Journal of Applied Mathematics

Volume 2012, Article ID 541203, 13 pages

http://dx.doi.org/10.1155/2012/541203

## Logarithmically Improved Blow up Criterion for Smooths Solution to the 3D Micropolar Fluid Equations

^{1}School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China^{2}Department of Mathematical and Physical Sciences, Henan Institute of Engineering, Zhengzhou 451191, China

Received 8 January 2012; Accepted 24 April 2012

Academic Editor: Mina Abd-El-Malek

Copyright © 2012 Yin-Xia Wang and Hengjun Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Blow-up criteria of smooth solutions for the 3D micropolar fluid equations are investigated. Logarithmically improved blow-up criteria are established in the Morrey-Campanto space.

#### 1. Introduction

This paper concerns the initial value problem for the micropolar fluid equations in with the initial value where , , and stand for the velocity field, microrotation field, and the scalar pressure, respectively. And is the Newtonian kinetic viscosity, is the dynamics micro-rotation viscosity, and are the angular viscosity (see, i.e., Lukaszewicz [1]).

The micropolar fluid equations were first proposed by Eringen [2]. It is a type of fluids which exhibits 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, for example, liquid crystals that are made up of dumbbell molecules, are of the same type. For more background, we refer to [1] and references therein. Besides their physical applications, the micropolar fluid equations are also mathematically significant. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained (see [3–8]). Regularity criterion of weak solutions to (1.1) and (1.2) in terms of the pressure was obtained (see [4]). Gala [5] established a Serrin-type regularity criterion for the weak solutions to (1.1) and (1.2) in Morrey-Campanato space. Wang and Chen [7] established the regularity criteria of weak solutions to (1.1) and (1.2) via the derivative of the velocity in one direction. A new logarithmically improved blow-up criterion of smooth solutions to (1.1) and (1.2) in an appropriate homogeneous Besov space is established by Wang and Yuan [8].

If and , then (1.1) reduces to be the Navier-Stokes equations. Besides its physical applications, the Navier-Stokes equations are also mathematically significant. In the last century, Leray [9] and Hopf [10] constructed weak solutions to the Navier-Stokes equations. The solution is called the Leray-Hopf weak solution. Later on, much effort has been devoted to establish 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 are established (see [11–26]). Regularity criteria of weak solutions to the Navier-Stokes equations in Morrey space were obtained in [13, 21].

The main aim of this paper is to establish two logarithmically blow-up criteria of smooth solution to (1.1), (1.2). Our results state as follows.

Theorem 1.1. *Let with . Assume that is a smooth solution to (1.1) and (1.2) on [0, ). If satisfies
**
then the solution can be extended beyond . *

We have the following corollary immediately.

Corollary 1.2. *Let with . Assume that is a smooth solution to (1.1) and (1.2) on [0, ). Suppose that is the maximal existence time, then
*

Theorem 1.3. *Let with . Assume that is a smooth solution to (1.1) and (1.2) on [0, ). If satisfies
**
then the solution can be extended beyond .*

One has the following corollary immediately.

Corollary 1.4. *Let with . Assume that is a smooth solution to (1.1) and (1.2) on [0, ). Suppose that is the maximal existence time, then
*

The paper is organized as follows. We first state 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 and Section 4, respectively.

#### 2. Preliminaries

Firstly, we recall the definition and some properties of the space that we are going to use. The space plays an important role in studying the regularity of solutions to nonlinear differential equations.

*Definition 2.1. *For , the Morrey-Campanato space is defined by
where denotes the ball of center with radius .

It is easy to verify that is a Banach space under the norm . Furthermore, it is easy to check the following:
Morrey-Campanato spaces can be seen as a complement to spaces. In fact, for , one has
one has the following comparison between Lorentz spaces and Morrey-Campanato spaces: for ,
where denotes the usual Lorentz (weak ) space.

In the proof of our main result, we need the following lemma which was given in [27].

Lemma 2.2. *For , the space is defined as the space of such that
**
Then if and only if with equivalence of norms. And the fact that
**
one has
**
where denotes the pointwise multiplier space from to .*

We need the following lemma that is basically established in [28]. For completeness, the proof will be also sketched here.

Lemma 2.3. *For , the inequality
**
holds, where is a positive constant that depends on . *

*Proof. *It follows from the definition of Besov spaces that
where . Choosing such that , from (2.9) we get
Therefore, we have completed the proof of Lemma 2.3.

The following Lemma comes from [29].

Lemma 2.4. *Assume that . For , and , , one has
**
where and .*

In order to prove Theorem 1.1, we need the following interpolation inequalities in three space dimensions.

Lemma 2.5. *In three space dimensions, the following inequalities
**
hold.*

#### 3. Proof of Theorem 1.1

*Proof. *Multiplying the first equation of (1.1) by and integrating with respect to over , using integration by parts, we obtain
Similarly, we get
Summing up (3.1) and (3.2), we deduce thats
Using integration by parts and Cauchy’s inequality, we obtain
Combining (3.3) and (3.4) yields
Integrating with respect to , we have

Taking to the first equation of (1.1), then multiplying the resulting equation by and using integration by parts, we obtain
Similarly, we get
Combining (3.7) and (3.8) yields
Using integration by parts and Cauchy’s inequality, we obtain
Using Hölder’s inequality, (2.8), and Young’s inequality, we obtain
Similarly, we have the following estimate:
Combining (3.9)-(3.12) yields
where we have used
For any , we set
Thus, from (3.13), we have
It follows from (3.8) and Gronwall’s inequality that
provided that
where .

Applying to the first equation of (1.1), then multiplying the resulting equation by and using integration by parts, we have
Likewise, from the second equation of (1.1), we obtain
Using and (3.19) and (3.20), we have
In what follows, for simplicity, we will set .

By Hölder’s inequality, (2.11), (2.12), and Young’s inequality, we obtain
It follows from integration by parts and Cauchy’s inequality that
Combining (3.21)-(3.24) yields
Taking small enough yields
for all .

Integrating (3.26) with respect to time from to , we have
Owing to (3.27), we get
For all , with help of Gronwall inequality and (3.28), we have
where depends on . From (3.29) and (3.5), we know that . Thus, can be extended smoothly beyond . We have completed the proof of Theorem 1.1.

#### 4. Proof of Theorem 1.3

We start to estimate every term on the right of (3.9). Using integration by parts, Hölder inequality, (2.8) and Young inequality, we obtain Similarly, we have the following estimate Thus from (3.9), (3.10), (4.1), and (4.2), we obtain It follows from (4.3) and Gronwall’s inequality that provided that where .

From (4.4), estimate for Theorem 1.3 is same as that for Theorem 1.1. Thus, Theorem 1.3 is proved.

#### Acknowledgments

This work was supported in part by the NNSF of China (Grant no. 11101144) and Research Initiation Project for High-level Talents (201031) of North China University of Water Resources and Electric Power.

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