# The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

Authors: Nan Li; Shaoyong Lai; Shuang Li; Meng Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev spaceHS(R) with s>3/2 is established via a limiting procedure. Provided that the initial value u0 satisfies the sign condition and u0∈Hs(R)(s>3/2), it is shown that there exists a unique
global solution for the equation in space C([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)).


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## Body

## 1. Introduction

Camassa and Holm [1] employed the Hamiltonian method to derive a completely integrable shallow water wave model(1.1)ut-utxx+2kux+3uux=2uxuxx+uuxxx,
which was alternatively established as a water wave equation in [2–4]. Equation (1.1) also models wave current interaction [5], while Dai [6] derived it as a model in elasticity (see [7]). In addition, it was pointed out in Lakshmanan [8] that the Camassa-Holm equation (1.1) could be relevant to the modeling of tsunami waves (see Constantin and Johnson [9]).After the birth of the Camassa-Holm equation (1.1), many works have been carried out to probe its dynamic properties. For k=0, (1.1) has travelling wave solutions of the form ce-|x-ct|, called peakons, which describes an essential feature of the travelling waves of largest amplitude (see [10–14]). For k>0, its solitary waves are stable solitons [15]. It is shown in [16–18] that the inverse spectral or scattering approach is a powerful tool to handle the Camassa-Holm equation and analyze its dynamics. It is worthwhile to mention that (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group [19–21], and this geometric illustration leads to a proof that the least action principle holds. Xin and Zhang [22] proved the global existence of the weak solution in the energy space H1(R) without any sign conditions on the initial value, and the uniqueness of this weak solution is obtained under some assumptions on the solution [23]. Coclite et al. [24] extended the analysis presented in [22, 23] and obtained many useful dynamic properties to other partial differential equations (see [25–28] for an alternative approach). Li and Olver [29] established the local well posedness in the Sobolev space Hs(R) with s>3/2 for (1.1) and gave conditions on the initial data that lead to finite time blowup of certain solutions. It is shown in Constantin and Escher [30] that the blowup occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. For other methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [31–39] and the references therein.Motivated by the work in Hakkaev and Kirchev [33] to investigate the generalization forms of the Camassa-Holm equation with high-order nonlinear terms, we study the following generalized Camassa-Holm equation:(1.2)ut-utxx+kumux+(m+3)um+1ux=(m+2)umuxuxx+um+1uxxx,
where m≥0 is a natural number and k≥0. Obviously, (1.2) reduces to (1.1) if we set m=0. As the Camassa-Holm equation (1.1) has been discussed by many mathematicians, we let the natural number m≥1 in this paper.The objective of this paper is to study (1.2). Its local well posedness of solutions in the Sobolev space Hs(R) with s>3/2 is developed by using the pseudoparabolic regularization method. Provided that (1-∂x2)u0+k/2(m+1)≥0 and u0∈Hs(s>3/2), the existence and uniqueness of the global solutions are established in space C([0,∞);Hs(R))⋂C1([0,∞);Hs-1(R)). It should be mentioned that the existence and uniqueness of global strong solutions for the nonlinear generalized Camassa-Holm models like (1.2) have never been investigated in the literatures.


## 2. Main Results

The space of all infinitely differentiable functionsϕ(t,x) with compact support in [0,+∞)×R is denoted by C0∞. Lp=Lp(R)(1≤p<+∞) is the space of all measurable functions h such that ∥h∥Lpp=∫R|h(t,x)|pdx<∞. We define L∞=L∞(R) with the standard norm ∥h∥L∞=inf⁡m(e)=0sup⁡x∈R∖e|h(t,x)|. For any real number s, Hs=Hs(R) denotes the Sobolev space with the norm defined by(2.1)‖h‖Hs=(∫R(1+|ξ|2)s|ĥ(t,ξ)|2dξ)1/2<∞,
where ĥ(t,ξ)=∫Re-ixξh(t,x)dx.ForT>0 and nonnegative number s, C([0,T);Hs(R)) denotes the Frechet space of all continuous Hs-valued functions on [0,T). We set Λ=(1-∂x2)1/2. For simplicity, throughout this paper, we let c denote any positive constant which is independent of parameter ε.We consider the Cauchy problem of (1.2), which has the equivalent form(2.2)ut-utxx=-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx,k≥0,m≥1,u(0,x)=u0(x).Now, we give our main results for problem (2.2).Theorem 2.1.
Suppose that the initial functionu0(x) belongs to the Sobolev space Hs(R) with s>3/2. Then there is a T>0,  which depends on ∥u0∥Hs, such that there exists a unique solution u(t,x) of the problem (2.2) and
(2.3)u(t,x)∈C([0,T];Hs(R))⋂C1([0,T];Hs-1(R)).Theorem 2.2.
Letu0(x)∈Hs,s>3/2 and (1-∂x2)u0+k/2(m+1)≥0 for all x∈R. Then problem (2.2) has a unique solution satisfying that
(2.4)u(t,x)∈C([0,∞);Hs(R))⋂C1([0,∞);Hs-1(R)).


## 3. Local Well-Posedness

In order to prove Theorem2.1, we consider the associated regularized problem(3.1)ut-utxx+εutxxxx=-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx,u(0,x)=u0(x),
where the parameter ε satisfies 0<ε<1/4.Lemma 3.1.
Letr and q be real numbers such that -r<q≤r.  Then
(3.2)‖uv‖Hq≤c‖u‖Hr‖v‖Hq,ifr>12,‖uv‖Hr+q-1/2≤c‖u‖Hr‖v‖Hq,ifr<12.This lemma can be found in [34, 40].Lemma 3.2.
Letu0(x)∈Hs(R) with s>3/2.  Then the Cauchy problem (3.1) has a unique solution u(t,x)∈C([0,T];Hs(R)) where T>0 depends on ∥u0∥Hs(R). If s≥2,  the solution u∈C([0,+∞);Hs) exists for all time.Proof.
Assuming thatD=(1-∂x2+ε∂x4)-1, we know that D:Hs→Hs+4 is a bounded linear operator. Applying the operator D on both sides of the first equation of system (3.1) and then integrating the resultant equation with respect to t over the interval (0,t) lead to
(3.3)u(t,x)=u0(x)+∫0tD[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx1m+2∂x3(um+2)]dt.
Suppose that both u and v are in the closed ball BM0(0) of radius M0 about the zero function in C([0,T];Hs(R)) and  A is the operator in the right-hand side of (3.3). For any fixed t∈[0,T], we get the following:
(3.4)‖∫0tD[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxxm+3m+2(um+2)x]dt-∫0tD[-km+1(vm+1)x-m+3m+2(vm+2)x+1m+2∂x3(vm+2)-(m+1)∂x(vmvx2)+vmvxvxxkm+1(vm+1)x]dt‖Hs≤TC1(sup⁡0≤t≤T‖um+1-vm+1‖Hs+sup⁡0≤t≤T‖um+2-vm+2‖Hs+sup⁡0≤t≤T‖D∂x[umux2-vmvx2]‖Hs+sup⁡0≤t≤T‖D[umuxuxx-vmvxvxx]‖Hs),
where C1 may depend on ε. The algebraic property of Hs0(R) with s0>1/2 derives
(3.5)‖um+2-vm+2‖Hs=‖(u-v)(um+1+umv+⋯+uvm+vm+1)‖Hs≤‖(u-v)‖Hs∑j=0m+1‖u‖Hsm+1-j‖v‖Hsj≤M0m+1‖(u-v)‖Hs,(3.6)‖um+1-vm+1‖Hs≤M0m‖(u-v)‖Hs,‖D∂x(umux2-vmvx2)‖Hs≤‖D∂x[um(ux2-vx2)]‖Hs+‖D∂x[vx2(um-vm)]‖Hs≤C(‖um(ux2-vx2)‖Hs-1+‖vx2(um-vm)‖Hs-1)≤CM0m+1‖u-v‖Hs.
Using the first inequality of Lemma 3.1, we have
(3.7)‖D[umuxuxx-vmvxvxx]‖Hs=‖12D[um(ux2)x-vm(vx2)x]‖Hs≤12(‖D[um(ux2-vx2)x]‖Hs+‖D[(vx2)x(um-vm)]‖Hs)≤C(‖um(ux2-vx2)x‖Hs-2+‖(vx2)x(um-vm)‖Hs-2)≤C(‖um‖Hs‖ux2-vx2‖Hs-1+‖vx2‖Hs-1‖um-vm‖Hs)≤CM0m+1‖u-v‖Hs,
where C may depend on ε. From (3.5)–(3.7), we obtain that
(3.8)‖Au-Av‖Hs≤θ‖u-v‖Hs,
where θ=TC2(M0m+M0m+1) and C2 is independent of 0<t<T. Choosing T sufficiently small such that θ<1, we know that A is a contraction. Applying the above inequality yields that
(3.9)‖Au‖Hs≤‖u0‖Hs+θ‖u‖Hs.
Choosing T sufficiently small such that θM0+∥u0∥Hs<M0, we deduce that A maps BM0(0) to itself. It follows from the contraction-mapping principle that the mapping A has a unique fixed-point u in BM0(0).
Fors≥2, using the first equation of system (3.1) derives
(3.10)ddt∫R(u2+ux2+εuxx2)dx=0,
from which we have the conservation law
(3.11)∫R(u2+ux2+εuxx2)dx=∫R(u02+u0x2+εu0xx2)dx.
The proof of the global existence result is a routine argument by using (3.11) (see Xin and Zhang [22]).Lemma 3.3 (Kato and Ponce [41]).
Ifr≥0,  then Hr⋂L∞ is an algebra. Moreover
(3.12)‖uv‖r≤c(‖u‖L∞‖v‖r+‖u‖r‖v‖L∞),
where c is a constant depending only on r.Lemma 3.4 (Kato and Ponce [41]).
Letr>0.  If u∈Hr⋂W1,∞ and v∈Hr-1⋂L∞,  then
(3.13)‖[Λr,u]v‖L2≤c(‖∂xu‖L∞‖Λr-1v‖L2+‖Λru‖L2‖v‖L∞).Lemma 3.5.
Lets≥2, and the function u(t,x) is a solution of problem (3.1) and the initial data u0(x)∈Hs.  Then the following inequality holds:
(3.14)‖u‖H12≤∫R(u2+ux2+εuxx2)dx=∫R(u02+u0x2+εu0xx2)dx.
Forq∈(0,s-1], there is a constant c independent of ε such that
(3.15)∫R(Λq+1u)2dx≤∫R[(Λq+1u0)2+ε(Λqu0xx)2]dx+c∫0t‖u‖Hq+12((‖u‖L∞m-1+‖u‖L∞m)‖ux‖L∞+‖u‖L∞m-1‖ux‖L∞2)dτ.
Forq∈[0,s-1], there is a constant c independent of ε such that
(3.16)(1-2ε)‖ut‖Hq≤c‖u‖Hq+1((‖u‖L∞m-1+‖u‖L∞m)‖u‖H1+‖u‖L∞m‖ux‖L∞+∥‖u‖L∞m-1‖ux‖L∞2)Proof.
The inequality∥u∥H12≤∫R(u2+ux2)dx and (3.11) derives (3.14).
Using∂x2=-Λ2+1 and the Parseval equality gives rise to
(3.17)∫RΛquΛq∂x3(um+2)dx=-(m+2)∫R(Λq+1u)Λq+1(um+1ux)dx+(m+2)∫R(Λqu)Λq(um+1ux)dx.
Forq∈(0,s-1], applying (Λqu)Λq to both sides of the first equation of system (3.1) and integrating with respect to x by parts, we have the identity
(3.18)12ddt∫R((Λqu)2+(Λqux)2+ε(Λquxx)2)dx=-km+1∫R(Λqu)Λq(um+1)xdx-(m+2)∫R(Λqu)Λq(um+1ux)dx-∫R(Λq+1u)Λq+1(um+1ux)dx+(m+1)∫R(Λqux)Λq(umux2)dx+∫RΛquΛq(umuxuxx)dx.
We will estimate the terms on the right-hand side of (3.18) separately. For the second term, by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have
(3.19)|∫R(Λqu)Λq(um+1ux)dx|=|∫R(Λqu)[Λq(um+1ux)-um+1Λqux]dx+∫R(Λqu)um+1Λquxdx|≤c‖u‖Hq((m+1)‖u‖L∞m‖ux‖L∞‖u‖Hq+‖ux‖L∞‖u‖L∞m‖u‖Hq)+m+12‖u‖L∞m‖ux‖L∞‖Λqu‖L22≤c‖u‖Hq2‖u‖L∞m‖ux‖L∞.
Similarly, for the first term in (3.18), we have
(3.20)|∫R(Λqu)Λq(umux)dx|≤c‖u‖Hq2‖u‖L∞m-1‖ux‖L∞.
Using the above estimate to the third term yields that
(3.21)|∫R(Λq+1u)Λq+1(um+1ux)dx|≤c‖u‖Hq+12‖u‖L∞m‖ux‖L∞.
For the fourth term, using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain that
(3.22)|∫R(Λqux)Λq(umux2)dx|≤‖Λqux‖L2‖Λq(umux2)‖L2≤c‖u‖Hq+1(‖umux‖L∞‖ux‖Hq+‖ux‖L∞‖umux‖Hq)≤c‖u‖Hq+12‖ux‖L∞‖u‖L∞m,
in which we have used ∥umux∥Hq≤c∥(um+1)x∥Hq≤c∥u∥L∞m∥u∥Hq+1.
For the last term in (3.18), using um(ux2)x=(umux2)x-(um)xux2 results in
(3.23)|∫R(Λqu)Λq(umuxuxx)dx|≤|∫RΛquxΛq(umux2)dx|+∫RΛquΛq[(um)xux2]dx=K1+K2.
For K1, it follows from (3.22) that
(3.24)K1≤c‖u‖Hq+12‖ux‖L∞‖u‖L∞m.
For K2, applying Lemma 3.3 derives
(3.25)K2≤c‖u‖Hq‖(um)xux2‖Hq≤c‖u‖Hq(‖(um)x‖L∞‖ux2‖Hq+‖(um)x‖Hq‖ux2‖L∞)≤c‖u‖Hq+12(‖u‖L∞m-1‖ux‖L∞2).
It follows from (3.19)–(3.25) that there exists a constant c depending only on m such that
(3.26)12ddt∫R[(Λqu)2+(Λqux)2+ε(Λquxx)2]dx≤c‖u‖Hq+12(‖ux‖L∞‖u‖L∞m+‖u‖L∞m-1‖ux‖L∞2).
Integrating both sides of the above inequality with respect to t results in (3.15).
To estimate the norm ofut, we apply the operator (1-∂x2)-1 to both sides of the first equation of system (3.1) to obtain the equation
(3.27)(1-ε)ut-εutxx=(1-∂x2)-1[-εut-kumux-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx].
Applying (Λqut)Λq to both sides of (3.27) for q∈[0,s-1] gives rise to
(3.28)(1-ε)∫R(Λqut)2dx+ε∫R(Λquxt)2dx=∫R(Λqut)Λq-2[-εut+∂x(-km+1um+1-m+3m+2um+2+1m+2∂x2um+2-(m+1)umux2)+umuxuxxkm+1]dτ.
For the right hand of (3.28), we have
(3.29)|∫R(Λqut)Λq-2(-εut)dx|≤ε‖ut‖Hq2,|∫R(Λqut)(1-∂x2)-1Λq∂x(-km+1um+1-m+3m+2um+2-(m+1)umux2)dx|≤c‖ut‖Hq(∫R(1+ξ2)q-1[∫R[-km+1um̂(ξ-η)û(η)-m+3m+2um+1̂(ξ-η)û(η)-(m+1)umux̂(ξ-η)ux̂(η)km+1]dη∫R]2)1/2≤c‖ut‖Hq‖u‖H1‖u‖Hq+1(‖u‖L∞m-1+‖u‖L∞m).
Since
(3.30)∫(Λqut)(1-∂x2)-1Λq∂x2(um+1ux)dx=-∫(Λqut)Λq(um+1ux)dx+∫(Λqut)(1-∂x2)-1Λq(um+1ux)dx,
using Lemma 3.3, ∥um+1ux∥Hq≤c∥(um+2)x∥Hq≤c(m+2)∥u∥L∞m+1∥u∥Hq+1 and ∥u∥L∞≤∥u∥H1, we have
(3.31)|∫(Λqut)Λq(um+1ux)dx|≤c‖ut‖Hq‖um+1ux‖Hq≤c‖ut‖Hq‖u‖L∞m‖u‖H1‖u‖Hq+1,|∫(Λqut)(1-∂x2)-1Λq(um+1ux)dx|≤c‖ut‖Hq‖u‖L∞m‖u‖H1‖u‖Hq+1.
Using the Cauchy-Schwartz inequality and Lemmas 3.1 and 3.3 yields that
(3.32)|∫R(Λqut)(1-∂x2)-1Λq(umuxuxx)dx|≤c‖ut‖Hq‖umuxuxx‖Hq-2≤c‖ut‖Hq‖um(ux2)x‖Hq-2≤c‖ut‖Hq‖[um(ux2)]x-(um)xux2‖Hq-2≤c‖ut‖Hq(‖umux2‖Hq-1+‖(um)xux2‖Hq-2)≤c‖ut‖Hq(‖umux2‖Hq+‖(um)xux2‖Hq)≤c‖ut‖Hq‖u‖Hq+1(‖u‖L∞m‖ux‖L∞+‖u‖L∞m-1‖ux‖L∞2),
in which we have used (3.25).
Applying (3.29)–(3.32) into (3.28) yields the inequality
(3.33)(1-2ε)‖ut‖Hq≤c‖u‖Hq+1((‖u‖L∞m-1+‖u‖L∞m)‖u‖H1+‖u‖L∞m‖ux‖L∞+‖u‖L∞m-1‖ux‖L∞2(‖u‖L∞m-1+‖u‖L∞m))
for a constant c>0. This completes the proof of Lemma 3.5.Remark 3.6.
In fact, lettingε=0 in problem (3.1), (3.14), (3.15), and (3.16) are still valid.Settingϕε(x)=ε-1/4ϕ(ε-1/4x) with 0<ε<1/4 and uε0=ϕε⋆u0, we know that uε0∈C∞ for any u0∈Hs,s>0. From Lemma 3.2, it derives that the Cauchy problem(3.34)ut-utxx+εutxxxx=-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxx,u(0,x)=uε0(x),x∈R
has a unique solution uε(t,x)∈C∞([0,∞);H∞).Furthermore, we have the following.Lemma 3.7.
Fors>0,u0∈Hs, it holds that
(3.35)‖uε0x‖L∞≤c‖u0x‖L∞,(3.36)‖uε0‖Hq≤cifq≤s,(3.37)‖uε0‖Hq≤cε(s-q)/4ifq>s,(3.38)‖uε0-u0‖Hq≤cε(s-q)/4ifq≤s,(3.39)‖uε0-u0‖Hs=o(1),
where c is a constant independent of ε.The proof of Lemma3.7 can be found in [38].Remark 3.8.
Fors≥1, using ∥uε∥L∞≤c∥uε∥H1/2+≤c∥uε∥H1, ∥uε∥H12≤c∫R(uε2+uεx2)dx, (3.14), (3.36), and (3.37), we know that
(3.40)‖uε‖L∞2≤c‖uε‖H1≤c∫R(uε02+uε0x2+εuε0xx2)dx≤c(‖uε0‖H12+ε‖uε0‖H22)≤c(c+cε×ε(s-2)/2)≤c0,
where c0 is independent of ε.Lemma 3.9.
Ifu0(x)∈Hs(R) with s≥1 such that ∥u0x∥L∞<∞.  Let uε0 be defined as in system (3.34). Then there exist two positive constants T and c,  which are independent of ε,  such that the solution uε of problem (3.34) satisfies ∥uεx∥L∞≤c for any t∈[0,T).Proof.
Using notationu=uε and differentiating both sides of the first equation of problem (3.34) or (3.27) with respect to x give rise to
(3.41)(1-ε)utx-εutxxx+1m+2∂2(um+2)-(m+12)(umux2)=km+1um+1+um+2-Λ-2[εutx+km+1um+1+um+2+(m+12)(umux2)+12∂x[(um)xux2]].
Letting p>0 be an integer and multiplying the above equation by (ux)2p+1 and then integrating the resulting equation with respect to x yield the equality
(3.42)1-ε2p+2ddt∫R(ux)2p+2dx-ε∫R(ux)2p+1utxxxdx+p-m2p+2∫R(ux)2p+3umdx=∫R(ux)2p+1(km+1um+1+um+2)dx-∫R(ux)2p+1Λ-2[εutx+km+1um+1+um+2+(m+12)(umux2)+12∂x[(um)xux2]]dx.
Applying the Hölder's inequality yields that
(3.43)1-ε2p+2ddt∫R(ux)2p+2dx≤{ε(∫R|utxxx|2p+2dx)1/(2p+2)+(∫R|um+1|2p+2dx)1/(2p+2)+(∫R|um+2|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)}×(∫R|ux|2p+2dx)(2p+1)/(2p+2)+|p-m2p+2|‖ux‖L∞‖u‖L∞m∫R|ux|2p+2dx
or
(3.44)(1-ε)ddt(∫R(ux)2p+2dx)1/(2p+2)≤{ε(∫R|utxxx|2p+2dx)1/(2p+2)+(∫R|um+1|2p+2dx)1/(2p+2)+(∫R|um+2|2p+2dx)1/(2p+2)+(∫R|G|2p+2dx)1/(2p+2)}+|p-m2p+2|‖ux‖L∞‖u‖L∞m(∫R|ux|2p+2dx)1/(2p+2),
where
(3.45)G=Λ-2[εutx+km+1um+1+um+2+(m+12)(umux2)+12∂x[(um)xux2]].
Since ∥f∥Lp→∥f∥L∞ as p→∞ for any f∈L∞⋂L2, integrating both sides of (3.44) with respect to t and taking the limit as p→∞ result in the estimate
(3.46)(1-ε)‖ux‖L∞≤(1-ε)‖u0x‖L∞+∫0t[12ε‖utxxx‖L∞+c(‖u‖L∞m+1+‖u‖L∞m+2+‖G‖L∞)+12‖u‖L∞m‖ux‖L∞2]dτ.
Using the algebraic property of Hs0(R) with s0>1/2 and (3.40) yields that
(3.47)‖u‖L∞m+2≤c‖u‖H1m+2≤c,(3.48)‖G‖L∞≤c‖G‖H1/2+=c‖Λ-2[εutx+um+1+um+2+(m+12)(umux2)+12∂x[(um)xux2]]‖H1/2+≤c(‖Λ-2uxt‖H1/2++‖Λ-2(umux2)‖H1/2++‖Λ-2∂x[(um)xux2]‖H1/2+)+c≤c(‖ut‖L2+‖umux2‖H0+‖(um)xux2‖H0)+c≤c(‖ut‖L2+‖u‖L∞m‖ux‖L∞‖u‖H1+‖ux‖L∞2‖u‖L∞m-1‖u‖H1)+c≤c(‖ut‖L2+‖ux‖L∞+‖ux‖L∞2)+c≤c(1+‖ux‖L∞2),
where we have used (3.16) and (3.40). Using (3.48), we have
(3.49)∫0t‖G‖L∞dτ≤c∫0t(1+‖ux‖L∞2)dτ,
where c is a constant independent of ε. Moreover, for any fixed r∈(1/2,1), there exists a constant cr such that ∥utxxx∥L∞≤cr∥utxxx∥Hr≤cr∥ut∥Hr+3. Using (3.16) and (3.40) yields that
(3.50)‖utxxx‖L∞≤c‖u‖Hr+4(1+‖ux‖L∞2).
Making use of the Gronwall's inequality to (3.15) with q=r+3, u=uε and (3.40) gives rise to
(3.51)‖u‖Hr+42≤(∫R(Λr+4u0)2+ε(Λr+3u0xx)2)exp⁡[c∫0t(1+‖ux‖L∞2)dτ].
From (3.36), (3.37), (3.50), and (3.51), one has
(3.52)‖utxxx‖L∞≤cε(s-r-4)/4(1+‖ux‖L∞2)exp⁡[c∫0t(1+‖ux‖L∞2)dτ].
For ε<1/4, it follows from (3.46), (3.49), and (3.52) that
(3.53)‖ux‖L∞≤‖u0x‖L∞+c∫0t[ε(s-r)/4(1+‖ux‖L∞2)exp⁡(c∫0τ(1+‖ux‖L∞2)dς)+1+‖ux‖L∞2]dτ.
It follows from the contraction mapping principle that there is aT>0 such that the equation
(3.54)‖W‖L∞=‖u0x‖L∞+c∫0t[(1+‖W‖L∞2)exp⁡(c∫0τ(1+‖W‖L∞2)dς)+1+‖W‖L∞2]dτ
has a unique solution W∈C[0,T]. Using the Theorem presented at page 51 in Li and Olver [29] or Theorem II in section I.1 presented in [42] yields that there are constants T>0 and c>0, which are independent of ε, such that ∥ux∥L∞≤W(t) for arbitrary t∈[0,T], which leads to the conclusion of Lemma 3.9.Lemma 3.10 (Li and Olver [29]).
Ifu and f are functions in Hq+1∩{∥ux∥L∞<∞}, then
(3.55)|∫RΛquΛq(uf)xdx|≤{cq‖f‖Hq+1‖u‖Hq2,q∈(12,1],cq(‖ux‖L∞‖u‖Hq2‖f‖Hq+1‖u‖Hq‖u‖L∞+‖fx‖L∞‖u‖Hq2+‖f‖Hq‖u‖Hq‖ux‖L∞),q∈(0,∞).Lemma 3.11.
Foru,v∈Hs(R) with s>3/2,  w=u-v, q>1/2,  and a natural number n,  it holds that
(3.56)|∫RΛswΛs(un+1-vn+1)xdx|≤c(‖w‖Hs‖w‖Hq‖v‖Hs+1+‖w‖Hs2).The proof of this Lemma can be found in [38].Lemma 3.12.
For problem (3.34), s>3/2 and  u0∈Hs(R),  there exist two positive constants c and M, which are independent of  ε, such that the following inequalities hold for any sufficiently small ε and t∈[0,T)(3.57)‖uε‖Hs≤Mect,(3.58)‖uε‖Hs+k1≤ε-k1/4Mect,k1>0,(3.59)‖uεt‖Hs+k1≤ε-(k1+1)/4Mect,k1>-1.Proof.
Ifs>3/2, u0∈Hs, we obtain that
(3.60)u0∈Hs1with1≤s1≤32,‖u0x‖L∞≤c‖u0x‖H1/2+≤c‖u0‖Hs≤c.
From Lemma 3.9, we know that there exist two constants T and c (both independent of ε) such that
(3.61)‖uεx‖L∞≤cforanyt∈[0,T).
Applying the inequality (3.15) with q+1=s and the bounded property of solution u (see (3.40) and (3.60)), we have
(3.62)∫R(Λsuε)2dx≤∫R[(Λsuε0)2+ε(Λs-1uε0xx)2]dx+c∫0t‖uε‖Hs2dτ,=A+c∫0t‖uε‖Hs2dτ,
where
(3.63)A=∫R[(Λsuε0)2+ε(Λs-1uε0xx)2]dx≤‖uε0‖Hs2+‖uε0‖Hs+12≤c+cεε-1/2≤2c,
in which we have used (3.36) and (3.37).
From (3.61) and (3.62) and using the Gronwall's inequality, we get the following:
(3.64)‖uε‖Hs≤2cect,
from which we know that (3.57) holds.
In a similar manner, forq+1=s+k1 and k1>0, applying (3.40) and (3.60) to (3.15), we have
(3.65)‖uε‖Hs+k12≤(cε-k1/2+cε-(k1+1)/2ε)+c∫0t‖uε‖Hs+k12dτ,
which results in (3.58) by using Gronwall's inequality.
From (3.16), for q=s+k1, we have
(3.66)(1-2ε)‖uεt‖Hs+k1≤c‖uε‖Hs+k1+1,
which leads to (3.59) by (3.58).Lemma 3.13.
If1/2<q<min⁡{1,s-1} and s>3/2,  then for any functions w,f defined on R,  it holds that
(3.67)|∫RΛqwΛq-2(wf)xdx|≤c‖w‖Hq2‖f‖Hq,(3.68)|∫RΛqwΛq-2(wxfx)xdx|≤c‖w‖Hq2‖f‖Hs.The proof of this lemma can be found in [38].Our next step is to demonstrate thatuε is a Cauchy sequence. Let uε and uδ be solutions of problem (3.34), corresponding to the parameters ε and δ, respectively, with 0<ε<δ<1/4, and let w=uε-uδ. Then w satisfies the problem(3.69)(1-ε)wt-εwxxt+(δ-ε)(uδt+uδxxt)=(1-∂x2)-1[[uεmuεxuεxx-uδmuδxuδxx]-εwt+(δ-ε)uδt-km+1∂x(uεm+1-uδm+1)-∂x(uεm+2-uδm+2)-∂x[∂x(uεm+1)∂xw+∂x(uεm+1-uεm+1)∂xuδ]+[uεmuεxuεxx-uδmuδxuδxx]km+1]-1m+2∂x(uεm+2-uδm+2),(3.70)w(x,0)=w0(x)=uε0(x)-uδ0(x).Lemma 3.14.
Fors>3/2,  u0∈Hs(R),  there exists T>0 such that the solution uε of (3.34) is a Cauchy sequence in C([0,T];Hs(R))⋂C1([0,T];Hs-1(R)).Proof.
Forq with 1/2<q<min⁡{1,s-1}, multiplying both sides of (3.69) by ΛqwΛq and then integrating with respect to x give rise to
(3.71)12ddt∫R[(1-ε)(Λqw)2+ε(Λqwx)2]dx=(ε-δ)∫R(Λqw)[(Λquδt)+(Λquδxxt)]dx-ε∫RΛqwΛq-2wtdx+(δ-ε)∫RΛqwΛq-2uδtdx-1m+2∫R(Λqw)Λq(uεm+2-uδm+2)xdx-km+1∫RΛqwΛq-2(uεm+1-uδm+1)xdx-∫RΛqwΛq-2(uεm+2-uδm+2)xdx-∫RΛqwΛq-2[∂x(uεm+1)∂xw]xdx-∫RΛqwΛq-2[∂x(uεm+1-uδm+1)∂xuδ]xdx+∫RΛqwΛq-2[uεmuεxuεxx-uδmuδxuδxx]dx.
It follows from the Schwarz inequality that
(3.72)ddt∫[(1-ε)(Λqw)2+ε(Λqwx)2]dx≤c{|∫RΛqwΛq-2[∂x(uεm+1-uδm+1)∂xuδ]xdx|‖Λqw‖L2[‖Λq-2uδt‖L2(δ-ε)(‖Λquδt‖L2+‖Λquδxxt‖L2)+ε‖Λq-2wt‖L2+(δ-ε)‖Λq-2uδt‖L2]+|∫RΛqwΛq(uεm+2-uδm+2)xdx||∫ΛqwΛq-2(uεm+1-uδm+1)xdx|+|∫ΛqwΛq-2(uεm+2-uδm+2)xdx|+|∫RΛqwΛq-2[∂x(uεm+1)∂xw]xdx|+|∫RΛqwΛq-2[∂x(uεm+1-uδm+1)∂xuδ]xdx|+|∫RΛqwΛq-2[uεmuεxuεxx-uδmuδxuδxx]dx|}.
Using the first inequality in Lemma3.10, we have
(3.73)|∫RΛqwΛq(uεm+2-uδm+2)xdx|=|∫RΛqwΛq(wgm+1)xdx|≤c‖w‖Hq2‖gm+1‖Hq+1,
where gm+1=∑j=0m+1uεm+1-juδj. For the last three terms in (3.72), using Lemmas 3.1 and 3.13, 1/2<q<min⁡{1,s-1}, s>3/2, the algebra property of Hs0 with s0>1/2 and (3.40), we have
(3.74)|∫RΛqwΛq-2(∂x(uεm+1)∂xw)xdx|≤c‖w‖Hq2‖uε‖Hsm+1,(3.75)|∫RΛqwΛq-2(∂x(uεm+1-uδm+1)∂xuδ)xdx|≤c‖w‖Hq‖uδ‖Hs‖uεm+1-uδm+1‖Hq≤c‖w‖Hq2‖uδ‖Hs,|∫RΛqwΛq-2[uεmuεxuεxx-uδmuδxuδxx]dx|≤c‖w‖Hq‖(uεm-uδm)(uεx2)x+uδm[uεx2-uδx2]x‖Hq-2≤c‖w‖Hq(‖(uεm-uδm)(uεx2)x‖Hq-1+‖uδm[uεx2-uδx2]x‖Hq-2)≤c‖w‖Hq(‖uεm-uδm‖Hq‖(uεx2)x‖Hq-1+‖uδm‖Hs‖[uεx2-uδx2]x‖Hq-2)≤c‖w‖Hq(‖w‖Hq‖gm-1‖Hq‖u‖Hs2+‖uδm‖Hs‖uεx+uδx‖Hq‖w‖Hq)≤c‖w‖Hq2(‖gm-1‖Hq‖u‖Hs2+‖uδm‖Hs‖uεx+uδx‖Hq).
Using (3.67), we derives that the inequality
(3.76)|∫RΛqwΛq-2(uεm+2-uδm+2)xdx|=|∫RΛqwΛq-2(wgm+1)xdx|≤c‖gm+1‖Hq‖w‖Hq2
holds for some constant c, where gm+1=∑j=0m+1uεm+1-juδj. Using the algebra property of Hq with q>1/2, q+1<s and Lemma 3.12, we have ∥gm∥Hq+1≤c for t∈(0,T̃]. Then it follows from (3.57)–(3.59) and (3.73)–(3.76) that there is a constant c depending on T̃ such that the estimate
(3.77)ddt∫R[(1-ε)(Λqw)2+ε(Λqwx)2]dx≤c(δγ‖w‖Hq+‖w‖Hq2)
holds for any t∈[0,T̃), where γ=1 if s≥3+q and γ=(1+s-q)/4 if s<3+q. Integrating (3.77) with respect to t, one obtains the estimate
(3.78)12‖w‖Hq2=12∫R(Λqw)2dx≤∫R[(1-ε)(Λqw)2+ε(Λqw)2]dx≤∫R[(Λqw0)2+ε(Λqw0x)2]dx+c∫0t(δγ‖w‖Hq+‖w‖Hq2)dτ.
Applying the Gronswall inequality, (3.37) and (3.39) yields that
(3.79)‖u‖Hq≤cδ(s-q)/4ect+δγ(ect-1),
for any t∈[0,T̃).
Multiplying both sides of (3.69) by ΛswΛs and integrating the resultant equation with respect to x, one obtains that
(3.80)12ddt∫R[(1-ε)(Λsw)2+ε(Λswx)2]dx=(ε-δ)∫R(Λsw)[(Λsuδt)+(Λsuδxxt)]dx-ε∫RΛswΛs-2wtdx+(δ-ε)∫RΛswΛs-2uδtdx-km+1∫R(Λsw)Λs(uεm+1-uδm+1)xdx-1m+2∫R(Λsw)Λs(uεm+2-uδm+2)xdx-∫RΛswΛs-2(uεm+2-uδm+2)xdx-∫RΛswΛs-2[∂x(uεm+1)∂xw]xdx-∫RΛswΛs-2[∂x(uεm+1-uδm+1)∂xuδ]xdx+∫RΛswΛs-2[uεmuεxuεxx-uδmuδxuδxx]dx.
From Lemma 3.13, we have
(3.81)|∫RΛswΛs-2(uεm+2-uδm+2)xdx|≤c3‖gm+1‖Hs‖w‖Hs2.
From Lemma 3.11, it holds that
(3.82)|∫RΛswΛs(uεm+2-uδm+2)xdx|≤c(‖w‖Hs‖w‖Hq‖uδ‖Hs+1+‖w‖Hs2).
Using the Cauchy-Schwartz inequality and the algebra property of Hs0 with s0>1/2, for s>3/2, we have
(3.83)|∫RΛswΛs-2[∂x(uεm+1)∂xw]xdx|=|∫RΛqwΛs-2[∂x(uεm+1)∂xw]xdx|≤c‖Λsw‖L2‖Λs-2[∂x(uεm+1)∂xw]x‖L2≤c‖w‖Hq‖∂x(uεm+1)∂xw‖Hs-1≤c‖uε‖Hsm+1‖w‖Hs2|∫RΛswΛs-2[∂x(uεm+1-uδm+1)∂xuδ]xdx|≤c‖w‖Hs‖Λs-2[∂x(uεm+1-uδm+1)∂xuδ]x‖L2≤c‖uδ‖Hs‖gm‖Hs‖w‖Hs2,(3.84)|∫RΛswΛs-2[uεmuεxuεxx-uδmuδxuδxx]dx|≤c‖w‖Hs(‖(uεm-uδm)(uεx2)x‖Hs-2+‖uδm[uεx2-uδx2]x‖Hs-2)≤c‖w‖Hs(‖(uεm-uδm)‖Hs‖(uεx2)x‖Hs-2+‖uδm‖Hs‖[uεx2-uδx2]x‖Hs-2)≤c‖w‖Hs2,
in which we have used Lemma 3.1 and the bounded property of ∥uε∥Hs and ∥uδ∥Hs (see Lemma 3.12). It follows from (3.80)–(3.84) and (3.57)–(3.59) and (3.79) that there exists a constant c depending on m such that
(3.85)ddt∫R[(1-ε)(Λsw)2+ε(Λswx)2]dx≤2δ(‖uδt‖Hs+‖uδxxt‖Hs+‖Λs-2wt‖L2+‖Λs-2uδt‖)‖w‖Hs+c(‖w‖Hs2+‖w‖Hq‖w‖Hs‖uδ‖Hs+1)≤c(δγ1‖w‖Hs+‖w‖Hs2),
where γ1=min⁡(1/4,(s-q-1)/4)>0. Integrating (3.85) with respect to t leads to the estimate
(3.86)12‖w‖Hs2≤∫R[(1-ε)(Λsw)2+ε(Λswx)2]dx≤∫R[(Λsw0)2+ε(Λsw0x)2]dx+c∫0t(δγ1‖w‖Hs+‖w‖Hs2)dτ.
It follows from the Gronwall inequality and (3.86) that
(3.87)‖w‖Hs≤(2∫R[(Λsw0)2+ε(Λsw0x)2]dx)1/2ect+δγ1(ect-1)≤c1(‖w0‖Hs+δ3/4)ect+δγ1(ect-1),
where c1 is independent of ε and δ.
Then (3.39) and the above inequality show that
(3.88)‖w‖Hs⟶0asε⟶0,δ→0.
Next, we consider the convergence of the sequence {uεt}. Multiplying both sides of (3.69) by Λs-1wtΛs-1 and integrating the resultant equation with respect to x, we obtain
(3.89)(1-ε)‖wt‖Hs-12+1m+2∫R(Λs-1wt)Λs-1(uεm+2-uδm+2)xdx+∫R[-ε(Λs-1wt)(Λs-1wxxt)+(δ-ε)(Λs-1wt)Λs-1(uδt+uδxxt)]dx=∫R(Λs-1wt)Λs-3[-εwt+(δ-ε)uδt-km+1∂x(uεm+1-uδm+1)-∂x(uεm+2-uδm+2)-∂x[∂x(uεm+1)∂xw+∂x(uεm+1-uεm+1)∂xuδ]+[uεmuεxuεxx-uδmuδxuδxx]km+1∂x(uεm+1-uδm+1)]dx.
It follows from (3.57)–(3.60) and the Schwartz inequality that there is a constant c depending on T̃ and m such that
(3.90)(1-ε)‖wt‖Hs-12≤c(δ1/2+‖w‖Hs+‖w‖s-1)‖wt‖Hs-1+ε‖wt‖Hs-12.
Hence,
(3.91)12‖wt‖Hs-12≤(1-2ε)‖wt‖Hs-12≤c(δ1/2+‖w‖Hs+‖w‖Hs-1)‖wt‖Hs-1,
which results in
(3.92)12‖wt‖Hs-1≤c(δ1/2+‖w‖Hs+‖w‖Hs-1).
It follows from (3.79) and (3.88) that wt→0 as ε, δ→0 in the Hs-1 norm. This implies that uε is a Cauchy sequence in the spaces C([0,T);Hs(R)) and C([0,T);Hs-1(R)), respectively. The proof is completed.Proof of Theorem2.1.
We consider the problem(3.93)(1-ε)ut-εutxx=(1-∂x2)-1[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxxm+3m+2],u(0,x)=uε0(x).
Letting u(t,x) be the limit of the sequence uε and taking the limit in problem (3.93) as ε→0, from Lemma 3.14, it is easy to see that u is a solution of the problem
(3.94)ut=(1-∂x2)-1[-km+1(um+1)x-m+3m+2(um+2)x+1m+2∂x3(um+2)-(m+1)∂x(umux2)+umuxuxxm+3m+2],u(0,x)=u0(x),
and hence u is a solution of problem (3.94) in the sense of distribution. In particular, if s≥4, u is also a classical solution. Let u and v be two solutions of (3.94) corresponding to the same initial data u0 such that u, v∈C([0,T);Hs(R)). Then w=u-v satisfies the Cauchy problem
(3.95)wt=(1-∂x2)-1{∂x[-km+1wgm-m+3m+2wgm+1+1m+2∂x2(wgm+1)-∂x(um+1)∂xw-∂x(um+1-vm+1)∂xvm+3m+2]+umuxuxx-vmvxvxx[-km+1wgm-m+3m+2wgm+1+1m+2∂x2(wgm+1)},w(0,x)=0.
For any1/2<q<min⁡{1,s-1}, applying the operator ΛqwΛq to both sides of equation (3.95) and integrating the resultant equation with respect to x, we obtain the equality
(3.96)12ddt‖w‖Hq2=∫R(Λqw)Λq-2{∂x[-km+1wgm-m+3m+2wgm+1+1m+2∂x2(wgm+1)-∂x(um+1)∂xw-∂x(um+1-vm+1)∂xvm+3m+2]+umuxuxx-vmvxvxx[-km+1wgm-m+3m+2wgm+1+1m+2∂x2(wgm+1)}dx.
By the similar estimates presented in Lemma 3.14, we have
(3.97)ddt‖w‖Hq2≤c̃‖w‖Hq2.
Using the Gronwall inequality leads to the conclusion that
(3.98)‖w‖Hq≤0×ec̃t=0
for t∈[0,T̃). This completes the proof.


## 4. Global Existence of Strong Solutions

We study the differential equation(4.1)pt=um+1(t,p),t∈[0,T),p(0,x)=x.Motivated by the Lagrangian viewpoint in fluid mechanics, by which one looks at the motion of individual fluid particles (see [43]), we state the following Lemma.Lemma 4.1.
Letu0∈Hs,s≥3 and let T>0 be the maximal existence time of the solution to problem (2.2). Then problem (4.1) has a unique solution p∈C1([0,T)×R).  Moreover, the map p(t,.) is an increasing diffeomorphism of R with px(t,x)>0 for (t,x)∈[0,T)×R.Proof.
From Theorem2.1, we have u(t,x)∈C([0,T);Hs(R))⋂C1([0,T);Hs-1(R)) and Hs(R)∈C1(R), where the Sobolev imbedding theorem is used. Thus, we conclude that both functions u(t,x) and ux(t,x) are bounded, Lipschitz in space and C1 in time. Using the existence and uniqueness theorem of ordinary differential equations derives that problem (4.1) has a unique solution p∈C1([0,T)×R).
Differentiating (4.1) with respect to x yields that
(4.2)ddtpx=(m+1)umux(t,p)px,t∈[0,T),b≠0,px(0,x)=1,
which leads to
(4.3)px(t,x)=exp⁡(∫0t(m+1)umux(τ,p(τ,x))dτ).
For every T′<T, using the Sobolev imbedding theorem yields that
(4.4)sup⁡(τ,x)∈[0,T′)×R|ux(τ,x)|<∞.
It is inferred that there exists a constantK0>0 such that px(t,x)≥e-K0t for (t,x)∈[0,T)×R. It completes the proof.The next Lemma is reminiscent of a strong invariance property of the Camassa-Holm equation (the conservation of momentum [44, 45]).Lemma 4.2.
Letu0∈Hs with s≥3,  and let T>0 be the maximal existence time of the problem (2.2), it holds that
(4.5)y(t,p(t,x))px2(t,x)=y0(x)e∫0tmumuxdτ,
where (t,x)∈[0,T)×R and y:=u-uxx+k/2(m+1).Proof.
We have(4.6)ddt[y(t,p(t,x))px2(t,x)]=ytpx2+2ypxpxt+yxptpx2=ytpx2+2y(m+1)umuxpx2+um+1yxpx2=[yt+kumux+(m+2)umuxy+yxum+1]px2+mumuxypx2=[ut-utxx+kumux+(m+2)umux(u-uxx)+um+1(ux-uxxx)]px2+mumuxypx2=mumuxypx2.
Using px(0,x)=1 and solving the above equation, we complete the proof of this lemma.Lemma 4.3.
Ifu0∈Hs,s≥3/2,  such that (1-∂x2)u0+k/2(m+1)≥0,  then the solution of problem (2.2) satisfies the following:
(4.7)‖ux‖L∞≤‖u‖L∞+k2(m+1)≤c.Proof.
Usingu0-u0xx+k/2(m+1)≥0, it follows from Lemma 4.2 that u-uxx+k/2(m+1)≥0. Letting Y1=u-uxx, we have
(4.8)u=12e-x∫-∞xeηY1(t,η)dη+12ex∫x∞e-ηY1(t,η)dη,
from which we obtain that
(4.9)∂xu(t,x)=-12(e-x∫-∞xeηY1(t,η)dη+ex∫x∞e-ηY1(t,η)dη)+ex∫x∞e-ηY1(t,η)dη=-u(t,x)+ex∫x∞e-ηY1(t,η)dη=-u(t,x)+ex∫x∞e-η(Y1(t,η)+k2(m+1))dη-k2(m+1)ex∫x∞e-ηdη=-u(t,x)+ex∫x∞e-η(y(t,η))dη-k2(m+1)≥-u(t,x)-k2(m+1).
On the other hand, we have
(4.10)∂xu(t,x)=12(e-x∫-∞xeηY1(t,η)dη+ex∫x∞e-ηY1(t,η)dη)-e-x∫-∞xeηY1(t,η)dη=u(t,x)-e-x∫-∞xeηY1(t,η)dη=u(t,x)-e-x∫-∞xeη(Y1(t,η)+k2(m+1))dη+k2(m+1)e-x∫-∞xeηdη=u(t,x)-e-x∫-∞xeηy(t,η)dη+k2(m+1)≤u(t,x)+k2(m+1).
The inequalities (3.40), (4.9), and (4.10) derive that (4.7) is valid.Proof of Theorem2.2.
Noting Remarks3.6 and 3.8, ∥u∥H1≤c and taking q+1=s in inequality (3.15), we have
(4.11)‖u‖Hs2≤‖u0‖Hs2+c∫0t‖u‖Hs2(‖ux‖L∞+‖ux‖L∞2)dτ,
from which we obtain that
(4.12)‖u‖Hs≤‖u0‖Hsec∫0t(‖ux‖L∞+‖ux‖L∞2)dτ.
Applying Lemma 4.3 derives
(4.13)‖u‖Hs≤‖u0‖Hse(c+c2)t.
From Theorem 2.1 and (4.13), we know that the result of Theorem 2.2 holds.



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*Source: 532369-2012-04-10.xml*