#### Abstract

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.

#### 1. Introduction

The compressible isentropic Navier-Stokes equations (CNS) with density-dependent viscosity coefficients can be written for as where , , and stand for the fluid density, velocity, and pressure, respectively, is the stress tensor, and and are the Lamé viscosity coefficients satisfying and for . The typical model (the so-called viscous Saint-Venant system) is rigorously derived [1, 2] and expressed as (1) with , , and .

When and are density-dependent viscosity, there are many important results made on the compressible Navier-Stokes equations. For example, the existence of solution to 2D shallow water equations was investigated by Bresch and Desjardins [3, 4]. The well posedness of solution to the free boundary value problem with initial finite mass and the flow density being connected with the infinite vacuum either continuously or via jump discontinuity was considered by many authors; refer to [5] and the references therein. The global existence of classical solution for was shown by Mellet and Vasseur [6]. The qualitative behaviors of global solution and dynamical asymptotic of vacuum states were also addressed, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behaviors of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [7] and the references therein.

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 (1) with constant viscosity coefficients; the readers can refer to [8–12] and the references therein. When is density-dependent viscosity, Lian et al. [13] proved that the strong solution exists globally in time and investigated the long time behaviors of the solution to one-dimensional case. However, for multidimensional case, there are a few results on the global existence and asymptotical behaviors of strong solution with density-dependent viscosity coefficients for 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 [14, 15] proved the uniqueness of the weak solution under certain condition, Fan and Jiang in [16] showed the global existence of weak solution, and Jiang and Zhang obtained the existence of strong solution for the nonisentropic case in [17]. When (a positive constant), Yao et al. [18] showed the global existence for the three-dimensional compressible Navier-Stokes equations. However, when and are both density-dependent viscosity, there is no result made on the existence of cylindrically symmetric solution to three-dimensional compressible Navier-Stokes equations.

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 with . We show that the unique global cylindrically symmetric strong solution to the IBVP (2)–(4) exists and tends to the equilibrium state exponentially as time grows up for initial data satisfied in Theorem 1.

The rest part of the paper is arranged as follows. In Section 2, the main results about the global existence of strong solution to the compressible Navier-Stokes equations are stated in detail. Then, some important a priori estimates will be given in Section 3 and Theorem 1 is proven in Section 4.

#### 2. Main Results

For simplicity, the viscosity terms are assumed to satisfy and with and . Then (1) become

Consider a flow between two circular coaxial cylinders and assume that the corresponding solution depends only on the radial variable in and the time variable . Then, the three-dimensional Navier-Stokes equations governing the flow reduce to the following system in the domain of the form supplemented with the initial and boundary conditions The velocity vector is given by the radial, angular, and axial velocities. For simplicity, we take here , since otherwise we can use to replace and .

We are interested in the global existence of the initial boundary value problem for (3)-(4). It is convenient to deal with the IBVP (3) in the Lagrangian coordinates. For simplicity, we assume that , which implies for and ; define the Lagrangian coordinates transformation which translates the domain into and satisfies The initial boundary value problem (3)-(4) is changed to for , with the initial data and boundary conditions given by where is defined by Before stating the main result, we assume that with and , and define Then, we have the following main results.

Theorem 1 (IBVP). *Let , . Assume that the initial data satisfies (13) and . Then, there exist positive constants and with so that the unique global strong solution to IBVP (9)–(12) exists and satisfies
**
In addition, the solution tends to the equilibrium state exponentially as time grows up; that is
**
where and denote two constants independent of the time. Furthermore, if , then
*

*Remark 2. *The initial constraint for does not always require that the perturbation of the initial data around the equilibrium state is small. Indeed, it can be large provided that the state is large enough or the value of is near .

*Remark 3. *Theorem 1 applies to the viscous Saint-Venant model for shallow water (which is (9) with , ).

#### 3. The A Priori Estimates

In this section, we establish the a priori estimates for any solution with to IBVP (9)–(12). Making use of similar arguments as in [19] with modifications, we can establish the following lemmas.

Lemma 4. *Let , , and with be any regular solution to the IBVP (9)–(12) for under the assumptions of Theorem 1. Then, it holds that
*

*Proof. *Taking the product of (9)_{2}, (9)_{3}, and (9)_{4} with , respectively, integrating on , and using (9)_{1}, we have
and by computation, we have the following:
which leads to (18) after the integration with respect to .

Lemma 5. *Under the same assumptions as Lemma 4, it holds that
**
In addition, it holds that
*

*Proof. *Differentiating (9)_{1} with respect to , rewriting it in the following form:
and substituting (23) into (9)_{2}, we have
Since , the above equation can also be rewritten as
Multiplying (25), (9)_{3}, and (9)_{4} with , , , respectively, integrating over , and using (9)_{1} and boundary conditions, we have
and then (21) follows.

Denote
It is easy to verify that and . In addition, it holds as that
and as that
We can choose two constants with and so that
which obviously satisfies
Thus, it follows from (18) and (21) that
from which we obtain (22) with and determined as above.

Lemma 6. *Under the same assumptions as Lemma 4, it holds that
**
where denotes a constant independent of time.*

*Proof. *To prove (33), taking the product of (9)_{2} with over and making use of (11), we can obtain after a tedious computation that
which implies
From (9)_{2}, (18), (21), (22), and Gagliardo-Nirenberg-Sobolev inequality, we can deduce that
where denotes a constant independent of time. The combination of (35) and (36) leads to
which together with (9)_{2}, (18), (21), and (22) implies
Using the same methods, we can obtain the following:
The combination of (38)-(39) and (9)_{1}–(9)_{4} leads to (33).

Lemma 7. *Under the same assumptions as Lemma 4, it holds that
**
In addition, it holds that
**
where denotes a constant independent of time.*

*Proof. *Differentiating (9)_{2} with respect to , multiplying the result by , and integrating the result with respect to over , we have
A complicated computation gives
From (9)_{1}, (9)_{2}, (18), (21), (22), and Lemma 6, it holds that
where denotes a constant independent of time, and from (44), we can find
With the same methods, we can obtain the following:
The combination of (45)-(46) and (9)_{1}–(9)_{4} leads to (40).

Now we turn to prove (41). It is easy to verify
This implies and the continuity of density . Indeed, it follows from (9)_{1} that
which together with (21) leads to the half of (41). From the (33) and (40), we can obtain the continuity of on . The proof is completed.

Lemma 8. *Under the same assumptions as Lemma 4, it holds that
**
where and denote two constants independent of time.*

*Proof. *Applying (20) and (26), we can obtain
Meanwhile, with Poincáre-Sobolev inequality and Lemmas 4–7, we may get after a complicated computation that
with
where is a constant independent of time, and (52) implies that
from which we obtain (49) immediately.

#### 4. Proof of the Main Results

*Proof. *The global existence of unique strong solution to the IBVP (9)–(12) can be established in terms of short time existence carried out as in [7], the uniform a priori estimates, and the analysis regularities, which indeed follow from Lemmas 4–7. We omit the details. The large time behaviors follow Lemma 8 directly. The proof of Theorem 1 is completed.

#### Conflict of Interests

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

#### Authors’ Contribution

All authors contributed to each part of this work equally.

#### Acknowledgments

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.