Abstract and Applied Analysis

Volume 2013 (2013), Article ID 253260, 10 pages

http://dx.doi.org/10.1155/2013/253260

Research Article

On the Incompressible Limit for the Compressible Flows of Liquid Crystals under Strong Stratification on Bounded Domains

Young-Sam Kwon

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 10 October 2012; Accepted 12 January 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Young-Sam Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the incompressible limit of weak solutions for the compressible flows of liquid crystals under strong stratification on bounded domains.

1. Introduction

Liquid crystals flows can be found in the natural world and in technological applications with a variety of examples such as many proteins, cell membranes, and solutions of soap and various related detergents, as well as the tobacco mosaic virus. We here consider the compressible flows of liquid crystals:

where is the vector field, is the density, is the direction field for the averaged macroscopic molecular orientations, is a given potential, and are viscosities. The smooth vector and the smooth scalar function are related by

where represents the penalty function and is the bulk part of the elastic energy verifying: there exists such that if ,

Note that the condition (5) is required for the global weak solution. We can see an example as follows:

where is a constant. Finally, the pressure is defined as follows

and the function is the unique positive solution of the following problem:

We use the following scaling used in Feireisl et al. [1] and Wang and Yu [2]:

and for the viscosity coefficients,

with the convergence of the viscosity coefficients

Finally for the pressure, we use

Following the above scalings, the system (1)–(7) reads

We now notice that the global-in-time existence solutions for system ((1)–(3)) have been studied by Wang and Yu [3] and Liu and Qing [4]. For the case of , incompressible limit problems have been investigated by many authors, starting with the work by Klainerman and Majda [5] for the Euler equations and Lions and Masmoudi [6] for the isentropic Navier Stokes equations. Similar results in the spirit of the analysis presented by Lions and Masmoudi [6] are the recent progress by Feireisl and Novotný [7, 8] and Kwon and Trivisa [9] for the full Navier-Stokes Fourier system. For the liquid crystals, there is one recent progress by Wang and Yu [2] based on the spectral analysis and Duhamel's principle to control difficulties arising in the boundary of bounded domains. There are some results of incompressible limit problems of Navier Stoke Fourier system under strong stratification by Feireisl and Novotný [7] on bounded domain and by Feireisl et al. on unbounded domains, which is extended to full magnetohydrodynamic flows on bounded domain by Novotný et al. [10] and by Lee et al. [11].

For this kind of application, the polymer precursor leads to a stratified structure of a solidified film of polymer and so we have a natural question: is such fluid almost incompressible such that it is strongly stratified when as the different models of compressible fluid? In this paper, we derive the rigorous result of the incompressible limit of the flow of liquid crystals with the similar idea used in the previous results [7, 10–12]. Formally, we will investigate the limit

as tends to in the suitable sense such that the given limit represents a solution of the following system:

Finally, in this paper, we will use the many parts of the presentation of Feireisl and Novotný [7] and Feireisl et al. [12] without modification.

The outline of this paper is as follows: In Section 2 we present two initial-boundary-value problems and introduce the notion of weak solutions for the compressible fluid of liquid crystals. In Section 3 we present the main results of the article on the low Mach number problems under strong stratification on bounded domains. In Section 4, we present the proof of the low Mach number problem for bounded domains.

2. Weak Solutions

2.1. An Initial-Boundary-Value Problem

Let be a bounded domain with the boundary of class

where stands for the outer normal vector. We also propose the boundary condition on the direction vector

Notice that Wang and Yu showed the global weak solution of the system (1)–(3) with the Dirichlet boundary condition on bounded domains but there will be no problem with the boundary conditions (18) and (19) for existence result.

2.2. Weak Solutions

Definition 1. We say that a quantity is a weak solution of the compressible flows of liquid crystals (13)–(15) supplemented with the initial data provided that the following hold.

(i) The density is a nonnegative function, , the velocity field , , and represents a renormalized solution of (1) on a time-space cylinder , that is, the integral identity

 holds for any test function and any such that

(ii)The balance of momentum holds in distributional sense, namely,

 for any test function satisfying , where is defined by

(iii)The total energy of the system holds:

 holds for a.e. , where

 with

(iv)The equation of direction field verifies

 for all .

Let us now discuss the static states which are solutions of system (13)–(15) with vanishing velocity field. In the present setting, the positive density must satisfy

From the statistic equation (28), we can easily derive

with

We now introduce the weak solutions of the target system.

Definition 2. A couple is said to be a weak solution of the target system of the compressible flows of liquid crystals with the potential , supplemented with the boundary conditions

on which belongs to , and the initial conditions

with ,  ,   if the following conditions hold:

(i) ,   a.e. on , in the sense of traces, and the integral identity

 holds for any test function

(ii) , and the integral identity

 holds for all .

3. Main Results

In this section we mention the main result as follows.

Theorem 3. Let be a bounded domain with a boundary of class and a family of weak solutions to the compressible of liquid crystals system verifying (5) in the sense of Definition 1 with . Assume that the initial condition is as follows:

Then, up to subsequence,

where solves a weak solution of the incompressible flows of liquid crystals in the sense of Definition 2 with the boundary condition and the initial data

where the Helmholtz projection and is defined in (81).

4. Proof of Theorem 3

4.1. Uniform Bounds

In this section we are going to derive some estimates on the sequence . Multiplying (13) by and adding the energy inequality (24), it follows that

where

For convenient presentation, we introduce the set of the essential and residual values

where , and is defined as follows:

where is the solution of (28) and

Notice that two assumptions of pressure in (7) and (12) imply that is a strict convex. Thus, thanks to (36), we get that is uniformly bounded for . Consequently, from the energy balance (40), we obtain

Using the estimate of (49) implies

Note that the static equation (29) holds

Due to the estimate of (51) and the convergence of (52), it follows that

In order that we now derive the estimate of the velocity, we first verify that

To do this, we use the following inequality due to two assumptions of pressure (7) and (12),

which deduce, thanks to (49),

Finally, we get

where we have used the above two estimates (57) and (58) and so the estimate (56) together with (59) shows (54).

We now derive the estimate of the velocity. Indeed, it is easy to show with a simple computation that

for a certain where we have used the Sobolev embedding inequality. Thus we get

and the estimates in (48) and (61) imply

4.2. Convergence of Anelastic Constraint

We will use the uniform estimate (62) to deduce

up to a subsequence of . In accordance with (53) and (54), we obtain

and so we can take the limit of in the continuity equation (20) to get

for all .

4.3. Convergence of Moment Equation

To begin with, using two estimates (63) and (64),

Hence

for a certain . Actually, we do not know due to the oscillations of the gradient component of the velocity field and we postpone this part to handle the oscillations of the gradient component in the next section.

We also need to get the uniform estimates for the directional field . Multiplying on (27) yields

and so applying the maximum principle for weak solutions provides

which implies

where we have used (47), the basic elliptic theory, and the following Gagliardo-Nirenberg inequality:

and thus we derive

for a certain . Applying the Aubin-Lions lemma applied to (27) together with (47), (70), (72), and (61) implies

Thus we get the boundary condition and

in the sense of distribution.

We are now able to identify the limit problem of the moment equation (22). To do this, we first rewrite the moment equation as follows:

From the previous estimates, we get

for any test function

if we show

where we have here used (8), (53), and (54). It remains to show that

for any

The proofs of (78) and (79) are provided in the next two sections.

4.4. Pressure Estimate

The next challenge is to establish uniform bounds on the pressure as well as on the internal energy in terms of a reflexive space , with . We here use the Bogovski operator. Let us take a test function and adapt this test function in the moment equation (2):

We will write the moment equation (2) with simple computations again:

where

and is defined by

We notice that all estimates in (84) are uniformly bounded due to the uniform estimates in the previous section if we take a special verifying

with a certain . In virtue of (57), it is easy to see

and so we get

for and the first integral of the right hand side of (82) is uniformly bounded. We now need to control the first and second integrals of the right hand side of (82). To do this, let us rewrite the first term into the following form with using the static problem (8)

where the last integral is uniformly bounded due to (86). Following estimates (53), (56), and (87) together with the -estimates for , one gets the uniform boundedness of the first and second terms of the right hand side of (88).

On the other hand, the estimates in (86) and (87) together with the -estimates for yield that the third term of the right hand side of (82) is uniformly bounded. Consequently, we deduce that

for a certain . We first split the integration of (78) into two parts to show (78):

For the first one of the right hand side, we get

where we have used the Taylor expansion of degree 2. We also use (57) and (89) to prove the second part of the right hand side.

4.5. Convergence of the Convective Term

In this section, our aim is to show that (79) holds. Before we prove (79), we will introduce the Helmholtz decomposition and the following material may be found in most of the text book of fluid mechanics. Let us denote by the Hilbert space with the inner product

and by the completion of with respect to the norm .

Theorem 4. For , a vector function is written as

where

and is uniquely determined as the following Neumann problem:

where is the outward unit normal to .

We now write

Let us first show

To do this, we adapt the following test function to the moment equation (22):

Taking into account the uniform estimates obtained in the previous sections, it follows that

precompact in and so the uniform estimates (45) and (64) with using the Sobolev embedding imply that

Notice that

where we have here used (100). In virtue of (64) and (101), one gets

Since the Helmholtz projections

map continuously the spaces and into itself for any , it is easily seen that

In order to show (79), we should prove

for any

We next study the acoustic equations. The acoustic equations are used to describe the time evolution of fast acoustic waves in the compressible models in order to handle the oscillation of . To begin with, we rewrite (1) and (2):

where the previous estimates provide

for a certain and

We now use the method of spectral analysis of the wave operator. Let us consider the eigenvalue problem:

with the boundary condition

Thus, the eigenvalue problem (112) and (113) can be written into the following form:

with

Following the eigenvalue problem (114) and (115), it is well known that there is an orthonormal basis of real eigenfunctions of the weighed Lebesgue space corresponding to eigenvalues such that

where stands for the multiplicity of . We also see that is an orthonormal basis in defined by

We now take and as test functions to the system (107), and then we obtain

for ., and , where ,   are defined by

and the estimates in (108) show that

Moreover, in virtue of section 5 in [7] for a finite number of modes, we set

We first note

as in where we here have used the Parseval identity and (121) (see section 6.6 of [7]). Thus it is sufficient to show (105) for instead of . Observe that, thanks to (114) and (121), it is easy to see

We are now ready to show (105) and so we now use (118) and (123) to rewrite the oscillation part as

which converges to as tends to where we can see this proof in section 6.6 of Feireisl and Novotný [7].

Acknowledgment

The work of Y.-S. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0003611).

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