Research

# Unique solvability for the non-Newtonian magneto-micropolar fluid

Changjia Wang

Author Affiliations

School of Science, Changchun University of Science and Technology, Changchun, 130022, P.R. China

Boundary Value Problems 2013, 2013:182  doi:10.1186/1687-2770-2013-182

 Received: 8 January 2013 Accepted: 24 July 2013 Published: 8 August 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

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

This paper is concerned about the existence and uniqueness of the weak solutions to the non-Newtonian magneto-micropolar fluid equations in , which are described by

(1.1)

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

(1.2)

(1.3)

where .

The theory of micropolar fluid was first proposed by Eringen [1] 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 [2] stated, without proof, the results of existence and uniqueness of strong solutions. Rojas-Medar [3] 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 [4] proved global existence of strong solutions with the small initial values. For weak solutions, Rojas-Medar and Boldrini [5] 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.[6] 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).

Definition 1.1 We say that is a weak solution to problems (1.1)-(1.3) if

(1.4)

satisfy

(1.5)

(1.6)

(1.7)

where the symbol denotes a generic duality pairing.

The following theorem gives the main results of this paper.

Theorem 1.1Letbe an open-bounded domain with a Lipschitz boundaryΩ. 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 [7]) (Korn’s inequality)

Let. Then there exists a constantsuch that

(1.8)

whereis open and bounded with a Lipschitz boundary.

Lemma 1.2 (See [8]) (On negative norm)

Let, and let. Then there exists a constantCsuch that

Lemma 1.3 (See [9])

Let. For each, there exists a constantsuch that

By using Hölder’s inequality and the imbedding inequality, we could arrive at

(1.9)

with

Here, if or . We will also apply the so-called multiplicative inequalities

(1.10)

with

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.

Setting in (1.5), in (1.6), in (1.7), and observing that , we obtain

Adding the identities above, noting that , and Korn’s inequality (1.8), we get

After choosing ε properly small, integrating over , , the Gronwall’s inequality yields that

(2.1)

where is a constant depending on the time T and , , .

Next, we derive the higher order estimates for ω and b. Setting in (1.6), we find

(2.2)

for the first term on the right hand side, we compute by the divergence free conditions

(2.3)

where Hölder’s, Young’s inequality and (1.9), (1.10) have been used.

Inserting (2.3) into (2.2), choosing and integrating over , , we have

(2.4)

since , we have

(2.5)

Gronwall’s inequality and estimate (2.1) now provide the bound

(2.6)

where is a constant depending on the time T, and .

Next, set in (1.7) to discover

(2.7)

Reasoning similar to (2.3), we could find

(2.8)

For the second term on the right hand side of (2.7), we compute

(2.9)

where we have used Hölder’s, Young’s inequality and (1.9), (1.10). Choosing ε and δ properly small, inserting (2.8)-(2.9) into (2.7) and integrating it over , , we find

(2.10)

Observing (2.5) and estimate (2.1), then Gronwall’s inequality yields

(2.11)

where is a constant depending on the time T, and .

Reasoning analogously to (2.6) and (2.11), it is easy to see that identity (1.6) with , (1.7) with , with the help of (2.6) and (2.11), guarantee the estimate

(2.12)

where , are both constants depending only on the time T and some norm of the initial values.

In fact (we here only take as an example), set in (1.7), we deduce that

(2.13)

Now, we compute, by using Hölder’s, Young’s inequality and (1.9), (1.10)

(2.14)

and

(2.15)

Combining (2.13)-(2.15), by choosing , we arrive at

noting that , so , and now estimate (2.1), (2.11) and Gronwall’s inequality imply the estimate of in (2.12).

In the following, we will derive the bound for . Setting in (1.5), we deduce that

Integrating it over , , by choosing and Korn’s inequality, we have

(2.16)

Now, we compute, by (2.11)

(2.17)

(2.18)

Inserting (2.17)-(2.18) into (2.16), by appealing to Korn’s inequality, it follows that

(2.19)

where depends on T and . Now, Gronwall’s inequality and (2.1) yield that

(2.20)

where depends on the time T, , and .

### 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 [7], the functions form a basis in the space , and . Setting and , we construct the Galerkin approximations being of the form

where , , solve the system of ordinary equations

(3.1)

(3.2)

(3.3)

Moreover, we require that , , satisfy the following initial conditions

(3.4)

The local solvability is guaranteed by the Carathéodory theorem, and the global unique solvability follows from the fact that

with upper bounds C that do not depend on k. Moreover, we have for , and the same estimates for all norms we have obtained in Section 2. More precisely, we have

(3.5)

with a constant C that does not depend on k.

Uniform estimates (3.5) imply that there exists a subsequence of , and (not relabeled) such that

where . Therefore, by making use of the Aubin-Lions lemma (see Lions [10], Theorem 1.5.1), we have

With the convergence above, it is easy to pass to the limit as in (3.1)-(3.3) to find

(3.6)

(3.7)

(3.8)

Next, to complete the existence proof, we need to verify that

(3.9)

By Lemma 1.3, we have

Considering of this identity together with (3.6) implies that

and thus (3.9) follows.

Having the estimates

(3.10)

we can now introduce the pressure from (1.5). For , define the functional as

We have

By using De Rahm’s theorem (see [11], 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

Let and be both solutions of the problem. Then, for their difference , , , we have

(4.1)

(4.2)

(4.3)

Taking in (4.1), by Lemma 1.3 and the fact that and , we obtain

for each , , it follows from Hölder’s, Young’s inequality and (1.9), (1.10) that

so, we have

(4.4)

Taking in (4.2) and noting that , it follows

and for , ,

thus, we obtain

(4.5)

Similarly, by taking in (4.3), reasoning analogous as above, we could get

(4.6)

Adding (4.4)-(4.6) and observing that , after choosing ε properly small, we finally get

with

Since for , then Gronwall’s inequality and the estimates obtained in Section 2 yield that

This completes the proof of the theorem.

### Competing interests

The author declares that they have no competing interests.

### Authors’ contributions

The author completed the paper. The author read and approved the final manuscript.

### Acknowledgements

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.

### References

1. Eringen, AC: Theory of micropolar fluids. J. Math. Mech.. 16, 1–18 (1966)

2. Galdi, GP, Rionero, S: A note on the existence and uniqueness of solutions of the micropolar fluid equations. Int. J. Eng. Sci.. 15, 105–108 (1977). PubMed Abstract | Publisher Full Text

3. Rojas-Medar, MA: Magneto-micropolar fluid motion: existence and uniqueness of strong solution. Math. Nachr.. 188, 301–319 (1997). Publisher Full Text

4. Ortega-Torres, EE, Rojas-Medar, MA: Magneto-micropolar fluid motion: global existence of strong solutions. Abstr. Appl. Anal.. 4, 109–125 (1999). Publisher Full Text

5. Rojas-Medar, MA, Boldrini, JL: Magneto-micropolar fluid motion: existence of weak solutions. Rev. Mat. Complut.. 11, 443–460 (1998)

6. 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.. 6, 462–482 (2004)

7. Málek, J, Něcas, J, Rokyta, M, Ružička, M: Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London (1996)

8. Bellout, H, Bloom, F, Nečas, J: Solutions for incompressible non-Newtonian fluids. C. R. Math. Acad. Sci. Paris, Sér. I. 317, 795–800 (1993)

9. Ebmeyer, C, Urbano, JM: Quasi-steady Stokes flow of multiphase fluids with shear-dependent viscosity. Eur. J. Appl. Math.. 18, 417–434 (2007). Publisher Full Text

10. Lions, JL: Quelques méthodes de Résolution de Problèmes aux limites non Linéaires, Dunod, Paris (1969).

11. Amrouche, C, Girault, V: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslov. Math. J.. 44, 109–140 (1994)