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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 673546, 8 pages
http://dx.doi.org/10.1155/2013/673546
Research Article

The Coupled Kuramoto-Sivashinsky-KdV Equations for Surface Wave in Multilayered Liquid Films

1Department of Mathematics, Weber State University, Ogden, UT 84408, USA
2Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
3Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Received 4 June 2013; Accepted 5 July 2013

Academic Editors: S. C. Lim and W.-H. Steeb

Copyright © 2013 Maomao Cai et al. 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 coupled Kuramoto-Sivashinsky-KdV equations describing the surface waves on multilayered liquid films. A priori energy estimates for linearized problem are derived and local existence of solutions for initial-value problem is established.

1. Introduction

This paper studies a two-dimensional coupled Kuramoto-Sivashinsky-Korteweg-de Vries equation. The model was introduced in [1] to describe the surface waves on multilayered liquid films, and the two-dimensional model was proposed in [2].

A generalized equation that combines conservative and dissipative effects is a mixed Kuramoto-Sivashinsky-Korteweg-de Vries (KS-KdV) equation, which was first introduced in [3] and is often called the Benney equation. This equation finds various applications in plasma physics, hydrodynamics, and other fields [4, 5].

Another version of the Benney equation was proposed in [1] for a real wave field , based on the KS-KdV equation, which is linearly coupled to an additional linear dissipative equation for an extra real wave field that provides for the stabilization of the zero background solution. The model is as follows: The system describes the propagation of surface waves in a two-layer liquid film with one layer being dominated by viscosity. Here the coefficients , the coupling parameters , and is a group-velocity mismatch between the two waves fields. The linear coupling via the first derivatives is the same as in known models of coupled internal waves propagating in multi-layered fluids [6]. Then, the linear dissipative equation in (2) implies that the substrate layer is essentially more viscous [1]. In [1], the stability of solutions in the system of (2) is investigated by treating the gain and the dissipation constants , , as small parameters and making use of the balance equation for the net momentum. In [7], an energy estimate has been derived for the linearized model of (2).

In this paper, we consider the following two-dimensional version of (2) for general viscous fluid without the smallness assumptions on , , :

The system (3) is proposed in [2] in the study of cylindrical solitary pulses. One immediately notices that the two space variables in (3) are not symmetric. This is because of the underlying nonsymmetric physics; see [2]. The stability of steady-state solutions is analyzed by perturbation theory and wave mode in [2, 8]. Global solutions for the coupled Kuramoto-Sivashinsky-KdV system are studied in [9]. However, the existence of local solution is not available. In this paper, we will use the energy estimate approach to study such solution and establish its linear stability and the local existence of such solution for initial-value problems.

The paper is arranged as follows. In Section 2, we investigate the linear stability of the solution to (3) by establishing the energy estimates for its linearization. In Section 3, we use the obtained energy estimate and continuation method to show the existence of the solution for such linearized problem. Finally in Section 4, the existence of solution for the nonlinear problem is obtained by linear iteration method.

2. Linear Stability and High-Order Estimate

Let be a small perturbation of a bounded solution of (3): with .

Substituting in (4) into (3) and omitting the higher-order terms of , we obtain a linearized system for :

The linear stability of solution is determined by the energy estimate for under small initial perturbation. For simplicity of notation, we will omit the over in the following. Hence, we will discuss the following initial-value problem for the linearized system:

Here, the coefficients , , , and   are all positive constants with possible smooth small perturbations. is a given bounded smooth function and is a given bounded smooth function.

Let denote the inner product in and let be the usual Sobolev space defined by the norm with and , where

Let be the Schwartz rapidly decaying function space [10]. We have the following energy estimate for the linearized problem (6).

Theorem 1. Any solution of (6) satisfies the estimate Here and in the following, always denotes a constant depending only on and coefficients of (6). In particular, the constant depends upon in the coefficients of (6) only in its norm.

Proof. Take inner product of the equations in (6) with over . Noticing that , we have
Integrating by parts in and noticing that for any we have Substituting (13) into (11), we obtain (9). To obtain (10), we apply Gronwall's theorem [10] to the inequality with Therefore, we have and hence and (10) follows readily. In particular, we notice that the constant in (12) depends upon only in its norm over .
Inequalities (9) and (10) in Theorem 1 are obtained for the Cauchy initial data. Obviously, it is also valid for the initial-boundary value problems with periodic boundary conditions studied for the periodic waves of KS-KdV system in [11].
Inequalities (9) and (10) in Theorem 1 can be improved. Taking inner product of the two equations in (6) with and integrating by parts in , we have
The right side of (18) can be controlled by with constant depending on only in norm. Combining (18) and (4) with (9) and noticing (12), we have the following: Indeed, we can further improve (20) by taking inner product of the first equation in (6) with to derive The right side of (21) can be controlled by
Combining (21),  (22), and (20), we have
Applying Gronwall’s inequality to (20) and (23), we can obtain or
We can also include the estimate for in (25) by using the equations in (6) to obtain It is easy to see that the constant in (26) depends upon in (6) only in its norm over . Since can be continuously imbedded into bounded by the Sobolev imbedding theorem [10], the constant in (26) hence depends only on in its norm.

We can obtain higher-order estimates by taking spatial derivative of (6) and applying (26) to the expanded system. Hence we have the following theorem.

Theorem 2. Any solution of (6) satisfies the estimate, for any integer , Here is a constant depending only on and coefficients of (6). In particular, the constant depends upon in the coefficients of (6) only in its norm.

Remark 3. For convenience, we will use the notation in the following to denote the product space of : if is a Banach space equipped with the norm

3. Existence of Solution for Linearized Problem

We will use the continuation method to prove the following existence and uniqueness of the solution to (6).

Theorem 4. In the initial-value problem (6), let be any integer and assume that (i) and and ; (ii) and ; (iii)all the coefficients in (6) are th order continuously differentiable and are constant outside a bounded domain. Then (6) has a unique solution in the space satisfying estimate (27).

Remark 5. By the Sobolev imbedding theorem, for , the solution is continuously differentiable in up to 4th and 2nd orders. If and , then by (6), we can derive that are also continuous. Therefore the solution in Theorem 4 is a classical solution.

First we rewrite (6) briefly as follows:

Here Consider the following one-parameter family of initial-value problems, denoted as :

Obviously, for , the problem in (32) is the same initial-value problem in (6). It is readily checked that energy estimate (27) is valid uniformly for the solution of (32) with the constant in (27) being independent of the parameter .

In the following we show that the conclusion on the existence of solution in Theorem 4 is true for (32) for all . In particular the conclusion in Theorem 4 is the case .

Let be such that, for , Theorem 4 is true for (32). To show , we need to prove that subset is not empty, and it is both closed and open.

(1) is not empty.

Actually, . The problem is the following two separate initial-value problems for and :

The existence of the solutions for the problem (33) is the standard result for the Cauchy problem of general parabolic equations; see, for example, [10, 12].

(2) is closed in .

Let and . Let be the solution of the following initial-value problem:

By (27), is uniformly bounded in . In particular

Let which satisfies

Applying (27) to the solution of (36), we have

Since , it follows that is a Cauchy sequence in and its limit is obviously the solution of (32) for . This shows is closed in .

(3) is open in .

Let and with .

Let be the solution of the following problem:

We can construct a sequence of solutions as the solution of the following problem:

Then satisfies

Since

therefore by (27), satisfies

Choose such that and is a Cauchy sequence with limit being the solution of (32) for . Hence is open.

This concludes the proof of Theorem 4.

To prepare the study of the nonlinear system (3) in the next section, we introduce the uniformly local Sobolev space ; see also [13].

Definition 6. The uniformly local Sobolev space is defined by if and only if, for all and ,
A corresponding norm is defined as
for a fixed with in .

It is readily verified that is a Hilbert space.

Then we have the following improved version of Theorem 4.

Theorem 7. In the initial-value problem (6), let be any integer. Under the same assumptions as in Theorem 4, except for the requirement on which is replaced by then the same conclusion in Theorem 4 is true.

To prove Theorem 7, we notice that estimate (27) can be obtained assuming that . Since is a Banach algebra, we have in Step (to show is closed) in the proof of Theorem 4 Hence (37) follows from (36) for Theorem 7.

The same argument also applies to the estimate of the right side of (40) in Step (to show is open) in the proof of Theorem 4 and we can obtain (42) correspondingly.

4. Existence of Solution for Nonlinear Problem

In this section, we prove the existence of solution for the following initial-value problem: The coefficients , , , , , and are all positive constants or positive constants outside a bounded domain with possible smooth perturbations in bounded domain. In particular, , are uniformly positive: .

Theorem 8. Let be a nonnegative integer and assume that (i)all the coefficients , , , , , and are positive constants with possible smooth perturbations in a bounded domain and ; (ii) with being constants and . Then there is a such that (47) has a unique solution satisfying For , such solution is also a classical solution and satisfies (47) in the classical sense.

Proof of Theorem 8. We use linear iteration to prove Theorem 8. First we construct an approximate solution of the form where is the solution for the initial-value problem of the linear system:
From Theorems 4 and 7, solution of (50) exists and satisfies Now we are looking for the solution of (47) in the form of Obviously is the solution of (47) if and only if is the solution of the following problem: with
From (51), and is controlled by norms in and consequently controlled by :
Denote the nonlinear differential operators on the left side of the equations in (53) as ; we can rewrite (53) briefly as
Take . By Theorem 7, one can obtain a sequence of solutions , for , in the product space by solving the linearized problem
From (27) of Theorem 2, satisfy
In particular, we notice that the constant in (58) depends upon only in its norm, and it is uniform in for small (say, ). So there is a constant such that if, for all , then the constant in (58) is uniform in .
Take such that Then (58) implies that (59) is satisfied for all . We conclude that for such , the sequence is uniformly bounded in the product space .
Let . satisfies
Again from (27), we have Choose such that . Then the sequence converges to which is the solution of (53). Then the solution of (47) is obtained by (52).
Inequality (48) is obtained from (51), (55), and (58). This completes the proof of Theorem 8.

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