A Regularity Criterion for the Nematic Liquid Crystal Flows

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 [4–6] 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 [7–10]. 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.