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.

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:

• (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)

• (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)

• (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. □