\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 78, pp. 1--23.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/78\hfil Blow-up solutions to the modified CH equation] {Blow-up of solutions to the rotation b-family system modeling equatorial water waves} \author[M. Zhu, Y. Wang \hfil EJDE-2018/78\hfilneg] {Min Zhu, Ying Wang} \address{Min Zhu \newline Department of Mathematics, Nanjing Forestry University, Nanjing, 210037, China} \email{zhumin@njfu.edu.cn} \address{Ying Wang \newline Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 611731, China} \email{nadine\_1979@163.com} \dedicatory{Communicated by Tuncay Aktosun} \thanks{Submitted December 4, 2017. Published March 20, 2018.} \subjclass[2010]{35B44, 35G25} \keywords{Rotation b-family system; blow up; wave-breaking} \begin{abstract} We consider the blow-up mechanism to the periodic generalized rotation b-family system (R-b-family system). This model can be derived from the f-plane governing equations for the geographical water waves with a constant underlying current in the equatorial water waves with effect of the Coriolis force. When $b=2$, it is a rotation two-component Camassa-Holm (R2CH) system. We consider the periodic R2CH system when linear dispersion is absent (which model is called r2CH system) and derive two finite-time blow-up results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} It is known that for the geophysical water waves the forces with primary influence are the gravity and the Coriolis force induced by the Earth's rotation. When considering waves propagating in the equatorial ocean regions throughout the extent of the Pacific Ocean, it is found however that the Equatorial Undercurrent is one essential feature and the effect of the Coriolis force is small, because of the smallness of the variation in latitude of the EUC in the equatorial region. There have recently appeared several works involving steady periodic rotational Equatorial water waves in the f-plane on topics like existence, regularity of free surface and of the stream lines, symmetry and stability. This paper is to study the following periodic generalized rotation b-family system (R-b-family system). \begin{equation}\label{1.1} \begin{gathered} \begin{aligned} &u_t-u_{xxt}-Au_x +(b+1)uu_x\\ &=\sigma(bu_xu_{xx}+uu_{xxx})-\mu u_{xxx}-(1-2\Omega A)\rho\rho_x +2\Omega \rho (\rho u)_x, \end{aligned}\\ \rho_t+(\rho u)_x=0,\\ u(t,x)=u(t,x+1),\rho(t,x)=\rho(t,x+1)\\ u(0,x)=u_0(x),\rho(0,x)=\rho_0(x). \end{gathered} \end{equation} for $t>0$ and $x \in \mathbb{S}$, where $u(x,t)$ is a horizontal velocity, $\rho(t,x)$ is related to the free surface elevation from equilibrium, the parameter $A$ characterizes a linear underlying shear flow, the real dimensionless constant $\sigma$ is a parameter which provides the competition, or balance, in fluid convection between nonlinear steepening and amplification due to stretching, $\mu$ is a non-dimensional parameter and $\Omega$ characterizes the constant rotational speed of the Earth. We can rewrite the periodic R-b-family system \eqref{1.1} into the system \begin{equation}\label{1.1a} \begin{gathered} \begin{aligned} u_t+(\sigma u-\mu)u_x &=-\partial_x G\ast \Big((\mu-A)u+\frac{b+1-\sigma}{2}u^2 + \frac{(b-1)\sigma}{2}u^2_x \\ &\quad +\frac{1-2\Omega A}{2}\rho^2-\Omega\rho^2u\Big) +\Omega G\ast (\rho^2 u_x), \end{aligned}\\ \rho_t+u\rho_x=-\rho u_x,\\ u(t,x)=u(t,x+1),\rho(t,x)=\rho(t,x+1),\\ u(0,x)=u_0(x),\rho(0,x)=\rho_0(x).\\ \end{gathered} \end{equation} for $t>0$ and $x \in \mathbb{S}$. We denote $G(x)=\frac{\cosh(x-[x]-\frac{1}{2})}{2\sinh(\frac{1}{2})}$, the fundamental solution of $1-\partial^2_x$ on $\mathbb{S}$, that is, $(1-\partial_x^2)^{-1}f =G\ast f$, we have the relation \begin{equation}\label{1.1b} G\ast f(t,x)=\int^1_0 \frac{\cosh \big((x-y)-[x-y]-{\frac{1}{2}}\big)} {2\sinh \big(\frac{1}{2}\big)}f(t,y)dy. \end{equation} The approach we adopt here to derive the R-b-family system is in the spirit of Ivanov's asymptotic perturbation analysis for the governing equations of two-dimensional rotational gravity water waves\cite{RI}. And there are two factors force us to do the asymptotic perturbation analysis. In the equatorial region there exist the shallow water waves, for which we mean that the shallow water parameter $\delta=h/\lambda<0.07$\cite{Con}, where $h$ is the mean depth of water and $\lambda$ is the wavelength. Actually, the equatorial region is characterized as a two-layer fluid with a shallow upper region of warmer and less dense water overlying a motionless deep region of cold water. The upper shallow water is less than 300 m deep \cite{SGH} and usually the wavelength of the surface waves can be 100S km or more. The equatorial Rossby waves whose wavelength can be 500 km are evidence examples. Besides, the westward travelling waves with a wavelength of 1000 km near 3 in the central and eastern Pacific Ocean has been observed, and the wavelengths slightly in excess of 2000 km can arise from the instabilities of surface Equatorial Currents. On the other hand, to ensure the earth's rotation to have a significant impact on the fluid motion, one expects the Rossby number $R_0=\frac{U}{\lambda \Omega}=O(1)$, where $\lambda$ is the typical horizontal length scale for the flow, $U$ is the typical horizontal length scale for the flow, $U$ is the typical horizontal velocity scale for the fluid motion, and the symbol $O(1)$ means that the term is of the order of magnitude of one, or less. This in turn implies that the smaller the characteristic velocity $U$ is, the smaller $L$ can be and yet it still enables us to consider large-scale waves. In fact, system \eqref{1.1} has significant relationship with several models describing the motion of waves at the free surface of shallow water under the influence of gravity. When $b=2$, it becomes the rotation Camassa-Holm system (R2CH system) \begin{equation}\label{1.4} \begin{gathered} \begin{aligned} u_t-u_{xxt}-Au_x +3uu_x &=\sigma(2u_xu_{xx}+uu_{xxx})-\mu u_{xxx}\\ &\quad -(1-2\Omega A)\rho\rho_x+2\Omega \rho (\rho u)_x, \end{aligned}\\ \rho_t+(\rho u)_x=0,\\ u(t,x)=u(t,x+1),\rho(t,x)=\rho(t,x+1),\\ u(0,x)=u_0(x),\rho(0,x)=\rho_0(x). \end{gathered} \end{equation} for $ t>0$ and $x \in \mathbb{S}$, Moreover, we can rewrite the periodic R2CH system as the system \begin{equation}\label{1.5} \begin{gathered} \begin{aligned} u_t+(\sigma u-\mu)u_x &=-\partial_x G\ast \Big((\mu-A)u+\frac{3-\sigma}{2}u^2\\ &\quad +\frac{\sigma}{2}u^2_x+\frac{1-2\Omega A}{2}\rho^2-\Omega\rho^2u\Big) +\Omega G\ast (\rho^2 u_x), \end{aligned}\\ \rho_t+u\rho_x=-\rho u_x,\\ u(t,x)=u(t,x+1),\rho(t,x)=\rho(t,x+1),\\ u(0,x)=u_0(x),\rho(0,x)=\rho_0(x). \end{gathered} \end{equation} for $t>0$ and $x \in \mathbb{S}$. If $\Omega=0$, without considering effect of the Earth's rotation, then the following functional is conserved $$ F(u,\rho)=\frac{1}{2}\int_\mathbb{R}(u^3+\sigma uu^2_x-Au^2-\mu u^2_x +2(\rho-1)u+u(\rho-1)^2)dx. $$ When $b=2,\Omega=0$, system \eqref{1.1} is the generalized DGH system \begin{equation} \begin{gathered} u_t-u_{xxt}-Au_x +3uu_x =\sigma(2u_xu_{xx}+uu_{xxx})-\mu u_{xxx}-\rho\rho_x,\\ \rho_t+(\rho u)_x=0, \end{gathered} \end{equation} in which $\sigma=1$, $\mu=0$. Then the equation recovers the standard two-component integrable Camassm-Holm system \begin{equation} \begin{gathered} u_t-u_{xxt}-Au_x +3uu_x+\rho\rho_x =2u_xu_{xx}+uu_{xxx},\\ \rho_t+(\rho u)_x=0, \end{gathered} \end{equation} Moreover, in the case $\rho=0$, \eqref{1.4} recovers the DGH equation and becomes the Camassa-Holm equation. The CH equation is completely integrable for a large class of initial data, for which it can be solved by the inverse scattering method \cite{C,C-Mc}. In contrast to the KdV equation, the CH equation has three remarkable distinctive properties. First, although CH is completely integrable, it can describe wave breaking phenomenon: the solution remains bounded while its slope becomes infinite in finite time. The second is the existence of peakons, which are nonanalytic solitary waves that are global weak solutions and interact cleanly like solitons. Indeed, the CH equation has the single peakon \cite{CaHo93} and the multi-peakon solutions \cite{holden}. It is significant that the peakons are orbitally stable: the shape is stable under small perturbations \cite{CoSt,Le}. These peakons capture a feature of the waves of greatest height for the free-boundary incompressible Euler equations \cite{Co2,CoEs4,Toland}. The last one is the variety of interesting geometric formulations of the CH equation \cite{CQ02,CoKo03, Ko99, Mi98}. Well-posedness and wave breaking of the CH equation were studied in a number of papers. It has been shown \cite{CoEs98-2,LiOl00,Ro01,No09} that the Cauchy problem is locally well-posed for initial data $u_0\in H^s({\mathbb R})$ with $s>3/2$ . Moreover, if the initial momentum density \begin{equation*}\label{m0} m_0(x) = m(0,x) = (1 - \partial_x^2) u_0 = u_0(x)-u_0''(x) \end{equation*} does not change sign, the Cauchy problem admits global solution for certain initial values \cite{Co00, CoEs98-2, CoEs98-3}, whereas solutions may blow up if their initial momentum density $m_0 $ changes sign \cite{Co00,CoEs98,CoEs98-2,CoEs98-3}. After blow-up, the solutions to CH can be continued uniquely as global weak solutions \cite{BrCo1,BrCo2}. Moreover, the existence of global weak solution was investigated in \cite{XinZh00, XinZh02}. It is observed that system \eqref{1.1} we derived is a generalization of system with the rotation of Earth-these effects feature significantly for such large scale phenomena as currents. It is found that the consideration of the Coriolis force has introduced a higher order nonlinear term into the generalized two-component b-family system, which has interesting implications for the fluid motion, particular in the relation to the wave breaking phenomena and the permanent waves. In our case, appearance of the Earth's rotation, however, introduces a cubic-order nonlinear term $\rho(\rho u)_x$ to R2CH system, which is difficult to estimate as usually by using the conservation laws. To deal with this higher nonlinearity, we reformulate the first equation in \eqref{1.4} into \eqref{1.5} with a nonlocal translate $\Omega \partial_xG\ast \rho^2$, and establish the Riccati differential inequality for $K_x=u_x+\Omega \partial_x G\ast\rho^2$. Then by solving the inequality and use the fact that the term $\Omega G\ast \rho^2$ is bounded, $u_x$ blows up if and only if $K_x$ blows up. But the advantage of considering $K$ is that in the equation for $K$ and $K_x$, the cubic terms can be bounded by the conservation laws, which enables one to carry out a standard procedure to reach a Riccati type inequality for $K_x$ $$ \frac{d}{dt}K_x\leq -K^2_x+C, $$ and thus by choosing $K_x$ sufficiently negative initially, the corresponding solution blows up in finite time. A crucial ingredient in this argument is the use of the ``global" information of solutions (like the conservation laws) in deriving various estimates. However the ``local" structure of solutions is under appreciated. On the other hand, the non-diffusive nature of the system indicates that the local structure of data may strongly affect the evolution of the solutions, in particular, the blow-ups. This has recently been evidenced in a class of CH-type equations in a series of works of Brandolese and Cortez \cite{Bra1,Bra2,Bra3}and Zhu \cite{zhu,zhu2,zhu3}, and later extended to some other quasilinear model equations with higher order nonlinearities. One of the main ideas lies in understanding of the interplay between the solution and its gradient. For this amounts to tracking the dynamics of $K\pm K_x$ along the characteristics. Due to the nonlocal character involved in $K$, the conservation law is still needed to establish the convolution estimate. However it is now much apparent to see how rotation affects the wave-breaking. In particular, when the Coriolis effect is turned off our wave-breaking criteria recovers the one for the classical CH equation. Our goal in the present paper to investigate the conditions for R2CH system to ensure the occurrence of the wave-breaking phenomena or permanent waves. In Section 2, we derived the R-b-family system as a model in the equatorial water waves. In Section 3, we recall some basic results concerning the formation of singularities in the R-b-family system and R2CH system. In Section 4, we obtain the r2CH system when linear dispersion is absent in R2CH system. Then we give two kinds of the wave-breaking criterion on which addresses the local structure of the solutions and also indicates explicitly how rotation is involved. Moreover, we further provide an upper bound of $u_x$ along each characteristics emanating from a vanishing point of $\rho_0$. \section{Derivation of the model} \subsection{Governing equations} Assume that the Earth to be a perfect sphere of radius 6371 km and with a constant rotational speed $\Omega=73\times 10^{-6}$ rad/s round the polar axis towards east. We choose a reference frame with the origin located at a point on the earth's surface and which is rotating with earth, setting $x$-axis horizontally due east, the $y$-axis horizontally due north and the $z$-axis upwards. We employ the f-plane approximation from the full geophysical governing equations \cite{Co}: \begin{equation}\label{2.1} \begin{gathered} u_t+uu_x+v u_y+\omega u_z+2\Omega \omega=-p_x\\ v_t+uv_x+vv_y+\omega u_z=-p_y,\\ \omega_t+u\omega_x+v\omega_y+\omega\omega_z-2\Omega u=-p_z-g, \end{gathered} \end{equation} here $(u,v,\omega)$ is the fluid velocity, $p$ is the pressure, $t$ represents time. At the wave surface, the pressure of the fluid matches the atmospheric pressure $p_{atm}$ and assume the fluid bed is impermeable, we impose the no-flux condition, then follow the idea in to derive the rotation-two-component CH system with effect of the Coriolis force. We use the undisturbed depth of the water $h$, as the vertical scale, a typical wavelength $\lambda$, as the horizontal scale, and a typical amplitude of the surface $a$, and we denote the dimensionless parameters $\varepsilon=a/h$ and $\delta=h/\lambda$. Let $z=h+\eta(t,x,y)$ be the surface of the ocean, and set $z=h$ to be the mean surface level for the flow, with $z=0$ we denote the lower boundary of the water. Under the assumption that the constant density of the water is one, the governing equations in the region $0\leq z\leq h+\eta(t,x,y)$ in the f-plane approximation comprise the Euler equation we summarize the above equations then we have the form \begin{equation} \label{2.2} \begin{gathered} u_t+uu_x+\omega u_x+2\Omega \omega=-p_x,\quad 00$. \section{Preliminaries} In this section, we recall some basic results concerning the formation of singularities in the R-b-family system \eqref{1.1} and the R2CH system \eqref{1.4}. Let $\sigma=1,\mu=0$, then the periodic of the R2CH system is been written as the system \begin{equation}\label{3.1} \begin{gathered} \begin{aligned} u_t+uu_x &=-\partial_x G\ast \Big(-Au+u^2+\frac{1}{2}u^2_x +\frac{1-2\Omega A}{2}\rho^2 -\Omega\rho^2u\Big) \\ &\quad +\Omega G\ast (\rho^2 u_x), \end{aligned}\\ \rho_t+u\rho_x=-\rho u_x,\\ u(0,x)=u(1,x),\\ \rho(0,x)=\rho(1,x). \end{gathered} \end{equation} for $t>0$ and $x \in \mathbb{S}$, which is called r2CH system. Similarity \cite{FG}, we can get the following the theorem about the periodic R-b-family system \eqref{1.1}. \begin{theorem} \label{thm3.1} Given $z_0=(u_0,\rho_0)\in H^s({\mathbb{S}})\times H^{s-1}({\mathbb{S}}),s>\frac{3}{2}$, there exist a maximal $T=T(z_0)>0$ and a unique solution $z=(u,\rho)$ to system \eqref{1.1} such that \begin{equation} z=z(\cdot;z_0)\in C([0,T);H^s({\mathbb{S}})\times H({\mathbb{S}})^{s-1})\cap C^1([0,T);H^{s-1}({\mathbb{S}}) \times H^{s-2}({\mathbb{S}})). \end{equation} Moreover, the solution depends continuously on the initial data, i.e. the mapping \begin{equation*} z_0\to z(\cdot;z_0):H^s({\mathbb{S}})\times H^{s-1}({\mathbb{S}})\to C([0,T);H^s({\mathbb{S}}) \times H^{s-1}({\mathbb{S}}))\cap C^1([0,T);H^{s-1}\times H^{s-2}) \end{equation*} is continuous. \end{theorem} Similar to the proof of \cite[Theorem 6.2]{QuLiLi11}. We have the following blow-up criterion for the periodic R-b-family system \eqref{1.1}. \begin{lemma}[Wave-breaking criteria] \label{lem3.2} Assume that $1-2\Omega A>0$. Let $(u_0,\rho_0)\in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S})$ with $s>3/2$, and $T>0$ be the maximal time of existence of the solution $(u,\rho)$ to system \eqref{1.1} with initial data $(u_0,\rho_0)$. Then the corresponding solution $(u,\rho)$ blows up in finite time $T<\infty$ if and only if $$ \lim_{t\to T^{-}}\;\sup_{x\in \mathbb{S}}\;|u_x(t,x)|=+\infty. $$ \end{lemma} \begin{lemma}[\cite{yin1}] \label{lem3.3} For every $f\in H^1(\mathbb{S})$, we have $$ \underset{x\in[0,1]}\max f^2(x)\le C\int_{\mathbb{S}}(f^2+\alpha^2f_x^2)dx, $$ where $$ C=\frac{\cosh(\frac{1}{2\alpha})}{2\alpha\sinh(\frac{1}{2\alpha})}. $$ Moreover $C$ is the minimum value. so in this sense, $C$ is the optimal constant which is obtained by the associated Green function. $$ G=\frac{\cosh(\frac{x}{\alpha}-\frac{[x]}{\alpha} -\frac{1}{2\alpha})}{2\alpha\sinh(\frac{1}{2\alpha})}. $$ \end{lemma} Note that when $\alpha=1$, the constant $C_1= \frac{e+1}{2(e-1)}$ is sharp. To study the wave-breaking problem, we now briefly give the needed results without proof to pursue our goal. We consider the following associated Lagrangian scales of the system \eqref{3.1}, \begin{equation}\label{3.3} \begin{gathered} \frac{\partial {q}}{\partial t}=u(t,q), \quad 00,\quad (t,x)\in [0,T)\times {\mathbb{S}}, $$ \end{lemma} The above lemmas indicate that $q(t,\cdot):\mathbb{S}\to \mathbb{S} $ is diffeomorphisms of the line for each $t\in[0,T)$. Hence, the $L^\infty$ norm of any function $u(t,\cdot)\in L^\infty(\mathbb{S})$ is preserved under the family of diffeomorphisms $q(t,\cdot)$ with $t\in[0,T)$, that is \begin{equation*} \|u(t,\cdot)\|_{L^\infty(\mathbb{S})}=\|u(t,q(t,\cdot))\|_{L^\infty(\mathbb{S})}, \quad t\in[0,T). \end{equation*} Similarly, we have \begin{gather*} \inf_{x\in \mathbb{S} } u(t,x)=\inf_{x\in \mathbb{S}} u(t,q(t,x)),\quad t\in[0,T),\\ \sup_{x\in \mathbb{S} } u(t,x)=\sup_{x\in \mathbb{S}} u(t,q(t,x)),\quad t\in[0,T). \end{gather*} \begin{lemma}]\cite{EsYue}] \label{lem3.5} For all $u\in H^1(\mathbb{S})$, the following inequality holds $$ G*(u^2+{1\over2}u^2_x)\geq \kappa u^2(x), $$ with $$ \kappa={1\over2}+{{\arctan\left(\sinh(1/2)\right)} \over{2\sinh(1/2)+2{\arctan\left(\sinh(1/2)\right)}\sinh^2(1/2)}}\approx 0.869 . $$ Moreover, $\kappa$ is the optimal constant obtained by the function $$ f_0={{1+\arctan\left(\sinh(x-[x]-{1/2})\right)\sinh(x-[x]-{1/2})} \over{1+\arctan\left(\sinh({1/2})\right)\sinh({1/2})}}. $$ \end{lemma} We then prove several useful conservation laws of strong solutions to r2CH system \eqref{3.1}. \begin{lemma} \label{lem3.6} Let $(u_0,\rho_0)\in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S}), s>3/2$, and $T$ be the maximal existence time of the solution $(u,\rho)$ in the periodic of the r2CH system \eqref{3.1}. Then for all $t\in[0,T)$, we have \begin{equation} \int_{\mathbb{S}}u^2+u^2_x+(1-2\Omega A)(\rho-1)^2dx =\int_{\mathbb{S}}u_0^2+u^2_{0x}+(1-2\Omega A)(\rho_0-1)^2dx \end{equation} which means that \[ E(u,\rho)=\frac{1}{2}\int_{\mathbb{S}} \left ( u^2+u_x^2+(1-2\Omega A)(\rho-1)^2 \right ) dx=E_0(u_0,\rho_0). \] \end{lemma} \begin{proof} Multiplying the first equation of r2CH \eqref{3.1} by $u$ and integrating by parts, in view of the periodicity of $u$ and $\rho$, we have \begin{gather*} \frac{1}{2}\frac{d}{dt}\int_{\mathbb{S}}(u^2+u^2_x)dx =-(1-2\Omega A)\int_{\mathbb{S}}u\rho\rho_x\,dx, \\ \frac{1}{2}\frac{d}{dt}\int_{\mathbb{S}}(1-2\Omega A)(\rho-1)^2dx =(1-2\Omega A)\int_{\mathbb{S}}u\rho\rho_x\,dx \end{gather*} Adding the above two equations, we obtain \begin{equation} \frac{d}{dt}\int_{\mathbb{S}}u^2+u^2_x+(1-2\Omega A)(\rho-1)^2dx=0\,. \end{equation} From the above equation, we obtain the statement of the lemma. \end{proof} \begin{lemma} \label{lem3.7} Let $(u_0,\rho_0)\in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S}), s>3/2$, and $T$ be the maximal existence time of the solution $(u,\rho)$ in the periodic of the r2CH system \eqref{3.1}. Then for all $t\in[0,T)$, we have \begin{gather} \int_{\mathbb{S}}u+\Omega(\rho-1)^2dx=\int_{\mathbb{S}}u_0+\Omega(\rho_0-1)^2dx,\\ \int_{\mathbb{S}}(\rho-1)dx=\int_{\mathbb{S}}(\rho_0-1)dx. \end{gather} which means that \begin{gather*} I_1(u,\rho)=\int_{\mathbb{S}}u+\Omega(\rho-1)^2dx =\int_{\mathbb{S}}u_0+\Omega(\rho_0-1)^2dx, \\ I_2(u,\rho)=\int_{\mathbb{S}}(\rho-1)dx=\int_{\mathbb{S}}(\rho_0-1)dx. \end{gather*} \end{lemma} \begin{proof} Integrating the first equation of \eqref{3.1} by parts, in view of the periodicity of $u$ and $G$, we obtain \begin{equation} \begin{aligned} &\frac{d}{dt}\int_{\mathbb{S}}u(t,x)dx\\ &=\int_{\mathbb{S}}-\partial_x G\ast (-Au+u^2+\frac{1}{2}u^2_x +\frac{1-2\Omega A}{2}\rho^2-\Omega\rho^2u)+\Omega G\ast (\rho^2 u_x)dx \end{aligned} \end{equation} Multiplying the second equation by $\Omega(\rho-1)$ and integrating by parts, we have \begin{equation} \frac{d}{dt}\int_{\mathbb{S}}\Omega(\rho-1)^2dx =\int_{\mathbb{S}}\Omega u (\rho^2)_x\,dx. \end{equation} Adding the above two equations and using the identity $G\ast f=f+\partial^2_x G* f$, we obtain $\frac{d}{dt}\int_{\mathbb{S}}{u+\Omega(\rho-1)^2}dx=0$. On the other hand, integrating the second equation, we obtain \begin{equation} \frac{d}{dt}\int_{\mathbb{S}}(\rho-1)dx=0. \end{equation} This completes the proof. \end{proof} \section{Blow-up criteria} In this subsection we address the problem of the blow-up criteria of the periodic r2CH system \eqref{3.1}. The following blow-up criterion can be proved easily by Lemma \ref{lem3.2}, so that we omit its proof. \begin{lemma}\label{Wave-breaking criteria} Assume that $1-2\Omega A>0$. Let $(u_0,\rho_0-1) \in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S})$, with $s>3/2$, and $T>0$ be the maximal time of existence of the solution $(u,\rho)$ to r2CH system \eqref{3.1} with initial data $(u_0,\rho_0)$. Then the corresponding solution $(u,\rho)$ blows up in finite time $T<\infty$ if and only if \begin{align*} \lim_{t\to T^-}\big\{{\inf_{x\in \mathbb{S}}\;{u_x(t,x)}}\big\}=-\infty. \end{align*} \end{lemma} For wave-breaking, one would like to choose some initial data such that $u_x$ approaches $-\infty$ in finite time. The difficulty in the analysis of the dynamics of $u_x$ sources from the last term $G_x \ast (\rho^2 u_x)$, which fails to be controlled by the conservation laws. Our idea is to absorb this term by considering the dynamics of the quantity $K:=u+\Omega G\ast \rho^2$. The following lemma is about $K$, which is important to Theorem \ref{thm4.3}. \begin{lemma} \label{lem4.2} Let $K:=u+\Omega G\ast \rho^2$. Then, R2CH system \eqref{1.4} can be written with the following equation, \begin{equation*} K_t+uK_x=\Omega(A-\mu)\partial_x G\ast \rho^2 +\Omega\sigma u \partial_xG\ast \rho^2-\partial_x G\ast\big((\mu-A)u +\frac{3-\sigma}{2}u^2+\frac{\sigma}{2}u_x^2+\frac{1}{2}\rho^2\big) \end{equation*} and \begin{align*} &K_{xt} +(\sigma u-\mu)K_{xx}\\ &=-\frac{\sigma}{2}(K_x-\Omega\partial_xG\ast\rho^2)^2 +\frac{1+2\Omega(\mu-A)-2\Omega\sigma u}{2}\rho^2 +\Omega\sigma uG\ast\rho^2\\ & \quad+(A-\mu)\partial^2_x G\ast u+\frac{3-\sigma}{2}u^2 -G\ast\big(\frac{3-\sigma}{2}u^2+\frac{\sigma}{2}u^2_x +\frac{1-2\Omega(A-\mu)}{2}\rho^2 \big). \end{align*} \end{lemma} \begin{proof} Recall the first equation in \eqref{1.4} in the form \begin{equation} \begin{aligned} &u_t-u_{xxt}-Au_x+3uu_x \\ &=\sigma (2u_xu_{xx}+uu_{xxx})-\mu u_{xxx}-(1-2\Omega A)\rho\rho_x +2\Omega\rho(\rho u)_x. \end{aligned} \end{equation} To deal with the high-order nonlinear term $\rho(\rho u)_x$, it is found that \begin{equation}\label{4.2} \begin{split} u_t-u_{xxt}+2\Omega\rho \rho_t & =A(u-u_{xx}+\Omega\rho^2)_x+(A-\mu)u_{xxx}-3uu_x\\ &\quad +\sigma(2u_xu_{xxx}+uu_{xxx})-\mu u_{xxx}-\rho\rho_x. \end{split} \end{equation} Applying the operator $(1-\partial^2_x)^{-1}$ to both sides of \eqref{4.2}, we have \begin{equation} \begin{split} &(u+\Omega(1-\partial^2_x)^{-1}\rho^2)_t-A(u+\Omega(1-\partial^2_x)^{-1}\rho^2)_x\\ &= (\mu-A)[u+\Omega (1-\partial^2_x)^{-1}\rho^2]_x-(\mu-A)\Omega \partial_x(1-\partial^2_x)^{-1}\rho^2 \\ &\quad -\sigma u[u+\Omega (1-\partial^2_x)^{-1}\rho^2]_x +\Omega \sigma u \partial_x (1-\partial^2_x)^{-1}\rho^2 \\ &\quad -\partial_x(1-\partial^2_x)^{-1}[(\mu-A)u_x +\frac{\sigma}{2}u^2_x+\frac{3-\sigma}{2}u^2+\frac{1}{2}\rho^2]. \end{split} \end{equation} Taking $K=u+\Omega p\ast\rho^2$, we deduce that \begin{equation}\label{4.4} \begin{split} &K_t-AK_x \\ &= (\mu-A)K_x-\sigma u K_x-\Omega(\mu-A)\partial_x(1-\partial^2_x)^{-1}\rho^2 +\Omega\sigma u\partial_x(1-\partial^2_x)^{-1}\rho^2\\ &\quad -\partial_x(1-\partial^2_x)^{-1}\big( (\mu-A)u+\frac{3-\sigma}{2}u^2 +\frac{\sigma}{2}u^2_x+\frac{1}{2}\rho^2 \big). \end{split} \end{equation} And using that $(1-\partial^2_x)^{-1}f=G\ast f$, we obtain the statement of Lemma. Now we take derivative to equation \eqref{4.4} with respect to $x$, and use $-\partial^2_x G\ast f=f-G\ast f$ to obtain \begin{align*} &K_{xt} +(\sigma u-\mu)K_{xx}\\ &=-\sigma u^2_x+\Omega(\mu-A)(\rho^2-G\ast \rho^2) +\Omega \sigma u(G\ast \rho^2-\rho^2)-(\mu-A)\partial^2_xG\ast u\\ &\quad +\frac{b+1-\sigma}{2}(u^2-G\ast u^2) +\frac{\sigma}{2}u^2_x-\frac{\sigma}{2}G\ast u^2_x +\frac{\rho^2}{2}-G\ast(\frac{\rho^2}{2}) \end{align*} A rearrangement of the equation leads to the lemma. This completes the proof. \end{proof} \begin{theorem}\label{thm4.3} Suppose that $1-2\Omega A>0$. Let $(u,\rho)$ be the solution of r2CH system \eqref{3.1} with initial data $(u_0,\rho_0-1)\in H^s(\mathbb{S})\times H^{s-1}(\mathbb{S})$ with $s>\frac{3}{2}$ and $T$ be the maximal time of existence. Assume there exists a $x_0$ such that \begin{equation} \rho_0(x_0)=0, \end{equation} and \begin{equation} u_{0,x}(x_0)< -|u_0(x_0)-\frac{A}{2}|-4\Omega C_1-\sqrt{\frac{2A^2}{e-1}+16\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)}}. \end{equation} where the constant $C_1$ is the following $$ C_1=\frac{e}{2(e-1)}\frac{E_0(u_0,\rho_0)}{2(1-2\Omega A)} +{\frac{e}{e-1}}\Big(\frac{E_0(u_0,\rho_0)}{2(1-2\Omega A)}\Big)^{1/2} +{\frac{e}{2(e-1)}}. $$ Then the corresponding solution $(u,\rho)$ to system \eqref{3.1} will blow up in finite time in the following sense, there is a $T_1$ with \begin{equation} 00$ for all $([0,T]\times \mathbb{S},\mathbb{S})$. First we take $\sigma=1,\mu=0$ then \begin{equation}\label{4.11} \begin{split} K_{xt}+uK_{xx} &= -\frac{1}{2}(K_x-\Omega \partial_xG\ast \rho^2)^2 +\frac{1-2\Omega A-2\Omega u}{2}\rho^2+\Omega u G\ast \rho^2\\ &\quad +A\partial^2_x G\ast u+u^2-G\ast(u^2+\frac{1}{2}u^2_x +\frac{1-2\Omega A}{2}\rho^2). \end{split} \end{equation} From now on, we abuse of notation by denoting \begin{gather*} u(t)=u(t,q(t,x_0), \quad u_x(t)=u_x(t,q(t,x_0)), \\ K(t)=K(t,q(t,x_0)),\quad K_x(t)=K_x(t,q(t,x_0)). \end{gather*} We further denote by $'$ the material derivative $\partial_t+u \partial_x$ along the characteristics $q(t,x_0)$. Let \[ 2G_{-}=G+G_{x}=\frac{e^x}{2\sinh(\frac{1}{2})}, \quad 2G_{+}=G-G_{x}=-\frac{e^{-x}}{2\sinh(\frac{1}{2})} \] for $-\frac{1}{2}0, \end{equation} which implies \begin{equation} [(K-\frac{A}{2})+K_x]'(0)<0,\quad [(K-\frac{A}{2})-K_x]'(0)>0. \end{equation} Hence at least for a short time $t$, $K(t)+K_x(t)$ is non-increasing and $K(t)-K_x(t)$ is non-decreasing, then we have \begin{equation}\label{4.25} \begin{gathered} \begin{aligned} &(K(0)-\frac{A}{2})+K_x(0) \\ &<-\Big(\frac{2A^2}{e-1} +16\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)}\Big)^{1/2}-2\Omega C_1, \end{aligned}\\ \begin{aligned} &(K(0)-\frac{A}{2})-K_x(0) \\ &>\Big(\frac{2A^2}{e-1} +16\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)}\Big)^{1/2}+2\Omega C_1. \end{aligned} \end{gathered} \end{equation} The short time monotonicity indicates that the above bounds continue to hold, at least for a short time. Therefore, we have \begin{equation}\label{4.26} \begin{gathered} (u(t)-\frac{A}{2})+u_x(t)<-\Big(\frac{2A^2}{e-1} +16\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E(0)}\Big)^{1/2}-2\Omega C_1,\\ (u(t)-\frac{A}{2})-u_x(t)>\Big(\frac{2A^2}{e-1} +16\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E(0)}\Big)^{1/2}+2\Omega C_1. \end{gathered} \end{equation} Then, plugging these to \eqref{4.22}, shows that the monotonicity of $(K-\frac{A}{2})+K_x$ persists and thus the bounds of the form in \eqref{4.25} continue to hold true, pushing the monotonicity even further in time. Hence, we always have $K(t)-\frac{A}{2}-K_x(t)<0$ is non-increasing, and $K(t)-\frac{A}{2}-K_x(t)>0$ is non-decreasing, which allows us to define the function \begin{equation} h(t)=\sqrt{K^2_x(t)-[K(t)-\frac{A}{2}]^2}>0. \end{equation} Computing the derivative of $h$ leads to \begin{equation} \begin{split} h'(t) & =\frac{-(K-\frac{A}{2}+K_x)'(K-\frac{A}{2}-K_x)-(K-\frac{A}{2}+K_x)(K-\frac{A}{2}-K_x)'}{2\sqrt{K^2_x(t)-[K(t)-\frac{A}{2}]^2}}\\ & \geq \Big(\frac{1}{2}[u^2_x-(u-\frac{A}{2})^2]-\frac{A^2}{4(e-1)} -2\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_(u_0,\rho_0)}\Big)\\ & \quad\times\frac{(K-\frac{A}{2}-K_x)-(K-\frac{A}{2}+K_x)}{2\sqrt{K^2_x(t) -[K(t)-\frac{A}{2}]^2}}\\ & \geq \frac{1}{2}[u^2_x-(u-\frac{A}{2})^2]-\frac{A^2}{4(e-1)} -2\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)}>0, \end{split} \end{equation} where we have used that \begin{equation} \frac{(K-\frac{A}{2}-K_x)-(K-\frac{A}{2}+K_x)}{2}\geq h. \end{equation} From \eqref{4.22} and \eqref{4.26}, it follows that \begin{equation} \begin{gathered} 0<(K-\frac{A}{2})-K_x\leq 2[(u-\frac{A}{2})-u_x],\\ 0<-(K-\frac{A}{2})-K_x\leq -(u-\frac{A}{2})-u_x. \end{gathered} \end{equation} Therefore, $$ K^2_x-(K-\frac{A}{2})^2\leq 2[u^2_x-(u-\frac{A}{2})^2], $$ and hence \begin{equation} h'\geq \frac{1}{4}h^2-\frac{A^2}{4(e-1)} -2\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)}. \end{equation} Evaluating \eqref{4.22} at initial time we have \begin{equation} \begin{split} (K(0)-\frac{A}{2})+K_x(0) &\leq (u_0(x_0)-\frac{A}{2})+u_{0,x}(x_0)+2\Omega C_1\\ & < -\Big(\frac{A^2}{2(e-1)}+4\Omega C_1\sqrt{\frac{e+1}{2(e-1)} E_0(u_0,\rho_0)}\Big)^{1/2},\\ (K(0)-\frac{A}{2})-K_x(0) &\geq (u_0(x_0)-\frac{A}{2})-u_{0,x}(x_0)\\ & >\Big(\frac{A^2}{2(e-1)}+4\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)} \Big)^{1/2}. \end{split} \end{equation} Therefore $$ h^2(0)>\frac{2A^2}{e-1}+16\Omega C_1\sqrt{\frac{e+1}{2(e-1)}E_0(u_0,\rho_0)}. $$ We see that $h$ is increasing and in fact we have $$ h'\geq \frac{1}{8}h^2. $$ This is enough to show that $h$ blows up in finite time. Indeed, we can solve the above equation to get \begin{equation} h(t)\geq \frac{8h(0)}{8-th(0)}. \end{equation} Therefore $$ h(t)\to +\infty,\quad t\to \frac{8}{h(0)}. $$ On the other hand, since $$ h(t)\leq -K_x=u_x-\Omega p_x\ast\rho^2 $$ and from \eqref{4.19} we know that \begin{equation} h(t)\leq -u_x+2C_1. \end{equation} Therefore $-u_x$ must blow up at time $T^\ast$ with \begin{equation} T^\ast\leq \frac{8}{h(0)} \end{equation} This completes the proof. \end{proof} Now we give the other blow-up condition about the r2CH system. The following method is an important method to obtain the blow-up time about the shallow wave system. \begin{theorem} \label{thm4.5} Suppose that $1-2\Omega A>0$. Let$(u,\rho)$ be the solution of r2CH\eqref{3.1} with initial data $(u_0,\rho_0-1)\in H^s\times H^{s-1}$ with $s>\frac{3}{2}$ and $T$ be the maximal time of existence. Assume there exists a $x_0$ such that \begin{gather} \rho_0(x_0)=0, \\ \label{4.36} u_{0,x}(x_0)< -C_3-2\Omega C_2. \end{gather} Then the corresponding solution $(u,\rho)$ to system \eqref{3.1} blows up in finite time in the following sense, there is a $T_2$ with \begin{equation} \begin{aligned} 0&0$. Let $(u_0,\rho_0-1)\in H^{s}\times H^{s-1}$ with $s>3/2$, and $T>0$ be the maximal time of existence of the solution $(u,\rho)$ to system r2CH with initial data $(u_0,\rho_0)$. Then for $x\in \{\Lambda:={x\in \mathbb{S}:\rho_0(x)=0 }\}$, we have that $u_x(t,q(t,x))$ is bounded from above for $t\in [0,T)$. \end{proposition} \begin{proof} We need only to prove this proposition for $s>3$. Given $x\in \mathbb{S}$. From Theorem \ref{thm4.5}, we have \begin{equation}\label{4.63} f\leq C^2_{E(0)}, \end{equation} where the $C_{E(0)}$ denotes a constant that depends only on $E(0)$. Given any $x\in \mathbb{S}$, let us define $$ P(t)=M_1(t)-\|u_{0,x}\|_{L^\infty}-2\Omega C_1-2C_{E(0)}, $$ where $C_1$ is defined by \eqref{4.19}. Observing $P(t)$ is a $C^1$-differentiable function in [0,t) and satisfies \begin{align*} P(0)&=M_1(0)-\|u_{0,x}\|_{L^\infty}-2\Omega C_1-2C_{E(0)}\\ &\leq u_{0,x}(x)+\Omega G_x\ast\rho^2(0,x)-\|u_{0,x}\|_L^\infty-2\Omega C_1 \leq 0 \end{align*} where we used the estimate \eqref{4.44}. We now claim that \begin{equation} \label{4.58} P(t)\leq 0,\quad \forall t\in [0,T). \end{equation} On the contrary assume that there is $t_0\in [0,T)$ such that $P(t_0)>0$. Let $t_1=\{\max {t