\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 14, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/14\hfil Camassa-Holm shallow water systems] {Global dissipative solutions for the two-component Camassa-Holm \\ shallow water system} \author[Y. Wang, Y. Song \hfil EJDE-2015/14\hfilneg] {Yujuan Wang, Yongduan Song} \address{Yujuan Wang \newline School of Automation, Institute of smart system and renewable energy, Chongqing University, Chongqing 400044, China } \email{iamwyj123456789@163.com} \address{Yongduan Song \newline School of Automation, Institute of smart system and renewable energy, Chongqing University, Chongqing 400044, China} \email{ydsong@cqu.edu.cn, Phone 0862365103001} \thanks{Submitted September 3, 2013. Published January 19, 2015} \subjclass[2000]{35L05, 35L65, 35L51} \keywords{Two-component Camassa-Holm system; global solutions; \hfill\break\indent dissipative solutions} \begin{abstract} This article presents a continuous semigroup of globally defined weak dissipative solutions for the two-component Camassa-Holm system. Such solutions are established by using a new approach based on characteristics a set of new variables overcoming the difficulties inherent in multi-component systems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} We consider the two-component Camassa-Holm shallow water system (see \cite{c2,c7,f1}) \begin{equation}\label{eq1} \begin{gathered} m_t +um_x +2u_x m-Au_x +\rho \rho _x =0,\quad t>0,\;x\in \mathbb{R}, \\ m=u-u_{xx} ,\quad t>0,\;x\in \mathbb{R}, \\ \rho _t +(u\rho )_x =0,\quad t>0,\;x\in \mathbb{R}. \end{gathered} \end{equation} Here $A>0$ characterizes a linear underlying shear flow so that \eqref{eq1} models wave-current interactions. The variable $u(x,t)$ represents the horizontal velocity of the fluid, and $\rho (x,t)$ is the scalar density. This system appears in \cite{o1}; it was also derived by Constantin and Ivanov in \cite{c7} in the context of shallow water theory. It is an extension of the Camassa--Holm (CH), is formally integrable \cite{c2,c7,f1}, and also has a bi-Hamiltonian structure with Hamiltonians \[ H_1 =\frac{1}{2}\int \big({um+({\rho -1})^2}\big) dx \] and \[ H_2 =\frac{1}{2}\int {\Big({u({\rho -1})^2+2u({\rho -1} )+u^3+uu_x^2 -Au^2}\Big)} dx. \] For $\rho \equiv 0$, one obtains the classical CH equation, which models the unidirectional propagation of shallow water waves over a flat bottom. It has a bi-Hamiltonian structure \cite{c3} and is completely integrable \cite{c1,c4}. The CH equation has attracted a lot of attention just because it has peaked solitons \cite{c1,c8} and models wave breaking \cite{c1,c6}. The presence of breaking waves means the solution remains bounded while its slope becomes unbounded in finite time \cite{c5,c6}. After wave breaking the solutions of the CH equation can be continued uniquely as either globally conservative \cite{b1} or globally dissipative solutions\cite{b2}. System \eqref{eq1} is an integrable multi-component generalization of the CH equation. System \eqref{eq1} has a physical interpretation\cite{c7}, just like the CH equation, has an integrable structure \cite{f1}, and can be expressed as a flow \cite{h1}. It has been shown that the two-component CH system is locally well-posed with initial data $({u_0 ,\;\rho _0 })\in H^s\times H^{s-1}$, $s>3/2$ \cite{g2}. The system also has global strong solutions which blow up in finite time \cite{e1}. More interestingly, it possesses a global continuous semigroup of weak conservative solutions \cite{w1,w2}. The goal of the present paper is to construct a global continuous semigroup of weak dissipative solutions for the two-component Camassa-Holm system \eqref{eq1}. It should be stressed that system \eqref{eq1} is a multiple (rather than single) component system in which the mutual effect between the components $u$ and $\rho $ exits, making it quite challenging to address properties of the solutions associated with the system. To circumvent the difficulties coming from the two-component coupling effect, we introduce a suitable characteristic and a new set of independent and dependent variables to transfer the system \eqref{eq1} into a semilinear hyperbolic system. By solving the corresponding semilinear system which contains a discontinuous non-local source term but has bounded directional variation, a global dissipative solution is derived. Then, by mapping the solution of the semilinear system into the solution of original system \eqref{eq1}, the problem is solved. Furthermore, it is proved that the solutions actually construct a semigroup. The remainder of this article is organized as follows. Section 2 is the introduction of the original system. In Section 3, a transformation from the original system to an equivalent semilinear system is conducted by applying a new set of variables. The unique global solution of the equivalent semilinear system is derived in Section 4 and then it is reversed to the weak dissipative solution of the original system in Section 5, which constructs a global continuous semigroup. \section{The original system} Let $G(x)=\frac{1}{2}e^{-| x |}$ and $\ast $ denotes the spatial convolution such that $G\ast f=({1-\partial ^2} )^{-1}f$ for all $f\in L^2(\mathbb{R})$. System \eqref{eq1} can thus be rewritten as a form of quasi-linear evolution equation \begin{gather*} u_t +uu_x +\partial _x G\ast ({u^2+\frac{1}{2}u_x^2 -Au+\frac{1}{2}\eta ^2+\eta })=0,\quad t>0,\;x\in \mathbb{R}, \\ \eta _t +u\eta _x +\eta u_x +u_x =0,\quad t>0,\;x\in \mathbb{R}, \end{gather*} which can be further represented in the form \begin{equation} \label{eq2} \begin{gathered} u_t +uu_x +P_x =0,\quad t>0,\;x\in \mathbb{R}, \\ \eta _t +u\eta _x +\eta u_x +u_x =0,\quad t>0,\;x\in \mathbb{R}, \end{gathered} \end{equation} where $\eta =\rho -1$ and $P=G\ast (u^2+u_x^2/2 -Au+\eta ^2 / 2 +\eta )$, with the initial condition $({u_0 ,\eta _0 })\in H^1\times U$ with $U=L^2\cap L^\infty $. For smooth solutions, the total energy \begin{equation} \label{eq3} E(t)=\int_R {u^2} +u_x^2 +\eta ^2\,dx \end{equation} is constant in time. Indeed, by using the identity $\partial _x^2 G\ast f=G\ast f-f$ and differentiating the two equations in \eqref{eq2} with respect to $x$ respectively, we have \begin{equation}\label{eq4} \begin{gathered} u_{xt} +uu_{xx} +u_x^2 -({u^2+\frac{1}{2}u_x^2 -Au+\frac{1}{2}\eta ^2+\eta })+P=0, \\ \eta _{xt} +2u_x \eta _x +({u\eta _{xx} +\eta u_{xx} +u_{xx} })=0. \end{gathered} \end{equation} Multiplying the first equation in \eqref{eq2} by $u$ and the second equation by $\eta $, and multiplying the first one in \eqref{eq4} by $u_x $, we obtain the following conservation laws \begin{gather} \label{eq5} \Big({\frac{u^2}{2}}\Big)_t +\Big({\frac{u^3}{3}}\Big)_x +uP_x =0, \\ \label{eq6} \Big({\frac{u_x^2 }{2}}\Big)_t +({\frac{1}{2}uu_x^2 -\frac{1}{3}u^3+\frac{1}{2}Au^2})_x -\frac{1}{2}\eta ^2u_x -\eta u_x +u_x P=0, \\ \label{eq7} \Big({\frac{\eta ^2}{2}}\Big)_t +\eta ^2u_x +\eta u_x +u\eta \eta _x =0. \end{gather} It then follows from \eqref{eq5}-\eqref{eq7} that \[ \frac{d}{dt}E(t)=\frac{d}{dt}\int_S {({u^2+u_x^2 +v^2} )} ({t,x})dx=0. \] Thus \eqref{eq2} possesses the $H^1$-norm conservation law given by \[ \| z \|_{H^1} =\Big({\,\int_R {[ {u^2+u_x^2 +\eta ^2} ]dx} }\Big)^{1/2}, \] where $z=({u,\eta })$. Since $z=({u,\eta })\in H^1\times U$, Young's inequality ensures $P\in H^1$. \begin{definition}\label{def1} \rm By a solution of the Cauchy problem \eqref{eq2} we mean a H\"{o}lder continuous function $z=z({t,x})$ defined on $[{0,T} ]\times R$ with the following properties: \begin{itemize} \item[(i)] $z({t,\;\cdot })\in H^1\times [ {L^2\cap L^\infty } ]$ for each fixed $t$. \item[(ii)] The map $t\to z({t,\cdot })$ is Lipschitz continuous from $[0,T]$ to $L^2$, satisfying \begin{equation}\label{eq8} \begin{gathered} z_t =-uz_x -f(z), \\ z({0,x})=\bar {z}(x), \end{gathered} \end{equation} where $z=({u,\eta })$, $z_x =({u_x ,\eta _x })$ and $f(z)=({P_x ,\;({\eta +1})u_x })$. \end{itemize} \end{definition} \begin{definition} \label{def2}\rm We call a solution of the Cauchy problem \eqref{eq2} a dissipative solution if it satisfies the Oleinik type inequality \[ u_x ({t,x}),\eta _x ({t,x})\le C({1+t^{-1}}), \quad t>0 \] for some constant $C$ depending only on the norm of the initial data $\| {\bar {z}} \|_{H^1} $ and its energy $E(t)$ in \eqref{eq3} is a non-increasing function of time. \end{definition} \section{The equivalent semilinear system} In this section, a transformation is conducted by introducing a characteristic and a new set of Lagrangian variables, with which the original system is transformed into an equivalent semilinear hyperbolic system. For given initial data $\bar {z}=({\bar {u},\bar {\eta }})\in H^1\times U$, we consider the following initial problem, \begin{equation}\label{eq9} \begin{gathered} \frac{\partial }{\partial t}q({t,\xi })=u({t,q({t,\xi })}),\quad t\in [ 0,T ], \\ q({0,\xi })=\bar {q}(\xi),\quad x\in \mathbb{R}, \end{gathered} \end{equation} where the solution $z=({u,\eta })$ to \eqref{eq2} remains Lipschitz continuous for $t\in [0,T]$, and the non-decreasing maps $\xi \mapsto \bar {q}(\xi)$ is defined as \begin{equation} \label{eq10} \int_0^{\bar {q}(\xi)} {\bar {u}_x^2 } dx=\xi . \end{equation} The following notation is used: \begin{gather*} u({t,\;\xi })=u({t,\;q({t,\xi })}), \quad \eta ({t,\;\xi })=\eta ({t,\;q({t,\xi })}), \quad P({t,\;\xi })=P({t,\;q({t,\xi })}), \\ u_x ({t,\;\xi })=u_x ({t,\;q({t,\xi })}),\quad \eta _x ({t,\;\xi })=\eta _x ({t,\;q({t,\xi })}), \quad P_x ({t,\;\xi })=P_x ({t,\;q({t,\xi })}). \end{gather*} Define the variables $\theta =\theta ({t,\xi })$ and $w=w({t,\xi })$ as \begin{equation} \label{eq11} \theta =2\operatorname{arcsec} u_x , \quad w=u_x^2 \cdot \frac{\partial q}{\partial \xi }. \end{equation} ($\theta $ in $[ {0,\pi })\cup ({\pi ,2\pi } ])$. We remark that the transformed variable $\theta $ used in this paper is of the form $\theta =2\operatorname{arcsec} u_x $, which makes the calculation much simple and convenient to set up the dissipative solution in contrast to the applied variable $v=2\arctan u_x $ in \cite{b1,b2}, which further overcomes the difficulties existing in the multi-component system. The following useful identities are prepared for later use from \eqref{eq9}-\eqref{eq11}, \begin{gather} \label{eq12} w({0,\xi })\equiv 1, \\ \label{eq13} u_x =\sec \frac{\theta }{2}, \quad \frac{1}{u_x^2 }=\cos ^2\frac{\theta }{2}, \\ \label{eq14} \frac{\partial q}{\partial \xi }=\frac{w}{u_x^2 } =\cos ^2\frac{\theta }{2}\cdot w. \end{gather} According to \eqref{eq14}, we obtain \begin{equation} \label{eq15} q({t,{\xi }'})-q({t,\xi })=\int_\xi ^{{\xi }'} {\cos ^2\frac{\theta }{2}({t,s})\cdot w} ({t,s})ds. \end{equation} By using the new variable $\xi $, we represent $P$ and $P_x $ as follows, \begin{align*} P(\xi)&=\frac{1}{2}\int_{-\infty }^{+\infty } {\exp \Big\{ {-\big| {\int_\xi ^{{\xi }'} {\cos ^2} \frac{\theta (s )}{2}\cdot w(s)ds} \big|} \Big\}} \\ &\times \big[ {\big({u^2-Au+\frac{1}{2}\eta ^2+\eta }\big)\cos ^2\frac{\theta }{2}+\frac{1}{2}} \big] w({{\xi }'})d{\xi}', \end{align*} \begin{equation} \label{eq16} \begin{aligned} P_x (\xi) &=\frac{1}{2}\Big({\int_\xi ^{+\infty } {-\int_{-\infty }^\xi } }\Big)\exp \Big\{ {-\big| {\int_\xi ^{{\xi }'} {\cos ^2} \frac{\theta (s)}{2} w(s)ds} \big|} \Big\}\\ &\quad\times \big[ {\big({u^2-Au+\frac{1}{2}\eta ^2+\eta }\big)\cos ^2\frac{\theta }{2}+\frac{1}{2}} \big] w({{\xi }'})d{\xi}', \end{aligned} \end{equation} System \eqref{eq2} can be further rewritten with the new variables $({t,\xi })$ as \begin{equation}\label{eq17} \begin{gathered} \frac{\partial }{\partial t}u({t,\xi })=u_t +uu_x =-P_x ({t,\xi }),\\ \frac{\partial }{\partial t}\eta ({t,\xi })=\eta _t +u\eta _x =-({\eta +1})u_x ({t,\xi }) \end{gathered} \end{equation} From \eqref{eq9}, \eqref{eq11} and \eqref{eq4}, we obtain \begin{equation}\label{eq18} \begin{aligned} \frac{\partial }{\partial t}\theta ({t,\xi }) &=\frac{2}{u_x \sqrt {u_x^2 -1} }({u_{xt} +uu_{xx} }) \\ &=-\csc \frac{\theta }{2}+({2u^2-2Au+\eta ^2+2\eta -2P})\cos \frac{\theta }{2}\cdot \cot \frac{\theta }{2}. \end{aligned} \end{equation} Furthermore, it follows from \eqref{eq9}, \eqref{eq11} and \eqref{eq6} that \begin{equation} \label{eq19} \frac{\partial }{\partial t}w({t,\xi })=({u_x^2 })_t +({uu_x^2 })_x =({2u^2-2Au+\eta ^2+2\eta -2P})\cos \frac{\theta }{2}\cdot w. \end{equation} The functions $P$ and $P_x $ used in the above \eqref{eq17}-\eqref{eq19} are given in \eqref{eq16}. Now the corresponding Cauchy problems \eqref{eq17}-\eqref{eq19} for the variables $({u,\eta ,\theta ,w})$ becomes the semilinear system \begin{equation} \label{eq20} \begin{gathered} \frac{\partial u}{\partial t}=-P_x , \\ \frac{\partial \eta }{\partial t}=-({\eta +1})\sec \frac{\theta }{2}, \\ \frac{\partial \theta }{\partial t}=-\csc \frac{\theta }{2}+( {2u^2-2Au+\eta ^2+2\eta -2P})\cos \frac{\theta }{2}\cdot \cot \frac{\theta }{2}, \\ \frac{\partial w}{\partial t}=({2u^2-2Au+\eta ^2+2\eta -2P} )\cos \frac{\theta }{2}\cdot w, \\ \end{gathered} \end{equation} with the initial condition \begin{equation} \label{eq21} \begin{gathered} u({0,\xi })=\bar {u}({\bar {q}(\xi)}), \\ \eta ({0,\xi })=\bar {\eta }({\bar {q}(\xi)}), \\ \theta ({0,\xi })=2\operatorname{arcsec} \bar {u}_x ({\bar {q}(\xi)}), \\ w({0,\xi })=1, \end{gathered} \end{equation} which can be regarded as an ordinary differential equation (ODE) in the Banach space \[ X=H^1\times [ {L^2\cap L^\infty } ]\times [ {L^2\cap L^\infty } ]\times L^\infty , \] endowed with the norm \[ \| {({u,\eta ,\theta ,w})} \|_X =\| u \|_{H^1} +\| \eta \|_{L^2} +\| \eta \|_{L^\infty } +\| \theta \|_{L^2} +\| \theta \|_{L^\infty } +\| w \|_{L^\infty } . \] In the dissipative case, we need modify the system \eqref{eq20} suitably. Suppose that, along a given characteristic $t\to q({t,\xi })$, the wave breaks at a first time $t=\tau (\xi)$. As $t\uparrow \tau (\xi)$, the variable $\theta =2\operatorname{arcsec} u_x $ implies that $u_x ({t,\xi })\to -\infty $. For all $t\ge \tau $, we set $\theta ({t,\xi })\equiv \pi $. Then the $P$ and $P_x $ in \eqref{eq16} are replaced by \begin{equation} \label{eq22} \begin{aligned} P(\xi)&=\frac{1}{2}\int_{\{ {\theta ({{\xi }'} )\ne \pi }\}} {\exp \Big\{ {-| {\int_{\{ {s\in [ {\xi ,{\xi }'} ],\theta (s)\ne \pi } \}} {\cos ^2\frac{\theta (s)}{2}\cdot w(s)ds} } |} \Big\}} \\ &\quad \times ({u^2+\frac{1}{2}u_x^2 -Au+\frac{1}{2}\eta ^2+\eta } ) \cos ^2\frac{\theta }{2}\cdot w({{\xi }'})d{\xi }' \end{aligned} \end{equation} and \begin{equation}\label{eq23} \begin{aligned} P_x (\xi) &=\frac{1}{2}\int_{\{ {{\xi }'>\xi ,\theta ({{\xi }'})\ne \pi } \}} \exp\Big\{ {-| {\int_{\{ {s\in [ {\xi ,{\xi }'} ],\theta (s)\ne \pi } \}} {\cos ^2\frac{\theta (s )}{2}\cdot w(s)ds} } |} \Big\} \\ &\quad \times ({u^2+\frac{1}{2}u_x^2 -Au+\frac{1}{2}\eta ^2+\eta } )\cdot \cos ^2\frac{\theta }{2}\cdot w({{\xi }'})d{\xi}' \\ &\quad -\frac{1}{2}\int_{\{ {{\xi }'<\xi ,\theta ({{\xi }'})\ne \pi }\}} \exp\Big\{ {-| {\int_{\{ {s\in [ {\xi ,{\xi }'} ],\theta (s)\ne \pi } \}} {\cos ^2\frac{\theta (s )}{2}\cdot w(s)ds} } |} \Big\} \\ &\quad \times ({u^2+\frac{1}{2}u_x^2 -Au+\frac{1}{2}\eta ^2+\eta } )\cdot \cos ^2\frac{\theta }{2}\cdot w({{\xi }'})d{\xi}', \end{aligned} \end{equation} System \eqref{eq20} can thus be rewritten in the form \begin{equation}\label{eq24} \begin{gathered} \frac{\partial u}{\partial t}=-P_x , \\ \frac{\partial \eta }{\partial t}=\begin{cases} -({\eta +1})\sec \frac{\theta }{2} &\text{if }\theta \ne \pi \\ 0 &\text{if }\theta =\pi , \end{cases} \\ \frac{\partial \theta }{\partial t}=\begin{cases} -\csc \frac{\theta }{2}+({2u^2-2Au+\eta ^2+2\eta -2P})\cos \frac{\theta }{2}\cdot \cot \frac{\theta }{2} &\text{if }\theta \ne \pi \\ 0 &\text{if }\theta =\pi, \end{cases} \\ \frac{\partial w}{\partial t}=\begin{cases} ({2u^2-2Au+\eta ^2+2\eta -2P})\cos \frac{\theta }{2}\cdot w &\text{if }\theta \ne \pi \\ 0 &\text{if }\theta =\pi . \end{cases} \end{gathered} \end{equation} where the right hand side is now discontinuous. The discontinuity occurs precisely when $\theta =\pi $. \section{Global solutions of the equivalent semilinear system} A unique local solution of the equivalent semilinear system defined on some time interval $[0,T]$ is first obtained, and then it is proved that this local solution can be globally extended for all times $t\ge 0$. Denote \begin{gather*} U=({u,\eta ,\theta ,w})\in \mathbb{R}^4, \\ F(U)=\begin{cases} \Big(0,\;-({\eta +1})\sec \frac{\theta }{2},\; -\csc \frac{\theta}{2}+({2u^2-2Au+\eta ^2+2\eta })\cos \frac{\theta }{2}\\ \times \cot \frac{\theta }{2}, ({2u^2-2Au+\eta ^2+2\eta })\cos \frac{\theta }{2}\cdot w\Big) &\text{if }\theta \ne \pi ,\\[4pt] (0, 0, 0, 0) &\text{if } \theta =\pi , \end{cases} \\ G({\xi ,U(\cdot)})=\begin{cases} \Big(-P_x , 0,-2P\cos \frac{\theta }{2}\cdot \cot \frac{\theta }{2},-2P\cos \frac{\theta }{2}\cdot w\Big)&\text{if }\theta \ne \pi , \\ ({-P_x , 0, 0, 0})&\text{if } \theta =\pi . \end{cases} \end{gather*} The Cauchy problem for \eqref{eq24} is rewritten in more compact form with this notation, \begin{equation} \label{eq25} \frac{\partial }{\partial t}U({t,\xi })=F({U({t,\xi})})+G({\xi ,U({t,\cdot })}), \quad \xi \in \mathbb{R} \end{equation} with the initial condition \[ U({0,\xi })=\bar {U}(\xi). \] After a solution $({u,\eta ,\theta ,w})$ of \eqref{eq25} is obtained, a solution of \eqref{eq24} will be soon provided by the mapping $({t,\xi })\to ({u,\eta ,\theta ,w})$. We regard \eqref{eq25} as an ODE on the space $L^\infty (\mathbb{R},\mathbb{R}^4)$. Observe that the vector field $F:R^4\to R^4$ is uniformly bounded and Lipschitz continuous as long as $u,\eta $ remain in a bounded set. However, the nonlocal operator $G$ is discontinuous. To prove the unique local solution of the system \eqref{eq25}, we begin with some assumptions. \textbf{Assumption 1.} Suppose $F$ and $G$ are given in \eqref{eq25}, there exists some constant $C>0$ and constant $\kappa ^\ast >0$ depending only on $C$ such that, for $U=({u,v,\theta ,w})\in L^\infty (\mathbb{R},\mathbb{R}^4)$, $\tilde {U}=({\tilde {u},\tilde {v},\tilde {\theta },\tilde {w}})\in L^\infty (\mathbb{R},\mathbb{R}^4)$, the following inequalities hold: \begin{gather} \label{eq26} \| u \|_{L^\infty } ,\| \eta \|_{L^\infty } ,\| {\tilde {u}} \|_{L^\infty } ,\| {\tilde {\eta }} \|_{L^\infty } \le C, \quad \frac{1}{C}\le w(\xi), \quad \tilde {w}(\xi)\le C, \\ \label{eq27} \operatorname{meas}({\{ {\xi ;\theta (\xi)\ne \pi ,| {\theta (\xi)-\pi } |\ge \frac{\pi }{2}} \}}) \le C, \\ \label{eq27b} \operatorname{meas}({\{ {\xi ;\tilde {\theta }(\xi)\ne \pi ,| {\tilde {\theta }(\xi)-\pi } |\ge \frac{\pi }{2}} \}})\le C, \\ \label{eq28} \| P \|_{L^\infty } +\| {P_x } \|_{L^\infty } \le \kappa ^\ast , \\ \label{eq28b} \| P \|_{L^1} +\| {P_x } \|_{L^1} \le \kappa ^\ast ({1+\| u \|_{L^1} +\| \eta \|_{L^1} +\| \theta \|_{L^1} }), \\ \label{eq29} \| {F(U)} \|_{L^\infty } ,\| {G(U)} \|_{L^\infty } \le \kappa ^\ast , \\ \label{eq30} \| {F(U)-F({\tilde {U}})} \|_{L^\infty } \le \kappa \| {U-\tilde {U}} \|_{L^\infty } , \\ \label{eq31} \begin{aligned} &\|{G(U)-G({\tilde {U}})} \|_{L^\infty}\\ &\le \kappa \big[ {\| {U-\tilde {U}} \|_{L^\infty } +\operatorname{meas}({\{ {\xi ;\theta \ne \pi ,\tilde {\theta }=\pi } \}} )+\operatorname{meas}({\{ {\xi ;\tilde {\theta }\ne \pi ,\theta =\pi } \}})} \big], \end{aligned} \end{gather} where $\kappa $ is a Lipschitz constant. \smallskip \textbf{Assumption 2.} Given initial data $\bar {z}=({\bar {u},\bar {\eta }})\in H^1\times L^2$, there exists a constant $C>0$ such that \[ \| u \|_{L^\infty } ,\| \eta \|_{L^\infty } \le \frac{C}{2}, \quad \operatorname{meas}({\{ {\xi ;\theta (\xi)\ne \pi ,| {\theta (\xi)-\pi } |\ge \frac{\pi }{4}} \}})\le \frac{C}{2}. \] Define the set $\Omega ^\delta =\{ {\xi \in \mathbb{R};\bar {\theta }(\xi )\in ({\pi ,\pi +\delta } ]\;} \}$, where $\delta >0$ is a constant small enough. By possibly reducing the size of $\delta >0$, thus we can assume that $\operatorname{meas}({\Omega ^\delta })\le 1/(8\kappa )$. Given $T>0$, let $D$ be the set of all continuous mappings $t\to U(t):[0,T]\to L^\infty ({R,R^4})$, with the following properties: \begin{gather*} U(0)=\bar {U}, \\ \| {U(t)-U(s)} \|_{L^\infty } \le 2k^\ast | {t-s} |, \\ \theta ({t,\xi })-\theta ({t,\xi })\le -\frac{t-s}{2}, \quad \xi \in \Omega ^\delta ,\; 0\le s0$ . \end{theorem} \begin{proof} We first show that $\Pi :D\to D$ defined above is a strict contraction. Choose $T>0$ sufficiently small and $U,\tilde {U}\in D$. Define \begin{gather*} \lambda =\mathop {\max }_{t\in [0,T]} \| {U( t)-\tilde {U}(t)} \|_{L^\infty } , \quad \tau (\xi)=\sup _{t\in [0,T]} \{ {t;\theta ({t,\xi })\ne \pi } \}, \\ \tilde {\tau }(\xi)=\sup _{t\in [0,T]} \{ {t;\tilde {\theta }({t,\xi })\ne \pi }\} \end{gather*} For each $\xi \in \Omega ^\delta $, we have $| {\tau (\xi )-\tilde {\tau }(\xi)} |\le 2\lambda $. For $t\in[0,T]$, we have \begin{align*} &\| {\Pi U(t)-\Pi \tilde {U}(t)}\|_{L^\infty }\\ &\le \int_0^t {\| {F({U(\tau)})-F( {\tilde {U}(\tau)})} \|_{L^\infty } d} \tau +\int_0^t {\| {G({U(\tau)})-G({\tilde {U}(\tau)})} \|_{L^\infty } d} \tau \\ &\le 2\kappa \int_0^t {\| {U(\tau)-\tilde {U}(\tau )} \|_{L^\infty } d} \tau +\kappa \int_0^t {\operatorname{meas}({\{ {\xi ;\theta \ne \pi ,\tilde {\theta }=\pi } \}})d\tau }\\ &\quad +\kappa \int_0^t {\operatorname{meas}({\{ {\xi ;\tilde {\theta } \ne \pi ,\theta =\pi } \}})d\tau } \\ &\le 2T\kappa \lambda +\kappa \int_{\Omega ^\delta } {| {\tau (\xi )-\tilde {\tau }(\xi)} |} d\xi \\ &\le 2\kappa T\lambda +2\kappa \operatorname{meas}({\Omega ^\delta } ) \lambda \le \frac{\lambda }{2}, \end{align*} where $T$ is chosen as $T\le 1 /(8\kappa)$. This shows that $\Pi $ is a strict contraction, which yields the desired local solution of Cauchy problem \eqref{eq25}. Next we show that the local solutions of the semilinear system \eqref{eq24} can be globally extended for all times $t\ge 0$. In the following, we prove that the ``extended energy'' \[ \tilde {E}(t)=\int_R {\Big({u^2\cos ^2\frac{\theta _1 }{2}+\eta ^2\cos ^2\frac{\theta _1 }{2}+1}\Big)w} ({t,\xi } )d\xi \] remains constant in time. We remark that the extended energy $\tilde {E}(t)$ is strictly larger than the total energy \[ E(t)=\int_{\{ {\theta ({t,\xi })\ne \pi } \}} {\Big({u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1}\Big)w({t,\xi })} ({t,\xi })d\xi \] in the sense that here the integration ranges over the entire real line. For future use, we show the following identities \begin{equation}\label{eq33} \begin{gathered} u_\xi =u_x \cdot \frac{\partial y}{\partial \xi }=\sec \frac{\theta }{2}\cdot \cos ^2\frac{\theta }{2}\cdot w=\cos \frac{\theta }{2}\cdot w, \\ P_\xi =P_x \cdot \frac{\partial y}{\partial \xi }=P_x \cdot \cos ^2\frac{\theta }{2}\cdot w\,, \end{gathered} \end{equation} hold for all times $t\ge 0$, as long as the solution is defined. Moreover, when $\theta =\pi $, a separate computation yields \[ u_\xi =0=\cos \frac{\pi }{2}\cdot w, \quad P_\xi =0=P_x \cdot \cos ^2\frac{\pi }{2}\cdot w. \] Thus the identity in \eqref{eq33} still holds for the cases $\theta =\pi $. Then we obtain \begin{gather*} ({uP})_\xi =u_\xi P+uP_\xi =w({P\cdot \cos \frac{\theta }{2}+uP_x \cdot \cos ^2\frac{\theta }{2}}),\\ ({u^3})_\xi =3u^2u_\xi =3wu^2\cdot \cos \frac{\theta }{2}, \\ ({u^2})_\xi =2uu_\xi =2uw\cdot \cos \frac{\theta }{2}. \end{gather*} Differentiating the extended energy $\tilde {E}(t)$ with respect to the variable $t$, we obtain \begin{equation} \label{eq34} \begin{aligned} &\frac{d}{dt}\int_R {\tilde {E}(t)} d\xi =\frac{d}{dt}\int_R {\Big({u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1} \Big)w} d\xi \\ &=\int_R \Big[ \Big({u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1}\Big)\frac{\partial w}{\partial t}+\Big({2uu_t \cdot \cos ^2\frac{\theta }{2}-u^2\cos \frac{\theta }{2}\sin \frac{\theta }{2}\frac{\partial \theta }{\partial t}}\Big)w \\ &\quad +\Big({2\eta \eta _t \cdot \cos ^2\frac{\theta }{2}-\eta ^2\cos \frac{\theta }{2}\sin \frac{\theta }{2}\frac{\partial \theta }{\partial t}} \Big)w\Big]d\xi \\ &=\int_{ \{\theta (\xi)\ne \pi \}} \Big\{2\Big({u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1}\Big) \Big({u^2-Au+\frac{\eta ^2}{2}+\eta -P}\Big) \\ &\quad -2uP_x \cdot \cos \frac{\theta }{2}-2\eta ({\eta +1})-( {u^2+\eta ^2})\sin \frac{\theta }{2} \\ &\quad\times \big[ {-\csc \frac{\theta }{2}+({2u^2-2Au+\eta ^2+2\eta -2P} )\cos \frac{\theta }{2}\cdot \cot \frac{\theta }{2}} \big]\Big\}\cos \frac{\theta }{2} wd\xi \\ &=\int_R {w\Big\{ {3u^2\cos \frac{\theta }{2}-2Au\cos \frac{\theta }{2}-2P\cos \frac{\theta }{2}-2uP_x \cos ^2\frac{\theta }{2}} \Big\}} d\xi \\ &=\int_R {\partial _\xi } ({u^3-Au^2-2uP})d\xi =0. \end{aligned} \end{equation} In the sense that $\cos \frac{\theta }{2}=0$ whenever $\theta =\pi $, thus we are again integrating over the entire real line $R$ on the fourth identity of \eqref{eq34}. This implies that the extended energy $\tilde {E}(t)$ is consistent, namely \begin{equation} \label{eq35} \tilde {E}(t)=\int_R {\Big({u^2\cos ^2\frac{\theta _1 }{2}+\eta ^2\cos ^2\frac{\theta _1 }{2}+1}\Big)w} ({t,\xi } )d\xi =\tilde {E}(0)=E_0 . \end{equation} From \eqref{eq33} and \eqref{eq35}, we can obtain the bound \begin{equation}\label{eq36} \sup _{\xi \in \mathbb{R}} | {u^2({t,\xi })} |\le 2\int_R {| {uu_\xi } |} d\xi \le 2\int_R {| u |\cdot | {\cos \frac{\theta }{2}} |} wd\xi \le E_0 . \end{equation} This provides a priori bound on $\| {u(t)} \|_{L^\infty } $, similarly we can derive an a priori bound on $\| {\eta (t)} \|_{L^\infty } $. Also from the estimation \eqref{eq35} and the definitions \eqref{eq21} and \eqref{eq22}, we obtain \begin{equation}\label{eq37} \begin{aligned} &\| {P(t)} \|_{L^\infty } , \| {P_x (t)} \|_{L^\infty }\\ &\le \| G \|_{L^\infty } \| {\big( {u^2+\frac{1}{2}u_x^2 +\eta ^2}\big)(t)} \|_{L^1} +\frac{A}{2}\big({\| G \|_{L^2}^2 +\| u \|_{L^2}^2 }\big) +\frac{1}{2}\big({\| G \|_{L^2}^2 +\| \eta \|_{L^2}^2 }\big)\\ &\le \frac{1}{2}E_0 +\frac{A}{2}({\frac{1}{4}+E_0 } )+\frac{1}{2}({\frac{1}{4}+E_0 })\\ &\le \frac{2+A}{2}E_0 +\frac{A+1}{8}. \end{aligned} \end{equation} Then by \eqref{eq36}, \eqref{eq37} and the fourth equation in \eqref{eq24}, we deduce that there exists a constant $B$, depending only on the total energy $E_0$, such that \[ | {\frac{\partial w}{\partial t}} |\le Bw, \] which yields \[ e^{-Bt}\le w(t)\le e^{Bt}. \] From the third equation in \eqref{eq24}, we know that $0\le \theta (t)\le 2\pi$. All the above estimates show that a priori bounds \eqref{eq26}-\eqref{eq28} which we needed to construct a local solution with a constant $C$ exist over any given time interval $[0,T]$. This completes the proof that the local solution can be extended globally for all times $t\ge 0$. \end{proof} \section{Global dissipative solutions for the two-component Camassa-Holm system} In this section, we show that the global solution of the system \eqref{eq24} yields a global dissipative solution of system \eqref{eq2}, in the original variables $({t,x})$. In the following, we shall show the continuous dependence of solutions to system \eqref{eq2}. Recalling that we have obtained the local existence theorem by representing the solution of \eqref{eq25} as the fixed point of a contraction in a suitable space, this yields uniqueness and continuous dependence with respect to convergence on the initial data in $L^\infty \times L^\infty $. \begin{theorem}\label{thm2} Let $\bar {z}_n =({\bar {u}_n ,\bar {\eta }_n })$ be a sequence of initial data with $\| {\bar {z}_n -\bar {z}} \|_{H^1} \to 0$. Then, for any T $>$ 0, the corresponding solutions $z_n ({t,\xi })=({u_n ,\eta _n })({t,\xi })$ converge to $z({t,\xi })=({u,\eta })({t,\xi })$ uniformly with $({t,\xi })\in [0,T]\times R$. \end{theorem} \begin{proof} Let $({u,\eta ,\theta ,w})$ and $({\tilde {u},\tilde {\eta },\tilde {\theta },\tilde {w}})$ be any two solutions of \eqref{eq24}, with the initial condition \eqref{eq21}. Let $E_0 $ be an upper bound for the energies of the two solutions. Suppose that at time $t=0$, there exists a constant $\delta _0 $, \[ \| {z(0)-\tilde {z}(0)} \|_{L^\infty } \le \delta _0 , \quad \| {\theta ({0,\xi })-\tilde {\theta }({0,\xi } )} \|_{L^2} \le \delta _0 . \] Next, for $t\in [0,T]$, we will establish an a-priori bound depending only on $\delta _0 $, $T$ and $E_0 $ on \begin{equation} \label{eq38} \| {z(t)-\tilde {z}(t)} \|_{L^\infty }. \end{equation} Define the set \[ \Lambda =\{ {\xi \in \mathbb{R};\theta ({T,\xi })=\pi } \}\cup \{ {\xi \in \mathbb{R};\tilde {\theta }({T,\xi })=\pi } \}, \] thus $\alpha ^\ast =\operatorname{meas}(\Lambda)$ is a uniformly bounded number depending only on $T$ and $E_0 $. Let $\tau (\xi)=\inf \{ {t\in [0,T];\min\{ {\theta ({t,\xi }), \tilde {\theta }({t,\xi } )} \}=\pi } \}$ such that $\tau (\xi)$ is the first time when one of the two solutions reaches the value $\pi $. We now construct a measure-preserving mapping: $[ {0,\alpha ^\ast } ]\to \Lambda $, which is denoted as $\alpha \to \xi (\alpha)$ with the additional property: \begin{equation} \label{e5.2} \alpha \le {\alpha }' \text{ if and only if } \tau ({\xi (\alpha )})\ge \tau ({\xi ({{\alpha }'})}). \end{equation} According to the mapping $[ {0,\alpha ^\ast } ]\to \Lambda $, we define the distance function \begin{equation}\label{eq39} \begin{aligned} &J({({u,\eta ,\theta ,w}),({\tilde {u},\tilde {\eta },\tilde {\theta },\tilde {w}})}) \\ &=({\| {u-\tilde {u}} \|_{L^\infty } +\| {\eta -\tilde {\eta }} \|_{L^\infty } +\| {\theta -\tilde {\theta }} \|_{L^2} +\| {w-\tilde {w}} \|_{L^2} }) \\ &\quad +K_0 \int_0^{\alpha ^\ast } {e^{K\alpha }} ({| {\theta ( {\xi (\alpha)})-\tilde {\theta }({\xi ( \alpha)})} |})d\alpha . \end{aligned} \end{equation} For convenience, we set \begin{equation} \label{eq40} J(t)=J({({u,v,\theta ,w}),({\tilde {u},\tilde {v},\tilde {\theta },\tilde {w}})})(t )=J^\ast (t)+K_0 J^\# (t), \end{equation} where \[ J^\ast (t)=\Big({\| {u-\tilde {u}} \|_{L^\infty } +\| {\eta -\tilde {\eta }} \|_{L^\infty } +\| {\theta -\tilde {\theta }} \|_{L^2} +\| {w-\tilde {w}} \|_{L^2} } \Big), \] \begin{equation} \label{eq41} J^\# (t)=\int_0^{\alpha ^\ast } {e^{K\alpha }} \big({| {\theta ({\xi (\alpha)})-\tilde {\theta }( {\xi (\alpha)})} |}\big)d\alpha . \end{equation} In the following we show that, for suitable constants $K_0 $, $K$, $M$ depending only on $T$ and $E_0 $, the inequality \begin{equation}\label{eq42} \frac{d}{dt}J(t)\le M J(t) \end{equation} holds. Moreover, this will imply \[ J(t)\le e^{Mt}J(0), \quad t\in [0,T], \] which provides an a-priori estimate on the distance at \eqref{eq38}. For each fixed $t\in [0,T]$, we define the sets \begin{gather*} \Gamma (t)=\{ {\xi \in \Lambda :\theta ({t,\xi } )\ne \pi ,\tilde {\theta }({t,\xi })=\pi } \}\cup \{ {\xi \in \Lambda :\tilde {\theta }({t,\xi })\ne \pi ,\theta ({t,\xi })=\pi } \}, \\ \Gamma ^+(t)=\{ {\xi \in \Lambda :\theta ({t,\xi } )=\tilde {\theta }({t,\xi })=\pi } \}, \\ \Gamma ^-(t)=\{ {\xi \in \Lambda :\theta ({t,\xi } ),\tilde {\theta }({t,\xi })\ne \pi } \}=\{ {\xi \in \Lambda :\tau (t)>t} \}, \end{gather*} with the following properties \[ \Gamma (t)\cap \Gamma ^+(t)=\Gamma (t )\cap \Gamma ^-(t)=\Gamma ^+(t)\cap \Gamma ^-(t)=\Phi , \quad \Gamma (t)\cup \Gamma ^+(t)\cup \Gamma ^-(t)=\Lambda \] for each $t\in [0,T]$. Set $m(t)=\operatorname{meas}({\Gamma ^-(t)})$, such that \begin{equation}\label{eq43} \Gamma ^-(t)=\{ {\xi (\alpha);\alpha \in [ {0,m(t)} ]} \}. \end{equation} From the equations in \eqref{eq24}, we have the estimate \begin{equation}\label{eq44} \begin{aligned} &\frac{d}{dt}\Big({\| {u-\tilde {u}} \|_{L^\infty } +\| {\eta -\tilde {\eta }} \|_{L^\infty } +\| {\theta -\tilde {\theta }} \|_{L^2} +\| {w-\tilde {w}} \|_{L^2} }\Big) \\ &\le \kappa \Big({\| {u-\tilde {u}} \|_{L^\infty } +\| {\eta -\tilde {\eta }} \|_{L^\infty } +\| {\theta -\tilde {\theta }} \|_{L^2} +\| {w-\tilde {w}} \|_{L^2} +\operatorname{meas}({\Gamma (t)})}\Big). \end{aligned} \end{equation} Moreover, from \eqref{eq43} we can deduce that \begin{equation} \label{eq45} \begin{aligned} &\frac{d}{dt}\int_0^{\alpha ^\ast } {e^{K\alpha }({| {\theta ({t,\xi (\alpha)})-\tilde {\theta }({t,\xi (\alpha)})} |})} d\alpha \\ &=\int_{\Gamma (t)\cup \Gamma ^+(t)\cup \Gamma ^-(t)} {e^{K\alpha (\xi)}\cdot \frac{\partial }{\partial t}({| {\theta ({t,\xi (\alpha)} )-\tilde {\theta }({t,\xi (\alpha)})} |})} d\alpha \\ &=\int_{\Gamma (t)} {e^{K\alpha (\xi)}\cdot \frac{\partial }{\partial t}({| {\theta ({t,\xi ( \alpha)})-\tilde {\theta }({t,\xi (\alpha)} )} |})} d\xi \\ &\quad +\int_0^{m(t)} {e^{K\alpha (\xi)}\cdot \frac{\partial }{\partial t}({| {\theta ({t,\xi ( \alpha)})-\tilde {\theta }({t,\xi (\alpha)} )} |})} d\alpha . \end{aligned} \end{equation} Indeed, the integral over $\Gamma ^+(t)$ is zero. Choosing $\delta >0$ which depends only on $T$, $E_0 $ sufficiently small, we have \[ | {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} |\le \delta \] for $\xi \in \Gamma (t)$, which implies \[ \frac{\partial }{\partial t}| {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} |\le -\frac{1}{2}. \] On the other hand, choosing a constant $\kappa $ large enough such that $| {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} |\ge \delta $, we obtain \[ \frac{\partial }{\partial t}| {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} |\le -\frac{1}{2}+\kappa | {\theta ({t,\xi })-\tilde {\theta }({t,\xi })}|. \] Finally, for $\xi \in \Gamma ^-(t)$, we have \begin{align*} \frac{\partial }{\partial t}| {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} | &\le \kappa \cdot (\| {u-\tilde {u}} \|_{L^\infty } +\| {\eta -\tilde {\eta }} \|_{L^\infty } +\| {\theta -\tilde {\theta }} \|_{L^2} +\| {w-\tilde {w}} \|_{L^2}\\ &\quad +\operatorname{meas}({\Gamma (t)})+| {\theta ({t,\xi } )-\tilde {\theta }({t,\xi })} |). \end{align*} Therefore, \begin{equation}\label{eq46} \begin{aligned} &\int_0^{m(t)} {e^{K\alpha }\cdot \frac{\partial }{\partial t}({| {\theta ({t,\xi (\alpha)})-\tilde {\theta }({t,\xi (\alpha)})} |})} d\alpha \\ & \le \kappa ({J^\ast (t)+\operatorname{meas}({\Gamma (t )})})\int_0^{m(t)} {e^{K\alpha }d\alpha } +\kappa \int_0^{m(t)} {e^{K\alpha }\Big({| {\theta ({t,\xi (\alpha)})-\tilde {\theta }({t,\xi (\alpha)})} |}\Big)d\alpha } \\ &\le \kappa ({J^\ast (t)+\operatorname{meas}({\Gamma (t )})})\int_0^{m(t)} {e^{K\alpha }d\alpha } +\kappa \int_{\Gamma ^-(t)} {e^{K\alpha (\xi )}\big({| {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} |}\big)} d\xi . \end{aligned} \end{equation} Now, \eqref{eq44} can be rewritten in the form \begin{equation}\label{eq47} \frac{d}{dt}J^\ast (t)\le \kappa \cdot ({J^\ast (t )+\operatorname{meas}({\Gamma (t)})}). \end{equation} Notice that $\xi \in \Gamma (t)$ implies $\alpha (\xi)\ge m(t)$, together \eqref{eq45} and \eqref{eq46} imply \begin{equation}\label{eq48} \begin{aligned} \frac{d}{dt}J^\# (t) &\le -\frac{1}{2}\int_{\Gamma (t)} {e^{K\alpha (\xi)}} d\xi +\kappa \int_{\Gamma (t)\cup \Gamma ^-(t)} {e^{K\alpha (\xi)}( {| {\theta ({t,\xi })-\tilde {\theta }({t,\xi })} |})} d\xi \\ &\quad +\kappa ({J^\ast (t)+\operatorname{meas}({\Gamma (t )})})\cdot \int_0^{m(t)} {e^{K\alpha}d\alpha } \\ &\le -\frac{1}{2}e^{Km(t)} \operatorname{meas}({\Gamma (t )})+\kappa J^\# (t)+\kappa J^\ast (t)\int_0^{\alpha ^\ast } {e^{K\alpha }d\alpha } \\ &\quad +\kappa \operatorname{meas}({\Gamma (t)})e^{Km(t )}\int_0^{m(t)} {e^{K({\alpha -m(t)} )}d\alpha } \\ & \le -\frac{1}{4}e^{Km(t)}\operatorname{meas}({\Gamma (t )})+\kappa J^\# (t)+\frac{\kappa }{K}e^{K\alpha^\ast }J^\ast (t). \end{aligned} \end{equation} We choose the constant $K=4\kappa $ in the above inequality such that \[ \kappa \int_0^{m(t)} {e^{K({\alpha -m(t)} )}d\alpha } \le \frac{\kappa }{K}=\frac{1}{4}. \] From \eqref{eq47} and \eqref{eq48}, choosing $K_0 =4k$, we obtain \begin{align*} &\frac{d}{dt}({J^\ast (t)+4\kappa J^\# (t)})\\ &\le \kappa \cdot ({J^\ast (t)+\operatorname{meas}({\Gamma(t)})}) +4\kappa ({-\frac{1}{4}\operatorname{meas}({\Gamma (t)} )+\kappa J^\# (t)+\frac{\kappa }{K}e^{K\alpha ^\ast }J^\ast (t)}) \\ &\le \kappa J^\ast (t)+4\kappa ^2J^\# (t)+\kappa e^{4\kappa \alpha ^\ast }J^\ast (t), \end{align*} with $J^\ast $ and $J^\# $ are defined in \eqref{eq41}. With $M=\kappa +\kappa e^{4\kappa \alpha ^\ast }$, our claim \eqref{eq42} is satisfied. \end{proof} Next we revert to the original variables $({t,x})$, and show that the global solution of system \eqref{eq24} yields a global dissipative solution of the original system \eqref{eq2}. Let us begin with a global solution $({u,\eta ,\theta ,w})$ of system \eqref{eq24}. Define \begin{equation}\label{eq49} q({t,\xi })=\bar {q}(\xi)+\int_0^t {u({\tau,\xi })d\tau } . \end{equation} Then for each fixed $\xi $, the function $t\mapsto q({t,\xi })$ provides a solution to the Cauchy problem \begin{equation}\label{eq50} \begin{gathered} \frac{\partial }{\partial t}q({t,\xi })=u({t,\xi }), \\ q({0,\xi })=\bar {q}(\xi). \end{gathered} \end{equation} We claim that, if $q({t,\xi })=x$, a solution of system \eqref{eq2} can be obtained by setting \begin{equation}\label{eq51} z({t,x})=z({t,\xi }), \end{equation} where $z({t,x})=({u,\;\eta })({t,x})$, $z({t,\xi })=({u,\;\eta })({t,\xi})$. The main result reads as follows. \begin{theorem}\label{thm3} If $({u,\eta ,\theta ,w})$ is a global solution to the Cauchy problem \eqref{eq24}-\eqref{eq21}, then the function $z=z({t,x})$ defined by \eqref{eq48}-\eqref{eq51} provides a global dissipative solution of system \eqref{eq2}. \end{theorem} \begin{proof} Using the uniform bound $| {u({t,\xi })}|\le E_0^{1/2}$, from \eqref{eq48} we obtain \[ \bar {q}(\xi)-E_0^{1/ 2} t\le q({t,\xi })\le \bar {q}(\xi)+E_0^{1/2} t, \quad t\ge 0. \] The definition of $\xi $ in \eqref{eq10} yields \[ \lim _{\xi \to \pm \infty } \bar {q}({t,\xi })=\pm \infty . \] Then the image of the continuous map $({t,\xi })\to ({t,q({t,\xi })})$ covers the entire plane $[{0,\infty } ]\times R$. It is clear that the map $\xi \mapsto q({t,\xi })$ is non-decreasing. Then the map $({t,x})\mapsto z({t,x})$ at \eqref{eq51} is well defined, for all $t\ge 0$ and $x\in \mathbb{R}$. For every fixed $t$, we claim \begin{align*} \operatorname{meas}({\{ {q({t,\xi });\theta ({t,\xi })=\pi } \}}) &=\int_{\{ {\theta ({t,\xi })=\pi } \}} {q_\xi ({t,\xi })d\xi }\\ &=\int_{\{ {\theta ({t,\xi })=\pi } \}} {w\cos ^2\frac{\theta }{2}({t,\xi })d\xi } =0, \end{align*} which implies that, in the $x$-variable, the image of the singular set, where $\theta =\pi $, has measure zero. By changing the variable of integration, we compute \begin{equation}\label{eq52} \begin{aligned} &\int_R {({u^2+u_x^2 +\eta ^2})({t,x})} dx \\ &=\int_{\{ {\cos \theta \ne -1} \}} {({u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1})} w({t,\xi })d\xi \le E_0 . \end{aligned} \end{equation} This implies the uniform H\"{o}lder continuity with exponent $1/ 2$ of $z$ as a function of $x$. From the first and second equations in \eqref{eq24} and the bounds on $\| P \|_{L^\infty } $, $\| {P_x } \|_{L^\infty } $, we can obtain that the map $t\mapsto z({t,q(t)})$ is uniformly Lipschitz continuous along the characteristic curve $t\mapsto q(t)$. Hence, $z=z({t,x})$ is globally H\"{o}lder continuous for $({t,x})\in \mathbb{R}^+\times \mathbb{R}$. We know that the map $t\to z(t)$ is Lipschitz continuous with values in $L^2(\mathbb{R})$. Since $L^2(\mathbb{R})$ is a reflexive space, we can deduce that the map $t\mapsto q(t)$ is differentiable for almost every (a.e.) time $t\ge 0$. Note that the right hand side of the first equation in \eqref{eq8} lies in $L^2(\mathbb{R})$, to establish the equality, one may proceed as follows. For each smooth function with compact support $\varphi \in C_c^\infty $, at a. e. time $t\ge 0$, we have \begin{equation} \label{eq53} \begin{aligned} &\frac{d}{dt}\int {u({t,x})\varphi (x)} dx =\int {({-uu_x -P_x })({t,x})\varphi (x)}dx \\ &=\int {[ {u^2({t,x}){\varphi }'(x)-P_x ( {t,x})\varphi (x)+u({t,x})u_x ({t,x})\varphi (x)} ]} dx. \end{aligned} \end{equation} Let us set \begin{equation}\label{eq54} \tau (\xi)=\inf \{ {t>0;\theta (t)=\pi }\} \end{equation} for each $\xi \in \mathbb{R}$. Note that, for a. e. time $t\ge 0$ \begin{equation}\label{eq55} \operatorname{meas}({\{ {\xi ;\tau (\xi)=t} \}})=0. \end{equation} Choosing a time $t$ such that \eqref{eq55} holds. Integrating with respect to the variable $\xi $ and thus we obtain from \eqref{eq13} that \begin{align*} &\frac{d}{dt}\int {u({t,\xi })\varphi ({q({t,\xi } )})({w\cdot \cos ^2\frac{\theta }{2}})} ({t,\xi })d\xi \\ &=\int \big\{ {u_t \varphi w\cos ^2\frac{\theta }{2} +u{\varphi }'q_t w\cos ^2\frac{\theta }{2}+u\varphi w_t \cos ^2\frac{\theta }{2}-u\varphi w\theta _t \frac{\sin \theta }{2}} \big\} d\xi \\ &=\int_{\theta ({t,\xi })\ne \pi } \Big\{ {-P_x \varphi w\cos ^2\frac{\theta }{2}} +u^2{\varphi }'w\cos ^2\frac{\theta}{2} +u\varphi ({2u^2-2Au+\eta ^2+2\eta -2P}) \\ &\quad\times \cos \frac{\theta}{2}w \cos ^2\frac{\theta }{2} -u\varphi w\big[ -\csc \frac{\theta}{2} +({2u^2-2Au+\eta ^2+2\eta -2P})\\ &\quad \cos \frac{\theta }{2}\cot \frac{\theta }{2} \big] \frac{\sin \theta }{2} \Big\} d\xi \\ &=\int_{\theta ({t,\xi })\ne \pi } {[ {-P_x \varphi +u^2{\varphi }'+uu_x \varphi } ]} w\cos ^2\frac{\theta }{2}d\xi . \end{align*} It can be seen \eqref{eq53} holds, and therefore we can conclude that $z=({u,\eta })$ is a global solution of the two-component Camassa-Holm system in the sense of Definitions \ref{def1} and \ref{def2}. \end{proof} Next we prove that global dissipative solutions of the two-component Camassa-Holm system \eqref{eq2} construct a semigroup. To do this some relevant properties are first given. \smallskip \textbf{Property 1.} The total energy is a non-increasing function of time, namely $\| {z(t)} \|_{H^1(\mathbb{R})} \le \|{z({{t}'})} \|_{H^1(\mathbb{R})} $ if $0\le {t}'\le t$. \begin{proof} By \eqref{eq52}, we have \begin{align*} \| {z(t)} \|_{H^1} &=\int_R {({u^2+u_x^2 +\eta ^2})({t,x})} dx \\ &=\int_{\{ {\cos \theta \ne -1} \}} \Big({u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1}\Big) w({t,\xi })d\xi \\ &=E_0 -\int_{\{ {\tau (\xi)\le t} \}} \Big( {u^2\cos ^2\frac{\theta }{2}+\eta ^2\cos ^2\frac{\theta }{2}+1}\Big) w({t,\xi })d\xi \\ &\le E_0 -\int_{\{ {\tau (\xi)\le t} \}} {w({t,\xi })} d\xi\\ &\le E_0 -\int_{\{ {\tau (\xi)\le {t}'} \}} {w({t,\xi })} d\xi =\| {z({{t}'})} \|_{H^1} \end{align*} with $\tau (\xi )$ given in \eqref{eq54}. \end{proof} \textbf{Property 2.} Given a sequence of initial data $\bar {z}_n $ such that $\bar {z}_n \to \bar {z}$ in $H^1\times L^2$, the corresponding solutions $z_n ({t,x})\to z({t,x})$ uniformly for $({t,x})$ in bounded sets. \begin{theorem}\label{thm4} Let $\bar {z}_n \in H^1\times [ {L^2\cap L^\infty } ]$ be an initial data, and $z(t)=S_t \bar {z}$ the corresponding global solution of system \eqref{eq2} constructed in Theorem \ref{thm3}. Then the mapping S: $H^1\times [ {L^2\cap L^\infty }]\times [ {0,\infty })\to H^1$ is a semigroup. \end{theorem} \begin{proof} Let $({t,\xi })\to ({u,\eta ,\theta ,w})({t,\xi })$ be the corresponding solutions to systems \eqref{eq24} and \eqref{eq21}. For the fixed $\tau >0$ and all $t\in \mathbb{R}^+$, one needs to prove that \[ S_t ({S_\tau \bar {z}})=S_{\tau +t} \bar {z}. \] A new energy variable $p$ is defined as \begin{equation}\label{eq56} \frac{d}{d\xi }p(\xi)=\begin{cases} w({\tau ,\xi })&\text{if }\theta ({\tau ,\xi })\ne \pi , \\ 0&\text{if } \theta ({\tau ,\xi })=\pi , \end{cases} \end{equation} with initial data \begin{equation}\label{eq57} p({\xi _0 })=0. \end{equation} Choose the value $\xi _0 $ such that $q({\tau ,\xi _0 })=0$. By setting $\hat {z}=S_\tau \bar {z}$, we define \begin{equation}\label{eq58} \begin{gathered} \hat {z}({t,p})=z({\tau +t,\xi (p)}), \\ \hat {\theta }({t,p})=\theta ({\tau +t,\xi (p)}), \\ \hat {w}({t,p})=\frac{w({\tau +t,\xi (p)})}{w({\tau ,\xi (p)})}, \\ \end{gathered} \end{equation} such that $p\to \xi (p)$ provides an a.e. inverse to the mapping in \eqref{eq56}-\eqref{eq57}, namely, \[ \xi ({p^\ast })=\sup \{ {s;p(s)\le p^\ast }\}. \] Recalling the identities \eqref{eq14} and \eqref{eq13}, one has \[ \frac{\partial }{\partial \xi }q({\tau ,\xi })\cdot u_x^2 ({\tau ,q({\tau ,p(\xi)})}) =w({\tau ,\xi })=\frac{d}{d\xi }p(\xi). \] By an integration and using \eqref{eq56}, one gets that \[ \int_0^{q({\tau ,\xi })} {u_x^2 ({\tau ,x})}dx=p(\xi). \] Now it can be shown that the functions in \eqref{eq58} provide a solution to system \eqref{eq24}. The identities $w({\tau +t,\xi })d\xi =\frac{\hat {w}({t,p(\xi)})}{w({t,p( \xi)})}\cdot \frac{dp(\xi)}{d\xi }\cdot dp=\hat {w}({t,p(\xi)})dp$ imply that the corresponding integral source terms in \eqref{eq24} satisfy \begin{equation}\label{eq59} \hat {P}({t,p})=P({\tau +t,\xi (p)}),\quad \hat {P}_x ({t,p})=P_x ({\tau +t,\xi (p)}). \end{equation} Since the last equation in \eqref{eq24} is linear with respect to the variable $w$, then we can draw the conclusion that the functions in \eqref{eq58} provide a solution to system \eqref{eq24}. In summary, the global dissipative solutions of system \eqref{eq2} construct a semigroup. \end{proof} \subsection*{Acknowledgements} This work was supported in part by the Major State Basic Research Development Program 973 (No. 2012CB215202), and by the National Natural Science Foundation of China (No.61134001). \begin{thebibliography}{00} \bibitem{a1} N. 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