1 Introduction and main results
here is an open-bounded domain with Lipschitz boundaries, and the unknowns u, ω, b, π denote the velocity of the fluid, the micro-rotational velocity, magnetic field and hydrostatic pressure, respectively. χ, μ, λ are positive numbers associated with properties of the material: χ is the vortex viscosity, μ is spin viscosity and is the magnetic Reynold. In (1.1), is the symmetric part of the velocity gradient, i.e.,
To (1.1) we append the following initial and boundary conditions
The theory of micropolar fluid was first proposed by Eringen  in 1966, which enabled us to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations for the viscous incompressible fluid, for example, the motion of animal blood, liquid crystals and dilute aqueous polymer solutions, etc.
If , system (1.1) reduces to the classical magneto-micropolar fluid equations, and there are many earlier results concerning the weak and strong solvability in a bounded domain . For strong solutions, Galdi and Rionero  stated, without proof, the results of existence and uniqueness of strong solutions. Rojas-Medar  studied it and established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. In 1999, Ortega-Torres and Rojas-Medar  proved global existence of strong solutions with the small initial values. For weak solutions, Rojas-Medar and Boldrini  proved the existence of weak solutions, and in the 2D case, also proved the uniqueness of the weak solutions.
On the other hand, there are few existence results about the non-Newtonian magneto-micropolar fluid, i.e., the case. In a recent work, Gunzburger et al. studied the reduced problem (with both and ), and gave the global unique solvability of the first initial-boundary value problem in a bounded two or three-dimensional domain. Improved results are proved for the periodic boundary condition case.
In this paper, we will prove the global existence and uniqueness of the weak solutions for the full system (1.1)-(1.3) under the condition that . These results are based on the Galerkin method and a series of uniform estimates, which do not depend on the parameters.
Throughout this work, we use a standard notation (normed ) for Lebesgue -spaces, as well as (normed ) for the usual Sobolev spaces. As usual, denotes the set of all -functions with the compact support in Ω. Given and a Banach space X, we denote by Bochner spaces, which are equipped with the norm
We also introduce the following functional vector spaces:
We next introduce the definition of a weak solution for problems (1.1)-(1.3).
The following theorem gives the main results of this paper.
For latter use, let us state some useful inequalities.
Lemma 1.1 (See ) (Korn’s inequality)
Lemma 1.2 (See ) (On negative norm)
Lemma 1.3 (See )
By using Hölder’s inequality and the imbedding inequality, we could arrive at
Finally, the paper is organized as follows. In Section 2, we focus on the derivation of the priori estimates for the smooth solutions. On the bases of these estimates, in Section 3, we get the existence result with the help of the Galerkin method. The aim of Section 4 is to give the uniqueness criterion.
2 The priori estimates
for the first term on the right hand side, we compute by the divergence free conditions
where Hölder’s, Young’s inequality and (1.9), (1.10) have been used.
Gronwall’s inequality and estimate (2.1) now provide the bound
Reasoning similar to (2.3), we could find
For the second term on the right hand side of (2.7), we compute
Observing (2.5) and estimate (2.1), then Gronwall’s inequality yields
Now, we compute, by using Hölder’s, Young’s inequality and (1.9), (1.10)
Now, we compute, by (2.11)
Inserting (2.17)-(2.18) into (2.16), by appealing to Korn’s inequality, it follows that
3 Approximate solutions and existence result
In this section, we show the existence of a weak solution to the system (1.1)-(1.3) via the Galerkin approximations. For this purpose, we take the set formed by the eigenvectors , , of the Stokes operator and the set formed by the eigenvectors , , of the Laplace operator. According to the Appendix of , the functions form a basis in the space , and . Setting and , we construct the Galerkin approximations being of the form
The local solvability is guaranteed by the Carathéodory theorem, and the global unique solvability follows from the fact that
with a constant C that does not depend on k.
where . Therefore, by making use of the Aubin-Lions lemma (see Lions , Theorem 1.5.1), we have
Next, to complete the existence proof, we need to verify that
By Lemma 1.3, we have
and thus (3.9) follows.
Having the estimates
By using De Rahm’s theorem (see , Lemma 2.7), we obtain a function such that
Moreover, due to estimates (3.10),
Then, by Lemma 1.2, there is a generic constant C, depending only on the data such that
Now, we complete the proof of the existence part of Theorem 1.1.
4 Uniqueness criterion
so, we have
thus, we obtain
This completes the proof of the theorem.
The author declares that they have no competing interests.
The author completed the paper. The author read and approved the final manuscript.
The author would like to thank the referees for valuable comments and suggestions for improving this paper. This work was partially supported by the National NSF (Grant No. 10971080) of China.
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