Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 395209 | https://doi.org/10.1155/2012/395209

Ruxu Lian, Lan Huang, Jian Liu, "Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity", Journal of Applied Mathematics, vol. 2012, Article ID 395209, 22 pages, 2012. https://doi.org/10.1155/2012/395209

Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity

Academic Editor: Nazim I. Mahmudov
Received20 Dec 2011
Revised21 Mar 2012
Accepted31 Mar 2012
Published05 Jul 2012

Abstract

We consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. For regular initial density, we show that there exists a unique global strong solution to the exterior problem or the initial boundary value problem, respectively. In particular, the strong solution tends to the equilibrium state as 𝑡+.

1. Introduction

The isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read as follows: 𝜌𝑡+div(𝜌𝐔)=0,(𝜌𝐔)𝑡+div(𝜌𝐔𝐔)+𝑃(𝜌)div(𝜇(𝜌)𝐷(𝐔))(𝜆(𝜌)div𝐔)=0,(1.1) where 𝑡(0,+) is the time and 𝐱𝑅𝑁, 𝑁 is the spatial coordinate, 𝜌>0 and 𝑢 denote the density and velocity, respectively. Pressure function is taken as 𝑃(𝜌)=𝜌𝛾 with 𝛾>1, and 𝐷(𝐔)=(𝐔)+𝑡(𝐔)2(1.2) is the strain tensor and 𝜇(𝜌), 𝜆(𝜌) are the Lamé viscosity coefficients satisfying 𝜇(𝜌)>0,𝜇(𝜌)+𝑁𝜆(𝜌)0.(1.3)

There are many important progress achieved recently on the compressible Navier-Stokes equations with density-dependent viscosity coefficient. For instance, the mathematical derivations are derived in the simulation of flow surface in shallow region [14]. The prototype model is the viscous Saint-Venant. The well posedness of solutions 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 is investigated by many authors, refer to [512] and references therein. Mellet and Vasseur showed the global existence of strong solutions for 𝛼(0,1/2) [13]. The qualitative behaviors of global solutions and dynamical asymptotics of vacuum states are also made, 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 [1417] and references therein.

In this present paper, we consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and focus on the regularities and dynamical behaviors of global strong solution, and so forth. As 𝛾>1, we show that the exterior problem and the initial boundary value problem with regular initial data both admit the unique global strong solution. In particular, the strong solution tends to the equilibrium state as 𝑡+.

The rest of the paper is arranged as follows. In Section 2, the main results about the dynamical behaviors of global strong solution for compressible Navier-Stokes equations are stated. Then, the theorems of the exterior problem and the initial boundary value problem are proved in Sections 3 and 4, respectively.

2. Notations and Main Results

For simplicity, we will take 𝜇(𝜌)=𝜌 and 𝜆(𝜌)=0 and 𝐷(𝐔)=𝐔 in (1.1). The isentropic compressible Navier-Stokes equations become 𝜌𝑡+div(𝜌𝐔)=0,(𝜌𝐔)𝑡+div(𝜌𝐔𝐔)+𝑃(𝜌)div(𝜌𝐔)=0.(2.1) Firstly, we consider the exterior problem, namely, the initial data and boundary conditions of (2.1) are imposed as follows: 𝜌(𝜌,𝐔)(𝐱,0)=0,𝐔0(𝐱),𝐱Ω,𝐔=0,on𝜕Ω,lim|𝐱|+(𝜌,𝑢)(𝐱,𝑡)=[],𝜌,0,𝑡0,𝑇(2.2) where Ω=𝑅3/Ω𝑟, Ω𝑟 is a ball of radius 𝑟 centered at the origin in 𝑅3, and 𝜌>0 is a constant.

We are concerned with the spherically symmetric solutions of system (2.1) in an spherically symmetric exterior domain Ω. To this end, we denote that 𝐱|𝐱|=𝑟,𝜌(𝐱,𝑡)=𝜌(𝑟,𝑡),𝐔(𝐱,𝑡)=𝑢(𝑟,𝑡)𝑟,(2.3) which leads to the following system of equations for 𝑟>0, 𝜌𝑡+(𝜌𝑢)𝑟+2𝜌𝑢𝑟=0,(𝜌𝑢)𝑡+𝜌𝑢2+𝜌𝛾𝑟+2𝜌𝑢2𝑟𝜌𝑢𝑟𝑟𝜌2𝑢𝑟𝑟=0,(2.4) with the initial data and boundary conditions 𝜌(𝜌,𝑢)(𝑟,0)=0,𝑢0𝑟(𝑟),𝑟,𝑢𝑟,+,𝑡=0,lim𝑟+(𝜌,𝑢)(𝑟,𝑡)=[],𝜌,0𝑡0,𝑇(2.5) and the initial data satisfies for some constants 𝜌 and 𝜌>0𝜌0𝜌𝐿1𝑟,+𝐿2𝑟,+,inf𝑟𝐱,+𝜌0>𝜌>0,𝑟𝜌0𝑟𝐿2𝑟,+,𝑟2𝑢0𝐻2𝑟.,+(2.6) Then, we define that 𝐸01=2𝑟+𝜌0𝑢20𝑟2𝑑𝑟+𝑟+𝜌01𝜌𝛾10𝛾1𝜌𝛾1+𝜌𝛾𝜌01𝜌1𝑟2𝐸𝑑𝑟,11=2𝑟+𝜌0𝑢0+𝜌01𝜌0𝑟2𝑟2𝑑𝑟+𝑟+𝜌01𝜌𝛾10𝛾1𝜌𝛾1+𝜌𝛾𝜌01𝜌1𝑟2𝑑𝑟,(2.7) and give the main results as follows.

Theorem 2.1. Let 𝛾>1. Assume that the initial data satisfies (2.6) and 𝐸0+𝐸1<𝜈𝑟2𝜌𝛾/2+1/2, 𝜈 is a positive constant. Then, there exist two positive constants 𝜌 and 𝜌 and a unique global strong solution (𝜌,𝑢) to the exterior problem (2.4) and (2.5), namely, satisfying 0<𝜌𝜌(𝑟,𝑡)𝜌,𝑟(𝑟,𝑡)×[],,+0,𝑇𝜌𝜌𝐿[]0,𝑇;𝐿2𝑟,+,𝜌𝑟𝐿[]0,𝑇;𝐿2𝑟,,+𝑢𝐿[]0,𝑇;𝐿2𝑟,+,𝑢𝑟𝐿[]0,𝑇;𝐿2𝑟.,+(2.8) Furthermore, the solution tends to the equilibrium state (𝜌,0)𝜌𝜌,𝑢(,𝑡)𝐿([𝑟,+))0,𝑡+.(2.9)

Then investigates the initial boundary value problem, and the initial data and boundary conditions of (2.1) are assumed as follow: 𝜌(𝜌,𝐔)(𝐱,0)=0,𝐔0[].(𝐱),𝐱Ω,𝐔(𝐱,𝑡)=0,𝐱𝜕Ω,𝑡0,𝑇(2.10) By (2.3), one considers (2.4) with the initial data and boundary conditions 𝜌(𝜌,𝑢)(𝑟,0)=0,𝑢0𝑟(𝑟),𝑟,𝑟+,𝑢𝑟𝑟,𝑡=𝑢+[],,𝑡=0,𝑡0,𝑇(2.11) and the initial data satisfies for some constant 𝜌>0𝜌0𝐿1𝑟,𝑟+𝑊1,𝑟,𝑟+,inf𝑟𝐱,𝑟+𝜌0>𝜌𝑟>0,2𝑢0𝐻2𝑟,𝑟+.(2.12)

Then, can give the main results as follows.

Theorem 2.2. Let 𝛾>1. Assume that the initial data satisfies (2.12), there exist two positive constants 𝜌 and 𝜌 and a unique global strong solution (𝜌,𝑢) to the initial boundary value problem (2.4) and (2.11), namely, satisfying 0<𝜌𝜌(𝑟,𝑡)𝜌,𝑟(𝑟,𝑡),𝑟+×[],0,𝑇𝜌𝐿[]0,𝑇;𝐿2𝑟,𝑟+,𝜌𝑟𝐿[]0,𝑇;𝐿2𝑟,𝑟+,𝑢𝐿[]0,𝑇;𝐿2𝑟,𝑟+,𝑢𝑟𝐿[]0,𝑇;𝐿2𝑟,𝑟+.(2.13) Furthermore, the solution (𝜌,𝑢) tends to the equilibrium state exponentially (𝜌̃𝜌,𝑢)(,𝑡)𝐿([𝑟,𝑟+])𝐶0𝑒𝐶1𝑡,(2.14) where 𝐶0 and 𝐶1 are positive constants independent of time and ̃𝜌=𝑟+𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟.

Remark 2.3. Theorems 2.1 and 2.2 hold for one-dimensional Saint-Venant’s model for shallow water, that is, 𝛾=2, 𝛼=1.

3. Proof of the Exterior Problem

3.1. The A Priori Estimates

It is convenient to make use of the Lagrangian coordinates so as to establish the uniformly a priori estimates. Take the Lagrange coordinates transform 𝑥=𝑟𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟,𝜏=𝑡,(3.1) which map (𝑟,𝑡)[𝑟,+)×𝑅+ into (𝑥,𝜏)[0,+)×𝑅+. The relation between Lagrangian and Eulerian coordinates are satisfied as 𝜕𝑥𝜕𝑟=𝜌𝑟2,𝜕𝑥𝜕𝑡=𝜌𝑢𝑟2.(3.2) The exterior problem (2.4) and (2.5) is reformulated to 𝜌𝜏+𝜌2𝑟2𝑢𝑥𝑟=0,2𝑢𝜏+(𝜌𝛾)𝑥=𝜌2𝑟2𝑢𝑥𝑥2𝜌𝑥𝑢𝑟,𝜌(𝜌,𝑢)(𝑥,0)=0,𝑢0[(𝑥),𝑥0,+),𝑢(0,𝑡)=0,lim𝑥+(𝜌,𝑢)=[𝜌,0,𝜏0,+),(3.3) where the initial data satisfies 𝜌0𝜌𝐿1([0,+))𝐿2([0,+)),inf[𝑥0,+)𝜌0>𝜌𝑟>0,2𝜌0𝑥𝐿2([10,+)),𝑟2𝜌0𝑟2𝑢0𝐿2([0,+)),𝑟2𝜌0𝑟2𝑢0𝑥𝐿2([10,+)),𝑟2𝜌0𝑟2𝜌𝑟2𝜌𝑟2𝑢0𝑥𝑥𝐿2([0,+)).(3.4) First, we will establish the a-priori estimates for the solution (𝜌,𝑢) to the exterior problem (3.3).

Lemma 3.1. Let 𝑇>0. Under the conditions in Theorem 2.1, it holds for any solution (𝜌,𝑢) to the exterior problem (3.3) that 120+𝑢2𝑑𝑥+0+1𝜌𝛾1𝛾1𝜌𝛾1+𝜌𝛾𝜌1𝜌1+𝑑𝑥𝜏00+2𝑢2𝑟2+𝜌2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠𝐸0[].,𝜏0,𝑇(3.5)

Proof. Multiplying (3.3) 2 by 𝑟2𝑢 and integrating the result with respect to 𝑥 over [0,+), making use of (3.3) 1 and (3.4), we have 12𝑑𝑑𝜏0+𝑢2𝑑𝑑𝑥+𝑑𝜏0+1𝜌𝛾1𝛾1𝜌𝛾1+𝜌𝛾𝜌1𝜌1𝑑𝑥+0+𝜌2𝑟2𝑢2𝑥𝑑𝑥=20+𝜌𝑟𝑢2𝑥𝑑𝑥,(3.6) integrating (3.6) with respect to 𝜏, we obtain 120+𝑢2𝑑𝑥+0+1𝜌𝛾1𝛾1𝜌𝛾1+𝜌𝛾𝜌1𝜌1𝑑𝑥+𝜏00+2𝑢2𝑟2+𝜌2𝑢2𝑥𝑟4=1𝑑𝑥𝑑𝑠20+𝑢20𝑑𝑥+0+1𝜌𝛾10𝛾1𝜌𝛾1+𝜌𝛾𝜌01𝜌1𝑑𝑥.(3.7) Lemma 3.1 can be obtained.

Lemma 3.2. Let 𝑇>0. Under the conditions in Theorem 2.1, it holds for any solution (𝜌,𝑢) to the exterior problem (3.3) that 120+𝑢+𝑟2𝜌𝑥2𝑑𝑥+0+1𝜌𝛾1𝛾1𝜌𝛾1+𝜌𝛾𝜌1𝜌1𝑑𝑥+𝛾𝜏00+𝜌𝛾1𝜌2𝑥𝑟4𝑑𝑥𝑑𝑠𝐸1[],,𝜏0,𝑇(3.8) and there exist two constants 0<𝜌<𝜌 such that 0<𝜌𝜌(𝑥,𝜏)𝜌[[].,(𝑥,𝜏)0,+)×0,𝑇(3.9)

Proof. Differentiating (3.3) 1 with respect to 𝑥, we have 𝜌𝑥𝜏+𝜌2𝑟2𝑢𝑥𝑥=0.(3.10) Summing (3.10) and (3.3) 2, we get 𝑟2𝑢+𝜌𝑥𝜏+(𝜌𝛾)𝑥=𝑟2𝜏𝑢2𝜌𝑥𝑢𝑟.(3.11) Note that 𝑟3(𝑥,𝜏)=𝑟3+3𝑥01𝜌(𝑧,𝜏)𝑑𝑧,(3.12) and so 𝜕𝑟=1𝜕𝜏𝑟2𝑥01𝜌𝑡1(𝑧,𝑡)𝑑𝑧=𝑟2𝑥0𝑟2𝑢𝑧(𝑧,𝑡)𝑑𝑧=𝑢(𝑥,𝜏),(3.13) which together with (3.11) yields 𝑟2𝑢+𝜌𝑥𝜏+(𝜌𝛾)𝑥=2𝑟3𝑢22𝜌𝑥𝑢𝑟.(3.14) Multiplying (3.14) by (𝑢+𝑟2𝜌𝑥)𝑟2 and integrating the result with respect to 𝑥 and 𝜏, we have 120+𝑢+𝑟2𝜌𝑥2𝑑𝑥+0+1𝜌𝛾1𝛾1𝜌𝛾1+𝜌𝛾𝜌1𝜌1𝑑𝑥+𝛾𝜏00+𝜌𝛾1𝜌2𝑥𝑟4=1𝑑𝑥𝑑𝑠20+𝑢0+𝑟2𝜌0𝑥2𝑑𝑥+0+1𝜌𝛾10𝛾1𝜌𝛾1+𝜌𝛾𝜌01𝜌1𝑑𝑥.(3.15)
Let 1𝜑(𝜌)=𝜌𝛾1𝛾1𝜌𝛾1+𝜌𝛾𝜌1𝜌1,𝜓(𝜌)=𝜌𝜌𝜑(𝜂)1/2𝑑𝜂.(3.16) It follows from (3.6) and (3.13) that ||𝜓||||||(𝜌)𝑦+𝜕𝑥||||𝜓(𝜌)𝑑𝑥𝑟2||||𝑦+𝜑(𝜌)1/2𝜌𝑥𝑟2||||𝑑𝑥𝑟2||||0+𝜑(𝜌)𝑑𝑦0+𝑟2𝜌𝑥2||||𝑑𝑥1/2𝑟2𝐸0+𝐸1.(3.17) We can verify that(1) As 𝜌+, it holds for 𝜉=𝜃𝜌+(1𝜃)𝜌, where 𝜃(0,1)lim𝜌+𝜓(𝜌)=lim𝜌+𝜌𝜌(𝛾2)𝜉𝛾3+2𝜌𝛾𝜉31/2||𝜂𝜌||𝑑𝜂lim𝜌+(𝛾2)𝜉𝛾3+2𝜌𝛾𝜉31/2𝜌𝜌𝜂𝜌𝑑𝜂=lim𝜌+12𝜃(𝛾2)𝜌+(1𝜃)𝜌𝛾3+2𝜌𝛾𝜃𝜌+(1𝜃)𝜌31/2𝜌𝜌2+.(3.18)(2) As 𝜌0, it holds for 𝜉=𝜃𝜌+(1𝜃)𝜌, where 𝜃(0,1)lim𝜌0𝜓(𝜌)=lim𝜌0𝜌𝜌(𝛾2)𝜉𝛾3+2𝜌𝛾𝜉31/2||𝜂𝜌||𝑑𝜂lim𝜌0(𝛾2)𝜉𝛾3+2𝜌𝛾𝜉31/2𝜌𝜌𝜂𝜌𝑑𝜂=lim𝜌012𝜃(𝛾2)𝜌+(1𝜃)𝜌𝛾3+2𝜌𝛾𝜃𝜌+(1𝜃)𝜌31/2𝜌𝜌2=𝜈𝜌𝛾/2+1/2.(3.19) Applying (3.17)–(3.19) and 𝐸0+𝐸1<𝜈𝑟2𝜌𝛾/2+1/2, where 𝜈 is a positive constant, we can prove (3.9).

Lemma 3.3. Let 𝑇>0. Under the conditions in Theorem 2.1, it holds for any solution (𝜌,𝑢) to the exterior problem (3.3) that 0+𝑟2𝑢2𝑥𝑑𝑥+0+𝑟2𝑢2𝜏𝑟4𝑑𝑥+𝜏00+𝑟2𝑢2𝑠𝑟4+𝑑𝑥𝑑𝑠𝜏00+𝜌2𝑟2𝑢2𝑥𝑠𝑑𝑥𝑑𝑠+𝜏00+𝑟2𝑢2𝑥𝑥[],𝑑𝑥𝑑𝑠𝐶,𝜏0,𝑇(3.20) where 𝐶>0 denotes a constant independent of time.

Proof. Multiplying (3.3)2 by 𝜌2(𝑟2𝑢)𝜏 and integrating the result with respect to 𝑥 over [0,+), making use of (3.4), we obtain 𝑑𝑑𝜏0+12𝑟2𝑢2𝑥𝜌𝛾2𝑟2𝑢𝑥𝑑𝑥+0+𝜌2𝑟2𝑢2𝜏𝑟4𝑑𝑥=(𝛾2)0+𝜌𝛾1𝑟2𝑢2𝑥𝑑𝑥20+𝜌𝛾3𝜌𝑥𝑟2𝑢𝜏𝑑𝑥+20+𝜌1𝜌𝑥𝑟2𝑢𝑥𝑟2𝑢𝜏𝑑𝑥+20+𝜌2𝑢2𝑟2𝑢𝜏𝑟3𝑑𝑥20+𝜌2𝜌𝑥𝑢𝑟2𝑢𝜏𝑟1𝑑𝑥,(3.21) which implies 0+𝑟2𝑢2𝑥𝑑𝑥+𝜏00+𝑟2𝑢2𝑠𝑟4𝑑𝑥𝑑𝑠𝐶+𝐶0+𝜌𝛾2𝜌𝛾22𝑑𝑥+𝐶𝜏00+𝑢2𝑟2+𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠+𝐶𝜏00+𝜌2𝑥𝑟4𝑑𝑥𝑑𝑠+𝐶𝜏00+𝜌2𝑥𝑟2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠+𝐶𝜏00+𝑢4𝑟2𝑑𝑥𝑑𝑠+𝐶𝜏00+𝜌2𝑥𝑢2𝑟2𝑑𝑥𝑑𝑠𝐶+𝐶𝜏00+𝜌2𝑥𝑟2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠+𝐶sup[]𝜏0,𝑇𝑢2𝐿.(3.22) From (3.3) 2, (3.5), (3.8), and (3.9), we can deduce that for some small 𝜖(0,1)𝜏00+𝜌2𝑥𝑟2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠𝜖𝜏00+𝜌2𝑥𝑟2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠+𝜖𝜏00+𝑟2𝑢2𝑥𝑠𝑟4𝑑𝑥𝑑𝑠+𝜖𝜏00+𝜌2𝑥𝑑𝑥𝑑𝑠+𝐶(𝜖)𝜏00+𝑢2𝑟2+𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠,(3.23)sup[]𝜏0,𝑇𝑢2𝐿𝜖sup[]𝜏0,𝑇0+𝑟2𝑢2𝑥𝑑𝑥+𝐶(𝜖)sup[]𝜏0,𝑇0+𝑢2𝑑𝑥,(3.24) using (3.22)–(3.24), we can obtain that 0+𝑟2𝑢2𝑥𝑑𝑥+𝜏00+𝑟2𝑢2𝑠𝑟4𝑑𝑥𝑑𝑠𝐶+𝐶𝜖𝜏00+𝑟2𝑢2𝑥𝑠𝑟4𝑑𝑥𝑑𝑠.(3.25)
Differentiating (3.3) 2 with respect to 𝜏, multiplying the result by (𝑟2𝑢)𝜏, and integrating the result with respect to 𝑥 over [0,+), we have 12𝑑𝑑𝜏0+𝑟2𝑢2𝜏𝑟4𝑑𝑥+0+𝜌2𝑟2𝑢2𝑥𝜏𝑑𝑥=20+𝑢𝑢𝜏𝑟2𝑢𝜏𝑟31𝑑𝑥20+𝑟4𝜏𝑟2𝑢2𝜏𝑑𝑥+20+𝑢𝑟1𝑢𝜏𝑟2𝑢𝜏𝑟2+𝑑𝑥0+(𝜌𝛾)𝜏𝑟2𝑢𝑥𝜏𝑑𝑥0+𝜌2𝜏𝑟2𝑢𝑥𝑟2𝑢𝑥𝜏𝑑𝑥0+2𝜌𝑥𝑢𝑟𝜏𝑟2𝑢𝜏𝑑𝑥+20+𝑟2𝜏𝑟1𝑢2𝑟2𝑢𝜏𝑑𝑥.(3.26) A complicated computation gives 𝑑𝑑𝜏0+𝑟2𝑢2𝜏𝑟4𝑑𝑥+0+𝜌2𝑟2𝑢2𝑥𝜏𝑑𝑥𝐶0+𝑟2𝑢2𝜏𝑟4𝑑𝑥+𝐶sup𝜏[0,𝑇]𝑟2𝑢2𝑥𝐿0+𝑟2𝑢2𝑥𝑑𝑥+𝐶0+𝑢2𝑟2+𝑢2𝑥𝑟4𝑑𝑥1+sup[]𝜏0,𝑇0+𝑟2𝑢2𝑥,𝑑𝑥(3.27) integrating (3.27) with respect to 𝜏, by means of (3.3) 2, (3.5), (3.8), (3.9), and (3.25), it holds that 0+𝑟2𝑢2𝜏𝑟4𝑑𝑥+𝜏00+𝜌2𝑟2𝑢2𝑥𝑠𝑑𝑥𝑑𝑠𝐶+𝐶𝜏00+𝑟2𝑢2𝑠𝑟4𝑑𝑥𝑑𝑠+𝐶sup[]𝜏0,𝑇𝑟2𝑢2𝑥𝐿+𝐶sup𝜏[0,𝑇]0+𝑟2𝑢2𝑥𝑑𝑥𝐶+𝐶𝜏00+𝑟2𝑢2𝑠𝑟4𝑑𝑥𝑑𝑠+𝐶sup[]𝜏0,𝑇0+𝑟2𝑢2𝑥𝑑𝑥1/20+𝑟2𝑢2𝑥𝑥𝑑𝑥1/2+𝐶sup[]𝜏0,𝑇0+𝑟2𝑢2𝑥𝑑𝑥𝐶+𝐶𝜖𝜏00+𝜌2𝑟2𝑢2𝑥𝑠𝑑𝑥𝑑𝑠+𝐶𝜖0+𝑟2𝑢2𝜏𝑟4𝑑𝑥,(3.28) choosing the constant 𝜖 small sufficiently, we can complete the proof of Lemma 3.3.

Remark 3.4. By Lemmas 3.13.3, the following inequality holds: 0+𝑢2𝑑𝑥+0+𝜌𝜌2𝑑𝑥+0+𝑢2𝑥𝑑𝑥+0+𝑢2𝜏𝑑𝑥+0+𝜌2𝑥+𝑑𝑥𝜏00+𝜌2𝑥𝑑𝑥𝑑𝑠+𝜏00+𝑢2𝑥𝑑𝑥𝑑𝑠+𝜏00+𝑢2𝑠+𝑑𝑥𝑑𝑠𝜏00+𝑢2𝑥𝑥𝑑𝑥𝑑𝑠+𝜏00+𝑢2𝑥𝜏𝑑𝑥𝑑𝑠𝐶.(3.29)

Lemma 3.5. Under the conditions in Theorem 2.1, it holds for any solution (𝜌,𝑢) to the exterior problem (3.3) that 𝜌𝜌,𝑢(,𝜏)𝐿([0,+))0,𝜏+,(3.30) where 𝐶>0 denotes a constant independent of time.

Proof. From Lemmas 3.13.3, we can obtain 0𝜌𝜌,𝑢𝑥2𝐿2([0,+))𝑑𝜏𝑟40𝜌𝜌,𝑢𝑥𝑟22𝐿2([0,+))𝑑𝜏𝐶,(3.31)0|||𝑑𝑑𝜏𝜌𝜌,𝑢𝑥2𝐿2([0,+))|||=𝑑𝜏0||||0+4𝜌𝜌2𝑥𝑟2𝑢𝑥2𝜌2𝜌𝑥𝑟2𝑢𝑥𝑥𝑑𝑥+20+𝑢𝑥𝑢𝑥𝜏||||𝑑𝑥𝑑𝜏𝐶00+𝜌2𝑥𝑑𝑥𝑑𝜏+𝐶00+𝑢2𝑟2+𝑢2𝑥𝑟4+𝑟2𝑢2𝑥𝑥+𝑢2𝑥𝜏𝑑𝑥𝑑𝜏𝐶,(3.32) which together with (3.31) implies 𝜌𝜌,𝑢𝑥2𝐿2([0,+))𝑊1,1𝑅+.(3.33) It holds from Gagliardo-Nirenberg-Sobolev inequality that 𝜌𝜌,𝑢𝐿([0,+))𝜌𝜌,𝑢𝐿1/22([0,+))𝜌𝜌,𝑢𝑥𝐿1/22([0,+)),(3.34) which together with (3.5), (3.9), and (3.33) implies this lemma.

3.2. Proof of Theorem 2.1

Proof. The global existence of unique strong solution to the exterior problem as (2.4) and (2.5) can be established in terms of the short-time existence carried out as in [6], the uniform a-priori estimates and the analysis of regularities, which indeed follow from Lemmas 3.13.3. We omit the details. The large time behaviors follow from Lemma 3.5 directly. The proof of Theorem 2.1 is completed.

4. Proof of the Initial Boundary Value Problem

4.1. The A-Priori Estimates

Take the Lagrange coordinates transform 𝑥=𝑟𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟,𝜏=𝑡.(4.1) By (4.1) and the conservation of mass for (𝜌,𝑢)𝑟+𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟=𝑟+𝑟𝜌0(𝑟)𝑟2𝑑𝑟=1,(4.2) the Lagrange coordinates transform (4.1) map (𝑟,𝑡)[𝑟,𝑟+]×𝑅+ into (𝑥,𝜏)[0,1]×𝑅+.

The relation between Lagrangian and Eulerian coordinates are satisfied as 𝜕𝑥𝜕𝑟=𝜌𝑟2,𝜕𝑥𝜕𝑡=𝜌𝑢𝑟2,(4.3) and the initial boundary value problem’s (2.4) and (2.11) are reformulated to 𝜌𝜏+𝜌2𝑟2𝑢𝑥𝑟=0,2𝑢𝜏+(𝜌𝛾)𝑥=𝜌2𝑟2𝑢𝑥𝑥2𝜌𝑥𝑢𝑟,𝜌(𝜌,𝑢)(𝑥,0)=0,𝑢0[],[(𝑥),𝑥0,1𝑢(0,𝑡)=𝑢(1,𝑡)=0,𝜏0,+),(4.4) where the initial data satisfies 𝜌0𝐿1([]0,1)𝑊1,([]0,1),inf[]𝑥0,1𝜌0>𝜌1>0,𝑟2𝜌0𝑟2𝑢0𝐿2([]0,1),𝑟2𝜌0𝑟2𝑢0𝑥𝐿2([]10,1),𝑟2𝜌0𝑟2𝜌𝑟2𝜌𝑟2𝑢0𝑥𝑥𝐿2([]0,1).(4.5) Then, we will establish the a-priori estimates for the solution (𝜌,𝑢) to the initial boundary value problem (4.4).

Lemma 4.1. Let 𝑇>0. Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (4.4) that 1012𝑢2+1𝜌𝛾1𝛾1𝑑𝑥+𝜏0102𝑢2𝑟2+𝜌2𝑢2𝑥𝑟4=𝑑𝑥𝑑𝑠1012𝑢20+1𝜌𝛾10𝛾1[].𝑑𝑥,𝜏0,𝑇(4.6)

Proof. Multiplying (4.4)2 by 𝑟2𝑢 and integrating the result with respect to 𝑥 over [0,1], using (4.4)1 and (4.5), we obtain 𝑑𝑑𝜏1012𝑢2+1𝜌𝛾1𝛾1𝑑𝑥+10𝜌2𝑟2𝑢2𝑥𝑑𝑥=210𝜌𝑟𝑢2𝑥𝑑𝑥,(4.7) and integrating (4.7) with respect to 𝜏, we obtain 1012𝑢2+1𝜌𝛾1𝛾1𝑑𝑥+𝜏0102𝑢2𝑟2+𝜌2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠=1012𝑢20+1𝜌𝛾10𝛾1𝑑𝑥.(4.8) Lemma 4.1 can be obtained.

Lemma 4.2. Let 𝑇>0. Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (4.4) that 1210𝑢+𝑟2𝜌𝑥21𝑑𝑥+𝛾110𝜌𝛾1𝑑𝑥+𝛾𝜏010𝜌𝛾1𝜌2𝑥𝑟4=1𝑑𝑥𝑑𝑠210𝑢0+𝑟2𝜌0𝑥21𝑑𝑥+𝛾110𝜌0𝛾1[],𝑑𝑥,𝜏0,𝑇(4.9) where 𝐶 is a positive constant independent of time.

Proof. Differentiating (4.4)1 with respect to 𝑥, we have 𝜌𝑥𝜏+𝜌2𝑟2𝑢𝑥𝑥=0,(4.10) which together with (4.4)2 and 𝜕𝑟/𝜕𝜏=𝑢 gives 𝑟2𝑢+𝜌𝑥𝜏+(𝜌𝛾)𝑥=2𝑟3𝑢22𝜌𝑥𝑢𝑟.(4.11) Multiplying (4.11) by (𝑢+𝑟2𝜌𝑥)𝑟2, and integrating the result with respect to 𝑥 and 𝜏, it holds that 1210𝑢+𝑟2𝜌𝑥21𝑑𝑥+𝛾110𝜌𝛾1𝑑𝑥+𝛾𝜏010𝜌𝛾1𝜌2𝑥𝑟4=1𝑑𝑥𝑑𝑠210𝑢0+𝑟2𝜌0𝑥21𝑑𝑥+𝛾110𝜌0𝛾1𝑑𝑥.(4.12) The proof of (4.9) is completed.

Lemma 4.3. Let 𝑇>0. Under the conditions in Theorem 2.2, there exists a constant 𝜌>0 such that 0<𝜌(𝑥,𝜏)𝜌[]×[].,(𝑥,𝜏)0,10,𝑇(4.13)

Proof. It follows from (4.6) and (4.9) that 𝜌(𝑥,𝜏)=𝜌(𝑟,𝑡)𝑟+𝑟𝜌(𝑟,𝑡)𝑑𝑟+𝑟+𝑟||𝜌𝑟||(𝑟,𝑡)𝑑𝑟𝑟2𝑟+𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟+𝑟2𝑟+𝑟𝜌||𝜌𝑟||(𝑟,𝑡)𝜌𝑟2𝑑𝑟𝐶+𝐶𝑟+𝑟1𝜌||𝜌𝑟||(𝑟,𝑡)2𝑟2𝑑𝑟𝐶+𝐶10||𝜌𝑥||(𝑥,𝜏)2𝑟4𝑑𝑥𝐶=𝜌.(4.14)

Lemma 4.4. Let 𝑇>0. Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (4.4) that 10𝑢2𝑛𝑑𝑥+𝜏010𝑢2𝑛𝑟2+𝜌2𝑢2𝑛2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠𝐶(𝑇),(4.15) for any positive integer 𝑛𝑁, where 𝐶(𝑇) is a positive constant dependent of time.

Proof. Multiplying (4.4)2 with 𝑢2𝑛1, integrating by parts over [0,1], we have 1𝑑2𝑛𝑑𝜏10𝑢2𝑛𝑑𝑥+10𝜌2𝑟2𝑢𝑥𝑟2𝑢2𝑛1𝑥𝑑𝑥=10𝜌𝛾𝑟2𝑢2𝑛1𝑥+𝜌2𝑟𝑢2𝑛𝑥𝑑𝑥.(4.16) Since it holds that 𝑟2𝑢𝑥𝑟2𝑢2𝑛1𝑥=2𝑢𝜌𝑟+𝑟2𝑢𝑥2𝑢2𝑛1𝜌𝑟+(2𝑛1)𝑟2𝑢2𝑛2𝑢𝑥=4𝑢2𝑛𝜌2𝑟2+(2𝑛1)𝑢2𝑛2𝑢2𝑥𝑟4+4𝑛𝑢2𝑛1𝑢𝑥𝑟𝜌,(4.17) it follows from (4.16) that 𝑑𝑑𝜏10𝑢2𝑛2𝑛𝑑𝑥+210𝑢2𝑛𝑟2𝑑𝑥+(2𝑛1)10𝜌2𝑢2𝑛2𝑢2𝑥𝑟4𝑑𝑥=210𝜌𝛾1𝑢2𝑛1𝑟𝑑𝑥+(2𝑛1)10𝜌𝛾𝑢2𝑛2𝑢𝑥𝑟2𝑑𝑥10𝑢2𝑛𝑟2𝑑𝑥+10𝜌2𝑢2𝑛2𝑢2𝑥𝑟4𝑑𝑥+𝐶𝜌𝐿2(𝛾1)10𝑢2𝑛2𝑑𝑥,(4.18) which together with (4.13) and Young’s inequality yields 𝑑𝑑𝜏10𝑢2𝑛𝑑𝑥+10𝑢2𝑛𝑟2𝑑𝑥+10𝜌2𝑢2𝑛2𝑢2𝑥𝑟4𝑑𝑥𝐶+10𝑢2𝑛𝑑𝑥,(4.19) and by applying the Gronwall’s inequality to (4.19), we can obtain (4.15).

Lemma 4.5. Let 𝑇>0, for 𝑛𝑁, and 𝑛>1/2(𝛾1). Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (4.4) that 𝜏0𝜌2𝑛(𝛾1)𝑢2𝑛𝐿([0,1])𝑑𝑠𝐶(𝑇),(4.20)𝜏0||(𝜌𝛾)𝑥𝑟2||2𝑛𝐿([0,1])[],𝑑𝑠𝐶(𝑇),𝜏0,𝑇(4.21) where 𝐶(𝑇) is a positive constant dependent of time.

Proof. By means of Sobolev imbedding theorem and Cauchy-Schwarz inequality, applying (4.6), (4.13), and (4.15), we get 𝜏0𝜌2𝑛(𝛾1)𝑢2𝑛𝐿([0,1])𝑑𝑠𝜏010||𝜌2𝑛(𝛾1)𝑢2𝑛||𝑑𝑥𝑑𝑠+𝜏010||𝜌2𝑛(𝛾1)𝑢2𝑛𝑥||𝑑𝑥𝑑𝑠𝐶(𝑇)+𝐶𝜏010𝜌2𝑛(𝛾1)1||𝜌𝑥||𝑢2𝑛𝑑𝑥𝑑𝑠+𝐶𝜏010𝜌2𝑛(𝛾1)||𝑢2𝑛1𝑢𝑥||𝑑𝑥𝑑𝑠𝐶(𝑇)+𝐶𝜏010||𝜌𝑥||2𝑟4𝑑𝑥𝑑𝑠+𝐶𝜏010𝜌2(2𝑛(𝛾1)1)𝑢4𝑛𝑟4+𝑑𝑥𝑑𝑠𝜏010𝜌2(2𝑛(𝛾1)1)𝑢2𝑛𝑟4𝑑𝑥𝑑𝑠+𝐶𝜏010𝜌2𝑢2𝑛2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠𝐶(𝑇).(4.22) Next, we find that 𝜌𝑥𝑟2=𝜌0𝑥𝑟20+𝑢0𝑢𝜏0(𝜌𝛾)𝑥𝑟2𝑑𝑠,(4.23) which together with (4.13), (4.15), and (4.22) gives 𝜏0||(𝜌𝛾)𝑥𝑟2||2𝑛𝐿𝑑𝑠=𝛾2𝑛𝜏0𝜌2𝑛(𝛾1)||𝜌𝑥𝑟2||2𝑛𝐿𝑑𝑠=𝛾2𝑛𝜏0𝜌2𝑛(𝛾1)||||𝜌0𝑥𝑟20+𝑢0𝑢𝑠0(𝜌𝛾)𝑥𝑟2||||𝑑𝑙2𝑛𝐿𝑑𝑠𝐶𝜏0𝜌2𝑛(𝛾1)𝜌2𝑛0𝑥+𝑢02𝑛+𝑢2𝑛𝐿𝑑𝑠+𝐶𝜏0𝜌2𝑛(𝛾1)𝑠0(𝜌𝛾)𝑥𝑟2𝑑l2𝑛𝐿𝑑𝑠𝐶(𝑇)+𝐶𝜏0𝜌2𝑛(𝛾1)𝑢2𝑛𝐿𝑑𝑠+𝐶(𝑇)𝜏0𝑠0||(𝜌𝛾)𝑥𝑟2||2𝑛𝐿𝑑𝑙𝑑𝑠,(4.24) applying the Gronwall’s inequality to (4.24), we obtain (4.21).

Lemma 4.6. Let 𝑇>0. Under the conditions in Theorem 2.2, there exists a constant 𝜌>0 such that 𝜌(𝑥,𝜏)𝜌[]×[].>0,(𝑥,𝜏)0,10,𝑇(4.25)

Proof. It is easy to verify that 𝜌(𝛾+1)/2(𝜏)=10𝜌(𝛾+1)/2𝑟(𝑥,𝜏)𝑑𝑥2𝑟2+𝑟+𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟1+(𝛾+1)/2=𝑟2𝑟2+𝑟+𝑟𝜌0(𝑟)𝑟2𝑑𝑟1+(𝛾+1)/2>0,(4.26)𝜌(𝛾+1)/2𝜌(𝛾+1)/2(𝜏)2𝐿2([])0,1𝑊1,1([]0,𝑇).(4.27) Indeed, it holds that 𝜏0𝜌(𝛾+1)/2𝜌(𝛾+1)/2(𝑠)2𝐿2([])0,1𝑑𝑠𝐶𝜏0𝜌(𝛾+1)/2𝑥𝑟22𝐿2([])0,1𝑑𝑠𝐶,(4.28) which together with (4.4), (4.9), and (4.13) gives 𝜏0|||𝑑𝜌𝑑𝑠(𝛾+1)/2𝜌(𝛾+1)/2(𝑠)2𝐿2([])0,1|||𝑑𝑠(𝛾+1)𝜏0||||10𝜌(𝛾+1)/2𝜌(𝛾+1)/2𝜌(𝑠)(𝛾+1)/21𝜌𝑠||||𝑑𝑥𝑑𝑠+2𝜏0||||10𝜌(𝛾+1)/2𝜌(𝛾+1)/2(𝑠)𝜌(𝛾+1)/2(𝑠)𝑠||||𝑑𝑥𝑑𝑠𝐶𝜏010𝜌(𝛾+1)/2𝜌(𝛾+1)/2(𝑠)2𝑑𝑥𝑑𝑠+𝜏010𝜌2𝑢2𝑥𝑟4𝑑𝑥𝑑𝑠𝐶.(4.29) By Gagliardo-Nirenberg-Sobolev inequality and (4.27), it follows that 𝜌(𝛾+1)/2𝜌(𝛾+1)/2(𝜏)2𝐿([])0,10as𝜏+.(4.30) Thus, there is a 𝑇0>0 and a constant 𝜌1>0 such that 𝜌(𝑥,𝜏)𝜌1[]𝑇,𝑥0,1,𝜏0,+.(4.31)
For 𝜏[0,𝑇0], denote 1𝑣(𝑥,𝜏)=𝜌(𝑥,𝜏)𝑟2(,𝑥,𝜏)(4.32) then from (4.4), we can obtain 𝑣𝜏=𝑟2𝑢𝑥𝑟22𝑣𝑢𝑟,(4.33) multiplying (4.33) by 4𝑣3 and integrating the result over [0,1]×[0,𝑇0], we have 10𝑣4𝑑𝑥=10𝑣40𝑑𝑥+4𝜏010𝑣3𝑟2𝑢𝑥𝑟2𝑑𝑥𝑑𝑠8𝜏010𝑣4𝑢𝑟=𝑑𝑥𝑑𝑠10𝑣40𝑑𝑥+12𝜏010𝑣4𝑢𝜌𝑥𝑟2𝑑𝑥𝑑𝑠+24𝜏010𝑣4𝑢𝑟=𝑑𝑥𝑑𝑠10𝑣40𝑑𝑥+12𝜏010𝑣4𝑢𝜌0𝑥𝑟20+𝑢0𝑢𝑠0(𝜌𝛾)𝑥𝑟2𝑑𝑙𝑑𝑥𝑑𝑠+24𝜏010𝑣4𝑢𝑟𝑑𝑥𝑑𝑠.(4.34) From (4.21), it holds that 10𝑣4𝑑𝑥+12𝜏010𝑣4𝑢2=𝑑𝑥𝑑𝑠10𝑣40𝑑𝑥+12𝜏010𝑣4𝑢𝜌0𝑥𝑟20𝑑𝑥𝑑𝑠+12𝜏010𝑣4𝑢𝑢0𝑑𝑥𝑑𝑠12𝜏010𝑣4𝑢𝑠0(𝜌𝛾)𝑥𝑟2𝑑𝑙𝑑𝑥𝑑𝑠+24𝜏010𝑣4𝑢𝑟𝑑𝑥𝑑𝑠6𝜏010𝑣4𝑢2𝑇𝑑𝑥𝑑𝑠+𝐶0𝜏010𝑣4𝑑𝑥𝑑𝑠,(4.35) which yields to 10𝑣4𝑇𝑑𝑥𝐶+𝐶0𝜏010𝑣4𝑑𝑥𝑑𝑠,(4.36) where 𝐶(𝑇0) is a positive constant dependent of time 𝑇0. By Gronwall’s inequality, (4.36) leads to 10𝑣4𝑑𝑥=101𝜌4𝑟8𝑇𝑑𝑥𝐶0.(4.37) It holds for (𝑥,𝜏)([0,1])×[0,𝑇0] that 1𝜌=101𝜌𝑑𝑥+10||||1𝜌𝑥||||𝑑𝑥𝐶+𝐶101𝜌4𝑟8𝑑𝑥+𝐶101𝜌4𝑟8𝑑𝑥1/210𝜌2𝑥𝑟4𝑑𝑥1/2𝑇𝐶0,(4.38) namely, 𝜌𝑇(𝑥,𝜏)𝐶0=𝜌2,[](𝑥,𝜏)(0,1)×0,𝑇0.(4.39) Therefore, we can choose 𝜌𝜌=min1,𝜌2,(4.40) to get 𝜌𝜌[][,(𝑥,𝜏)(0,1)×0,+).(4.41)

Lemma 4.7. Let 𝑇>0. Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (4.4) that 10𝑟2𝑢2𝑥𝑑𝑥+10𝑟2𝑢2𝜏𝑟4𝑑𝑥+𝜏010𝑟2𝑢2𝑠𝑟4+𝑑𝑥𝑑𝑠𝜏010𝜌2𝑟2𝑢2𝑥𝑠𝑑𝑥𝑑𝑠+𝜏010𝑟2𝑢2𝑥𝑥[],𝑑𝑥𝑑𝑠𝐶,𝜏0,𝑇(4.42) where 𝐶>0 denotes a constant independent of time.

Proof. The proof is similar to Lemma 3.3. We omit here.

Remark 4.8. By Lemmas 4.14.7, the following inequality holds: 10𝑢2𝑑𝑥+10𝜌𝜌2𝑑𝑥+10𝑢2𝑥𝑑𝑥+10𝑢2𝜏𝑑𝑥+10𝜌2𝑥+𝑑𝑥𝜏010𝜌2𝑥𝑑𝑥𝑑𝑠+𝜏010𝑢2𝑥𝑑𝑥𝑑𝑠+𝜏010𝑢2𝑠+𝑑𝑥𝑑𝑠𝜏010𝑢2𝑥𝑥𝑑𝑥𝑑𝑠+𝜏010𝑢2𝑥𝜏𝑑𝑥𝑑𝑠𝐶.(4.43)

Lemma 4.9. Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problems (2.4) and (2.11) that (𝜌̃𝜌,𝑢)(,𝑡)𝐿([𝑟,𝑟+])𝐶0𝑒𝐶1𝑡,𝑡>0,(4.44) where 𝐶0 and 𝐶1 denote the constants independent of time and ̃𝜌=𝑟+𝑟𝜌(𝑟,𝑡)𝑟2𝑑𝑟.

Proof. In a similar argument to show (4.8) and (4.12) with modification, we can obtain the following 𝑑𝑑𝑡𝑟+𝑟12𝜌𝑢2+1𝜌𝛾1𝛾̃𝜌𝛾𝛾̃𝜌𝛾1𝑟(𝜌̃𝜌)2𝑑𝑟+𝑟+𝑟2𝜌𝑢2+𝜌𝑢2𝑟𝑟2𝑑𝑟=0,(4.45)12𝑑𝑑𝑡𝑟+𝑟𝜌𝑢+𝜌1𝜌𝑟2𝑟21𝑑𝑟+𝑑𝛾1𝑑𝑡𝑟+𝑟𝜌𝛾̃𝜌𝛾𝛾̃𝜌𝛾1𝑟(𝜌̃𝜌)2𝑑𝑟+𝛾𝑟+𝑟𝜌𝛾2𝜌2𝑟𝑟2𝑑𝑟=0.(4.46) It holds from (4.9), (4.13), and (4.25) that 𝑟+𝑟(𝜌̃𝜌)2𝑟2𝑑𝑟𝐶𝑟+𝑟𝜌𝛾2𝜌2𝑟𝑟2𝑑𝑟.(4.47) Denote 𝐸(𝑡)=𝑟+𝑟𝜌𝑢2+𝑢+𝜌1𝜌𝑟2𝑟2𝑑𝑟+𝑟+𝑟𝜌𝛾̃𝜌𝛾𝛾̃𝜌𝛾1𝑟(𝜌̃𝜌)2𝑑𝑟.(4.48) By (4.45)–(4.48), a complicated computation gives rise to 𝑑𝑑𝑡𝐸(𝑡)+𝐶𝐸(𝑡)0,(4.49) which gives 𝐸(𝑡)𝐸(0)𝑒𝐶𝑡.(4.50) By the fact that 𝐸(𝑡)𝑐𝑢2𝐿2𝑟,𝑟++𝜌̃𝜌2𝐿2𝑟,𝑟++𝜌𝑟2𝐿2𝑟,𝑟+,(4.51) where 𝑐>0 is a constant independent of time and Gagliardo-Nirenberg-Sobolev inequality (𝜌̃𝜌,𝑢)𝐿([𝑟,𝑟+])𝐶(𝜌̃𝜌,𝑢)𝐿1/22([𝑟,𝑟+])(𝜌̃𝜌,𝑢)𝑟𝐿1/22𝑟,𝑟+,(4.52) we obtain (4.44).

4.2. Proof of Theorem 2.2

Proof. The proof of Theorem 2.2 is similar to Theorem 2.1. We omit the details.

Acknowledgments

The research of R. Lian is partially supported by NNSFC no. 11101145. The research of L. Huang is partially supported by NNSFC no. 11126323.

References

  1. D. Bresch and B. Desjardins, “Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,” Communications in Mathematical Physics, vol. 238, no. 1-2, pp. 211–223, 2003. View at: Google Scholar | Zentralblatt MATH
  2. D. Bresch and B. Desjardins, “On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models,” Journal de Mathématiques Pures et Appliquées Neuvième Série, vol. 86, no. 4, pp. 362–368, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. J.-F. Gerbeau and B. Perthame, “Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation,” Discrete and Continuous Dynamical Systems Series B, vol. 1, no. 1, pp. 89–102, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. F. Marche, “Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects,” European Journal of Mechanics B, vol. 26, no. 1, pp. 49–63, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. Z. Guo, Q. Jiu, and Z. Xin, “Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients,” SIAM Journal on Mathematical Analysis, vol. 39, no. 5, pp. 1402–1427, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. S. Jiang, Z. Xin, and P. Zhang, “Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity,” Methods and Applications of Analysis, vol. 12, no. 3, pp. 239–251, 2005. View at: Google Scholar | Zentralblatt MATH
  7. T.-P. Liu, Z. Xin, and T. Yang, “Vacuum states for compressible flow,” Discrete and Continuous Dynamical Systems, vol. 4, no. 1, pp. 1–32, 1998. View at: Google Scholar | Zentralblatt MATH
  8. M. Okada, Š. M. Nečasová, and T. Makino, “Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity,” Annali dell'Università di Ferrara Nuova Serie, vol. 48, pp. 1–20, 2002. View at: Google Scholar | Zentralblatt MATH
  9. S.-W. Vong, T. Yang, and C. Zhu, “Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. II,” Journal of Differential Equations, vol. 192, no. 2, pp. 475–501, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. T. Yang, Z. Yao, and C. Zhu, “Compressible Navier-Stokes equations with density-dependent viscosity and vacuum,” Communications in Partial Differential Equations, vol. 26, no. 5-6, pp. 965–981, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. T. Yang and H. Zhao, “A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity,” Journal of Differential Equations, vol. 184, no. 1, pp. 163–184, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. T. Yang and C. Zhu, “Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum,” Communications in Mathematical Physics, vol. 230, no. 2, pp. 329–363, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. A. Mellet and A. Vasseur, “Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations,” SIAM Journal on Mathematical Analysis, vol. 39, no. 4, pp. 1344–1365, 2007/08. View at: Publisher Site | Google Scholar
  14. Q. Jiu and Z. Xin, “The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients,” Kinetic and Related Models, vol. 1, no. 2, pp. 313–330, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. Q. Jiu, Y. Wang, and Z. Xin, “Stability of rarefaction waves to the 1D compressible Navier-Stokes equations with density-dependent viscosity,” Communications in Partial Differential Equations, vol. 36, no. 4, pp. 602–634, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. H.-L. Li, J. Li, and Z. Xin, “Vanishing of vacuum states and blow-up ph