# A wave breaking criterion for a modified periodic two-component Camassa-Holm system

## Abstract

In this paper, a wave-breaking criterion of strong solutions is acquired in the Soblev space $$H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})$$ with $$s>\frac{3}{2}$$ by employing the localization analysis in the transport equation theory, which is different from that of the two-component Camassa-Holm system.

## Introduction

The classical two-component Camassa-Holm system takes the form

$$\left \{ \textstyle\begin{array}{@{}l} (1-\partial^{2}_{x})u_{t}+u(1-\partial^{2}_{x})u_{x}+2u_{x}(1-\partial ^{2}_{x})u+\rho\rho_{x}=0,\quad t>0, x\in\mathbb{R},\\ \rho_{t}+(u\rho)_{x}=0, \quad t>0, x\in\mathbb{R}, \end{array}\displaystyle \right .$$
(1)

where the variable $$u(t,x)$$ represents the horizontal velocity of the fluid, and $$\rho(t,x)$$ is related to the free surface elevation from equilibrium with the boundary assumptions $$u\rightarrow0$$ and $$\rho\rightarrow1$$ as $$| x|\rightarrow\infty$$. System (1) was found originally in , but it was firstly derived rigorously by Constantin and Ivanov . The system has bi-Hamiltonian structure and is completely integrable. Since the birth of the system, a large number of literature was devoted to the study of the two-component Camassa-Holm system. Some mathematical and physical properties of the system have been obtained. Chen et al.  established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of the AKNS hierarchy. Escher et al.  used Kato’s theory to establish local well-posedness for the two-component system and presented some precise blow-up scenarios for strong solutions of the system. In , Constantin and Ivanov described sufficient conditions for wave-breaking and global solution to the system. Dynamics in the periodic case for system (1) were considered in . It is worth mentioning that the wave-breaking criteria of strong solutions is determined in the lowest Soblev space $$H^{s}$$ with $$s>\frac{3}{2}$$ by applying the localization analysis in the transport equation theory . The other results related to the system can be found in .

Inspired by the works mentioned, in this article, we consider a modified periodic two-component Camassa-Holm system on the circle $$\mathbb{S}$$ with $$\mathbb{S=R/Z}$$ (the circle of unit lengh):

$$\left \{ \textstyle\begin{array}{@{}l} m_{t}+um_{x}+2u_{x}m+\rho\rho_{x}=0,\quad t>0, x\in\mathbb{R},\\ \rho_{t}+(u\rho)_{x}=0, \quad t>0, x\in\mathbb{R},\\ m(0,x)=m_{0}(x), \quad x\in\mathbb{R},\\ \rho(0,x)=\rho_{0}(x), \quad x\in\mathbb{R},\\ m(t,x+1)=m(t,x), \quad t>0, x\in\mathbb{R},\\ \rho(t,x+1)=\rho(t,x),\quad t>0, x\in\mathbb{R}, \end{array}\displaystyle \right .$$
(2)

where $$m=(1-\partial^{2}_{x})^{2}u$$, and $$\mathbb{R}$$ is the real number set. In fact, system (2) is a two-component generalization of the equation (if $$\rho=0$$ in system (2))

\begin{aligned} m_{t}+um_{x}+2u_{x}m=0, \qquad m=\bigl(1- \partial^{2}_{x}\bigr)^{2}u. \end{aligned}
(3)

Equation (3) was first derived as the Euler-Poincaré differential equation on the Bott-Virasoro group with respect to the $$H^{2}$$ metric , and it is known as a modified Camassa-Holm equation and also viewed as a geodesic equation on some diffeomorphism group . It is shown in  that the well-posedness and dynamics of Eq. (3) on the unit circle $$\mathbb{S}$$ are significantly different from that of the Camassa-Holm equation. For example, Eq. (3) does not conform with blow-up solution in finite time.

As we know, differently from the Camassa-Holm equation, Eq. (3) has not blow-up solution. The motivation of the present paper is to find out whether or not system (2) has some similar dynamics as the classical two-component Camassa-Holm equation and Eq. (3) mathematically, for example, wave-breaking and global solution. One of the difficulties is the acquisition of the a priori estimates of $$\| u_{xx}\|_{L^{\infty}}$$ and $$\| u_{xxx}\|_{L^{\infty}}$$. This difficulty has been overcome by Lemmas 3.4 and 3.5. We mainly use the ideas of  to derive a wave-breaking criterion (see Theorem 1) of strong solutions for system (2) in the low Sobolev spaces $$H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})$$ with $$s>\frac{3}{2}$$, where a new conservation law is necessary. We need to point out that in the Sobolev spaces $$H^{s}(\mathbb{R})\times H^{s-1}(\mathbb{R})$$ with $$s>\frac{3}{2}$$, the wave-breaking of the solution for system (1) only depends on the slope of the component u of the solution . However, since the slope of the component u of the solution is bounded by the Sobolev imbedding theorem $$H^{1}\hookrightarrow L^{\infty}$$, the wave-breaking of the solution for system (2) is determined only by the slope of the component ρ of solution definitely in the low Sobolev spaces $$H^{s}(\mathbb {S})\times H^{s-1}(\mathbb{S})$$ with $$s>\frac{3}{2}$$ (see Theorem 1). This implies that there exists some difference between system (2) and the two-component Camassa-Holm equation. Moreover, this is quite different from Eq. (3) because Eq. (3) does not admit a blow-up solution in infinite time.

## The main results

We denote by the convolution. Note that if $$g(x):=1+2\sum^{\infty}_{n=1}\frac{1}{1+2n^{2}+n^{4}}\cos(nx)$$, then $$(1-\partial^{2}_{x})^{-2}f=g\ast f$$ for all $$f\in L^{2}(\mathbb{R})$$, and $$g\ast m=u$$. We let C denote all of different positive constants that depend on initial data. To investigate dynamics of system (2), we can rewrite system (2) in the form

$$\left \{ \textstyle\begin{array}{@{}l} u_{t}+uu_{x}+\partial_{x}g\ast[u^{2}+u^{2}_{x}-\frac {7}{2}u^{2}_{xx}-3u_{x}u_{xxx}+\frac{1}{2}\rho^{2}],\quad t>0, x\in\mathbb{R},\\ \rho_{t}+(u\rho)_{x}=0,\quad t>0, x\in\mathbb{R},\\ u(0,x)=u_{0}(x), \quad x\in\mathbb{R},\\ \rho(0,x)=\rho_{0}(x), \quad x\in\mathbb{R},\\ u(t,x+1)=u(t,x),\quad t>0, x\in\mathbb{R},\\ \rho(t,x+1)=\rho(t,x),\quad t>0, x\in\mathbb{R}. \end{array}\displaystyle \right .$$
(4)

The main result of the present paper is as follows.

### Theorem 1

Let $$z_{0}=(u_{0},\rho_{0})\in H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})$$, $$s>\frac{3}{2}$$, and T be the maximal existence time of the solution $$z=(u,\rho)$$ to system (4). Assume that $$m_{0}\in L^{2}(\mathbb{S})$$ and $$T<\infty$$. Then

$$\int^{T}_{0}\bigl\| \partial_{x}\rho(\tau) \bigr\| _{L^{\infty}(\mathbb{S})}\,d\tau=\infty.$$

## Preliminaries

In order to prove Theorem 1, we first give some lemmas.

### Lemma 3.1

([6, 17]) (1-D Moser-type estimates)

The following estimates hold:

1. (i)

For $$s\geq0$$,

\begin{aligned} \| fg\|_{H^{s}}\leq C\bigl(\| f\| _{H^{s}}\| g\|_{L^{\infty}}+ \| f \|_{L^{\infty}}\| g\|_{H^{s}}\bigr). \end{aligned}
(5)
2. (ii)

For $$s>0$$,

\begin{aligned} \| f\partial_{x}g\|_{H^{s}}\leq C\bigl(\| f\| _{H^{s+1}}\| g \|_{L^{\infty}}+ \| f\|_{L^{\infty}}\| \partial_{x}g \|_{H^{s}}\bigr). \end{aligned}
(6)
3. (iii)

For $$s_{1}\leq\frac{1}{2}$$, $$s_{2}>\frac{1}{2}$$, and $$s_{1}+s_{2}>0$$,

\begin{aligned} \| f\partial_{x}g\|_{H^{s_{1}}}\leq C\| f\| _{H^{s_{1}}}\| g \|_{H^{s_{2}}}, \end{aligned}
(7)

where C is a constant independent of f and g.

### Lemma 3.2

([17, 18])

Suppose that $$s>-\frac{d}{2}$$. Let v be a vector field such that v belongs to $$L^{1}([0,T]; H^{s-1})$$ if $$s>1+\frac{d}{2}$$ or to $$L^{1}([0,T]; H^{\frac{d}{2}}\cap L^{\infty})$$ otherwise. Suppose also that $$f_{0}\in H^{s}$$, $$F\in L^{1}([0,T]; H^{s})$$, and that $$f\in L^{\infty}([0,T];H^{s})\cap C([0,T]; S')$$ solves the d-dimensional linear transport equation

$$\left \{ \textstyle\begin{array}{@{}l} f_{t}+v\cdot\nabla f=F, \\ f|_{t=0}=f_{0}. \end{array}\displaystyle \right .$$
(8)

Then $$f\in C([0,T]; H^{s})$$. More precisely, there exists a constant C depending only s, p, and d, and such that the following statements hold:

(1) If $$s\neq1+\frac{d}{2}$$, then

\begin{aligned} \| f\|_{H^{s}}\leq\| f_{0}\|_{H^{s}} +C \int_{0}^{t} \bigl\| F(\tau)\bigr\| _{H^{s}}\,d\tau+C \int_{0}^{t} V'(\tau)\bigl\| f(\tau) \bigr\| _{H^{s}}\,d\tau, \end{aligned}
(9)

or

\begin{aligned} \| f\|_{H^{s}}\leq e^{CV(t)}\biggl(\| f_{0}\| _{H^{s}}+ \int_{0}^{t} e^{-CV(t)}\bigl\| F(\tau) \bigr\| _{H^{s}}\,d\tau\biggr) \end{aligned}
(10)

with $$V(t)=\int_{0}^{t}\|\nabla v(\tau)\|_{H^{\frac{d}{2}}\cap L^{\infty}}\,d\tau$$ if $$s<1+\frac{d}{2}$$ and $$V(t)=\int_{0}^{t}\|\nabla v(\tau)\|_{H^{s-1}}\,d\tau$$ else.

(2) If $$f=v$$, then for all $$s>0$$, estimates (9) and (10) hold with $$V(t)=\int_{0}^{t}\|\partial_{x}u (\tau)\|_{L^{\infty}}\,d\tau$$.

### Lemma 3.3

()

Let $$0<\sigma<1$$. Suppose that $$f_{0}\in H^{\sigma}$$, $$g\in L^{1}([0,T]; H^{\sigma})$$, $$\nu, \partial_{x}\nu\in L^{1}([0,T]; L^{\infty})$$, and $$f\in L^{\infty}([0,T]; H^{\sigma})\cap C([0,T];S')$$ solves the 1-dimensional linear transport equation

$$\left \{ \textstyle\begin{array}{@{}l} f_{t}+\nu\partial_{x}f=g, \\ f|_{t=0}=f_{0}. \end{array}\displaystyle \right .$$
(11)

Then $$f\in C([0,T]; H^{\sigma})$$. More precisely, there exists a constant C depending only σ and such that the following statement holds:

\begin{aligned} \| f\|_{H^{\sigma}}\leq\| f_{0}\|_{H^{\sigma }} +C \int_{0}^{t} \bigl\| g(\tau)\bigr\| _{H^{\sigma}}\,d\tau+C \int_{0}^{t} V'(\tau )\bigl\| f(\tau) \bigr\| _{H^{\sigma}}\,d\tau, \end{aligned}
(12)

or

\begin{aligned} \| f\|_{H^{\sigma}}\leq e^{CV(t)}\biggl(\| f_{0}\| _{H^{\sigma}}+ \int_{0}^{t} C\bigl\| g(\tau)\bigr\| _{H^{\sigma}}\,d\tau \biggr) \end{aligned}
(13)

with $$V(t)=\int_{0}^{t}(\| \nu(\tau)\|_{L^{\infty}}+\| \partial_{x}\nu(\tau)\|_{L^{\infty}})\,d\tau$$.

### Lemma 3.4

For all $$x\in\mathbb{R}$$, the following statements hold:

\begin{aligned} (\mathrm{i})\quad \bigl\| \partial^{2}_{x}g\bigr\| _{L^{\infty}(\mathbb{R})} \leq1+\frac{\pi }{4} \end{aligned}
(14)

and

\begin{aligned} (\mathrm{ii})\quad \bigl\| \partial^{3}_{x}g\bigr\| _{L^{\infty}(\mathbb{R})} \leq2+\ln2+\pi. \end{aligned}
(15)

### Proof

Let $$g(x)$$ be the Green function for the operator $$(1-\partial^{2}_{x})^{2}$$. Then from

$$\bigl(1-2\partial^{2}_{x}+\partial^{4}_{x} \bigr)g(x)=\delta(x)=\sum^{\infty }_{n=-\infty}e^{inx}$$

we get

$$g(x)=\sum^{\infty}_{n=-\infty}\frac{1}{1+2n^{2}+n^{4}}e^{inx} =1+2\sum^{\infty}_{n=1}\frac{1}{1+2n^{2}+n^{4}}\cos(n x).$$

Hence,

$$g_{xx}(x) =-2\sum^{\infty}_{n=1} \frac{n^{2}}{1+2n^{2}+n^{4}}\cos(n x),$$

which results in

$$\bigl| g_{xx}(x)\bigr| \leq2\sum^{\infty}_{n=1} \frac{n^{2}}{1+2n^{2}+n^{4}}\bigl|\cos(n x)\bigr|\leq2\sum^{\infty}_{n=1} \frac{n^{2}}{1+2n^{2}+n^{4}}.$$

From Cauchy integral test we have

$$\sum^{\infty}_{n=2}\frac{n^{2}}{1+2n^{2}+n^{4}} \leq\lim _{n\rightarrow\infty} \int^{n}_{1}\frac {x^{2}}{(1+x^{2})^{2}}\,dx=\frac{1}{4}+ \frac{\pi}{8}.$$

It follows that

$$|g_{xx}|\leq2\sum^{\infty}_{n=1} \frac{n^{2}}{1+2n^{2}+n^{4}} \leq1+\frac{\pi}{4}.$$

Now, we prove (ii). From the Fourier series we have

$$h(x)=\sum^{\infty}_{n=1}\frac{\sin(n x)}{n}= \frac{\pi}{2}\biggl(1-\frac{x}{\pi}\biggr) \quad\mbox{for } 0< x< 2\pi,$$

from which we get

\begin{aligned} \bigl|2h(x)-g_{xxx}\bigr| =&\Biggl| 2\sum^{\infty}_{n=1} \biggl(\frac{1}{n}-\frac{n^{3}}{1+2n^{2}+n^{4}}\biggr)\sin(n x)\Biggr| \\ \leq& 2\sum^{\infty}_{n=1}\biggl( \frac{1}{n}-\frac{n^{3}}{1+2n^{2}+n^{4}}\biggr) \\ =&2\sum^{\infty}_{n=1}\biggl( \frac{1}{n(1+n^{2})}+\frac {n}{(1+n^{2})^{2}}\biggr). \end{aligned}

On the other hand,

\begin{aligned} \sum^{\infty}_{n=2}\biggl(\frac{1}{n(1+n^{2})}+ \frac{n}{(1+n^{2})^{2}}\biggr) \leq& \lim_{n\rightarrow\infty} \int^{n}_{1}\biggl(\frac{1}{x}- \frac {x}{1+x^{2}}+\frac{x}{(1+x^{2})^{2}}\biggr)\,dx\\ =&\frac{1}{2}\ln2+\frac {1}{4}. \end{aligned}

Hence, we have

$$\bigl\| \partial^{3}_{x}g\bigr\| _{L^{\infty}}\leq2+\ln2+\pi.$$

□

### Lemma 3.5

Let $$z_{0}=(u_{0},\rho_{0})\in H^{s}(\mathbb{S})\times H^{s-1}(\mathbb{S})$$ with $$s>\frac{3}{2}$$. Suppose that T is the maximal existence time of solution $$z=(u,\rho)$$ of system (4) with the initial data $$z_{0}$$. Then, for all $$t\in[0, T)$$, the following conservation law holds:

\begin{aligned} H= \int_{\mathbb{S}}\bigl(u^{2}+2u_{x}^{2}+u^{2}_{xx}+ \rho^{2}\bigr)\,dx= \int _{\mathbb{S}}\bigl(u^{2}_{0}+2u_{0x}^{2}+u^{2}_{0xx}+ \rho^{2}_{0}\bigr)\,dx. \end{aligned}
(16)

Moreover, assume that $$m_{0}\in L^{2}$$. Then

\begin{aligned} \| u_{xx}\|_{L^{\infty}(\mathbb{S})}\leq{}&\biggl(1+ \frac{\pi }{4}\biggr) \biggl(\| m_{0}\|^{2}_{L^{2}} +\| z_{0}\|_{H^{2}\times L^{2}} \int^{t}_{0} \|\rho_{x} \|_{L^{\infty}}\,d\tau\biggr)^{\frac{1}{2}} \\ &{} \times\exp\biggl[\frac{1}{2}\| z_{0}\| _{H^{2}\times L^{2}} \int^{t}_{0}\bigl(3+ \|\rho_{x} \|_{L^{\infty}}\bigr)\,d\tau\biggr] \\ \triangleq{}& L(t) \end{aligned}
(17)

and

\begin{aligned} \| u_{xxx}\|_{L^{\infty}(\mathbb{S})} \leq&(2+\ln2+\pi ) \biggl(\| m_{0}\|^{2}_{L^{2}} +\| z_{0} \|_{H^{2}\times L^{2}} \int^{t}_{0} \|\rho_{x} \|_{L^{\infty}}\,d\tau\biggr)^{\frac{1}{2}} \\ &{} \times\exp\biggl[\frac{1}{2}\| z_{0}\| _{H^{2}\times L^{2}} \int^{t}_{0}\bigl(3+ \|\rho_{x} \|_{L^{\infty}}\bigr)\,d\tau\biggr] \\ \triangleq& M(t). \end{aligned}
(18)

### Proof

Multiplying the first equation of system (2) by u and integrating by parts, we get

\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{\mathbb {S}}\bigl(u^{2}+2u^{2}_{x}+u^{2}_{xx} \bigr)\,dx+ \int_{\mathbb{S}}u\rho\rho _{x}\,dx=0. \end{aligned}
(19)

Multiplying the second equation of system (2) by ρ and integrating by parts, we get

\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{\mathbb{S}}\rho^{2}\,dx- \int_{\mathbb {S}}u\rho\rho_{x}\,dx=0, \end{aligned}
(20)

which, together with (19), yields

\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{\mathbb {S}}\bigl(u^{2}+2u^{2}_{x}+u^{2}_{xx}+ \rho^{2}\bigr)\,dx=0, \end{aligned}
(21)

which implies (16).

Next, we prove (17). Multiplying the first equation of system (2) by m and integrating by parts, we have

\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{\mathbb{S}}m^{2}\,dx=- \int_{\mathbb {S}}umm_{x}\,dx-2 \int_{\mathbb{S}}u_{x}m^{2}\,dx- \int_{S}m\rho\rho _{x}\,dx, \end{aligned}
(22)

which results in

\begin{aligned} \frac{d}{dt} \int_{\mathbb{S}}m^{2}\,dx=-3 \int_{\mathbb {S}}u_{x}m^{2}\,dx-2 \int_{\mathbb{S}}m\rho\rho_{x}\,dx. \end{aligned}
(23)

By the Hölder inequality we get from (23) that

\begin{aligned} \frac{d}{dt}\| m\|^{2}_{L^{2}(\mathbb{S})} \leq& 3\| u_{x} \|_{L^{\infty}} \| m\|^{2}_{L^{2}}+2\| m\|_{L^{2}}\| \rho\|_{L^{2}} \|\rho_{x}\|_{L^{\infty}} \\ \leq&3\| u_{x}\|_{L^{\infty}} \| m\|^{2}_{L^{2}}+ \bigl(1+\| m\| ^{2}_{L^{2}}\bigr)\|\rho\|_{L^{2}} \| \rho_{x}\|_{L^{\infty}} \\ \leq&\| m\|^{2}_{L^{2}}\bigl(3\| u_{x}\|_{L^{\infty}}+ \|\rho\|_{L^{2}} \|\rho_{x}\|_{L^{\infty}}\bigr)+\|\rho \|_{L^{2}} \|\rho_{x}\|_{L^{\infty}}. \end{aligned}

Applying Gronwall’s inequality, we obtain

$$\| m\|^{2}_{L^{2}}\leq\biggl(\| m_{0}\| ^{2}_{L^{2}}+ \int^{t}_{0}\|\rho\|_{L^{2}} \| \rho_{x}\|_{L^{\infty}}\,d\tau\biggr)\exp\biggl[ \int ^{t}_{0}\bigl(3\| u_{x} \|_{L^{\infty}}+\|\rho\|_{L^{2}} \|\rho_{x}\|_{L^{\infty}}\bigr)\,d\tau\biggr],$$

which, together with (16), yields

\begin{aligned} \| m\|^{2}_{L^{2}} \leq&\biggl(\| m_{0}\|^{2}_{L^{2}} +\| z_{0} \|_{H^{2}\times L^{2}} \int^{t}_{0} \|\rho_{x} \|_{L^{\infty}}\,d\tau\biggr) \\ &{}\times\exp\biggl[\| z_{0}\|_{H^{2}\times L^{2}} \int^{t}_{0}\bigl(3+ \|\rho_{x} \|_{L^{\infty}}\bigr)\,d\tau\biggr]. \end{aligned}
(24)

On the other hand, from Lemma 3.4 we deduce

\begin{aligned} \| u_{xx}\|_{L^{\infty}}=\| g_{xx}\ast m\|_{L^{\infty}} \leq\| g_{xx} \|_{L^{\infty}}\| m\|_{L^{1}}\leq\biggl(1+ \frac{\pi}{4}\biggr)\| m\|_{L^{2}}. \end{aligned}
(25)

It follows from (24) that

\begin{aligned} \| u_{xx}\|_{L^{\infty}} \leq&\biggl(1+ \frac{\pi }{4}\biggr) \biggl(\| m_{0}\|^{2}_{L^{2}} +\| z_{0}\|_{H^{2}\times L^{2}} \int^{t}_{0} \|\rho_{x} \|_{L^{\infty}}\,d\tau\biggr)^{\frac{1}{2}} \\ &{} \times\exp\biggl[\frac{1}{2}\| z_{0}\| _{H^{2}\times L^{2}} \int^{t}_{0}\bigl(3+ \|\rho_{x} \|_{L^{\infty}}\bigr)\,d\tau\biggr]. \end{aligned}
(26)

Similarly, we can obtain (18).

This completes the proof of Lemma 3.5. □

## Proof of main theorem

### Proof of Theorem 1

Using the maximal principle to the transport equation about ρ,

$$\rho_{t}+u\rho_{x}=-u_{x}\rho,$$

we have

$$\bigl\| \rho(t)\bigr\| _{L^{\infty}(\mathbb{S})}\leq \| \rho_{0}\|_{L^{\infty}(\mathbb{S})}+C \int^{t}_{0}\bigl\| \partial_{x}u(\tau) \bigr\| _{L^{\infty}}\bigl\| \rho(\tau)\bigr\| _{L^{\infty}}\,d\tau.$$

Applying Gronwall’s inequality yields

$$\bigl\| \rho(t)\bigr\| _{L^{\infty}(\mathbb{S})}\leq\|\rho _{0}\|_{L^{\infty}}\exp \biggl[C \int^{t}_{0}\bigl\| \partial_{x}u(\tau) \bigr\| _{L^{\infty}}\,d\tau\biggr].$$

Using the Sobolev embedding theorem $$H^{s}\hookrightarrow L^{\infty}$$ ($$s>\frac{1}{2}$$), we get from Lemma 3.5 that

$$\| u_{x}\|_{L^{\infty}(\mathbb{S})}\leq C\bigl(\| u_{0}\|_{H^{2}}+ \| \rho_{0}\|_{L^{2}}\bigr).$$

Therefore, we have

\begin{aligned} \bigl\| \rho(t)\bigr\| _{L^{\infty}(\mathbb{S})}\leq\|\rho _{0}\|_{L^{\infty}} e^{Ct(\| u_{0}\|_{H^{2}}+\| \rho_{0}\|_{L^{2}})}=\|\rho_{0}\|_{L^{\infty}} e^{CT\| z_{0}\|_{H^{2}\times L^{2}}}. \end{aligned}
(27)

Next, we split the remaining proof of Theorem 1 into five steps.

Step 1. For $$s\in(\frac{3}{2},2)$$, applying Lemma 3.3 to the second equation, we have

\begin{aligned} \|\rho\|_{H^{s-1}(\mathbb{S})} \leq&\| \rho_{0}\|_{H^{s-1}} +C \int^{t}_{0}\| u_{x}\rho \|_{H^{s-1}}\,d\tau \\ &{} +C \int^{t}_{0}\|\rho\|_{H^{s-1}} \bigl(\| u \|_{L^{\infty}}+\| \partial_{x}u\|_{L^{\infty}}\bigr)\,d\tau. \end{aligned}

From Lemma 3.1 (5) we get

\begin{aligned} \|\rho u_{x}\|_{H^{s-1}(\mathbb{S})}\leq C\bigl(\| u_{x} \|_{H^{s-1}} \|\rho\|_{L^{\infty}}+\|\rho\|_{H^{s-1}} \| u_{x}\|_{L^{\infty}}\bigr). \end{aligned}
(28)

From (28) we obtain

\begin{aligned} \|\rho\|_{H^{s-1}(\mathbb{S})} \leq&\| \rho_{0} \|_{H^{s-1}} +C \int^{t}_{0}\| u\|_{H^{s}}\|\rho\| _{L^{\infty}}\,d\tau \\ &{}+C \int^{t}_{0}\|\rho\|_{H^{s-1}} \bigl(\| u \|_{L^{\infty}}+\| \partial_{x}u\|_{L^{\infty}}\bigr)\,d\tau. \end{aligned}
(29)

On the other hand, using Lemma 3.2, we get from the first equation of system (4) that

\begin{aligned} \bigl\| u(t)\bigr\| _{H^{s}(\mathbb{S})} \leq& C \int^{t}_{0}\biggl\| \partial_{x}g\ast \biggl[u^{2}+u^{2}_{x}-\frac {7}{2}u^{2}_{xx}-3u_{x}u_{xxx} +\frac{1}{2}\rho^{2}\biggr]\biggr\| _{H^{s}}\,d\tau \\ &{} +\| u_{0}\|_{H^{s}}+C \int ^{t}_{0}\bigl\| u(t)\bigr\| _{H^{s}} \bigl\| \partial_{x}u(\tau)\bigr\| _{L^{\infty}}\,d\tau. \end{aligned}

From Lemma 3.4(b) of , we have

\begin{aligned} &\biggl\| \partial_{x}g\ast\biggl[u^{2}+u^{2}_{x}- \frac{7}{2}u^{2}_{xx}-3u_{x}u_{xxx} +\frac{1}{2}\rho^{2}\biggr]\biggr\| _{H^{s}} \\ &\quad \leq C\biggl\| u^{2}+u^{2}_{x}- \frac{7}{2}u^{2}_{xx}-3u_{x}u_{xxx} +\frac{1}{2}\rho^{2}\biggr\| _{H^{s-3}} \\ &\quad\leq C\bigl(\| u\|_{H^{s-3}}\| u\|_{L^{\infty}} +\| u_{x} \|_{H^{s-3}}\| u_{x}\|_{L^{\infty}} +\| u_{xx} \|_{H^{s-3}}\| u_{xx}\|_{L^{\infty}} \\ &\qquad{} +\| u_{xxx}\|_{H^{s-3}}\| u_{x} \|_{L^{\infty}} +\|\rho\|_{H^{s-3}}\|\rho\|_{L^{\infty }}\bigr). \end{aligned}

Hence, we get

\begin{aligned} \bigl\| u(t)\bigr\| _{H^{s}(\mathbb{S})} \leq&\| u_{0} \|_{H^{s}(\mathbb{S})}+ C \int^{t}_{0}\bigl\| \rho(\tau)\bigr\| _{H^{s-1}} \bigl\| \rho( \tau)\bigr\| _{L^{\infty}}\,d\tau \\ &{}+C \int^{t}_{0}\| u\| _{H^{s}}\bigl(\| u \|_{L^{\infty}}+\| u_{x}\| _{L^{\infty}} +\| u_{xx} \|_{L^{\infty}}\bigr)\,d\tau, \end{aligned}
(30)

which, together with (29), ensures that

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}(\mathbb{S})}+\bigl\| \rho (t)\bigr\| _{H^{s-1}(\mathbb{S})} \\ &\quad\leq\| u_{0}\|_{H^{s}(\mathbb{S})} +\|\rho_{0} \|_{H^{s-1}(\mathbb{S})} +C \int^{t}_{0}\bigl(\| u\|_{H^{s}} +\bigl\| \rho(t) \bigr\| _{H^{s-1}}\bigr) \\ &\qquad{}\times\bigl(\| u\|_{L^{\infty}}+\| u_{x}\|_{L^{\infty}} +\| u_{xx}\|_{L^{\infty}}+\|\rho\| _{L^{\infty}}\bigr)\,d\tau. \end{aligned}
(31)

Using Gronwall’s inequality, we have

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}(\mathbb{S})} +\bigl\| \rho (t)\bigr\| _{H^{s-1}(\mathbb{S})} \\ &\quad\leq\bigl(\| u_{0}\|_{H^{s}(\mathbb{S})} +\|\rho_{0} \|_{H^{s-1}(\mathbb{S})}\bigr) \\ &\qquad{}\times\exp\biggl[C \int ^{t}_{0}\bigl(\| u\|_{L^{\infty}}+\| u_{x}\|_{L^{\infty}} +\| u_{xx}\|_{L^{\infty}}+\| \rho \|_{L^{\infty}}\bigr)\,d\tau\biggr]. \end{aligned}
(32)

From (27) and Lemma 3.5 we get

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}(\mathbb{S})}+\bigl\| \rho (t)\bigr\| _{H^{s-1}(\mathbb{S})} \\ &\quad\leq\bigl(\| u_{0}\|_{H^{s}(\mathbb{S})} +\|\rho_{0} \|_{H^{s-1}(\mathbb{S})}\bigr) \\ &\qquad{} \times\exp \biggl(C \int^{t}_{0}\bigl(L(t)+\| z_{0} \|_{H^{2}\times L^{2}}+\| \rho_{0}\|_{L^{\infty}}e^{CT\| z_{0}\|_{H^{2}\times L^{2}}}\bigr)\,d\tau \biggr). \end{aligned}
(33)

Therefore, if the maximal existence time $$T<\infty$$ satisfies $$\int^{t}_{0}\| \rho_{x}\|_{L^{\infty}}\,d\tau<\infty$$, then we get from (33) that

\begin{aligned} \limsup_{t\rightarrow T}\bigl(\bigl\| u(t)\bigr\| _{H^{s}(\mathbb {S})}+\bigl\| \rho(t) \bigr\| _{H^{s-1}(\mathbb{S})}\bigr)< \infty , \end{aligned}
(34)

which completes the proof of Theorem 1 for $$s\in(\frac{3}{2},2)$$.

Step 2. For $$s\in[2,\frac{5}{2})$$, applying Lemma 3.2 to the second equation of system (4), we get

\begin{aligned} \|\rho\|_{H^{s-1}(\mathbb{S})} \leq&\| \rho_{0}\|_{H^{s-1}} +C \int^{t}_{0}\| u_{x}\rho \|_{H^{s-1}}\,d\tau \\ &{}+C \int^{t}_{0}\|\rho\|_{H^{s-1}} \| \partial_{x}u\|_{L^{\infty}\cap H^{\frac{1}{2}}}\,d\tau. \end{aligned}

Using (28) results in

\begin{aligned} \|\rho\|_{H^{s-1}(\mathbb{S})} \leq&\| \rho_{0}\|_{H^{s-1}} +C \int^{t}_{0}\| u_{x}\|_{H^{s-1}}\| \rho\|_{L^{\infty}}\,d\tau +C \int^{t}_{0}\|\rho\|_{H^{s-1}} \| \partial_{x}u\|_{L^{\infty}\cap H^{\frac{1}{2}}}\,d\tau, \end{aligned}

which, together with (30), yields

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}} +C \int^{t}_{0}\bigl(\| u\|_{H^{s}} +\bigl\| \rho(t) \bigr\| _{H^{s-1}}\bigr) \\ &\qquad{}\times(\| u\|_{L^{\infty}}+\| u\|_{ H^{\frac{3}{2}+\varepsilon}} +\| u_{xx} \|_{L^{\infty}}+\|\rho\| _{L^{\infty}})\,d\tau, \end{aligned}
(35)

where $$\varepsilon\in(0,\frac{1}{2})$$, and we used the fact that $$H^{\frac{1}{2}+\varepsilon}\hookrightarrow L^{\infty}\cap H^{\frac{1}{2}}$$.

Using Gronwall’s inequality, we have

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr)\exp\biggl[C \int^{t}_{0}\bigl(\| u\|_{L^{\infty}}+\| u \|_{H^{\frac{3}{2}+\varepsilon}} +\| u_{xx}\|_{L^{\infty}}+\| \rho\|_{L^{\infty}}\bigr)\,d\tau\biggr]. \end{aligned}
(36)

From (27) and Lemma 3.5 we get

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr) \\ &\qquad{} \times\exp \biggl(C \int^{t}_{0}\bigl(L(t)+\| z_{0} \|_{H^{2}\times L^{2}}+\| \rho_{0}\|_{L^{\infty}}e^{CT\| z_{0}\|_{H^{2}\times L^{2}}}\bigr)\,d\tau \biggr). \end{aligned}
(37)

Applying the argument as in step 1, we complete the proof of Theorem 1 for $$s\in[2,\frac{5}{2})$$.

Step 3. For $$s\in(2,3)$$, differentiating once the second equation of system (4) with respect to x, we have

\begin{aligned} \partial_{t}\rho_{x}+u\partial_{x} \rho_{x}+2u_{x}\rho_{x}+u_{xx}\rho=0. \end{aligned}
(38)

Using Lemma 3.3, we get

\begin{aligned} \|\rho_{x}\|_{H^{s-2}(\mathbf{S})} \leq&\| \rho_{0x} \|_{H^{s-2}} +C \int^{t}_{0}\| u\|_{H^{s}}\| \rho \|_{L^{\infty}}\,d\tau \\ &{} +C \int^{t}_{0}\|\rho\|_{H^{s-1}} \bigl(\| u \|_{L^{\infty}}+\| \partial_{x}u\|_{L^{\infty}}\bigr)\,d\tau, \end{aligned}
(39)

where we used the estimates

$$\| u_{x}\rho_{x}\|_{H^{s-2}}\leq C\bigl(\| u_{x}\|_{H^{s-1}} \|\rho\|_{L^{\infty}}+ \|\rho_{x} \|_{H^{s-2}}\| u_{x}\|_{L^{\infty}}\bigr)$$

and

$$\|\rho u_{xx}\|_{H^{s-2}}\leq C\bigl(\|\rho\|_{H^{s-1}} \| u_{x}\|_{L^{\infty}}+ \| u_{xx}\|_{H^{s-2}}\| \rho \|_{L^{\infty}}\bigr),$$

where Lemma 3.1 (6) was used.

Using (39), (30), and (29) (where $$s-1$$ is replaced by $$s-2$$) yields

\begin{aligned} \bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \leq{}&\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}} +C \int^{t}_{0}\bigl(\| u\|_{H^{s}} +\bigl\| \rho(t) \bigr\| _{H^{s-1}}\bigr) \\ &{}\times\bigl(\| u\|_{L^{\infty}}+\| u_{x}\|_{L^{\infty}} +\| u_{xx}\|_{L^{\infty}}+\|\rho\| _{L^{\infty}}\bigr)\,d\tau. \end{aligned}
(40)

Applying Gronwall’s inequality, we have

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr)\exp\biggl[C \int^{t}_{0}\bigl(\| u\|_{L^{\infty}}+\| u_{x}\|_{L^{\infty}} +\| u_{xx}\|_{L^{\infty}}+\| \rho \|_{L^{\infty}}\bigr)\,d\tau\biggr]. \end{aligned}
(41)

From (27) and Lemma 3.5 we get

\begin{aligned}[b] &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}}\\ &\quad\leq\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr)\\ &\qquad{} \times\exp \biggl(C \int^{t}_{0}\bigl(L(t)+\| z_{0} \|_{H^{2}\times L^{2}}+\| \rho_{0}\|_{L^{\infty}}e^{CT\| z_{0}\|_{H^{2}\times L^{2}}}\bigr)\,d\tau \biggr). \end{aligned}
(42)

Using the argument as in step 1, we complete the proof of Theorem 1 for $$s\in (2,3)$$.

Step 4. For $$s=k\in\mathbf{N}$$, $$k\geq3$$, differentiating $$k-2$$ times the second equation of system (4) with respect to x, we obtain

\begin{aligned} (\partial_{t}+u\partial_{x})\partial^{k-2}_{x} \rho+\sum_{l_{1}+l_{2}=k-3,l_{1},l_{2}\geq0} C_{l_{1},l_{2}}\partial^{l_{1}+1}_{x}u \partial^{l_{2}+1}_{x}\rho+\rho \partial_{x}\bigl( \partial^{k-2}_{x}u\bigr)=0. \end{aligned}
(43)

Using Lemma 3.2, we get from (43) that

\begin{aligned} \bigl\| \partial^{k-2}_{x}\rho\bigr\| _{H^{1}} \leq&\bigl\| \partial ^{k-2}_{x}\rho_{0}\bigr\| _{H^{1}} +C \int^{t}_{0}\bigl\| \partial^{k-2}_{x} \rho\bigr\| _{H^{1}}\| \partial_{x}u\|_{H^{\frac{1}{2}}\cap L^{\infty}}\,d\tau \\ &{} +C \int^{t}_{0}\biggl\| \sum_{l_{1}+l_{2}=k-3,l_{1},l_{2}\geq0} C_{l_{1},l_{2}}\partial^{l_{1}+1}_{x}u\partial^{l_{2}+1}_{x} \rho+\rho \partial^{k-1}_{x}u\biggr\| _{H^{1}}\,d\tau. \end{aligned}
(44)

Since $$H^{1}$$ is an algebra, we have

$$\bigl\| \rho\partial^{k-1}_{x}u\bigr\| _{H^{1}}\leq C\|\rho \|_{H^{1}} \bigl\| \partial^{k-1}_{x}u\bigr\| _{H^{1}}\leq C \|\rho \|_{H^{1}} \| u\|_{H^{s}}$$

and

$$\biggl\| \sum_{l_{1}+l_{2}=k-3,l_{1},l_{2}\geq0} C_{l_{1},l_{2}}\partial^{l_{1}+1}_{x}u \partial^{l_{2}+1}_{x}\rho \biggr\| _{H^{1}}\leq C\|\rho \|_{H^{s-1}} \| u\|_{H^{s-1}}.$$

It follows that

\begin{aligned} \bigl\| \partial^{k-2}_{x}\rho\bigr\| _{H^{1}} \leq&\bigl\| \partial ^{k-2}_{x}\rho_{0}\bigr\| _{H^{1}} +C \int^{t}_{0}\bigl(\| u\|_{H^{s}}+\|\rho \|_{H^{s-1}}\bigr) \\ &{} \times\bigl(\| u\|_{H^{s-1}}+\|\rho\|_{H^{1}}\bigr)\,d\tau. \end{aligned}
(45)

From the Gagliardo-Nirenberg inequality we have that, for $$\sigma\in(0,1)$$,

\begin{aligned} &\|\rho\|_{H^{s-1}}\leq C\bigl(\| \rho\|_{H^{\sigma}}+\bigl\| \partial^{k-2}_{x}\rho\bigr\| _{H^{1}}\bigr). \end{aligned}
(46)

On the other hand, for $$\sigma\in(0,1)$$, rewrite (29) as

\begin{aligned} \|\rho\|_{H^{\sigma}(\mathbf{S})} \leq&\| \rho_{0} \|_{H^{\sigma}} +C \int^{t}_{0}\| u\|_{H^{\sigma+1}}\|\rho \|_{L^{\infty}}\,d\tau \\ &{} +C \int^{t}_{0}\|\rho\|_{H^{\sigma}} \bigl(\| u \|_{L^{\infty}}+\| \partial_{x}u\|_{L^{\infty}}\bigr)\,d\tau, \end{aligned}
(47)

which, together with (45), yields

\begin{aligned} \|\rho\|_{H^{s-1}} \leq& C\|\rho_{0} \|_{H^{s-1}}+C \int^{t}_{0}\bigl(\| u\|_{H^{s}}+\|\rho \|_{H^{s-1}}\bigr) \\ &{}\times\bigl(\| u\|_{H^{s-1}}+\|\rho\|_{H^{1}}\bigr)\,d\tau, \end{aligned}
(48)

where (46) was used.

Using Lemma 3.1 (5), we get

\begin{aligned} \bigl\| u(t)\bigr\| _{H^{s}(\mathbf{S})} \leq&\| u_{0} \|_{H^{s}}+ C \int^{t}_{0}\| u\|_{H^{s}}\bigl(\| u\| _{L^{\infty}}+\| u_{x}\|_{L^{\infty}} +\| u_{xx} \|_{L^{\infty}} \\ &{} +\| u_{xxx}\|_{L^{\infty}}\bigr)\,d\tau +C \int^{t}_{0}\bigl\| \rho(\tau)\bigr\| _{H^{s-1}} \bigl\| \rho( \tau)\bigr\| _{L^{\infty}}\,d\tau, \end{aligned}
(49)

which, together with (48), results in

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq C\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr) +C \int^{t}_{0}\bigl(\| u\|_{H^{s}} +\bigl\| \rho(t) \bigr\| _{H^{s-1}}\bigr) \\ &\qquad{} \times\bigl(\| u\|_{H^{s-1}}+\| \rho\|_{H^{1}} +\| u_{xx}\|_{L^{\infty}}+\| u_{xxx}\| _{L^{\infty}}\bigr)\,d\tau. \end{aligned}
(50)

Using Gronwall’s inequality, we get

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq C\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr) \\ &\qquad{}\times\exp\biggl[C \int^{t}_{0}\bigl(\| u\|_{H^{s-1}}+\| \rho \|_{H^{1}} +\| u_{xx}\|_{L^{\infty}}+\| u_{xxx}\| _{L^{\infty}}\bigr)\,d\tau\biggr]. \end{aligned}
(51)

If $$T<\infty$$ satisfies $$\int^{T}_{0}\| \rho_{x}\|_{L^{\infty}}\,d\tau<\infty$$, applying step 2 and the induction assumption, we obtain from Lemma 3.5 that $$\| u\|_{H^{s-1}}+\| \rho\|_{H^{1}} +\| u_{xx}\|_{L^{\infty}}+\| u_{xxx}\| _{L^{\infty}}$$ is uniformly bounded. From (51) we get

$$\limsup_{t\rightarrow T}\bigl(\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho (t) \bigr\| _{H^{s-1}}\bigr)< \infty,$$

which contradicts the assumption that $$T<\infty$$ is the maximal existence time. This completes the proof of Theorem 1 for $$s=k\in N$$ and $$k\geq3$$.

Step 5. For $$s\in(k,k+1)$$, $$k\in N$$, and $$k\geq3$$, differentiating $$k-1$$ times the second equation of system (4) with respect to x, we obtain

\begin{aligned} (\partial_{t}+u\partial_{x})\partial^{k-1}_{x} \rho+\sum_{l_{1}+l_{2}=k-2,l_{1},l_{2}\geq0} C_{l_{1},l_{2}}\partial^{l_{1}+1}_{x}u \partial^{l_{2}+1}_{x}\rho+\rho \partial_{x}\bigl( \partial^{k-1}_{x}u\bigr)=0. \end{aligned}
(52)

Using Lemma 3.3 with $$s-k\in(0,1)$$, we get from (52) that

\begin{aligned} \bigl\| \partial^{k-1}_{x}\rho\bigr\| _{H^{s-k}} \leq{}&\bigl\| \partial^{k-1}_{x}\rho_{0} \bigr\| _{H^{s-k}} +C \int^{t}_{0}\bigl\| \partial^{k-1}_{x} \rho\bigr\| _{H^{s-k}}\bigl(\| u\|_{L^{\infty}} +\|\partial_{x}u \|_{L^{\infty}}\bigr)\,d\tau \\ &{} +C \int^{t}_{0}\biggl\| \sum_{l_{1}+l_{2}=k-2,l_{1},l_{2}\geq0} C_{l_{1},l_{2}}\partial^{l_{1}+1}_{x}u\partial^{l_{2}+1}_{x} \rho+\rho \partial^{k}_{x}u\biggr\| _{H^{s-k}}\,d\tau. \end{aligned}
(53)

For each $$\varepsilon\in(0,\frac{1}{2})$$, using Lemma 3.1 (6) and the fact that $$H^{\frac{1}{2}+\varepsilon}\hookrightarrow L^{\infty}$$, we have

\begin{aligned}[b] \bigl\| \rho\partial^{k}_{x}u\bigr\| _{H^{s-k}}& \leq C\bigl(\|\rho\|_{H^{s-k+1}} \bigl\| \partial^{k-1}_{x}u \bigr\| _{L^{\infty}}+\bigl\| \partial ^{k}_{x}u\bigr\| _{H^{s-k}} \| \rho\|_{L^{\infty}}\bigr)\\ &\leq C\bigl(\|\rho\|_{H^{s-k+1}} \| u\|_{H^{k-\frac{1}{2}+\varepsilon}}+\| u\|_{H^{s-1}} \| \rho\|_{L^{\infty}}\bigr) \end{aligned}
(54)

and

\begin{aligned} &\biggl\| \sum_{l_{1}+l_{2}=k-2,l_{1},l_{2}\geq0} C_{l_{1},l_{2}} \partial^{l_{1}+1}_{x}u\partial^{l_{2}+1}_{x}\rho \biggr\| _{H^{s-k}} \\ &\quad\leq C\sum_{l_{1}+l_{2}=k-2,l_{1},l_{2}\geq0} C_{l_{1},l_{2}} \bigl(\bigl\| \partial^{l_{1}+1}_{x}u\bigr\| _{H^{s-k+1}}\bigl\| \partial^{l_{2}}_{x} \rho\bigr\| _{L^{\infty}} \\ &\qquad{} +\bigl\| \partial^{l_{1}+1}_{x}u\bigr\| _{L^{\infty}}\bigl\| \partial^{l_{2}+1}_{x}\rho\bigr\| _{H^{s-k}} \bigr) \\ &\quad\leq C\bigl(\| u\|_{H^{s}}\|\rho\|_{H^{k-\frac{3}{2}+\varepsilon }}+\| u\|_{H^{k-\frac{1}{2}+\varepsilon}} \|\rho\| _{H^{s-1}}\bigr). \end{aligned}
(55)

Therefore, from (53), (54), and (55) we get

\begin{aligned} \bigl\| \partial_{x}^{k-1}\rho\bigr\| _{H^{s-k}} \leq&\bigl\| \partial_{x}^{k-1}\rho_{0}\bigr\| _{H^{s-k}} +C \int^{t}_{0}\bigl(\| u\|_{H^{s}}+\| \rho \|_{H^{s-1}}\bigr) \\ &{}\times\bigl(\| u\|_{H^{k-\frac{3}{2}+\varepsilon}} + \| \rho\|_{H^{k-\frac{1}{2}+\varepsilon}}\bigr)\,d\tau. \end{aligned}
(56)

Applying Lemma 3.2 to the first equation of system (4) for $$s\in(k,k+1)$$ with $$k\geq3$$, we obtain

\begin{aligned} \bigl\| u(t)\bigr\| _{H^{s}(\mathbf{S})} \leq&\| u_{0} \|_{H^{s}}+C \int^{t}_{0}\bigl\| \rho(\tau)\bigr\| _{H^{s-1}} \bigl\| \rho( \tau)\bigr\| _{L^{\infty}}\,d\tau \\ &{} +C \int^{t}_{0}\| u\| _{H^{s}}\bigl(\| u \|_{L^{\infty}}+\| u_{x}\| _{L^{\infty}} +\| u_{xx} \|_{L^{\infty}}\bigr)\,d\tau, \end{aligned}
(57)

which, together with (56) and (29) (where $$s-1$$ is replaced by $$s-k$$), gives

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq C\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr) +C \int^{t}_{0}\bigl(\| u\|_{H^{s}} +\bigl\| \rho(t) \bigr\| _{H^{s-1}}\bigr) \\ &\qquad{} \times\bigl(\| u\|_{H^{k-\frac{1}{2}+\varepsilon }}+\| \rho\|_{H^{k-\frac{3}{2}+\varepsilon}}\bigr)\,d\tau. \end{aligned}
(58)

Using Gronwall’s inequality, we get

\begin{aligned} &\bigl\| u(t)\bigr\| _{H^{s}}+\bigl\| \rho(t)\bigr\| _{H^{s-1}} \\ &\quad\leq C\bigl(\| u_{0}\|_{H^{s}} +\|\rho_{0} \|_{H^{s-1}}\bigr)\exp\biggl[C \int^{t}_{0}\bigl(\| u\|_{H^{k-\frac{1}{2}+\varepsilon}}+\| \rho \|_{H^{k-\frac{3}{2}+\varepsilon}}\bigr)\,d\tau\biggr]. \end{aligned}
(59)

Noting that $$k-\frac{1}{2}+\varepsilon< k$$, $$k-\frac{3}{2}+\varepsilon< k-1$$, and $$k\geq3$$ and applying step 4, we obtain that $$\| u\|_{H^{k-\frac{1}{2}+\varepsilon}}+\| \rho\|_{H^{k-\frac{3}{2}+\varepsilon}}$$ is uniformly bounded. Therefore, we complete the proof of Theorem 1 for $$s\in (k,k+1)$$, $$k\in N$$, and $$k\geq3$$.

So, the proof of Theorem 1 is completed. □

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## Acknowledgements

The author thanks the referees for their valuable comments and suggestions. This work was supported by Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing [grant number 2013QZJ02], [grant number 2014QYJ03], Scientific Research Foundation of the Education Department of Sichuan province Project [grant number 16ZA0265], SUSER [grant number 2014RC03].

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Correspondence to Ying Wang. 