# On the Global Well-Posedness of the Viscous Two-Component Camassa-Holm System

Authors: Xiuming Li
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We establish the local well-posedness for the viscous two-component Camassa-Holm system. Moreover, applying the energy identity, we obtain a global
existence result for the system with(u0,η0)∈H1(ℝ)×L2(ℝ).


---

## Body

## 1. Introduction

We are interested in the global well-pose dness of the initial value problesm associated to the viscous version of the two-component Camassa-Holm shallow water system [1–3], namely,(1.1)mt+umx+2uxm-Aux+ρρx=0,t>0,x∈R,m=u-uxx,,t>0,x∈R,ρt+(uρ)x=0,t>0,x∈R,
where the variable u(t,x) represents the horizontal velocity of the fluid or the radial stretch related to a prestressed state, and ρ(t,x) is related to the free surface elevation from equilibrium or scalar density with the boundary assumptions, u→0 and ρ→1 as |x|→∞. The parameter A>0 characterizes a linear underlying shear flow, so that (1.1) models wave-current interactions [4–6]. All of those are measured in dimensionless units.Setp(x):=(1/2)e-|x|, x∈ℝ. Then (1-∂x2)-1f=p*f for all f∈L2(ℝ), where * denotes the spatial convolution. Let η=ρ-1, (1.1) can be rewritten as a quasilinear nonlocal evolution system of the type(1.2)ut+uux=-∂x(1-∂x2)-1(u2+12ux2-Au+12η2+η),t>0,x∈R,ηt+uηx+ηux+ux=0,t>0,x∈R.The system (1.1) without vorticity, that is, A=0, was also rigorously justified by Constantin and Ivanov [1] to approximate the governing equations for shallow water waves. The multipeakon solutions of the same system have been constructed by Popivanov and Slavova [7], and the corresponding integral surface is partially ruled. Chen et al. [8] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of AKNS hierarchy. More recently, Holm et al. [9] proposed a modified two-component Camassa-Holm system which possesses singular solutions in component ρ. Mathematical properties of (1.1) with A=0 have been also studied further in many works. For example, Escher et al. [10] investigated local well-posedness for the two-component Camassa-Holm system with initial data (u0,ρ0)∈Hs×Hs-1 with s≥2 and derived some precise blow-up scenarios for strong solutions to the system. Constantin and Ivanov [1] provided some conditions of wave breaking and small global solutions. Gui and Liu [11] recently obtained results of local well-posedness in the Besov spaces and wave breaking for certain initial profiles. More recently, Gui and Liu [12] studied global existence and wave-breaking criteria for the system (1.2) with initial data (u0,ρ0-1)∈Hs×Hs-1 with s>(3/2).In this paper, we consider the global well-posedness of the viscous two-component Camassa-Holm system(1.3)ut+uux-uxx=-∂x(1-∂x2)-1(u2+12ux2-Au+12η2+η),t>0,x∈R,ηt+uηx+ηux+ux=0,t>0,x∈R,u(0,x)=u0(x),x∈R,η(0,x)=η0(x),x∈R.The goal of the present paper is to study global existence of solutions for (1.3) to better understand the properties of the two-component Camassa-Holm system (1.2). We state the main result as follows.Theorem 1.1.
For(u0,η0)∈H1(ℝ)×L2(ℝ), there exists a unique global solution  (u,η) of (1.3) such that
(1.4)u(t,x)∈C([0,∞);Lx2(R))∩C((0,∞);Hx1(R)),η(t,x)∈C([0,∞);Lx2(R)).To proof Theorem1.1, we will first establish global well-posedness of the following regularized two-component system with ε>0 given:(1.5)ut+uux-uxx=-∂x(1-∂x2)-1(u2+12ux2-Au+12η2+η),t>0,x∈R,ηt-εηxx+uηx+ηux+ux=0,t>0,x∈R,u(0,x)=u0(x),x∈R,η(0,x)=η0(x),x∈R,that is,(1.6)mt-mxx+umx+2uxm-Aux+ηηx+ηx=0,t>0,x∈R,ηt-εηxx+uηx+ηux+ux=0,t>0,x∈R,m=u-uxx,t≥0,x∈R,u(0,x)=u0(x),η(0,x)=η0(x),x∈R.
Due to the Duhamel’s principle, we can also rewrite (1.6) as an integral equation(1.7)u(t,x)=et∂x2u0+∫0te(t-τ)∂x2f(u,∂xu,η)dτ,η(t,x)=eεt∂x2η0+∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)dτ,
whereg(u,∂xu,η,∂xη)=-u∂xη-η∂xu-∂xu,
(1.8)f(u,∂xu,η)=-∂x(1-∂x2)-1(u2+12(∂xu)2-Au+12η2+η)-12∂x(u2),et∂x2u0=(e-4π2tξu0̂(ξ))∨, eεt∂x2η0=(e-4π2εtξη0̂(ξ))∨, here and in what follows, we denote the Fourier (or inverse Fourier) transform of a function f by f̂ (or f∨).The remainder of the paper is organized as follows. In Section2, we will set up and introduce some estimates for the nonlinear part of (1.5). In Section 3, we will get the local well-posedness of (1.3) by constructing the global well-posedness of (1.5) using the contraction argument and energy identity. The last section is devoted to the proof of Theorem 1.1.


## 2. Preliminaries

We will list some lemmas needed in Section3. First, we state the following lemma which consists of the crucial inequality involving the operator ∂x(1-∂x2)-1.Lemma 2.1 (see [13]).
Forg,h∈L2(ℝ),
(2.1)‖∂x(1-∂x2)-1(gh)‖Lx2≤c‖g‖Lx2‖h‖Lx2,
or more generally
(2.2)‖|∂x|s(1-∂x2)-1(gh)‖Lx2≤c‖g‖Lx2‖h‖Lx2,
for all s<3/2.The next two lemmas are regarding the nonlinear part of (1.5).Lemma 2.2.
Consider the following:(2.3)‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT∞Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),(2.4)‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤C(‖∂xu‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2).Proof.
Let us prove (2.3) firstly. Thanks to Lemma 2.1, the Sobolev embedding theorem Hx1(ℝ)↪Lx∞(ℝ), and the Hölder’s inequality, we have
(2.5)‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT∞Lx2≤sup⁡t∈[0,T]∫0t‖e(t-τ)∂x2f(u,∂xu,η)(τ,x)‖Lx2dτ≤∫0T‖u‖Lx22dt+12∫0T‖∂xu‖Lx22dt+A∫0T‖u‖Lx2dt+12∫0T‖η‖Lx22dt+∫0T‖η‖Lx2dt+∫0T‖u∂xu‖Lx2dt,
which yields that
(2.6)‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2≤∫0T‖u‖Lx22dt+12∫0T‖∂xu‖Lx22dt+AT1/2(∫0T‖u‖Lx22dt)1/2+12∫0T‖η‖Lx22dt+T1/2(∫0T‖η‖Lx22dt)1/2+∫0T‖u‖Hx1‖∂xu‖Lx2dt≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)).
Similarly, we can get
(2.7)‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤sup⁡t∈[0,T]∫0t‖eε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)‖Lx2dτ≤∫0T‖u∂xη‖Lx2dt+∫0T‖η∂xu‖Lx2dt+∫0T‖∂xu‖Lx2dt≤∫0T‖u‖Lx∞‖∂xη‖Lx2dt+∫0T‖η‖Lx∞‖∂xu‖Lx2dt+T1/2(∫0T‖∂xu‖Lx22dt)1/2,
and then
(2.8)‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤∫0T‖u‖Hx1‖∂xη‖Lx2dt+∫0T‖η‖Hx1‖∂xu‖Lx2dt+T1/2(∫0T‖∂xu‖Lx22dt)1/2≤∫0T‖∂xu‖Lx2‖∂xη‖Lx2dt+∫0T‖∂xη‖Lx2‖∂xu‖Lx2dt+T1/2(∫0T‖∂xu‖Lx22dt)1/2≤2∫0T‖∂xu‖Lx22dt+2∫0T‖∂xη‖Lx22dt+T1/2(∫0T‖∂xu‖Lx22dt)1/2,
which implies (2.4).Lemma 2.3.
Consider the following:(2.9)‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2≤CT1/2(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),(2.10)‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT2Lx2≤CT1/2(‖∂xu‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2).Proof.
We mainly prove (2.9). For this, we have that
(2.11)‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2≤(∫0T‖∫R(∫0te(t-τ)∂x2f(τ,x)dτ)2dx‖LT∞dt)1/2≤T1/2‖∫0te(t-τ)∂x2f(τ,x)dτ‖LT∞Lx2.
Therefore, applying Lemma 2.2, we can easily obtain (2.9).
Similarly, we can also obtain (2.10).Let us state the following lemma, which was obtained in [13] (up to a slight modification).Lemma 2.4.
For anyu0∈H1(ℝ) and  δ>0, there exists  T1=T1(u0)>0 such that
(2.12)‖∂xet∂x2u0‖LT2Lx2=(∫0T1∫R|∂xet∂x2u0|2dxdt)1/2≤δ.
For any η0∈L2(ℝ), ε>0, and δ>0, there exists T2=T2(η0,ε)>0 such that
(2.13)‖∂xeεt∂x2η0‖LT2Lx2=(∫0T2∫R|∂xeεt∂x2η0|2dxdt)1/2≤δ.Next, we consider the nonlinear part of (1.7). When written as v=u-et∂x2u0, μ=η-eεt∂x2η0, then we have(2.14)∂tv=∂x2v+f(u,∂xu,η),∂tμ=ε∂x2μ+g(u,∂xu,η,∂xη),v(x,0)=0,μ(x,0)=0,
that is,(2.15)v(x,t)=∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ,μ(x,t)=∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ.First, we have the following basic estimates.Lemma 2.5.
Consider the following:(2.16)‖f‖LT1Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),(2.17)‖g‖LT1Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2).Proof.
First, let us prove (2.16),
(2.18)‖f‖LT1Lx2=‖12∂x(u2)+∂x(1-∂x2)-1(u2+12(∂xu)2-Au+12η2+η)‖LT1Lx2≤∫0T‖12∂x(u2)+∂x(1-∂x2)-1(u2+12(∂xu)2-Au+12η2+η)‖Lx2dt≤12∫0T‖∂x(1-∂x2)-1(∂xu)2‖Lx2dt+∫0T‖∂x(1-∂x2)-1(u2)‖Lx2dt+∫0T‖∂x(1-∂x2)-1(Au)‖Lx2dt+12∫0T‖∂x(1-∂x2)-1(η2)‖Lx2dt+∫0T‖u∂xu‖Lx2dt+∫0T‖∂x(1-∂x2)-1(η)‖Lx2dt,
which implies
(2.19)‖f‖LT1Lx2≤12∫0T‖∂xu‖Lx22dt+∫0T‖u‖Lx22dt+∫0T‖u‖Lx∞‖∂xu‖Lx2dt+A∫0T‖u‖Lx2dt+12∫0T‖η‖Lx22dt+∫0T‖η‖Lx2dt,
then we can get that
(2.20)‖f‖LT1Lx2≤12∫0T‖∂xu‖Lx22dt+C∫0T‖u‖Lx22dt+∫0T‖u‖Hx1‖∂xu‖Lx2dt+12∫0T‖η‖Lx22dt+T1/2(∫0T‖η‖Lx22dt)1/2+AT1/2(∫0T‖u‖Lx22dt)1/2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),
where we applied Lemma 2.1, Sobolev embedding theorem Hx1(ℝ)↪Lx∞(ℝ), and Hölder’s inequality. This proves (2.16).
Next, we prove (2.17),
(2.21)‖g‖LT1Lx2=‖-u∂xη-η∂xu-∂xu‖LT1Lx2≤∫0T‖u∂xη+η∂xu+∂xu‖Lx2dt≤∫0T‖u∂xη‖Lx2dt+∫0T‖η∂xu‖Lx2dt+∫0T‖∂xu‖Lx2dt≤∫0T‖u‖Lx∞‖∂xη‖Lx2dt+∫0T‖η‖Lx∞‖∂xu‖Lx2dt+∫0T‖∂xu‖Lx2dt,
which yields that
(2.22)‖g‖LT1Lx2≤∫0T‖u‖Hx1‖∂xη‖Lx2dt+∫0T‖η‖Hx1‖∂xu‖Lx2dt+∫0T‖∂xu‖Lx2dt≤C(∫0T‖u‖Lx22dt+∫0T‖∂xu‖Lx22dt+∫0T‖η‖Lx22dt+∫0T‖∂xη‖Lx22dt+∫0T‖∂xu‖Lx2dt)≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2),
and this ends the proof of (2.17).Then we have some estimates for∂xv and ∂xμ.Lemma 2.6.
Consider the following:(2.23)‖∂xv‖LT2Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),(2.24)‖∂xμ‖LT2Lx2≤Cε-1/2(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2).Proof.
We mainly prove (2.24). We have that
(2.25)‖μ‖LT∞Lx2=‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤∫0T‖g‖Lx2dt=‖g‖LT1Lx2.
Multiplyμ to the second equation of (2.14) and integrate with respect to x over ℝ. After integration by parts, we have
(2.26)ddt∫Rμ22dx+ε∫R(∂xμ)2dx=∫Rμgdx,12∫Rμ2(T)dx-12∫Rμ2(0)dx+ε∫0T∫R(∂xμ)2dxdt=∫0T∫Rμgdxdt,
for any ε>0, which implies
(2.27)ε∫0T∫R(∂xμ)2dxdt≤∫0T∫Rμgdxdt,‖∂xμ‖LT2Lx22≤1ε∫0T‖μ‖Lx2‖g‖Lx2dt≤1ε‖μ‖LT∞Lx2‖g‖LT1Lx2.
By (2.25), together with (2.27), we have that
(2.28)‖∂xμ‖LT2Lx2≤1ε1/2‖g‖LT1Lx2.
By Lemma 2.5, (2.24) follows.
Similarly, we can also obtain (2.23).


## 3. Local Well-Posedness

Letz∶=(uη), A(z)∶=(et∂x2u0eεt∂x2η0),(3.1)B(z)∶=(∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ),D1∶=C([0,T);Lx2(R))∩C((0,T);Hx1(R)),D2∶=C([0,T);Lx2(R)),XaT∶={z∈D1×D2:⟦z⟧=‖z-A(z)‖LT∞Lx2+‖∂xz‖LT2Lx2+‖z‖LT2Lx2≤a},
and define the mapping Φ:XaT→XaT by(3.2)Φ(z)=A(z)+B(z).Theorem 3.1.
For anyε>0, there exist  T=Tε>0 and  a>0 such that  Φ(XaT)⊆XaT. In addition,  Φ:XaT→XaT is a contraction mapping.Proof.
We first need to show that the map is well defined for some appropriatea and T. Let z∈XaT, then we have
(3.3)⟦Φz⟧=‖Φz-A(z)‖LT∞Lx2+‖Φz‖LT2Lx2+‖∂x(Φz)‖LT2Lx2.
Considering the terms in (3.3) one by one, from Lemma 2.2, the first term in (3.3) can be estimated as follows:
(3.4)‖Φz-A(z)‖LT∞Lx2=‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT∞Lx2+‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+‖∂xη‖LT2Lx22)+CT1/2(‖u‖LT2Lx2+‖∂xu‖LT2Lx2+‖η‖LT2Lx2)≤C⟦z⟧2+CT1/2⟦z⟧.
From Lemma 2.3, the second term in (3.3) can be estimated as follows:
(3.5)‖Φz‖LT2Lx2=‖et∂x2u0+∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2+‖eεt∂x2η0+∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT2Lx2≤‖et∂x2u0‖LT2Lx2+‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2+‖eεt∂x2η0‖LT2Lx2+‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT2Lx2,
which implies
(3.6)‖Φz‖LT2Lx2≤T1/2‖u0‖Lx2+‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2+T1/2‖η0‖Lx2+‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT2Lx2≤T1/2‖z0‖Lx2+CT1/2⟦z⟧2+CT⟦z⟧.
From Lemma 2.6, the third term in (3.3) can be estimated as follows:
(3.7)‖∂x(Φz)‖LT2Lx2=‖∂xet∂x2u0+∂x∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT2Lx2+‖∂xeεt∂x2η0+∂x∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT2Lx2≤‖∂xet∂x2u0‖LT2Lx2+‖∂xv‖LT2Lx2+‖∂xeεt∂x2η0‖LT2Lx2+‖∂xμ‖LT2Lx2≤2δ+C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2))+Cε1/2(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2)≤2δ+C(1+1ε1/2)⟦z⟧2+CT1/2(1+1ε1/2)⟦z⟧.
Combining (3.4)–(3.7), we have that
(3.8)⟦Φz⟧≤2δ+T1/2‖z0‖Lx2+C⟦z⟧2+CT1/2⟦z⟧2+C(1+1ε1/2)⟦z⟧2+CT⟦z⟧+CT1/2(2+1ε1/2)⟦z⟧≤2δ+T1/2‖z0‖Lx2+C(2+T1/2)(1+1ε1/2)a2+CTa+CT1/2(2+1ε1/2)a.
With appropriate values ofδ, a, and T, we are able to have that ⟦Φz⟧≤a, that is, Φ:XaT→XaT is well defined.
Similar to the above argument, we can show thatΦ:XaT→XaT is a contraction mapping,
(3.9)⟦Φz1-Φz2⟧≤C′⟦z1-z2⟧,
where C′=C′(T,a,ε,∥z1∥LT2Lx2,∥z2∥LT2Lx2,∥∂xz1∥LT2Lx2,∥∂xz2∥LT2Lx2) can be chosen as 0<C′<1 with appropriate values of T and a.Theorem 3.2.
For anyε>0 and  (u0,η0)∈H1(ℝ)×L2(ℝ), there exist a  T=T(u0,η0,ε)>0 and a unique solution  (uε,ηε) of (1.5) such that
(3.10)uε(x,t)∈C([0,T);Lx2(R))∩C((0,T);Hx1(R)),ηε(x,t)∈C([0,T);Lx2(R)).Proof.
Theorem3.2 is merely Theorem 3.1 with a standard uniqueness argument.Theorem 3.3.
For anyε>0 and  (u0,η0)∈H1(ℝ)×L2(ℝ), there exists a unique global solution  (uε,ηε) of (1.5) such that
(3.11)uε(x,t)∈C([0,∞);Lx2(R))∩C((0,∞);Hx1(R)),ηε(x,t)∈C([0,∞);Lx2(R)).Proof.
To prove Theorem3.3, we need only to establish the a priori energy identity. Multiplying the first equation in (1.6) by u and integrating by parts (with respect to x over ℝ), we have that
(3.12)12ddt∫R(u2+ux2)dx+∫R(ux2+uxx2)dx-12∫Rη2uxdx+∫Ruηxdx=0,
where we used the relation m=u-uxx and ∫ℝu2uxdx=0. Multiplying the first equation in (1.6) by η and integrating by parts (with respect to x over ℝ), we have that
(3.13)12ddt∫Rη2dx+ε∫Rηx2dx+12∫Rη2uxdx-∫Ruηxdx=0.
From (3.12) and (3.13), we obtain the energy identity
(3.14)12ddt∫R(u2+ux2+η2)dx+∫R(ux2+uxx2)dx+ε∫Rηx2dx=0,
which gives rise to the following inequality independent of ε and T:
(3.15)sup⁡t∈[0,T)(‖u(t,⋅)‖H12+‖η(t,⋅)‖L22)+2‖ux‖LT2(H1)2≤‖u0‖H12+‖η0‖L22.
According to Theorem 3.2 and the energy inequality (3.15), one can extend the local solution to the global one by a standard contradiction argument, which completes the proof of Theorem 3.3.From Theorem3.3, one has the local well-posedness of system (1.3).Theorem 3.4.
For(u0,η0)∈H1(ℝ)×L2(ℝ), there exist a  T=T(u0,η0)>0 and a unique local solution  (u,η) of (1.3) such that
(3.16)u(x,t)∈C([0,T);Lx2(R))∩C((0,T);Hx1(R)),η(x,t)∈C([0,T);Lx2(R)).Similar to the proof of Theorem  4.1 in [12] (up to a slight modification), we may get the following.Theorem 3.5.
Letz0=(u0,η0)∈Hs×Hs-1,  s≥1, and let  z=(u,η) be the corresponding solution to (1.3). Assume that   Tz0*>0 is the maximal time of existence, then
(3.17)Tz0*<∞⟹∫0Tz0*‖∂xu(τ)‖L∞dτ=∞.


## 4. Proof of Theorem1.1

Proof of Theorem1.1.
From Theorem3.4, we have got the local solution of (1.3). On the other hand, according to the second equation of (1.3), we have
(4.1)12ddt‖η‖Lx22≤2‖∂xu‖Lx∞‖η‖Lx22+‖∂xu‖Lx2‖η‖Lx2≤4‖∂xu‖Lx∞‖η‖Lx22+‖∂xu‖Lx2‖η‖Lx2≤4‖∂xu‖Lx∞‖η‖Lx22+‖η‖Lx22+14‖∂xu‖Lx22=(4‖∂xu‖Lx∞+1)‖η‖Lx22+14‖∂xu‖Lx22.
An application of Gronwall’s inequality yields
(4.2)‖η‖Lx22≤(‖η0‖Lx22+14‖∂xu‖LT2Lx22)e∫0t(1+4‖∂xu‖Lx∞)dτ.
Similar to the proof of Theorem 3.3, we may get the energy identity
(4.3)12ddt∫R(u2+ux2+η2)dx+ddt∫R(ux2+uxx2)dx=0,
which implies
(4.4)sup⁡t∈[0,T)(‖u(t,⋅)‖H12+‖η(t,⋅)‖L22)+2‖ux‖LT2(H1)2≤‖u0‖H12+‖η0‖L22.
Due to the Sobolev embedding theorem H1(ℝ)↪L∞(ℝ) and (4.4), we obtain that for any T<+∞,
(4.5)∫0T‖∂xu‖Lx∞dt≤∫0T‖∂xu‖Hx1dt≤T1/2‖∂xu‖LT2(Hx1)<+∞.
Therefore, from Theorem 3.5, we deduce that the maximal existence time T=+∞. This proves Theorem 1.1.



---

*Source: 327572-2012-05-03.xml*