1 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 [1], but it was firstly derived rigorously by Constantin and Ivanov [2]. 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. [3] established a reciprocal transformation between the two-component Camassa-Holm system and the first negative flow of the AKNS hierarchy. Escher et al. [4] 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 [2], 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 [5]. 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 [6]. The other results related to the system can be found in [7–15].

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 [16], and it is known as a modified Camassa-Holm equation and also viewed as a geodesic equation on some diffeomorphism group [16]. It is shown in [16] 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 [6] 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 [6]. 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.

2 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

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

3 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

([6])

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. □

4 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 [19], 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. □