Regularity Criterion for Weak Solution to the 3 D Micropolar Fluid Equations

Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for some T > 0 , if ∫ 0 T ∥ v x 3 ∥ L ϱ ρ d t ∞ , where 3 / ϱ + 2 / ρ ≤ 1 and ϱ ≥ 3 , then the solution ( v , w ) can be extended smoothly beyond t = T . The derivative v x 3 can be substituted with any directional derivative of v .


Introduction
In the paper, we investigate the initial value problem for the micropolar fluid equations in R 3 : with the initial value t 0 : v v 0 x , w w 0 x , 1.2 where v t, x , w t, x , and π t, x stand for the divergence free velocity field, nondivergence free microrotation field angular velocity of the rotation of the particles of the fluid , the scalar pressure, respectively ν > 0 is the Newtonian kinetic viscosity, κ > 0 is the dynamics microrotation viscosity, and α, β, γ > 0 are the angular viscosity see, e.g., Lukaszewicz 1 .

Journal of Applied Mathematics
The micropolar fluid equations was first proposed by Eringen 2 .It is a type of fluids which exhibits the microrotational effects and microrotational inertia and can be viewed as a non-Newtonian fluid.Physically, micropolar fluid may represent fluids that consists of rigid, randomly oriented or spherical particles suspended in a viscous medium, where the deformation of fluid particles is ignored.It can describe many phenomena appeared in a large number of complex fluids such as the suspensions, animal blood, and liquid crystals which cannot be characterized appropriately by the Navier-Stokes equations, and that is important to the scientists working with the hydrodynamic fluid problems and phenomena.For more background, we refer to 1 and references therein.Besides their physical applications, the micropolar fluid equations are also mathematically significant.The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero 3 and Yamaguchi 4 , respectively.The convergence of weak solutions of the micropolar fluids in bounded domains of R n was investigated see 5 .When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research, and many interesting results have been obtained see 6-8 .A Beale-Kato-Madja criterion see 9 of smooth solutions to a related model with 1.1 was established in 10 .
If κ 0 and w 0, 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 11 and Hopf 12 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 13-31 .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 then the solution v, w can be extended smoothly beyond t T .The paper is organized as follows.We first state some important inequalities in Section 2, which play an important roles in the proof of our main result.Then, we give definition of weak solution and state main results in Section 3 and then prove main result in Section 4.

Preliminaries
In order to prove our main result, we need the following Lemma, which may be found in 32 see also 33, 34 .For the convenience of the readers, the proof of the Lemmas are provided.Lemma 2.1.Assume that μ, λ, ι ∈ R and satisfy Then, there exists a positive constant such that Especially, when λ 2, there exists a positive constant C C μ such that

2.4
Then, we obtain

2.5
Integrating with respect to x 1 and using H ölder inequality, we have

2.6
Integrating with respect to x 2 , x 3 and using H ölder inequality, we obtain

2.7
It follows from H ölder inequality that By the above inequality, we get 2.2 .

2.9
Proof.Using the interpolating inequality, we obtain .

Main Results
Before stating our main results, we introduce some function spaces.Let The subspace Before stating our main results, we give the definition of weak solution to 1.1 , 1.2 see 6 .
A measurable R 3 -valued triple v, w is said to be a weak solution to 1.1 , 1.2 on 0, T if the following conditions hold the following.

3.4
2 Equations 1.1 , 1.2 are satisfied in the sense of distributions; that is, for every

3.6
3 The energy inequality, that is, where then the solution v, w can be extended smoothly beyond t T .

Proof of Theorem 3.2
Proof.Multiplying the first equation of 1.1 by v and integrating with respect to x on R 3 , using integration by parts, we obtain 1 2 Similarly, we get 1 2 Summing up 4.1 -4.2 , we deduce that 1 2

4.3
By integration by parts and Cauchy inequality, we obtain 2κ Integrating with respect to t, we have

4.6
Journal of Applied Mathematics 7 Differentiating 1.1 with respect to x 3 , we obtain Taking the inner product of v x 3 with the first equation of 4.7 and using integration by parts yields 1 2

Journal of Applied Mathematics
In what follows, we estimate I j j 1, 2 . . ., 5 .By integration by parts and H ölder inequality, we obtain where It follows from the interpolating inequality that

4.19
From H ölder inequality, we obtain

4.22
From Gronwall inequality, we get Multiplying the first equation of 1.1 by −Δv and integrating with respect to x on R 3 , then using integration by parts, we obtain 1 2 Similarly, we get 1 2 Journal of Applied Mathematics Thanks to integration by parts and Cauchy inequality, we get

4.27
It follows from 4.26 -4.27 and integration by parts that 1 2