Logarithmically Improved Regularity Criterion for the 3 D Micropolar Fluid Equations

where u(x, t) is the velocity field, w(x, t) is the microrotational velocity field, andp = p(x, t) is the scalar pressure field, while (u 0 , w 0 ) are the given initial data with ∇ ⋅ u 0 = 0 in the sense of distribution. Micropolar fluid system was firstly developed by Eringen [1, 2]. It is a type of fluids which exhibits microrotational effects and microrotational inertia and can be viewed as a non-Newtonian fluid. It can describe many phenomena that appear 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 system and that is important to the scientists working with the hydrodynamic-fluid problems and phenomena. The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [3] and Yamaguchi [4], respectively. The uniqueness of strong solutions to themicropolar flows and themagnetomicropolar flows either local for large data or global for small data is considered in [5, 6] and references therein. The purpose of this paper is to study the regularity of weak solutions to themicropolar fluid system (1). Bymeans of the Littlewood-Paley decomposition methods and function decomposition technique, Dong and Zhang [7, 8] recently prove the regularity of weak solutions under the velocity condition and the pressure condition in Besov spaces. Yuan proved [9] some classical regularity criteria of weak solutions to the Navier-Stokes equation which also holds for the micropolar fluid equations. Particularly, the well-known Beale-Kato-Majda’s criterion is also established [10]. If (u, w) satisfies the condition


Introduction
This paper focuses on the incompressible micropolar fluid equations in R 3      + ( ⋅ ∇)  − Δ + ∇ − ∇ ×  = 0,    − Δ − ∇ (∇ ⋅ ) + 2 +  ⋅ ∇ − ∇ ×  = 0, ∇ ⋅  = 0,  (, 0) =  0 () , (, 0) =  0 () , where (, ) is the velocity field, (, ) is the microrotational velocity field, and  = (, ) is the scalar pressure field, while ( 0 ,  0 ) are the given initial data with ∇ ⋅  0 = 0 in the sense of distribution.Micropolar fluid system was firstly developed by Eringen [1,2].It is a type of fluids which exhibits microrotational effects and microrotational inertia and can be viewed as a non-Newtonian fluid.It can describe many phenomena that appear 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 system and that is important to the scientists working with the hydrodynamic-fluid problems and phenomena.
The existences of weak and strong solutions for micropolar fluid equations were treated by Galdi and Rionero [3] and Yamaguchi [4], respectively.The uniqueness of strong solutions to the micropolar flows and the magnetomicropolar flows either local for large data or global for small data is considered in [5,6] and references therein.
The purpose of this paper is to study the regularity of weak solutions to the micropolar fluid system (1).By means of the Littlewood-Paley decomposition methods and function decomposition technique, Dong and Zhang [7,8] recently prove the regularity of weak solutions under the velocity condition and the pressure condition in Besov spaces.
Yuan proved [9] some classical regularity criteria of weak solutions to the Navier-Stokes equation which also holds for the micropolar fluid equations.Particularly, the well-known Beale-Kato-Majda's criterion is also established [10].
If (, ) satisfies the condition then the solution (, ) can be extended smoothly beyond  = .Motivated by the ideas of [11][12][13][14], this paper is to establish logarithmically improved regularity criterion in terms of the vorticity.

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International Journal of Analysis Theorem 1.Let (, ) be a smooth solution to (1) with initial data ( 0 ,  0 ) ∈  3 (R 3 ).Suppose that the corresponding vorticity field satisfies then the solution can be smoothly extended after time .
We have the following corollary immediately.
Remark 5. Throughout the paper,  stands for a constant and changes from line to line; ‖ ⋅ ‖  denotes the norm of the Lebesgue space   (R 3 ) and ‖ ⋅ ‖   denotes the norm of the Lebesgue space   (R 3 ).

Proof of Theorem 1
Before going to the proof, we recall the following two inequalities established in [15,16].
By choosing  =  =  = ∞, ] =  =  =  2 = 2, and  1 =  = 0, we have where we used the following relations: Proof of Theorem 1. Multiplying the first equation of ( 1) by || 2 , after integration by parts, we have Similarly, multiplying the second equation of ( Adding (10) to (11), one has that We estimate above terms one by one, using the following relation: We have where we have used the fact |∇||| ≤ |∇|.
Applying Holder inequality and Cauchy inequality, we get Similarly, for  3 , we have In the same way, for  4 one can deduce In order to estimate  2 , we first establish an estimate between the pressure and the velocity.Taking the operator div on both sides of the first equation of (1), Applying   (1 <  < ∞) boundedness of the singular operators yields Inequality (19), together with Lemma 6, shows that Combining ( 12), ( 15), ( 16), (17), and (20) yields For the right hand side of (21), we have where we used the inequality ‖∇‖ Ḃ 0 ∞,∞ ≤ ‖∇ × ‖ Ḃ 0 ∞,∞ .Due to (3), one can show that, for any small constant  > 0, there exists  * <  such that For any  * <  ≤ , we set ) To estimate  1 , we integrate by parts and apply Holder's inequality to obtain Due to the incompressible condition ∇ ⋅  = 0, we obtain By integrating by parts and applying Holder's inequality and Young's inequality, we have From the above computation, we have (31) Applying Gronwall's inequality and (25), we have sup We should point out that the constant  * also changes from line to line.Applying ∇ 3 to (1) and taking the  2 inner product of the resulting equation with (∇ 3 , ∇ 3 ) with help of integrating by parts, we have = − ∫ Now, we introduce the following commutator estimate according to Kato and Ponce [17]: where we used the following inequalities It should be clear that applying Gronwall's inequality to (39), we can obtain () ≤  provided that  is small enough.This completes the proof of the theorem.