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

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, 𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=−𝛼𝑢_𝑥𝑥−𝛾𝑢_{𝑥𝑥𝑥𝑥}, (1) 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: 𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥=−𝛼𝑢_𝑥𝑥−𝛾𝑢_{𝑥𝑥𝑥𝑥}+𝜖₁𝑣_𝑥,𝑣_𝑡+𝑎₁𝑣_𝑥=Γ𝑣_𝑥𝑥+𝜖₂𝑢_𝑥. (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 𝛼,𝛾,Γ>0, the coupling parameters 𝜖₁,𝜖₂>0, and 𝑎₁ 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 𝛼, 𝛾, 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 𝛼, 𝛾, Γ: 𝑢_𝑡+𝑢𝑢_𝑥+Δ𝑢_𝑥=−𝛼𝑢_𝑥𝑥−𝛾Δ²𝑢+𝜖₁𝑣_𝑥,𝑣_𝑡+𝑎₁𝑣_𝑥=ΓΔ𝑣+𝜖₂𝑢_𝑥. (3)

The system (3) is proposed in [2] in the study of cylindrical solitary pulses. One immediately notices that the two space variables (𝑥,𝑦) 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 (𝑢,𝑣) be a small perturbation of a bounded 𝐶^∞ solution (𝑢⁰,𝑣⁰) of (3): 𝑢∼𝑢⁰+𝜖𝑢,𝑣∼𝑣⁰+𝜖𝑣, (4) with 𝜖≪1.

Substituting (𝑢,𝑣) in (4) into (3) and omitting the higher-order terms of 𝜖, we obtain a linearized system for (𝑢,𝑣): 𝑢_𝑡+𝑢⁰𝑢_𝑥+Δ𝑢_𝑥=−𝛼𝑢_𝑥𝑥−𝛾Δ²𝑢+𝜖₁𝑣_𝑥𝑣+𝑓,_𝑡+𝑎₁𝑣_𝑥=ΓΔ𝑣+𝜖₂𝑢_𝑥+𝑔. (5)

The linear stability of solution (𝑢⁰,𝑣⁰) is determined by the energy estimate for (𝑢,𝑣) under small initial perturbation. For simplicity of notation, we will omit the ̃‌ over (𝑢,𝑣) in the following. Hence, we will discuss the following initial-value problem for the linearized system: 𝑢_𝑡+𝑢⁰𝑢_𝑥+Δ𝑢_𝑥=−𝛼𝑢_𝑥𝑥−𝛾Δ²𝑢+𝜖₁𝑣_𝑥𝑣+𝑓,_𝑡+𝑎₁𝑣_𝑥=ΓΔ𝑣+𝜖₂𝑢_𝑥𝑢+𝑔,𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣𝑥,𝑦,0=𝑣₀𝑥,𝑦. (6)

Here, the coefficients 𝛼, 𝛾, Γ, 𝜖₁,and  𝜖₂ are all positive constants with possible smooth small perturbations. 𝑢⁰ is a given bounded smooth function and 𝑎₁ is a given bounded smooth function.

Let ⟨⋅,⋅⟩ denote the 𝐿² inner product in 𝑅² and let 𝐻^𝑘(𝑅²) be the usual Sobolev space defined by the norm 𝑓²_𝑘=_{𝑅2|𝑗|≤𝑘}𝐷^𝑗𝑓²𝑑𝑥𝑑𝑦 (7) with 𝐻⁰(𝑅²)=𝐿²(𝑅²) and ‖𝑓‖=‖𝑓‖₀, where 𝐷^𝑗𝜕𝑓=^|𝑗|𝑓𝜕^𝑗1_𝑥𝜕^𝑗2_𝑦. (8)

Let 𝑆(𝑅²) be the Schwartz rapidly decaying function space [10]. We have the following energy estimate for the linearized problem (6).

Theorem 1. Any solution (𝑢,𝑣)∈𝐶^∞([0,𝑇];𝑆(𝑅²)) of (6) satisfies the estimate 𝜕_𝑡‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²+‖𝑢(𝑡)‖²₂+‖𝑣(𝑡)‖²₁≤𝐶‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²+𝑓(𝑡)²+𝑔(𝑡)²,∀𝑡∈[0,𝑇],sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²+^𝑇₀‖𝑢(𝑠)‖²₂+‖𝑣(𝑠)‖²₁𝑢𝑑𝑠≤𝐶₀²+𝑣₀²+^𝑇₀𝑓(𝑠)²+𝑔(𝑠)²𝑑𝑠. (9)(10) Here and in the following, 𝐶 always denotes a constant depending only on 𝑇 and coefficients of (6). In particular, the constant 𝐶 depends upon 𝑢⁰ in the coefficients of (6) only in its 𝐿^∞ norm.

Proof. Take 𝐿² inner product of the equations in (6) with (𝑢,𝑣) over 𝑅². Noticing that 𝑤𝜕_𝑡𝑤=(1/2)𝜕_𝑡𝑤², we have 12𝜕_𝑡‖𝑢‖²+‖𝑣‖²+𝛾Δ²𝑢,𝑢−⟨ΓΔ𝑣,𝑣⟩+𝛼𝑢_𝑥𝑥𝑢,𝑢=−⁰𝑢_𝑥,𝑢−𝑎₁𝑣_𝑥,𝑣−Δ𝑢_𝑥+𝜖,𝑢₁𝑣_𝑥,𝑢+𝑓,𝑢+𝜖₁𝑢_𝑥,𝑣+𝑔,𝑣. (11)
Integrating by parts in (𝑥,𝑦) and noticing that for any 𝛿>0‖𝑢‖₂∼‖Δ𝑢‖+‖𝑢‖,‖𝑢‖²₁≤14𝛿‖𝑢‖²₂+𝐶_𝛿‖𝑢‖²₀, (12) we have 𝛾Δ²𝑢,𝑢−⟨ΓΔ𝑣,𝑣⟩≥𝛿‖Δ𝑢‖²+‖∇𝑣‖²,𝛼𝑢_𝑥𝑥,𝑢+𝑎𝑢_𝑥,𝑢+𝑐𝑣_𝑥,𝑣+Δ𝑢_𝑥+𝑣,𝑢_𝑥,𝑢+𝑢_𝑥≤1,𝑣2𝛿‖Δ𝑢‖²+‖∇𝑣‖²+𝐶‖𝑢‖²+‖𝑣‖²,𝑓,𝑢+𝑔,𝑣≤𝑓²+𝑔²+‖𝑢‖²+‖𝑣‖². (13) Substituting (13) into (11), we obtain (9). To obtain (10), we apply Gronwall's theorem [10] to the inequality 𝜕_𝑡𝑃(𝑡)≤𝐶𝑃(𝑡)+𝑀(𝑡) (14) with 𝑃(𝑡)=‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²+^𝑡₀‖𝑢(𝑠)‖²₂+‖𝑣(𝑠)‖²₁𝑀(𝑡)=𝑓(𝑡𝑑𝑠,)²+𝑔(𝑡)². (15) Therefore, we have 𝑒^−𝐶𝑡𝑃(𝑡)−𝑃(0)≤^𝑡₀𝑒^−𝐶𝑠𝑀(𝑠)𝑑𝑠 (16) and hence 𝑃(𝑡)≤𝐶₁(𝑡)𝑃(0)+^𝑡₀𝑀(𝑠)𝑑𝑠, (17) and (10) follows readily. In particular, we notice that the constant 𝐶 in (12) depends upon 𝑢⁰ only in its 𝐿^∞ norm over [0,𝑇]×𝑅².
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 𝐿² inner product of the two equations in (6) with (Δ𝑢,Δ𝑣) and integrating by parts in (𝑥,𝑦), we have 12𝜕_𝑡‖∇𝑢‖²+‖∇𝑣‖²+𝛿‖∇Δ𝑢‖²+‖Δ𝑣‖²≤−𝑢⁰𝑢_𝑥,Δ𝑢−Δ𝑢_𝑥,Δ𝑢−𝛼𝑢_𝑥𝑥,Δ𝑢+𝜖₁𝑣_𝑥+,Δ𝑢𝑓,Δ𝑢−𝑎₁𝑣_𝑥,Δ𝑣+𝜖₂𝑢_𝑥,Δ𝑣+𝑔,Δ𝑣. (18)
The right side of (18) can be controlled by 12𝛿‖∇Δ𝑢(𝑡)‖²+‖Δ𝑣(𝑡)‖²+𝐶‖𝑢(𝑡)‖²₂+‖𝑣(𝑡)‖²₁+𝑓(𝑡)²+𝑔(𝑡)² (19) with constant 𝐶 depending on 𝑢⁰ only in 𝐿^∞ norm. Combining (18) and (4) with (9) and noticing (12), we have the following: 𝜕_𝑡‖𝑢(𝑡)‖²₁+‖𝑣(𝑡)‖²₁+‖𝑢(𝑡)‖²₃+‖𝑣(𝑡)‖²₂≤𝐶‖𝑢(𝑡)‖²₁+‖𝑣(𝑡)‖²₁+𝑓(𝑡)²+𝑔(𝑡)²,∀𝑡∈[0,𝑇]. (20) Indeed, we can further improve (20) by taking 𝐿² inner product of the first equation in (6) with Δ²𝑢 to derive 12𝜕_𝑡‖Δ𝑢‖²+𝛾Δ²𝑢²≤−𝑢⁰𝑢_𝑥,Δ²𝑢−Δ𝑢_𝑥,Δ²𝑢−𝛼𝑢_𝑥𝑥,Δ²𝑢+𝜖₁𝑣_𝑥,Δ²𝑢+𝑓,Δ²𝑢. (21) The right side of (21) can be controlled by 12𝛾Δ²𝑢²+𝐶‖𝑢‖²₃+‖𝑣‖²₁+𝑓². (22)
Combining (21),  (22), and (20), we have 𝜕_𝑡‖𝑢(𝑡)‖²₁+‖Δ𝑢‖²+‖𝑣(𝑡)‖²₁+‖𝑢(𝑡)‖²₄+‖𝑣(𝑡)‖²₂≤𝐶‖𝑢(𝑡)‖²₂+‖𝑣(𝑡)‖²₁+𝑓(𝑡)²+𝑔(𝑡)²,∀𝑡∈[0,𝑇]. (23)
Applying Gronwall’s inequality to (20) and (23), we can obtain sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²₁+‖𝑣(𝑡)‖²₁+^𝑇₀‖𝑢(𝑠)‖²₃+‖𝑣(𝑠)‖²₂𝑢𝑑𝑠≤𝐶₀²₁+𝑣₀²₁+^𝑇₀𝑓(𝑠)²+𝑔(𝑠)²𝑑𝑠 (24) or sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²₂+‖𝑣(𝑡)‖²₁+^𝑇₀‖𝑢(𝑠)‖²₄+‖𝑣(𝑠)‖²₂𝑢𝑑𝑠≤𝐶₀²₂+𝑣₀²₁+^𝑇₀𝑓(𝑠)²+𝑔(𝑠)²𝑑𝑠. (25)
We can also include the estimate for (𝑢_𝑡,𝑣_𝑡) in (25) by using the equations in (6) to obtain sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²₂+‖𝑣(𝑡)‖²₁+^𝑇₀‖𝑢(𝑠)‖²₄+‖𝑣(𝑠)‖²₂+𝑣_𝑡(𝑠)²₀+𝑢_𝑡(𝑠)²₀𝑢𝑑𝑠≤𝐶₀²₂+𝑣₀²₁+^𝑇₀𝑓(𝑠)²+𝑔(𝑠)²𝑑𝑠. (26) It is easy to see that the constant 𝐶 in (26) depends upon 𝑢⁰ in (6) only in its 𝐿^∞ norm over [0,𝑇]×𝑅². Since 𝐻²(𝑅²) can be continuously imbedded into bounded 𝐶(𝑅²) by the Sobolev imbedding theorem [10], the constant 𝐶 in (26) hence depends only on 𝑢⁰ in its 𝐿^∞([0,𝑇],𝐻²(𝑅²)) 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 (𝑢,𝑣)∈𝐶^∞([0,𝑇];𝑆(𝑅²)) of (6) satisfies the estimate, for any integer 𝑘≥0, sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²_𝑘+2+‖𝑣(𝑡)‖²_𝑘+1+^𝑇₀‖𝑢(𝑠)‖²_𝑘+4+‖𝑣(𝑠)‖²_𝑘+2+𝑣_𝑡(𝑠)²_𝑘+𝑢_𝑡(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘𝑢₀²_𝑘+2+𝑣₀²_𝑘+1+^𝑇₀𝑓(𝑠)²_𝑘+𝑔(𝑠)²_𝑘𝑑𝑠. (27) Here 𝐶_𝑘 is a constant depending only on 𝑇 and coefficients of (6). In particular, the constant 𝐶_𝑘 depends upon 𝑢⁰ in the coefficients of (6) only in its 𝐿^∞([0,𝑇],𝐻^𝑘+2(𝑅²)) norm.

Remark 3. For convenience, we will use the notation Π^𝑘_𝑇 in the following to denote the product space of (𝑢,𝑣): (𝑢,𝑣)∈Π^𝑘_𝑇 if 𝑢∈𝐶[0,𝑇],𝐻^𝑘+2𝑅²∩𝐿²[0,𝑇],𝐻^𝑘+4𝑅²∩𝐻¹[0,𝑇],𝐻^𝑘𝑅²,𝑣∈𝐶[0,𝑇],𝐻^𝑘+1𝑅²∩𝐿²[0,𝑇],𝐻^𝑘+2𝑅²∩𝐻¹[0,𝑇],𝐻^𝑘𝑅². (28)Π^𝑘_𝑇 is a Banach space equipped with the norm sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²_𝑘+2+‖𝑣(𝑡)‖²_𝑘+1+^𝑇₀‖𝑢(𝑠)‖²_𝑘+4+‖𝑣(𝑠)‖²_𝑘+2+𝑣_𝑡(𝑠)²_𝑘+𝑢_𝑡(𝑠)²_𝑘𝑑𝑠. (29)

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 𝑘≥0 be any integer and assume that (i)𝑢₀∈𝐻^𝑘+2(𝑅²) and 𝑣₀∈𝐻^𝑘+1(𝑅²) and 𝑓,𝑔∈𝐿²([0,𝑇],𝐻^𝑘(𝑅²)); (ii)𝛾≥𝛿>0 and Γ≥𝛿>0; (iii)all the coefficients 𝜉≡(𝛼,𝑎₁,𝛾,Γ,𝜖₁,𝜖₂,𝑢⁰) in (6) are 𝑘th order continuously differentiable and are constant outside a bounded domain. Then (6) has a unique solution (𝑢,𝑣) in the space Π^𝑘_𝑇 satisfying estimate (27).

Remark 5. By the Sobolev imbedding theorem, for 𝑘≥2, the solution (𝑢,𝑣) is continuously differentiable in (𝑥,𝑦) up to 4th and 2nd orders. If 𝑘≥4 and 𝑓,𝑔∈𝐻¹([0,𝑇],𝐻^𝑘(𝑅²)), then by (6), we can derive that (𝑢_𝑡,𝑣_𝑡) are also continuous. Therefore the solution (𝑢,𝑣) in Theorem 4 is a classical solution.

First we rewrite (6) briefly as follows: 𝑢_𝑡+ℒ₁()𝑣𝑢,𝑣=𝑓,_𝑡+ℒ₂()𝑢𝑢,𝑣=𝑔,𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣𝑥,𝑦,0=𝑣₀𝑥,𝑦. (30)

Here ℒ₁(𝑢,𝑣)=𝑢⁰𝑢_𝑥+Δ𝑢_𝑥+𝛼𝑢_𝑥𝑥−𝜖₁𝑣_𝑥+𝛾Δ²ℒ𝑢,₂(𝑢,𝑣)=𝑎₁𝑣_𝑥−𝜖₂𝑢_𝑥−ΓΔ𝑣. (31) Consider the following one-parameter family of initial-value problems, denoted as 𝒜_𝜆(𝜆∈[0,1]): 𝑢_𝑡+𝜆ℒ₁(𝑢,𝑣)+(1−𝜆)Δ²𝑣𝑢=𝑓,_𝑡+𝜆ℒ₂()−()𝑢𝑢,𝑣1−𝜆Δ𝑣=𝑔,𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣𝑥,𝑦,0=𝑣₀𝑥,𝑦. (32)

Obviously, for 𝜆=1, the problem 𝒜₁ in (32) is the same initial-value problem in (6). It is readily checked that energy estimate (27) is valid uniformly for the solution (𝑢,𝑣) of (32) with the constant 𝐶_𝑘 in (27) being independent of the parameter 𝜆∈[0,1].

In the following we show that the conclusion on the existence of solution in Theorem 4 is true for (32) for all 𝜆∈[0,1]. In particular the conclusion in Theorem 4 is the case 𝜆=1.

Let ℬ⊂[0,1] be such that, for 𝜆∈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.

(1) ℬ is not empty.

Actually, 0∈ℬ. The problem 𝒜₀ is the following two separate initial-value problems for 𝑢 and 𝑣: 𝑢_𝑡+Δ²𝑢𝑢=𝑓,𝑥,𝑦,0=𝑢₀,𝑣𝑥,𝑦_𝑡𝑣−Δ𝑣=𝑔,𝑥,𝑦,0=𝑣₀𝑥,𝑦. (33)

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

(2) ℬ is closed in [0,1].

Let 𝜆_𝑗∈ℬ and 𝜆_𝑗→𝜆₀. Let (𝑢_𝑗,𝑣_𝑗) be the solution of the following initial-value problem: 𝑢_𝑗𝑡+𝜆_𝑗ℒ₁𝑢_𝑗,𝑣_𝑗+1−𝜆_𝑗Δ²𝑢_𝑗𝑣=𝑓,_𝑗𝑡+𝜆_𝑗ℒ₂𝑢_𝑗,𝑣_𝑗−1−𝜆_𝑗Δ𝑣_𝑗𝑢=𝑔,_𝑗𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣_𝑗𝑥,𝑦,0=𝑣₀𝑥,𝑦. (34)

By (27), (𝑢_𝑗,𝑣_𝑗) is uniformly bounded in Π^𝑘_𝑇. In particular ^𝑇₀𝑢_𝑗(𝑠)²_𝑘+4+𝑣_𝑗(𝑠)²_𝑘+2𝑑𝑠≤𝐶_𝑘𝑢₀²_𝑘+2+𝑣₀²_𝑘+1+^𝑇₀𝑓(𝑠)²_𝑘+𝑔(𝑠)²_𝑘𝑑𝑠≤𝐾. (35)

Let (𝑢_𝑗,𝑣_𝑗)=(𝑢_𝑗−𝑢_𝑗−1,𝑣_𝑗−𝑣_𝑗−1)(𝑗=2,3,…) which satisfies 𝑢_𝑗𝑡+𝜆_𝑗ℒ₁𝑢_𝑗,𝑣_𝑗+1−𝜆_𝑗Δ²𝑢_𝑗=−𝜆_𝑗−𝜆_𝑗−1ℒ₁𝑢_𝑗−1,𝑣_𝑗−1−Δ²𝑢_𝑗−1,𝑣_𝑗𝑡+𝜆_𝑗ℒ₂𝑢_𝑗,𝑣_𝑗−1−𝜆_𝑗Δ𝑣_𝑗=−𝜆_𝑗−𝜆_𝑗−1ℒ₂𝑢_𝑗−1,𝑣_𝑗−1+Δ𝑣_𝑗−1,𝑢_𝑗𝑥,𝑦,0=0,𝑣_𝑗𝑥,𝑦,0=0. (36)

Applying (27) to the solution (𝑢_𝑗,𝑣_𝑗) of (36), we have sup_{0≤𝑡≤𝑇}𝑢_𝑗(𝑡)²_𝑘+2+𝑣_𝑗(𝑡)²_𝑘+1+^𝑇₀𝑢_𝑗(𝑠)²_𝑘+4+𝑣_𝑗(𝑠)²_𝑘+2𝑑𝑠≤𝐶_𝑘𝜆_𝑗−𝜆_𝑗−1^2𝑇₀𝑢_𝑗−1(𝑠)²_𝑘+4+𝑣_𝑗−1(𝑠)²_𝑘+2𝑑𝑠≤𝐶_𝑘𝜆_𝑗−𝜆_𝑗−1²𝐾. (37)

Since 𝜆_𝑗→𝜆₀, it follows that (𝑢_𝑗,𝑣_𝑗) is a Cauchy sequence in Π^𝑘_𝑇 and its limit (𝑢,𝑣) is obviously the solution of (32) for 𝜆₀. This shows ℬ is closed in [0,1].

(3) ℬ is open in [0,1].

Let 𝜆₀∈ℬ and 𝜆∈[0,1] with |𝜆−𝜆₀|≤𝜖.

Let (𝑢₁,𝑣₁) be the solution of the following problem: 𝑢_1𝑡+𝜆₀ℒ₁𝑢₁,𝑣₁+1−𝜆₀Δ²𝑢₁𝑣=𝑓,_1𝑡+𝜆₀ℒ₂𝑢₁,𝑣₁−1−𝜆₀Δ𝑣₁𝑢=𝑔,₁𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣₁𝑥,𝑦,0=𝑣₀𝑥,𝑦. (38)

We can construct a sequence of solutions (𝑢_𝑗,𝑣_𝑗)(𝑗=2,3,…) as the solution of the following problem: 𝑢_𝑗𝑡+𝜆₀ℒ₁𝑢_𝑗,𝑣_𝑗+1−𝜆₀Δ²𝑢_𝑗=𝑓+𝜆₀−𝜆ℒ₁𝑢_𝑗−1,𝑣_𝑗−1−Δ²𝑢_𝑗−1,𝑣_𝑗𝑡+𝜆ℒ₂𝑢_𝑗,𝑣_𝑗−1−𝜆₀Δ𝑣_𝑗=𝑔+𝜆₀−𝜆ℒ₂𝑢_𝑗−1,𝑣_𝑗−1−Δ𝑣_𝑗−1,𝑢_𝑗𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣_𝑗𝑥,𝑦,0=𝑣₀𝑥,𝑦. (39)

Then (𝑢_𝑗,𝑣_𝑗)=(𝑢_𝑗−𝑢_𝑗−1,𝑣_𝑗−𝑣_𝑗−1)(𝑗=2,3,…) satisfies 𝑢_𝑗𝑡+𝜆₀ℒ₁𝑢_𝑗,𝑣_𝑗+1−𝜆₀Δ²𝑢_𝑗=𝜆₀−𝜆ℒ₁𝑢_𝑗−1,𝑣_𝑗−1−Δ²𝑢_𝑗−1,𝑣_𝑗𝑡+𝜆ℒ₂𝑢_𝑗,𝑣_𝑗−(1−𝜆)Δ𝑣_𝑗=𝜆₀−𝜆ℒ₂𝑢_𝑗−1,𝑣_𝑗−1−Δ𝑣_𝑗−1,𝑢_𝑗𝑥,𝑦,0=0,𝑣_𝑗𝑥,𝑦,0=0. (40)

Since ℒ₁𝑢_𝑗−1,𝑣_𝑗−1−Δ²𝑢_𝑗−1²_𝑘+ℒ₂𝑢_𝑗−1,𝑣_𝑗−1−Δ𝑣_𝑗−1²_𝑘≤𝑀𝑢_𝑗−1(𝑠)²_𝑘+4+𝑣_𝑗−1(𝑠)²_𝑘+2, (41)

therefore by (27), (𝑢_𝑗,𝑣_𝑗) satisfies sup_{0≤𝑡≤𝑇}𝑢_𝑗(𝑡)²_𝑘+2+𝑣_𝑗(𝑡)²_𝑘+1+^𝑇₀𝑢_𝑗(𝑠)²_𝑘+4+𝑣_𝑗(𝑠)²_𝑘+2𝑑𝑠≤𝐶_𝑘𝑀𝜆−𝜆₀^2𝑇₀𝑢_𝑗−1(𝑠)²_𝑘+4+𝑣_𝑗−1(𝑠)²_𝑘+2𝑑𝑠. (42)

Choose 𝜖≪1 such that 𝐶_𝑘𝑀𝜖²<1/2 and (𝑢_𝑗,𝑣_𝑗) is a Cauchy sequence with limit (𝑢,𝑣) being the solution of (32) for 𝜆. Hence ℬ 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 𝐻^𝑠_ul(𝑅²); see also [13].

Definition 6. The uniformly local Sobolev space 𝐻^𝑠_ul(𝑅^𝑛) is defined by𝑓∈𝐻^𝑠_ul(𝑅^𝑛) if and only if, for all 𝜙∈𝐶^∞₀(𝑅^𝑛) and 𝜙_𝑋0(𝑋)≡𝜙(𝑋−𝑋₀), sup_{𝑋0∈𝑅𝑛}𝑓𝜙_{𝑋0𝐻𝑠(𝑅𝑛)}≤∞. (43)
A corresponding norm is defined as 𝑓_{𝐻𝑠(𝑅2),ul}=sup_{𝑋0∈𝑅2}𝑓𝜙_{𝑋0𝐻𝑠(𝑅2)} (44)
for a fixed 𝜙∈𝐶^∞₀(𝑅²) with 𝜙(𝑋)=1 in |𝑋|≤1.

It is readily verified that 𝐻^𝑠_ul is a Hilbert space.

Then we have the following improved version of Theorem 4.

Theorem 7. In the initial-value problem (6), let 𝑘≥0 be any integer. Under the same assumptions as in Theorem 4, except for the requirement on 𝑢⁰ which is replaced by 𝑢⁰∈𝐶[0,𝑇];𝐻^𝑘+2_ul𝑅², (45) then the same conclusion in Theorem 4 is true.

To prove Theorem 7, we notice that estimate (27) can be obtained assuming that 𝑢⁰∈𝐶([0,𝑇];𝐻^𝑘+2_ul(𝑅²)). Since 𝐻^𝑘+2_ul(𝑅²) is a Banach algebra, we have in Step (2) (to show ℬ is closed) in the proof of Theorem 4𝑢⁰𝑢_{(𝑗−1)𝑥}²_𝑘≤𝑢⁰𝑢_(𝑗−1)²_𝑘+2≤𝐶𝑢⁰²_𝑘+2𝑢_(𝑗−1)²_𝑘+2. (46) Hence (37) follows from (36) for Theorem 7.

The same argument also applies to the estimate of the right side of (40) in Step (3) (to show ℬ 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: 𝑢_𝑡+𝑢𝑢_𝑥+Δ𝑢_𝑥=−𝛼𝑢_𝑥𝑥−𝛾Δ²𝑢+𝜖₁𝑣_𝑥,𝑣_𝑡+𝑎₁𝑣_𝑥=ΓΔ𝑣+𝜖₂𝑢_𝑥,𝑢𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣𝑥,𝑦,0=𝑣₀𝑥,𝑦. (47) The coefficients 𝛾, Γ, 𝛼, 𝑎₁, 𝜖₁, and 𝜖₂ are all positive constants or positive constants outside a bounded domain with possible smooth perturbations in bounded domain. In particular, 𝛾, Γ are uniformly positive: 𝛾≥𝛿>0,Γ≥𝛿>0.

Theorem 8. Let 𝑘 be a nonnegative integer and assume that (i)all the coefficients 𝛾, Γ, 𝛼, 𝑎₁, 𝜖₁, and 𝜖₂ are positive constants with possible smooth perturbations in a bounded domain and 𝛾,Γ≥𝛿>0; (ii)(𝑢₀,𝑣₀)=(𝑈₀+𝑢₀,𝑉₀+𝑣₀) with (𝑈₀,𝑉₀) being constants and (𝑢₀,𝑣₀)∈𝐻^𝑘+2(𝑅²)×𝐻^𝑘+1(𝑅²). Then there is a 𝑇>0 such that (47) has a unique solution (𝑢,𝑣)=(𝑈₀+𝑢,𝑉₀+𝑣)∈𝐶([0,𝑇];𝐻^𝑘+2_𝑢𝑙(𝑅²)×𝐻^𝑘+1_𝑢𝑙(𝑅²) satisfying sup_{0≤𝑡≤𝑇}‖𝑢(𝑡)‖²_𝑘+2+‖𝑣(𝑡)‖²_𝑘+1+^𝑇₀‖𝑢(𝑠)‖²_𝑘+4+‖𝑣(𝑠)‖²_𝑘+2+𝑢_𝑡(𝑠)²_𝑘+𝑣_𝑡(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘𝑢₀²_𝑘+2+𝑣₀²_𝑘+1. (48) For 𝑘≥2, 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 (𝑢_𝑎,𝑣_𝑎) of the form 𝑢_𝑎,𝑣_𝑎=𝑈₀+𝑢_𝑎,𝑉₀+𝑣_𝑎, (49) where (𝑢_𝑎,𝑣_𝑎) is the solution for the initial-value problem of the linear system: 𝑢_𝑎𝑡+𝑈₀+𝑢₀𝑢_𝑎𝑥+Δ𝑢_𝑎𝑥+𝛼𝑢_𝑎𝑥𝑥+𝛾Δ²𝑢_𝑎−𝜖₁𝑣_𝑎𝑥𝑣=0,_𝑎𝑡+𝑎₁𝑣_𝑎𝑥−ΓΔ𝑣_𝑎−𝜖₂𝑢_𝑎𝑥𝑢=0,_𝑎𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑣_𝑎𝑥,𝑦,0=𝑣₀𝑥,𝑦. (50)
From Theorems 4 and 7, solution (𝑢_𝑎,𝑣_𝑎)∈Π^𝑘_𝑇 of (50) exists and satisfies sup_{0≤𝑡≤𝑇}𝑢_𝑎(𝑡)²_𝑘+2+𝑣_𝑎(𝑡)²_𝑘+1+^𝑇₀𝑢_𝑎(𝑠)²_𝑘+4+𝑣_𝑎(𝑠)²_𝑘+2+𝑢_𝑎𝑡(𝑠)²_𝑘+𝑣_𝑎𝑡(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘𝑢₀²_𝑘+2+𝑣₀²_𝑘+1. (51) Now we are looking for the solution of (47) in the form of (𝑢,𝑣)=𝑢_𝑎+𝑢̇,𝑣_𝑎+𝑣̇. (52) Obviously (𝑢,𝑣) is the solution of (47) if and only if (𝑢̇,𝑣)̇ is the solution of the following problem: 𝑢̇_𝑡+𝑢_𝑎+𝑢̇𝑢̇_𝑥+Δ𝑢̇_𝑥+𝛼𝑢̇_𝑥𝑥+𝛾Δ²𝑢̇−𝜖₁𝑣̇_𝑥+𝑢_𝑎𝑥̇𝑣̇𝑢=𝑓,_𝑡+𝑎₁𝑣̇_𝑥−ΓΔ𝑣̇−𝜖₂𝑢̇_𝑥=𝑢̇𝑣̇𝑔,𝑥,𝑦,0=0,𝑥,𝑦,0=0, (53) with 𝑓=−𝑢_𝑎𝑡+𝑢_𝑎𝑢_𝑎𝑥+Δ𝑢_𝑎𝑥+𝛼𝑢_𝑎𝑥𝑥+𝛾Δ²𝑢_𝑎−𝜖₁𝑣_𝑎𝑥+𝑢_𝑎𝑢_𝑎𝑥,𝑔=−𝑣_𝑎𝑡+𝑎₁𝑣_𝑎𝑥−ΓΔ𝑣_𝑎−𝜖₂𝑢_𝑎𝑥. (54)
From (51), (𝑓,𝑔)∈𝐿²([0,𝑇],𝐻^𝑘(𝑅²)) and ∫^𝑇₀(‖𝑓(𝑠)‖²_𝑘+‖𝑔(𝑠)‖²_𝑘)𝑑𝑠 is controlled by (𝑢_𝑎,𝑣_𝑎) norms in Π^𝑘_𝑇 and consequently controlled by (‖𝑢₀‖²_𝑘+2+‖𝑣₀‖²_𝑘+1): ^𝑇₀𝑓(𝑠)²_𝑘+𝑔(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘𝑢₀²_𝑘+2+𝑣₀²_𝑘+1. (55)
Denote the nonlinear differential operators on the left side of the equations in (53) as ℳ(𝑢̇); we can rewrite (53) briefly as ℳ(𝑢̇̇𝑣)̇=𝑔,𝑢̇𝑣̇)(𝑢,𝑓,𝑥,𝑦,0=0,𝑥,𝑦,0=0. (56)
Take 𝑢̇₀(𝑥,𝑦,𝑡)=𝑣̇₀(𝑥,𝑦,𝑡)=0. By Theorem 7, one can obtain a sequence of solutions (𝑢̇_𝑗,𝑣̇_𝑗), for 𝑗=1,2,3,…, in the product space Π^𝑘_𝑇 by solving the linearized problem ℳ𝑢̇_𝑗−1𝑢̇_𝑗,𝑣̇_𝑗=𝑔,𝑢̇𝑓,_𝑗𝑥,𝑦,0=0,𝑣̇_𝑗𝑥,𝑦,0=0. (57)
From (27) of Theorem 2, (𝑢̇_𝑗,𝑣̇_𝑗) satisfy sup_{0≤𝑡≤𝑇}𝑢̇_𝑗(𝑡)²_𝑘+2+𝑣̇_𝑗(𝑡)²_𝑘+1+^𝑇₀𝑢̇_𝑗(𝑠)²_𝑘+4+𝑣̇_𝑗(𝑠)²_𝑘+2+𝑣̇_𝑗𝑡(𝑠)²_𝑘+𝑢̇_𝑗𝑡(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘,𝑗^𝑇₀𝑓(𝑠)²_𝑘+𝑔(𝑠)²_𝑘𝑑𝑠. (58)
In particular, we notice that the constant 𝐶_𝑘,𝑗 in (58) depends upon 𝑢̇_𝑗−1 only in its 𝐿^∞([0,𝑇],𝐻^𝑘+2(𝑅²)) norm, and it is uniform in 𝑇 for small 𝑇 (say, 𝑇≤1). So there is a constant 𝐾 such that if, for all 𝑗≥1, sup_{0≤𝑡≤𝑇}𝑢̇_𝑗−1(𝑡)²_𝑘+2≤𝐾, (59) then the constant 𝐶_𝑘,𝑗=𝐶_𝑘 in (58) is uniform in 𝑗.
Take 𝑇₀≪1 such that 𝐶_𝑘^𝑇0₀𝑓(𝑠)²_𝑘+𝑔(𝑠)²_𝑘𝑑𝑠≤𝐾. (60) Then (58) implies that (59) is satisfied for all 𝑗. We conclude that for such 𝑇₀, the sequence (𝑢̇_𝑗,𝑣̇_𝑗) is uniformly bounded in the product space Π^𝑘_𝑇0.
Let (𝑤_𝑗,𝑧_𝑗)=(𝑢̇_𝑗+1−𝑢̇_𝑗,𝑣̇_𝑗+1−𝑣̇_𝑗). (𝑤_𝑗,𝑧_𝑗) satisfies ℳ𝑢̇_𝑗𝑤_𝑗,𝑧_𝑗=−ℳ𝑢̇_𝑗−ℳ𝑢̇_𝑗−1𝑢̇_𝑗,𝑣̇_𝑗=−𝑤_𝑗−1𝜕_𝑥𝑢̇_𝑗,𝑤,0_𝑗𝑥,𝑦,0=𝑧_𝑗𝑥,𝑦,0=0. (61)
Again from (27), we have sup_{0≤𝑡≤𝑇0}𝑤_𝑗(𝑡)²_𝑘+2+𝑧_𝑗(𝑡)²_𝑘+1+^𝑇0₀𝑤_𝑗(𝑠)²_𝑘+4+𝑧_𝑗(𝑠)²_𝑘+2+𝑤_𝑗𝑡(𝑠)²_𝑘+𝑧_𝑗𝑡(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘^𝑇0₀𝑤_𝑗−1(𝑠)𝑢̇_𝑗𝑥(𝑠)²_𝑘𝑑𝑠≤𝐶_𝑘sup_{𝑡∈0,𝑇0}𝑢̇_𝑗𝑘+2^𝑇0₀𝑤_𝑗−1(𝑠)²_𝑘+2𝑑𝑠≤𝐶_𝑘𝐾𝑇₀sup_{𝑡∈0,𝑇0}𝑤_𝑗−1(𝑡)²_𝑘+2. (62) Choose 𝑇₀≪1 such that 𝐶_𝑘𝐾𝑇₀<1. Then the sequence (𝑢̇_𝑗,𝑣̇_𝑗) converges to (𝑢̇,𝑣)̇ which is the solution of (53). Then the solution (𝑢,𝑣) of (47) is obtained by (52).
Inequality (48) is obtained from (51), (55), and (58). This completes the proof of Theorem 8.