Global Regularity Criterion for the Magneto-Micropolar Fluid Equations

ω(t, x) ∈ Rand b(t, x) ∈ R, and p(t, x) ∈ R denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. μ, χ, κ, γ, and ] are positive numbers associated with properties of the material: μ is the kinematic viscosity, χ is the vortex viscosity, κ and γ are spin viscosities, and 1/] is the magnetic Reynold. u 0 , ω 0 , and b 0 are initial data for the velocity, the angular velocity, and the magnetic field with properties div u 0 = 0 and div b 0 = 0. It is well known that the question of global existence or finite time blowup of smooth solutions for the 3D incompressible Euler orNavier-Stokes equations has been one of the most outstanding open problems in applied analysis, as well as that for the 3D incompressible magneto-micropolar fluid equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the global regularity criterion of the solutions for system (1). But there are few theories about regularity and blow-up criteria of magneto-micropolar fluid equations. Some blow-up criterion are obtained by Yuan [1] in 2010. His paper implies that most classical blow-up criteria of smooth solutions toNavierStokes or magneto-hydrodynamic equations also hold for magneto-micropolar fluid equations. In particular, using Fourier frequency localization, Yuan proved the Beale-KatoMajda criterion only relying on ∇u; that is, if

It is well known that the question of global existence or finite time blowup of smooth solutions for the 3D incompressible Euler or Navier-Stokes equations has been one of the most outstanding open problems in applied analysis, as well as that for the 3D incompressible magneto-micropolar fluid equations.This challenging problem has attracted significant attention.Therefore, it is interesting to study the global regularity criterion of the solutions for system (1).But there are few theories about regularity and blow-up criteria of magneto-micropolar fluid equations.Some blow-up criterion are obtained by Yuan [1] in 2010.His paper implies that most classical blow-up criteria of smooth solutions to Navier-Stokes or magneto-hydrodynamic equations also hold for magneto-micropolar fluid equations.In particular, using Fourier frequency localization, Yuan proved the Beale-Kato-Majda criterion only relying on ∇; that is, if then the solution (, , ) can be extended past time  = .In 2008, Yuan [2] obtain the following blow-up criteria: if then the solution (, , ) can be extended  = .Recently, Xu [3] also studied the regularity of weak solutions to magneto-micropolar fluid equations in Besov spaces.
In this paper, we establish a refined global regularity criterion by means of Osgood norm which improves the results (2) and (3).As we know, Osgood condition plays an important role in solving uniqueness of solutions to the incompressible fluids equations.This induces us to apply it to global regularity criterion problems of smooth solution.To achieve this goal, taking full advantage of Fourier frequency localization method and using the low-high decomposition technique, we show the following main results.Theorem 1. Suppose that for  > 1/2, (, , ) ∈ ([0, );   ) ∩  1 ((0, );   ) ∩ ((0, );  +2 ) is the smooth solution to (1)

Preliminaries
The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition.So, we first introduce the Littlewood-Paley decomposition and review the so-called Beinstein estimate and commutator estimate, which are to be used in the proof of our theorem.Let S(R 3 ) be the Schwartz class of rapidly decreasing functions.Given  ∈ S(R 3 ), the Fourier transform of  is defined by We consider that ,  ∈ S(R 3 ), respectively, support in Setting Let ℎ =  −1  and h =  −1 ; the dyadic blocks are defined as follows: Informally, Δ  =  +1 −   is a frequency projection to the annulus || ≈ 2  , while   is frequency projection to the ball || ≲ 2  .The details of Littlewood-Paley decomposition can be found in Triebel [4] and Chemin [5].Now Besov spaces in R 3 can be defined as follows: where Z  denotes the dual space of Z = { ∈ S;   f(0) = 0; ∀ ∈ N  multi-index}.Now we introduce well-known Bernstetin's Lemma and commutator estimate, the proof is omitted here, and we can find the details in Chemin [5], Chemin and Lerner [6], and Kato and Ponce [7].

Proof of the Theorem 1
First we go on with the  1 estimates of the solution (, , ) under the condition (3).Denote that  = ∇ × ,  = ∇ × , and  = ∇ ×  we take curl on both sides of (1); we get the following equation: with which uses the fact ∇ × ∇ div  = 0.
Multiplying the three equations with (, , ), respectively, integrating by parts over R 3 about the variable , then adding the resulting equations yields that where we use the following fact due to the divergence free condition of , : Let us begin with estimating  1 and  3 .Using Littlewood-Paley decomposition to ∇, we have For the terms  1 1 and  2 1 , using Hölder's inequality, Beinstein's inequality, and (12), (13), we obtain where we use the interpolation inequality Summing up ( 26)-( 27), we have 3 can be treated in the same way, and we decompose it as then we obtain Now we study  2 , we decompose  by using Littlewood-Paley theory; that is, Using Young's inequality, the term  6 can be written as Summing up (29)-( 35) and taking the sum into (23), by Young's inequality, we get If we let 2 −(N/2) (‖‖ where [] stands for the integral parts of  ∈ R, log + () = log( + ), then we have where Using Gronwall's inequality, we have On the other hand, by multiplying (, , ), it can be easily derived from magneto-micropolar fluid equation (1) that Equation (39) along with (40) implies that the  1 estimate of solution (, , ).
Next, we will show how to deduce   estimates based on the  1 estimates.We apply operator Λ  on the two sides of (1), multiply (Λ  , Λ  , Λ  ) by the resulting equations and integrate the final form over R 3 , and ge where we use the fact Furthermore, the divergence free conditions of (, ) imply that   Hence by ( 39) and (51), we can get the   regularity at time  = ; that is, the smooth solution (, , ) can be extended past time .