In this paper we prove some blow-up criteria for two 3D density-dependent nematic liquid crystal models in a bounded domain.
MSC: 35Q30, 76D03, 76D09.
Keywords:liquid crystal; blow-up criterion; bounded domain
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 density-dependent incompressible nematic liquid crystal model as follows [1-4]:
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 , Guillén-González et al. proved the blow-up criterion
It is easy to prove that the problem (1.1)-(1.6) has a unique local-in-time 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.
Remark 1.2 By similar calculations as those in , 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 global-in-time strong solution by the same method as that in , 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  proved that the problem has a unique local strong solution. When , Fan et al. 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.
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
Summing up (2.2) and (2.3), we have the well-known energy inequality
Next, we prove the following estimate:
In the following calculations, we use the following Gauss-Green formula :
Taking ∇ to (1.4)i, we deduce that
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
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
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
Summing up (2.2) and (3.2), we have the well-known energy inequality
We still have (2.10) and (2.11).
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).
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.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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.