1 Introduction

It is well known that the compressible isentropic Navier–Stokes equations which describe the motion of compressible fluids can be written in Eulerian coordinates as

$$\begin{aligned}& \rho_{t}+\mathrm {div}(\rho \mathbf{U})=0, \end{aligned}$$
(1.1)
$$\begin{aligned}& (\rho {\mathbf{U}})_{t}+\mathrm {div}(\rho {\mathbf{U}}\otimes { \mathbf{U}})-\mathrm {div}\bigl( \mu (\rho )D({\mathbf{U}}) \bigr) -{\nabla \bigl(\lambda ( \rho )\mathrm {div}\mathbf{U} \bigr)}+ \nabla {P(\rho )}=0, \end{aligned}$$
(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 the strain tensor is given by

$$ D({\mathbf{U}})= \frac{\nabla {\mathbf{U}}+(\nabla {\mathbf{U}})^{T}}{2}. $$
(1.3)

The Lamé viscosity coefficients \(\mu (\rho )\), \(\lambda (\rho )\) satisfy the natural restrictions

$$ \mu (\rho )>0,\quad\quad \mu (\rho )+N\lambda (\rho )\geq 0. $$
(1.4)

For simplicity of the presentation, we consider only the viscosity terms \(\mu (\rho )=\rho \), \(\lambda (\rho )=0\) and \(D({\mathbf{U}})=\nabla {\mathbf{U}}\). Then (1.1)–(1.2) become

$$\begin{aligned}& \rho_{t}+\mathrm {div}(\rho \mathbf{U})=0, \end{aligned}$$
(1.5)
$$\begin{aligned}& {(\rho \mathbf{U})}_{t}+\mathrm {div}({\rho \mathbf{U}\otimes \mathbf{U})}+ \nabla P(\rho )-{\mathrm {div}(\rho \nabla \mathbf{U})}=0. \end{aligned}$$
(1.6)

We are concerned with the spherically symmetric solutions of system (1.5)–(1.6) in bounded annular domains \(G=\{{\mathbf{x}} \in R^{3}, 0< a< \vert {\mathbf{x}} \vert <b<+\infty \}\). To this end, we denote

$$ { \vert \mathbf{x} \vert }=r, \quad\quad \rho ({\mathbf{x}},t)=\rho (r,t), \quad\quad {\mathbf{U}}( {\mathbf{x}},t)=u(r,t)\frac{\mathbf{x}}{r} , $$
(1.7)

which leads to the following system of equations for \(r>0\):

$$\begin{aligned}& \rho_{t}+(\rho u)_{r}+\frac{2\rho u}{r}=0, \end{aligned}$$
(1.8)
$$\begin{aligned}& (\rho u)_{t}+ \bigl(\rho u^{2}+\rho^{\gamma } \bigr)_{r}+\frac{2\rho u^{2}}{r}-( \rho u)_{r}-\rho \biggl( \frac{2u}{r} \biggr)_{r}=0. \end{aligned}$$
(1.9)

We shall consider problem (1.8)–(1.9) in the region G subject to the initial data

$$ (\rho , u) (r,0)=(\rho_{0},u_{0}) (r), \quad r\in [a,b], $$
(1.10)

and the boundary condition

$$ u(a,t)=u(b,t)=0, \quad t\in [0,T]. $$
(1.11)

First we find it convenient to transfer problems (1.8)–(1.11) into that in Lagrangian coordinates and draw the desired results. We introduce the following coordinate transformation:

$$ x(r,t)= \int_{a}^{r}\rho (s,\tau ) s^{2}\,ds,\quad t=\tau , $$
(1.12)

then the boundaries \(r=a\) and \(r=b\) become

$$ x=0,\quad\quad x= \int_{a}^{b}\rho (s,\tau ) s^{2}\,ds= \int_{a}^{b}\rho_{0}(s)s ^{2} \,ds , $$
(1.13)

where \(\int_{a}^{b}\rho_{0}(s)s^{2}\,ds\) is the total initial mass and, without loss of generality, we can normalize it to 1. So in terms of Lagrangian coordinates, the domain G becomes \(\Omega =(0,1)\). The relations between Lagrangian and Eulerian coordinates are satisfied by

$$ \frac{\partial x}{\partial r}=\rho r^{2},\quad\quad \frac{\partial x}{ \partial t}=\rho u r^{2}. $$
(1.14)

The initial boundary value problem (1.8)–(1.11) are changed to

$$\begin{aligned}& \rho_{t}+\rho^{2} \bigl(r^{2}u \bigr)_{x}=0, \end{aligned}$$
(1.15)
$$\begin{aligned}& \frac{u_{t}}{r^{2}}+ \bigl(\rho^{\gamma } \bigr)_{x}= \bigl( \rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr)_{x}- \frac{2 \rho_{x}u}{r}, \end{aligned}$$
(1.16)
$$\begin{aligned}& (\rho ,u) (x,0)=(\rho_{0},u_{0}) (x),\quad x\in [0,1], \end{aligned}$$
(1.17)
$$\begin{aligned}& u(0,t)=u(1,t)=0,\quad t>0. \end{aligned}$$
(1.18)

Much progress was achieved recently on the compressible Navier–Stokes equations with density-dependent viscosity coefficient. Firstly let us recall some well-known results as regards the one-dimensional compressible isentropic Navier–Stokes equations with the flow density being connected with the infinite vacuum, [19, 24, 25] for the local well-posedness and the global existence of weak solutions to an initial boundary value problem with the viscous gas being connected vacuum states with jump discontinuities, [9, 10] for the global behavior with the initial density being piecewise smooth; [28–31, 34] for the local existence, the global existence, the asymptotic behavior and the uniqueness of weak solutions with a viscous gas being connected vacuum states with continuous density.

In spatial multi-dimension, there is a huge literature as regards the global existence, the regularity and the asymptotic behavior of a solution to system (1.1)–(1.2) with constant viscosity, we refer the reader to [2, 3, 11–14, 16, 20–23, 33] and the references therein. For the 3-D flow of a compressible fluid with cylindrical symmetry, the global existence and the large-time behavior of generalized solutions have been proved in [1, 4, 5, 7, 15, 23, 26, 27, 32] for the isentropic and the nonisentropic cases. The corresponding study of the regularity of a solution for any given initial datum has been carried out in [17]. For the 3-D flow of compressible fluid with spherical symmetry, there are some interesting results, [18] for the global well-posedness of classical solutions with large oscillations and vacuum; [33] for the global existence and uniqueness of the weak solution without a solid core; [14] for the structure of the solution; [21] for the global existence of the exterior problem and the initial boundary value problem. Besides, we would like to refer to [6, 8] as regards the existence and regularity of solutions for micropolar fluid with spherical symmetry in the three-dimensional case.

In the paper, we shall analyze the spherical symmetric model and focus on the initial boundary problem of an isentropic compressible fluid. We show the regularity in \(H^{2}\) and \(H^{4}\) under certain assumptions imposed on the initial data.

The notation in this paper will be as follows:

  • \(L^{p}\), \(1\leq p\leq +\infty \), \(W^{m,p}\), \(m\in N\), \(H^{1}=W^{1,2}\), \(H_{0} ^{1}=W_{0}^{1,2}\) denote the usual (Sobolev) spaces on \([0, 1]\). To denote various constants, we use \(C_{i}\) (\(i=1,2,4\)) to denote the generic positive constant depending only on the \(H^{i}\) norm of initial datum \((\rho_{0},u_{0})\), \({\min_{x\in [0,1]}\rho_{0}(x)}\) and variable t, respectively. In addition, \(\Vert \cdot \Vert \) denotes the norm in the space \(L^{2}\).

The basic assumption of this paper is the following:

$$ \inf_{[0,1]}\rho_{0}>\underline{\rho }, $$
(1.19)

for some constant \(\underline{\rho }>0\).

Theorem 1.1

Let \(\gamma >1\). Assume that the initial data \((\rho_{0},u_{0})\in H ^{2}(\Omega )\times H^{2}(\Omega )\) and (1.19) hold, then there exists a unique generalized global solution \((\rho (t),u(t))\in (H ^{2}(\Omega ))^{2}\) to the problem (1.15)–(1.18) verifying that, for any \(T>0\),

$$\begin{aligned}& \rho \in L^{\infty } \bigl([0,T], H^{2}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{2}( \Omega ) \bigr), \end{aligned}$$
(1.20)
$$\begin{aligned}& u\in L^{\infty } \bigl([0,T], H^{2}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{3}( \Omega ) \bigr), \end{aligned}$$
(1.21)
$$\begin{aligned}& u_{t}\in L^{\infty } \bigl([0,T], L^{2}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{1}( \Omega ) \bigr). \end{aligned}$$
(1.22)

Theorem 1.2

Let \(\gamma >1\). Assume that the initial data satisfies (1.19) and \((\rho_{0},u_{0})\in H^{4}(\Omega )\times H^{4}(\Omega )\), then there exists a unique generalized global solution \((\rho (t),u(t))\in (H ^{4}(\Omega ))^{2}\) to the problem (1.15)–(1.18) verifying that, for any \(T>0\),

$$\begin{aligned}& \rho \in L^{\infty } \bigl([0,T], H^{4}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{4}( \Omega ) \bigr), \end{aligned}$$
(1.23)
$$\begin{aligned}& u\in L^{\infty } \bigl([0,T], H^{4}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{4}( \Omega ) \bigr), \end{aligned}$$
(1.24)
$$\begin{aligned}& u_{t}\in L^{\infty } \bigl([0,T], H^{2}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{3}( \Omega ) \bigr), \end{aligned}$$
(1.25)
$$\begin{aligned}& u_{tt}\in L^{\infty } \bigl([0,T], L^{2}(\Omega ) \bigr)\cap L^{2} \bigl([0,T], H^{1}( \Omega ) \bigr). \end{aligned}$$
(1.26)

Corollary 1.1

Under assumptions of Theorem 1.2, (1.23)–(1.24) imply \((\rho (t),u(t))\) is the classical solution verifying that, for any \(t>0\),

$$ \bigl\Vert \rho (t) \bigr\Vert _{C^{3+1/2}}+ \bigl\Vert u(t) \bigr\Vert _{C^{3+1/2}}\leq C_{4}. $$
(1.27)

2 Proof of Theorem 1.1

This section is devoted to deriving the estimates of the solutions to prove Theorem 1.1 which will be presented in a sequence of lemmas. We begin with the following lemma.

Lemma 2.1

(Theorem 2.2 in [21])

Under the assumptions in Theorem 1.1, then there exist positive constants \(\rho_{*}>0\) and \(\rho^{*}>0\) so that the unique global solution \((\rho (t),u(t))\) to problem (1.15)–(1.18) exists and satisfies, for any \(T>0\),

$$\begin{aligned}& 0< \rho_{*}\leq \rho (x,t)\leq \rho^{*}, \end{aligned}$$
(2.1)
$$\begin{aligned}& \int_{0}^{1} \bigl( u^{2}+(\rho -\bar{ \rho })^{2}+u_{x}^{2}+ u_{t} ^{2}+\rho_{x}^{2} \bigr) (x,t)\,dx+ \int_{0}^{t} \int_{0}^{1} \bigl(\rho_{x}^{2}+u _{x}^{2} \\& \quad{} +u_{t}^{2}+u_{xx}^{2}+u_{xt}^{2} \bigr) (x,s)\,dx\,ds\leq C_{1} ,\quad t\in [0,T], \end{aligned}$$
(2.2)

where \(\bar{\rho }=\frac{1}{b-a}\int_{a}^{b}\rho (s,t)s^{2}\,ds\).

Lemma 2.2

Under the conditions in Theorem 1.1, for any \(T>0\),

$$\begin{aligned}& \bigl\Vert \rho_{xx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert \rho_{xx}(s) \bigr\Vert ^{2}\,ds\leq C_{2}, \quad t\in [0,T], \end{aligned}$$
(2.3)
$$\begin{aligned}& \bigl\Vert u_{xx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \int_{0}^{1}u_{xxx}^{2}(x,s)\,dx \,ds \leq C_{2},\quad t\in [0,T]. \end{aligned}$$
(2.4)

Proof

We infer from (1.16) that

$$ \frac{u_{t}}{r^{2}}=-\gamma \rho^{\gamma -1}\rho_{x}+2\rho \rho_{x} \bigl(r ^{2}u \bigr)_{x}+ \rho^{2} \bigl(r^{2}u_{xx}+4rr_{x}u_{x}+2r_{x}^{2}u+2rr_{xx}u \bigr)-\frac{2 \rho_{x}u}{r}. $$
(2.5)

Multiplying (2.5) by \(u_{xx}\) in \(L^{2}(\Omega )\), we deduce

$$ \begin{aligned}[b] \int_{0}^{1}\rho^{2} r^{2}u_{xx}^{2} \,dx & = \int_{0}^{1}u_{xx} \biggl( \frac{u_{t}}{r ^{2}}+\gamma \rho^{\gamma -1}\rho_{x}-2\rho \rho_{x} \bigl(r^{2}u \bigr)_{x} \\ &\quad{} -\rho ^{2} \bigl(4rr_{x}u_{x}+2r_{x}^{2}u+2rr_{xx}u \bigr)+\frac{2\rho_{x}u}{r} \biggr)\,dx. \end{aligned} $$
(2.6)

Using Young’s inequality, Sobolev’s embedding theorem, and Lemma 2.1, we deduce from (2.6) that

$$\begin{aligned} \int_{0}^{1}u_{xx}^{2}\,dx \leq & C_{1} \int_{0}^{1} \bigl(u_{t}^{2}+ \rho_{x} ^{2}+\rho_{x}^{2}u_{x}^{2} \bigr)\,dx+\frac{1}{4} \int_{0}^{1}u_{xx}^{2}\,dx \\ \leq & C_{2}+C_{1} \Vert u_{x} \Vert ^{2}_{L^{\infty }} \int_{0}^{1}\rho_{x} ^{2}+ \frac{1}{4} \int_{0}^{1}u_{xx}^{2}\,dx \\ \leq & C_{2}+\frac{1}{2} \int_{0}^{1}u_{xx}^{2}\,dx, \end{aligned}$$

whence

$$ \int_{0}^{1}u_{xx}^{2}\,dx\leq C_{2}. $$
(2.7)

Differentiating (1.16) with respect to x, and exploiting (1.15), we have

$$ \biggl( \frac{u_{t}}{r^{2}} \biggr) _{x}= \bigl( -\rho^{\gamma }+ \rho^{2} \bigl(r ^{2}u \bigr)_{x} \bigr) _{xx}- \biggl( \frac{2\rho_{x}u}{r} \biggr) _{x} , $$
(2.8)

which gives

$$ \rho_{txx}+\gamma \rho^{\gamma -1}\rho_{xx}=E_{0}(x,t) $$
(2.9)

with

$$\begin{aligned}& E_{0}(x,t)=\frac{2\rho_{x}ur_{x}}{r^{2}}-\frac{2\rho_{xx}u+2\rho_{x}u _{x}}{r} -\gamma (\gamma -1) \rho^{\gamma -2}\rho_{x}^{2}+\frac{2u_{t}r _{x}}{r^{3}}- \frac{u_{tx}}{r^{2}}. \end{aligned}$$

Multiplying (2.9) by \(\rho_{xx}\), integrating the resultant over \([0,1]\), we deduce

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert \rho_{xx}(t) \bigr\Vert ^{2}+ \int_{0}^{1}\gamma \rho^{\gamma -1} \rho_{xx}^{2}\,dx \\& \quad = \int_{0}^{1} \biggl( \frac{2\rho_{x}ur_{x}}{r^{2}}- \frac{2\rho_{xx}u+2 \rho_{x}u_{x}}{r} -\gamma (\gamma -1)\rho^{\gamma -2}\rho_{x}^{2}+ \frac{2u _{t}r_{x}}{r^{3}}-\frac{u_{tx}}{r^{2}} \biggr) \rho_{xx}\,dx \\& \quad \leq C_{1} \Vert \rho_{xx} \Vert \bigl( \Vert u_{tx} \Vert + \Vert u_{t}r_{x} \Vert + \bigl\Vert \rho_{x}^{2} \bigr\Vert + \Vert \rho_{x}u_{x} \Vert + \Vert \rho_{x}ur_{x} \Vert \bigr)-2 \int_{0}^{1}\frac{u}{r}\rho_{xx} ^{2}\,dx. \end{aligned}$$

Using the Young inequality and the interpolation inequality to the above inequality, then we get

$$ \begin{aligned}[b] \frac{d}{dt} \bigl\Vert \rho_{xx}(t) \bigr\Vert ^{2}&\leq C_{1} \bigl( \Vert u_{tx} \Vert ^{2}+ \Vert u_{t} \Vert ^{2}+ \Vert \rho_{x} \Vert ^{2}+ \Vert u_{x} \Vert ^{2}+ \Vert u \Vert ^{2} \bigr) \\ &\quad{} +C_{1} \Vert \rho_{xx} \Vert ^{2} \bigl(1+ \Vert u \Vert ^{2}_{L^{\infty }} \bigr). \end{aligned} $$
(2.10)

Integrating (2.10) with respect to t, using initial condition \(\rho_{0}\in H^{2}\) and Lemma 2.1, we have

$$\begin{aligned}& \bigl\Vert \rho_{xx}(t) \bigr\Vert ^{2}\leq C_{2}+C_{1} \int_{0}^{t} \bigl\Vert \rho_{xx}(s) \bigr\Vert ^{2}\,ds, \quad \forall t\in [0,T]. \end{aligned}$$

Then using the Gronwall inequality to the above inequality, we can get (2.3).

Differentiating (1.16) with respect to x, we can obtain

$$\begin{aligned} u_{xxx} =&-\frac{1}{\rho^{2}r^{2}} \biggl[(4ru\rho_{x}^{2}r_{x}+2r^{2} \rho_{x}^{2}u_{x}+2\rho \rho_{xx} \bigl(2rr_{x}u+r^{2}u_{x} \bigr)+4\rho \rho_{x} \bigl(2r _{x}^{2}u \\ & {} +4rr_{x}u_{x}+2rr_{xx}u+r^{2}u_{xx} \bigr) +\rho^{2} \bigl(6r_{x}r_{xx}u+6r_{x} ^{2}u_{x}+6rr_{xx}u+r^{2}u_{xx} \bigr) \\ & {} +\rho^{2} \bigl(6r_{x}r_{xx}u+6r_{x}^{2}u_{x}+6rr_{xx}u_{x}+6rr_{x}u_{xx}+2rr _{xxx}u \bigr)+\frac{2\rho_{xx}u+2\rho_{x}u_{x}}{r} \\ & {} -\frac{2\rho_{x}ur_{x}}{r^{2}}+\frac{u_{tx}}{r^{2}}-\frac{u_{t}r _{x}}{r^{3}} +\gamma \rho^{\gamma -1}\rho_{xx}+\gamma (\gamma -1) \rho^{\gamma -2} \rho_{x}^{2} \biggr]. \end{aligned}$$
(2.11)

Integrating (2.11) over x and t, applying the embedding theorem, Lemma 2.1, (2.7) and (2.3), we conclude, for any \(t\in [0,T]\),

$$\begin{aligned} \int_{0}^{t} \int_{0}^{1}u_{xxx}^{2}\,dx\,ds \leq & C_{1} \int_{0}^{t} \int _{0}^{1} \bigl(u_{tx}^{2}+ \rho_{x}^{4}+\rho_{xx}^{2}+ \rho_{x}^{4}u_{x}^{2}+ \rho_{x}^{2}u_{x}^{2} + \rho_{xx}^{2}u_{x}^{2}+u_{t}^{2} \bigr) (x,s)\,dx\,ds \\ \leq &C_{1}+C_{1} \int_{0}^{t} \Vert u_{x} \Vert _{L^{\infty }}^{2} \int_{0}^{1} \bigl( \rho_{xx}^{2}+ \rho_{x}^{4} \bigr)\,dx\,ds+C_{1} \int_{0}^{t} \Vert \rho_{x} \Vert _{L^{ \infty }}^{2} \Vert u_{xx} \Vert ^{2}\,ds \\ \leq & C_{2}, \end{aligned}$$

which, along with (2.7), gives (2.4). The proof is complete. □

Proof of Theorem 1.1

Clearly, (1.20)–(1.22) follow from Lemmas 2.1–2.2. This completes the proof of Theorem 1.1. □

3 Proof of Theorem 1.2

In this section, we shall complete the proof of Theorem 1.2. To this end, we assume that in this section that all assumptions in Theorem 1.2 hold. We begin with the following lemma.

Lemma 3.1

The following estimate holds for any \(T>0\):

$$ \bigl\Vert u_{tt}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert u_{ttx}(s)\bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int _{0}^{t} \bigl\Vert u_{txx}(s) \bigr\Vert ^{2}\,ds, \quad t\in [0,T]. $$
(3.1)

Proof

We easily infer from (1.16) and (2.1)–(2.4) that

$$ \bigl\Vert u_{t}(t) \bigr\Vert \leq C_{1} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert + \bigl\Vert u_{x}(t) \bigr\Vert + \bigl\Vert u_{xx}(t) \bigr\Vert \bigr). $$
(3.2)

Differentiating (1.16) with respect to x and using Lemmas 2.1–2.2, we have

$$ \bigl\Vert u_{tx}(t) \bigr\Vert \leq C_{2} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert + \bigl\Vert \rho_{xx}(t) \bigr\Vert + \bigl\Vert u_{x}(t) \bigr\Vert + \bigl\Vert u_{xx}(t) \bigr\Vert + \bigl\Vert u_{xxx}(t) \bigr\Vert \bigr), $$
(3.3)

or

$$ \bigl\Vert u_{xxx}(t) \bigr\Vert \leq C_{2} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert + \bigl\Vert \rho_{xx}(t) \bigr\Vert + \bigl\Vert u_{x}(t) \bigr\Vert + \bigl\Vert u_{xx}(t) \bigr\Vert + \bigl\Vert u_{tx}(t) \bigr\Vert \bigr). $$
(3.4)

Differentiating (1.16) with respect to x twice and using the Cauchy–Schwarz inequality, we have

$$\begin{aligned}& \bigl\Vert u_{txx}(t) \bigr\Vert \leq C_{2} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{2}}+ \bigl\Vert u_{x}(t) \bigr\Vert _{H ^{3}} \bigr), \end{aligned}$$
(3.5)
$$\begin{aligned}& \bigl\Vert u_{xxxx}(t) \bigr\Vert \leq C_{2} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{2}}+ \bigl\Vert u_{x}(t) \bigr\Vert _{H ^{2}}+ \bigl\Vert u_{txx}(t) \bigr\Vert \bigr). \end{aligned}$$
(3.6)

After differentiating (1.16) with respect to t, using (3.2)–(3.3) and (3.5), we can get

$$\begin{aligned} \bigl\Vert u_{tt}(t) \bigr\Vert \leq & C_{2} \bigl( \bigl\Vert u_{tx}(t) \bigr\Vert + \bigl\Vert u_{txx}(t) \bigr\Vert + \bigl\Vert u_{t}(t) \bigr\Vert + \bigl\Vert \rho_{x}(t) \bigr\Vert + \bigl\Vert u_{x}(t) \bigr\Vert + \bigl\Vert u_{xx}(t) \bigr\Vert \bigr) \end{aligned}$$
(3.7)
$$\begin{aligned} \leq & C_{2} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{2}}+ \bigl\Vert u_{x}(t) \bigr\Vert _{H^{3}} \bigr). \end{aligned}$$
(3.8)

Now differentiating (1.16) with respect to t twice, multiplying the resulting equation by \((\frac{u}{r^{2}})_{tt}\) in \(L^{2}(\Omega )\), and using integration by parts and (1.18), we conclude

$$\begin{aligned} \frac{1}{2}\frac{d}{dt} \int_{0}^{1} \biggl( \frac{u}{r^{2}} \biggr) _{tt} ^{2}\,dx =&- \int_{0}^{1} \bigl( -\rho^{\gamma }+ \rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr) _{tt} \biggl( \frac{u}{r^{2}} \biggr) _{ttx}\,dx \\ & {}- \int_{0}^{1} \biggl( \frac{2\rho_{x}u}{r}- \frac{u^{2}}{r^{3}} \biggr) _{tt} \biggl( \frac{u}{r^{2}} \biggr) _{tt}\,dx=:A_{1}+A_{2}. \end{aligned}$$
(3.9)

We use (2.1)–(2.4), (3.2)–(3.8), (1.16) and the interpolation inequality to deduce that, for any small \(\varepsilon \in (0,1)\),

$$\begin{aligned}& A_{1} =- \int_{0}^{1} \bigl( -\rho^{\gamma }+ \rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr) _{tt} \biggl( \frac{u}{r^{2}} \biggr) _{ttx}\,dx \\& \hphantom{ A_{1}}\leq - \int_{0}^{1}\rho^{2}u_{ttx}^{2} \,dx+C_{2} \bigl( \Vert \rho_{tt}u_{x} \Vert + \bigl\Vert \rho_{t}^{2}u_{x} \bigr\Vert + \Vert \rho_{t}u_{t} \Vert + \Vert \rho_{t}u_{x} \Vert + \Vert \rho_{t}u _{xt} \Vert + \Vert u_{x} \Vert + \Vert u_{t} \Vert \\& \hphantom{ A_{1}}\quad{} + \Vert u_{x}u_{t} \Vert + \Vert u_{tx} \Vert + \Vert u_{tt} \Vert \bigr) \bigl( \Vert u_{ttx} \Vert + \Vert u_{tt} \Vert + \Vert u _{x}u_{t} \Vert + \Vert u_{tx} \Vert + \Vert u_{t} \Vert + \Vert u_{x} \Vert + \bigl\Vert u^{3} \bigr\Vert \bigr) \\& \hphantom{ A_{1}}\leq - \int_{0}^{1}\rho^{2}u_{ttx}^{2} \,dx +\varepsilon \bigl\Vert u_{ttx}(t) \bigr\Vert ^{2}+C_{2} \bigl( \bigl\Vert \rho_{tt}(t) \bigr\Vert _{H^{1}}^{2} + \bigl\Vert \rho_{t}(t) \bigr\Vert ^{2} \\& \hphantom{ A_{1}}\quad{} + \bigl\Vert u_{t}(t) \bigr\Vert _{H^{1}}^{2}+ \bigl\Vert u_{tt}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{x}(t) \bigr\Vert _{H^{1}} ^{2}+ \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{1}}^{2} \bigr) \\& \hphantom{ A_{1}}\leq -C_{2}^{-1} \bigl\Vert u_{ttx}(t) \bigr\Vert ^{2}+ C_{2} \bigl( \bigl\Vert u_{x}(t) \bigr\Vert _{H^{2}}^{2}+ \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{1}}^{2}+ \bigl\Vert u_{t}(t) \bigr\Vert ^{2}_{H^{2}} \bigr), \end{aligned}$$
(3.10)
$$\begin{aligned}& A_{2} =- \int_{0}^{1} \biggl( \frac{2\rho_{x}u}{r}- \frac{u^{2}}{r^{3}} \biggr) _{tt} \biggl( \frac{u}{r^{2}} \biggr) _{tt}\,dx \\& \hphantom{ A_{2}}\leq C_{2} \bigl( \Vert \rho_{ttx} \Vert + \Vert \rho_{tx}u_{t} \Vert + \Vert \rho_{x}u_{tt} \Vert + \bigl\Vert \rho_{tx}u^{2} \bigr\Vert + \bigl\Vert u_{t}^{2} \bigr\Vert + \Vert u_{tt} \Vert + \Vert u_{t} \Vert + \Vert \rho_{xt}u \Vert \\& \hphantom{ A_{2}}\quad{} + \Vert \rho_{x}u_{t} \Vert + \bigl\Vert \rho_{x}u^{3} \bigr\Vert + \Vert u_{t} \Vert + \bigl\Vert u^{3} \bigr\Vert \bigr) \bigl( \Vert u_{tt} \Vert + \Vert u_{t} \Vert + \Vert u_{t}r_{t} \Vert + \Vert ur_{tt} \Vert + \bigl\Vert ur_{t}^{2} \bigr\Vert \bigr) \\& \hphantom{ A_{2}}\leq C_{2} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{tt}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{txx}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{tx}(t) \bigr\Vert ^{2}+ \bigl\Vert \rho_{x}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{x}(t) \bigr\Vert ^{2} \bigr). \end{aligned}$$
(3.11)

Integrating (3.9) with respect to t, and using (2.1)–(2.4), initial condition (1.16) and (3.10)–(3.11), then we obtain (3.1). □

Lemma 3.2

For any \(T>0\) and \(\varepsilon \in (0,1)\), the following estimate holds:

$$ \bigl\Vert u_{tx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert u_{txx}(s) \bigr\Vert ^{2}\leq C_{4}+C_{2}^{-1} \varepsilon^{2} \int_{0}^{t} \bigl\Vert u_{ttx}(s) \bigr\Vert ^{2}\,ds, \quad t\in [0,T]. $$
(3.12)

Proof

Differentiating (1.16) with respect to t and x, then multiplying the resultant by \((\frac{u}{r^{2}})_{tx}\) in \(L^{2}[0,1]\), and integrating by parts, we know that

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{0}^{1} \biggl( \frac{u}{r} \biggr) _{tx} ^{2}\,dx \\& \quad = \bigl( -\rho^{\gamma }+\rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr) _{tx} \biggl( \frac{u}{r ^{2}} \biggr) _{tx}\bigg\vert _{0}^{1} - \int_{0}^{1} \bigl( -\rho^{\gamma }+ \rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr) _{tx} \biggl( \frac{u}{r^{2}} \biggr) _{txx}\,dx \\& \qquad {} + \int_{0}^{1} \biggl( \frac{u^{2}}{r^{3}}- \frac{2\rho_{x}u}{r} \biggr) _{tx} \biggl( \frac{u}{r^{2}} \biggr) _{tx}\,dx \\& \quad =B_{0}(x,t)+B_{1}(t)+B_{2}(t), \end{aligned}$$
(3.13)

where

$$\begin{aligned}& B_{0}(x,t)= \bigl( -\rho^{\gamma }+\rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr) _{tx} \biggl( \frac{u}{r^{2}} \biggr) _{tx}\bigg\vert _{0}^{1}, \\& B_{1}(t)=- \int_{0}^{1} \bigl( -\rho^{\gamma }+ \rho^{2} \bigl(r^{2}u \bigr)_{x} \bigr) _{tx} \biggl( \frac{u}{r^{2}} \biggr) _{txx}\,dx, \\& B_{2}(t)= \int_{0}^{1} \biggl( \frac{u^{2}}{r^{3}}- \frac{2\rho_{x}u}{r} \biggr) _{tx} \biggl( \frac{u}{r^{2}} \biggr) _{tx}\,dx. \end{aligned}$$

Now using Young’s inequality several times, and employing (2.1)–(2.4) and the interpolation inequality, after some calculation, we have, for any \(\varepsilon \in (0,1)\),

$$\begin{aligned}& B_{0}(x,t) \leq C_{2} \bigl( \Vert \rho_{tx} \Vert _{L^{\infty }}+ \Vert \rho_{t}\rho_{x} \Vert _{L^{\infty }}+ \Vert \rho_{x}\rho_{t}u_{x} \Vert _{L^{\infty }} + \Vert \rho_{tx}u _{x} \Vert _{L^{\infty }}+ \Vert \rho_{t}u_{xx} \Vert _{L^{\infty }} \\& \hphantom{B_{0}(x,t)}\quad{} + \Vert u_{txx} \Vert _{L^{\infty }}+ \Vert u_{tx} \Vert _{L^{\infty }} \bigr) \Vert u_{tx} \Vert _{L ^{\infty }} \\& \hphantom{B_{0}(x,t)}\leq C_{2} \bigl( \Vert u_{x} \Vert _{H^{2}}+ \Vert \rho_{x} \Vert _{H^{1}} + \Vert u_{tx} \Vert ^{ \frac{1}{2}} \Vert u_{txx} \Vert ^{\frac{1}{2}} + \Vert u_{txx} \Vert ^{\frac{1}{2}} \Vert u _{txxx} \Vert ^{\frac{1}{2}} \bigr) \Vert u_{tx} \Vert ^{\frac{1}{2}} \Vert u_{txx} \Vert ^{ \frac{1}{2}} \\& \hphantom{B_{0}(x,t)}\leq C_{2}^{-1}\varepsilon^{2} \bigl( \Vert u_{txx} \Vert ^{2}+ \Vert u_{txxx} \Vert ^{2} \bigr)+C _{2}\varepsilon^{-6} \bigl( \Vert u_{tx} \Vert ^{2}+ \Vert u_{x} \Vert _{H^{2}}+ \Vert \rho_{x} \Vert _{H^{1}} \bigr), \end{aligned}$$
(3.14)
$$\begin{aligned}& B_{1}(t) \leq - \int_{0}^{1}\rho^{2}u_{txx}^{2} \,dx+ \varepsilon \int _{0}^{1}\rho^{2}u_{txx}^{2} \,dx+C_{2} \bigl( \bigl\Vert \rho_{tx}(t) \bigr\Vert _{H^{1}}^{2}+ \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{1}}^{2} \\& \hphantom{B_{1}(t)}\quad {} + \bigl\Vert u_{x}(t) \bigr\Vert _{H^{1}}^{2}+ \bigl\Vert u_{t}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{tx}(t) \bigr\Vert ^{2} \bigr), \end{aligned}$$
(3.15)
$$\begin{aligned}& B_{2}(t) \leq C_{2} \bigl( \Vert u_{x}u_{t} \Vert + \Vert u_{tx} \Vert + \Vert u_{x} \Vert + \Vert u_{t} \Vert + \bigl\Vert u ^{3} \bigr\Vert + \Vert \rho_{txx} \Vert + \Vert \rho_{tx}u_{x} \Vert + \Vert \rho_{xx}u_{t} \Vert \\& \hphantom{B_{2}(t)}\quad {} + \Vert \rho_{x}u_{tx} \Vert + \Vert \rho_{xx} \Vert + \Vert \rho_{x}u_{x} \Vert + \Vert \rho_{tx} \Vert + \Vert \rho_{x}u_{t} \Vert + \Vert \rho_{x} \Vert \bigr) \bigl( \Vert u_{tx} \Vert + \Vert u_{x} \Vert \bigr) \\& \hphantom{B_{2}(t)}\leq C_{2} \bigl( \Vert u_{x} \Vert ^{2}_{H^{1}}+ \Vert u_{t} \Vert _{H^{1}}^{2}+ \Vert \rho_{x} \Vert ^{2}_{H^{1}}+ \Vert \rho_{tx} \Vert ^{2}+ \Vert \rho_{txx} \Vert ^{2} \bigr) \\& \hphantom{B_{2}(t)}\leq C_{2} \bigl( \Vert u_{x} \Vert ^{2}_{H^{2}}+ \Vert u_{t} \Vert ^{2}_{H^{1}}+ \Vert \rho_{x} \Vert ^{2}_{H^{1}} \bigr). \end{aligned}$$
(3.16)

Differentiating (1.16) with respect to x and t, and using Lemmas 2.1–2.2 and (3.7)–(3.8), we conclude

$$ \bigl\Vert u_{txxx}(t) \bigr\Vert \leq C_{2} \bigl( \bigl\Vert u_{x}(t) \bigr\Vert _{H^{2}}+ \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{1}}+ \bigl\Vert u_{tx}(t) \bigr\Vert _{H^{1}}+ \bigl\Vert u_{ttx}(t) \bigr\Vert \bigr). $$
(3.17)

We integrate (3.13) with respect to t, use (3.3), (3.14)–(3.16) and Lemmas 2.1–2.2 to obtain (3.12). The proof is complete. □

Lemma 3.3

The following estimates hold for any \(T>0\):

$$\begin{aligned}& \bigl\Vert u_{tt}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{tx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl( \Vert u_{txx} \Vert ^{2}+ \Vert u _{ttx} \Vert ^{2} \bigr) (s)\leq C_{4}, \quad t\in [0,T], \end{aligned}$$
(3.18)
$$\begin{aligned}& \bigl\Vert \rho_{xxx}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{xxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl( \Vert \rho_{xxx} \Vert ^{2}+ \Vert u_{xxxx} \Vert ^{2} \bigr) (s) \,ds\leq C_{4}, \quad t\in [0,T]. \end{aligned}$$
(3.19)

Proof

We insert (3.1) into (3.12) and pick ε small enough to get (3.18).

Differentiating (2.9) with respect to x, we have

$$ \rho_{txxx}+\gamma \rho^{\gamma -1}\rho_{xxx}=E_{1}(x,t), $$
(3.20)

where

$$ E_{1}(x,t)=E_{0x}(x,t)-\gamma (\gamma -1) \rho^{\gamma -2}\rho_{x}\rho _{xx}. $$

Taking into account estimate (3.18), from (2.1)–(2.4) we can get

$$\begin{aligned} \bigl\Vert E_{1}(t) \bigr\Vert \leq & C_{2} \bigl( \Vert E_{0x} \Vert + \Vert \rho_{x}\rho_{xx} \Vert \bigr) \\ \leq &C_{2} \bigl( \Vert \rho_{x}u_{x} \Vert + \Vert \rho_{xx} \Vert + \Vert \rho_{xx}u_{x} \Vert + \Vert \rho_{xxx} \Vert + \Vert \rho_{x}u_{xx} \Vert + \Vert \rho_{xx}\rho_{x} \Vert \\ & {} + \Vert u_{tx} \Vert + \Vert u_{t} \rho_{x} \Vert + \Vert u_{txx} \Vert \bigr) \\ \leq &C_{2} \bigl( \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{2}}+ \bigl\Vert u_{x}(t) \bigr\Vert _{H^{2}}+ \bigl\Vert u_{t}(t) \bigr\Vert _{H^{2}} \bigr), \end{aligned}$$

which, along with Lemmas 2.1–2.2 and (3.18), implies

$$ \int_{0}^{t} \bigl\Vert E_{1}(s) \bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int_{0}^{t} \bigl\Vert \rho_{xxx}(s) \bigr\Vert ^{2}\,ds. $$
(3.21)

After multiplying (3.20) by \(\rho_{xxx}\) in \(L^{2}[0,1]\), we deduce

$$ \frac{1}{2}\frac{d}{dt} \Vert \rho_{xxx} \Vert ^{2}+\gamma \int_{0}^{1} \rho^{\gamma -1} \rho_{xxx}^{2}\,dx\leq C_{1} \bigl\Vert E_{1}(t) \bigr\Vert ^{2}, $$
(3.22)

which implies

$$ \frac{d}{dt} \Vert \rho_{xxx} \Vert ^{2}\leq C_{1} \bigl\Vert E_{1}(t) \bigr\Vert ^{2}. $$
(3.23)

Integrating (3.23) with respect to t, using (3.21), we conclude

$$ \Vert \rho_{xxx} \Vert ^{2}\leq C_{4}+C_{2} \int_{0}^{t} \bigl\Vert \rho_{xxx}(s) \bigr\Vert ^{2}\,ds, $$

which, after applying the Gronwall inequality and (3.22), yields

$$ \bigl\Vert \rho_{xxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert \rho_{xxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4}, \quad \forall t\in [0,T]. $$
(3.24)

By (3.4), (3.6), (3.18) and Lemmas 2.1–2.2, we conclude

$$\begin{aligned}& \bigl\Vert u_{xxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert u_{xxxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4},\quad \forall t\in [0,T], \end{aligned}$$

which, along with (3.24), gives (3.19). The proof is complete. □

Lemma 3.4

The following estimates hold for any \(T>0\):

$$\begin{aligned}& \bigl\Vert \rho_{xxxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert \rho_{xxxx}(s) \bigr\Vert ^{2}\,ds\leq C _{4}, \quad t\in [0,T], \end{aligned}$$
(3.25)
$$\begin{aligned}& \bigl\Vert u_{txx}(t) \bigr\Vert ^{2}+ \bigl\Vert u_{xxxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert u_{xxxxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4}, \quad t\in [0,T]. \end{aligned}$$
(3.26)

Proof

Differentiating (1.16) with respect to t, and using Lemmas 2.1–2.2 and Lemma 3.3, we can get

$$\begin{aligned} \bigl\Vert u_{txx}(t) \bigr\Vert \leq & C_{1} \bigl( \bigl\Vert u_{t}(t) \bigr\Vert _{H^{1}}+ \bigl\Vert \rho_{x}(t) \bigr\Vert + \bigl\Vert u _{x}(t) \bigr\Vert _{H^{1}}+ \bigl\Vert u_{tt}(t) \bigr\Vert \bigr) \\ \leq &C_{4},\quad \forall t\in [0,T]. \end{aligned}$$
(3.27)

Differentiating (3.20) with respect to x, we have

$$ \rho_{txxxx}+\gamma \rho^{\gamma -1}\rho_{xxxx}=E_{2}(x,t), $$
(3.28)

where

$$ E_{2}(x,t)=E_{1x}(x,t)-\gamma (\gamma -1)\rho_{x} \rho_{xxx} $$
(3.29)

and

$$ E_{1x}(x,t)=E_{0xx}(x,t)-\gamma (\gamma -1) \bigl( \rho^{\gamma -2}\rho_{x} \rho_{xx} \bigr)_{x}. $$

An easy calculation with the interpolation inequality, (3.2)–(3.8) and Lemmas 2.1–2.2 and Lemma 3.3 gives

$$\begin{aligned}& \bigl\Vert E_{0x}(t) \bigr\Vert \leq C_{2} \bigl( \Vert \rho_{x}u_{x} \Vert + \Vert \rho_{xxx}u_{x} \Vert + \Vert \rho_{xx}u_{xx} \Vert + \bigl\Vert \rho_{x}^{3} \bigr\Vert + \Vert u_{tx}r_{x} \Vert + \Vert u_{t}r_{xx} \Vert ++ \Vert u_{txx} \Vert \bigr) \\& \hphantom{\bigl\Vert E_{0x}(t) \bigr\Vert }\leq C_{2} \bigl( \Vert \rho_{x} \Vert _{H^{3}}+ \Vert u_{x} \Vert _{H^{2}}+ \Vert u_{tx} \Vert + \Vert u _{txx} \Vert \bigr), \end{aligned}$$
(3.30)
$$\begin{aligned}& \bigl\Vert E_{0xx}(t) \bigr\Vert \leq C_{2} \bigl( \Vert \rho_{xxxx} \Vert + \Vert \rho_{xx}u_{x} \Vert + \Vert \rho _{x}u_{xx} \Vert + \Vert \rho_{xxx}u_{x} \Vert + \Vert \rho_{xx}u_{xx} \Vert \\& \hphantom{\bigl\Vert E_{0xx}(t) \bigr\Vert }\quad {} + \Vert u_{txx} \Vert + \Vert u_{tx}u_{x} \Vert + \Vert u_{txxx} \Vert \bigr) \\& \hphantom{\bigl\Vert E_{0xx}(t) \bigr\Vert }\leq C_{2} \bigl( \Vert \rho_{x} \Vert _{H^{3}}+ \Vert u_{x} \Vert _{H^{2}}+ \Vert u_{tx} \Vert _{H ^{2}} \bigr), \end{aligned}$$
(3.31)
$$\begin{aligned}& \bigl\Vert E_{1x}(t) \bigr\Vert \leq C_{2} \bigl( \bigl\Vert E_{0xx}(t) \bigr\Vert + \bigl\Vert \rho_{x}^{2} \rho_{xx} \bigr\Vert + \bigl\Vert \rho_{xx}^{2} \bigr\Vert + \Vert \rho_{x}\rho_{xxx} \Vert \bigr) \\& \hphantom{\bigl\Vert E_{1x}(t) \bigr\Vert } \leq C_{2} \bigl( \Vert \rho_{x} \Vert _{H^{3}}+ \Vert u_{x} \Vert _{H^{2}}+ \Vert u_{tx}\Vert _{H ^{2}} \Vert \bigr). \end{aligned}$$
(3.32)

Taking into account estimate (3.30)–(3.32), from (3.29), we obtain

$$\begin{aligned}& \bigl\Vert E_{2}(t) \bigr\Vert \leq C_{4} \bigl( \bigl\Vert u_{x}(t) \bigr\Vert _{H^{3}}+ \bigl\Vert \rho_{x}(t) \bigr\Vert _{H^{3}}+ \bigl\Vert u_{tx}(t) \bigr\Vert _{H^{2}} \bigr). \end{aligned}$$
(3.33)

Inserting (3.17) into (3.33), and integrating (3.33) with respect to t, using Lemmas 2.1–2.2 and Lemma 3.3, we have

$$ \int_{0}^{t} \bigl\Vert E_{2}(s) \bigr\Vert ^{2}\,ds\leq C_{4}+C_{2} \int_{0}^{t} \bigl\Vert \rho_{xxxx}(s) \bigr\Vert ^{2}\,ds,\quad \forall t\in [0,T]. $$
(3.34)

Multiplying (3.28) by \(\rho_{xxxx}\) in \(L^{2}[0,1]\), we can get

$$ \frac{1}{2}\frac{d}{dt} \Vert \rho_{xxxx} \Vert ^{2}+\gamma \int_{0}^{1} \rho^{\gamma -1} \rho_{xxxx}^{2}\,dx\leq C_{2} \bigl\Vert E_{2}(t) \bigr\Vert ^{2} , $$
(3.35)

which implies

$$ \frac{d}{dt} \Vert \rho_{xxxx} \Vert ^{2}\leq C_{2} \bigl\Vert E_{2}(t) \bigr\Vert ^{2}. $$
(3.36)

Integrating (3.36) with respect to t, using (3.34), we conclude

$$ \Vert \rho_{xxxx} \Vert ^{2}\leq C_{4}+C_{2} \int_{0}^{t} \bigl\Vert \rho_{xxxx}(s) \bigr\Vert ^{2}\,ds, \quad t\in [0,T] $$
(3.37)

which, after using Gronwall’s inequality, yields

$$ \bigl\Vert \rho_{xxxx}(t) \bigr\Vert ^{2}\leq C_{4}, \quad t\in [0,T]. $$
(3.38)

Thus, we can obtain (3.25) by virtue of (3.37)–(3.38).

By (3.6), (3.17), (3.25), Lemmas 2.1–2.2 and Lemma 3.3, we have

$$\begin{aligned}& \bigl\Vert u_{xxxx}(t) \bigr\Vert ^{2}+ \int_{0}^{t} \bigl\Vert u_{txxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4}, \quad t\in [0,T]. \end{aligned}$$
(3.39)

On the other hand, we differentiate (1.16) with respect to x three times, use Lemmas 2.1–2.2 and (3.25), (3.27) to conclude, for any \(t\in [0,T]\),

$$ \bigl\Vert u_{xxxxx}(t) \bigr\Vert \leq C_{4} \bigl( \bigl\Vert u_{txxx}(t) \bigr\Vert + \bigl\Vert u_{x}(t) \bigr\Vert _{H^{3}}+ \bigl\Vert \rho _{x}(t) \bigr\Vert _{H^{3}} \bigr). $$
(3.40)

Thus we deduce from (3.38)–(3.40) that

$$\begin{aligned}& \int_{0}^{t} \bigl\Vert u_{xxxxx}(s) \bigr\Vert ^{2}\,ds\leq C_{4},\quad \forall t\in [0,T] \end{aligned}$$

which, along with (3.39), gives (3.26). This completes the proof of the lemma. □

Proof of Theorem 1.2

Applying Lemmas 2.1–2.2 and Lemmas 3.1–3.4, we readily get estimates (1.23)–(1.26). This completes the proof of Theorem 1.2. □

4 Conclusions

In this paper, we have established the regularity of global solutions for the spherically symmetric compressible fluid with density-dependent viscosity in \(H_{2}\) and \(H_{4}\). The biggest difference from other papers is that our domain has spherical symmetry and the viscosity coefficients are density dependent.