# Periodic Conservative Solutions for a Modified Two-Component Camassa-Holm System with Peakons

Authors: Yujuan Wang; Yongduan Song
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper investigates the periodically weak conservative solutions for a modified periodic two-component Camassa-Holm system arisen from shallow water waves moving. The underlying approach is based on a system reformulation by introducing a skillfully defined characteristic and a set of newly defined variables. Such treatment makes it possible to overcome the difficulties inherent in the multicomponent system, leading to the establishment of the periodic conservative solutions for the original system.


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## Body

## 1. Introduction

We consider in this paper a modified periodic two-component Camassa-Holm system with peakons [1], namely,
(1)mt+2mux+mxu+(mv)x+nvx=0,cccct>0,x∈R,nt+2nvx+nxv+(nu)x+mux=0,cccct>0,x∈R,
with the initial and periodic conditions
(2)u(0,x)=u0(x),x∈R,v(0,x)=v0(x),x∈R,u(t,x+1)=u(x),t>0,x∈R,v(t,x+1)=v(x),t>0,x∈R,
where m=u-uxx, n=v-vxx are periodic on the x-variable. System (1) is a modified version of the coupled two-component Camassa-Holm system in [2],
(3)mt=2mux+mxu+(mv)x+nvx,t>0,x∈R,nt=2nvx+nxv+(nu)x+mux,t>0,x∈R,
which has been established by Fu and Qu to allow for peakon solitons in the form of a superposition of multipeakons. By parameterizing t~=-t for system (3), it then takes the form of (1), which can be rewritten as a Hamiltonian system,
(4)∂∂t(mn)=-(∂m+m∂∂m+n∂∂n+m∂∂n+n∂)(δHδm=uδHδn=v)
with the Hamiltonian H=(1/2)∫(mG*m+nG*n)dx, where G*m=u, G*n=v, and G=(1/2)e-|x|. Furthermore, it has the following conserved quantities:
(5)∫Sudx,∫Svdx,∫Smdx,∫Sndx,∫S(u2+ux2+v2+vx2)dx.
A remarkable fact for the two types of coupled two-component Camassa-Holm system is that they have an infinite number of peakons, which take the following form in the periodic case that
(6)u(t,x)=1sinh(1/2)∑i=1n(pi(t)cosh(x-qi(t)-[x-qi(t)]-12)),v(t,x)=1sinh(1/2)∑i=1n(ri(t)cosh(x-qi(t)-[x-qi(t)]-12)),
where (pi(t),ri(t),qi(t)) satisfy the explicit system of ordinary differential equations
(7)p˙i=∑j=1,i≠jn(pipj+rirj)sgn(qj-qi)e-|qi-qj|,r˙i=∑j=1,i≠jn(pipj+rirj)sgn(qj-qi)e-|qi-qj|,q˙i=-∑j=1n(pj+rj)e-|qi-qj|.
In particular, when u=0 (or v=0), system (1) has the same peakon solitons as the Camassa-Holm (CH) equation.Moreover, whenu=v, system (1) is reduced to the scalar Camassa-Holm equation,
(8)mt+4mux+2mxu=0.
The CH equation which models the unidirectional propagation of shallow water waves over a flat bottom has attracted considerable attention because it has a bi-Hamiltonian structure [3] and is completely integrable [4, 5]. Another interesting property of the CH equation is that the solutions experience wave breaking [4, 6, 7]. The presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [7, 8]. After wave breaking the solutions of the CH equation can be continued uniquely as either global conservative [9–12] or global dissipative solutions [13].The important issue to address is the Cauchy problem for system (1) in the periodic case on S. It has been shown that system (1) is locally well posed with periodic initial value and there exist global strong solutions which blow up in finite time for certain class of initial data [1]. Moreover, an existence result for a class of lower regularity of solutions [1] and the global attractor with the periodic boundary condition [14] were also given.It should be noted that extending the solutions beyond wave breaking imposes significant challenge, as reflected in the case of multipeakons given by (6). Peakons interact in a way similar to that of solitons of the CH equation, and wave breaking may appear when at least two of the qi coincide. However, the solutions (6) are not smooth even with continuous functions (pi(t),ri(t),qi(t)). One possible way to interpret (6) as a weak solution of system (1) is to rewrite (6) as
(9)ut+(u+v)ux+(1-∂x2)-1×[uvx+(u2+12ux2+uxvx+12v2-12vx2)x]=0,t>0,x∈R,vt+(u+v)vx+(1-∂x2)-1×[uxv+(v2+12vx2+uxvx+12u2-12ux2)x]=0,t>0,x∈R.
In our recent work [15, 16], we investigate the continuity of the solutions for system (1) beyond collision time on the full line, leading to uniquely to either global conservative solutions where the energy is conserved for almost all times or global dissipative solutions where energy loss occurs only through wave breaking.Motivated by our recent work [15–17], we study in this paper the continuation of the solutions for system (1) beyond wave breaking in the periodic case and present a novel approach from a Lagrangian point of view. The approach we applied here is mainly based on the reformulation by introducing a newly defined characteristic and a set of variables, where the associated energy is introduced as an additional variable to obtain a well-posed initial-value problem. The argument is inspired mainly by [11] in the study of the periodic conservative solution of the CH equation. However, one problematic issue is that we here deal with a coupled system where the mutual effect between two components makes the analysis more complicated than a single one. The newly introduced characteristic and the skillfully defined variables are crucial to overcome the above-mentioned problems. By overcoming the problems, we obtain the global periodic conservative solutions of (1).The outline of this paper is as follows. The original system is transformed into a Lagrangian equivalent semilinear system by introducing a set of newly defined variables in Section2 and the global solutions of the equivalent system are obtained in Section 3. Section 4 is devoted to establishing the periodic conservative solutions for the original system, which construct a global continuous semigroup. The paper is just closed in Section 5.


### 1.1. Notation

Throughout this paper, we identify all spaces of periodic functions with function spaces over the unit circleS in R2; that is, S=R/Z. We denote by * the spatial convolution.


## 1.1. Notation

Throughout this paper, we identify all spaces of periodic functions with function spaces over the unit circleS in R2; that is, S=R/Z. We denote by * the spatial convolution.


## 2. The Original System and the Lagrangian Equivalent System

### 2.1. Part A: The Original System

Let us introduce an operatorΛ=(1-∂x2)-1, associated with its Green function G(x)=(cosh(x-[x]-1/2))/(2sinh(1/2)), where [x] stands for the integer part of x, such that
(10)Λf(x)=G*f(x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)f(x′)dx′,f∈L2(S).
Thus we can rewrite (1) as a form of a quasilinear evolution equation,
(11)ut+(u+v)ux+G*(uvx)+∂xG*(u2+12ux2+uxvx+12v2-12vx2)=0,t>0,x∈R,vt+(u+v)vx+G*(uxv)+∂xG*(v2+12vx2+uxvx+12u2-12ux2)=0,t>0,x∈R.
If we define A(t,x), B(t,x), C(t,x), and D(t,x) as follows:
(12)A(t,x)=G*(uvx)A(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)(uvx)(t,x′)dx′,B(t,x)=G*(u2+12ux2+uxvx+12v2-12vx2)B(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)B(t,x)×(u2+12ux2+uxvx+12v2-12vx2)(t,x′)dx′,C(t,x)=G*(vux)C(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)(vux)(t,x′)dx′,D(t,x)=G*(v2+12vx2+uxvx+12u2-12ux2)D(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)D(t,x)×(v2+12vx2+uxvx+12u2-12ux2)(t,x′)dx′,
then (1) can be rewritten as
(13)ut+(u+v)ux+A+Bx=0,t>0,x∈R,vt+(u+v)vx+C+Dx=0,t>0,x∈R.
For smooth solutions, we have that the total energy
(14)E(t)=∫S(u2+ux2+v2+vx2)(t,x)dx
is constant in time. Indeed, differentiating the two equations in (13), respectively, with respect to x and using the identity ∂x2G*f=G*f-f, we have
(15)uxt+ux2+uxvx+uuxx+vuxx+Ax+B-(u2+ux22+uxvx+v22-vx22)=0,vxt+vx2+uxvx+vvxx+uvxx+Cx+D-(v2+vx22+uxvx+u22-ux22)=0.
Multiplying the first equation in (13) by u and the second by v and multiplying the first equation in (15) by ux and the second by vx, we obtain the following four equalities:(16a)(u22)t+(u33)x+uvux+uA+uBx=0,(16b)(v22)t+(v33)x+uvvx+vC+vDx=0,(16c)(ux22)t+(uux22-u33)x+vuxuxx+u2vx-uA+uxB-v2ux2+vx2ux2=0,(16d)(vx22)t+(vvx22-v33)x+uvxvxx+v2ux-vC+vxD-u2vx2+ux2vx2=0.Using the four identities in (16a), (16b), (16c), and (16d), we can get that
(17)ddtE(t)=ddt∫S(u2+ux2+v2+vx2)(t,x)dx=0.
Thus (13) possesses the Hper1-norm conservation law defined as
(18)∥z∥Hper1=∥u∥Hper1+∥v∥Hper1=(∫[0,1][u2+ux2+v2+vx2]dx)1/2,
where z(t,x)=(u,v)(t,x) denotes the solution of system (13). Since z∈Hper1, Young’s inequality ensures A,B,C,D∈Hper1.


### 2.2. Part B: The Lagrangian Equivalent System

We reformulate system (13) as follows. We first introduce the corresponding characteristic y:R→W1, t→y(t,·) as the solution of
(19)yt(t,ξ)=(u+v)(t,y(t,ξ)),
where the space W1 is defined by W1={f∈Hloc1(R)∣f(ξ+1)=f(ξ)+1forallξ∈R}. We define the cumulative energy distribution H as
(20)H(t,ξ)=∫y(t,0)y(t,ξ)(u2+ux2+v2+vx2)(t,x)dx.
It is not hard to check that
(21)(u2+ux2+v2+vx2)t+((u+v)(u2+ux2+v2+vx2))x=(u3-2uB+v3-2vD)x.
Then it follows from (19) and (21) that
(22)dHdt=[(u3-2uB+v3-2vD)(t,y(t,ξ))]0ξ.
Using the fact that y∈W1 and the periodicity of u and v, it follows from (20) and (22) that
(23)H(t,ξ+1)-H(t,ξ)=H(t,1)-H(t,0)=H(0,1)-H(0,0)
for all ξ∈R. Let us introduce the vector space W given by
(24)W={f∈Hloc1(R)∣f(ξ+1)=f(ξ)+αforsomeα∈R,allξ∈RHloc1}
endowed with the norm ∥f∥W=∥f∥H1([0,1]), which is a Banach space. Then we have that H∈W.Throughout the following, we use the notation(25)U(t,ξ)=u(t,y(t,ξ)),V(t,ξ)=v(t,y(t,ξ)),Ux(t,ξ)=ux(t,y(t,ξ)),Vx(t,ξ)=vx(t,y(t,ξ)).
After the change of variables x=y(t,ξ) and x′=y(t,ξ′), we obtain the following expressions for A, Ax, B, Bx, C, Cx, D, and Dx, namely,
(26)A(ξ)=12∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)A(ξ)×[(UVx)yξ](ξ′)dξ′,Ax(ξ)=12∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Ax(ξ)×[(UVx)yξ](ξ′)dξ′,B(ξ)=14∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)B(ξ)·[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,Bx(ξ)=14∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Bx(ξ)·[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,C(ξ)=12∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)C(ξ)×[(VUx)yξ](ξ′)dξ′,Cx(ξ)=12∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Cx(ξ)×[(VUx)yξ](ξ′)dξ′,D(ξ)=14∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)D(ξ)·[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′,Dx(ξ)=14∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Dx(ξ)·[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′,
where the variable t is omitted for simplification and in the above we used the fact that Hξ=(u2+ux2+v2+vx2)∘yyξ and took for granted that y is an increasing function for any fixed time t (the validity will be proved later). In view of the periodicity of Hξ, U, and V, these expressions can be rewritten as
(27)A(ξ)=1e-1∫01cosh(y(ξ)-y(ξ′))[(UVx)yξ](ξ′)dξ′A(ξ)+12∫01e-|y(ξ)-y(ξ′)|[(UVx)yξ](ξ′)dξ′,Ax(ξ)=1e-1∫01sinh(y(ξ)-y(ξ′))[(UVx)yξ](ξ′)dξ′Ax(ξ)-12∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Ax(ξ)×[(UVx)yξ](ξ′)dξ′,B(ξ)=12(e-1)B(ξ)×∫01cosh(y(ξ)-y(ξ′))B(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′B(ξ)+14∫01e-|y(ξ)-y(ξ′)|B(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,Bx(ξ)=12(e-1)Bx(ξ)×∫01sinh(y(ξ)-y(ξ′))Bx(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′Bx(ξ)-14∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Bx(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,C(ξ)=1e-1∫01cosh(y(ξ)-y(ξ′))[(VUx)yξ](ξ′)dξ′C(ξ)+12∫01e-|y(ξ)-y(ξ′)|[(VUx)yξ](ξ′)dξ′,Cx(ξ)=1e-1∫01sinh(y(ξ)-y(ξ′))[(VUx)yξ](ξ′)dξ′Cx(ξ)-12∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Cx(ξ)×[(VUx)yξ](ξ′)dξ′,D(ξ)=12(e-1)D(ξ)×∫01cosh(y(ξ)-y(ξ′))D(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′D(ξ)+14∫01e-|y(ξ)-y(ξ′)|D(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′,Dx(ξ)=12(e-1)Dx(ξ)×∫01sinh(y(ξ)-y(ξ′))Dx(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′Dx(ξ)-14∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Dx(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′.
From the definition of the characteristic, the evolution equations for U and V in the new variables (t,ξ) take the form
(28)Ut(t,ξ)=ut(t,y)+ux(t,y)yt(t,ξ)=(-A-Bx)∘y(t,ξ),Vt(t,ξ)=vt(t,y)+vx(t,y)yt(t,ξ)=(-C-Dx)∘y(t,ξ).
With these new variables, we now derive an equivalent system of (13), namely,
(29)yt=U+V,Ut=-A-Bx,Vt=-C-Dx,Ht=[U3-2UB+V3-2VD]0ξ,
with A, Ax, B, Bx, C, Cx, D, and Dx given in (26). Differentiating (29) with respect to ξ yields
(30)yξt=Uξ+Vξ,Uξt=12Hξ+(12U2+UxVx-Vx2-B-Ax)yξ,Vξt=12Hξ+(12V2+UxVx-Ux2-D-Cx)yξ,Hξt=(3U2-2B)Uξ-2UBxyξ+(3V2-2D)Vξ-2VDxyξ,
which is semilinear with respect to the variables yξ, Uξ, Vξ, and Hξ.


## 2.1. Part A: The Original System

Let us introduce an operatorΛ=(1-∂x2)-1, associated with its Green function G(x)=(cosh(x-[x]-1/2))/(2sinh(1/2)), where [x] stands for the integer part of x, such that
(10)Λf(x)=G*f(x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)f(x′)dx′,f∈L2(S).
Thus we can rewrite (1) as a form of a quasilinear evolution equation,
(11)ut+(u+v)ux+G*(uvx)+∂xG*(u2+12ux2+uxvx+12v2-12vx2)=0,t>0,x∈R,vt+(u+v)vx+G*(uxv)+∂xG*(v2+12vx2+uxvx+12u2-12ux2)=0,t>0,x∈R.
If we define A(t,x), B(t,x), C(t,x), and D(t,x) as follows:
(12)A(t,x)=G*(uvx)A(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)(uvx)(t,x′)dx′,B(t,x)=G*(u2+12ux2+uxvx+12v2-12vx2)B(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)B(t,x)×(u2+12ux2+uxvx+12v2-12vx2)(t,x′)dx′,C(t,x)=G*(vux)C(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)(vux)(t,x′)dx′,D(t,x)=G*(v2+12vx2+uxvx+12u2-12ux2)D(t,x)=12∫01cosh(x-x′-[x-x′]-1/2)sinh(1/2)D(t,x)×(v2+12vx2+uxvx+12u2-12ux2)(t,x′)dx′,
then (1) can be rewritten as
(13)ut+(u+v)ux+A+Bx=0,t>0,x∈R,vt+(u+v)vx+C+Dx=0,t>0,x∈R.
For smooth solutions, we have that the total energy
(14)E(t)=∫S(u2+ux2+v2+vx2)(t,x)dx
is constant in time. Indeed, differentiating the two equations in (13), respectively, with respect to x and using the identity ∂x2G*f=G*f-f, we have
(15)uxt+ux2+uxvx+uuxx+vuxx+Ax+B-(u2+ux22+uxvx+v22-vx22)=0,vxt+vx2+uxvx+vvxx+uvxx+Cx+D-(v2+vx22+uxvx+u22-ux22)=0.
Multiplying the first equation in (13) by u and the second by v and multiplying the first equation in (15) by ux and the second by vx, we obtain the following four equalities:(16a)(u22)t+(u33)x+uvux+uA+uBx=0,(16b)(v22)t+(v33)x+uvvx+vC+vDx=0,(16c)(ux22)t+(uux22-u33)x+vuxuxx+u2vx-uA+uxB-v2ux2+vx2ux2=0,(16d)(vx22)t+(vvx22-v33)x+uvxvxx+v2ux-vC+vxD-u2vx2+ux2vx2=0.Using the four identities in (16a), (16b), (16c), and (16d), we can get that
(17)ddtE(t)=ddt∫S(u2+ux2+v2+vx2)(t,x)dx=0.
Thus (13) possesses the Hper1-norm conservation law defined as
(18)∥z∥Hper1=∥u∥Hper1+∥v∥Hper1=(∫[0,1][u2+ux2+v2+vx2]dx)1/2,
where z(t,x)=(u,v)(t,x) denotes the solution of system (13). Since z∈Hper1, Young’s inequality ensures A,B,C,D∈Hper1.


## 2.2. Part B: The Lagrangian Equivalent System

We reformulate system (13) as follows. We first introduce the corresponding characteristic y:R→W1, t→y(t,·) as the solution of
(19)yt(t,ξ)=(u+v)(t,y(t,ξ)),
where the space W1 is defined by W1={f∈Hloc1(R)∣f(ξ+1)=f(ξ)+1forallξ∈R}. We define the cumulative energy distribution H as
(20)H(t,ξ)=∫y(t,0)y(t,ξ)(u2+ux2+v2+vx2)(t,x)dx.
It is not hard to check that
(21)(u2+ux2+v2+vx2)t+((u+v)(u2+ux2+v2+vx2))x=(u3-2uB+v3-2vD)x.
Then it follows from (19) and (21) that
(22)dHdt=[(u3-2uB+v3-2vD)(t,y(t,ξ))]0ξ.
Using the fact that y∈W1 and the periodicity of u and v, it follows from (20) and (22) that
(23)H(t,ξ+1)-H(t,ξ)=H(t,1)-H(t,0)=H(0,1)-H(0,0)
for all ξ∈R. Let us introduce the vector space W given by
(24)W={f∈Hloc1(R)∣f(ξ+1)=f(ξ)+αforsomeα∈R,allξ∈RHloc1}
endowed with the norm ∥f∥W=∥f∥H1([0,1]), which is a Banach space. Then we have that H∈W.Throughout the following, we use the notation(25)U(t,ξ)=u(t,y(t,ξ)),V(t,ξ)=v(t,y(t,ξ)),Ux(t,ξ)=ux(t,y(t,ξ)),Vx(t,ξ)=vx(t,y(t,ξ)).
After the change of variables x=y(t,ξ) and x′=y(t,ξ′), we obtain the following expressions for A, Ax, B, Bx, C, Cx, D, and Dx, namely,
(26)A(ξ)=12∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)A(ξ)×[(UVx)yξ](ξ′)dξ′,Ax(ξ)=12∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Ax(ξ)×[(UVx)yξ](ξ′)dξ′,B(ξ)=14∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)B(ξ)·[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,Bx(ξ)=14∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Bx(ξ)·[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,C(ξ)=12∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)C(ξ)×[(VUx)yξ](ξ′)dξ′,Cx(ξ)=12∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Cx(ξ)×[(VUx)yξ](ξ′)dξ′,D(ξ)=14∫01cosh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)D(ξ)·[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′,Dx(ξ)=14∫01sinh(y(ξ)-y(ξ′)-[y(ξ)-y(ξ′)]-1/2)sinh(1/2)Dx(ξ)·[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′,
where the variable t is omitted for simplification and in the above we used the fact that Hξ=(u2+ux2+v2+vx2)∘yyξ and took for granted that y is an increasing function for any fixed time t (the validity will be proved later). In view of the periodicity of Hξ, U, and V, these expressions can be rewritten as
(27)A(ξ)=1e-1∫01cosh(y(ξ)-y(ξ′))[(UVx)yξ](ξ′)dξ′A(ξ)+12∫01e-|y(ξ)-y(ξ′)|[(UVx)yξ](ξ′)dξ′,Ax(ξ)=1e-1∫01sinh(y(ξ)-y(ξ′))[(UVx)yξ](ξ′)dξ′Ax(ξ)-12∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Ax(ξ)×[(UVx)yξ](ξ′)dξ′,B(ξ)=12(e-1)B(ξ)×∫01cosh(y(ξ)-y(ξ′))B(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′B(ξ)+14∫01e-|y(ξ)-y(ξ′)|B(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,Bx(ξ)=12(e-1)Bx(ξ)×∫01sinh(y(ξ)-y(ξ′))Bx(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′Bx(ξ)-14∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Bx(ξ)×[Hξ+(U2+2UxVx-Vx2)yξ](ξ′)dξ′,C(ξ)=1e-1∫01cosh(y(ξ)-y(ξ′))[(VUx)yξ](ξ′)dξ′C(ξ)+12∫01e-|y(ξ)-y(ξ′)|[(VUx)yξ](ξ′)dξ′,Cx(ξ)=1e-1∫01sinh(y(ξ)-y(ξ′))[(VUx)yξ](ξ′)dξ′Cx(ξ)-12∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Cx(ξ)×[(VUx)yξ](ξ′)dξ′,D(ξ)=12(e-1)D(ξ)×∫01cosh(y(ξ)-y(ξ′))D(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′D(ξ)+14∫01e-|y(ξ)-y(ξ′)|D(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′,Dx(ξ)=12(e-1)Dx(ξ)×∫01sinh(y(ξ)-y(ξ′))Dx(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′Dx(ξ)-14∫01sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|Dx(ξ)×[Hξ+(V2+2UxVx-Ux2)yξ](ξ′)dξ′.
From the definition of the characteristic, the evolution equations for U and V in the new variables (t,ξ) take the form
(28)Ut(t,ξ)=ut(t,y)+ux(t,y)yt(t,ξ)=(-A-Bx)∘y(t,ξ),Vt(t,ξ)=vt(t,y)+vx(t,y)yt(t,ξ)=(-C-Dx)∘y(t,ξ).
With these new variables, we now derive an equivalent system of (13), namely,
(29)yt=U+V,Ut=-A-Bx,Vt=-C-Dx,Ht=[U3-2UB+V3-2VD]0ξ,
with A, Ax, B, Bx, C, Cx, D, and Dx given in (26). Differentiating (29) with respect to ξ yields
(30)yξt=Uξ+Vξ,Uξt=12Hξ+(12U2+UxVx-Vx2-B-Ax)yξ,Vξt=12Hξ+(12V2+UxVx-Ux2-D-Cx)yξ,Hξt=(3U2-2B)Uξ-2UBxyξ+(3V2-2D)Vξ-2VDxyξ,
which is semilinear with respect to the variables yξ, Uξ, Vξ, and Hξ.


## 3. Global Solutions of the Lagrangian Equivalent System

We consider the Cauchy problem for (29) in the following form:
(31)ςt=U+V,Ut=-A-Bx,Vt=-C-Dx,σt=[U3-2UB+V3-2VD]0ξ,ht=0,
in the Banach space E given by
(32)E=Hper1×Hper1×Hper1×Hper1×R,
where
(33)Hper1={f∈Hloc1(R)∣f(ξ+1)=f(ξ)∀ξ∈R}
with the norm ∥f∥Hper1=∥f∥H1 (here we denote H1([0,1]) by H1 for simplification and the same for other norms in the following), ς=y-I, σ=H-hI, and h=H(t,1)-H(t,0) (I denotes identity mapping; i.e., I(ξ)=ξ). Note that there is a bijection (ς,U,V,σ,h)→(y,U,V,H) between E and W1×Hper1×Hper1×W. The advantage of the modification in (31) is that E is a Banach space, which is important for the contraction argument, while W1×Hper1×Hper1×W is not Banach space. In the following, we use the Banach space E and the set of variables (ς,U,V,σ,h).Lemma 1.
For any givenX=(ς,U,V,σ,h)∈E, one has that A, Ax, B, Bx, C, Cx, D, and Dx defined by (26) are locally Lipschitz continuous from E to Hper1. Moreover, one has
(34)Axξ=(A-UVx)(1+ςξ),Aξ=Ax(1+ςξ),Bxξ=-12(σξ+h)+(B-12U2-UxVx+Vx2)(1+ςξ),Bξ=Bx(1+ςξ),Cxξ=(C-VUx)(1+ςξ),Cξ=Cx(1+ςξ),Dxξ=-12(σξ+h)+(D-12V2-UxVx+Ux2)(1+ςξ),Dξ=Dx(1+ςξ).Proof.
We firstly prove thatBx given in (26) is locally Lipschitz continuous from E to Hper1, and the others follow in the same way. Let X=(ς,U,V,σ,h) and X~=(ς~,U~,V~,σ~,h~) be two elements of BM where BM={X=(ς,U,V,σ,h)∣∥X∥E≤M}. We have
(35)∥y∥L∞=∥I+ς∥L∞≤1+C∥ς∥H1≤1+CM
and ∥y~∥L∞≤1+CM. Since coshx is locally Lipschitz on {x∈R∣|x|≤2(1+CM)}, we have
(36)|cosh(y(ξ)-y(ξ′))-cosh(y~(ξ)-y~(ξ′))|≤C|y(ξ)-y~(ξ)-y(ξ′)+y~(ξ′)|≤C∥ς-ς~∥L∞,
where C is a generic constant depending only on M, for all ξ,ξ′∈[0,1]. It follows that
(37)∥cosh(y(ξ)-y(·))U2yξ(·)-cosh(y~(ξ)-y~(·))U~2y~ξ(·)∥L2≤C(∥ς-ς~∥L∞+∥U-U~∥L∞+∥ςξ-ς~ξ∥L2),
where we have used the fact that ∥U2-U~2∥L∞≤C∥U-U~∥L∞ in BM. We handle the other terms of B in the same way and then prove that B is Lipschitz from BM to Lper∞. Similarly, one proves that Bx is Lipschitz from BM to Lper∞. Direct computation by the chain rule gives us the expressions (34) for the derivatives of A, Ax, B, Bx, C, Cx, D, and Dx. Using the derivative Bxξ of Bx, we have
(38)∥Bxξ-B~xξ∥L2≤C(∥B-B~∥L∞+∥U-U~∥L∞+∥Ux-U~x∥L2+∥Vx-V~x∥L2+∥ςξ-ς~ξ∥L2+∥σξ-σ~ξ∥L2+|h-h~|∥U-U~∥L∞)≤∥X-X~∥E.
Thus we have proved that Bx given in (26) is Lipschitz from BM to Hper1. Similarly, A, Ax, B, C, Cx, D, and Dx defined in (26) are locally Lipschitz continuous from BM to Hper1.Theorem 2.
Let any initial dataX-=(ς-,U-,V-,σ-,h-)∈E be given. The system (31) admits a unique local solution defined on some time interval [0,T], where T depends only on ∥X-∥E.Proof.
To establish the local existence of solutions, one proceeds as in Lemma1 and obtain that F(X) given by
(39)F(X)=([U3-2UB+V3-2VD]0ξU+V,-A-Bx,-C-Dx,[U3-2UB+V3-2VD]0ξ,0),
with X=(ς,U,V,σ,h), is Lipchitz continuous on any bounded set of E. The solutions of system (31) then can be rewritten as
(40)X(t)=X-+∫0tF(X(τ))dτ.
Since E is a Banach space, the theorem then follows from the standard contraction argument.It remains to show that the local solution can be extended globally in time. Note that global solutions of (31) may not exist for all initial data in E. However, they exist when the initial data X-=(ς-,U-,V-,σ-,h-) belongs to the set Γ which is defined as follows.Definition 3.
The setΓ is composed of all (ς,U,V,σ,h)∈E such that(41a)(i)(ς,U,V,σ,h)∈[W1,∞(R)]4×R,(41b)(ii)yξ≥0,Hξ≥0,yξ+Hξ>0almosteverywhere,(41c)(iii)yξHξ=yξ2U2+Uξ2+yξ2V2+Vξ2almosteverywhere,
withy(ξ)=ς(ξ)+ξ, H=σ+hI, where Wper1,∞={f∈Wloc1,∞∣f(ξ+1)=f(ξ)forallξ∈R}, which is a Banach space with the norm ∥f∥Wper1,∞=∥f∥W1,∞.
Theorem2 gives us the existence of local solutions of (31) for initial data in E. In the following, we will only consider initial data that belongs to E∩[Wper1,∞]4×R. To obtain that the solution of (31) belongs to E∩[Wper1,∞]4×R, we have to specify the initial condition for (30). Define
(42)Ω={ξ∈R∣|ς-ξ(ξ)|≤∥ς-ξ∥L∞,|U-ξ(ξ)|≤∥U-ξ∥L∞,|V-ξ(ξ)|≤∥V-ξ∥L∞,|σ-ξ(ξ)|≤∥σ-ξ∥L∞}.
It is clear that meas(Ωc)=0. We take (ςξ,Uξ,Vξ,σξ)(0,ξ)=(ς-ξ,U-ξ,V-ξ,σ-ξ)(ξ) for ξ∈Ω, while (ςξ,Uξ,Vξ,σξ)(0,ξ)=(0,0,0,0) for ξ∈Ωc.
The global existence of the solution for initial data inΓ relies essentially on the fact that the set Γ is preserved by the flow as the next lemma shows.Lemma 4.
For initial dataX-=(ς-,U-,V-,σ-,h-)∈Γ and some T>0, one considers the local solution X(t)=(ς,U,V,σ,h)(t)∈C([0,T],E) of system (31). Then(i)
X(t)∈Γ  for all  t∈[0,T],(ii)
yξ(t,ξ)>0  for almost every  t∈[0,T] and almost every  ξ∈R.Proof.
(i) LetΩ be defined as before. For any given initial data X-∈E∩[Wper1,∞]4×R, we have that the local solution X(t)∈C([0,T],E) of (31) belongs to [Wper1,∞]4×R, which satisfies (41a) for all t∈[0,T]. We now show that (41c) holds for any ξ∈Ω and therefore almost everywhere. Consider a fixed ξ∈Ω and drop it in the notation. On the one hand, it follows from (30) that
(43)(yξHξ)t=yξtHξ+yξHξt=(Uξ+Vξ)Hξ+yξ×[(3U2-2B)Uξ+(3V2-2D)Vξ-2UBxyξ-2VDxyξ]=UξHξ+VξHξ+3U2Uξyξ-2BUξyξ+3V2Vξyξ-2DVξyξ-2UBxyξ2-2VDxyξ2,
and on the other hand,
(44)(yξ2U2+Uξ2+yξ2V2+Vξ2)t=2yξyξtU2+2yξ2UUt+2UξUξt+2yξyξtV2+2yξ2VVt+2VξVξt=UξHξ+VξHξ+3U2Uξyξ+3V2Vξyξ-2BUξyξ-2DVξyξ-2UBxyξ2-2VDxyξ2.
Thus (yξHξ)t=(yξ2U2+Uξ2+yξ2V2+Vξ2)t. Notice that yξHξ(0)=(yξ2U2+Uξ2+yξ2V2+Vξ2)(0), and then yξHξ(t)=(yξ2U2+Uξ2+yξ2V2+Vξ2)(t) for all t∈[0,T] and (41c) has been proved. It remains to prove that the inequalities in (41b) hold. Define  t*=sup{t∈[0,T]∣yξ(t′)≥0forallt′∈[0,t]}. Assume that t*<T. Since yξ(t) is continuous with respect to t, we have yξ(t*)=0. It follows from (41c) that Uξ(t*)=Vξ(t*)=0. Furthermore, (30) implies yξt(t*)=Uξ(t*)+Vξ(t*)=0 and yξtt(t*)=(Uξt+Vξt)(t*)=Hξt(t*). If Hξt(t*)=0, then (yξ,Uξ,Vξ,Hξ)(t*)=(0,0,0,0) which implies (yξ,Uξ,Vξ,Hξ)(t)=0 for all t∈[0,T] by the uniqueness of the solution of system (30). This contradicts the fact that yξ(0)+Hξ(0)>0 for all ξ∈Ω. If Hξt(t*)<0, then yξtt(t*)<0. Since yξ(t*)=yξt(t*)=0, there exists a neighborhood ϖ of t* such that yξ(t)<0 for all t∈ϖ/{t*}. This contradicts the definition of t*. Hence, Hξt(t*)>0. We now have yξtt(t*)>0, which conversely implies yξ(t)>0 for all t∈ϖ/{t*}, which contradicts the fact that t*<T. Thus we have proved yξ(t)≥0 for all t∈[0,T]. We now prove that Hξ≥0 for all t∈[0,T]. This follows from (41c) when yξ(t)>0. If yξ(t)=0, then Uξ(t)=Vξ(t)=0 from (41c). As we have seen, Hξ<0 would imply that yξ(t′)<0 for some t′ in a punctured neighborhood of t, which is impossible. Hence, Hξ≥0 for all t∈[0,T]. Now we have that yξ(t)+Hξ(t)≥0 for all t∈[0,T]. If yξ(t′)+Hξ(t′)=0 for some t′, it then follows that (yξ,Uξ,Vξ,Hξ)(t′)=0 which implies (yξ,Uξ,Vξ,Hξ)(t)=0 for all t∈[0,T], which contradicts the fact that yξ(0)+Hξ(0)>0 for all ξ∈Ω. Thus yξ(t)+Hξ(t)>0.
(ii) Define the setΘ={(t,ξ)∈[0,T]×R∣yξ(t,ξ)=0}. It follows from Fubini’s theorem that
(45)meas(Θ)=∫Rmeas(Θξ)dξ=∫[0,T]meas(Θt)dt,
where Θξ={t∈[0,T]∣yξ(t,ξ)=0} and Θt={ξ∈R∣yξ(t,ξ)=0}.   From the above proof, we know that, for all ξ∈Ω, Θξ consists of isolated points that are countable. This means that meas(Θξ)=0. It follows from (45) and since meas(Ωc)=0 that
(46)meas(Θt)=0foralmosteveryt∈[0,T].
This implies yξ(t,ξ)>0 for almost all t and therefore y(t,ξ) is strictly increasing and invertible w. r. t. ξ.Theorem 5.
For any initial dataX-=(y-,U-,V-,σ-,h-)∈Γ, system (31) admits a unique global solution X(t)=(y,U,V,σ,h)(t)∈C1(R+,E). Furthermore, for all t≥0, one has X(t)∈Γ, which constructs a continuous semigroup.Proof.
To ensure that the local solutionX=(ς,U,V,σ,h)∈C([0,T],E) of system (31) can be extended to a global solution, it suffices to show that
(47)supt∈[0,T)∥ς(t,·),U(t,·),V(t,·),σ(t,·),h(t)∥E<∞.
We now consider a fixed t∈[0,T) and drop it for simplification. We have
(48)∥H∥L∞≤H(1)=H(1)-H(0)=h,
where we have used the fact that H(0)=0 and H is an increasing function as Hξ≥0. It is clear that h(t)=h- for all t. Since σ=H-hI, we have ∥σ∥L∞≤2h and therefore supt∈[0,T)∥σ(t,·)∥L∞(R)≤2h-. For ξ,ξ′∈[0,1], we have |y(ξ)-y(ξ′)|≤y(1)-y(0)=1 as y is increasing. Since Hξ=(U2+Ux2+V2+Vx2)yξ, we have
(49)|(U2+2UxVx-Vx2)yξ|≤(U2+Ux2+Vx2+Vx2)yξ≤2Hξ,
which implies that
(50)|Bx|≤12(e-1)|∫01sinh(y(ξ)-y(ξ′))3Hξ(ξ′)dξ′|+14|∫01e-|y(ξ)-y(ξ′)|3Hξ(ξ′)dξ′|≤C(H(1)-H(0))=Ch=Ch-,
for some constant C and then supt∈[0,T)∥Bx(t,·)∥L∞(R)<∞. Similarly, one can prove that the bounds hold for A, Ax, B, C, Cx, D, and Dx. Since Ut=-A-Bx and Vt=-C-Dx, we get
(51)supt∈[0,T)∥U(t,·)∥L∞(R)<∞,supt∈[0,T)∥V(t,·)∥L∞(R)<∞,
and therefore supt∈[0,T)∥ς(t,·)∥L∞(R)<∞ as ςt=U+V. Let
(52)Z(t)=∥U(t,·)∥L2+∥Uξ(t,·)∥L2+∥V(t,·)∥L2+∥Vξ(t,·)∥L2+∥ςξ(t,·)∥L2+∥Hξ(t,·)∥L2.
After taking the L2-norms on both sides of (29) and (30), we have
(53)Z(t)≤Z(0)+C∫0tZ(τ)dτ.
Gronwall’s lemma implies that supt∈[0,T)Z(t)<∞, and therefore (47) holds as ςξ=yξ-I and σ=H-hI. It then follows that the map St:Γ→Γ×R+ given by St(X-)=X(t) generates a continuous semigroup from standard ODE theory.


## 4. Global Conservative Solutions of the Original System

In this section, we show that the global solution of the equivalent system (31) yields a global conservative solution of the original system (13), which constructs a continuous semigroup.It suffices to establish the correspondence between the Lagrangian equivalent system and the original system.We first introduce the subsetsF and Fα of Γ defined by
(54)F={X=(y,U,V,H)∈Γ∣11+h(y+H)∈G},Fα={X=(y,U,V,H)∈Γ∣11+h(y+H)∈Gα},
where h=H(ξ+1)-H(ξ)=H(1)-H(0), and the set of relabeling functions G is given by
(55)G={f∣f∈Wloc1,∞(R),f(ξ+1)=f(ξ)+1∀ξ∈R,f-I,f-1-I∈Wper1,∞}.
For any α>1, we denote the subsets of G by Gα as
(56)Gα={f∈G∣∥f-I∥Wper1,∞+∥f-1-I∥Wper1,∞≤α},
with a useful characterization: if f∈Gα (α≥0), then 1/(1+α)≤fξ≤1+α almost everywhere. Conversely, if f∈G and there exists c≥1 such that 1/c≤fξ≤c almost everywhere, then f∈Gα for some α depending only on c. With this useful characterization of Gα, it is not hard to prove that the space F is preserved by the governing equation (29).We denote byG~ the product group G×R, where the group operation is given by (f1,γ1)·(f2,γ2)=(f2∘f1,γ1+γ2) for any (f1,γ1),(f2,γ2)∈G~. We define the map Φ:G~×F→F as
(57)Φ{(f,γ),(y,U,V,H)}=(y∘f,U∘f,V∘f,H∘f+γ).
The map Φ defines a group action of G~ on F. We then consider the quotient space F/G~ of F with respect to the group action. We denote the projection Π:F→F/G~ by Π(X)=[X]. Let us introduce the subset F~0 of F0 defined as
(58)F~0={(y,U,V,H)∈F0∣∫01y(ξ)dξ=0}.
It turns out that there exists a bijection between F~0 and F/G~. To prove this, we introduce the composition map K:F→F~0 such as K(X)=K2∘(K1(X)), where K1:F→F0 is given by
(59)K1(X)=X∘f-1
with X=(y,U,V,H) and f=(1/(1+h))(y+H)∈F, and K2:F0→F~0 is given by
(60)K2(y,U,V,H)=(y(ξ-a),U(ξ-a),V(ξ-a),H(ξ-a)+(1+h)a)
with a=∫01y(ξ)dξ. It is not hard to check that K1(X)∈F0,   that is, (1/(1+h-))(y-+H-)=I, and K2(y,U,V,H)∈F~0. Thus K(X) belongs to the same equivalence class as X, and there corresponds to K a map K~:F/G~→F~0 given by K~([X])=K(X), where [X]∈F/G~ denotes the equivalence class of X∈F. It is not hard to check that K|F~0=I|F~0. Thus, K~∘Π|F~0=I|F~0 and therefore K~ is a bijection from F/G~ to F~0. Note that any topology defined on F~0 is naturally transported into F/G~ by the bijection K~. That is, if we equip F~0 with the metric induced by the E-norm, that is dF~0(X,X′)=∥X-X′∥E for all X,X′∈F~0,   which is complete, then the topology on F/G~ is defined by a complete metric given by dF/G~([X],[X′])=∥K~(X)-K~(X′)∥E for any [X],[X′]∈F/G~.For any initial dataX-∈F, we denote by S:F×R+→F the continuous semigroup with the solution X(t) of (29). Equation (13) is invariant with respect to relabeling. That is, for any X∈F and (f,γ)∈G~, we have that St(Φ{(f,γ),X})=Φ{(f,γ),St(X)}. Therefore, the map S~t:F/G~→F/G~ given by S~t([X])=[StX] is well defined, which generates a continuous semigroup.To obtain a semigroup of solution for (13), we have to consider the space Ξ, which characterizes the solutions in the original system:
(61)Ξ={(z,μ)∣z∈Hper1×Hper1andμac=(u2+ux2+v2+vx2)dxHper1},
where z=(u,v) and μ is a positive periodic Radon measure with μac being its absolute continuous part.In the following, we establish a bijection betweenF/G~ and Ξ to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in F/G~) into the original system (functions in Ξ).Let us first introduce the mappingM, which transforms the Lagrangian equivalent system into the original system. In the other direction, we obtain the energy density μ in the original system, by pushing forward by y the energy density Hξdξ in Lagrangian equivalent system, where the pushforward f#ν of a measure ν by a measurable function f is defined as
(62)f#ν(B)=ν(f-1(B))
for all Borel set B. Let (z,μ) be defined as(63a)z(x)=Z(ξ)foranyξsuchthatx=y(ξ),(63b)μ=y#(Hξdξ),withz(x)=(u,v)(x) and Z(ξ)=(U,V)(ξ). We have that (z,μ)∈Ξ, which does not depend on the representative X=(y,U,V,H)∈F of [X] that we choose. We denote by M:F/G~→Ξ the mapping to any [X]∈F/G~ and (z,μ)∈Ξ given by ((63a) and (63b)), which transforms the Lagrangian equivalent system into the original system.We are led to the mappingL:Ξ→F/G~, which corresponds to the transformation from the original system into the Lagrangian equivalent system defined as follows.Definition 6.
For any(z,μ)∈Ξ, let
(64)h=μ([0,1)),(65)y(ξ)=sup{y∣Fμ(y)+y<(1+h)ξ},(66)U(ξ)=u∘y(ξ),V(ξ)=v∘y(ξ),(67)H(ξ)=(1+h)ξ-y(ξ),
where z=(u,v). We define L(z,μ)∈F/G~ as the equivalence class of (y,U,V,H). Here, the function Fμ associated with any positive periodic Radon measure μ is given by
(68)Fμ(x)={μ([0,x))ifx>0,0ifx=0,-μ([x,0))ifx<0,
which is lower-semicontinuous, increasing and moreover, Fμ(b)-Fμ(a)=μ([a,b)) for all a<b in R.Remark 7.
From the definition ofy, U,V, H in ((64), (65), (66), and (67)), we can check that X=(y,U,V,H)∈E and X=(y,U,V,H) satisfies (41a), (41b), and (41c), respectively. Moreover, by definition (67), we have (1/(1+h))(y+H)=I. Thus X=(y,U,V,H)∈F0.
We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.Theorem 8.
The mappingsM and L are invertible; that is,
(69)L∘M=IF/G~,M∘L=IΞ.Proof.
Let[X]∈F/G~ be given. We denote by X=(y,U,V,N,H)=K~([X]) a representative of [X] and (z,μ) given by ((63a) and (63b)) for this particular X. It follows from the definition of K~ that X∈F~0. Let X-=(y-,U-,V-,H-) be the representative of L(z,μ) in F0 given by (64), (65), (66), and (67). It suffices to show that (y-,U-,V-,H-)=(y,U,V,H) and therefore L∘M=IF/G~. Set
(70)g(x)=sup{ξ∈R∣y(ξ)<x}.
Using the fact that y is increasing and continuous, it is not hard to know that
(71)y(g(x))=x
and y-1([a,b))=[g(a),g(b)) for any a<b in R. It follows from (63b) that
(72)μ([a,b))=∫y-1([a,b))Hξdξ=∫g(a)g(b)Hξdξ=H(g(b))-H(g(a)).
Since X∈F0 and y+H=(1+h)I, we have
(73)H(g(b))=(1+h)g(b)-b,H(g(a))=(1+h)g(a)-a.
From (72) and the definition of Fμ, we get
(74)Fμ(x)+x=(1+h)(g(x)-g(0)).
It then follows from the definition of y- that
(75)y-(ξ)=sup{x∈R∣g(x)<ξ+g(0)}.
For any given ξ∈R, using the fact that y is increasing and (71), we have y-(ξ)≤y(ξ+g(0)). Assume y-(ξ)<y(ξ+g(0)), and there then exists x such that y-(ξ)<x<y(ξ+g(0)) and (75) implies that g(x)≥ξ+g(0). Therefore, x≥y(ξ+g(0)), which contradicts the fact that x<y(ξ+g(0)). Hence, we have
(76)y-(ξ)=y(ξ+g(0)).
Since X-∈F0, H-(ξ)=(1+h)ξ-y-(ξ), and (y+H)(ξ+g(0))=(1+h)(ξ+g(0)), we have
(77)H-(ξ)=H(ξ+g(0))-(1+h)g(0).
It is not hard to know that
(78)U-(ξ)=U(ξ+g(0)),V-(ξ)=V(ξ+g(0)).
It then follows that L∘M=IF/G~.Given(z,μ) in Ξ, we denote by (y,U,V,H) the representative of L(z,μ) in F0 given by (64), (65), (66), and (67). Then, let (z-,μ-)=M∘L(z,μ). Let g be defined as before by (70). The same computation that leads to (74) now gives
(79)Fμ-(x)+x=(1+h)(g(x)-g(0)).Givenξ∈R, we consider an increasing sequence xi converging to y(ξ) which is guaranteed by (65), such that Fμ(xi)+xi<(1+h)ξ. Let i tend to infinity and since Fμ is lower semicontinuous, we have Fμ(y(ξ))+y(ξ)≤(1+h)ξ. Take ξ=g(x) and then we get
(80)Fμ(x)+x≤(1+h)g(x).
By the definition of g, there exists an increasing sequence ξi converging to g(x) such that y(ξi)<x. It follows from the definition of y in (65) that Fμ(x)+x≥(1+h)ξi. Passing to the limit, we obtain Fμ(x)+x≥(1+h)g(x) which, together with (80), yields
(81)Fμ(x)+x=(1+h)g(x).
We obtain that μ-=μ by comparing (81) and (79). It is clear from the definitions that z-=z. Hence, (z-,μ-)=(z,μ) and M∘L=IΞ.Now we have established a bijection between the two equivalent systems, and thus the topology defined inF/G~ can be transported into Ξ. Let us define the distance dΞ on Ξ as
(82)dΞ((z,μ),(z-,μ-))=dF/G~(L(z,μ),L(z-,μ-)),
which makes the bijection L between Ξ and F/G~ into an isometry. Since F/G~ equipped with dF/G~ is a complete metric space, it is not hard to check that Ξ equipped with the metric dΞ is also a complete metric space. For each t∈R, we define the mapping Tt:Ξ→Ξ as
(83)Tt=MS~tL.Our main results are given as follows.Theorem 9.
Let(z-,μ-)∈Ξ be given. If one denotes by t→(z(t),μ(t))=Tt(z-,μ-) the corresponding trajectory, z=(u,v) is a weak solution of the modified coupled two-component Camassa-Holm system (13), which constructs a continuous semigroup. Moreover, μ is a weak solution of the following transport equation for the energy density
(84)μt+[(u+v)μ]x=(u3-2Bu+v3-2Dv)x.
Furthermore, one has
(85)μ(t)([0,1))=μ(0)([0,1))∀t,
and for almost all t, there holds
(86)μ(t)([0,1))=μac(t)([0,1))=∥z(t)∥Hper12=∥u(t)∥Hper12+∥v(t)∥Hper12=μ(0)([0,1)).
Thus, the unique solution described here is a conservative weak solution of the system (13).Proof.
To provez=(u,v) is a weak solution of the original system (13), it suffices to show that,   for all ϕ∈C∞(R+×R) with compact support,
(87)∫R+×R[-uϕt+(u+v)uxϕ](t,x)dxdt=-∫R+×R[(A+Bx)ϕ](t,x)dxdt,∫R+×R[-vϕt+(u+v)vxϕ](t,x)dxdt=-∫R+×R[(C+Dx)ϕ](t,x)dxdt,
where A, Bx, C, and Dx are given by (13). Let the solution (y,U,V,H)(t) of (29) be a representative of L(z(t),μ(t)). Since y(t,ξ) is Lipschitz and invertible w. r. t. ξ for almost all t, we can use the change of variables x=y(t,ξ) and get
(88)∫R+×R[-uϕt+(u+v)uxϕ](t,x)dxdt=∫R+×R[-(Uyξ)(t,ξ)ϕt(t,y(t,ξ))+((U+V)Uξ)×(t,ξ)ϕ(t,y(t,ξ))((+V)Uξ)]dξdt.
Using the identities yt=U+V and yξt=Uξ+Vξ, it then follows from (29) that
(89)∫R+×R[-Uyξϕt(t,y)+(U+V)Uξϕ(t,y)]dξdt=12∫R+×R2{-UVxyξ+12sgn(ξ-ξ′)×[Hξ+(U2+2UxVx-Vx2)yξ]12}(ξ′)·e-|y(ξ)-y(ξ′)|ϕ(t,y(ξ))yξ(ξ)dξ′dξdt.
On the other hand, using the change of variables x=y(t,ξ) and x′=y(t,ξ′) and since y is an increasing function, we obtain that
(90)-∫R+×R[(A+Bx)ϕ](t,x)dxdt=12∫R+×R2[(u2+12ux2+uxvx+12v2-12vx2)-uvx+sgn(ξ-ξ′)×(u2+12ux2+uxvx+12v2-12vx2)]×(t,y(ξ′))·e-|y(ξ)-y(ξ′)|ϕ(t,y(ξ))yξ(ξ′)yξ(ξ)dξ′dξdt.
It then follows from (41c) that
(91)-∫R+×R[(A+Bx)ϕ](t,x)dxdt=12∫R+×R2{-UVxyξ+12sgn(ξ-ξ′)×[Hξ+(U2+2UxVx-Vx2)yξ]12}(ξ′)·e-|y(ξ)-y(ξ′)|ϕ(t,y(ξ))yξ(ξ)dξ′dξdt.
By comparing (89) and (91), we can see that
(92)∫R+×R[-Uyξϕt(t,y)+(U+V)Uξϕ(t,y)]dξdt=-∫R+×R[(A+Bx)ϕ](t,x)dxdt.
Hence, the first identity in (87) holds. The second identity in (87) follows in the same way. One can easily check that μ(t) is solution of (84). It follows from the definitiong in (70) that
(93)y-1([0,1))=[g(0),g(1)),g(x+1)=g(x)+1.
Since Hξ is periodic, (63b) implies that
(94)μ(t)([0,1))=∫[g(0),g(0)+1)Hξdξ=∫[0,1)Hξdξ=H(t,1)-H(t,0),
which is constant in time from (29). Thus, we have proved (85). Since yξ(t,ξ)>0 almost everywhere for almost every ξ∈R, then it follows from (41c) that
(95)μ(t)(B)=∫y-1(B)Hξdξ=∫y-1(B)(U2+Uξ2yξ2+V2+Vξ2yξ2)yξdξ
for any Borel set B. Using the fact that y is one-to-one and ux∘yyξ=Uξ, vx∘yyξ=Vξ almost everywhere, (95) implies that
(96)μ(t)(B)=∫B(u2+ux2+v2+vx2)(t,x)dx.
Hence, (86) holds. This completes the proof.


## 5. Conclusion

We consider in this paper the periodic conservative property of the solutions for the modified coupled two-component Camassa-Holm system with peakons. By introducing a skillfully defined characteristic, together with a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which periodic conservative solution is obtained. It is the transformation together with the associated properties that overcomes the difficulties inherent in the multicomponent system and allows for the continuity of the solution beyond collision time, leading to the desired results.



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*Source: 437473-2013-12-05.xml*