Abstract
1 Introduction and main results
This paper is concerned about the existence and uniqueness of the weak solutions to the nonNewtonian magnetomicropolar fluid equations in , which are described by
here is an openbounded domain with Lipschitz boundaries, and the unknowns u, ω, b, π denote the velocity of the fluid, the microrotational 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 [1] in 1966, which enabled us to consider some physical phenomena that cannot be treated by the classical NavierStokes 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 magnetomicropolar 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. RojasMedar [3] studied it and established the local in time existence and uniqueness of strong solutions by the spectral Galerkin method. In 1999, OrtegaTorres and RojasMedar [4] proved global existence of strong solutions with the small initial values. For weak solutions, RojasMedar 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 nonNewtonian magnetomicropolar 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 initialboundary value problem in a bounded two or threedimensional 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
satisfy
where the symbol denotes a generic duality pairing.
The following theorem gives the main results of this paper.
Theorem 1.1Letbe an openbounded 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 [7]) (Korn’s inequality)
Let. Then there exists a constantsuch that
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
with
Here, if or . We will also apply the socalled multiplicative inequalities
with
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
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
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.
Inserting (2.3) into (2.2), choosing and integrating over , , we have
Gronwall’s inequality and estimate (2.1) now provide the bound
where is a constant depending on the time T, and .
Next, set in (1.7) to discover
Reasoning similar to (2.3), we could find
For the second term on the right hand side of (2.7), we compute
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
Observing (2.5) and estimate (2.1), then Gronwall’s inequality yields
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
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
Now, we compute, by using Hölder’s, Young’s inequality and (1.9), (1.10)
and
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
Now, we compute, by (2.11)
Inserting (2.17)(2.18) into (2.16), by appealing to Korn’s inequality, it follows that
where depends on T and . Now, Gronwall’s inequality and (2.1) yield 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 [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
Moreover, we require that , , satisfy the following initial conditions
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
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 AubinLions 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
Next, to complete the existence proof, we need to verify that
By Lemma 1.3, we have
Considering of this identity together with (3.6) implies that
and thus (3.9) follows.
Having the estimates
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
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
Taking in (4.2) and noting that , it follows
thus, we obtain
Similarly, by taking in (4.3), reasoning analogous as above, we could get
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.
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