Journal of Applied Mathematics

Volume 2013 (2013), Article ID 262131, 5 pages

http://dx.doi.org/10.1155/2013/262131

Research Article

A New Pressure Regularity Criterion of the Three-Dimensional Micropolar Fluid Equations

Junbai Ren

School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China

Received 30 August 2013; Accepted 29 October 2013

Academic Editor: Suh-Yuh Yang

Copyright © 2013 Junbai Ren. 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.

This paper concerns the regularity criterion of the weak solutions to the three-dimensional (3D) micropolar fluid equations in terms of the pressure. It is proved that if one of the partial derivatives of pressure satisfies with , then the weak solution of the micropolar fluid equations becomes regular on .

1. Introduction

In the past ten years, the mathematical models of fluid dynamics attract more and more attention. As a classic fluid dynamical model, Naiver-Stokes equations [1] are proved as an accurate model in many practical situations, which presume that the derivatives of the components of the velocity are small. However, for certain anisotropic fluids, for example, liquid crystals, which are made up of dumbbell molecules, and some polymeric fluids or fluids containing certain additives in narrow films [2], the constructive relations do not satisfy Stoke Law. In 1960s, Eringen [3] introduced viscous incompressible micropolar fluid flows, a non-Newtonian fluid model with asymmetric stress tensor. From the viewpoint of mathematics, micropolar fluid model is coupled with the incompressible Navier-Stokes equations, micro-rotational effects, and microrotational inertia. The three-dimensional (3D) viscous incompressible micropolar fluid equations are written as

associated with the initial prescribed data

Here , , and stand for the divergence-free velocity vector field, the scalar pressure field, and nondivergence free microrotation vector field, respectively. is the Newtonian kinetic viscosity, is the dynamics microrotation viscosity, and are the angular viscosity.

Due to their importance in mathematics, there is large literature on the well-posedness and large time behaviors for weak solutions of micropolar fluid equations [4–10]. However, the question of global regularity or uniqueness of weak solutions of three-dimensional micropolar fluid equations is still a challenge open problem. Therefore, it is interesting and natural to consider the regularity criteria for weak solutions of micropolar fluid equations by imposing some growth conditions on the velocity or the pressure. As for the velocity regularity criteria, Dong and Chen [11] (see also [12]) obtained that if the velocity fields satisfy one of the following conditions:

then the weak solutions of the micropolar fluid equations (1) and (2) are regular. Here the critical spaces , are Lorentz spaces and Multiplier space. The results were further refined by many authors [13–15] to some large critical spaces such as Besov spaces and Triebel-Lizorkin spaces.

On the other hand, as for the pressure regularity criteria of the micropolar fluid equations, Yuan [12] showed that the weak solution becomes regular if the pressure satisfies

Dong et al. [16, 17] improved the regularity criteria by imposing the growth conditions in the critical Besov spaces

Recently, some interesting logarithmical pressure regularity criteria [18, 19] of micropolar fluid equations are studied. In particular, Jia et al. [20] refined this question by imposing the following regularity criterion condition:

Compared with the results (4) and (6), it is natural to consider whether or not the growth condition of the partial derivative of the pressure can be released. It should be mentioned that the optimal result is that the regularity for weak solutions of three-dimensional micropolar fluid equations is valid if

This is an open problem on the pressure regularity criterion of micropolar fluid equations. The aim of this paper is to understand this challenge problem. More precisely, we will show the regularity of weak solutions to three-dimensional micropolar fluid equations if one of the partial derivatives of the pressure, say, , satisfies

One may also refer to some important results on the regularity criteria of some mathematical models in fluid dynamics. For example, Cao and Titi [21], Chen and Zhang [22], Fan et al. [23], and Zhou [24] investigated the regularity criteria for the classic Navier-Stokes equations, Chen et al. [25], He and Xin [26], and Jia and Zhou [27, 28] for MHD equations, Dong et al. [29, 30] for quasigeostrophic equation, and so on.

2. Preliminaries and Main Results

In this paper, we use the following usual notations. is the abstract constant which may change from line to line. is the scalar or vector Lebesgue space of all integral functions associated with the norm

The following anisotropic Sobolev inequality is due to Cao and Wu [31].

Lemma 1 (Cao and Wu [31]). Suppose , , and and for three constants satisfy

then there exists a constant such that the following anisotropic Sobolev inequality:

is valid.

To aid the introduction of our main results, let us recall the definition of the weak solutions of the 3D micropolar fluid flows (1) and (2) (see Łukaszewicz [9]).

Definition 2. Let and . A pair of vector fields is termed as a weak solution to the 3D micropolar fluid equations (1) and (2) on if satisfies the following properties:

(i);(ii) verifies (1) in the sense of distribution.

The following existence result of micropolar fluid equations is useful for our results.

Lemma 3 (Dong et al. [16]). Assume and with in the sense of distributions. Then there exist a constant and a unique strong solution of the 3D micropolar fluid equations (1) and (2) such that

Theorem 4. Suppose and and in the sense of distributions and assume is a weak solution of the 3D micropolar fluid equations (1) and (2) on . If one of the partial derivatives of the pressure, say, , satisfies

then is regular on .

3. Proof of Theorem 4

We first establish some fundamental estimates between the pressure and the velocity of the micropolar fluid equations (1) and (2). Taking the operator div to both sides of the first equation of (1) and noting the fact of the divergence-free velocity, one shows that

together with Calderon-Zygmund inequality, implies that for any

Similarly, acting the operator on both sides of the second equation of (1), we have

In order to prove the main result, we also need some auxiliary estimates of the weak solutions of the micropolar fluid equations (1). Thanks to the divergence-free velocity fields and application to the integration by parts, we have the following estimates:

In particular, by direct computation, we also have

We now begin to prove Theorem 4.

Taking the inner product of the first equation of (1) with and the second equation of (1) with , respectively, it follows that

Applying Hölder inequality, Young inequality, and integration by parts, one shows that

Summing up (19) and (20), we have

Now we estimate the right hand side of (21). Employing Hölder inequality and Young inequality firstly yields

Applying interpolation inequality and Lemma 1 together with (14) and (16), it follows that

where we have used interpolation inequality

(let , , and in Lemma 1).

That is to say, we may check as

With the aid of Hölder inequality and Young inequality, is now further estimated as

We choose

that is,

Thus

which together with (27) gives

Collecting (21), (22), and (31), we derive

or

Taking Gronwall inequality into consideration yields

By the definitions of the weak solutions, we have

Letting , we could obtain

which exactly satisfies the growth condition (13) in Theorem 4; therefore, we derive the uniform bounds of

together with Lemma 3; we now complete the proof of Theorem 4.

Acknowledgments

The author would like to express his sincere thanks to the editors and the referees for their valuable comments and suggestions. This work is partially supported by the NNSF of China grants nos. 11361026 and 11161022.

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