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Regularity for compressible isentropic Navier-Stokes equations with cylinder symmetry

  • Lan Huang
  • Ruxu Lian
Open Access
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  • 645 Downloads

Abstract

This paper is concerned with the regularity of the global solutions in \(H^{4}\) to the compressible isentropic Navier-Stokes equations with the cylinder symmetry in \(R^{3}\). Such a circular coaxial cylinder symmetric domain is an unbounded domain, but we assume that the corresponding solution depends only on one radial variable, r in \(G=\{r\in R^{+},0< a\leq r\leq b<+\infty\}\), in which the related domain G is a bounded domain. Some new ideas and more delicate estimates are introduced to prove these results.

Keywords

compressible Navier-Stokes equations cylinder symmetry density-dependent viscosity regularity 

1 Introduction

The compressible isentropic Navier-Stokes equations with density-dependent viscosity coefficients can be written for \(t>0\) as
$$\begin{aligned}& \rho_{t}+\operatorname {div}(\rho\mathbf{U})=0, \end{aligned}$$
(1.1)
$$\begin{aligned}& (\rho{\mathbf{U}})_{t}+\operatorname {div}(\rho{\mathbf{U}}\otimes{\mathbf {U}})- \operatorname {div}\bigl(\mu(\rho)D({\mathbf{U}})\bigr) -\nabla\bigl(\lambda(\rho)\operatorname {div}{ \mathbf{U}}\bigr)+\nabla{P(\rho)}=0, \end{aligned}$$
(1.2)
where \(\rho({\mathbf{x}},t)\), \({\mathbf{U}}({\mathbf{x}},t)\) and \(P(\rho)=\rho^{\gamma}\) (\(\gamma>1\)) stand for the fluid density, velocity, and pressure, respectively, and
$$ D({\mathbf{U}})=\frac{\nabla{\mathbf{U}}+(\nabla{\mathbf {U}})^{T}}{2} $$
(1.3)
is the strain tensor and \(\mu(\rho)\), \(\lambda(\rho)\) are the Lamé viscosity coefficients satisfying
$$ \mu(\rho)>0, \qquad \mu(\rho)+N\lambda(\rho)\geq0. $$
(1.4)
In this paper we establish the regularity of global solutions to the compressible Navier-Stokes equations with cylinder symmetry in \(R^{3}\). We will pay attention to the flows between two circular coaxial cylinders. We assume that the corresponding solutions depend only on the radial variable \(r\in G=\{r|0< a\leq r\leq b<+\infty\}\) and the time variable \(t\in[0,T]\). For simplicity, we will take \(\mu(\rho )=\rho^{\alpha}\) and \(\lambda(\rho)=\rho\mu'(\rho)-\mu(\rho)=(\alpha-1)\rho ^{\alpha}\) with \(\alpha>{1/2}\) and \(D({\mathbf{U}})=\nabla{\mathbf {U}}\). Then system (1.1)-(1.2) reduce to the following form:
$$\begin{aligned}& \rho_{t}+\frac{1}{r}(r\rho u)_{r}=0, \end{aligned}$$
(1.5)
$$\begin{aligned}& r\rho u_{t}+r\rho uu_{r}-\rho v^{2}+r\bigl( \rho^{\gamma}\bigr)_{r}-\bigl(r\rho^{\alpha }u_{r} \bigr)_{r} \\& \quad {} -(\alpha-1)r \biggl(\frac{1}{r}\rho^{\alpha}(ru)_{r} \biggr)_{r}+\frac {1}{r}\rho^{\alpha}u=0, \end{aligned}$$
(1.6)
$$\begin{aligned}& r\rho v_{t}+r\rho uv_{r}+\rho uv- \bigl(r \rho^{\alpha}v_{r} \bigr)_{r}+\frac{1}{r} \rho^{\alpha} v=0, \end{aligned}$$
(1.7)
$$\begin{aligned}& r\rho w_{t}+r\rho uw_{r}-\bigl(r\rho^{\alpha}w_{r} \bigr)_{r}=0. \end{aligned}$$
(1.8)
We consider the initial boundary value problem (1.5)-(1.8) subject to the following initial and boundary conditions:
$$\begin{aligned}& (\rho,u,v,w) (r,0)=(\rho_{0},u_{0},v_{0},w_{0}) (r),\quad r\in[a,b], \end{aligned}$$
(1.9)
$$\begin{aligned}& u(a)=u(b)=0,\qquad v(a)=v(b)=0,\qquad w(a)=w(b)=0. \end{aligned}$$
(1.10)
First we find it convenient to transfer problems (1.5)-(1.10) into that in Lagrangian coordinates and present the desired results. It is well known that Eulerian coordinates \((r,t)\) are connected to the Lagrangian coordinates \((\xi,t)\) by the following relation:
$$ r(\xi,t)=r_{0}(\xi )+ \int_{0}^{t}\tilde{u}(\xi,\tau)\,d\tau, $$
(1.11)
where \(\tilde{u}(\xi,t)=u(r(\xi,t),t)\) and
$$ r_{0}(\xi)=\eta ^{-1}(\xi),\qquad \eta(r)= \int_{a}^{r}s\rho_{0}(s)\,ds,\quad r\in G. $$
(1.12)
It follows from (1.5) and the boundary condition (1.10) that
$$ \frac{\partial}{\partial t} \biggl( \int_{a}^{r(\xi,t)}s\rho (s,t)\,ds \biggr)=\rho ru+ \int_{a}^{r}s \rho_{t}\,ds=0. $$
(1.13)
Thus,
$$ \int_{a}^{r}s\rho(s,t)\,ds= \int_{a}^{r_{0}}s\rho_{0}(s)\,ds=\xi $$
(1.14)
and G is transformed to \(\Omega=(0,1)\) with
$$ 1= \int_{a}^{b}s\rho (s,t)\,ds= \int_{a}^{b}s\rho_{0}(s)\,ds. $$
(1.15)
Moreover, we have
$$ \partial_{\xi}r(\xi,t)=\bigl[r(\xi,t)\rho\bigl(r(\xi,t),t\bigr) \bigr]^{-1}. $$
(1.16)
For a function \(\phi(r,t)\), if we write \(\tilde{\phi}(\xi,t)=\phi (r(\xi,t),t)\), by virtue of (1.11), (1.16), and the chain rule, we have
$$\begin{aligned}& \partial_{t}\tilde{\phi}(\xi,t)=\partial_{t}\phi (r,t)+v \partial_{r}\phi(r,t), \end{aligned}$$
(1.17)
$$\begin{aligned}& \partial_{\xi}\tilde{\phi}(\xi,t)=\partial_{r}\phi(r,t) \partial _{\xi} r(\xi,t)=\bigl(r^{2}\rho(r,t) \bigr)^{-1}\partial_{r}\phi(r,t). \end{aligned}$$
(1.18)
In the following, without danger of confusion we denote \((\tilde{\rho},\tilde{u},\tilde{v},\tilde{w})\) still by \((\rho,u, v,w)\) and \((\xi,t)\) by \((x,t)\). Therefore, (1.5)-(1.8) in the Eulerian coordinates can be written in the Lagrangian coordinates in the new variables \((x,t)\) as follows (see also [1]):
$$\begin{aligned}& \rho_{t}+\rho^{2}(ru)_{x}=0, \end{aligned}$$
(1.19)
$$\begin{aligned}& \frac{u_{t}}{r}-\frac{v^{2}}{r^{2}}+\bigl(\rho^{\gamma} \bigr)_{x}=\alpha \bigl(\rho^{\alpha+1}(ru)_{x} \bigr)_{x}-\frac{(\rho^{\alpha})_{x}u}{r}, \end{aligned}$$
(1.20)
$$\begin{aligned}& \frac{v_{t}}{r}+\frac{uv}{r^{2}}= \bigl(\rho^{\alpha+1}(rv)_{x} \bigr)_{x}-\frac{(\rho^{\alpha})_{x}v}{r}, \end{aligned}$$
(1.21)
$$\begin{aligned}& w_{t}-\bigl(r^{2}\rho^{1+\alpha} w_{x} \bigr)_{x}=0, \end{aligned}$$
(1.22)
subject to the following initial and boundary conditions:
$$\begin{aligned}& (\rho,u,v,w) (x,0)=(\rho_{0},u_{0},v_{0},w_{0}) (x),\quad x\in\Omega, \end{aligned}$$
(1.23)
$$\begin{aligned}& (u,v,w) (0,t)=(u,v,w) (1,t)=0, \quad t\geq0, \end{aligned}$$
(1.24)
where \(r(x,t)\) is determined by
$$ \begin{aligned} &r_{t}(x,t)=u(x,t), \qquad r(x,t)r_{x}(x,t)= \frac{1}{\rho}, \\ &r|_{t=0}=r_{0}(x)= \biggl[a^{2}+2 \int_{0}^{x}\frac{1}{\rho_{0}(y,t)}\,dy \biggr]^{1/2}. \end{aligned} $$
(1.25)

The compressible Navier-Stokes system has been noticed academically by physicists and mathematicians for a relatively long time. We are interested in the case that the viscosity is density-dependent. Now let us first recall the related results in this direction. In the one-dimensional case, for the initial boundary value problems in a bounded domain, there have been many works (see, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) on the existence, uniqueness, and asymptotic behavior of weak solutions, based on the initial finite mass and the flow density being connected with the infinite vacuum either continuously or by a jump discontinuity. For the one-dimensional Cauchy problem, see, e.g., [12] and references therein for the results on global existence based on the density-dependent viscosity.

In two or three dimensions, the global existence and large time behavior of solution to compressible Navier-Stokes equations (1.1)-(1.2) with constant viscosity and density-dependent viscosity have been investigated for initial boundary value problems, we refer the reader to [1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] and references therein. Among them, Bresch and Desjardins [13, 14] obtained the global existence of 2D shallow water equations. For the spherically symmetric problem to a three-dimensional compressible isentropic Navier-Stokes problem, Guo et al. [19] analyzed the structure of the solution; Huang et al. [28] studied the global well-posedness of the classical solutions with large oscillations and vacuum; Lian et al. [22] obtained the global existence of spherically symmetric solution for the exterior problem and the initial boundary value problem; Zhang and Fang [27] investigated the global existence and uniqueness of the weak solution without a solid core. For the cylindrically symmetric problem to the three-dimensional compressible Navier-Stokes equations, when the viscosity coefficients are both constants, the uniqueness of the weak solutions was proved in [17, 18], the global existence of isentropic compressible cylindrically symmetric solution was established in [29]; this result was later generalized to the nonisentropic case in [21]. Recently, Cui and Yao [15] proved the asymptotic behavior of a compressible pth power Newtonian fluid with cylinder symmetry; Qin [24] established the exponential stability in \(H^{1}\) and \(H^{2}\) for an ideal fluid. Later on, Qin and Jiang [25] proved the global existence and the exponential stability in \(H^{4}\). Jiang and Zhang [21] established a boundary layer effect and the convergence rate as the shear viscosity μ goes to zero. When the viscosity coefficient \(\mu(\rho)\) is density-dependent and \(\lambda(\rho)\) is a positive constant, the global existence was obtained in [26]. When viscosity coefficients μ and λ are density-dependent, Liu and Lian [1] established the global existence and asymptotic behavior of cylindrically symmetric solutions, however, there is no result on the regularity for this system.

It is noticed that the above analysis concerns the existence of solution in \(H^{1}[0,1]\), the regularity in \(H^{4}[0,1]\) has never been investigated for the three-dimensional isentropic compressible Navier-Stokes equations. Therefore, we continue the work by Liu and Lian [1] and study the regularity of the solutions in \(H^{4}\). In order to obtain a higher regularity of global strong solutions, there are many complicated estimates on higher derivations of the solution involved; this is our difficulty. To overcome this difficulty, we shall use some proper embedding theorems, and the interpolation techniques as well as many delicate estimates.

The notation in this paper will be as follows: \(L^{\bar{p}}\), \(1\leq\bar{p}\leq+\infty\), \(W^{m,\bar{p}}\), \(m\in N\), \(H^{1}=W^{1,2}\), \(H_{0}^{1}=W_{0}^{1,2}\) denote the usual (Sobolev) spaces on \([0, 1]\). In addition, \(\Vert \cdot \Vert _{B}\) denotes the norm in the space B; we also put \(\Vert \cdot \Vert = \Vert \cdot \Vert _{L^{2}}\). Subscripts t and x denote the (partial) derivatives with respect to t and x, respectively. We use \(C_{i}\) (\(i=1,2,4\)) to denote the generic positive constant depending only on the \(H^{i}\) norm of the initial data \((\rho_{0},u_{0},v_{0},w_{0})\) and the variable t.

Before stating the main result, we assume the initial data
$$ (\rho_{0}-\bar{\rho}, u_{0},v_{0},w_{0}) \in\bigl(H^{4}[0,1]\bigr)^{4}, $$
(1.26)
with \(\rho_{0}>0\) and \(\bar{\rho}=\frac{1}{b-a}\int_{a}^{b}\rho_{0}r\,dr\), and we define
$$\begin{aligned}& H_{0}= \int_{0}^{1} \biggl(\frac{1}{2} \bigl(u_{0}^{2}+v_{0}^{2}+w_{0}^{2} \bigr)+\frac{1}{\gamma -1}\bigl(\rho_{0}^{\gamma-1} -\bar{\rho}^{\gamma-1}\bigr)+\bar{\rho}^{\gamma}\biggl(\frac{1}{\rho_{0}}- \frac {1}{\bar{\rho}}\biggr) \biggr)\,dx, \\& H_{1}= \int_{0}^{1} \biggl(\frac{1}{2}\bigl( \bigl(u_{0}+r\bigl(\rho_{0}^{\alpha } \bigr)_{x}\bigr)^{2}+v_{0}^{2}+w_{0}^{2} \bigr)+\frac{1}{\gamma-1}\bigl(\rho_{0}^{\gamma-1} -\bar{\rho}^{\gamma-1}\bigr)+\bar{\rho}^{\gamma}\biggl(\frac{1}{\rho_{0}}- \frac {1}{\bar{\rho}}\biggr) \biggr)\,dx. \end{aligned}$$

Now we are in a position to state our main results.

Theorem 1.1

Let \(\gamma>1\), \(\alpha>1/2\). Assume that the initial data satisfies (1.26) and \(H_{0}(H_{0}+H_{1})< a^{2}\alpha^{2}(2\alpha -1)^{-2}\bar{\rho}^{\gamma+2\alpha-1}\). Then there exists a unique generalized global solution \((\rho(t),u(t),v(t),w(t))\in(H^{4}[0,1])^{4}\) to the problem (1.19)-(1.24) verifying that, for \(T>0\),
$$\begin{aligned}& \rho-\bar{\rho}\in L^{\infty}\bigl([0,T], H^{4}[0,1]\bigr)\cap L^{2}\bigl([0,T], H^{4}[0,1]\bigr), \end{aligned}$$
(1.27)
$$\begin{aligned}& (u,v,w)\in L^{\infty}\bigl([0,T], H^{4}[0,1]\bigr)\cap L^{2}\bigl([0,T], H^{5}[0,1]\bigr), \end{aligned}$$
(1.28)
$$\begin{aligned}& (u_{t},v_{t},w_{t})\in L^{\infty} \bigl([0,T], H^{2}[0,1]\bigr)\cap L^{2}\bigl([0,T], H^{3}[0,1]\bigr), \end{aligned}$$
(1.29)
$$\begin{aligned}& (u_{tt},v_{tt},w_{tt})\in L^{\infty} \bigl([0,T], L^{2}[0,1]\bigr)\cap L^{2}\bigl([0,T], H^{1}[0,1]\bigr). \end{aligned}$$
(1.30)

Corollary 1.1

Under the assumptions of Theorem  1.1, (1.27)-(1.28) implies \((\rho(t),u(t), v(t),w(t))\) is the classical solution verifying, for any \(t>0\),
$$ \bigl\Vert \rho(t)-\bar{\rho}\bigr\Vert _{C^{3+1/2}}+\bigl\Vert u(t)\bigr\Vert _{C^{3+1/2}}+\bigl\Vert v(t)\bigr\Vert _{C^{3+1/2}}+\bigl\Vert w(t)\bigr\Vert _{C^{3+1/2}}\leq C_{4}. $$
(1.31)

The rest of the paper is arranged as follows. Section 2 is concerned with the proof of the regularity for a cylindrically symmetric solution to the compressible Navier-Stokes problem in detail.

2 Proof of Theorem 1.1

We will complete the proof of Theorem 1.1 and assume that the assumptions in Theorem 1.1 are valid. We begin with the following lemma.

Lemma 2.1

Under the assumptions in Theorem  1.1, there exist positive constants \(\rho_{*}>0\) and \(\rho^{*}>0\) with \(\rho_{*}<\bar{\rho}<\rho^{*}\) so that the unique global solution \((\rho(t),u(t),v(t),w(t))\) to problem (1.19)-(1.24) exists and satisfies, for any \(T>0\),
$$\begin{aligned}& 0< \rho_{*}\leq\rho(x,t)\leq\rho^{*}, \end{aligned}$$
(2.1)
$$\begin{aligned}& (\rho-\bar{\rho})_{x}\in L^{\infty}\bigl([0,T],L^{2}[0,1] \bigr)\cap L^{2}\bigl([0,T],L^{2}[0,1]\bigr), \end{aligned}$$
(2.2)
$$\begin{aligned}& (u,v,w)\in L^{\infty}\bigl([0,T], H^{2}[0,1]\bigr)\cap L^{2}\bigl([0,T],H^{3}[0,1]\bigr), \end{aligned}$$
(2.3)
$$\begin{aligned}& \bigl\Vert (u_{t},v_{t},w_{t}) (t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert (u_{t},v_{t},w_{t}) (s)\bigr\Vert ^{2}_{H^{1}}\,ds\leq C_{2}, \quad \forall t \in[0,T], \end{aligned}$$
(2.4)
where ρ̄ is the same as in Theorem  1.1.

Proof

Estimates (2.1)-(2.4) were obtained in Ref. [1], the proof is complete. □

Lemma 2.2

Under the assumptions in Theorem  1.1, the following estimate holds for any \(T>0\):
$$ \bigl\Vert \rho_{xx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \rho _{xx}(s) \bigr\Vert ^{2}\,ds\leq C_{2},\quad t\in[0,T]. $$
(2.5)

Proof

Differentiating (1.20) with respect to x, exploiting (1.19), we have
$$ \frac{u_{tx}}{r}= \bigl(-\rho^{\gamma}+\alpha \rho^{\alpha+1}(ru)_{x} \bigr)_{xx}+ \biggl(\frac{v^{2}}{r^{2}}-\frac {(\rho^{\alpha})_{x}u}{r} \biggr)_{x} +\frac{u_{t}}{\rho r^{3}}, $$
(2.6)
which gives
$$ \alpha\bigl(\rho^{\alpha-1}\rho_{xx}\bigr)_{t}+\gamma \rho^{\gamma-1}\rho _{xx}=M_{0}(x,t) $$
(2.7)
with
$$\begin{aligned} M_{0}(x,t) =&\frac{2vv_{x}}{r^{2}} -\frac{2v^{2}r_{x}}{r^{3}}-\alpha(\alpha-1) \bigl(2\rho^{\alpha-2}\rho _{x}\rho_{tx}-(\alpha-2) \rho^{\alpha-3}\rho_{t}\rho_{x}^{2} \bigr) \\ & {}-\gamma(\gamma-1)\rho^{\gamma-2}\rho_{x}^{2}- \frac{(\rho^{\alpha })_{xx}u+(\rho^{\alpha})_{x}u}{r}-\frac{(\rho^{\alpha})_{x}u}{\rho r^{3}} -\frac{u_{tx}}{r}+\frac{u_{t}}{\rho r^{3}}. \end{aligned}$$
Multiplying (2.7) by \(\rho^{\alpha-1}\rho_{xx}\), integrating the result over \([0,1]\), we deduce
$$\begin{aligned} &\frac{d}{dt}\bigl\Vert \rho^{\alpha -1}\rho_{xx}(t)\bigr\Vert ^{2}+ \int_{0}^{1}\gamma\rho^{\gamma+\alpha-2}\rho _{xx}^{2}\,dx \\ &\quad \leq C_{1} \int_{0}^{1}\Vert \rho_{xx}\Vert \bigl(\Vert v_{x}\Vert +\bigl\Vert v^{2}\bigr\Vert + \Vert \rho_{x}\rho_{tx}\Vert +\bigl\Vert \rho_{t}\rho_{x}^{2}\bigr\Vert +\bigl\Vert \rho_{x}^{2}\bigr\Vert \\ &\qquad {} +\Vert \rho_{x}\Vert +\Vert u_{tx}\Vert +\Vert u_{t}\Vert \bigr)- \int_{0}^{1}\alpha\frac{u}{r}\rho ^{2\alpha-2}\rho_{xx}^{2}\,dx, \end{aligned}$$
which, by Young’s inequality and the interpolation inequality, implies
$$ \frac{d}{dt}\bigl\Vert \rho^{\alpha-1}\rho_{xx}(t)\bigr\Vert ^{2}\leq C_{1}\bigl(\Vert \rho _{xx} \Vert ^{2}+\Vert \rho_{x}\Vert ^{2}+\Vert u_{x}\Vert _{H^{1}}^{2}+\Vert u_{t} \Vert ^{2}_{H^{1}}\bigr). $$
(2.8)
Integrating (2.8) with respect to t over \([0,T]\), using initial condition (1.26) and Lemma 2.1, we derive
$$\bigl\Vert \rho_{xx}(t)\bigr\Vert ^{2}\leq C_{2}+C_{1} \int_{0}^{t}\bigl\Vert \rho_{xx}(s)\bigr\Vert ^{2}\,ds,\quad \forall t\in[0,T], $$
which, by virtue of Gronwall’s inequality, gives (2.5). The proof is complete. □

Lemma 2.3

Under the assumptions in Theorem  1.1, the following estimates hold for any \(T>0\):
$$\begin{aligned}& \bigl\Vert u_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert u_{ttx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int_{0}^{t}\bigl(\Vert u_{txx}\Vert ^{2}+\Vert v_{txx}\Vert ^{2}\bigr) (s)\,ds, \quad t \in[0,T], \end{aligned}$$
(2.9)
$$\begin{aligned}& \bigl\Vert v_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert v_{ttx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int_{0}^{t}\bigl(\Vert v_{txx}\Vert ^{2}+\Vert u_{txx}\Vert ^{2}\bigr) (s)\,ds,\quad t \in[0,T], \end{aligned}$$
(2.10)
$$\begin{aligned}& \bigl\Vert w_{tt}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert w_{ttx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}, \quad t\in[0,T]. \end{aligned}$$
(2.11)

Proof

We infer from (1.20)-(1.22) and Lemmas 2.1-2.2 that
$$\begin{aligned}& \bigl\Vert u_{t}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.12)
$$\begin{aligned}& \bigl\Vert v_{t}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert +\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}\bigr), \end{aligned}$$
(2.13)
$$\begin{aligned}& \bigl\Vert w_{t}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert w_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \rho_{x}(t)\bigr\Vert \bigr). \end{aligned}$$
(2.14)
Differentiating (1.20)-(1.22) with respect to x, respectively, and exploiting Lemmas 2.1-2.2, we have
$$\begin{aligned}& \bigl\Vert u_{tx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(2.15)
$$\begin{aligned}& \bigl\Vert v_{tx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{x}(t)\bigr\Vert +\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(2.16)
$$\begin{aligned}& \bigl\Vert w_{tx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(2.17)
or
$$\begin{aligned}& \bigl\Vert u_{xxx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{tx}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.18)
$$\begin{aligned}& \bigl\Vert v_{xxx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{x}(t)\bigr\Vert +\bigl\Vert v_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{tx}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.19)
$$\begin{aligned}& \bigl\Vert w_{xxx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{tx}(t) \bigr\Vert \bigr). \end{aligned}$$
(2.20)
Differentiating (1.20)-(1.22) with respect to x twice, respectively, using Lemmas 2.1-2.2, we get
$$\begin{aligned}& \bigl\Vert u_{txx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert u_{x}(t)\bigr\Vert _{H^{3}}\bigr), \end{aligned}$$
(2.21)
$$\begin{aligned}& \bigl\Vert v_{txx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}}\bigr), \end{aligned}$$
(2.22)
$$\begin{aligned}& \bigl\Vert w_{txx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert w_{x}(t)\bigr\Vert _{H^{3}}\bigr), \end{aligned}$$
(2.23)
or
$$\begin{aligned}& \bigl\Vert u_{xxxx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(2.24)
$$\begin{aligned}& \bigl\Vert v_{xxxx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert u_{x}(t) \bigr\Vert _{H^{1}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}\bigr), \end{aligned}$$
(2.25)
$$\begin{aligned}& \bigl\Vert w_{xxxx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert w_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert w_{txx}(t) \bigr\Vert \bigr). \end{aligned}$$
(2.26)
Differentiating (1.20)-(1.22) with respect to t, respectively, we deduce
$$\begin{aligned}& \bigl\Vert u_{tt}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert u_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{t}(t)\bigr\Vert +\bigl\Vert \rho_{x}(t)\bigr\Vert + \bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}\bigr), \end{aligned}$$
(2.27)
$$\begin{aligned}& \bigl\Vert v_{tt}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert v_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert u_{t}(t)\bigr\Vert +\bigl\Vert \rho_{x}(t)\bigr\Vert + \bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}\bigr), \end{aligned}$$
(2.28)
$$\begin{aligned}& \bigl\Vert w_{tt}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert u_{tx}(t)\bigr\Vert _{H^{1}}+\bigl\Vert \rho_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}\bigr). \end{aligned}$$
(2.29)
Now differentiating (1.20) with respect to t twice, multiplying the resulting equation by \((\frac{u}{r})_{tt}\) in \(L^{2}[0,1]\), and using integration by parts and (1.19), we conclude
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{0}^{1} \biggl(\frac{u}{r} \biggr)_{tt}^{2}\,dx =&{-} \int_{0}^{1} \bigl(-\rho^{\gamma}+\alpha \rho^{\alpha +1}(\rho u)_{x} \bigr)_{tt} \biggl( \frac{u}{r} \biggr)_{ttx}\,dx \\ &{}+ \int_{0}^{1}\biggl(\frac{v^{2}-u^{2}}{r^{2}} \biggr)_{tt} \biggl(\frac{u}{r} \biggr)_{tt}\,dx - \int_{0}^{1} \biggl(\frac{(\rho^{\alpha})_{x}u}{r} \biggr)_{tt} \biggl(\frac{u}{r} \biggr)_{tt}\,dx \\ =&A_{1}+A_{2}+A_{3}. \end{aligned}$$
(2.30)
Employing Lemmas 2.1-2.2, (2.27)-(2.28), (1.19), and the interpolation inequality, we get, for any small \(\varepsilon\in(0,1)\),
$$\begin{aligned}& \begin{aligned}[b] A_{1}={}&{-} \int_{0}^{1} \bigl(-\rho^{\gamma}+\alpha \rho^{\alpha+1}(\rho u)_{x} \bigr)_{tt} \biggl( \frac{u}{r} \biggr)_{ttx}\,dx \\ \leq{}&{-}\alpha \int_{0}^{1}\frac{\rho^{\alpha +2}}{r}u_{ttx}^{2} \,dx+\varepsilon\bigl\Vert u_{ttx}(t)\bigr\Vert ^{2}+C_{2} \bigl(\bigl\Vert \rho_{tt}(t)\bigr\Vert _{H^{1}}^{2} +\bigl\Vert \rho_{t}(t)\bigr\Vert _{H^{1}}^{2} \\ &{}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}^{2}+ \bigl\Vert u_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}^{2}\bigr) \\ \leq{}&{-}C_{2}^{-1}\bigl\Vert u_{ttx}(t)\bigr\Vert ^{2}+ C_{2}\bigl(\bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}}^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert u_{t}(t)\bigr\Vert ^{2}_{H^{2}}\bigr), \end{aligned} \end{aligned}$$
(2.31)
$$\begin{aligned}& \begin{aligned}[b] A_{2}={}& \int_{0}^{1}\biggl(\frac{v^{2}-u^{2}}{r^{2}} \biggr)_{tt} \biggl(\frac{u}{r} \biggr)_{tt}\,dx \\ \leq{}& C_{2}\bigl(\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert u_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert v_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert u_{x}(t)\bigr\Vert ^{2}\bigr) \\ \leq{}& C_{2}\bigl(\bigl\Vert u_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \rho_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}\bigr), \end{aligned} \end{aligned}$$
(2.32)
$$\begin{aligned}& \begin{aligned}[b] A_{3}={}&{-} \int_{0}^{1} \biggl(\frac{(\rho^{\alpha})_{x}u}{r} \biggr)_{tt} \biggl(\frac{u}{r} \biggr)_{tt}\,dx \\ \leq{}& C_{2}\bigl(\bigl\Vert u_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}^{2}+ \bigl\Vert \rho_{tx}(t)\bigr\Vert ^{2}+\bigl\Vert \rho_{ttx}(t)\bigr\Vert ^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert ^{2}\bigr) \\ \leq{}& C_{2}\bigl(\bigl\Vert u_{t}(t)\bigr\Vert _{H^{2}}^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert ^{2}+\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}^{2} \bigr). \end{aligned} \end{aligned}$$
(2.33)
Integrating (2.30) with respect to t, applying Lemmas 2.1-2.2, initial condition (1.26), and (2.31)-(2.33), we obtain (2.9).
Differentiating (1.21) with respect to t twice, multiplying the resulting equation by \((\frac{v}{r})_{tt}\) in \(L^{2}[0,1]\), and using integration by parts, we have
$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{0}^{1} \biggl(\frac{v}{r} \biggr)_{tt}^{2}\,dx =&- \int_{0}^{1} \bigl(\rho^{\alpha+1}(\rho v)_{x} \bigr)_{tt} \biggl(\frac{v}{r} \biggr)_{ttx}\,dx - \int_{0}^{1} \biggl(\frac{(\rho^{\alpha})_{x}v}{r} \biggr)_{tt} \biggl(\frac{v}{r} \biggr)_{tt}\,dx \\ =&B_{1}+B_{2}, \end{aligned}$$
(2.34)
where
$$B_{1}(t)=- \int_{0}^{1} \bigl(\rho^{\alpha+1}(\rho v)_{x} \bigr)_{tt} \biggl(\frac{v}{r} \biggr)_{ttx}\,dx,\qquad B_{2}(t)=- \int_{0}^{1} \biggl(\frac{(\rho ^{\alpha})_{x}v}{r}- \frac{2uv}{r^{2}} \biggr)_{tt} \biggl(\frac {v}{r} \biggr)_{tt}\,dx. $$
Using the interpolation inequality, Lemmas 2.1-2.2 and (2.27)-(2.28), we obtain for \(\epsilon\in(0,1)\),
$$\begin{aligned}& \begin{aligned}[b] B_{1}(t)\leq{}&{-} \int_{0}^{1}\frac{\rho^{\alpha+2}}{r}v_{ttx}^{2} \,dx+\epsilon \bigl\Vert v_{ttx}(t)\bigr\Vert ^{2} +C_{2}\bigl(\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert v_{tt}(t)\bigr\Vert ^{2} \\ &{}+\bigl\Vert \rho_{tx}(t)\bigr\Vert ^{2}+\bigl\Vert \rho_{ttx}(t)\bigr\Vert ^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert ^{2}+\bigl\Vert v_{x}(t) \bigr\Vert ^{2}\bigr) \\ \leq{}&{-} \int_{0}^{1}\frac{\rho^{\alpha+2}}{r}v_{ttx}^{2} \,dx+\epsilon\bigl\Vert v_{ttx}(t)\bigr\Vert ^{2} +C_{2}\bigl(\bigl\Vert v_{x}(t)\bigr\Vert ^{2}_{H^{1}}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{1}}^{2} \\ &{}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{2}}^{2}+ \bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}^{2}\bigr), \end{aligned} \end{aligned}$$
(2.35)
$$\begin{aligned}& \begin{aligned}[b] B_{2}(t)\leq{}&C_{2}\bigl(\bigl\Vert v_{tt}(t) \bigr\Vert ^{2}+\bigl\Vert v_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert u_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert \rho_{ttx}(t)\bigr\Vert ^{2} \\ &{}+\bigl\Vert \rho_{tx}(t)\bigr\Vert ^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}^{2} +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}^{2}\bigr) \\ \leq{}& C_{2}\bigl(\bigl\Vert v_{t}(t)\bigr\Vert _{H^{2}}^{2}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{2}}^{2}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}^{2} +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}^{2}\bigr). \end{aligned} \end{aligned}$$
(2.36)
Integrating (2.34) with respect to t, using Lemmas 2.1-2.2 and (2.35)-(2.36), and picking ϵ small enough, we conclude (2.10).
Similarly, differentiating (1.22) with respect to t twice, multiplying the resulting equation by \(w_{tt}\) in \(L^{2}[0,1]\), and using integration by parts, we deduce
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int _{0}^{1}w_{tt}^{2}(x,t) \,dx \\& \quad =- \int _{0}^{1} \bigl(r^{2} \rho^{1+\alpha}w_{x} \bigr)_{tt}w_{ttx}\,dx \\& \quad \leq- \int_{0}^{1}r^{2}\rho^{1+\alpha}w_{ttx}^{2}(x,t) \,dx+C_{2}\bigl(\Vert w_{x}\Vert +\Vert u_{t}w_{x}\Vert +\Vert \rho_{t}w_{x} \Vert +\Vert \rho_{t}w_{tx}\Vert \\& \qquad {}+\Vert \rho_{tt}w_{x}\Vert +\Vert \rho_{t}w_{tx}\Vert \bigr)\Vert w_{ttx}\Vert \\& \quad \leq-C_{1}^{-1}\Vert w_{ttx}\Vert ^{2}+C_{2}\bigl(\bigl\Vert w_{x}(t)\bigr\Vert ^{2}+\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}^{2}+ \bigl\Vert w_{t}(t)\bigr\Vert _{H^{1}}^{2}+\bigl\Vert u_{x}(t)\bigr\Vert ^{2}\bigr). \end{aligned}$$
(2.37)
Integrating (2.37) with respect to t, using Lemmas 2.1-2.2, we derive the estimate (2.11). The proof is complete. □

Lemma 2.4

Under the assumptions in Theorem  1.1, the following estimates hold for any \(T>0\) and \(\varepsilon\in(0,1)\):
$$\begin{aligned}& \bigl\Vert u_{tx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert u_{txx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2}^{-1} \varepsilon^{2} \int_{0}^{t}\bigl\Vert u_{ttx}(s)\bigr\Vert ^{2}\,ds,\quad t\in[0,T], \end{aligned}$$
(2.38)
$$\begin{aligned}& \bigl\Vert v_{tx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert v_{txx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2}^{-1} \varepsilon^{2} \int_{0}^{t}\bigl\Vert v_{ttx}(s)\bigr\Vert ^{2}\,ds, \quad t\in[0,T], \end{aligned}$$
(2.39)
$$\begin{aligned}& \bigl\Vert w_{tx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert w_{txx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}, \quad t\in[0,T]. \end{aligned}$$
(2.40)

Proof

Differentiating (1.20) with respect to t and x, then multiplying the result by \((\frac{u}{r})_{tx}\) in \(L^{2}[0,1]\), and integrating by parts, we deduce that
$$\begin{aligned}& \frac{1}{2}\frac {d}{dt} \int_{0}^{1} \biggl(\frac{u}{r} \biggr)_{tx}^{2}\,dx \\& \quad = \bigl(-\rho^{\gamma}+\alpha\rho^{\alpha+1}(\rho u)_{x} \bigr)_{tx} \biggl(\frac{u}{r} \biggr)_{tx} \bigg|_{0}^{1} - \int_{0}^{1} \bigl(-\rho^{\gamma}+\alpha \rho^{\alpha+1}(\rho u)_{x} \bigr)_{tx} \biggl( \frac{u}{r} \biggr)_{txx}\,dx \\& \qquad {} + \int_{0}^{1} \biggl(\frac{v^{2}-u^{2}}{r^{2}}- \frac{(\rho^{\alpha })_{x}u}{r} \biggr)_{tx} \biggl(\frac{u}{r} \biggr)_{tx}\,dx \\& \quad =D_{0}(x,t)+D_{1}(t)+D_{2}(t), \end{aligned}$$
(2.41)
where
$$\begin{aligned}& D_{0}(x,t)= \bigl(-\rho^{\gamma}+\alpha\rho^{\alpha +1}( \rho u)_{x} \bigr)_{tx} \biggl(\frac{u}{r} \biggr)_{tx} \bigg|_{0}^{1}, \\& D_{1}(t)=- \int_{0}^{1} \bigl(-\rho^{\gamma}+\alpha \rho^{\alpha +1}(\rho u)_{x} \bigr)_{tx} \biggl( \frac{u}{r} \biggr)_{txx}\,dx, \\& D_{2}(t)= \int_{0}^{1} \biggl(\frac{v^{2}-u^{2}}{r^{2}}- \frac{(\rho^{\alpha })_{x}u}{r} \biggr)_{tx} \biggl(\frac{u}{r} \biggr)_{tx}\,dx. \end{aligned}$$
Now employing Lemmas 2.1-2.2 and the interpolation inequality, using Young’s inequality several times, we have, for \(\varepsilon\in(0,1)\),
$$\begin{aligned}& \begin{aligned}[b] D_{0}(x,t)\leq{}&C_{2}\bigl(\Vert \rho_{x} \Vert ^{2}_{L^{\infty}}+\Vert \rho_{x}u_{x} \Vert _{L^{\infty}}+\Vert u_{xx}\Vert _{L^{\infty}} +\bigl\Vert \rho_{x}^{2}u_{x}\bigr\Vert _{L^{\infty}} \\ &{}+\bigl\Vert \rho_{x}u_{x}^{2}\bigr\Vert _{L^{\infty}}+\Vert u_{x}u_{xx}\Vert _{L^{\infty}} + \Vert \rho_{x}u_{xx}\Vert _{L^{\infty}}+\Vert \rho_{x}u_{tx}\Vert _{L^{\infty }} \\ &{}+\bigl\Vert u_{x}^{2}\bigr\Vert _{L^{\infty}}+ \Vert u_{txx}\Vert _{L^{\infty}}\bigr)\Vert u_{tx} \Vert _{L^{\infty}} \\ \leq{}&C_{2}\bigl(\Vert u_{x}\Vert _{H^{2}}+ \Vert \rho_{x}\Vert _{H^{1}} +\Vert u_{tx}\Vert ^{\frac{1}{2}}\Vert u_{txx}\Vert ^{\frac{1}{2}} \\ &{} +\Vert u_{txx}\Vert ^{\frac{1}{2}}\Vert u_{txxx} \Vert ^{\frac{1}{2}}\bigr)\Vert u_{tx}\Vert ^{\frac{1}{2}} \Vert u_{txx}\Vert ^{\frac{1}{2}} \\ \leq{}& C_{2}^{-1}\varepsilon^{2}\bigl(\Vert u_{txx}\Vert ^{2}+\Vert u_{txxx}\Vert ^{2}\bigr)\\ &{}+C_{2}\varepsilon^{-6}\bigl(\Vert u_{tx}\Vert ^{2}+\Vert u_{x}\Vert ^{2}_{H^{2}}+\Vert \rho_{x}\Vert ^{2}_{H^{1}}\bigr), \end{aligned} \end{aligned}$$
(2.42)
$$\begin{aligned}& \begin{aligned}[b] D_{1}(t)\leq{}& {-}\alpha \int_{0}^{1}\rho^{\alpha+1}u_{txx}^{2} \,dx+\varepsilon \bigl\Vert u_{txx}(t)\bigr\Vert ^{2}\\ &{}+C_{2} \bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}^{2}+ \bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}}^{2} \\ &{}+\bigl\Vert u_{tx}(t)\bigr\Vert ^{2}+\bigl\Vert u_{t}(t)\bigr\Vert ^{2}\bigr), \end{aligned} \end{aligned}$$
(2.43)
$$\begin{aligned}& \begin{aligned}[b] D_{2}(t)\leq{}&C_{2}\bigl(\Vert u_{x}u_{t} \Vert +\Vert u_{t}\Vert _{H^{1}}+\Vert v_{t} \Vert _{H^{1}}+\Vert u_{x}\Vert +\Vert v_{x} \Vert +\bigl\Vert u^{3}\bigr\Vert +\bigl\Vert v^{2} \bigr\Vert \\ &{}+\Vert \rho_{tx}\Vert _{H^{1}}+\Vert \rho_{tx}u_{x}\Vert +\Vert \rho_{xx}u_{t} \Vert +\Vert \rho_{x}u_{tx}\Vert +\Vert \rho_{xx}\Vert \\ &{}+\Vert \rho_{x}u_{x}\Vert \bigr) \bigl(\Vert u_{tx}\Vert +\Vert u_{t}\Vert +\Vert u_{x} \Vert +\bigl\Vert u^{2}\bigr\Vert \bigr) \\ \leq{}& C_{2}\bigl(\Vert u_{x}\Vert ^{2}_{H^{2}}+ \Vert \rho_{x}\Vert ^{2}_{H^{1}}+\Vert v_{t}\Vert ^{2}_{H^{1}}+\Vert u_{t} \Vert ^{2}_{H^{1}}+\Vert v_{x}\Vert ^{2}_{H^{1}}\bigr). \end{aligned} \end{aligned}$$
(2.44)
On the other hand, we differentiate (1.20)-(1.22) with respect to x and t, and use Lemmas 2.1-2.2 and (2.12)-(2.29) to conclude
$$\begin{aligned}& \bigl\Vert u_{txxx}(t)\bigr\Vert \leq C_{2}\bigl(\bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{t}(t) \bigr\Vert _{H^{2}}+\bigl\Vert u_{ttx}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.45)
$$\begin{aligned}& \bigl\Vert v_{txxx}(t)\bigr\Vert \leq C_{2}\bigl(\bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{t}(t) \bigr\Vert _{H^{2}}+\bigl\Vert v_{ttx}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.46)
$$\begin{aligned}& \bigl\Vert w_{txxx}(t)\bigr\Vert \leq C_{2}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{tx}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{ttx}(t) \bigr\Vert \bigr). \end{aligned}$$
(2.47)

Integrating (2.41) with respect to t, using (2.42)-(2.45) and Lemmas 2.1-2.2, we obtain (2.38).

Analogously, differentiating (1.21) with respect to t and x, then multiplying the resultant by \((\frac{v}{r})_{tx}\) in \(L^{2}[0,1]\), and integrating by parts, we deduce that
$$\begin{aligned}& \frac{1}{2}\frac {d}{dt} \int_{0}^{1} \biggl(\frac{v}{r} \biggr)_{tx}^{2}\,dx \\& \quad = \bigl(\rho^{\alpha+1}(\rho v)_{x} \bigr)_{tx} \biggl(\frac {v}{r} \biggr)_{tx}\bigg|_{0}^{1} - \int_{0}^{1} \bigl(\rho^{\alpha+1}(\rho v)_{x} \bigr)_{tx} \biggl(\frac {v}{r} \biggr)_{txx}\,dx \\& \qquad {} + \int_{0}^{1} \biggl(\frac{2uv}{r^{2}}- \frac{(\rho^{\alpha })_{x}v}{r} \biggr)_{tx} \biggl(\frac{v}{r} \biggr)_{tx}\,dx \\& \quad =E_{0}(x,t)+E_{1}(t)+E_{2}(t), \end{aligned}$$
(2.48)
where
$$\begin{aligned}& E_{0}(x,t)= \bigl(\rho^{\alpha+1}(\rho v)_{x} \bigr)_{tx} \biggl(\frac{v}{r} \biggr)_{tx} \bigg|_{0}^{1}, \qquad E_{1}(t)=- \int_{0}^{1} \bigl(\rho^{\alpha+1}(\rho v)_{x} \bigr)_{tx} \biggl(\frac{v}{r} \biggr)_{txx}\,dx, \\& E_{2}(t)= \int_{0}^{1} \biggl(\frac{2uv}{r^{2}}- \frac{(\rho^{\alpha })_{x}v}{r} \biggr)_{tx} \biggl(\frac{v}{r} \biggr)_{tx}\,dx. \end{aligned}$$
Now we apply the interpolation inequality and Young’s inequality to estimate \(E_{0}(x,t)\), \(E_{1}(t)\), \(E_{2}(t)\) for any \(\varepsilon>0\),
$$\begin{aligned}& \begin{aligned}[b] E_{0}(x,t)\leq{}& C_{2}\bigl(\Vert v_{txx}\Vert _{L^{\infty}}+\Vert v_{tx}\Vert _{L^{\infty }}+\Vert v_{t}\rho_{x}\Vert _{L^{\infty}}+\Vert v_{t} \Vert _{L^{\infty}}+\Vert u_{xx}\Vert _{L^{\infty}}+\Vert u_{x}v_{x}\Vert _{L^{\infty}} \\ &{} +\Vert v_{xx}\Vert _{L^{\infty}}+\Vert u_{x} \Vert _{L^{\infty}}+\Vert \rho_{x}\Vert _{L^{\infty }}\bigr) \Vert v_{tx}\Vert _{L^{\infty}} \\ \leq{}&C_{2}\bigl(\Vert u_{x}\Vert _{H^{2}}+ \Vert v_{x}\Vert _{H^{2}}+\Vert \rho_{x}\Vert _{H^{1}}+\Vert v_{t}\Vert _{H^{1}} \\ &{} +\Vert v_{tx}\Vert ^{\frac{1}{2}}\Vert v_{txx} \Vert ^{\frac{1}{2}} +\Vert v_{txx}\Vert ^{\frac{1}{2}}\Vert v_{txxx}\Vert ^{\frac{1}{2}}\bigr)\Vert v_{tx}\Vert ^{\frac{1}{2}}\Vert v_{txx}\Vert ^{\frac{1}{2}} \\ \leq{}& C_{2}^{-1}\varepsilon^{2}\bigl(\bigl\Vert v_{txx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{txxx}(t) \bigr\Vert ^{2}\bigr)+C_{2}\varepsilon^{-6}\bigl( \bigl\Vert v_{tx}(t)\bigr\Vert ^{2}+\bigl\Vert u_{x}(t)\bigr\Vert _{H^{2}} \\ &{}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{1}}\bigr), \end{aligned} \end{aligned}$$
(2.49)
$$\begin{aligned}& \begin{aligned}[b] E_{1}(t)\leq{}&{-} \int_{0}^{1}\rho^{\alpha+1}v_{txx}^{2} \,dx+\frac{1}{2} \int _{0}^{1}\rho^{\alpha+1}v_{txx}^{2} \,dx +C_{2}\bigl(\Vert u_{x}\Vert ^{2}_{H^{2}}+ \Vert v_{x}\Vert ^{2}_{H^{1}} \\ &{}+\Vert \rho_{x}\Vert ^{2}_{H^{1}}+\Vert v_{t}\Vert ^{2}+\Vert v_{tx}\Vert ^{2}\bigr), \end{aligned} \end{aligned}$$
(2.50)
$$\begin{aligned}& \begin{aligned}[b] E_{2}(t)\leq{}&C_{2}\bigl(\Vert u_{tx}\Vert + \Vert u_{t}v_{x}\Vert +\Vert u_{x}v_{t} \Vert +\Vert v_{tx}\Vert +\Vert u_{t}\Vert +\Vert v_{t}\Vert +\Vert u_{x}\Vert \\ &{}+\Vert v_{x}\Vert +\Vert \rho_{txx}\Vert +\Vert \rho_{xt}v_{x}\Vert +\Vert \rho_{xx}v_{t} \Vert +\Vert \rho_{x}v_{tx}\Vert +\Vert \rho_{xt}\Vert +\Vert \rho_{x}v_{t}\Vert \\ &{}+\Vert \rho_{xx}\Vert +\Vert \rho_{x}v_{x} \Vert +\Vert \rho_{x}u_{x}\Vert \bigr) \bigl(\Vert v_{tx}\Vert +\Vert v_{t}\Vert +\Vert u_{x} \Vert +\Vert v_{x}\Vert \bigr) \\ \leq{}& C_{2}\bigl(\Vert u_{t}\Vert _{H^{1}}^{2}+ \Vert v_{t}\Vert _{H^{1}}^{2}+\Vert u_{x}\Vert _{H^{1}}^{2}+\Vert \rho_{x} \Vert _{H^{1}}^{2}+\Vert v_{x}\Vert _{H^{1}}^{2}\bigr). \end{aligned} \end{aligned}$$
(2.51)

Inserting (2.46) into (2.49), integrating (2.48) with respect to t, and using (2.49)-(2.51), we can get (2.39).

Differentiating (1.22) with respect to x and t, multiplying the resulting equation by \(w_{tx}\) in \(L^{2}[0,1]\), integrating by parts, we have
$$ \begin{aligned}[b] \frac{1}{2}\frac{d}{dt}\Vert w_{tx}\Vert ^{2}&=\bigl(r^{2}\rho ^{1+\alpha }w_{x} \bigr)_{tx}w_{tx}\big|_{0}^{1}- \int_{0}^{1}\bigl(r^{2} \rho^{1+\alpha }w_{x}\bigr)_{tx}w_{txx}\,dx \\ &=F_{0}(x,t)+F_{1}(t). \end{aligned} $$
(2.52)
Employing Lemmas 2.1-2.2, the interpolation inequality and (2.47), we infer for any \(\varepsilon\in(0,1)\) that
$$\begin{aligned}& \begin{aligned}[b] F_{0}(x,t)={}&\bigl(r^{2}\rho^{1+\alpha}w_{x} \bigr)_{tx}w_{tx}\big|_{0}^{1} \\ \leq{}& C_{2}\bigl(\Vert w_{x}\Vert _{L^{\infty}}+ \Vert w_{x}u_{x}\Vert _{L^{\infty}}+\Vert \rho _{x}w_{x}\Vert _{L^{\infty}}+\Vert w_{xx} \Vert _{L^{\infty}} \\ &{}+\Vert w_{tx}\Vert _{L^{\infty}}+\Vert w_{txx} \Vert _{L^{\infty}}\bigr)\Vert w_{tx}\Vert _{L^{\infty}} \\ \leq{}&\varepsilon^{2}\bigl(\Vert w_{txx}\Vert ^{2}+\Vert w_{txxx}\Vert ^{2} \bigr)+C_{2}\bigl(\Vert w_{x}\Vert _{H^{2}}^{2}+ \Vert \rho_{x}\Vert _{H^{1}}^{2}+\Vert u_{x}\Vert _{H^{1}}^{2}+\Vert w_{tx} \Vert ^{2}\bigr) \\ \leq{}& \varepsilon^{2}\bigl(\Vert w_{txx}\Vert ^{2}+\Vert w_{ttx}\Vert ^{2} \bigr)+C_{2}\bigl(\Vert w_{x}\Vert _{H^{2}}^{2}+ \Vert \rho_{x}\Vert _{H^{1}}^{2}+\Vert u_{x}\Vert _{H^{1}}^{2}+\Vert w_{tx} \Vert ^{2}\bigr), \end{aligned} \end{aligned}$$
(2.53)
$$\begin{aligned}& \begin{aligned}[b] F_{1}(t)={}&{-} \int_{0}^{1}\bigl(r^{2} \rho^{1+\alpha}w_{x}\bigr)_{tx}w_{txx}\,dx \\ \leq{}&{-} \int_{0}^{1}r^{2}\rho^{1+\alpha}w_{txx}^{2} \,dx+C_{2}\bigl(\Vert w_{x}\Vert +\Vert w_{x}u_{x}\Vert +\Vert w_{x}\rho_{x} \Vert +\Vert w_{xx}\Vert \\ &{}+\Vert w_{xt}\Vert +\Vert \rho_{x}w_{tx} \Vert \bigr)\Vert w_{txx}\Vert \\ \leq{}&{-}C_{1}^{-1}\Vert w_{txx}\Vert ^{2}+C_{2}\bigl(\Vert w_{x}\Vert _{H^{1}}^{2}+\Vert w_{t}\Vert _{H^{1}}^{2}+ \Vert \rho_{x}\Vert _{H^{1}}^{2}\bigr). \end{aligned} \end{aligned}$$
(2.54)
Integrating (2.52) with respect to t, picking ε small enough, using Lemmas 2.1-2.3 and (2.53)-(2.54), we derive the estimate (2.40). The proof is complete. □

Lemma 2.5

Under the assumptions in Theorem  1.1, the following estimates hold for any \(T>0\):
$$\begin{aligned}& \bigl\Vert u_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert u_{tx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl(\Vert u_{txx}\Vert ^{2}+\Vert u_{ttx}\Vert ^{2}\bigr) (s)\leq C_{4}, \quad t\in[0,T], \end{aligned}$$
(2.55)
$$\begin{aligned}& \bigl\Vert v_{tt}(t)\bigr\Vert ^{2}+\bigl\Vert v_{tx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl(\Vert v_{ttx}\Vert ^{2}+\Vert v_{txx}\Vert ^{2}\bigr) (s)\leq C_{4},\quad t\in[0,T], \end{aligned}$$
(2.56)
$$\begin{aligned}& \bigl\Vert \rho_{xxx}(t)\bigr\Vert ^{2}+\bigl\Vert u_{xxx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{xxx}(t) \bigr\Vert ^{2}+\bigl\Vert w_{xxx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl(\Vert \rho_{xxx}\Vert ^{2}+\Vert u_{xxxx}\Vert ^{2} \\& \quad {} +\Vert v_{xxxx}\Vert ^{2}+\Vert w_{xxxx} \Vert ^{2}\bigr) (s)\,ds\leq C_{4}, \quad t\in[0,T]. \end{aligned}$$
(2.57)

Proof

Adding (2.38) and (2.39), using (2.9)-(2.10) and picking ε small enough, we easily obtain (2.55)-(2.56).

Differentiating (2.7) with respect to x, we have
$$ \alpha\bigl(\rho ^{\alpha-1}\rho_{xxx}\bigr)_{t}+\gamma \rho^{\gamma-1}\rho _{xxx}=M_{1}(x,t), $$
(2.58)
where
$$M_{1}(x,t)=M_{0x}(x,t)-(\alpha-1) \bigl(\rho^{\alpha-2} \rho_{x}\rho _{xx}\bigr)_{t}-\gamma(\gamma-1) \rho_{x}\rho_{xx}. $$
An easy calculation with the interpolation inequality, Lemmas 2.1-2.2, and (2.55)-(2.56) gives
$$\begin{aligned} \bigl\Vert M_{1}(t)\bigr\Vert \leq{}& C_{2}\bigl(\Vert M_{0x}\Vert +\Vert \rho _{x}\rho_{xx}\Vert + \Vert \rho_{xt}\rho_{xx}\Vert +\Vert \rho_{x} \rho_{txx}\Vert \bigr) \\ \leq{}&C_{2}\bigl(\bigl\Vert v_{x}^{2}\bigr\Vert +\Vert v_{xx}\Vert +\Vert \rho_{x} \rho_{txx}\Vert +\Vert \rho_{xx}\rho _{tx}\Vert +\bigl\Vert \rho_{x}^{2}\rho_{tx}\bigr\Vert + \Vert \rho_{t}\rho_{x}\rho_{xx}\Vert \\ &{}+\Vert \rho_{x}\rho_{xx}\Vert +\Vert \rho_{xxx}\Vert +\Vert \rho_{xx}u_{x}\Vert + \Vert u_{tx}\Vert +\Vert u_{txx}\Vert +\Vert u_{t}\Vert +\Vert u_{t}\rho_{x}\Vert \bigr) \\ \leq{}&C_{2}\bigl(\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{2}}^{2}+\bigl\Vert u_{x}(t)\bigr\Vert ^{2}_{H^{2}}+\bigl\Vert v_{x}(t)\bigr\Vert ^{2}_{H^{1}}+\bigl\Vert u_{t}(t)\bigr\Vert ^{2}_{H^{2}}\bigr), \end{aligned}$$
which, along with Lemmas 2.1-2.2 and (2.55)-(2.56), implies
$$ \int _{0}^{t}\bigl\Vert M_{1}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int_{0}^{t}\bigl\Vert \rho_{xxx}(s)\bigr\Vert ^{2}\,ds,\quad \forall t\in[0,T]. $$
(2.59)
Multiplying (2.58) by \(\rho^{\alpha-1}\rho_{xxx}\) in \(L^{2}[0,1]\), we deduce
$$ \frac{\alpha}{2}\frac{d}{dt}\bigl\Vert \rho^{\alpha-1}\rho _{xxx}\bigr\Vert ^{2}+\gamma \int_{0}^{1}\rho^{\gamma+\alpha-2}\rho_{xxx}^{2} \,dx\leq C_{1}\bigl\Vert M_{1}(t)\bigr\Vert ^{2}, $$
(2.60)
which implies
$$ \frac{d}{dt}\bigl\Vert \rho^{\alpha-1}\rho_{xxx}\bigr\Vert ^{2}\leq C_{1}\bigl\Vert M_{1}(t)\bigr\Vert ^{2}. $$
(2.61)
Integrating (2.61) with respect to t, using (2.59), we conclude
$$\bigl\Vert \rho^{\alpha-1}\rho_{xxx}\bigr\Vert ^{2} \leq C_{4}+C_{2} \int_{0}^{t}\bigl\Vert \rho _{xxx}(s) \bigr\Vert ^{2}\,ds, $$
which, by virtue of Gronwall’s inequality and (2.60), gives
$$ \bigl\Vert \rho _{xxx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \rho_{xxx}(s)\bigr\Vert ^{2}\,ds\leq C_{4},\quad \forall t\in [0,T]. $$
(2.62)
By (2.18)-(2.20), (2.55)-(2.56), and Lemmas 2.1-2.4, we conclude
$$\begin{aligned} \bigl\Vert u_{xxx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{xxx}(t)\bigr\Vert ^{2}+\bigl\Vert w_{xxx}(t) \bigr\Vert ^{2}\leq C_{4}, \quad \forall t\in [0,T]. \end{aligned}$$
(2.63)
By virtue of (2.24)-(2.26), (2.55)-(2.56), (2.62), and Lemmas 2.1-2.4, we can get
$$\int_{0}^{t}\bigl(\Vert u_{xxxx}\Vert ^{2}+\Vert v_{xxxx}\Vert ^{2}+\Vert w_{xxxx}\Vert ^{2}\bigr) (s)\,ds\leq C_{4},\quad t \in[0,T], $$
which, along with (2.62)-(2.63), gives (2.57). The proof is complete. □

Lemma 2.6

Under the assumptions in Theorem  1.1, the following estimates hold for any \(T>0\):
$$\begin{aligned}& \bigl\Vert u_{txx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{txx}(t)\bigr\Vert ^{2}+\bigl\Vert w_{txx}(t) \bigr\Vert ^{2}\leq C_{4},\quad t\in[0,T], \end{aligned}$$
(2.64)
$$\begin{aligned}& \bigl\Vert \rho_{xxxx}(t)\bigr\Vert ^{2}+ \int_{0}^{t}\bigl\Vert \rho_{xxxx}(s)\bigr\Vert ^{2}\,ds\leq C_{4},\quad t\in [0,T], \end{aligned}$$
(2.65)
$$\begin{aligned}& \bigl\Vert u_{xxxx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{xxxx}(t)\bigr\Vert ^{2}+\bigl\Vert w_{xxxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t}\bigl(\Vert u_{xxxxx}\Vert ^{2} \\& \quad {} +\Vert v_{xxxxx}\Vert ^{2}+\Vert w_{xxxxx} \Vert ^{2}\bigr) (s)\,ds\leq C_{4},\quad t\in[0,T]. \end{aligned}$$
(2.66)

Proof

Differentiating (1.20)-(1.22) with respect to t, respectively, we deduce
$$\begin{aligned}& \bigl\Vert u_{txx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert u_{t}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{t}(t)\bigr\Vert +\bigl\Vert \rho_{x}(t)\bigr\Vert + \bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{tt}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.67)
$$\begin{aligned}& \bigl\Vert v_{txx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert v_{t}(t)\bigr\Vert _{H^{1}}+\bigl\Vert u_{t}(t)\bigr\Vert +\bigl\Vert \rho_{x}(t)\bigr\Vert + \bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert v_{tt}(t)\bigr\Vert \bigr), \end{aligned}$$
(2.68)
$$\begin{aligned}& \bigl\Vert w_{txx}(t)\bigr\Vert \leq C_{1}\bigl(\bigl\Vert w_{tx}(t)\bigr\Vert +\bigl\Vert \rho_{x}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{x}(t)\bigr\Vert _{H^{1}}+\bigl\Vert w_{tt}(t) \bigr\Vert \bigr). \end{aligned}$$
(2.69)
By virtue of Lemmas 2.1-2.5 and estimates (2.67)-(2.69), we conclude (2.64).
Differentiating (2.58) with respect to x, we have
$$ \alpha\bigl(\rho ^{\alpha-1}\rho_{xxxx}\bigr)_{t}+\gamma \rho^{\gamma-1}\rho _{xxxx}=M_{2}(x,t), $$
(2.70)
where
$$M_{2}(x,t)=M_{1x}(x,t)-(\alpha-1) \bigl(\rho^{\alpha-2} \rho_{x}\rho _{xxx}\bigr)_{t}-\gamma(\gamma-1) \rho_{x}\rho_{xxx} $$
and
$$M_{1x}(x,t)=M_{0xx}(x,t)-\gamma(\gamma-1) (\rho_{x} \rho _{xx})_{x}-(\alpha-1) \bigl(\rho^{\alpha-2} \rho_{x}\rho_{xx}\bigr)_{tx}. $$
Using the interpolation inequality, and the embedding theorem, Lemmas 2.1-2.5, we can deduce that
$$ \bigl\Vert M_{2}(t)\bigr\Vert \leq C_{4}\bigl(\bigl\Vert u_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \rho_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert u_{tx}(t) \bigr\Vert _{H^{2}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{2}}\bigr). $$
(2.71)
Inserting (2.45) into (2.71), and integrating (2.71) with respect to t over \([0,T]\), using Lemmas 2.1-2.5, we have
$$ \int_{0}^{t}\bigl\Vert M_{2}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int_{0}^{t}\bigl\Vert \rho_{xxxx}(s)\bigr\Vert ^{2}\,ds,\quad \forall t\in [0,T]. $$
(2.72)
Multiplying (2.70) by \(\rho^{\alpha-1}\rho_{xxxx}\) in \(L^{2}[0,1]\), we can get
$$ \frac{\alpha}{2}\frac{d}{dt}\bigl\Vert \rho^{\alpha-1}\rho _{xxxx}\bigr\Vert ^{2}+\gamma \int_{0}^{1}\rho^{\gamma+\alpha-2}\rho _{xxxx}^{2}\,dx\leq C_{2}\bigl\Vert M_{2}(t)\bigr\Vert ^{2}, $$
(2.73)
which implies
$$ \frac{d}{dt}\bigl\Vert \rho^{\alpha-1}\rho_{xxxx}\bigr\Vert ^{2}\leq C_{2}\bigl\Vert M_{2}(t)\bigr\Vert ^{2}. $$
(2.74)
Integrating (2.74) with respect to t over \([0,T]\), using (2.72), we conclude
$$ \bigl\Vert \rho^{\alpha-1}\rho_{xxxx}\bigr\Vert ^{2} \leq C_{4}+C_{2} \int_{0}^{t}\bigl\Vert \rho _{xxxx}(s) \bigr\Vert ^{2}\,ds,\quad t\in[0,T], $$
(2.75)
which, by virtue of Gronwall’s inequality, gives
$$ \bigl\Vert \rho _{xxxx}(t)\bigr\Vert ^{2}\leq C_{4},\quad t\in[0,T]. $$
(2.76)
Thus, we can obtain (2.65) by virtue of (2.75)-(2.76).
By (2.24)-(2.26), (2.64)-(2.65), (2.45)-(2.47), and Lemmas 2.1-2.5, we deduce that
$$\begin{aligned}& \bigl\Vert u_{xxxx}(t)\bigr\Vert ^{2}+\bigl\Vert v_{xxxx}(t)\bigr\Vert ^{2}+\bigl\Vert w_{xxxx}(t) \bigr\Vert ^{2} \\& \quad {}+ \int_{0}^{t}\bigl(\Vert u_{txxx}\Vert ^{2}+\Vert v_{txxx}\Vert ^{2}+\Vert w_{txxx}\Vert ^{2}\bigr) (s)\,ds\leq C_{4},\quad t \in[0,T]. \end{aligned}$$
(2.77)
On the other hand, we differentiate (1.20)-(1.22) with respect to x three times, use Lemmas 2.1-2.5 and (2.64)-(2.65) to conclude, for any \(t\in[0,T]\),
$$\begin{aligned}& \bigl\Vert u_{xxxxx}(t)\bigr\Vert \leq C_{4}\bigl(\bigl\Vert u_{txxx}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \rho _{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}} \bigr), \end{aligned}$$
(2.78)
$$\begin{aligned}& \bigl\Vert v_{xxxxx}(t)\bigr\Vert \leq C_{4}\bigl(\bigl\Vert v_{txxx}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \rho _{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert v_{x}(t)\bigr\Vert _{H^{3}} \bigr), \end{aligned}$$
(2.79)
$$\begin{aligned}& \bigl\Vert w_{xxxxx}(t)\bigr\Vert \leq C_{4}\bigl(\bigl\Vert w_{txxx}(t)\bigr\Vert +\bigl\Vert u_{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert \rho _{x}(t)\bigr\Vert _{H^{3}}+\bigl\Vert w_{x}(t)\bigr\Vert _{H^{3}} \bigr). \end{aligned}$$
(2.80)
Thus we conclude from (2.77)-(2.80), (2.64)-(2.65), and Lemmas 2.1-2.5 that
$$\int_{0}^{t}\bigl(\Vert u_{xxxxx}\Vert ^{2}+\Vert v_{xxxxx}\Vert ^{2}+\Vert w_{xxxxx}\Vert ^{2}\bigr) (s)\,ds\leq C_{4},\quad \forall t\in[0,T] $$
which, combined with (2.77), implies (2.66). The proof is complete. □

Proof of Theorem 1.1.

Applying Lemmas 2.1-2.6, we readily get estimate (1.27)-(1.30) and complete the proof of Theorem 1.1. □

Notes

Acknowledgements

This research was supported in part by NSFC (No. 11501199) and the Natural Science Foundation of Henan Province (No. 14B110037).

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Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceNorth China University of Water Resources and Electric PowerZhengzhouP.R. China

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