\documentclass[reqno]{amsart} \usepackage[notref,notcite]{showkeys} %\usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 148, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/148\hfil Two-component Camassa-Holm system] {Initial data problems for the two-component Camassa-Holm system} \author[X. Wang \hfil EJDE-2014/148\hfilneg] {Xiaohuan Wang} % in alphabetical order \address{Xiaohuan Wang \newline College of Mathematics and Information Science, Henan University, Kaifeng 475001, China} \email{xhwangmaths@163.com, Phone 86+15226038672} \thanks{Submitted June 4, 2014. Published June 24, 2014.} \thanks{This work was supported by PRC Grant NSFC 11301146.} \subjclass[2000]{35G25, 35B30, 35L05} \keywords{Non-uniform dependence; Camassa-Holm system; well-posedness; \hfill\break\indent energy estimates; initial value problem} \begin{abstract} This article concerns the study of some properties of the two-component Camassa-Holm system. By constructing two sequences of solutions of the two-component Camassa-Holm system, we prove that the solution map of the Cauchy problem of the two-component Camassa-Holm system is not uniformly continuous in $H^s(\mathbb{R})$, $s>5/2$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Many authors have studied shallow water equations, of which a typical example is Camassa-Holm (CH) equation. This equation has been extended to a two-component integrable system (CH2) by combining its integrability property with compressibility, or free-surface elevation dynamics in its shallow-water interpretation \cite{CI,HNT}: \begin{equation} \label{1.1} \begin{gathered} m_t+um_x+2m u_x+\sigma\rho\rho_x=0,\quad t>0,\; x\in\mathbb{R},\\ \rho_t+(\rho u)_x=0, \quad t>0,\; x\in\mathbb{R}, \end{gathered} \end{equation} where $m=u-u_{xx}$ and $\sigma=\pm1$. We remark that $\sigma=1$ is the hydrodynamically relevant choice, see the discussion in \cite{CI}. Local well-posedness of \eqref{1.1} with $\sigma=1$ was obtained by \cite{CI,ELY}. The precise blow-up scenarios and blow-up phenomena of strong solution for \eqref{1.1} was established by \cite{CI,ELY,FQ,GY,GZ,GL1}. Guan-Yin obtained the existence of global weak solution to \eqref{1.1}. Just recently, Gui and Liu \cite{GL2} studied \eqref{1.1} with $\sigma=1$ in Besov space and they obtained the local well-posedness. In this paper, we consider the Cauchy problem of \eqref{1.1} and study the some properties of it. If $\rho\equiv0$, then \eqref{1.1} becomes the well-known Camassa-Holm equation \cite{CH}. In the past decade, the Camassa-Holm equation has attracted much attention because of its integrability and the existence of multi-peakon solutions, see \cite{BC1}-\cite{CE4} and \cite{YLF}-\cite{ZC} for the details. The Cauchy problem and initial boundary value problem of the Camassa-Holm equation have been studied extensively \cite{CE1,EY1}. It has been shown that the Camassa-Holm equation is locally well-posedness \cite{CE1} for initial data $u_0\in H^s(\mathbb{R})$, $s>3/2$. Moreover, it has global strong solutions \cite{CE1} and finite time blow-up solutions \cite{CE1,CE2,CE3}. On the other hand, it has global weak solution in $H^1(\mathbb{R})$ \cite{BC1,BC2,CH,CE4}. The advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solutions and models wave breaking (i.e. the solution remains bounded while its slope becomes unbounded in finite time \cite{CH,CE1,CE2,TSD}). Here peaked solutions are actually peaked traveling waves, similar to the waves of greatest height encountered in classical hydrodynamics, see the discussion in the papers \cite{C2006,CE2007,T1996}. Moreover, there is a rich geometric structure underlying the Camassa-Holm equation, see the discussion in the papers \cite{K2007,K2008}. Recently, some properties of solutions to the Camassa-Holm equation have been studied by many authors. Himonas et al. \cite{HMPZ} studied the persistence properties and unique continuation of solutions of the Camassa-Holm equation. They showed that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time, see \cite{ZC,FL} for the similar properties of solutions to other shallow water equation. Just recently, Himonas-Kenig \cite{HK} and Himonas et al. \cite{HKM} considered the non-uniform dependence on initial data for the Camassa-Holm equation on the line and on the circle, respectively. Lv et al. \cite{LPW} obtained the non-uniform dependence on initial data for $\mu$-$b$ equation. Lv-Wang \cite{LWjmp} considered the \eqref{1.1} with $\rho=\gamma-\gamma_{xx}$ and obtained the non-uniform dependence on initial data. Wang \cite{Wxh} obtained the non-uniform dependence on initial data of periodic Camassa-Holm system. Tang-Wang \cite{TW} obtained the H\"{o}lder continuous of Camassa-Holm system. In this paper, we consider the non-uniform dependence on initial data for \eqref{1.1}. We remark that there is significant difference between \eqref{1.1} and \eqref{1.1} with $\rho=\gamma-\gamma_{xx}$. It is easy to see that when $\rho=\gamma-\gamma_{xx}$, there are some similar properties between the two equations in \eqref{1.1}. Thus the proof of non-uniform dependence on initial data to \eqref{1.1} with $\rho=\gamma-\gamma_{xx}$ is similar to the single equation, for example, Camassa-Holm equation. But in \eqref{1.1}, $\rho$ and $u$ have different properties, see Theorem \ref{t2.1}. This needs construct different asymptotic solution, see section 3. Besides, the results in this paper are different from those in \cite{LPW} because of the difference of the two operators $1-\partial_{xx}$ and $\mu-\partial_{xx}$. This article is organized as follows. In section 2, we recall the well-posedness result of Constantin-Ivanov \cite{CI} and Escher et al. \cite{ELY} and use it to prove the basic energy estimate from which we derive a lower bound for the lifespan of the solution as well as an estimate of the $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})$ norm of the solution $(u(t,x),\rho(t,x))$ in terms of $H^s(\mathbb{R})\times H^{s-1}(\mathbb{R})$ norm of the initial data $(u_0,\rho_0)$. In section 3, we construct approximate solutions, compute the error and estimate the $H^1$-norm of this error. In section 4, we estimate the difference between approximate and actual solutions, where the exact solution is a solution to \eqref{1.1} with initial data given by the approximate solutions evaluated at time zero. The non-uniform dependence on initial data for \eqref{1.1} is established in section 5 by constructing two sequences of solutions to \eqref{1.1} in a bounded subset of the Sobolev space $H^s(\mathbb{R})$, whose distance at the initial time is converging to zero while at any later time it is bounded below by a positive constant. \textbf{Notation.} In the following, we denote by $\ast$ the spatial convolution. Given a Banach space $Z$, we denote its norm by $\|\cdot\|_Z$. Since all space of functions are over $\mathbb{R}$, for simplicity, we drop $\mathbb{R}$ in our notations of function spaces if there is no ambiguity. Let $[A,B]=AB-BA$ denotes the commutator of linear operator $A$ and $B$. Set $\|z\|_{H^s\times H^{s-1}}^2=\|u\|_{H^s}^2+\|\rho\|_{H^{s-1}}^2$, where $z=(u,\rho)$. \section{Local well-posedness} In this section we first recall the known results of Constantin-Ivanov \cite{CI} and Escher et al. \cite{ELY} and give a new estimate of the solution to \eqref{1.1}. Let $\Lambda=(1-\partial^2_x)^{1/2}$. Then the operator $\Lambda^{-2}$ acting on $L^2(\mathbb{R})$ can be expressed by its associated Green's function $G(x)=\frac{1}{2}e^{-|x|}$ as \[ \Lambda^{-2}f(x)=(G\ast f)(x)=\frac{1}{2}\int_{-\infty}^\infty e^{-|x-y|} f(y){\rm d}y, \quad f\in L^2(\mathbb{R}). \] Hence \eqref{1.1} is equivalent to the system \begin{equation} \label{2.1} \begin{gathered} u_t+uu_x=-\partial_x\Lambda^{-2}\big(u^2+\frac{1}{2}u^2_x+\frac{1}{2}\rho^2 \big),\quad t>0,\; x\in\mathbb{R},\\ \rho_t+u\rho_x=-u_x\rho,\quad t>0, \; x\in\mathbb{R}, \end{gathered} \end{equation} with initial data \begin{equation} u(0,x)=u_0(x),\quad \rho(0,x)=\rho_0(x), \quad x\in\mathbb{R}. \label{2.1a} \end{equation} The following result is given by Constantin-Ivanov \cite{CI} and Escher et al. \cite{ELY}. \begin{theorem}\label{t2.1} Given $z_0=(u_0,\rho_0)\in H^s\times H^{s-1}$, $s\geq2$. Then there exists a maximal existence time $T=T(\|z_0\|_{H^s\times H^{s-1}})>0$ and a unique solution $z=(u,\rho)$ to \eqref{2.1} with \eqref{2.1a} such that \[ z=z(\cdot,z_0)\in C([0,T);H^s\times H^{s-1})\cap C^1([0,T);H^{s-1}\times H^{s-2}). \] Moreover, the solution depends continuously on the initial data, i.e. the mapping \[ z_0\mapsto z(\cdot,z_0): H^s\times H^{s-1}\to C([0,T);H^s\times H^{s-1})\cap C^1([0,T);H^{s-1}\times H^{s-2}) \] is continuous. \end{theorem} Next, we will give an explicit estimate for the maximal existence time $T$. Also, we will show that at any time $t$ in the time interval $[0,T_0]$ the $H^s$-norm of the solution $z(t,x)$ is dominated by the $H^s$-norm of the initial data $z_0(x)$. In order to do this, we need the following lemmas. \begin{lemma}[\cite{IM}] \label{l2.3} If $r>0$, then \[ \|[\Lambda^r, f]g\|_2\leq C(\|f_x\|_\infty\|\Lambda^{r-1}g\|_2+\|\Lambda^rf\|_2\|g\|_\infty), \] where $C$ is a positive constant depending only on $r$. \end{lemma} \begin{theorem}\label{t2.2} Let $s>5/2$. If $z=(u,\rho)$ is a solution of {\rm\eqref{2.1}} with initial data $z_0$ described in Theorem \ref{t2.1}, then the maximal existence time $T$ satisfies \begin{equation} T\geq T_0:=\frac{1}{2C_s\|z_0\|_{H^s\times H^{s-1}}}, \label{2.2} \end{equation} where $C_s$ is a constant depending only on $s$. Also, we have \begin{equation} \|z(t)\|_{H^s\times H^{s-1}}\leq 2\|z_0\|_{H^s\times H^{s-1}}, \quad 0\leq t\leq T_0. \label{2.3} \end{equation} \end{theorem} \begin{proof} The derivation of the lower bound for the maximal existence time \eqref{2.2} and the solution size estimate \eqref{2.3} is based on the following differential inequality for the solution $z$: \begin{equation} \frac{1}{2}\frac{d}{dt}\|z(t)\|_{H^s\times H^{s-1}}^2\leq C_s\|z(t)\|_{H^s\times H^{s-1}}^3, \quad 0\leq t0,\; x\in\mathbb{R}, \\ (J_\varepsilon\rho)_t+J_\varepsilon(u\rho_x)=- J_\varepsilon(u_x\rho),\quad t>0,\; x\in\mathbb{R}, \end{gathered} \label{2.5} \end{equation} where for each $\varepsilon\in(0,1]$ the operator $J_\varepsilon$ is the Friedrichs mollifier defined by \[ J_\varepsilon f(x)=J_\varepsilon(f)(x)=j_\varepsilon\ast f. \] Here $j_\varepsilon(x)=\frac{1}{\varepsilon}j(\frac{x}{\varepsilon})$, and $j(x)$ is a $C^\infty$ function supported in the interval $[-1,1]$ such that $j(x)\geq0,\,\int_\mathbb{R}j(x){\rm d}x=1$. Applying the operator $\Lambda^s$ and $\Lambda^{s-1}$ to the first and second equations of \eqref{2.5} respectively, then multiplying the resulting equations by $\Lambda^sJ_\varepsilon u$ and $\Lambda^{s-1}J_\varepsilon\rho$, respectively, and integrating them with respect to $x\in\mathbb{R}$, we obtain \begin{gather} \begin{aligned} \frac{1}{2}\frac{d}{dt}\|J_\varepsilon u\|^2_{H^s} &= -\int_\mathbb{R}\Lambda^sJ_\varepsilon (u u_x)\Lambda^sJ_\varepsilon u{\rm d}x \\ &\quad -\int_\mathbb{R}\partial_x\Lambda^{s-2}\partial_x\Lambda^{-2} \Big(J_\varepsilon u^2+\frac{1}{2}J_\varepsilon u_x^2 +\frac{1}{2}J_\varepsilon\rho^2\Big)\Lambda^sJ_\varepsilon u{\rm d}x, \end{aligned} \label{2.6} \\ \frac{1}{2}\frac{d}{dt}\|J_\varepsilon\rho\|^2_{H^{s-1}} =-\int_\mathbb{R}\Lambda^{s-1}J_\varepsilon (u\rho_x)\Lambda^{s-1}J_\varepsilon \rho{\rm d}x-\int_\mathbb{R}\Lambda^{s-1}J_\varepsilon(u_x\rho) \Lambda^{s-1}J_\varepsilon \rho{\rm d}x. \label{2.7} \end{gather} Similar to \cite{Wxh}, we can estimate the right-hand sides of \eqref{2.6} and \eqref{2.7}. We obtain \begin{gather*} \frac{1}{2}\frac{d}{dt}\|J_\varepsilon u\|^2_{H^s} \leq C_s(\|u\|_\infty+\|\rho\|_\infty+\| u_x\|_\infty +\|\rho_x\|_\infty)(\|u\|^2_{H^s} +\|\rho\|^2_{H^{s-1}}), \\ \frac{1}{2}\frac{d}{dt}\|J_\varepsilon \rho\|^2_{H^{s-1}} \leq C_s(\|u\|_\infty+\|\rho\|_\infty+\| u_x\|_\infty +\|\rho_x\|_\infty)(\|u\|^2_{H^s} +\|\rho\|^2_{H^{s-1}}). \end{gather*} Consequently, \begin{align*} &\frac{1}{2}\frac{d}{dt}\left(\|J_\varepsilon u\|^2_{H^s}+\|J_\varepsilon \rho\|^2_{H^{s-1}}\right)\\ & \leq C_s(\|u\|_\infty+\|\rho\|_\infty+\| u_x\|_\infty +\|\rho_x\|_\infty)(\|u\|^2_{H^s} +\|\rho\|^2_{H^{s-1}}). \end{align*} Then, letting $\varepsilon$ aproach $0$, we have \[ \frac{1}{2}\frac{d}{dt}\left(\|u\|^2_{H^s}+\| \rho\|^2_{H^{s-1}}\right)\leq C_s(\|u\|_\infty+\|\rho\|_\infty+\| u_x\|_\infty +\|\rho_x\|_\infty)(\|u\|^2_{H^s} +\|\rho\|^2_{H^{s-1}}), \] or \begin{equation} \frac{1}{2}\frac{d}{dt}\|z(t)\|^2_{H^s\times H^{s-1}}\leq C_s(\|u(t)\|_{C^1}+\|\rho\|_{C^1})\|z(t)\|^2_{H^s\times H^{s-1}}. \label{2.19} \end{equation} Since $s>5/2$, using Sobolev's inequality we have that \[ \|u(t)\|_{C^1}\leq C_s\|u(t)\|_{H^s},\quad \|\rho(t)\|_{C^1}\leq C_s\|\rho(t)\|_{H^{s-1}}. \] From \eqref{2.19} we obtain the desired inequality \eqref{2.4}. This completes the proof of Theorem \ref{t2.2}. \end{proof} Recall that $\|z(t)\|_{H^s\times H^{s-1}}^2=\|u(t)\|_{H^s}^2+\|\rho(t)\|_{H^{s-1}}^2$, where $z(t)=(u(t),\rho(t))$. It follows from Theorem \ref{t2.2} that \begin{equation} \|u(t)\|_{H^s},\,\|\rho(t)\|_{H^{s-1}}\leq\|z(t)\|_{H^s\times H^{s-1}}\leq2\|z_0\|_{H^s\times H^{s-1}},\quad 0\leq t\leq T_0. \label{2.20} \end{equation} \begin{remark}\label{r2.1} \rm Comparing Theorem \ref{t2.2} with that in \cite{LWjmp}, we will see that there exists a significant different between \eqref{1.1} and \eqref{1.1} with $\rho=\gamma-\gamma_{xx}$. In the other words, we require $s>5/2$ because of the Sobolev embedding Theorem. But in paper \cite{LWjmp}, since $u$ and $\gamma$ have the same property, we assume that $s>3/2$. \end{remark} \section{Approximate solutions} In this section we first construct a two-parameter family of approximate solutions by using a similar method to \cite{HK}, then compute the error and last estimate the $H^1$-norm of the error. Following \cite{HK}, our approximate solutions $u^{\omega,\lambda}=u^{\omega,\lambda}(t,x)$ and $\rho^{\omega,\lambda}=\rho^{\omega,\lambda}(t,x)$ to \eqref{2.1} will consist of a low frequency and a high frequency part, i.e. \[ u^{\omega,\lambda}=u_l+u^h,\quad \rho^{\omega,\lambda}=\rho_l+\rho^h, \] where $\omega$ is in a bounded set of $\mathbb{R}$ and $\lambda>0$. The high frequency part is given by \begin{equation} \label{3.1} \begin{gathered} u^h=u^{h,\omega,\lambda}(t,x)=\lambda^{-\frac{1}{2}\delta-s} \phi\big(\frac{x}{\lambda^\delta}\big) \cos(\lambda x-\omega t),\\ \rho^h=\rho^{h,\omega,\lambda}(t,x)=\lambda^{-\frac{1}{2}\delta-s+1} \psi\big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x-\omega t), \end{gathered} \end{equation} where $\phi$ and $\psi$ are $C^\infty$ cut-off functions such that \[ \phi(x)=\begin{cases} 1 & \text{if } |x|<1,\\ 0 & \text{if } |x|\geq2, \end{cases}\quad \psi(x)=\begin{cases} 1 &\text{if } |x|<1,\\ 0 &\text{if } |x|\geq2. \end{cases} \] The low frequency part $(u_l,\rho_l)=(u_{l,\omega,\lambda}(t,x),\rho_{l,\omega,\lambda}(t,x))$ is the solution to \eqref{2.1} with initial data \begin{equation} u_l(0,x)=\omega\lambda^{-1}\tilde\phi\big(\frac{x}{\lambda^\delta}\big), \quad \rho_l(0,x)=\omega\lambda^{-1}\tilde\psi\big(\frac{x}{\lambda^\delta}\big), \quad x\in\mathbb{R}, \label{3.2} \end{equation} where $\tilde\phi$ and $\tilde\psi$ are $C_0^\infty(\mathbb{R})$ functions such that \[ \tilde\phi(x)=1 \quad\text{if } x\in \operatorname{supp}\phi\cup \operatorname{supp}\psi. \] We first study the properties of $(u_l,\rho_l)$ and $(u^h,\rho^h)$. The high frequency part $(u^h,\rho^h)$ defined by \eqref{3.1} satisfies \[ \|u^h(t)\|_{H^s}\approx O(1), \quad \|\rho^h(t)\|_{H^{s-1}}\approx O(1) \quad\text{for }\lambda\gg1 \] because of the following result. \begin{lemma}[\cite{HK}] \label{l3.1} Let $\psi\in \mathcal{S}(\mathbb{R})$, $1<\delta<2$ and $\alpha\in\mathbb{R}$. Then for any $s\geq0$ we have that \begin{equation} \lim _{\lambda\to\infty}\lambda^{-\frac{1}{2}\delta-s}\|\psi \big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x-\alpha)\|_{H^s}=\frac{1}{\sqrt{2}}\|\psi\|_2. \label{3.3} \end{equation} Relation \eqref{3.3} is also true if $\cos$ is replaced by $\sin$. \end{lemma} For the low frequency part $(u_l,\rho_l)$, we have the following result. \begin{lemma}\label{l3.2} Let $\omega$ belong to a bounded set of $\mathbb{R}$, $1<\delta<2$ and $\lambda\gg1$. Then the initial-value problem \eqref{2.1}-\eqref{3.2} has a unique solution $(u_l,\rho_l)\in C([0,T);H^s)\times C([0,T);H^{s-1})$, for all $s>5/2$, satisfying the estimates \[ \|u_l(t)\|_{H^s}\leq C_s\lambda^{-1+\frac{1}{2}\delta}, \quad \|\rho_l(t)\|_{H^{s-1}}\leq C_{s-1}\lambda^{-1+\frac{1}{2}\delta}. \] \end{lemma} \begin{proof} The existence and uniqueness of local a solution can be derived from Theorem \ref{t2.1} for $s>5/2$. It follows from \cite[Lemma 5]{HK} that \[ \|\psi\big(\frac{x}{\lambda^\delta}\big)\|_{H^s} \leq\lambda^{\delta/2}\|\psi\|_{H^s}, \] where $s\geq0$ and $\psi\in \mathcal {S}(\mathbb{R})$. Using the above inequality, we have that the initial data $(u_l(0,x),\rho_l(0,x))$ satisfies the estimate \[ \|u_l(0)\|_{H^s}\leq|\omega|\lambda^{-1+\frac{1}{2}\delta}\|\tilde\phi\|_{H^s},\quad \|\rho_l(0)\|_{H^{s-1}}\leq|\omega|\lambda^{-1+\frac{1}{2}\delta} \|\tilde\psi\|_{H^{s-1}}, \] which decay if $\delta<2$ and $\omega$ is in a bounded set of $\mathbb{R}$. Recall that $\|z_l(t)\|_{H^s\times H^{s-1}}^2=\|u_l(t)\|_{H^s}^2+\|\rho_l(t)\|_{H^{s-1}}^2$, we obtain \[ \|z_l(0)\|_{H^s\times H^{s-1}}=(\|u_l(0)\|_{H^s}^2+\|\rho_l(0)\|_{H^{s-1}}^2)^{1/2}\leq |\omega|\lambda^{-1+\frac{1}{2}\delta} (\|\tilde\phi\|^2_{H^s}+\|\tilde\psi\|^2_{H^{s-1}})^{1/2}. \] It follows from \eqref{3.2} that $z_l(0)\in H^s\times H^{s-1}$ for all $s>5/2$. If $s>5/2$, then from estimate \eqref{2.2} of Theorem \ref{t2.2}, we have \begin{gather*} \|u_l(t)\|_{H^s}\leq C_s\|u_l(0)\|_{H^{s}}\leq C_s\lambda^{-1+\frac{1}{2}\delta},\\ \|\rho_l(t)\|_{H^{s-1}}\leq C_s\|\rho_l(0)\|_{H^{s-1}}\leq C_{s-1}\lambda^{-1+\frac{1}{2}\delta}. \end{gather*} The proof is complete. \end{proof} Now we compute the error. Substituting the approximate solution $(u^{\omega,\lambda},\rho^{\omega,\lambda})$ into the first and second equation of \eqref{2.1}, we obtain the error \begin{gather*} \begin{aligned} E&=u^h_t+u_lu^h_x+u^h u_{lx}+u^hu^h_x +\partial_x\Lambda^{-2}\Big((u^h)^2+k_1u_lu^h\\ &\quad +\frac{1}{2}(u^h_x)^2 +u_{lx}u^h_x+\frac{1}{2}(\rho^h)^2+\rho_l\rho^h\Big), \end{aligned}\\ F=\rho^h_t+u_l\rho^h_x+u^h\rho_{lx} +u^h\rho^h_x +\rho^h u_{lx}+\rho_lu^h_x+\rho^hu^h_x, \end{gather*} where we have used that $(u_l,\rho_l)$ solves \eqref{3.2}. Direct calculation shows that \begin{gather*} u^h_t(t,x)=\omega\lambda^{-\frac{1}{2}\delta-s} \phi\big(\frac{x}{\lambda^\delta}\big)\sin(\lambda x-\omega t),\\ \rho^h_t(t,x)=\omega \lambda^{-\frac{1}{2}\delta-s+1} \psi\big(\frac{x}{\lambda^\delta}\big)\sin(\lambda x-\omega t). \end{gather*} Since $\tilde\phi=1$ if $x\in\operatorname{supp}\phi\cup\operatorname{supp}\psi$, we can write $ u^h_t$ and $\rho^h_t$ in the form \begin{equation} \begin{aligned} u^h_t(t,x) &= \omega\tilde\phi\big(\frac{x}{\lambda^\delta}\big) \lambda^{-\frac{1}{2}\delta-s}\phi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t) \\ &= \lambda u_l(0,x)\lambda^{-\frac{1}{2}\delta-s}\phi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t), \\ \rho^h_t(t,x)&= \omega \tilde\phi\big(\frac{x}{\lambda^\delta}\big) \lambda^{-\frac{1}{2}\delta-s+1}\psi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t) \\ &= \lambda u_l(0,x)\lambda^{-\frac{1}{2}\delta-s+1}\psi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t). \end{aligned}\label{3.4} \end{equation} Computing the spacial derivatives of $u^h$ and $\rho^h$, we have \begin{equation} \begin{aligned} u^h_x(t,x) &= -\lambda\lambda^{-\frac{1}{2}\delta-s}\phi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t)+\lambda^{-\frac{3}{2}\delta-s}\phi'\big(\frac{x}{\lambda^\delta}\big) \cos(\lambda x-\omega t), \\ \rho^h_x(t,x) &= -\lambda\lambda^{-\frac{1}{2}\delta-s+1}\psi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t)+\lambda^{-\frac{3}{2}\delta-s+1}\psi'\big(\frac{x}{\lambda^\delta}\big) \cos(\lambda x-\omega t). \end{aligned}\label{3.5} \end{equation} Combining \eqref{3.4} with \eqref{3.5}, we obtain \begin{align*} u^h_t(t,x)+u_lu^h_x(t,x) &= \lambda [u_l(0,x)-u_l(t,x)]\lambda^{-\frac{1}{2}\delta-s}\phi\left(\frac{x}{\lambda^\delta} \right)\sin(\lambda x-\omega t)\\ &\quad +u_l(t,x)\lambda^{-\frac{3}{2}\delta-s}\phi' \big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x-\omega t), \\ \rho^h_t(t,x)+u_l\rho^h_x(t,x) &= \lambda [u_l(0,x)-u_l(t,x)]\lambda^{-\frac{1}{2}\delta-s+1} \psi\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t)\\ &\quad +u_l(t,x)\lambda^{-\frac{3}{2}\delta-s+1}\psi' \left (\frac{x}{\lambda^\delta}\right)\cos(\lambda x-\omega t). \end{align*} Therefore, we can rewrite the error $E$ and $F$ as \[ E=E_1+E_2+\dots+E_8,\quad F=F_1+F_2+\dots+F_6, \] where \begin{gather*} E_1 = -\lambda[u_l(0,x)-u_l(t,x)]\lambda^{-\frac{1}{2}\delta-s} \phi\big(\frac{x}{\lambda^\delta}\big)\sin(\lambda x+\omega t),\\ E_2 = u_l(t,x)\lambda^{-\frac{3}{2}\delta-s}\phi'\big(\frac{x}{\lambda^\delta}\big) \cos(\lambda x+\omega t),\\ E_3 = -u^h u_{lx},\quad E_4 =-u^hu^h_x,\\ E_5 = -\partial_x \Lambda^{-2}\Big(\frac{k_1}{2}(u^h)^2 +\frac{k_2}{2}(\rho^h)^2\Big),\quad E_6 =-\partial_x \Lambda^{-2} \left(k_1u_lu^h+k_2\rho_l\rho^h\right), \\ E_7 = -(3-k_1)\partial_x \Lambda^{-2} (u_{lx}u^h_x) ,\ \ E_8 =\frac{3-k_1}{2}\partial_x \Lambda^{-2}\left((u^h_x)^2\right),\\ F_1 = -k_3\lambda[u_l(0,x)-u_l(t,x)]\lambda^{-\frac{1}{2}\delta-s+1} \psi\big(\frac{x}{\lambda^\delta}\big)\sin(\lambda x+\omega t),\\ F_2 = k_3u_l(t,x)\lambda^{-\frac{3}{2}\delta-s+1} \psi'\big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x+\omega t),\\ F_3 = -k_3u^h\rho_{lx},\ \ F_4=-k_3u^h\rho^h_x,\\ F_5 = -k_3\left(\rho^h u_{lx}+\rho_lu^h_x +\rho^hu^h_x\right). \end{gather*} Now we are ready to estimate the $H^1$-norm of each error $E_i$ and the $L^2$-norm of each error $F_j$ $(i=1,\dots,8,j=1,\dots,6)$. Let $C$ be a generic positive constant. For any positive quantities $P$ and $Q$, we write $P\lesssim Q$ $(P\gtrsim Q)$ means that $P\leq CQ$ $(P\geq CQ)$ in the following. \textbf{Estimates of $\|E_1\|_{H^1}$ and $\|F_1\|_{L^2}$.} Note that \[ \|fg\|_{H^1}\leq\sqrt{2}\|f\|_{C^1}\|g\|_{H^1}, \quad \forall f\in C^1,\; g\in H^1, \] and $\|\phi\big(\frac{x}{\lambda^\delta}\big)\sin(\lambda x-\omega t)\|_{C^1}=\lambda\|\phi\|_\infty$, we have \begin{equation} \begin{aligned} \|E_1\|_{H^1} &= \lambda^{1-\frac{1}{2}\delta-s}\|\phi\left(\frac{x} {\lambda^\delta}\right)\sin(\lambda x-\omega t) [u_l(0,x)-u_l(t,x)]\|_{H^1} \\ &\lesssim \lambda^{1-\frac{1}{2}\delta-s}\|\phi\left(\frac{x}{\lambda^\delta} \right)\sin(\lambda x-\omega t)\|_{C^1}\|u_l(0,x)-u_l(t,x)\|_{H^1} \\ &\lesssim \lambda^{2-\frac{1}{2}\delta-s}\|u_l(0,x)-u_l(t,x)\|_{H^1}. \end{aligned} \label{3.6} \end{equation} To estimate the $H^1$-norm of the difference $u_l(0,x)-u_l(t,x)$, we apply the fundamental theorem of calculus in time variable to obtain \[ \|u_l(0,x)-u_l(t,x)\|_{H^1}=\int_0^t\| u_{lt}(\tau)\|_{H^1}{\rm d}\tau. \] It follows from the first equation of \eqref{3.2} that \begin{equation} \begin{aligned} \| u_{lt}(t)\|_{H^1} &\leq \|u_l u_{lx}\|_{H^1}+ \|\partial_x\Lambda^{-2}\big(u_l^2+\frac{1}{2}u^2_{lx}+\frac{1}{2}\rho_l^2 \big)\|_{H^1} \\ &\leq \|u_l\|_{H^1}\|u_l\|_{H^2}+\|u_l^2+\frac{1}{2}u^2_{lx}+\frac{1}{2}\rho_l^2\|_2 \\ &\lesssim \|u_l\|_{H^2}^2+\|u_l\|_\infty\|u_l\|_2+ \| u_{lx}\|_\infty\|u_l\|_{H^1}+ \|\rho_l\|_\infty\|\rho_l\|_2 \\ &\lesssim \|u_l\|_{H^2}^2+\|u_l\|_{H^1}^2+\|\rho_l\|_{H^2}^2 \\ &\lesssim \|u_l\|_{H^3}^2+\|\rho_l\|_{H^3}^2 \\ &\lesssim \lambda^{-2+\delta},\quad \lambda\gg 1, \end{aligned} \label{3.7} \end{equation} where we have used Lemma \ref{l3.2} and the Sobolev embedding Theorem $H^s\hookrightarrow L^\infty$ for $s>3/2$. Combining \eqref{3.6} and \eqref{3.7}, we obtain \[ \|E_1\|_{H^1}\lesssim\lambda^{-s+\frac{1}{2}\delta}, \quad \lambda\gg1. \] Similarly, \[ \|F_1\|_{L^2}\lesssim\lambda^{-s+\frac{1}{2}\delta}, \quad \lambda\gg1. \] \textbf{Estimates of $\|E_i\|_{H^1}$ and $\|F_j\|_{H^1}$, $i=2,\dots,8,j=2,3$.} In \cite{LWjmp}, the authors obtained the following estimates \begin{gather*} \|E_2\|_{H^1} \lesssim \lambda^{-s-\delta},\\ \|E_3\|_{H^1},\,\|E_6\|_{H^1},\,\|E_7\|_{H^1} \lesssim \lambda^{-\frac{1}{2}\delta-s+1}\lambda^{-1+\frac{1}{2}\delta}, \\ \|E_4\|_{H^1},\,\|E_5\|_{H^1},\,\|E_8\|_{H^1} \lesssim \lambda^{-\frac{1}{2} \delta-2s+2} \end{gather*} Similar to the estimate of $\|E_2\|_{H^1}$, we have \[ \|F_2\|_{L^2}\lesssim\lambda^{-s-\delta},\quad \lambda\gg 1. \] Direct calculation shows that \[ \|F_3\|_{L^2}=\|u^h\rho_{lx}\|_{L^2}\lesssim\|u^h\|_{L^\infty}\|\rho_{lx}\|_{H^1} \lesssim\lambda^{-\frac{1}{2}\delta-s}\lambda^{-1+\frac{1}{2}\delta},\quad \lambda\gg 1. \] \textbf{Estimates of $\|F_4\|_{L^2}$.} It follows from \eqref{3.1} that \begin{equation} \|u^h_x(t)\|_\infty\lesssim\lambda^{-\frac{1}{2} \delta-s+1},\quad \|\rho^h_x(t)\|_\infty\lesssim\lambda^{-\frac{1}{2} \delta-s+2},\quad \lambda\gg 1. \label{3.8} \end{equation} By using Lemma \ref{l3.1}, we have \begin{equation} \begin{aligned} \|u^h(t)\|_{H^k} &= \lambda^{-\frac{1}{2}\delta-s}\|\phi \big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x-\omega t) \|_{H^k} \\ &= \lambda^{-s+k}\lambda^{-\frac{1}{2}\delta-k}\|\phi \big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x-\omega t)\|_{H^k} \\ &\lesssim \lambda^{-s+k},\quad \lambda\gg 1. \end{aligned} \label{3.9} \end{equation} The above inequality also holds for $\rho^h(t)$. Combining \eqref{3.8} and \eqref{3.9}, we obtain that, for $\lambda\gg1$, \[ \|F_4\|_{L^2}=\|u^h\rho^h_x\|_{L^2} \lesssim \|u^h\|_\infty\|\rho^h\|_{H^1}\lesssim\lambda^{-\frac{1}{2} \delta-s} \lambda^{-s+2}\lesssim\lambda^{-\frac{1}{2} \delta-2s+2}. \] \textbf{Estimate of $\|F_5\|_{L^2}$.} It follows from \eqref{3.8} and \eqref{3.9} that \begin{align*} \|F_5\|_{L^2} &= \|\left(\rho^h u_{lx}+\rho_lu^h_x +\rho^hu^h_x\right)\|_{L^2}\\ &\leq \left(\|\rho^h\|_\infty\| u_{lx}\|_{H^1}+\|u^h_x\|_\infty\|\rho_l\|_{H^1}+ \|\rho^h\|_\infty\|u^h_x\|_{L^2}\right)\\ &\lesssim \|\rho^h\|_\infty\|u_l\|_{H^2}+\|u^h_x\|_\infty\|\rho_l\|_{H^2}+ \|\rho^h\|_\infty\|u^h_x\|_{H^1}\\ &\lesssim \lambda^{-\frac{1}{2} \delta-s}\lambda^{-1+\frac{1}{2}\delta}+ \lambda^{-\frac{1}{2}\delta-s+1}\lambda^{-1+\frac{1}{2}\delta} +\lambda^{-\frac{1}{2} \delta-s+1}\lambda^{-s+1}, \end{align*} which gives $\|F_5\|_{H^1}\lesssim\lambda^{-\frac{1}{2}\delta-2s+2}$, $\lambda\gg 1$. Collecting all error estimates together, we have the following theorem. \begin{theorem}\label{t3.1} Let $s>5/2$ and $1<\delta<2$. When $\omega$ is in a bounded set of $\mathbb{R}$ and $\lambda\gg1$, we have that \begin{equation} \|E\|_{H^1}\lesssim\lambda^{-r_s},\quad \|F\|_{L^2}\lesssim\lambda^{-r_s},\quad \text{for } \lambda\gg1,\; 00$. \end{theorem} \section{Difference between approximate and actual solutions} In this section, we estimate the difference between the approximate and actual solutions. Let $(u_{\omega,\lambda}(t,x),\rho_{\omega,\lambda}(t,x))$ be the solution to \eqref{2.1} with initial data the value of the approximate solution $(u^{\omega,\lambda}(t,x),\rho^{\omega,\lambda}(t,x))$ at time zero, that is, $(u_{\omega,\lambda}(t,x),\rho_{\omega,\lambda}(t,x))$ satisfies \begin{equation} \begin{gathered} \partial_tu_{\omega,\lambda}-u_{\omega,\lambda}\partial_xu_{\omega,\lambda} -\partial_x\Lambda^{-2} (u_{\omega,\lambda}^2+\frac{1}{2}(\partial_xu_{\omega,\lambda})^2 +\frac{1}{2}\rho_{\omega,\lambda}^2 )=0,\quad t>0,\; x\in\mathbb{R},\\ \partial_t\rho_{\omega,\lambda}-u_{\omega,\lambda}\partial_x\rho_{\omega,\lambda}- (\partial_xu_{\omega,\lambda}\rho_{\omega,\lambda} +\partial_x\rho_{\omega,\lambda}u_{\omega,\lambda})=0,\quad t>0,\; x\in\mathbb{R},\\ u_{\omega,\lambda}(0,x)=u^{\omega,\lambda}(0,x) =\omega\lambda^{-1}\tilde\phi\big(\frac{x}{\lambda^\delta}\big)+ \lambda^{-\frac{1}{2}\delta-s}\phi\big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x), \quad x\in\mathbb{R},\\ \rho_{\omega,\lambda}(0,x)=\rho^{\omega,\lambda}(0,x)=\omega\lambda^{-1} \tilde\psi\big(\frac{x}{\lambda^\delta}\big)+ \lambda^{-\frac{1}{2}\delta-s+1}\psi \big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x), \quad x\in\mathbb{R}. \end{gathered} \label{4.1} \end{equation} Note that $(u_{\omega,\lambda}(0,x),\rho_{\omega,\lambda}(0,x))\in H^s\times H^{s-1}$, $s\geq2$, it follows from Lemma \ref{l3.2} and \eqref{3.9} that \begin{gather*} \|u_{\omega,\lambda}(0,x)\|_{H^s} \leq \|u_l(0)\|_{H^s}+\|u^h(0)\|_{H^s}\lesssim\lambda^{-1+\frac{1}{2}\delta}+1, \quad \lambda\gg1,\\ \|\rho_{\omega,\lambda}(0,x)\|_{H^{s-1}} \leq \|\rho_l(0)\|_{H^{s-1}}+\|\rho^h(0)\|_{H^{s-1}} \lesssim\lambda^{-1+\frac{1}{2}\delta}+1,\quad \lambda\gg1. \end{gather*} Therefore, if $s>5/2$, by using Theorem \ref{t2.1} and \ref{t2.2}, we have that for any $\omega$ in a bounded set and $\lambda\gg1$, problem \eqref{4.1} has a unique solution $z_{\omega,\lambda}\in C([0,T];H^s)\times C([0,T];H^{s-1})$ with \begin{equation} T\gtrsim \frac{1}{\|z_{\omega,\lambda}(0)\|_{H^s\times H^{s-1}}}\gtrsim\frac{1}{1+\lambda^{-1+\frac{1}{2}\delta}}\gtrsim1. \label{a.1} \end{equation} To estimate the difference between the approximate and actual solutions, we let \[ v=u^{\omega,\lambda}-u_{\omega,\lambda},\quad \sigma=\rho^{\omega,\lambda}-\rho_{\omega,\lambda}. \] Then $(v,\sigma)$ satisfies \begin{equation} \begin{gathered} \begin{aligned} &v_t-v v_x+u^{\omega,\lambda} v_x+v u^{\omega,\lambda}_x- \partial_x\Lambda^{-2}\Big[v^2 +\frac{1}{2}v_x^2\\ &+\frac{1}{2}\sigma^2 -2u^{\omega,\lambda}v- u^{\omega,\lambda}_x v_x-\rho^{\omega,\lambda}\sigma \Big]=\tilde E,\quad t>0,\; x\in\mathbb{R}, \end{aligned}\\ \sigma_t-v\sigma_x+u^{\omega,\lambda}\sigma_x+v\rho^{\omega,\lambda}_x- \big(\sigma v_x- u^{\omega,\lambda}\sigma- \rho^{\omega,\lambda} v_x\big)=\tilde F, \quad t>0,\; x\in\mathbb{R},\\ v(0,x)=\sigma(0,x)=0,\quad x\in\mathbb{R}, \end{gathered} \label{4.2} \end{equation} where \begin{gather*} \tilde E= u_t^{\omega,\lambda}+u^{\omega,\lambda} u^{\omega,\lambda}_x+\partial_x\Lambda^{-2} \Big((u^{\omega,\lambda})^2+ \frac{1}{2}( u^{\omega,\lambda}_x)^2+\frac{1}{2}(\rho^{\omega,\lambda})^2 \Big),\\ \tilde F= \rho_t^{\omega,\lambda}+u^{\omega,\lambda}\rho^{\omega,\lambda}_x+ +\rho^{\omega,\lambda} u^{\omega,\lambda}_x, \end{gather*} Similar to the prove of Theorem \ref{t3.1}, $\tilde E$ and $\tilde F$ satisfy the $H^1$-norm estimation \eqref{3.10}. Now we prove that the $H^1$-norm of difference decays. \begin{theorem}\label{t4.1} Let $1<\delta<2$ and $s>5/2$, then \[ \|v(t)\|_{H^1}\lesssim\lambda^{-r_s},\quad \|\sigma(t)\|_{L^2}\lesssim\lambda^{-r_s}, \quad 0\leq t\leq T, \; \lambda\gg 1, \] where $r_s=s-\frac{1}{2}\delta>0$. \end{theorem} \begin{proof} Note that \begin{gather} \frac{1}{2}\frac{d}{dt}\|v(t)\|^2_{H^1} = \int_\mathbb{R}(vv_t+ v_x v_{xt}){\rm d}x,\label{4.3}\\ \frac{1}{2}\frac{d}{dt}\|\sigma(t)\|^2_{L^2} = \int_\mathbb{R} \sigma\sigma_t{\rm d}x. \label{4.4} \end{gather} Applying the operator $1-\partial_x^2=\Lambda^2$ to both sides of the first equations of \eqref{4.2}, we have \begin{gather} \begin{aligned} v_t&= \Lambda^2\tilde E-\Lambda^2(u^{\omega,\lambda} v_x- v u^{\omega,\lambda}_x)-(2u^{\omega,\lambda}v+ u^{\omega,\lambda}_x v_x +\rho^{\omega,\lambda}\sigma)_x \\ &\quad +\frac{1}{2} (\sigma^2)_x+3v v_x- 2v_x v_{xx}-v v_{xxx}+v_{xxt}, \end{aligned} \label{4.5}\\ \sigma_t= \tilde F-(u^{\omega,\lambda}\sigma_x + v\rho^{\omega,\lambda}_x)-(u_x^{\omega,\lambda}\sigma+ \rho^{\omega,\lambda} v_x)+(v\sigma)_x. \label{4.6} \end{gather} Substituting \eqref{4.5} and \eqref{4.6} into \eqref{4.3} and \eqref{4.4}, respectively, we obtain \begin{gather} \begin{aligned} \frac{1}{2}\frac{d}{dt}\|v(t)\|^2_{H^1} &= \int_\mathbb{R}v\Lambda^2\tilde E{\rm d}x -\int_\mathbb{R}v\Lambda^2(u^{\omega,\lambda} v_x+ v u^{\omega,\lambda}_x){\rm d}x \\ &\quad -\int_\mathbb{R}v(2u^{\omega,\lambda}v+ u^{\omega,\lambda}_x v_x+\rho^{\omega,\lambda}\sigma)_x{\rm d}x +\frac{1}{2}\int_\mathbb{R}v(\sigma^2 )_x{\rm d}x \\ &\quad +\int_\mathbb{R}(v(3v v_x-2 v_x v_{xx}-v v_{xxx} +v_{xxt})+ v_x v_{xt}) {\rm d}x, \end{aligned}\label{4.7}\\ \begin{aligned} \frac{1}{2}\frac{d}{dt}\|\sigma(t)\|^2_{L^2} &= \int_\mathbb{R}\sigma\tilde F{\rm d}x-\int_\mathbb{R} \sigma(u^{\omega,\lambda}\sigma_x+ v\rho^{\omega,\lambda}_x){\rm d}x \\ &\quad -\int_\mathbb{R}\sigma(\rho^{\omega,\lambda} v_x +\sigma u^{\omega,\lambda}_x){\rm d}x+ \int_\mathbb{R}\sigma(v\sigma)_x {\rm d}x. \end{aligned} \label{4.8} \end{gather} A direct calculation yields \begin{align*} &\int_\mathbb{R}(v(3v v_x-2 v_x v_{xx}-v v_{xxx} +v_{xxt})+ v_x v_{xt}) {\rm d}x\\ &=\int_\mathbb{R}[(v^3)_x-(v^2 v_{xx})_x+(v v_{xt})_x] {\rm d}x=0. \end{align*} Substituting the above equalities in \eqref{4.7}, and adding the resulting equations, we obtain \begin{align*} &\frac{1}{2}\frac{d}{dt}\left(\|v(t)\|^2_{H^1}+\|\sigma(t)\|^2_{L^2}\right)\\ &= \int_\mathbb{R}v\Lambda^2 \tilde E{\rm d}x+\int_\mathbb{R}\sigma\tilde F{\rm d}x-\int_\mathbb{R} v\Lambda^2(u^{\omega,\lambda} v_x+ v u^{\omega,\lambda}_x){\rm d}x\\ &\quad -\int_\mathbb{R} \sigma(u^{\omega,\lambda}\sigma_x+ v\rho^{\omega,\lambda}_x){\rm d}x-\int_\mathbb{R}v (2u^{\omega,\lambda}v+ u^{\omega,\lambda}_x v_x +\rho^{\omega,\lambda}\sigma)_x{\rm d}x \\ &\quad -\int_\mathbb{R} \sigma(\rho^{\omega,\lambda} v_x +\sigma u^{\omega,\lambda}_x) {\rm d}x+\int_\mathbb{R} \big[\frac{1}{2}v(\sigma^2)_x+ \sigma( v\sigma)_x\big]{\rm d}x\\ &:= I_1+I_2+\dots+I_7. \end{align*} We first look at the last term $I_7$. Integrating by parts gives \[ I_7=\int_\mathbb{R} \big[\frac{1}{2}v(\sigma^2)_x+ \sigma( v\sigma)_x\big]{\rm d}x=0. \] \textbf{Estimates of integrals $I_1$ and $I_2$.} Integrating by parts and applying the Cauchy-Schwarz inequality, we have \begin{gather*} \Big|\int_\mathbb{R}v\Lambda^2 \tilde E{\rm d}x\Big|=\Big|\int_\mathbb{R}(v\tilde E- v_x\tilde E_x){\rm d}x\Big|\leq\|\tilde E\|_{H^1}\|v(t)\|_{H^1},\\ \Big|\int_\mathbb{R}\sigma\tilde F{\rm d}x\Big|\leq\|\tilde F\|_{L^2}\|\sigma(t)\|_{L^2}. \end{gather*} \textbf{Estimates of integrals $I_3$-$I_6$.} Similar to that in \cite{LWjmp}, we obtain \begin{align*} \sum_{i=3}^6I_i &\lesssim (\|u^{\omega,\lambda}(t)\|_\infty +\| u^{\omega,\lambda}_x(t)\|_\infty+\| u^{\omega,\lambda}_{xx}(t)\|_\infty+ \|\rho^{\omega,\lambda}(t)\|_\infty)\\ &\quad \times (\|v(t)\|^2_{H^1}+\|\sigma(t)\|^2_{L^2}). \end{align*} Combining the estimations for $I_1$--$I_7$, we have \begin{equation} \begin{aligned} &\frac{1}{2}\frac{d}{dt}(\|v(t)\|^2_{H^1}+\|\sigma(t)\|^2_{L^2}) \\ &\lesssim (\|\tilde E\|_{H^1}+\|\tilde F\|_{H^1})(\|v(t)\|_{H^1} +\|\sigma(t)\|_{L^2}) \\ &\quad +(\|u^{\omega,\lambda}(t)\|_\infty+\| u^{\omega,\lambda}_x(t) \|_\infty+\|u^{\omega,\lambda}_{xx}(t)\|_\infty +\|\rho^{\omega,\lambda}(t)\|_\infty+\|\rho_x^{\omega,\lambda}(t)\|_\infty )\\ &\quad \times (\|v(t)\|^2_{H^1}+\|\sigma(t)\|^2_{H^1}). \end{aligned} \label{4.9} \end{equation} It follows from \eqref{3.1} that \begin{gather*} u^h_x= -\lambda^{-\frac 32\delta-s}\phi'\left(\frac x{\lambda^\delta}\right)\cos(\lambda x-\omega t)-\lambda^{-\frac\delta 2-s+1}\phi\left(\frac x{\lambda^\delta}\right)\sin(\lambda x-\omega t),\\ \begin{aligned} u^h_{xx}&= \lambda^{-\frac{5}{2}\delta-s}\phi'' \big(\frac{x}{\lambda^\delta}\big)\cos(\lambda x-\omega t) -2\lambda^{-\frac{3}{2}\delta-s+1}\phi'\big(\frac{x}{\lambda^\delta}\big) \sin(\lambda x-\omega t)\\ &\quad -2\lambda^{-\frac{1}{2}\delta-s+2}\phi\big(\frac{x}{\lambda^\delta}\big) \cos(\lambda x-\omega t). \end{aligned} \end{gather*} Hence \[ \|u^h(t)\|_\infty+\|u^h_x(t)\|_\infty+\| u^h_{xx}(t)\|_\infty \lesssim\lambda^{-(\frac{1}{2}\delta+s-2)}, \quad \lambda\gg1. \] By using Lemma \ref{l3.2}, we have \[ \|u_l(t)\|_\infty+\| u_{lx}(t)\|_\infty+\|u_{lxx}(t)\|_\infty \lesssim\lambda^{-(1-\frac{1}{2}\delta)}, \quad \lambda\gg1. \] Therefore, \begin{equation} \|u^{\omega,\lambda}(t)\|_\infty+\| u_x^{\omega,\lambda}(t)\|_\infty +\|u^{\omega,\lambda}_{xx}(t)\|_\infty \lesssim\lambda^{-\rho_s}, \quad \lambda\gg1, \label{4.10} \end{equation} where $\rho_s=\min\{\frac{1}{2}\delta+s-2,1-\frac{1}{2}\delta\}>0$ for any $s>1$ if $\delta$ is chosen appropriately in the interval $(1,2)$. Similarly, we can prove that \begin{equation} \|\rho^{\omega,\lambda}(t)\|_\infty \lesssim\lambda^{-s},\ \ \|\rho_x^{\omega,\lambda}(t)\|_\infty \lesssim\lambda^{-\rho_s} \ \ \lambda\gg1. \label{4.11} \end{equation} Let $\tilde z(t,x)=(v(t,x),\sigma(t,x))$ and $\|\tilde z(t)\|_{H^1\times L^2}^2=\|v(t)\|_{H^1}^2+\|\sigma(t)\|_{L^2}^2$, then by \eqref{4.9}-\eqref{4.11}, we obtain that \begin{align*} \frac{1}{2}\frac{d}{dt}\|\tilde z(t)\|_{H^1\times L^2}^2 &\lesssim (\|\tilde E\|_{H^1}+\|\tilde F\|_{L^2})\|\tilde z(t)\|_{H^1\times L^2}+\lambda^{-\rho_s}\|\tilde z(t)\|_{H^1}^2\\ &\lesssim \lambda^{-r_s}\|\tilde z(t)\|_{H^1\times L^2}+\lambda^{-\rho_s} \|\tilde z(t)\|_{H^1\times L^2}^2, \quad \lambda\gg1, \end{align*} where we have used Theorem \ref{t3.1}. Consequently, \begin{equation} \frac{d}{dt}\|\tilde z(t)\|_{H^1\times L^2}\lesssim\lambda^{-\rho_s} \|\tilde z(t)\|_{H^1\times L^2}+\lambda^{-r_s}, \quad \lambda\gg1. \label{4.12} \end{equation} Since $\|\tilde z(0)\|_{H^1\times L^2}=(\|v(0)\|_{H^1}^2+\|\sigma(0)\|_{L^2}^2)^{1/2}=0$ and for $s>1$, we can choose $\delta\in(1,2)$ such that $\rho_s\geq0$, by \eqref{4.12} and Gronwall's inequality, we obtain \[ \|\tilde z(t)\|_{H^1\times L^2}\lesssim\lambda^{-r_s},\quad 0\leq t\leq T, \quad \lambda\gg1. \] Note that \[ \|v(t)\|_{H^1},\,\,\|\sigma(t)\|_{L^2}\leq\|\tilde z(t)\|_{H^1\times L^2}, \] we see that \[ \|v(t)\|_{H^1},\; \|\sigma(t)\|_{L^2}\lesssim\lambda^{-r_s},\quad 0\leq t\leq T, \; \lambda\gg1. \] This completes the proof. \end{proof} \section{Non-uniform dependence} In this section, we prove non-uniform dependence for \eqref{2.1} by taking advantage of the information provided by Theorem \ref{t2.1}-\ref{t2.2}, Theorem \ref{t3.1} and Theorem \ref{t4.1}. Our main result is the following. \begin{theorem}\label{t5.1} If $s>5/2$, then the data-to-solution $z(0)\to z(t)$ for \eqref{2.1} is not uniformly continuous from any bounded subset of $H^s\times H^{s-1}$ into $C([-T,T];H^s)\times C([-T,T];H^{s-1})$, where $z(0)=(u_0(x),\rho_0(x))$ and $z(t)=(u(t,x),\rho(t,x))$. More precisely, there exist two sequences of solutions $(u_\lambda(t),\rho_\lambda(t))$ and $(\tilde u_\lambda(t),\tilde\rho_\lambda(t))$ to the differential equations of {\rm\eqref{2.1}} in $C([-T,T];H^s)\times C([-T,T];H^{s-1})$ such that \begin{gather} \|u_\lambda(t)\|_{H^s}+\|\tilde u_\lambda(t)\|_{H^s} +\|\rho_\lambda(t)\|_{H^{s-1}}+\|\tilde\rho_\lambda(t)\|_{H^{s-1}}\lesssim1, \\ \lim _{\lambda\to\infty}\|u_\lambda(0)-\tilde u_\lambda(0)\|_{H^s}=\lim _{\lambda\to\infty}\|\rho_\lambda(0) -\tilde\rho_\lambda(0)\|_{H^{s-1}}=0, \label{5.2} \\ \liminf_{\lambda\to\infty}\left(\|u_\lambda(t)-\tilde u_\lambda(t)\|_{H^s}+\|\rho_\lambda(t)-\tilde \rho_\lambda(t)\|_{H^{s-1}}\right)\gtrsim\sin t,\quad |t|