A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity
© Yu-Zhu Wang et al. 2011
Received: 18 February 2011
Accepted: 7 March 2011
Published: 15 March 2011
We study the incompressible magneto-micropolar fluid equations with partial viscosity in . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.
where , , and denote the velocity of the fluid, the microrotational velocity, magnetic field, and hydrostatic pressure, respectively. is the kinematic viscosity, is the vortex viscosity, and are spin viscosities, and is the magnetic Reynold.
The incompressible magneto-micropolar fluid equation (1.1) has been studied extensively (see [1–7]). In , the authors have proven that a weak solution to (1.1) has fractional time derivatives of any order less than in the two-dimensional case. In the three-dimensional case, a uniqueness result similar to the one for Navier-Stokes equations is given and the same result concerning fractional derivatives is obtained, but only for a more regular weak solution. Rojas-Medar  established local existence and uniqueness of strong solutions by the Galerkin method. Rojas-Medar and Boldrini  also proved the existence of weak solutions by the Galerkin method, and in 2D case, also proved the uniqueness of the weak solutions. Ortega-Torres and Rojas-Medar  proved global existence of strong solutions for small initial data. A Beale-Kato-Majda type blow-up criterion for smooth solution to (1.1) that relies on the vorticity of velocity only is obtained by Yuan . For regularity results, refer to Yuan  and Gala .
If , (1.1) reduces to micropolar fluid equations. The micropolar fluid equations was first developed by Eringen . 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 consisting 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 that 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 it is important to the scientists working with the hydrodynamic-fluid problems and phenomena. For more background, we refer to  and references therein. The existences of weak and strong solutions for micropolar fluid equations were proved by Galdi and Rionero  and Yamaguchi , respectively. Regularity criteria of weak solutions to the micropolar fluid equations are investigated in . In , the authors gave sufficient conditions on the kinematics pressure in order to obtain regularity and uniqueness of the weak solutions to the micropolar fluid equations. The convergence of weak solutions of the micropolar fluids in bounded domains of was investigated (see ). When the viscosities tend to zero, in the limit, a fluid governed by an Euler-like system was found.
If both and , then (1.1) reduces to be the magneto-hydrodynamic (MHD) equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to MHD systems (see [15–23]). Zhou  established Serrin-type regularity criteria in term of the velocity only. Logarithmically improved regularity criteria for MHD equations were established in [16, 23]. Regularity criteria for the 3D MHD equations in term of the pressure were obtained . Zhou and Gala  obtained regularity criteria of solutions in term of and in the multiplier spaces. A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field in Morrey-Campanato spaces was established (see ). In , a regularity criterion for the 2D MHD system with zero magnetic diffusivity was obtained.
Regularity criteria for the generalized viscous MHD equations were also obtained in . Logarithmically improved regularity criteria for two related models to MHD equations were established in  and , respectively. Lei and Zhou  studied the magneto-hydrodynamic equations with and . Caflisch et al.  and Zhang and Liu  obtained blow-up criteria of smooth solutions to 3-D ideal MHD equations, respectively. Cannone et al.  showed a losing estimate for the ideal MHD equations and applied it to establish an improved blow-up criterion of smooth solutions to ideal MHD equations.
In the absence of global well-posedness, the development of blow-up/non blow-up theory is of major importance for both theoretical and practical purposes. For incompressible Euler and Navier-Stokes equations, the well-known Beale-Kato-Majda's criterion  says that any solution is smooth up to time under the assumption that . Beale-Kato-Majdas criterion is slightly improved by Kozono and Taniuchi  under the assumption . In this paper, we obtain a Beale-Kato-Majda type blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2).
Now we state our results as follows.
then the solution can be extended beyond .
We have the following corollary immediately.
The paper is organized as follows. We first state some preliminaries on functional settings and some important inequalities in Section 2 and then prove the blow-up criterion of smooth solutions to the magneto-micropolar fluid equations (1.2) in Section 3.
Thereafter, we will use the fact .
In what follows, we will make continuous use of Bernstein inequalities, which comes from .
hold, where and are positive constants independent of and .
The following inequality is well-known Gagliardo-Nirenberg inequality.
holds for all .
The following lemma comes from .
holds for all vectors with .
It follows from (2.12), (2.14), and (2.15) that (2.10) holds. Thus, the lemma is proved.
In order to prove Theorem 1.1, we need the following interpolation inequalities in two and three space dimensions.
Taking and and , respectively. From (2.21), we immediately get the last inequality in (2.16) and (2.17). Thus, we have completed the proof of Lemma 2.5.
3. Proof of Main Results
Proof of Theorem 1.1.
where we have used and .
where depends on , while is an absolute positive constant.
In what follows, for simplicity, we will set .
for all .
where depends on .
Noting that (3.2) and the right hand side of (3.25) is independent of for , we know that . Thus, Theorem 1.1 is proved.
This work was supported by the NNSF of China (Grant no. 10971190).
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