(ΔU (t)2L2 + ΔV (t)2L2 ) dt

= −4ν ΔV · ΔU dx + 2 (V · ∇)U · ΔU dx + 2 (U · ∇)V · ΔV dx R3

R3

R3

 I1 + I2 + I3 .

(4.3)

With the help of Cauchy-Schwarz inequality, we obtain I1 ≤ 2|ν|(ΔU (t)2L2 + ΔV (t)2L2 ).

(4.4)

Using integration by parts and ∇ · U = 0, we have I2 = −2 = −2 +2

3 3 i=1 R

3 3 i=1 R 3

i=1

R3

∂i V · ∇U · ∂i U dx ˜ · ∂i U dx − 2 ∂i V˜ · ∇U

3 i=1

R3

˜ · ∂i U ˜ dx ∂i V3 ∂3 U

˜ ·U ˜ ∂i U3 dx ∂i V3 ∇

 I21 + I22 + I23 .

(4.5)

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In what follows, we make estimates for I2,j (j = 1, 2, 3). Using integration by parts, the H¨ older inequality, and (2.2), we obtain I21 = 2 ≤2

3 i=1 3

R3

˜ i U · ∂i U dx + 2 V˜ · ∇∂

˜ i U L2 + 2 V˜ ∂i U L2 ∇∂

i=1

3 R3

i=1 3

˜ · ∂i2 U dx V˜ · ∇U

˜ L2 ∂i2 U L2 V˜ ∇U

i=1

≤ CV˜ M˙

∇U B˙ r ΔU L2

≤ CV˜ M˙

1+r ∇U 1−r L2 ΔU L2

2, 3 r 2, 3 r

2,1

2

1−r ≤ εΔU 2L2 + CV˜ M ∇U 2L2 . ˙

(4.6)

2, 3 r

It follows from integration by parts, the H¨ older inequality, (2.2), and Young inequality that I22 = 2

3 R3

i=1

≤2

3

≤C

˜ · ∂i U ˜ dx + 2 ∂i ∂3 V3 U

3 i=1

˜ · ∂i U ˜ L2 ∂i ∂3 V3 L2 + 2 U

i=1 3

R3

3

˜ · ∂i ∂3 U ˜ dx ∂i V3 U

˜ L2 ∂i ∂3 U ˜ L2 ∂i V U

i=1

˜ ˙ U M

2,

i=1

˜ ˙ ≤ CU M

2, 3 r

˜ ˙ r ∂i ∂3 V3 L2 + C ∂i U B 3 r

2,1

3

˜ ˙ U M

2, 3 r

i=1

˜ L2 ∂i V3 B˙ r ∂i ∂3 U 2,1

1−r r r 2 2 (∇U 1−r L2 ΔU L2 ΔV L + ∇V L2 ΔV L2 ΔU L ) 2

˜  1−r (∇U 2 2 + ∇V 2 2 ). ≤ ε(ΔU 2L2 + ΔV 2L2 ) + CU L L ˙ M

(4.7)

2, 3 r

From integration by parts, the H¨ older inequality, (2.2), and Young inequality, we deduce that I23 = −2

3 R3

i=1

≤2

3

≤C

˜ 3U ˜ ∂i U ˜3 dx − 2 ∂i ∇V

3 i=1

˜ · ∂i U ˜3 L2 ∂i ∇V ˜ 3 L2 + 2 U

i=1 3

R3

3

˜ · ∂i ∇ ˜U ˜3 dx ∂i V3 U

˜ L2 ∂i ∇U ˜ 3 L2 ∂i V3 U

i=1

˜ ˙ U M

2,

i=1

˜ ˙ ≤ CU M

2, 3 r

˜3  ˙ r ∂i ∇V ˜ 3 L2 + C ∂i U B 3 r

2,1

3

˜ ˙ U M

2, 3 r

i=1

˜ 3 L2 ∂i V3 B˙ r ∂i ∇U 2,1

1−r r r 2 2 (∇U 1−r L2 ΔU L2 ΔV L + ∇V L2 ΔV L2 ΔU L ) 2

˜  1−r (∇U 2 2 + ∇V 2 2 ). ≤ ε(ΔU 2L2 + ΔV 2L2 ) + CU L L ˙ M 2, 3 r

(4.8)

Combining (4.5)–(4.8) yields 2

2

˜  1−r + V˜  1−r )(∇U 2 2 + ∇V 2 2 ). I2 ≤ 3εΔU 2L2 + 2εΔV 2L2 + C(U L L ˙ ˙ M M 2, 3 r

In what follows, we make estimate for I3 .

2, 3 r

(4.9)

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It follows from integration by parts and ∇ · V = 0 that I3 = −2 = −2 +2

3 3 i=1 R 3 3 i=1 R

3

i=1

R3

∂i U · ∇V · ∂i V dx ˜ · ∇V ˜ · ∂i V dx − 2 ∂i U

3 R3

i=1

∂i U3 ∂3 V˜ · ∂i V˜ dx

˜ · V˜ ∂i V3 dx ∂i U3 ∇

 I31 + I32 + I33 .

(4.10)

Similar to the estimates of I2,j (j = 1, 2, 3), respectively, we obtain 2

˜  1−r ∇V 2 2 , I31 ≤ εΔV 2L2 + CU L ˙ M 2, 3 r

2

1−r I32 ≤ ε(ΔU 2L2 + ΔV 2L2 ) + CV˜ M (∇U 2L2 + ∇V 2L2 ), ˙ 2, 3 r

and

2

1−r (∇U 2L2 + ∇V 2L2 ). I33 ≤ ε(ΔU 2L2 + ΔV 2L2 ) + CV˜ M ˙ 2, 3 r

Combining these estimates and (4.10) yields 2

2

˜  1−r + V˜  1−r )(∇U 2 2 + ∇V 2 2 ). I3 ≤ 2εΔU 2L2 + 3εΔV 2L2 + C(U L L ˙ ˙ M M 2, 3 r

Taking 0 < ε ≤

μ−|ε| 5

2, 3 r

(4.11)

and combining (4.3), (4.4), (4.9), and (4.11), we obtain

d (∇U (t)2L2 + ∇V (t)2L2 ) + (μ − |ν|)(ΔU (t)2L2 + ΔV (t)2L2 ) dt 2 2 ˜  1−r + V˜  1−r )(∇U 2 2 + ∇V 2 2 ). ≤ C(U ˙ 3 M 2, r

˙ 3 M 2,

L

L

r

Thus, by Gronwall inequality, we obtain, for any t ∈ [0, T ]

t (ΔU (τ )2L2 + ΔV (τ )2L2 )dτ ∇U (t)2L2 + ∇V (t)2L2 + (μ − |ν|) 0  t  2 2 ˜ (τ ) 1−r + V˜ (τ ) 1−r )dτ , (U ≤ C(∇U0 2 2 + ∇V0 2 2 ) exp L

L

0

˙ 3 M 2, r

˙ 3 M 2, r

which implies that (u, B) ∈ L∞ (0, T ; H 1(R3 )). Thus, according to the regularity results in [15], (u, B) is smooth on [0, T ]. We have completed the proof of Theorem 3.1.

5

Proof of Theorem 3.2 Multiplying the first equation of (1.3) by −ΔU and using integration by parts, we obtain

1 d ∇U (t)2L2 + μΔU (t)2L2 = −ν ΔV · ΔU dx + (V · ∇)U · ΔU dx. (5.1) 2 dt R3 R3

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Similarly, we obtain 1 d ∇V (t)2L2 + μΔV (t)2L2 = −ν 2 dt

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R3

ΔV · ΔU dx +

R3

(U · ∇)V · ΔV dx.

(5.2)

Summing up (5.1) and (5.2), we deduce that d (∇U (t)2L2 + ∇V (t)2L2 ) + 2μ(ΔU (t)2L2 + ΔV (t)2L2 ) dt

= −4ν ΔV · ΔU dx + 2 (V · ∇)U · ΔU dx + 2 (U · ∇)V · ΔV dx R3

R3

R3

 I1 + I2 + I3 .

(5.3)

With the help of Cauchy-Schwarz inequality, we obtain I1 ≤ 2|ν|(ΔU (t)2L2 + ΔV (t)2L2 ).

(5.4)

Using integration by parts and ∇ · U = 0, we have 3 ∂i V · ∇U · ∂i U dx I2 = −2 3 i=1 R

3

= −2

i=1

R3

i=1

R3

3

+2

˜ · ∂i U dx − 2 ∂i V˜ · ∇U

3 i=1

R3

˜ · ∂i U ˜ dx ∂i V3 ∂3 U

˜ ·U ˜ ∂i U3 dx ∂i V3 ∇

 I21 + I22 + I23 .

(5.5)

By H¨ older inequality, (2.2), and Young inequality, we obtain I21 ≤ C

3

˜ L2 ∂i U L2 ≤ C ∂i V˜ · ∇U

i=1

3

∂i V˜ M˙

2, 3 r

i=1

≤ C∇V˜ M˙

2, 3 r

˜  ˙ r ∂i U L2 ∇U B 2,1

˜ 1−r ˜ r 2 ∇U L2 ∇∇U L2 ∇U L 2

2−r ≤ εΔU 2L2 + C∇V˜ M ∇U 2L2 , ˙

(5.6)

2, 3 r

I22 ≤ C

3

˜ L2 ∂i U ˜ L2 ≤ C ∂i V3 ∂3 U

i=1

3

˜ ˙ ∂3 U M

2, 3 r

i=1

˜ ˙ ≤ C∂3 U M

2, 3 r

˜ L2 ∂i V3 B˙ r ∂i U 2,1

r 2 ∇V 1−r L2 ΔV L2 ∇U L 2

˜  2−r (∇U 2 2 + ∇V 2 2 ), ≤ εΔV 2L2 + C∇U L L ˙ M

(5.7)

2, 3 r

and I23 ≤ C

3

˜ ·U ˜ L2 ∂i U3 L2 ≤ C ∂i V3 ∇

i=1

3

˜ · U ˜ ˙ ∇ M

i=1

˜ ˙ ≤ C∇U M

2, 3 r

2, 3 r

∂i V3 B˙ r ∂i U3 L2 2,1

r 2 ∇V 1−r L2 ΔV L2 ∇U L 2

˜  2−r (∇U 2 2 + ∇V 2 2 ). ≤ εΔV 2L2 + C∇U L L ˙ M 2, 3 r

(5.8)

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Combining (5.5)–(5.8) yields 2

2

˜  2−r + ∇V˜  2−r )(∇U 2 2 + ∇V 2 2 ). I2 ≤ εΔU 2L2 + 2εΔV 2L2 + C(∇U L L ˙ ˙ M M 2, 3 r

2, 3 r

(5.9)

It follows from integration by parts and ∇ · V = 0 that I3 = −2 = −2 +2

3 3 i=1 R

3 3 i=1 R 3

i=1

R3

∂i U · ∇V · ∂i V dx ˜ · ∇V ˜ · ∂i V dx − 2 ∂i U

3

∂i U3 ∂3 V˜ · ∂i V˜ dx

R3

i=1

˜ · V˜ ∂i V3 dx ∂i U3 ∇

 I31 + I32 + I33 .

(5.10)

Similar to the estimates of I2,j (j = 1, 2, 3), we obtain 2

˜  1−r ∇V 2 2 , I31 ≤ εΔV 2L2 + C∇U L ˙ M 2, 3 r

2

1−r (∇U 2L2 + ∇V 2L2 ), I32 ≤ εΔU 2L2 + C∇V˜ M ˙ 2, 3 r

and

2

1−r I33 ≤ εΔU 2L2 + C∇V˜ M (∇U 2L2 + ∇V 2L2 ). ˙ 2, 3 r

Combining these estimates and (5.10) yields 2

2

˜  1−r + ∇V˜  1−r )(∇U 2 2 + ∇V 2 2 ). I3 ≤ 2εΔU 2L2 + εΔV 2L2 + C(∇U L L ˙ ˙ M M 2, 3 r

Taking 0 < ε ≤

μ−|ν| 3

2, 3 r

(5.11)

and combining (5.3), (5.4), (5.9), and (5.11), we obtain

d (∇U (t)2L2 + ∇V (t)2L2 ) + (μ − |ν|)(ΔU (t)2L2 + ΔV (t)2L2 ) dt 2 2 ˜  1−r + ∇V˜  1−r )(∇U 2 2 + ∇V 2 2 ). ≤ C(∇U ˙ 3 M 2, r

L

˙ 3 M 2,

L

r

Thus, by Gronwall inequality, we obtain, for every t ∈ [0, T ], ∇U (t)2L2

t

+ (μ − |ν|) (ΔU (τ )2L2 + ΔV (τ )2L2 )dτ 0  t  2 2 2−r 2−r 2 ˜ ˜ + ∇V0 L2 ) exp (∇U (τ )M˙ + ∇V (τ )M˙ )dτ ,

+ ∇V

≤ C(∇U0 2L2

(t)2L2

0

2, 3 r

2, 3 r

which implies that (u, B) ∈ L∞ (0, T ; H 1(R3 )). Thus, according to the regularity results in [15], (u, B) is smooth on [0, T ]. Thus, Theorem 3.2 is proved.

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ACTA MATHEMATICA SCIENTIA

Vol.32 Ser.B

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