Abstract
In this paper, the author establishes a blowup criterion of strong solutions to 3D
compressible viscous magnetomicropolar fluids. It is shown that if the density and
the velocity satisfy
MSC: 76N10, 35B44, 35B45.
Keywords:
compressible magnetomicropolar fluids; blowup criterion; strong solution; vacuum1 Introduction
In this paper, we consider the following 3D compressible viscous magnetomicropolar fluids:
where
System (1.1)(1.2) describing the motion of aggregates of small solid ferromagnetic particles relative to viscous magnetic fluids, such as water, hydrocarbon, ester, fluorocarbon, etc., in which they are immersed, covers a wide range of heat and mass transfer phenomena, under the action of magnetic fields, and is of great importance in practical and mathematics applications (see [1]). Indeed, (1.1) is composed of the balance laws of mass, momentum, moment of momentum and magnetohydrodynamic, respectively. Due to its importance in mathematics and physics, there is a lot of literature devoted to the mathematical theory of the compressible viscous magnetomicropolar system (see [24]).
For the incompressible magnetomicropolar fluid models where
In particular, if the effect of angular velocity field of the particle’s rotation
is omitted, i.e.,
where
If
where r, s satisfy (1.3).
If
or
where r, s satisfy (1.3).
In this paper, our main purpose is to establish a blowup criterion of strong solutions for system (1.1) with the following conditions:
To proceed, we introduce the following notations. For
To present the main result, we first give the following local existence and uniqueness of strong solutions to the Cauchy problem (1.1), (1.2) and (1.4) with initial vacuum (without proof), which can be obtained by the same method developed by ChoeKim in [22] (see also FanYu [11] and Chen [19] for MHD and compressible micropolar fluids, respectively).
Theorem 1.1Assume that for some
and the compatibility conditions
with some
Motivated by [20,21] and [12], we have the main purpose in this paper to prove a blowup criterion for the problem (1.1), (1.2) and (1.4). More precisely, the main result in this paper reads as follows.
Theorem 1.2Assume that the initial data
for anyrandssatisfying (1.3).
Remark 1.3 Theorem 1.1 proves that the strong solutions of (1.1), (1.2) and (1.4) can exist
only in a small time
Remark 1.4 There is no any additional growth condition on the microrotational velocity w and magnetic field H. This reveals that the density and the linear velocity play a more important role compared to the angular velocity of rotation of particles and the magnetic field in the regularity theory of solutions to 3D compressible magnetomicropolar fluid flows.
The rest of the paper is devoted to completing the proof of Theorem 1.2.
2 Proof of Theorem 1.2
First, we give the following wellknown GagliardoNirenberg inequality that will be used frequently.
Lemma 2.1For
The following BKM’s type inequality which will be used to estimate
Lemma 2.2For
The proof of Theorem 1.2 is based on the contradiction arguments. Let
where r, s satisfy (1.3) and
One can easily deduce from the following energy estimate (1.1), (1.2) and (1.4).
Lemma 2.3It holds that
Here and hereafter, C denotes a generic positive constant which may depend onμ,
We denote the material derivative of f by
Since
Thus, from the standard
Lemma 2.4Under the condition (2.4), it holds that
Proof In view of standard
which, combined with (2.8), yields (2.9) immediately. □
The next lemma is concerned with the higher integrability of H under the assumption (2.4).
Lemma 2.5Under the condition (2.4), it holds for any
where
The proof is similar to Lemma 3.3 in [12] and is omitted here.
With the help of (2.4) and Lemmas 2.32.5, we can prove the following key lemma.
Lemma 2.6Under the condition (2.4), it holds that for any
Proof Multiplying (1.1)_{2}, (1.1)_{3} and (1.1)_{4} by
To estimate the first term on the righthand side of (2.12), we observe that P satisfies
Hence, using (2.4), (2.5) and (2.10) yields that
where we have used Young’s inequality and (2.1).
For the second term, we have, after integration by parts, that
and by the CauchySchwarz inequality, we have
Similarly, integrating by parts and using the fact
For the last three terms on the righthand side of (2.12), one has from (2.4) that
Thus, putting (2.13)(2.17) into (2.12) and choosing
For any r, s satisfying (1.3), we have by the Hölder and Sobolev inequalities that
for some
Taking
By the standard
Furthermore, it follows from (1.1)_{4} and Sobolev’s embedding inequality that
putting (2.21) and (2.22) into (2.20), such that
which, together (2.21) and (2.18), choosing
It is easily seen that
Taking this into account, we conclude from (2.4), (2.24) and Gronwall’s inequality
that part of (2.11) holds for any
Next we prove the boundedness of
Lemma 2.7Under the condition (2.4), it holds that for any
Proof Applying the operator
We get after integration by parts that
Similarly, we also have
After integration by parts, using (1.1)_{1} and (2.11), we obtain
Using the definition of the material derivation and integrating by parts, we deduce from (2.1), (2.5) and (2.11) that
and, similarly,
and
The ninth term on the righthand side of (2.26) can be estimated as follows, integrating by parts, using (2.1), (2.5), (2.10), (2.11) and Hölder’s inequality:
In a similar manner, one also has
Putting (2.27)(2.34) into (2.26), using the CauchySchwarz inequality and choosing
To estimate
Integrating by parts and using (1.1)_{5}, (2.1), (2.10) and (2.11), then we deduce
for some positive constants
Putting the estimates of
Then, combining (2.35) and (2.37), using Young’s inequality, and choosing
Firstly, we use (2.4)(2.6), (2.9), (2.10), (2.11), (2.1) and (2.2) to infer from
the standard
and
Moreover, by the standard
and hence,
Combining (2.39)(2.41), we obtain
Now, putting (2.41) and (2.42) into (2.38), one has
from which and (2.11), we immediately obtain (2.25) by Gronwall’s inequality, (1.6)
and (1.7). As a result of (2.41), we can also deduce the boundedness of
The next lemma is used to bound the density gradient and
Lemma 2.8Under the condition (2.4), it holds that for any
for any
Proof Differentiating (1.1)_{1} with respect to
It follows from (2.1), (2.4), (2.6)(2.9), (2.25) and the interpolation inequality
that for any
where (2.2) and (2.25) were also used to get that
We now estimate
Hence, using the standard
From (1.1)_{3} and the standard
and then
This, together with Lemmas 2.2 and 2.6, gives
Now, if we set
then it is seen from (2.45) and (2.48) that
due to
On the other hand, since
we thus deduce from (2.11), (2.25), (2.4), (2.8), (2.9) and (2.2) that
As a result, it follows from (2.49) and Gronwall’s inequality that
and consequently,
From this and (2.25), (2.48), (2.50), one obtains
Taking
Moreover, the standard
Similar to the proof of (2.47), there are
where we have used (2.1), (2.11), (2.46), (2.51) and (2.54). From this, together with (2.25), (2.46) and (2.51)(2.54), we can deduce (2.43). □
As a consequence of Lemmas 2.62.8, we have the following lemma.
Lemma 2.9Under the condition (2.4), it holds that for any
The proof is the same as that of Lemma 3.6 in [20] and is omitted here.
With the help of Lemmas 2.3, 2.62.9 and the local existence theorem, we can complete
the proof of Theorem 1.2 by the contradiction arguments. In fact, in view of Lemmas
2.3, 2.62.9, it is easy to see that the functions
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 11QZR16), the National Natural Science Foundation of China (Grant No. 11001090).
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