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A Regularity Criterion for the Nematic Liquid Crystal Flows
Journal of Inequalities and Applications volume 2010, Article number: 589697 (2010)
A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.
is the velocity field of the flow. is the (averaged) macroscopic/continuum molecular orientations vector in . is a scalar function representing the pressure (including both the hydrostatic part and the induced elastic part from the orientation field). is a positive viscosity constant. The constant represents the competition between kinetic energy and potential energy. The constant is the microscopic elastic relaxation time (Deborah number) for the molecular orientation field. . For simplicity, we will take . The matrix is defined by . is the usual Kronecker multiplication, for example, for .
Very recently, results for the local existence of classical solutions for the problems (1.1)–(1.4) were presented in . The aim of this paper is to establish a regularity criterion for it. We will prove the following.
Let with in . Suppose that a local smooth solution satisfies
Then can be extended beyond .
Equation (1.5) can be regarded as a logarithmically improved regularity criterion of the form with . Condition (1.5) only involves the velocity field , which plays a dominant role in regularity theorem. Similar phenomenon already appeared in the studies of MHD equations (see [4–6] for details).
A natural region for in (1.5) should be , but we only can prove it for here. We are unable to establish any other regularity criterion in terms of or .
2. Proof of Theorem 1.1
Since we deal with the regularity conditions of the local smooth solutions, we only need to establish the needed a priori estimates. We mainly will follow the method introduced in .
First, it has been proved in  that
Multiplying (1.3) by , integration by parts yields
Thanks to (2.1), (2.2), and the Gronwall inequality, we get
Let and , then the th () component of satisfies
Multiplying (2.5) by , after integration by parts, summing over , and using (1.2), we find that
Applying on (1.3), multiplying it by , and using (1.2), we have
Combining (2.6) and (2.7) together, noting that , , we deduce that
We do estimates for () as follows:
Here we have used the following Gagliardo-Nirenberg inequality:
Similarly, by using (2.10), we have
is simply bounded as follows:
for any .
When or , can be estimated easily and hence omitted here. If , we do estimates as follows:
for any . Here we have used the Gagliardo-Nirenberg inequality:
Finally, we omit the trivial term
Now, putting the above estimates for s into (2.8) and taking small enough, we obtain
Due to the integrability of (1.5), we conclude that for any small constant , there exists a time such that
Easily, from (2.16) and (2.17) it follows that
which implies that for ,
We are going to do the estimate for and . To this end, we introduce the following commutator estimates due to the work of Kato and Ponce :
where , for , and .
Applying to (2.5) and multiplying it by , after integration by parts, and summing over yield
Applying to (1.3), multiplying it by , we deduce that
Summing up (2.22) and (2.23), using , we have
Now we estimate each term as follows.
By using (2.20), we estimate as
here we used the following Gagliardo-Nirenberg inequalities:
Using (2.21), we estimate as
for any . Here we have used the following Gagliardo-Nirenberg inequalities:
only involves lower derivatives of and is easy to handle, so we omit it here:
for any . Here we have used
for any . Where we have used the following inequality
By using (2.20), we estimate as follows:
for any . Here we have used
The term is trivial, and we omit it here:
for any . Where we have used the following inequality:
Finally, using (2.26), can be bounded as follows:
for any . Now, inserting the above estimates for s into (2.24), using (2.19), and taking be small enough, we get
This completes the proof.
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The authors thank the referee for his/her careful reading and helpful suggestions. This work is partially supported by Zhejiang Innovation Project (Grant no. T200905), NSF of Zhejiang (Grant no. R6090109), and NSF of China (Grant no. 10971197).
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Zhou, Y., Fan, J. A Regularity Criterion for the Nematic Liquid Crystal Flows. J Inequal Appl 2010, 589697 (2010). https://doi.org/10.1155/2010/589697
- Natural Region
- Local Existence
- Crystal Material
- Regularity Criterion
- Part Yield