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

Authors: Maomao Cai; Dening Li; Chontita Rattanakul
Journal: ISRN Mathematical Physics
 (2013)

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## 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.


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## Body

## 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,(1)ut+uux+uxxx=-αuxx-γuxxxx,
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:
(2)ut+uux+uxxx=-αuxx-γuxxxx+ϵ1vx,vt+a1vx=Γvxx+ϵ2ux.
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:
(3)ut+uux+Δux=-αuxx-γΔ2u+ϵ1vx,vt+a1vx=ΓΔv+ϵ2ux.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 Section2, 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):
(4)u~u0+ϵu~,v~v0+ϵv~,
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~):
(5)u~t+u0u~x+Δu~x=-αu~xx-γΔ2u~+ϵ1v~x+f,v~t+a1v~x=ΓΔv~+ϵ2u~x+g.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:
(6)ut+u0ux+Δux=-αuxx-γΔ2u+ϵ1vx+f,vt+a1vx=ΓΔv+ϵ2ux+g,u(x,y,0)=u0(x,y),v(x,y,0)=v0(x,y).Here, the coefficientsα, 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 Hk(R2) be the usual Sobolev space defined by the norm
(7)∥f∥k2=∫R2‍∑|j|≤k‍|Djf|2dxdy
with H0(R2)=L2(R2) and ∥f∥=∥f∥0, where
(8)Djf=∂|j|f∂xj1∂yj2.LetS(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
(9)∂t(∥u(t)∥2+∥v(t)∥2)+∥u(t)∥22+∥v(t)∥12≤C(∥u(t)∥2+∥v(t)∥2+∥f(t)∥2+∥g(t)∥2),∀t∈[0,T],(10)sup0≤t≤T(∥u(t)∥2+∥v(t)∥2)+∫0T(∥u(s)∥22+∥v(s)∥12)ds≤C(∥u0∥2+∥v0∥2+∫0T(∥f(s)∥2+∥g(s)∥2)ds).
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.
TakeL2 inner product of the equations in (6) with (u,v) over R2. Noticing that w∂tw=(1/2)∂tw2, we have
(11)12∂t(∥u∥2+∥v∥2)+〈γΔ2u,u〉-〈ΓΔv,v〉+〈αuxx,u〉=-〈u0ux,u〉-〈a1vx,v〉-〈Δux,u〉+〈ϵ1vx,u〉+〈f,u〉+〈ϵ1ux,v〉+〈g,v〉.
Integrating by parts in(x,y) and noticing that for any δ>0(12)∥u∥2~∥Δu∥+∥u∥,∥u∥12≤14δ∥u∥22+Cδ∥u∥02,
we have
(13)〈γΔ2u,u〉-〈ΓΔv,v〉≥δ(∥Δu∥2+∥∇v∥2),|〈αuxx,u〉+〈aux,u〉+〈cvx,v〉+〈Δux,u〉+〈vx,u〉+〈ux,v〉|≤12δ(∥Δu∥2+∥∇v∥2)+C(∥u∥2+∥v∥2),|〈f,u〉+〈g,v〉|≤∥f∥2+∥g∥2+∥u∥2+∥v∥2.
Substituting (13) into (11), we obtain (9). To obtain (10), we apply Gronwall's theorem [10] to the inequality
(14)∂tP(t)≤CP(t)+M(t)
with
(15)P(t)=∥u(t)∥2+∥v(t)∥2+∫0t‍(∥u(s)∥22+∥v(s)∥12)ds,M(t)=∥f(t)∥2+∥g(t)∥2.
Therefore, we have
(16)e-CtP(t)-P(0)≤∫0t‍e-CsM(s)ds
and hence
(17)P(t)≤C1(t)(P(0)+∫0t‍M(s)ds),
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
(18)12∂t(∥∇u∥2+∥∇v∥2)+δ(∥∇Δu∥2+∥Δv∥2)≤-〈u0ux,Δu〉-〈Δux,Δu〉-〈αuxx,Δu〉+〈ϵ1vx,Δu〉+〈f,Δu〉-a1〈vx,Δv〉+〈ϵ2ux,Δv〉+〈g,Δv〉.
The right side of (18) can be controlled by
(19)12δ(∥∇Δu(t)∥2+∥Δv(t)∥2)+C(∥u(t)∥22+∥v(t)∥12+∥f(t)∥2+∥g(t)∥2)
with constant C depending on u0 only in L∞ norm. Combining (18) and (4) with (9) and noticing (12), we have the following:
(20)∂t(∥u(t)∥12+∥v(t)∥12)+∥u(t)∥32+∥v(t)∥22≤C(∥u(t)∥12+∥v(t)∥12+∥f(t)∥2+∥g(t)∥2),bbbbbbbbbbbbbbbbbbbbbb00bbbbbbb∀t∈[0,T].
Indeed, we can further improve (20) by taking L2 inner product of the first equation in (6) with Δ2u to derive
(21)12∂t∥Δu∥2+γ∥Δ2u∥2≤-〈u0ux,Δ2u〉-〈Δux,Δ2u〉-〈αuxx,Δ2u〉+〈ϵ1vx,Δ2u〉+〈f,Δ2u〉.
The right side of (21) can be controlled by
(22)12γ∥Δ2u∥2+C(∥u∥32+∥v∥12+∥f∥2).
Combining (21),  (22), and (20), we have
(23)∂t(∥u(t)∥12+∥Δu∥2+∥v(t)∥12)+∥u(t)∥42+∥v(t)∥22≤C(∥u(t)∥22+∥v(t)∥12+∥f(t)∥2+∥g(t)∥2),000000000000000000000000000000000∀t∈[0,T].
Applying Gronwall’s inequality to (20) and (23), we can obtain
(24)sup0≤t≤T(∥u(t)∥12+∥v(t)∥12)+∫0T‍(∥u(s)∥32+∥v(s)∥22)ds≤C(∥u0∥12+∥v0∥12+∫0T‍(∥f(s)∥2+∥g(s)∥2)ds)
or
(25)sup0≤t≤T(∥u(t)∥22+∥v(t)∥12)+∫0T‍(∥u(s)∥42+∥v(s)∥22)ds≤C(∥u0∥22+∥v0∥12+∫0T‍(∥f(s)∥2+∥g(s)∥2)ds).
We can also include the estimate for(ut,vt) in (25) by using the equations in (6) to obtain
(26)sup0≤t≤T(∥u(t)∥22+∥v(t)∥12)+∫0T‍(∥u(s)∥42+∥v(s)∥22+∥vt(s)∥02+∥ut(s)∥02)ds≤C(∥u0∥22+∥v0∥12+∫0T‍(∥f(s)∥2+∥g(s)∥2)ds).
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,
(27)sup0≤t≤T(∥u(t)∥k+22+∥v(t)∥k+12)+∫0T‍(∥u(s)∥k+42+∥v(s)∥k+22+∥vt(s)∥k2+∥ut(s)∥k2)ds≤Ck(∥u0∥k+22+∥v0∥k+12+∫0T‍(∥f(s)∥k2+∥g(s)∥k2)ds).
Here Ck is a constant depending only on T and coefficients of (6). In particular, the constant Ck depends upon u0 in the coefficients of (6) only in its L∞([0,T],Hk+2(R2)) norm.Remark 3.
For convenience, we will use the notationΠTk in the following to denote the product space of (u,v): (u,v)∈ΠTk if
(28)u∈C([0,T],Hk+2(R2))∩L2([0,T],Hk+4(R2))∩H1([0,T],Hk(R2)),v∈C([0,T],Hk+1(R2))∩L2([0,T],Hk+2(R2))∩H1([0,T],Hk(R2)).ΠTk is a Banach space equipped with the norm
(29)sup0≤t≤T(∥u(t)∥k+22+∥v(t)∥k+12)+∫0T‍(∥u(s)∥k+42+∥v(s)∥k+22+∥vt(s)∥k2+∥ut(s)∥k2)ds.


## 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∈Hk+2(R2) and v0∈Hk+1(R2) and f,g∈L2([0,T],Hk(R2));(ii)
γ≥δ>0 and Γ≥δ>0;(iii)
all the coefficientsξ≡(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 ΠTk satisfying estimate (27).Remark 5.
By the Sobolev imbedding theorem, fork≥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],Hk(R2)), then by (6), we can derive that (ut,vt) are also continuous. Therefore the solution (u,v) in Theorem 4 is a classical solution.First we rewrite (6) briefly as follows:
(30)ut+ℒ1(u,v)=f,vt+ℒ2(u,v)=g,u(x,y,0)=u0(x,y),v(x,y,0)=v0(x,y).Here(31)ℒ1(u,v)=u0ux+Δux+αuxx-ϵ1vx+γΔ2u,ℒ2(u,v)=a1vx-ϵ2ux-ΓΔv.
Consider the following one-parameter family of initial-value problems, denoted as 𝒜λ(λ∈[0,1]):
(32)ut+λℒ1(u,v)+(1-λ)Δ2u=f,vt+λℒ2(u,v)-(1-λ)Δv=g,u(x,y,0)=u0(x,y),v(x,y,0)=v0(x,y).Obviously, forλ=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 Ck in (27) being independent of the parameter λ∈[0,1].In the following we show that the conclusion on the existence of solution in Theorem4 is true for (32) for all λ∈[0,1]. In particular the conclusion in Theorem 4 is the case λ=1.Letℬ⊂[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:
(33)ut+Δ2u=f,u(x,y,0)=u0(x,y),vt-Δv=g,v(x,y,0)=v0(x,y).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].Letλj∈ℬ and λj→λ0. Let (uj,vj) be the solution of the following initial-value problem:
(34)ujt+λjℒ1(uj,vj)+(1-λj)Δ2uj=f,vjt+λjℒ2(uj,vj)-(1-λj)Δvj=g,uj(x,y,0)=u0(x,y),vj(x,y,0)=v0(x,y).By (27), (uj,vj) is uniformly bounded in ΠTk. In particular
(35)∫0T‍(∥uj(s)∥k+42+∥vj(s)∥k+22)ds≤Ck(∫0T‍∥u0∥k+22+∥v0∥k+120000000000+∫0T‍(∥f(s)∥k2+∥g(s)∥k2)ds)≤K.Let(u-j,v-j)=(uj-uj-1,vj-vj-1)(j=2,3,…) which satisfies
(36)u-jt+λjℒ1(u-j,v-j)+(1-λj)Δ2u-j=-(λj-λj-1)[ℒ1(uj-1,vj-1)-Δ2uj-1],v-jt+λjℒ2(u-j,v-j)-(1-λj)Δv-j=-(λj-λj-1)[ℒ2(uj-1,vj-1)+Δvj-1],u-j(x,y,0)=0,v-j(x,y,0)=0.Applying (27) to the solution (u-j,v-j) of (36), we have
(37)sup0≤t≤T(∥u-j(t)∥k+22+∥v-j(t)∥k+12)+∫0T‍(∥u-j(s)∥k+42+∥v-j(s)∥k+22)ds≤Ck|(λj-λj-1)|2∫0T‍(∥uj-1(s)∥k+420000000000000000000000000+∥vj-1(s)∥k+22)ds≤Ck|(λj-λj-1)|2K.Sinceλj→λ0, it follows that (uj,vj) is a Cauchy sequence in ΠTk 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].Letλ0∈ℬ and λ∈[0,1] with |λ-λ0|≤ϵ.Let(u1,v1) be the solution of the following problem:
(38)u1t+λ0ℒ1(u1,v1)+(1-λ0)Δ2u1=f,v1t+λ0ℒ2(u1,v1)-(1-λ0)Δv1=g,u1(x,y,0)=u0(x,y),v1(x,y,0)=v0(x,y).We can construct a sequence of solutions(uj,vj)(j=2,3,…) as the solution of the following problem:
(39)ujt+λ0ℒ1(uj,vj)+(1-λ0)Δ2uj=f+(λ0-λ)(ℒ1(uj-1,vj-1)-Δ2uj-1),vjt+λℒ2(uj,vj)-(1-λ0)Δvj=g+(λ0-λ)(ℒ2(uj-1,vj-1)-Δvj-1),uj(x,y,0)=u0(x,y),vj(x,y,0)=v0(x,y).Then(u-j,v-j)=(uj-uj-1,vj-vj-1)(j=2,3,…) satisfies
(40)u-jt+λ0ℒ1(u-j,v-j)+(1-λ0)Δ2u-j=(λ0-λ)(ℒ1(u-j-1,v-j-1)-Δ2u-j-1),v-jt+λℒ2(u-j,v-j)-(1-λ)Δv-j=(λ0-λ)(ℒ2(u-j-1,v-j-1)-Δv-j-1),u-j(x,y,0)=0,v-j(x,y,0)=0.Since(41)∥ℒ1(u-j-1,v-j-1)-Δ2u-j-1∥k2+∥ℒ2(u-j-1,v-j-1)-Δv-j-1∥k2≤M(∥u-j-1(s)∥k+42+∥v-j-1(s)∥k+22),therefore by (27), (u-j,v-j) satisfies
(42)sup0≤t≤T(∥u-j(t)∥k+22+∥v-j(t)∥k+12)+∫0T‍(∥u-j(s)∥k+42+∥v-j(s)∥k+22)ds≤CkM|(λ-λ0)|2∫0T‍(∥u-j-1(s)∥k+42+∥v-j-1(s)∥k+22)ds.Chooseϵ≪1 such that CkMϵ2<1/2 and (uj,vj) is a Cauchy sequence with limit (u,v) being the solution of (32) for λ. Hence ℬ is open.This concludes the proof of Theorem4.To prepare the study of the nonlinear system (3) in the next section, we introduce the uniformly local Sobolev space Huls(R2); see also [13].Definition 6.
The uniformly local Sobolev spaceHuls(Rn) is defined byf∈Huls(Rn) if and only if, for all ϕ∈C0∞(Rn) and ϕX0(X)≡ϕ(X-X0),
(43)supX0∈Rn∥fϕX0∥Hs(Rn)≤∞.
A corresponding norm is defined as(44)∥f∥Hs(R2),ul=supX0∈R2∥fϕX0∥Hs(R2)
for a fixedϕ∈C0∞(R2) with ϕ(X)=1 in |X|≤1.It is readily verified thatHuls is a Hilbert space.Then we have the following improved version of Theorem4.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
(45)u0∈C([0,T];Hulk+2(R2)),
then the same conclusion in Theorem 4 is true.To prove Theorem7, we notice that estimate (27) can be obtained assuming that u0∈C([0,T];Hulk+2(R2)). Since Hulk+2(R2) is a Banach algebra, we have in Step (2) (to show ℬ is closed) in the proof of Theorem 4(46)∥u0u(j-1)x∥k2≤∥u0u(j-1)∥k+22≤C∥u0∥k+22∥u(j-1)∥k+22.
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:(47)ut+uux+Δux=-αuxx-γΔ2u+ϵ1vx,vt+a1vx=ΓΔv+ϵ2ux,u(x,y,0)=u0(x,y),v(x,y,0)=v0(x,y).
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.
Letk be a nonnegative integer and assume that   (i)
all the coefficientsγ, 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+u~0,V0+v~0) with (U0,V0) being constants and (u~0,v~0)∈Hk+2(R2)×Hk+1(R2). Then there is a T>0 such that (47) has a unique solution (u,v)=(U0+u~,V0+v~)∈C([0,T];Hulk+2(R2)×Hulk+1(R2) satisfying
(48)sup0≤t≤T(∥u~(t)∥k+22+∥v~(t)∥k+12)+∫0T‍(∥u~(s)∥k+42+∥v~(s)∥k+22+∥u~t(s)∥k2+∥v~t(s)∥k2)ds≤Ck(∥u~0∥k+22+∥v~0∥k+12).
For k≥2, such solution is also a classical solution and satisfies (47) in the classical sense.Proof of Theorem8.
We use linear iteration to prove Theorem8. First we construct an approximate solution (ua,va) of the form
(49)(ua,va)=(U0+u~a,V0+v~a),
where (u~a,v~a) is the solution for the initial-value problem of the linear system:
(50)u~at+(U0+u~0)u~ax+Δu~ax+αu~axx+γΔ2u~a-ϵ1v~ax=0,v~at+a1v~ax-ΓΔv~a-ϵ2u~ax=0,u~a(x,y,0)=u~0(x,y),v~a(x,y,0)=v~0(x,y).
From Theorems4 and 7, solution (ua,va)∈ΠTk of (50) exists and satisfies
(51)sup0≤t≤T(∥u~a(t)∥k+22+∥v~a(t)∥k+12)+∫0T‍(∥u~a(s)∥k+42+∥v~a(s)∥k+22+∥u~at(s)∥k2+∥v~at(s)∥k2)ds≤Ck(∥u~0∥k+22+∥v~0∥k+12).
Now we are looking for the solution of (47) in the form of
(52)(u,v)=(ua+u˙,va+v˙).
Obviously (u,v) is the solution of (47) if and only if (u˙,v˙) is the solution of the following problem:
(53)u˙t+(ua+u˙)u˙x+Δu˙x+αu˙xx+γΔ2u˙-ϵ1v˙x+uaxu˙=f~,v˙t+a1v˙x-ΓΔv˙-ϵ2u˙x=g~,u˙(x,y,0)=0,v˙(x,y,0)=0,
with
(54)f~=-(γΔ2u~a-ϵ1v~ax+uauaxu~at+uau~ax+Δu~ax+αu~axx000000+γΔ2u~a-ϵ1v~ax+uauax),g~=-(v~at+a1v~ax-ΓΔv~a-ϵ2u~ax).
From (51), (f~,g~)∈L2([0,T],Hk(R2)) and ∫0T‍(∥f~(s)∥k2+∥g~(s)∥k2)ds is controlled by (u~a,v~a) norms in ΠTk and consequently controlled by (∥u~0∥k+22+∥v~0∥k+12):
(55)∫0T‍(∥f~(s)∥k2+∥g~(s)∥k2)ds≤Ck(∥u~0∥k+22+∥v~0∥k+12).
Denote the nonlinear differential operators on the left side of the equations in (53) as ℳ(u˙); we can rewrite (53) briefly as
(56)ℳ(u˙)(u˙,v˙)=(f~,g~),u˙(x,y,0)=0,v˙(x,y,0)=0.
Takeu˙0(x,y,t)=v˙0(x,y,t)=0. By Theorem 7, one can obtain a sequence of solutions (u˙j,v˙j), for j=1,2,3,…, in the product space ΠTk by solving the linearized problem
(57)ℳ(u˙j-1)(u˙j,v˙j)=(f~,g~),u˙j(x,y,0)=0,v˙j(x,y,0)=0.
From (27) of Theorem 2, (u˙j,v˙j) satisfy
(58)sup0≤t≤T(∥u˙j(t)∥k+22+∥v˙j(t)∥k+12)+∫0T‍(∥u˙j(s)∥k+42+∥v˙j(s)∥k+2200000000000+∥v˙jt(s)∥k2+∥u˙jt(s)∥k2)ds≤Ck,j∫0T‍(∥f~(s)∥k2+∥g~(s)∥k2)ds.
In particular, we notice that the constantCk,j in (58) depends upon u˙j-1 only in its L∞([0,T],Hk+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,
(59)sup0≤t≤T∥u˙j-1(t)∥k+22≤K,
then the constant Ck,j=Ck in (58) is uniform in j.
TakeT0≪1 such that
(60)Ck∫0T0‍(∥f~(s)∥k2+∥g~(s)∥k2)ds≤K.
Then (58) implies that (59) is satisfied for all j. We conclude that for such T0, the sequence (u˙j,v˙j) is uniformly bounded in the product space ΠT0k.
Let(wj,zj)=(u˙j+1-u˙j,v˙j+1-v˙j). (wj,zj) satisfies
(61)ℳ(u˙j)(wj,zj)=-(ℳ(u˙j)-ℳ(u˙j-1))(u˙j,v˙j)=(-wj-1∂xu˙j,0),wj(x,y,0)=zj(x,y,0)=0.
Again from (27), we have
(62)sup0≤t≤T0(∥wj(t)∥k+22+∥zj(t)∥k+12)+∫0T0‍(∥wj(s)∥k+42+∥zj(s)∥k+2200000000000+∥wjt(s)∥k2+∥zjt(s)∥k2)ds≤Ck∫0T0‍∥wj-1(s)u˙jx(s)∥k2ds≤Cksupt∈[0,T0]∥u˙j∥k+2∫0T0‍∥wj-1(s)∥k+22ds≤CkKT0supt∈[0,T0]∥wj-1(t)∥k+22.
Choose T0≪1 such that CkKT0<1. Then the sequence (u˙j,v˙j) converges to (u˙,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.



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*Source: 673546-2013-08-07.xml*