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

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 u (x, t), based on the KS-KdV equation, which is linearly coupled to an additional linear dissipative equation for an extra real wave field V (x, t) that provides for the stabilization of the zero background solution. The model is as follows:


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 V (, ) that provides for the stabilization of the zero background solution.The model is as follows: (2) The system describes the propagation of surface waves in a two-layer liquid film with one layer being dominated by viscosity.Here the coefficients , , Γ > 0, the coupling parameters  1 ,  2 > 0, and  1 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 , , Γ 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 , , Γ: (3) 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 2 ISRN Mathematical Physics 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.
Substituting (, V) in ( 4) into (3) and omitting the higherorder terms of , we obtain a linearized system for (ũ, Ṽ): The linear stability of solution ( 0 , V 0 ) is determined by the energy estimate for (ũ, Ṽ) under small initial perturbation.For simplicity of notation, we will omit the ̃over (, V) in the following.Hence, we will discuss the following initial-value problem for the linearized system: Here, the coefficients , , Γ,  1 , and  2 are all positive constants with possible smooth small perturbations. 0 is a given bounded smooth function and  1 is a given bounded smooth function.
Let ⟨⋅, ⋅⟩ denote the  2 inner product in  2 and let   ( 2 ) be the usual Sobolev space defined by the norm with  0 ( 2 ) =  2 ( 2 ) and ‖‖ = ‖‖ 0 , where Let  ( 2 ) be the Schwartz rapidly decaying function space [10].We have the following energy estimate for the linearized problem (6).( Here and in the following,  always denotes a constant depending only on  and coefficients of (6).In particular, the constant  depends upon  0 in the coefficients of (6) only in its  ∞ norm.
Proof.Take  2 inner product of the equations in ( 6) with (, V) over  2 .Noticing that    = (1/2)   2 , we have Integrating by parts in (, ) and noticing that for any  > 0 Substituting ( 13) into (11), we obtain (9).To obtain (10), we apply Gronwall's theorem [10] to the inequality with (15) Therefore, we have and hence and (10) follows readily.In particular, we notice that the constant  in ( 12) depends upon  0 only in its  ∞ norm over [0, ] ×  2 .Inequalities ( 9) and ( 10) in Theorem 1 are obtained for the Cauchy initial data.Obviously, it is also valid for the initialboundary 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  2 inner product of the two equations in ( 6) with (Δ, ΔV) and integrating by parts in (, ), we have The right side of (18) can be controlled by with constant  depending on  0 only in  ∞ norm.Combining ( 18) and ( 4) with ( 9) and noticing (12), we have the following: Indeed, we can further improve (20) by taking  2 inner product of the first equation in ( 6) with Δ 2  to derive The right side of (21) can be controlled by Combining ( 21), ( 22), and (20), we have We can also include the estimate for (  , V  ) in (25) by using the equations in (6) 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 (, V) ∈  ∞ ([0, ];  ( 2 )) of (6) satisfies the estimate, for any integer  ≥ 0, Here   is a constant depending only on  and coefficients of (6).In particular, the constant   depends upon  0 in the coefficients of (6) only in its  ∞ ([0, ],  +2 ( 2 )) norm.
Remark 3.For convenience, we will use the notation Π   in the following to denote the product space of (, V): (28) is a Banach space equipped with the norm sup

Existence of Solution for Linearized Problem
We will use the continuation method to prove the following existence and uniqueness of the solution to (6).
First we rewrite (6) briefly as follows: Here Consider the following one-parameter family of initial-value problems, denoted as A  ( ∈ [0, 1]): Obviously, for  = 1, the problem A 1 in (32) is the same initial-value problem in (6).It is readily checked that energy estimate ( 27) is valid uniformly for the solution (, V) of (32) with the constant   in (27) being independent of the parameter  ∈ [0, 1].
In the following we show that the conclusion on the existence of solution in Theorem 4 is true for (32) for all  ∈ [0, 1].In particular the conclusion in Theorem 4 is the case  = 1.
Let B ⊂ [0, 1] be such that, for  ∈ B, Theorem 4 is true for (32).To show B = [0, 1], we need to prove that subset B is not empty, and it is both closed and 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   ul ( 2 ); see also [13].
Definition 6.The uniformly local Sobolev space   ul (  ) is defined by  ∈   ul (  ) if and only if, for all  ∈  ∞ 0 (  ) and   0 () ≡ ( −  0 ), A corresponding norm is defined as It is readily verified that   ul is a Hilbert space.Then we have the following improved version of Theorem 4.

Theorem 7.
In the initial-value problem (6), let  ≥ 0 be any integer.Under the same assumptions as in Theorem 4, except for the requirement on  0 which is replaced by then the same conclusion in Theorem 4 is true.

Existence of Solution for Nonlinear Problem
In this section, we prove the existence of solution for the following initial-value problem: The coefficients , Γ, ,  1 ,  1 , and  2 are all positive constants or positive constants outside a bounded domain with possible smooth perturbations in bounded domain.In particular, , Γ are uniformly positive:  ≥  > 0, Γ ≥  > 0.