Open Access

On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative

Boundary Value Problems20122012:139

DOI: 10.1186/1687-2770-2012-139

Received: 1 August 2012

Accepted: 12 November 2012

Published: 27 November 2012

Abstract

In this paper, we establish two new regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative of the velocity or of the pressure and the magnetic field.

MSC:35Q35, 76W05, 35B65.

Keywords

magneto-micropolar fluid equations regularity criteria

1 Introduction

In this paper, we consider the Cauchy problem of the 3D incompressible magneto-micropolar fluid equations
{ t u + u u b b + ( π + | b | 2 ) χ × w = ( μ + χ ) Δ u , t 0 , x R 3 , t w + u w κ div w + 2 χ w χ × u = γ Δ w , t 0 , x R 3 , t b + u b b u = ν Δ b , t 0 , x R 3 , div u = div b = 0 , t 0 , x R 3 , u ( x , 0 ) = u 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , b ( x , 0 ) = b 0 ( x ) , x R 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ1_HTML.gif
(1.1)

where u is the fluid velocity, w is the micro-rotational velocity, b is the magnetic field and π is the pressure. Equations (1.1) describe the motion of a micropolar fluid which is moving in the presence of a magnetic field (see [1]). The positive parameters μ, χ, γ, κ and ν in (1.1) are associated with the properties of the materials: μ is the kinematic viscosity, χ is the vortex viscosity, ν and κ are the spin viscosities and 1 ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq1_HTML.gif is the magnetic Reynolds number.

Recently, Yuan [2] investigated the local existence and uniqueness of the strong solutions to the magneto-micropolar fluid equations (1.1) (see also [36] for the bounded domain cases). Thus, the further problem at the center of the mathematical theory concerning equations (1.1) is whether or not it has a global in time smooth solution for any prescribed smooth initial data, which is still a challenging open problem. In the absence of a global well-posedness theory, the development of regularity criteria is of major importance for both theoretical and practical purposes. We would like to recall some related results in this direction.

Note that if the micro-rotation effects and the magnetic filed are not taken into account, i.e., w = b = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq2_HTML.gif, equations (1.1) reduce to the classical Navier-Stokes equations. The global regularity issue has been thoroughly investigated for the 3D Navier-Stokes equations and many important regularity criteria have been established (see [716] and the references therein). In particular, the first well-known regularity criterion is due to Serrin [14]: if the Leray-Hopf weak solution u of the 3D Navier-Stokes equations satisfies
0 T u ( , t ) L p q d t < with  2 q + 3 p = 1  and  3 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equa_HTML.gif
then u is regular on ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq3_HTML.gif. Beirao da Veiga [8] and Penel and Pokorny [13] established another regularity criteria by replacing the above conditions with the following ones:
0 T u ( , t ) L p q d t < with  2 q + 3 p = 2  and  3 2 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equb_HTML.gif
or
0 T 3 u ( , t ) L p q d t < with  2 q + 3 p 3 2  and  2 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equc_HTML.gif
More recently, Cao and Titi [17] established a regularity criterion in terms of only one of the nine components of the gradient of a velocity field, that is, the solution u is regular on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq4_HTML.gif if
0 T k u j ( , t ) L p q d t < with  2 q + 3 p p + 3 2 p  and  3 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equd_HTML.gif
where k , j = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq5_HTML.gif and k j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq6_HTML.gif, or
0 T j u j ( , t ) L p q d t < with  2 q + 3 p 3 ( p + 2 ) 4 p  and  2 < p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Eque_HTML.gif
This result on j u j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq7_HTML.gif is stronger than a similar result of Zhou and Pokorny [18] in the sense of allowing for much smaller values of p. These regularity criteria are of physical relevance since experimental measurements are usually obtained for quantities of the form k u j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq8_HTML.gif. The regularity criterion by imposing the growth conditions on the pressure field are also examined by, for example, Berselli and Galdi [9], Chae and Lee [10] and Zhou [15, 16], i.e., if
0 T π ( , t ) L p q d t < with  2 q + 3 p = 2  and  3 2 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equf_HTML.gif
or
0 T π ( , t ) L p q d t < with  2 q + 3 p = 3  and  1 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equg_HTML.gif
then the solution u is regular on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq4_HTML.gif (see also [14, 17] for the Besov spaces cases). For the 3D Navier-Stokes equations with boundary conditions, Cao and Titi first introduced a regularity criterion in terms of only one component of the pressure gradient based on the breakthrough of the global regularity of the 3D primitive equations [19]. Recently, Cao and Titi [20] established a similar regularity criterion for the Cauchy problem of the 3D Navier-Stokes equations, that is, the solution u is regular on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq4_HTML.gif if
0 T 3 π ( , t ) L p q d t < with  2 q + 3 p < 20 7 , p > 20 16  and  q 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equh_HTML.gif
When the micro-rotation effects are neglected, i.e., w = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq9_HTML.gif, equations (1.1) become the usual magnetohydrodynamic (MHD) equations. Some of the regularity criteria established for the Navier-Stokes equations can be extended to the 3D MHD equations by making assumptions on both u and b (see [21, 22]). Moreover, He and Xin [23, 24] showed that the velocity field u plays a dominant role in the regularity issue and derived a criterion in terms of the velocity field u alone (see also [25, 26] for the Besov spaces cases). Recently, Cao and Wu [27] further proved that if
0 T 3 u ( , t ) L p q d t < with  2 q + 3 p 1  and  p 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equi_HTML.gif
or
0 T 3 π ( , t ) L p q d t < with  2 q + 3 p 7 4  and  p 12 7 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equj_HTML.gif

then ( u , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq10_HTML.gif is regular on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq4_HTML.gif. More recently, Liu, Zhao and Cui [28] have adapted the method of [27] to establish a similar regularity criterion for the 3D nematic liquid crystal flow.

If we ignore the magnetic filed, i.e., b = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq11_HTML.gif, equations (1.1) reduce to the micropolar fluid equations. The theory of micropolar fluid has attracted more and more scholars’ attention in recent years. In particular, Dong, Jia and Chen [29] recently established a regularity criterion via the pressure field, which says that if
0 T π ( , t ) L p q d t < with  2 q + 3 p = 3  and  1 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equk_HTML.gif

then ( u , w ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq12_HTML.gif is regular on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq4_HTML.gif (see also [30, 31] for the Lorentz spaces cases).

For the full magneto-micropolar fluid equations (1.1), Yuan [32] recently showed that the solution ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif is regular on ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq3_HTML.gif if
0 T u ( , t ) L p q d t < with  2 q + 3 p 1  and  3 < p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ2_HTML.gif
(1.2)
or
0 T u ( , t ) L p q d t < with  2 q + 3 p 2  and  3 2 < p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ3_HTML.gif
(1.3)

For other regularity criteria of equations (1.1), we refer to Gala [33], Geng, Chen and Gala [34], Wang, Hu and Wang [35], Yuan [2] and Zhang, Yao and Wang [36].

In this paper, we establish two new regularity criteria for the 3D magneto-micropolar fluid equations (1.1) in terms of one directional derivative of the velocity u or of the pressure π and the magnetic field b by adapting the method of [27]. Without loss of generality, we set the viscous coefficients μ + χ = γ = ν = κ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq14_HTML.gif.

We now state our main results as follows.

Theorem 1.1 Assume that ( u 0 , w 0 , b 0 ) H 3 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq15_HTML.gif with div u 0 = div b 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq16_HTML.gif. Let ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif be the corresponding local smooth solution to the magneto-micropolar fluid equations (1.1) on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq17_HTML.gif for some T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq18_HTML.gif. If the velocity u satisfies
0 T 3 u ( , t ) L p q d t : = M ( T ) < with  2 q + 3 p 1  and  p 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ4_HTML.gif
(1.4)

then ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif can be extended beyond T.

Note that when p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq19_HTML.gif, q = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq20_HTML.gif and thus the corresponding assumption in (1.4) should be understood as esssup 0 t T 3 u ( , t ) L 3 : = M ( T ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq21_HTML.gif.

Remark 1.1 Theorem 1.1 improves the regularity criterion in [32] (see (1.3)) in the sense that it depends only on one directional derivative of the velocity u.

Theorem 1.2 Assume that ( u 0 , w 0 , b 0 ) H 1 ( R 3 ) L 4 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq22_HTML.gif with div u 0 = div b 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq23_HTML.gif. Let ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif be the corresponding local smooth solution to the magneto-micropolar fluid equations (1.1) on [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq17_HTML.gif for some T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq18_HTML.gif. If the pressure π and the magnetic field b satisfy
0 T 3 ( π ( , t ) + | b | 2 ( , t ) ) L p q d t : = M ( T ) < with  2 q + 3 p 7 4  and  12 7 p 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ5_HTML.gif
(1.5)

then ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif can be extended beyond T.

Remark 1.2 When b = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq24_HTML.gif, we also obtain a new regularity criterion for the micropolar equations determined by one direction derivative of the pressure π alone.

We shall prove our results in the next section. For simplicity, we denote by p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq25_HTML.gif the L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq26_HTML.gif norm and by ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq27_HTML.gif the L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq28_HTML.gif inner product throughout the paper. The letter C denotes an inessential constant which might vary from line to line, but does not depend on particular solutions or functions.

2 Proof of the main results

In this section, we give the proof of Theorem 1.1 and Theorem 1.2. The following lemma plays an important role in our arguments. Its proof can be found in [37] or [27].

Lemma 2.1 Let the parameters r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq29_HTML.gif, r 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq30_HTML.gif, r 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq31_HTML.gif and r satisfy
1 r 1 , r 2 , r 3 , r < and 1 + 3 r = 1 r 1 + 1 r 2 + 1 r 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equl_HTML.gif
and suppose that i φ L r i ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq32_HTML.gif ( i = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq33_HTML.gif). Then there exists a constant C = C ( r 1 , r 2 , r 3 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq34_HTML.gif such that
φ r C 1 φ r 1 1 3 2 φ r 2 1 3 3 φ r 3 1 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equm_HTML.gif
In particular, when r 1 = r 2 = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq35_HTML.gif and r 3 = p [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq36_HTML.gif, there exists a constant C = C ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq37_HTML.gif such that
φ 3 p C 1 φ 2 1 3 2 φ 2 1 3 3 φ p 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equn_HTML.gif

for any φ satisfying 1 φ , 2 φ L 2 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq38_HTML.gif and 3 φ L p ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq39_HTML.gif.

Proof of Theorem 1.1 Observe that for any ( u 0 , w 0 , b 0 ) H 3 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq40_HTML.gif with div u 0 = div b 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq41_HTML.gif, there exists a unique local smooth solution to equations (1.1) (see [2]). Let T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq42_HTML.gif be the maximum existence time. To prove Theorem 1.1, it is sufficient to show that the assumption (1.4) implies T < T 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq43_HTML.gif. Indeed, we shall prove that under the condition (1.4), there exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq44_HTML.gif such that
lim sup t T ( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ6_HTML.gif
(2.1)

which implies that T is not the maximum existence time and thus the solution ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif can be extended beyond T by the standard arguments of continuation of local solutions.

Firstly, we derive the energy inequality. For this purpose, we take the L 2 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq45_HTML.gif inner product of u, w and b with equations (1.1), respectively, sum the resulting equations and then integrate by parts to obtain
1 2 d d t ( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) + ( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) + div w ( t ) 2 2 + 2 χ w ( t ) 2 2 = χ ( u , × w ) + χ ( w , × u ) 1 2 ( u ( t ) 2 2 + w ( t ) 2 2 ) + C ( u ( t ) 2 2 + w ( t ) 2 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equo_HTML.gif
where we used div u = div b = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq46_HTML.gif in the first equality and Hölder’s inequality in the last inequality. Thus,
d d t ( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) + ( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) + div w ( t ) 2 2 + χ w ( t ) 2 2 C ( u ( t ) 2 2 + w ( t ) 2 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equp_HTML.gif
It follows from Gronwall’s inequality that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ7_HTML.gif
(2.2)

Now we split the proof of the estimates (2.1) into two steps.

Step 1: Estimates for 0 T ( 3 u ( , t ) 2 2 + 3 w ( , t ) 2 2 + 3 b ( , t ) 2 2 ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq47_HTML.gif.

To this end, differentiating the first three equations in (1.1) with respect to x 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq48_HTML.gif, taking the L 2 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq45_HTML.gif inner product of 3 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq49_HTML.gif, 3 w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq50_HTML.gif and 3 b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq51_HTML.gif with the resulting equations, respectively, and then performing a space integration by parts, we get
1 2 d d t 3 u ( t ) 2 2 + 3 u 2 2 = ( 3 u , 3 u u ) + ( 3 u , 3 b b ) + ( 3 u , b 3 b ) + χ ( 3 u , × 3 w ) , 1 2 d d t 3 w ( t ) 2 2 + 3 w 2 2 + div 3 w 2 2 + 2 χ 3 w 2 2 = ( 3 w , 3 u w ) + χ ( 3 w , × 3 u ) , 1 2 d d t 3 b ( t ) 2 2 + 3 b 2 2 = ( 3 b , 3 u b ) + ( 3 b , 3 b u ) + ( 3 b , b 3 u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equq_HTML.gif
where we used the facts
( 3 u , u 3 u ) = ( 3 u , 3 ( π + | b | 2 ) ) = ( 3 w , u 3 w ) = ( 3 b , u 3 b ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equr_HTML.gif
by div u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq52_HTML.gif. Noticing that
( 3 u , × 3 w ) = ( 3 w , × 3 u ) and ( 3 u , b 3 b ) + ( 3 b , b 3 u ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equs_HTML.gif
by div b = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq53_HTML.gif, we can sum the above equations to obtain
1 2 d d t ( 3 u ( t ) 2 2 + 3 w ( t ) 2 2 + 3 b ( t ) 2 2 ) + ( 3 u 2 2 + 3 w 2 2 + 3 b 2 2 ) + div 3 w 2 2 + 2 χ 3 w 2 2 = ( 3 u , 3 u u ) + ( 3 u , 3 b b ) ( 3 w , 3 u w ) + 2 χ ( 3 w , × 3 u ) ( 3 b , 3 u b ) + ( 3 b , 3 b u ) : = I 1 + I 2 + I 3 + I 4 + I 5 + I 6 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equt_HTML.gif
We now estimate the above terms one by one. To bound I 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq54_HTML.gif, we first integrate by parts and then apply Hölder’s inequality to obtain
| I 1 | = | ( u , 3 u 3 u ) | 3 u 2 3 u 6 p 3 p 2 u 3 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ8_HTML.gif
(2.3)
It follows from the Gagliardo-Nirenberg inequality that
3 u 6 p 3 p 2 C 3 u 2 1 p 3 u 2 1 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equu_HTML.gif
and from Lemma 2.1 that
u 3 p C 1 u 2 1 3 2 u 2 1 3 3 u p 1 3 C u 2 2 3 3 u p 1 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equv_HTML.gif
Substituting these two estimates into (2.3) and then using Young’s inequality, we see that for p > 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq55_HTML.gif
| I 1 | C 3 u 2 1 + 1 p 3 u 2 1 1 p u 2 2 3 3 u p 1 3 1 4 3 u 2 2 + C 3 u 2 2 u 2 4 p 3 ( p 1 ) 3 u p 2 p 3 ( p 1 ) 1 4 3 u 2 2 + C 3 u 2 2 ( u 2 2 + 3 u p 2 p p 3 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ9_HTML.gif
(2.4)
and that for p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq56_HTML.gif
| I 1 | C 3 u 2 4 3 3 u 2 2 3 u 2 2 3 3 u 3 1 3 1 4 3 u 2 2 + C 3 u 2 2 ( u 2 2 3 u 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ10_HTML.gif
(2.5)
For I 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq57_HTML.gif, by Hölder’s inequality, the Gagliardo-Nirenberg inequality and Young’s inequality, we have for p > 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq55_HTML.gif
| I 2 | b 2 3 u p 3 b 2 p p 2 C b 2 3 u p 3 b 2 1 3 p 3 b 2 3 p 1 8 3 b 2 2 + C b 2 2 p 2 p 3 3 u p 2 p 2 p 3 3 b 2 2 p 6 2 p 3 1 8 3 b 2 2 + C ( b 2 2 + 3 u p 2 p p 3 ) 3 b 2 2 p 6 2 p 3 1 8 3 b 2 2 + C ( b 2 2 + 3 u p 2 p p 3 ) ( 1 + 3 b 2 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ11_HTML.gif
(2.6)
and for p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq56_HTML.gif
| I 2 | b 2 3 u 3 3 b 6 C b 2 3 u 3 3 b 2 1 8 3 b 2 2 + C b 2 2 3 u 3 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ12_HTML.gif
(2.7)
Applying similar procedure to I 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq58_HTML.gif and I 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq59_HTML.gif, we have for p < 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq60_HTML.gif
| I 3 | 1 2 3 w 2 2 + C ( w 2 2 + 3 u p 2 p p 3 ) ( 1 + 3 w 2 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ13_HTML.gif
(2.8)
and
| I 5 | 1 8 3 b 2 2 + C ( b 2 2 + 3 u p 2 p p 3 ) ( 1 + 3 b 2 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ14_HTML.gif
(2.9)
and for p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq19_HTML.gif
| I 3 | 1 8 3 w 2 2 + C w 2 2 3 u 3 2 , | I 5 | 1 8 3 b 2 2 + C b 2 2 3 u 3 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ15_HTML.gif
(2.10)
For the term I 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq61_HTML.gif, by using Hölder’s inequality and Young’s inequality, it can be bounded as follows:
| I 4 | = 2 χ ( 3 w , × 3 u ) 1 4 3 u 2 2 + C 3 w 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ16_HTML.gif
(2.11)
Finally, we can follow the steps as in the bound of I 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq54_HTML.gif to estimate I 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq62_HTML.gif. Precisely, by integrations by parts and Hölder’s inequality, we have
| I 6 | = | ( 3 b , 3 b u ) | = | ( u , 3 b 3 b ) | 3 b 2 3 b 6 p 3 p 2 u 3 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equw_HTML.gif
Then the Gagliardo-Nirenberg inequality, Lemma 2.1 and Young’s inequality yield that for p < 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq60_HTML.gif
| I 6 | C 3 b 2 1 + 1 p 3 b 2 1 1 p u 2 2 3 3 u p 1 3 1 4 3 b 2 2 + C 3 b 2 2 u 2 4 p 3 ( p 1 ) 3 u p 2 p 3 ( p 1 ) 1 4 3 b 2 2 + C 3 b 2 2 ( u 2 2 + 3 u p 2 p p 3 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ17_HTML.gif
(2.12)
and for p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq19_HTML.gif
| I 6 | C 3 b 2 4 3 3 b 2 2 3 u 2 2 3 3 u 3 1 3 1 4 3 b 2 2 + C 3 b 2 2 ( u 2 2 3 u 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ18_HTML.gif
(2.13)
Combining the estimates (2.4)-(2.12), we see that for p > 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq63_HTML.gif
d d t ( 3 u ( t ) 2 2 + 3 w ( t ) 2 2 + 3 b ( t ) 2 2 ) + ( 3 u 2 2 + 3 w 2 2 + 3 b 2 2 ) + div 3 w 2 2 + χ 3 w 2 2 C ( 1 + 3 u 2 2 + 3 w 2 2 + 3 b 2 2 ) ( 1 + u 2 2 + w 2 2 + b 2 2 + 3 u p 2 p p 3 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equx_HTML.gif
and that for p = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq19_HTML.gif
d d t ( 3 u ( t ) 2 2 + 3 w ( t ) 2 2 + 3 b ( t ) 2 2 ) + ( 3 u 2 2 + 3 w 2 2 + 3 b 2 2 ) + div 3 w 2 2 + χ 3 w 2 2 C ( 1 + 3 u 2 2 + 3 w 2 2 + 3 b 2 2 ) ( 1 + u 2 2 + w 2 2 + b 2 2 ) ( 1 + 3 u 3 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equy_HTML.gif
Thus, Gronwall’s inequality together with the energy inequality (2.2) and the assumption (1.4) implies that for p > 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq63_HTML.gif
( 3 u ( t ) 2 2 + 3 w ( t ) 2 2 + 3 b ( t ) 2 2 ) ( 1 + 3 u 0 2 2 + 3 w 0 2 2 + 3 b 0 2 2 ) e C 0 t ( 1 + u ( τ ) 2 2 + w ( τ ) 2 2 + b ( τ ) 2 2 + 3 u ( τ ) p 2 p p 3 ) d τ ( 1 + 3 u 0 2 2 + 3 w 0 2 2 + 3 b 0 2 2 ) e C e C t ( 1 + u 0 2 2 + w 0 2 2 + b 0 2 2 ) + C M ( t ) q / q t 1 q / q : = G p ( M ( t ) ) < ( t T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equz_HTML.gif
with q = 2 p / ( p 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq64_HTML.gif, and
( 3 u ( t ) 2 2 + 3 w ( t ) 2 2 + 3 b ( t ) 2 2 ) ( 1 + 3 u 0 2 2 + 3 w 0 2 2 + 3 b 0 2 2 ) e C 0 t ( 1 + u ( τ ) 2 2 + w ( τ ) 2 2 + b ( τ ) 2 2 ) ( 1 + 3 u ( τ ) 3 2 ) d τ ( 1 + 3 u 0 2 2 + 3 w 0 2 2 + 3 b 0 2 2 ) e C e C t ( 1 + u 0 2 2 + w 0 2 2 + b 0 2 2 ) ( t + M ( t ) 2 ) : = G 3 ( M ( t ) ) < ( t T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equaa_HTML.gif
Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ19_HTML.gif
(2.14)

which is the desired estimates.

Step 2: Estimates for ( u ( t ) 2 + w ( t ) 2 + b ( t ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq65_HTML.gif.

For this purpose, taking the L 2 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq45_HTML.gif inner product of Δu, Δw and Δb with the first three equations in (1.1), respectively, and then performing a space integration by parts, we have
1 2 d d t u ( t ) 2 2 + Δ u 2 2 = ( Δ u , u u ) ( Δ u , b b ) χ ( Δ u , × w ) , 1 2 d d t w ( t ) 2 2 + Δ w 2 2 + div w 2 2 + 2 χ w 2 2 = ( Δ w , u w ) χ ( Δ w , × u ) , 1 2 d d t b ( t ) 2 2 + Δ b 2 2 = ( Δ b , u b ) ( Δ b , b u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equab_HTML.gif
Noticing ( Δ u , × w ) = ( Δ w , × u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq66_HTML.gif, we sum the above equations and integrate by parts to obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ20_HTML.gif
(2.15)
By using the interpolation inequality and taking p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq67_HTML.gif in Lemma 2.1, we have
u 3 3 C ( u 2 1 2 u 6 1 2 ) 3 C ( u 2 1 2 h u 2 1 3 3 u 2 1 6 ) 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equac_HTML.gif
where h = ( 1 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq68_HTML.gif. Then Young’s inequality yields
u 3 3 1 4 h u 2 2 + C u 2 3 3 u 2 1 4 h u 2 2 + C ( u 2 2 + 3 u 2 2 ) u 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equad_HTML.gif
Similarly,
3 u 3 b 3 2 + u 3 w 3 2 4 u 3 3 + 3 b 3 3 + w 3 3 1 4 ( h u 2 2 + h b 2 2 + h w 2 2 ) + C ( u 2 2 + 3 u 2 2 ) u 2 2 + C ( b 2 2 + 3 b 2 2 ) b 2 2 + C ( w 2 2 + 3 w 2 2 ) w 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equae_HTML.gif
Substituting the above two estimates into (2.15), we have
d d t ( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) + ( Δ u 2 2 + Δ w 2 2 + Δ b 2 2 ) + div w 2 2 + χ w 2 2 C ( 1 + u 2 2 + 3 u 2 2 + w 2 2 + 3 w 2 2 + b 2 2 + 3 b 2 2 ) × ( u 2 2 + w 2 2 + b 2 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equaf_HTML.gif
By using Gronwall’s inequality, the energy inequality (2.2) and the estimate (2.14), we conclude that
( u ( t ) 2 2 + w ( t ) 2 2 + b ( t ) 2 2 ) + 0 t ( Δ u ( τ ) 2 2 + Δ w ( τ ) 2 2 + Δ b ( τ ) 2 2 + div w ( τ ) 2 2 + χ w ( τ ) 2 2 ) d τ ( u 0 2 2 + w 0 2 2 + b 0 2 2 ) × e C 0 t ( 1 + u ( τ ) 2 2 + 3 u ( τ ) 2 2 + w ( τ ) 2 2 + 3 w ( τ ) 2 2 + b ( τ ) 2 2 + 3 b ( τ ) 2 2 ) d τ C G ˜ ( M ( t ) ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equag_HTML.gif

for any t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq69_HTML.gif, which implies that the desired estimates (2.1) hold and thus the solution ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq70_HTML.gif can be extended beyond T. □

Now we turn our attention to proving Theorem 1.2. We will first transform equations (1.1) into a symmetric form.

Proof of Theorem 1.2 Following from Serrin type criteria (1.2) with p = 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq71_HTML.gif and q = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq72_HTML.gif on the 3D magneto-micropolar fluid equations (1.1), it is sufficient to prove that
lim t T ( u ( t ) 4 + w ( t ) 4 + b ( t ) 4 ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ21_HTML.gif
(2.16)
To do this, we set
v + = u + b , v = u b , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equah_HTML.gif
and then equations (1.1) are converted to the following symmetric form:
{ t v + + v v + + ( π + | b | 2 ) χ × w = Δ v + , t 0 , x R 3 , t w + 1 2 ( v + + v ) w div w + 2 χ w χ 2 × ( v + + v ) = Δ w , t 0 , x R 3 , t v + v + v + ( π + | b | 2 ) χ × w = Δ v , t 0 , x R 3 , div v + = div v = 0 , t 0 , x R 3 , v + ( x , 0 ) = u 0 ( x ) + b 0 ( x ) , w ( x , 0 ) = w 0 ( x ) , v ( x , 0 ) = u 0 ( x ) b 0 ( x ) , x R 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equ22_HTML.gif
(2.17)

Firstly, taking the L 2 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq45_HTML.gif inner product of v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq73_HTML.gif, w and v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq74_HTML.gif with the above equations, respectively, and integrating by parts, we can obtain the energy estimates similar to (2.2).

Next we take the L 2 ( R 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq45_HTML.gif inner product of | v + | 2 v + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq75_HTML.gif, | w | 2 w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq76_HTML.gif and | v | 2 v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq77_HTML.gif with the first three equations in (2.17), respectively, and then integrate by parts to obtain
1 4 d d t ( v + ( t ) 4 4 + w ( t ) 4 4 + v ( t ) 4 4 ) + 1 2 ( | v + | 2 2 2 + | w | 2 2 2 + | v | 2 2 2 ) + ( | v + | v + 2 2 + | w | w 2 2 + | v | v 2 2 ) + 2 χ w 4 4 + | w | div w 2 2 = R 3 ( π + | b | 2 ) ( v + | v + | 2 + v | v | 2 ) d x R 3 ( div w ) ( w | w | 2 ) d x + χ R 3 | v + | 2 v + ( × w ) d x + χ 2 R 3 | w | 2 w ( × ( v + + v ) ) d x + χ R 3 | v | 2 v ( × w ) d x : = I I 1 + I I 2 + I I 3 + I I 4 + I I 5 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equai_HTML.gif
We now bound the above terms one by one. For I I 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq78_HTML.gif, we have
| I I 2 | | w | w 2 | w | 2 2 1 2 | w | 2 2 2 + 1 2 | w | w 2 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equaj_HTML.gif
It follows from the integration by parts, we see
| I I 3 | = χ | R 3 w ( × | v + | 2 v + ) d x | C w 4 | v + | v + 2 v + 4 1 2 | v + | v + 2 2 + C w 4 4 + C v + 4 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equak_HTML.gif
Similarly, we have
| I I 4 | 1 2 | w | w 2 2 + C w 4 4 + C v + 4 4 + C v 4 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equal_HTML.gif
and
| I I 5 | 1 2 | v | v 2 2 + C w 4 4 + C v 4 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equam_HTML.gif
The process for estimating I I 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq79_HTML.gif is more subtle. It follows from Hölder’s inequality and Lemma 2.1 that
| I I 1 | ( π + | b | 2 ) 4 ( v + 4 | v + | 2 2 + v 4 | v | 2 2 ) 3 ( π + | b | 2 ) p 1 3 ( π + | b | 2 ) 8 p 7 p 4 2 3 ( v + 4 | v + | 2 2 + v 4 | v | 2 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equan_HTML.gif
To estimate the term involving ( π + | b | 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq80_HTML.gif, we take the divergence of the first equation of (2.17) and find
π + | b | 2 = ( Δ ) 1 ( v v + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equao_HTML.gif
by div v + = div ( × w ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq81_HTML.gif. Then the Calderón-Zygmund inequality, Hölder’s inequality and the interpolation inequality imply that
( π + | b | 2 ) 8 p 7 p 4 = ( Δ ) 1 ( v v + ) 8 p 7 p 4 v v + 8 p 7 p 4 C v + 2 v 8 p 3 p 4 = C v + 2 | v | 2 4 p 3 p 4 1 2 C v + 2 | v | 2 2 7 p 12 8 p | v | 2 2 12 3 p 8 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equap_HTML.gif
Similarly, we have
( π + | b | 2 ) 8 p 7 p 4 C v 2 | v + | 2 2 7 p 12 8 p | v + | 2 2 12 3 p 8 p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equaq_HTML.gif
If p > 12 7 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq82_HTML.gif, combining the above two estimates, we see
| I I 1 | C 3 ( π + | b | 2 ) p 1 3 v + 2 2 3 v 4 7 p 12 6 p | v | 2 2 4 p 4 p v + 4 | v + | 2 2 + C 3 ( π + | b | 2 ) p 1 3 v 2 2 3 v + 4 7 p 12 6 p | v + | 2 2 4 p 4 p v 4 | v | 2 2 1 4 | v + | 2 2 2 + 1 4 | v + | 2 2 2 + C 3 ( π + | b | 2 ) p 8 p 3 ( 5 p 4 ) v + 2 16 p 3 ( 5 p 4 ) v 4 4 ( 7 p 12 ) 3 ( 5 p 4 ) v + 4 8 p 5 p 4 + C 3 ( π + | b | 2 ) p 8 p 3 ( 5 p 4 ) v 2 16 p 3 ( 5 p 4 ) v + 4 4 ( 7 p 12 ) 3 ( 5 p 4 ) v 4 8 p 5 p 4 1 4 | v + | 2 2 2 + 1 4 | v + | 2 2 2 + C ( 3 ( π + | b | 2 ) p 8 p 7 p 12 + v + 2 2 ) ( v 4 4 ( 7 p 12 ) 3 ( 3 p 4 ) + v + 4 4 ) + C ( 3 ( π + | b | 2 ) p 8 p 7 p 12 + v 2 2 ) ( v + 4 4 ( 7 p 12 ) 3 ( 3 p 4 ) + v 4 4 ) 1 4 | v + | 2 2 2 + 1 4 | v + | 2 2 2 + C ( 3 ( π + | b | 2 ) p 8 p 7 p 12 + v + 2 2 + v 2 2 ) ( 1 + v 4 4 + v + 4 4 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equar_HTML.gif

The case p = 12 7 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq83_HTML.gif can be similarly dealt with.

Summarily, we conclude that
1 4 d d t ( v + ( t ) 4 4 + w ( t ) 4 4 + v ( t ) 4 4 ) C ( 1 + 3 ( π + | b | 2 ) p 8 p 7 p 12 + v + 2 2 + v 2 2 ) ( 1 + v 4 4 + w 4 4 + v + 4 4 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_Equas_HTML.gif

Thus, Gronwall’s inequality together with the assumption (1.5) and the energy estimates gives the desired L 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq84_HTML.gif estimates (2.12) and thus the solution ( u , w , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-139/MediaObjects/13661_2012_Article_253_IEq13_HTML.gif can be extended beyond T. □

Declarations

Acknowledgements

The authors would like to thank the referees for their valuable comments and remarks. This work was partially supported by the NNSF of China (No. 11101068), the Sichuan Youth Science & Technology Foundation (No. 2011JQ0003), the SRF for ROCS, SEM, and the Fundamental Research Funds for the Central Universities (ZYGX2009X019).

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Electronic Science and Technology of China

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© Xiang and Yang; licensee Springer. 2012

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