Abstract
In this paper, a motion of an incompressible nonNewtonian magnetomicropolar fluid
is considered. We assume that the stress tensor has a pstructure, and we establish the global in time existence and uniqueness of the weak
solutions with
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
here
To (1.1) we append the following initial and boundary conditions
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 NavierStokes equations for the viscous incompressible fluid, for example, the motion of animal blood, liquid crystals and dilute aqueous polymer solutions, etc.
If
On the other hand, there are few existence results about the nonNewtonian magnetomicropolar
fluid, i.e., the
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
Throughout this work, we use a standard notation
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
satisfy
where the symbol
The following theorem gives the main results of this paper.
Theorem 1.1Let
Remark 1.1 If (1.4)(1.5) hold, it could be easy to introduce the pressure
For latter use, let us state some useful inequalities.
Lemma 1.1 (See [7]) (Korn’s inequality)
Let
where
Lemma 1.2 (See [8]) (On negative norm)
Let
Lemma 1.3 (See [9])
Let
By using Hölder’s inequality and the imbedding inequality, we could arrive at
with
Here,
with
If
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
Setting
Adding the identities above, noting that
After choosing ε properly small, integrating over
where
Next, we derive the higher order estimates for ω and b. Setting
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
since
Gronwall’s inequality and estimate (2.1) now provide the bound
where
Next, set
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
Observing (2.5) and estimate (2.1), then Gronwall’s inequality yields
where
Reasoning analogously to (2.6) and (2.11), it is easy to see that identity (1.6) with
where
In fact (we here only take
Now, we compute, by using Hölder’s, Young’s inequality and (1.9), (1.10)
and
Combining (2.13)(2.15), by choosing
noting that
In the following, we will derive the bound for
Integrating it over
Now, we compute, by (2.11)
Inserting (2.17)(2.18) into (2.16), by appealing to Korn’s inequality, it follows that
where
where
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
where
Moreover, we require that
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
with a constant C that does not depend on k.
Uniform estimates (3.5) imply that there exists a subsequence of
where
With the convergence above, it is easy to pass to the limit as
Next, to complete the existence proof, we need to verify that
By Lemma 1.3, we have
Considering
and thus (3.9) follows.
Having the estimates
we can now introduce the pressure from (1.5). For
We have
By using De Rahm’s theorem (see [11], Lemma 2.7), we obtain a function
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
Taking
for each
so, we have
Taking
and for
thus, we obtain
Similarly, by taking
Adding (4.4)(4.6) and observing that
with
Since
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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