^{1}

^{2}

^{1}

^{2}

This paper is concerned with the initial boundary value problem for the three-dimensional Navier-Stokes equations with density-dependent viscosity. The cylindrically symmetric strong solution is shown to exist globally in time and tend to the equilibrium state exponentially as time grows up.

The compressible isentropic Navier-Stokes equations (CNS) with density-dependent viscosity coefficients can be written for

When

It should be mentioned here that important progress has been obtained on global existence and asymptotical behaviors of strong solution to compressible Navier-Stokes equations (

Recently, there have been some results on the existence of cylindrically symmetric solution to three-dimensional compressible Navier-Stokes equations. When viscosity coefficients are both constants, Frid and Shelukhin [

In the present paper, we consider the initial boundary value problem (IBVP) for the three-dimensional isentropic compressible Navier-Stokes equations and focus on the existence and time asymptotic behavior of the global strong solution. For simplicity, we deal with the case

The rest part of the paper is arranged as follows. In Section

For simplicity, the viscosity terms are assumed to satisfy

Consider a flow between two circular coaxial cylinders and assume that the corresponding solution depends only on the radial variable

We are interested in the global existence of the initial boundary value problem for (

Let

The initial constraint

Theorem

In this section, we establish the a priori estimates for any solution

Let

Taking the product of (_{2}, (_{3}, and (_{4} with _{1}, we have

Under the same assumptions as Lemma

Differentiating (_{1} with respect to _{2}, we have
_{3}, and (_{4} with _{1} and boundary conditions, we have

Denote

Under the same assumptions as Lemma

To prove (_{2} with _{2}, (_{2}, (_{1}–(_{4} leads to (

Under the same assumptions as Lemma

Differentiating (_{2} with respect to _{1}, (_{2}, (_{1}–(_{4} leads to (

Now we turn to prove (_{1} that

Under the same assumptions as Lemma

Applying (

The global existence of unique strong solution to the IBVP (

The authors declare that there is no conflict of interests regarding the publication of this paper.

All authors contributed to each part of this work equally.

The authors are grateful to Professor Hai-Liang Li for his helpful discussions and suggestions about the problem. The research of Jian Liu is partially supported by NNSFC no. 11326140 and the Doctoral Starting up Foundation of Quzhou University nos. BSYJ201314 and XNZQN201313. The research of Ruxu Lian is partially supported by NNSFC no. 11101145, the China Postdoctoral Science Foundation no. 2012M520360, and the Doctoral Foundation of North China University of Water Sources and Electric Power no. 201032.