Open Access

A Regularity Criterion for the Nematic Liquid Crystal Flows

Journal of Inequalities and Applications20102010:589697

https://doi.org/10.1155/2010/589697

Received: 25 September 2009

Accepted: 16 April 2010

Published: 24 May 2010

Abstract

A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.

1. Introduction

We consider the following hydrodynamical systems modeling the flow of nematic liquid crystal materials ([1, 2]):
(1.1)
(1.2)
(1.3)
(1.4)
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 [3]. The aim of this paper is to establish a regularity criterion for it. We will prove the following.

Theorem 1.1.

Let with in . Suppose that a local smooth solution satisfies
(1.5)

Then can be extended beyond .

Remark 1.2.

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 [46] for details).

Remark 1.3.

When in (1.1), then (1.1) and (1.2) are the well-known Navier-Stokes equations. Similar conditions to (1.5) have been established in [710]. But previous methods can not be used here.

Remark 1.4.

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 [9].

First, it has been proved in [3] that
(2.1)
Hence
(2.2)
Multiplying (1.3) by , integration by parts yields
(2.3)
Thanks to (2.1), (2.2), and the Gronwall inequality, we get
(2.4)
Let and , then the th ( ) component of satisfies
(2.5)
Multiplying (2.5) by , after integration by parts, summing over , and using (1.2), we find that
(2.6)
Applying on (1.3), multiplying it by , and using (1.2), we have
(2.7)
Combining (2.6) and (2.7) together, noting that , , we deduce that
(2.8)
We do estimates for ( ) as follows:
(2.9)
Here we have used the following Gagliardo-Nirenberg inequality:
(2.10)
Similarly, by using (2.10), we have
(2.11)
is simply bounded as follows:
(2.12)

for any .

When or , can be estimated easily and hence omitted here. If , we do estimates as follows:
(2.13)
for any . Here we have used the Gagliardo-Nirenberg inequality:
(2.14)
Finally, we omit the trivial term
(2.15)
Now, putting the above estimates for s into (2.8) and taking small enough, we obtain
(2.16)
Due to the integrability of (1.5), we conclude that for any small constant , there exists a time such that
(2.17)
Easily, from (2.16) and (2.17) it follows that
(2.18)
which implies that for ,
(2.19)
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 [11]:
(2.20)
(2.21)

where , for , and .

Applying to (2.5) and multiplying it by , after integration by parts, and summing over yield
(2.22)
Applying to (1.3), multiplying it by , we deduce that
(2.23)
Summing up (2.22) and (2.23), using , we have
(2.24)

Now we estimate each term as follows.

By using (2.20), we estimate as
(2.25)
here we used the following Gagliardo-Nirenberg inequalities:
(2.26)
Using (2.21), we estimate as
(2.27)
for any . Here we have used the following Gagliardo-Nirenberg inequalities:
(2.28)
only involves lower derivatives of and is easy to handle, so we omit it here:
(2.29)
for any . Here we have used
(2.30)
for any . Where we have used the following inequality
(2.31)
By using (2.20), we estimate as follows:
(2.32)
for any . Here we have used
(2.33)
The term is trivial, and we omit it here:
(2.34)
for any . Where we have used the following inequality:
(2.35)
Finally, using (2.26), can be bounded as follows:
(2.36)
for any . Now, inserting the above estimates for s into (2.24), using (2.19), and taking be small enough, we get
(2.37)

This completes the proof.

Declarations

Acknowledgments

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).

Authors’ Affiliations

(1)
Department of Mathematics, Zhejiang Normal University
(2)
Department of Applied Mathematics, Nanjing Forestry University
(3)
Department of Mathematics, Hokkaido University

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Copyright

© Y. Zhou and J. Fan. 2010

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