A Regularity Criterion for the Magneto-Micropolar Fluid Equations in ̇B − 1

Zhihao Tang, Gang Wang, and Haiwa Guan 1 Department of Fundamentals, Henan Polytechnic Institute, Nanyang, Henan 473009, China 2 Shandong Transport Vocational College, Weifang, Shandong 261206, China 3Department of Public Teaching, Wenzhou Vocational College of Science and Technology, Wenzhou, Zhejiang 325000, China Correspondence should be addressed to Haiwa Guan; haiwaguan1234@126.com Received 16 January 2013; Accepted 2 March 2013 Academic Editor: Fuyi Xu Copyright © 2013 Zhihao Tang et al. 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. The paper is dedicated to study of the Cauchy problem for the magneto-micropolar fluid equations in three-dimensional spaces. A new logarithmically improved regularity criterion for themagneto-micropolar fluid equations is established in terms of the pressure in the homogeneous Besov space ̇ B−1

As we know, the problem of global regularity or finite time singularity for the weak solutions of the magneto-micropolar fluid equations model with large initial data still remains unsolved since (1) includes the 3D Navier-Stokes equations.It is of interest that the regularity of the weak solutions is under preassumption of certain growth conditions.There are a lot of lectures to study the regularity of weak solutions of the magneto-micropolar fluid equations (see, [4][5][6]).The purpose of this paper is to establish a new logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure in Besov space Ḃ −1 ∞,∞ .Now we state the main results as follows.

Preliminaries and Lemmas
Throughout this paper, we introduce some function spaces, notations, and important inequalities.

Proof of Theorem 1
For given initial data (V 0 ,  0 ,  0 ) ∈  1 (R 3 ), the weak solution is the same as the local strong solution (V, , ) in a local interval (0, ) as in the discussion of Navier-Stokes equations.
For the uniqueness and existence of local strong solution, we refer to [1].Thus, it proves that Theorem 1 is reduced to establish a priori estimates uniformly in (0, ) for strong solutions.With the use of the a priori estimates, the local strong solution (V, , ) can be continuously extended to  =  by a standard process to obtain global regularity of the weak solution.Therefore, we assume that the solution (V, , ) is sufficiently smooth on (0, ).
Proof of Theorem 1.We show that Theorem 1 holds under condition (1).To prove the theorem, we need the  4 -estimate.
For this purpose, taking the inner product of the first equation of (1) with || 2  and integrating by parts, it can be deduced that where we used the following relations by the divergence-free condition div  = 0: Similarly, taking the inner product of the second equation of (1) with || 2  and integrating by parts, it can be inferred that Using an argument similar to that used in deriving the estimate ( 11)-( 13), it can be obtained for the third equation of ( 1) that Adding up ( 11), (13), and ( 14), then we obtain Applying the Hölder inequality and the Young inequality for  2 , it follows that Arguing similarly to above, it can be derived for  3 that Considering the term  1 , by virtue of the Cauchy inequality, we have Let us bound the integral Applying the divergence operator div to the first equation of ( 1), one formally has  = ∑ 3 ,=1     (    −     ), where   denotes the th Riesz operator.By the Calderon-Zygmund inequality, we have With the help of ( 8) and ( 19), by the Hölder inequality and the Young inequality, we deduce that So the term  1 can be estimated as Next we have the following estimate for the term  4 : Since  ∈  2 (R 3 ) ∩  6 (R The last term of (15) can be treated in the same way as where () is defined by Applying Gronwall's inequality on (25) for the interval [ 0 , ], one has sup provided that where  0 is a positive constant depending on  0 .Next we will estimate the  2 -norm of ∇V, ∇, and ∇.We multiply both sides of the first equation of ( 1) by (−ΔV), the second equation of ( 1) by (−Δ), and the third equation of ( 1) by (−Δ), by integration by parts over R 3 , we get with  > 0, Λ  = (−Δ) /2 and (1/) = (1/ 1 ) + (1/ 1 ) = (1/ 2 ) + (1/ 2 ).Taking the operation Λ 3 on both sides of (1), then multiplying them by Λ 3 V, Λ 3 , and Λ 3 , and integrating by parts over R 3 , we have Hence  1 can be estimated as where we used (33) with  = 3,  = 3/2,  1 =  1 =  2 =  2 = 3 and the following inequalities: (3/4)+(39/2) +  0 ( +  ()) . (41) Gronwall's inequality implies the boundness of  3 -norm of V, , and  provided that 39 < (1/2), which can be achieved by the absolute continuous property of integral (2).This completes the proof of Theorem 1.