Abstract
In this paper we prove some blowup criteria for two 3D densitydependent nematic liquid crystal models in a bounded domain.
MSC: 35Q30, 76D03, 76D09.
Keywords:
liquid crystal; blowup criterion; bounded domain1 Introduction
Let be a bounded domain with smooth boundary ∂Ω, and let ν be the unit outward normal vector on ∂Ω. We consider the regularity criterion to the densitydependent incompressible nematic liquid crystal model as follows [14]:
in with initial and boundary conditions
where ρ denotes the density, u the velocity, π the pressure, and d represents the macroscopic molecular orientations, respectively. The symbol denotes a matrix whose th entry is , and it is easy to find that .
When d is a given constant vector, then (1.1)(1.3) represent the wellknown densitydependent NavierStokes system, which has received many studies; see [57] and references therein.
When , GuillénGonzález et al.[8] proved the blowup criterion
It is easy to prove that the problem (1.1)(1.6) has a unique localintime strong solution [6,9], and thus we omit the details here. The aim of this paper is to consider the regularity criterion; we will prove the following theorem.
Theorem 1.1Let, , withand, in Ω andon∂Ω. We also assume that the following compatibility condition holds true: such that
Letbe a local strong solution to the problem (1.1)(1.6). Ifusatisfies
and, then the solutioncan be extended beyond.
Remark 1.1 When , our result improves (1.7) to (1.8).
Remark 1.2 By similar calculations as those in [6], we can replace norm in (1.8) by norm, and thus we omit the details here.
Remark 1.3 When the space dimension , we can prove that the problem (1.1)(1.6) has a unique globalintime strong solution by the same method as that in [10], and thus we omit the details here.
Next we consider another liquid model: (1.1), (1.2), (1.3), (1.5), (1.6) and
with in . Li and Wang [9] proved that the problem has a unique local strong solution. When , Fan et al.[11] proved a regularity criterion. The aim of this paper is to study the regularity criterion of the problem in a bounded domain. We will prove the following theorem.
Theorem 1.2Let the initial data satisfy the same conditions in Theorem 1.1 andin Ω. Letbe a local strong solution to the problem (1.1)(1.3), (1.5), (1.6) and (1.9). Ifuand ∇dsatisfy
2 Proof of Theorem 1.1
We only need to establish a priori estimates.
Below we shall use the notation
First, thanks to the maximum principle, it follows from (1.1) and (1.2) that
Testing (1.3) by u and using (1.1) and (1.2), we see that
Testing (1.4) by and using (1.1), we find that
Summing up (2.2) and (2.3), we have the wellknown energy inequality
Next, we prove the following estimate:
Without loss of generality, we assume that . Multiplying (1.4) by 2d, we get
with and and on . Then (2.5) follows from (2.6) by the maximum principle.
In the following calculations, we use the following GaussGreen formula [12]:
and the following estimate [13,14]:
Taking ∇ to (1.4)_{i}, we deduce that
Testing the above equation by (), using (1.1), (2.7), (2.8), (2.5) and summing over i, we derive
which gives
Therefore,
Testing (1.3) by , using (1.1), (1.2), (2.1) and (2.9), we have
By the regularity theory of the Stokes system, it follows from (1.3) that
which yields
Testing (1.4) by , using (2.5) and (2.9), we obtain
On the other hand, by the regularity theory of the elliptic equation, from (1.4), (2.5) and (2.9) we infer that
which gives
Combining (2.11) and (2.13), we have
Putting (2.14) into (2.10) and (2.12) and summing up, we arrive at
which leads to
It follows from (2.14), (2.15) and (2.16) that
Taking to (1.3), testing by , using (1.1), (1.2) and (2.15), we have
Taking to (1.4), testing by , using (2.5), (2.15), (2.16) and (2.17), we arrive at
Combining (2.18) and (2.19), we have
It follows from (1.4), (2.21) and (2.16) that
It follows from (2.14), (2.15), (2.20) and (2.21) that
It follows from (1.3), (2.20) and (2.23) that
from which it follows that
Applying ∇ to (1.2), testing by () and using (2.25), we have
which implies
and therefore
This completes the proof.
3 Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. We only need to establish a priori estimates.
First, we still have (2.1) and (2.2).
Next, we easily infer that
Testing (1.9) by and using (1.1) and (3.1), we find that
Summing up (2.2) and (3.2), we have the wellknown energy inequality
Taking ∇ to (1.9)_{i}, testing by (), using (1.1), (2.7), (2.8) and (3.1), similarly to (2.9), we deduce that
which yields
We still have (2.10) and (2.11).
Similarly to (2.12), testing (1.9) by , using (3.1) and (3.4), we get
Similarly to (2.13), we have
Combining (2.11) and (3.6), we have
Putting (3.7) into (3.5) and (2.10) and using the Gronwall inequality, we still have (2.15), (2.16), (2.17) and (2.18).
Similarly to (2.19), applying to (1.9), testing by and using (3.4), we have
Combining (2.18) and (3.8) and using the Gronwall inequality, we still obtain (2.20) and (2.21).
By similar calculations as those in (2.22)(2.27), we still arrive at (2.22)(2.27).
This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
This work is partially supported by the Zhejiang Innovation Project (Grant No. T200905), the ZJNSF (Grant No. R6090109) and the NSFC (Grant No. 11171154). The authors are indebted to the referee for some helpful suggestions.
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