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Regularity criterion for a weak solution to the three-dimensional magneto-micropolar fluid equations

Yinxia Wang

Author Affiliations

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China

Boundary Value Problems 2013, 2013:58  doi:10.1186/1687-2770-2013-58

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/58


Received:30 January 2013
Accepted:4 March 2013
Published:25 March 2013

© 2013 Wang; 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, a regularity criterion for the 3D magneto-micropolar fluid equations is investigated. A sufficient condition on the derivative of the velocity field in one direction is obtained. More precisely, we prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M1">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4">View MathML</a>, then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a> is regular.

MSC: 35K15, 35K45.

Keywords:
magneto-micropolar fluid equations; weak solution; regularity criterion

1 Introduction

In the paper we investigate the initial value problem for magneto-micropolar fluid equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M7">View MathML</a>

(1.1)

with the initial value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M8">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M12">View MathML</a> denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. μ is the kinematic viscosity, χ is the vortex viscosity, γ and κ are spin viscosities and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M13">View MathML</a> is the magnetic Reynold.

The incompressible magneto-micropolar fluid equations (1.1) have been studied extensively (see [1-6] and [7-10]). The existence and uniqueness of local strong solutions is proved by the Galerkin method in [5]. In [4], the author proved global existence of a strong solution with the small initial data. The existence of weak solutions and the uniqueness of weak solutions in 2D case were established in [6]. Yuan [8] obtained a Beale-Kato-Majda type blow-up criterion for a smooth solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a> to the Cauchy problem for (1.1) that relies on the vorticity of velocity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M15">View MathML</a> only. Wang et al.[10] established a Beale-Kato-Majda blow-up criterion of smooth solutions to the 3D magneto-micropolar fluid equation with partial viscosity. Fundamental mathematical issues such as the regularity of weak solutions have generated extensive research and many interesting results have been established (see [1,7] and [9]).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M16">View MathML</a>, (1.1) reduces to micropolar fluid equations. The micropolar fluid equations were first proposed by Eringen [11] (see also [12]). The existence of weak and strong solutions for micropolar fluid equations was obtained by Galdi and Rionero [13] and Yamaguchi [14], respectively. Dong and Chen [15] established regularity criteria of weak solutions to the three-dimensional micropolar fluid equations. In [3], the authors gave sufficient conditions on the kinematics pressure in order to obtain the regularity and uniqueness of weak solutions to the micropolar fluid equations. For more details on regularity criteria, see [16,17] and [18].

If both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M18">View MathML</a>, then equations (1.1) reduce to be magneto-hydrodynamic(MHD) equations. Magnetohydrodynamics (MHD), the science of motion of an electrically conducting fluid in the presence of a magnetic field, consists essentially of the interaction between the fluid velocity and the magnetic field (see [19]). Besides their physical applications, the MHD equations are also mathematically significant. The local existence of solutions to the Cauchy problem (1.1), (1.2) in the usual Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M19">View MathML</a> was established in [20] for any given initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M21">View MathML</a>. But whether the local solution can be extended to a global solution is a challenging open problem in the mathematical fluid mechanics. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (1.1), (1.2) (see [21-28] and [29-32]).

The purpose of this paper is to establish the regularity criteria of weak solutions to (1.1), (1.2) via the derivative of the velocity in one direction. It is proved that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4">View MathML</a>, then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a> can be extended smoothly beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M26">View MathML</a>.

The paper is organized as follows. We first state some important inequalities in Section 2. Then we give the definition of a weak solution and state main results in Section 3, and then we prove the main result in Section 4.

2 Preliminaries

In order to prove our main result, we need the following lemma, which may be found in [33] (see also [21,34] and [35]).

Lemma 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M27">View MathML</a>and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M28">View MathML</a>

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M30">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M31">View MathML</a>. Then there exists a positive constant such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M32">View MathML</a>

(2.1)

Especially, when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M33">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M34">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M35">View MathML</a>

(2.2)

which holds for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M31">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M38">View MathML</a>.

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M39">View MathML</a>and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M40">View MathML</a>. Then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M41">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M42">View MathML</a>

(2.3)

Proof

It follows from the interpolating inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M43">View MathML</a>

(2.4)

Using (2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M44">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M45">View MathML</a>

(2.5)

Combining (2.4) and (2.5) immediately yields (2.3). □

3 Main results

Before stating our main results, we introduce some function spaces. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M46">View MathML</a>

The subspace

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M47">View MathML</a>

is obtained as the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M48">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M49">View MathML</a>-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M50">View MathML</a> . <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M51">View MathML</a> is the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M52">View MathML</a> with respect to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M53">View MathML</a>-norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M54">View MathML</a>

Before stating our main results, we give the definition of a weak solution to (1.1), (1.2) (see [1,7] and [9]).

Definition 3.1 (Weak solutions)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M57">View MathML</a>. A measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6">View MathML</a>-valued triple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a> is said to be a weak solution to (1.1), (1.2) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M60">View MathML</a> if the following conditions hold:

1.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M61">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M62">View MathML</a>

2. (1.1), (1.2) is satisfied in the sense of distributions, i.e., for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M64">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M65">View MathML</a>, the following hold:

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M67">View MathML</a>

3. The energy inequality, that is,

(3.1)

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M69">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M70">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a>is a weak solution to (1.1), (1.2) on some interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M72">View MathML</a>. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M73">View MathML</a>

(3.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M74">View MathML</a>

then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a>can be extended smoothly beyond<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M26">View MathML</a>.

4 Proof of Theorem 3.1

Proof Multiplying the first equation of (1.1) by u and integrating with respect to x on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6">View MathML</a>, using integration by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M78">View MathML</a>

(4.1)

Similarly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M79">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M80">View MathML</a>

(4.3)

Summing up (4.1)-(4.3), we deduce that

(4.4)

By integration by parts and the Cauchy inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M82">View MathML</a>

(4.5)

Using integration by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M83">View MathML</a>

(4.6)

Combining (4.4)-(4.6) yields

Integrating with respect to t, we have

(4.7)

Differentiating (1.1) with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M86">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M87">View MathML</a>

(4.8)

Taking the inner product of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M1">View MathML</a> with the first equation of (4.8) and using integration by parts yield

(4.9)

Similarly, we get

(4.10)

and

(4.11)

Combining (4.9)-(4.11) yields

(4.12)

Using integration by parts and the Cauchy inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M93">View MathML</a>

(4.13)

Using integration by parts, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M94">View MathML</a>

(4.14)

Combining (4.12)-(4.14) yields

(4.15)

In what follows, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M97">View MathML</a>). By integration by parts and the Hölder inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M98">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M99">View MathML</a>

It follows from the interpolating inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M100">View MathML</a>

From (2.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M101">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M102">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M104">View MathML</a>, and the application of the Young inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M105">View MathML</a>

(4.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M106">View MathML</a>

From integration by parts and the Hölder inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M107">View MathML</a>

(4.17)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M108">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M109">View MathML</a>

(4.18)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M110">View MathML</a>

(4.19)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M111">View MathML</a>

By integration by parts and the inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M112">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M113">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M4">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M104">View MathML</a>, and the application of the Young inequality yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M116">View MathML</a>

(4.20)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M117">View MathML</a>

Combining (4.15)-(4.20) yields

From the Gronwall inequality, we get

(4.21)

Multiplying the first equation of (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M120">View MathML</a> and integrating with respect to x on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M6">View MathML</a>, and then using integration by parts, we obtain

(4.22)

Similarly, we get

(4.23)

and

(4.24)

Collecting (4.22)-(4.24) yields

(4.25)

Thanks to integration by parts and the Cauchy inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M126">View MathML</a>

(4.26)

It follows from (4.25)-(4.26) and integration by parts that

(4.27)

In what follows, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M128">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M129">View MathML</a>).

By (2.3) and the Young inequality, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M130">View MathML</a>

(4.28)

By (2.3) and the Young inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M131">View MathML</a>

(4.29)

Similarly, we obtain

(4.30)

(4.31)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M134">View MathML</a>

(4.32)

Combining (4.27)-(4.32) yields

(4.33)

From (4.33), the Gronwall inequality, (4.7) and (4.21), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M136">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M5">View MathML</a> can be extended smoothly beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/58/mathml/M26">View MathML</a>. We have completed the proof of Theorem 3.1. □

Competing interests

The author declares that she has no competing interests.

Author’s contributions

The author completed the paper herself. The author read and approved the final manuscript.

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