Nonlinear Functional Difference Equations with ApplicationsView this Special Issue
Global Regularity Criterion for the Magneto-Micropolar Fluid Equations
We are concerned with the magneto-micropolar fluid equations in . Using Littlewood-Paley decomposition, we obtain an Osgood type global regularity criterion for the system.
In this paper, we consider the following magneto-micropolar fluid equations in : where denotes the velocity of the fluid at a point , , and denote, respectively, the microrotational velocity, the magnetic field, and the hydrostatic pressure. are positive numbers associated with properties of the material: is the kinematic viscosity, is the vortex viscosity, and are spin viscosities, and is the magnetic Reynold. are initial data for the velocity, the angular velocity, and the magnetic field with properties and .
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  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  obtain the following blow-up criteria: if then the solution can be extended . Recently, Xu  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 , is the smooth solution to (1). If the following Osgood type condition: then the solution can be extended past time . Here one denotes that .
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 be the Schwartz class of rapidly decreasing functions. Given , the Fourier transform of is defined by
We consider that , respectively, support in and , such that Setting , then if and if . Let and ; the dyadic blocks are defined as follows: Informally, is a frequency projection to the annulus , while is frequency projection to the ball . The details of Littlewood-Paley decomposition can be found in Triebel  and Chemin . Now Besov spaces in can be defined as follows: where denotes the dual space of .
Lemma 3 (Bernstein's lemma). Let . Assume that , then there exist constants independent of , such that
Remark 4. From the above Beinstein estimate, we easily know that in , for the Reisz transform , it has for If suppose vector valued funtion be divergence free, by Biot Savard law with and the boundedness of Reisz transform on , we have, there exist constants independent such that If the frequency of is restricted to annulus , then (11) implies that
Now we denote that , which satisfies can be defined in the same way as follows: Using the perivious notation, we define the norm of Sobolev space especially by Fourier transform, and can be defined as where
Lemma 5 (Commutator estimate). Let ; assume that , then there exists a constant independent of , such that with , such that Here .
3. Proof of the Theorem 1
Multiplying the three equations with , respectively, integrating by parts over 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 and . Using Littlewood-Paley decomposition to , we have For the terms and , using Hölder's inequality, Beinstein's inequality, and (12), (13), we obtain for , we get where we use the interpolation inequality Summing up (26)-(27), we have can be treated in the same way, and we decompose it as then we obtain Now we study , we decompose by using Littlewood-Paley theory; that is, then Similarly for , and , we have Using Young's inequality, the term can be written as Summing up (29)–(35) and taking the sum into (23), by Young's inequality, we get If we let , that is, if we choose where stands for the integral parts of , , 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 estimate of solution .
Next, we will show how to deduce estimates based on the estimates. We apply operator on the two sides of (1), multiply by the resulting equations and integrate the final form over , and ge where we use the fact Furthermore, the divergence free conditions of imply that then By Lemma 5, Hölder's inequality, Gagliardo-Nirenberg's inequality and Young's inequality we deduce that Similarly, we estimate as follows: Finally, we estimate the last term Summing up (47)–(49) with (44), we obtain Using Gronwall's inequality we obtain Hence by (39) and (51), we can get the regularity at time ; that is, the smooth solution can be extended past time .