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A logarithmically improved blow-up criterion for smooth solutions to the micropolar fluid equations in weak multiplier spaces

Juan Zhao

Author Affiliations

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

Boundary Value Problems 2013, 2013:122  doi:10.1186/1687-2770-2013-122


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/122


Received:11 March 2013
Accepted:26 April 2013
Published:10 May 2013

© 2013 Zhao; 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 study the initial value problem for the three-dimensional micropolar fluid equations. A new logarithmically improved blow-up criterion for the three-dimensional micropolar fluid equations in a weak multiplier space is established.

MSC: 35K15, 35K45.

Keywords:
micropolar fluid equations; smooth solution; blow-up criterion

1 Introduction

In the paper, we consider the initial value problem for the micropolar fluid equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M2">View MathML</a>

(1.1)

with the initial value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M3">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M6">View MathML</a> represent the divergence free velocity field, non-divergence free micro-rotation field and the scalar pressure, respectively. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M7">View MathML</a> is the Newtonian kinetic viscosity and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M8">View MathML</a> is the dynamics micro-rotation viscosity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M9">View MathML</a> are the angular viscosity (see [1]).

The micropolar fluid equations were first proposed by Eringen [2]. The micropolar fluid equations are a generalization of the Navier-Stokes model. It takes into account the microstructure of the fluid, by which we mean the geometry and microrotation of particles. It is a type of fluids which exhibit the micro-rotational effects and micro-rotational inertia, and can be viewed as a non-Newtonian fluid. Physically, it may represent adequately the fluids consisting of bar-like elements. Certain anisotropic fluids, e.g., liquid crystals that are made up of dumbbell molecules, are of the type. For more background, we refer to [1] and references therein.

Due to its importance in mathematics and physics, there is lots of literature devoted to the mathematical theory of the 3D micropolar fluid equations. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [3-12]). The regularity of weak solutions is examined by imposing some critical growth conditions only on the pressure field in the Lebesgue space, Morrey space, multiplier space, BMO space and Besov space, respectively (see [4]). A new logarithmically improved blow-up criterion for the 3D micropolar fluid equations in an appropriate homogeneous Besov space was obtained by Wang and Yuan [9]. A Serrin-type regularity criterion for the weak solutions to the micropolar fluid equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1">View MathML</a> in the critical Morrey-Campanato space was built [10]. Wang and Zhao [12] established logarithmically improved blow-up criteria of a smooth solution to (1.1), (1.2) in the Morrey-Campanto space.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M12">View MathML</a>, then equations (1.1) reduce to be the Navier-Stokes equations. The Leray-Hopf weak solution was constructed by Leray [13] and Hopf [14], respectively. Later on, much effort has been devoted to establishing the global existence and uniqueness of smooth solutions to the Navier-Stokes equations. Different criteria for regularity of the weak solutions have been proposed and many interesting results were established (see [15-24]).

Without loss of generality, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M14">View MathML</a> in the rest of the paper. The purpose of this paper is to establish a new logarithmically improved blow-up criterion to (1.1), (1.2) in a weak multiplier space. Now we state our results as follows.

Theorem 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M16">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M17">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18">View MathML</a>be a smooth solution to equations (1.1), (1.2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M19">View MathML</a>. Ifvsatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M20">View MathML</a>

(1.3)

then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18">View MathML</a>can be extended beyond<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M22">View MathML</a>.

We have the following corollary immediately.

Corollary 1.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M16">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M17">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18">View MathML</a>be a smooth solution to equations (1.1), (1.2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M19">View MathML</a>. Suppose thatTis the maximal existence time, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M28">View MathML</a>

(1.4)

The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2, which play an important role in the proof of our main result. Then we prove the main result in Section 3.

2 Preliminaries

Definition 2.1[25]

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M30">View MathML</a> is a Banach space of all distributions f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1">View MathML</a> such that there exists a constant C such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M32">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M33">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M34">View MathML</a>

where we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M35">View MathML</a> the completion of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M36">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M37">View MathML</a>.

The norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M38">View MathML</a> is given by the operator norm of pointwise multiplication

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M39">View MathML</a>

The following lemma comes from [26].

Lemma 2.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M40">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M41">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M43">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M44">View MathML</a>

(2.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M46">View MathML</a>.

We also need the following interpolation inequalities in three space dimensions.

Lemma 2.3In three space dimensions, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M47">View MathML</a>

(2.2)

3 Proof of Theorem 1.1

Multiplying the first equation of (1.1) by v and integrating with x respect to on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1">View MathML</a>, using integration by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M49">View MathML</a>

(3.1)

Similarly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M50">View MathML</a>

(3.2)

Summing up (3.1)-(3.2), we deduce that

(3.3)

We apply integration by parts and the Cauchy inequality. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M52">View MathML</a>

(3.4)

Substituting (3.3) into (3.4) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M53">View MathML</a>

Integrating with respect to t, we have

(3.5)

Multiplying the first equation of (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M55">View MathML</a>, then integrating the resulting equation with respect to x over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1">View MathML</a> and using integrating by parts, we obtain

(3.6)

We multiply the second equation of (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M58">View MathML</a>, then integrate the resulting equation with respect to x over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M1">View MathML</a> and use integrating by parts. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M60">View MathML</a>

(3.7)

where we have used

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M61">View MathML</a>

Equations (3.6) and (3.7) give

(3.8)

Making use of the Young inequality, we have

(3.9)

Applying the divergence operator ∇⋅ to the first equation of (1.1) produces the expression of the pressure

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M64">View MathML</a>

(3.10)

It follows from (3.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M65">View MathML</a>

(3.11)

By integration by parts and (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M66">View MathML</a>

(3.12)

Combining (3.8), (3.9), (3.12) and (3.5) yields

(3.13)

where we have used

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M68">View MathML</a>

By (1.3), we know that for any small constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M69">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M70">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M71">View MathML</a>

(3.14)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M72">View MathML</a>

(3.15)

The Gronwall inequality and (3.13)-(3.15) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M73">View MathML</a>

(3.16)

Applying ∇ to the first equation, then multiplying the resulting equation by ∇v and using integration by parts, the Hölder inequality, (2.2) and the Young inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M74">View MathML</a>

(3.17)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M75">View MathML</a>

(3.18)

Adding (3.17) and (3.18), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M76">View MathML</a>

(3.19)

Equations (3.5), (3.16), (3.19) and the Gronwall inequality give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M77">View MathML</a>

(3.20)

Applying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M78">View MathML</a> to the first equation in (1.1), then taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M79">View MathML</a> inner product of the resulting equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M80">View MathML</a> and using integration by parts, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M81">View MathML</a>

(3.21)

Similarly, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M82">View MathML</a>

(3.22)

Summing (3.21), (3.22) and using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M83">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M84">View MathML</a>

(3.23)

In what follows, for simplicity, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M85">View MathML</a>.

By the Hölder inequality, (2.1), (2.2) and the Young inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M86">View MathML</a>

(3.24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M87">View MathML</a>

(3.25)

From the Young inequality and (2.2), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M88">View MathML</a>

(3.26)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M89">View MathML</a>

(3.27)

Inserting (3.24)-(3.27) into (3.23) and taking ε small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M90">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M91">View MathML</a>

(3.28)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M92">View MathML</a>.

Integrating (3.28) with respect to time from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M93">View MathML</a> to τ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M94">View MathML</a>

(3.29)

We get from (3.29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M95">View MathML</a>

(3.30)

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M96">View MathML</a>, with the help of Gronwall inequality and (3.30), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M97">View MathML</a>

(3.31)

where C depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M98">View MathML</a>. (3.31) and (3.5) imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M99">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M18">View MathML</a> can be extended smoothly beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/122/mathml/M101">View MathML</a>. We have completed the proof of Theorem 1.1.

Competing interests

The author declares that she has no competing interests.

Authors’ contributions

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

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