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).
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,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.
Set p(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 typeut+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 systemut+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
u(t,x)∈C([0,∞);Lx2(R))∩C((0,∞);Hx1(R)),η(t,x)∈C([0,∞);Lx2(R)).
To proof Theorem 1.1, we will first establish global well-posedness of the following regularized two-component system with ε>0 given: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,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 equationu(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,
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 Section 2, 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 Section 3. First, we state the following lemma which consists of the crucial inequality involving the operator ∂x(1-∂x2)-1.
Lemma 2.1 (see [13]).
For g,h∈L2(ℝ),
‖∂x(1-∂x2)-1(gh)‖Lx2≤c‖g‖Lx2‖h‖Lx2,
or more generally
‖|∂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: ‖∫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)),‖∫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
‖∫0te(t-τ)∂x2f(u,∂xu,η)(τ,x)dτ‖LT∞Lx2≤supt∈[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
‖∫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
‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤supt∈[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
‖∫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: ‖∫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)),‖∫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
‖∫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 any u0∈H1(ℝ) and δ>0, there exists T1=T1(u0)>0 such that
‖∂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
‖∂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∂tv=∂x2v+f(u,∂xu,η),∂tμ=ε∂x2μ+g(u,∂xu,η,∂xη),v(x,0)=0,μ(x,0)=0,
that is,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: ‖f‖LT1Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),‖g‖LT1Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+‖∂xη‖LT2Lx22+T1/2‖∂xu‖LT2Lx2).
Proof.
First, let us prove (2.16),
‖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
‖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
‖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),
‖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
‖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: ‖∂xv‖LT2Lx2≤C(‖u‖LT2Lx22+‖∂xu‖LT2Lx22+‖η‖LT2Lx22+T1/2(‖u‖LT2Lx2+‖η‖LT2Lx2)),‖∂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
‖μ‖LT∞Lx2=‖∫0teε(t-τ)∂x2g(u,∂xu,η,∂xη)(τ,x)dτ‖LT∞Lx2≤∫0T‖g‖Lx2dt=‖g‖LT1Lx2.
Multiply μ to the second equation of (2.14) and integrate with respect to x over ℝ. After integration by parts, we have
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
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
‖∂xμ‖LT2Lx2≤1ε1/2‖g‖LT1Lx2.
By Lemma 2.5, (2.24) follows.
Similarly, we can also obtain (2.23).
3. Local Well-Posedness
Let z∶=(uη), A(z)∶=(et∂x2u0eεt∂x2η0),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 byΦ(z)=A(z)+B(z).
Theorem 3.1.
For any ε>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 appropriate a and T. Let z∈XaT, then we have
⟦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:
‖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:
‖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
‖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:
‖∂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
⟦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 of δ, 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 that Φ:XaT→XaT is a contraction mapping,
⟦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 any ε>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
uε(x,t)∈C([0,T);Lx2(R))∩C((0,T);Hx1(R)),ηε(x,t)∈C([0,T);Lx2(R)).
Proof.
Theorem 3.2 is merely Theorem 3.1 with a standard uniqueness argument.
Theorem 3.3.
For any ε>0 and (u0,η0)∈H1(ℝ)×L2(ℝ), there exists a unique global solution (uε,ηε) of (1.5) such that
uε(x,t)∈C([0,∞);Lx2(R))∩C((0,∞);Hx1(R)),ηε(x,t)∈C([0,∞);Lx2(R)).
Proof.
To prove Theorem 3.3, 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
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
12ddt∫Rη2dx+ε∫Rηx2dx+12∫Rη2uxdx-∫Ruηxdx=0.
From (3.12) and (3.13), we obtain the energy identity
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:
supt∈[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 Theorem 3.3, 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
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.
Let z0=(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
Tz0*<∞⟹∫0Tz0*‖∂xu(τ)‖L∞dτ=∞.
4. Proof of Theorem 1.1Proof of Theorem 1.1.
From Theorem 3.4, we have got the local solution of (1.3). On the other hand, according to the second equation of (1.3), we have
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
‖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
12ddt∫R(u2+ux2+η2)dx+ddt∫R(ux2+uxx2)dx=0,
which implies
supt∈[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<+∞,
∫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.
Acknowledgments
The work of the author is supported in part by the NSF of China under Grant no. 11001111 and no. 11141003. The author would like to thank the referees for constructive suggestions and comments.
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