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

Zhaoyin Xiang* and Huizhi Yang

Author Affiliations

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, P.R. China

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Boundary Value Problems 2012, 2012:139  doi:10.1186/1687-2770-2012-139

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/139


Received:1 August 2012
Accepted:12 November 2012
Published:27 November 2012

© 2012 Xiang and Yang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M1">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M2">View MathML</a> 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 [3-6] 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., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M3">View MathML</a>, 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 [7-16] 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M4">View MathML</a>

then u is regular on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M5">View MathML</a>. Beirao da Veiga [8] and Penel and Pokorny [13] established another regularity criteria by replacing the above conditions with the following ones:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M6">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M7">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M11">View MathML</a>, or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M12">View MathML</a>

This result on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M13">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M14">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M15">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M16">View MathML</a>

then the solution u is regular on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8">View MathML</a> (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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M19">View MathML</a>

When the micro-rotation effects are neglected, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M20">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M21">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M22">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M23">View MathML</a> is regular on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8">View MathML</a>. 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., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M25">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M26">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M27">View MathML</a> is regular on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M8">View MathML</a> (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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a> is regular on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M5">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M31">View MathML</a>

(1.2)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M32">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M33">View MathML</a>.

We now state our main results as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M34">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M35">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a>be the corresponding local smooth solution to the magneto-micropolar fluid equations (1.1) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M37">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M38">View MathML</a>. If the velocityusatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M39">View MathML</a>

(1.4)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a>can be extended beyondT.

Note that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M42">View MathML</a> and thus the corresponding assumption in (1.4) should be understood as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M43">View MathML</a>.

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 1.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M44">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M45">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a>be the corresponding local smooth solution to the magneto-micropolar fluid equations (1.1) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M37">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M38">View MathML</a>. If the pressureπand the magnetic fieldbsatisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M49">View MathML</a>

(1.5)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a>can be extended beyondT.

Remark 1.2 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M51">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M52">View MathML</a> the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M53">View MathML</a> norm and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M54">View MathML</a> the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M55">View MathML</a> 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 2.1Let the parameters<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M58">View MathML</a>andrsatisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M59">View MathML</a>

and suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M60">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M61">View MathML</a>). Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M62">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M63">View MathML</a>

In particular, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M64">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M65">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M66">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M67">View MathML</a>

for anyφsatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M69">View MathML</a>.

Proof of Theorem 1.1 Observe that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M70">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M71">View MathML</a>, there exists a unique local smooth solution to equations (1.1) (see [2]). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M72">View MathML</a> be the maximum existence time. To prove Theorem 1.1, it is sufficient to show that the assumption (1.4) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M73">View MathML</a>. Indeed, we shall prove that under the condition (1.4), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M74">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M75">View MathML</a>

(2.1)

which implies that T is not the maximum existence time and thus the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77">View MathML</a> inner product of u, w and b with equations (1.1), respectively, sum the resulting equations and then integrate by parts to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M78">View MathML</a>

where we used <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M79">View MathML</a> in the first equality and Hölder’s inequality in the last inequality. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M80">View MathML</a>

It follows from Gronwall’s inequality that

(2.2)

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

Step 1: Estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M82">View MathML</a>.

To this end, differentiating the first three equations in (1.1) with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M83">View MathML</a>, taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77">View MathML</a> inner product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M87">View MathML</a> with the resulting equations, respectively, and then performing a space integration by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M88">View MathML</a>

where we used the facts

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M89">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M90">View MathML</a>. Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M91">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M92">View MathML</a>, we can sum the above equations to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M93">View MathML</a>

We now estimate the above terms one by one. To bound <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M94">View MathML</a>, we first integrate by parts and then apply Hölder’s inequality to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M95">View MathML</a>

(2.3)

It follows from the Gagliardo-Nirenberg inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M96">View MathML</a>

and from Lemma 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M97">View MathML</a>

Substituting these two estimates into (2.3) and then using Young’s inequality, we see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M98">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M99">View MathML</a>

(2.4)

and that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M100">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M101">View MathML</a>

(2.5)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M102">View MathML</a>, by Hölder’s inequality, the Gagliardo-Nirenberg inequality and Young’s inequality, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M98">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M104">View MathML</a>

(2.6)

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M100">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M106">View MathML</a>

(2.7)

Applying similar procedure to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M108">View MathML</a>, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M109">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M110">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M111">View MathML</a>

(2.9)

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M113">View MathML</a>

(2.10)

For the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M114">View MathML</a>, by using Hölder’s inequality and Young’s inequality, it can be bounded as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M115">View MathML</a>

(2.11)

Finally, we can follow the steps as in the bound of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M94">View MathML</a> to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M117">View MathML</a>. Precisely, by integrations by parts and Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M118">View MathML</a>

Then the Gagliardo-Nirenberg inequality, Lemma 2.1 and Young’s inequality yield that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M109">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M120">View MathML</a>

(2.12)

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M122">View MathML</a>

(2.13)

Combining the estimates (2.4)-(2.12), we see that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M123">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M124">View MathML</a>

and that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M41">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M126">View MathML</a>

Thus, Gronwall’s inequality together with the energy inequality (2.2) and the assumption (1.4) implies that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M123">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M128">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M129">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M130">View MathML</a>

Then

(2.14)

which is the desired estimates.

Step 2: Estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M132">View MathML</a>.

For this purpose, taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M134">View MathML</a>

Noticing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M135">View MathML</a>, we sum the above equations and integrate by parts to obtain

(2.15)

By using the interpolation inequality and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M137">View MathML</a> in Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M138">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M139">View MathML</a>. Then Young’s inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M140">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M141">View MathML</a>

Substituting the above two estimates into (2.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M142">View MathML</a>

By using Gronwall’s inequality, the energy inequality (2.2) and the estimate (2.14), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M143">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M144">View MathML</a>, which implies that the desired estimates (2.1) hold and thus the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M145">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M147">View MathML</a> on the 3D magneto-micropolar fluid equations (1.1), it is sufficient to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M148">View MathML</a>

(2.16)

To do this, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M149">View MathML</a>

and then equations (1.1) are converted to the following symmetric form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M150">View MathML</a>

(2.17)

Firstly, taking the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77">View MathML</a> inner product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M152">View MathML</a>, w and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M153">View MathML</a> with the above equations, respectively, and integrating by parts, we can obtain the energy estimates similar to (2.2).

Next we take the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M77">View MathML</a> inner product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M156">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M157">View MathML</a> with the first three equations in (2.17), respectively, and then integrate by parts to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M158">View MathML</a>

We now bound the above terms one by one. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M159">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M160">View MathML</a>

It follows from the integration by parts, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M161">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M162">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M163">View MathML</a>

The process for estimating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M164">View MathML</a> is more subtle. It follows from Hölder’s inequality and Lemma 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M165">View MathML</a>

To estimate the term involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M166">View MathML</a>, we take the divergence of the first equation of (2.17) and find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M167">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M168">View MathML</a>. Then the Calderón-Zygmund inequality, Hölder’s inequality and the interpolation inequality imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M169">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M170">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M171">View MathML</a>, combining the above two estimates, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M172">View MathML</a>

The case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M173">View MathML</a> can be similarly dealt with.

Summarily, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M174">View MathML</a>

Thus, Gronwall’s inequality together with the assumption (1.5) and the energy estimates gives the desired <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M175">View MathML</a> estimates (2.12) and thus the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/139/mathml/M29">View MathML</a> can be extended beyond T. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZX wrote the first draft and HY corrected and improved it. Both authors read and approved the final draft.

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).

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