Unique solvability for the non-Newtonian magneto-micropolar fluid
© Wang; licensee Springer 2013
Received: 8 January 2013
Accepted: 24 July 2013
Published: 8 August 2013
In this paper, a motion of an incompressible non-Newtonian magneto-micropolar fluid is considered. We assume that the stress tensor has a p-structure, and we establish the global in time existence and uniqueness of the weak solutions with in three dimensions.
1 Introduction and main results
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.
We next introduce the definition of a weak solution for problems (1.1)-(1.3).
where the symbol denotes a generic duality pairing.
The following theorem gives the main results of this paper.
Theorem 1.1 Let be an open-bounded domain with a Lipschitz boundary ∂ Ω. Assume that , , . Then, for , there exists a unique weak solution to problem (1.1)-(1.3) in the sense of Definition 1.1.
Remark 1.1 If (1.4)-(1.5) hold, it could be easy to introduce the pressure , . This will be done at the end of Section 3.
For latter use, let us state some useful inequalities.
Lemma 1.1 (See ) (Korn’s inequality)
where is open and bounded with a Lipschitz boundary.
Lemma 1.2 (See ) (On negative norm)
Lemma 1.3 (See )
If or , then .
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
Let be a smooth solution to system (1.1)-(1.3). The goal of this section is to derive some priori estimates about it. In all the following sections, we always assume that holds.
where is a constant depending on the time T and , , .
where Hölder’s, Young’s inequality and (1.9), (1.10) have been used.
where is a constant depending on the time T, and .
where is a constant depending on the time T, and .
where , are both constants depending only on the time T and some norm of the initial values.
noting that , so , and now estimate (2.1), (2.11) and Gronwall’s inequality imply the estimate of in (2.12).
where depends on the time T, , and .
3 Approximate solutions and existence result
with a constant C that does not depend on k.
and thus (3.9) follows.
Now, we complete the proof of the existence part of Theorem 1.1.
4 Uniqueness criterion
This completes the proof of the theorem.
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.
- Eringen AC: Theory of micropolar fluids. J. Math. Mech. 1966, 16: 1-18.MathSciNetGoogle Scholar
- Galdi GP, Rionero S: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int. J. Eng. Sci. 1977, 15: 105-108. 10.1016/0020-7225(77)90025-8MathSciNetView ArticleGoogle Scholar
- Rojas-Medar MA: Magneto-micropolar fluid motion: existence and uniqueness of strong solution. Math. Nachr. 1997, 188: 301-319. 10.1002/mana.19971880116MathSciNetView ArticleGoogle Scholar
- Ortega-Torres EE, Rojas-Medar MA: Magneto-micropolar fluid motion: global existence of strong solutions. Abstr. Appl. Anal. 1999, 4: 109-125. 10.1155/S1085337599000287MathSciNetView ArticleGoogle Scholar
- Rojas-Medar MA, Boldrini JL: Magneto-micropolar fluid motion: existence of weak solutions. Rev. Mat. Complut. 1998, 11: 443-460.MathSciNetView ArticleGoogle Scholar
- Gunzburger MD, Ladyzhenskaya OA, Peterson JS: On the global unique solvability of initial-boundary value problems for the coupled modified Navier-Stokes and Maxwell equations. J. Math. Fluid Mech. 2004, 6: 462-482.MathSciNetGoogle Scholar
- Málek J, Něcas J, Rokyta M, Ružička M: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, London; 1996.View ArticleGoogle Scholar
- Bellout H, Bloom F, Nečas J: Solutions for incompressible non-Newtonian fluids. C. R. Math. Acad. Sci. Paris, Sér. I 1993, 317: 795-800.Google Scholar
- Ebmeyer C, Urbano JM: Quasi-steady Stokes flow of multiphase fluids with shear-dependent viscosity. Eur. J. Appl. Math. 2007, 18: 417-434. 10.1017/S0956792507006948MathSciNetView ArticleGoogle Scholar
- Lions JL: Quelques méthodes de Résolution de Problèmes aux limites non Linéaires. Dunod, Paris; 1969.Google Scholar
- Amrouche C, Girault V: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslov. Math. J. 1994, 44: 109-140.MathSciNetGoogle Scholar