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

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: (𝜌𝐔)_𝑡𝜌_𝑡+div(𝜌𝐔)=0,+div(𝜌𝐔⊗𝐔)+∇𝑃(𝜌)−div(𝜇(𝜌)𝐷(𝐔))−∇(𝜆(𝜌)div𝐔)=0,(1.1) where 𝑡∈(0,+∞) is the time and 𝐱∈𝑅^𝑁, 𝑁 is the spatial coordinate, 𝜌>0 and 𝑢 denote the density and velocity, respectively. Pressure function is taken as 𝑃(𝜌)=𝜌^𝛾 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 [1–4]. 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 [5–12] 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 [14–17] 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 (𝜌𝐔)_𝑡𝜌_𝑡+div(𝜌𝐔)=0,+div(𝜌𝐔⊗𝐔)+∇𝑃(𝜌)−div(𝜌∇𝐔)=0.(2.1) Firstly, we consider the exterior problem, namely, the initial data and boundary conditions of (2.1) are imposed as follows: 𝐔=0,on(𝜕𝜌Ω,𝐔,)(𝐱,0li)m=_{|𝐱|→+∞}𝜌₀,𝐔₀(𝐱),𝐱∈Ω,(𝜌,𝑢)(𝐱,𝑡)=𝜌,0,𝑡∈[0,𝑇],(2.2) where Ω∶=𝑅³/Ω_𝑟−, Ω_𝑟− 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|=𝑟,𝜌(𝐱,𝑡)=𝜌(𝑟,𝑡),𝐔(𝐱,𝑡)=𝑢(𝑟,𝑡)𝐱𝑟,(2.3) which leads to the following system of equations for 𝑟>0, 𝜌_𝑡+(𝜌𝑢)_𝑟+2𝜌𝑢(𝜌𝑢)_𝑡𝑟=0,+𝜌𝑢²+𝜌^𝛾_𝑟+2𝜌𝑢²𝑟−𝜌𝑢^𝑟_𝑟−𝜌2𝑢𝑟_𝑟=0,(2.4) with the initial data and boundary conditions 𝑢𝑟(𝜌,𝑢)(𝑟,0)=𝜌₀,𝑢₀(𝑟),𝑟∈𝑟₋,+∞₋,,𝑡=0,lim_𝑟→+∞(𝜌,𝑢)(𝑟,𝑡)=𝜌,0𝑡∈[0,𝑇],(2.5) and the initial data satisfies for some constants 𝜌 and 𝜌>0𝜌₀−𝜌∈𝑟𝜌𝐿¹_0𝑟𝑟₋,+∞∩𝐿²𝑟₋,+∞,inf_{𝑟𝐱∈−,+∞}𝜌₀>𝜌>0,∈𝐿²𝑟₋,+∞,𝑟²𝑢₀∈𝐻²𝑟₋,+∞.(2.6) Then, we define that 𝐸₀∶=12^𝑟+∞₋𝜌₀𝑢²⁰𝑟²𝑑𝑟+^𝑟+∞₋𝜌₀𝛾−11𝜌^0𝛾−1𝐸−𝜌^𝛾−1+𝜌^𝛾𝜌⁰⁻¹−𝜌⁻¹𝑟²𝑑𝑟,₁∶=12^𝑟+∞₋𝜌₀𝑢₀+𝜌⁰⁻¹𝜌₀_𝑟²𝑟²𝑑𝑟+^𝑟+∞₋𝜌₀𝛾−11𝜌^0𝛾−1−𝜌^𝛾−1+𝜌^𝛾𝜌⁰⁻¹−𝜌⁻¹𝑟²𝑑𝑟,(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<𝜈𝑟²⁻𝜌^𝛾/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 𝜌−0<𝜌_∗≤𝜌(𝑟,𝑡)≤𝜌^∗,(𝑟,𝑡)∈𝑟₋,+∞×[0,𝑇],𝑢∈𝜌∈𝐿_∞𝐿^∞[0,𝑇];𝐿²𝑟₋,+∞,𝜌_𝑟∈𝐿^∞[0,𝑇];𝐿²𝑟₋,+∞,[0,𝑇];𝐿²𝑟₋,+∞,𝑢_𝑟∈𝐿^∞[0,𝑇];𝐿²𝑟₋,+∞.(2.8) Furthermore, the solution tends to the equilibrium state (𝜌,0)‖‖𝜌−𝜌,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: (𝜌𝐔,(𝐔𝐱),(𝑡)𝐱,=0)0=,𝐱𝜌₀∈,𝐔𝜕₀Ω,(𝐱𝑡)∈,[𝐱0,∈𝑇]Ω.,(2.10) By (2.3), one considers (2.4) with the initial data and boundary conditions (𝜌𝑢,𝑢)𝑟₋(𝑟,0)=𝜌₀,𝑢₀(𝑟),𝑟∈𝑟₋,𝑟₊,,𝑡=𝑢𝑟₊,𝑡=0,𝑡∈[0,𝑇],(2.11) and the initial data satisfies for some constant 𝜌>0𝜌₀∈𝐿¹𝑟₋,𝑟₊𝑟∩𝑊^1,∞𝑟₋,𝑟₊,inf_{𝑟𝐱∈−,𝑟+}𝜌₀>𝜌>0,₂𝑢₀∈𝐻²𝑟₋,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 𝜌∈𝐿_∞0<𝜌_∗≤𝜌(𝑟,𝑡)≤𝜌^∗,(𝑟,𝑡)∈𝑟₋,𝑟₊×[0,𝑇],𝑢∈𝐿_∞[0,𝑇];𝐿²𝑟₋,𝑟₊,𝜌_𝑟∈𝐿^∞[0,𝑇];𝐿²𝑟₋,𝑟₊,[0,𝑇];𝐿²𝑟₋,𝑟₊,𝑢_𝑟∈𝐿^∞[0,𝑇];𝐿²𝑟₋,𝑟₊.(2.13) Furthermore, the solution (𝜌,𝑢) tends to the equilibrium state exponentially ‖(𝜌−𝜌̃,𝑢)(⋅,𝑡)‖_𝐿^∞_{([𝑟−,𝑟+])}≤𝐶₀𝑒^−𝐶1𝑡,(2.14) where 𝐶₀ and 𝐶₁ are positive constants independent of time and 𝜌̃=∫^𝑟+_𝑟−𝜌(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=^𝑟𝑟₋𝜌(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 𝑟𝜌_𝜏+𝜌²𝑟²𝑢_𝑥=0,₋₂𝑢_𝜏+(𝜌^𝛾)_𝑥=𝜌²𝑟²𝑢_𝑥_𝑥−2𝜌^𝑥𝑢(𝜌,𝑢)(𝑥,0)=𝜌₀𝑟,𝑢(0,𝑡)=0,lim_𝑥→+∞,𝑢₀(𝑥),𝑥∈[0,+∞),(𝜌,𝑢)=𝜌,0,𝜏∈[0,+∞),(3.3) where the initial data satisfies 𝜌₀−𝑟𝜌∈𝐿¹([0,+∞))∩𝐿²([0,+∞)),inf_{[𝑥∈0,+∞)}𝜌₀>𝜌₋>0,₂𝜌_0𝑥∈𝐿²([0,+∞)),1√𝑟²𝜌₀𝑟²𝑢₀∈𝐿²([0,+∞)),𝑟²𝜌₀𝑟²𝑢₀_𝑥∈𝐿²([0,+∞)),1√𝑟²𝜌₀𝑟²𝜌𝑟²𝜌𝑟²𝑢₀_𝑥_𝑥∈𝐿²([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 12^0+∞𝑢²𝑑𝑥+^0+∞𝛾−11𝜌^𝛾−1+−𝜌^𝛾−1+𝜌^𝛾𝜌⁻¹−𝜌⁻¹𝑑𝑥_𝜏0^0+∞2𝑟𝑢²²+𝜌²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐸⁰,𝜏∈[0,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 12𝑑𝑑𝜏^0+∞𝑢²𝑑𝑥+𝑑𝑑𝜏^0+∞𝛾−11𝜌^𝛾−1=2_0+∞−𝜌^𝛾−1+𝜌^𝛾𝜌⁻¹−𝜌⁻¹𝑑𝑥+^0+∞𝜌²𝑟²𝑢^2𝑥𝑑𝑥𝜌𝑟𝑢²_𝑥𝑑𝑥,(3.6) integrating (3.6) with respect to 𝜏, we obtain 12^0+∞𝑢²𝑑𝑥+^0+∞𝛾−11𝜌^𝛾−1−𝜌^𝛾−1+𝜌^𝛾𝜌⁻¹−𝜌⁻¹𝑑𝑥+^𝜏0^0+∞2𝑟𝑢²²+𝜌²𝑢^2𝑥𝑟⁴=1𝑑𝑥𝑑𝑠2^0+∞𝑢²⁰𝑑𝑥+^0+∞𝛾−11𝜌^0𝛾−1−𝜌^𝛾−1+𝜌^𝛾𝜌⁰⁻¹−𝜌⁻¹𝑑𝑥.(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 12^0+∞𝑢+𝑟²𝜌_𝑥²𝑑𝑥+^0+∞𝛾−11𝜌^𝛾−1+𝛾_𝜏0−𝜌^𝛾−1+𝜌^𝛾𝜌⁻¹−𝜌⁻¹𝑑𝑥^0+∞𝜌^𝛾−1𝜌^2𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐸₁,𝜏∈[0,𝑇],(3.8) and there exist two constants 0<𝜌_∗<𝜌^∗ such that 0<𝜌_∗≤𝜌(𝑥,𝜏)≤𝜌^∗,(𝑥,𝜏)∈[0,+∞)×[0,𝑇].(3.9)

Proof. Differentiating (3.3) ₁ with respect to 𝑥, we have 𝜌_𝑥𝜏+𝜌²𝑟²𝑢_𝑥_𝑥=0.(3.10) Summing (3.10) and (3.3) ₂, we get 𝑟⁻²𝑢+𝜌_𝑥_𝜏+(𝜌^𝛾)_𝑥=𝑟⁻²_𝜏𝑢−2𝜌^𝑥𝑢𝑟.(3.11) Note that 𝑟³(𝑥,𝜏)=𝑟³⁻+3^𝑥01𝜌(𝑧,𝜏)𝑑𝑧,(3.12) and so 𝜕𝑟𝜕𝜏=𝑟1²^𝑥0𝜌1_𝑡(𝑧,𝑡)𝑑𝑧=1𝑟²^𝑥0𝑟²𝑢_𝑧(𝑧,𝑡)𝑑𝑧=𝑢(𝑥,𝜏),(3.13) which together with (3.11) yields 𝑟⁻²𝑢+𝜌_𝑥_𝜏+(𝜌^𝛾)_𝑥=−2𝑟⁻³𝑢²−2𝜌^𝑥𝑢𝑟.(3.14) Multiplying (3.14) by (𝑢+𝑟²𝜌_𝑥)𝑟² and integrating the result with respect to 𝑥 and 𝜏, we have 12^0+∞𝑢+𝑟²𝜌_𝑥²𝑑𝑥+^0+∞𝛾−11𝜌^𝛾−1+𝛾_𝜏0−𝜌^𝛾−1+𝜌^𝛾𝜌⁻¹−𝜌⁻¹𝑑𝑥^0+∞=1𝜌^𝛾−1𝜌^2𝑥𝑟⁴𝑑𝑥𝑑𝑠2^0+∞𝑢₀+𝑟²𝜌_0𝑥²𝑑𝑥+^0+∞𝛾−11𝜌^0𝛾−1−𝜌^𝛾−1+𝜌^𝛾𝜌⁰⁻¹−𝜌⁻¹𝑑𝑥.(3.15)
Let 𝜑(𝜌)∶=1𝛾−1𝜌^𝛾−1𝜓(𝜌)∶=−∫_𝜌𝜌𝜌^𝛾−1+𝜌^𝛾𝜌⁻¹−𝜌⁻¹,𝜑(𝜂)^1/2𝑑𝜂.(3.16) It follows from (3.6) and (3.13) that ||𝜓(𝜌)||≤||||^𝑦+∞𝜕_𝑥𝜓(𝜌)𝑑𝑥||||≤𝑟⁻⁻²||||^𝑦+∞≤𝑟₋₋₂𝜑(𝜌)^1/2𝜌_𝑥𝑟²𝑑𝑥||||||||^0+∞𝜑(𝜌)𝑑𝑦^0+∞𝑟²𝜌_𝑥²𝑑𝑥||||^1/2≤𝑟⁻⁻²𝐸₀+𝐸₁.(3.17) We can verify that(1) As 𝜌→+∞, it holds for 𝜉=𝜃𝜌+(1−𝜃)𝜌, where 𝜃∈(0,1)lim_𝜌→+∞𝜓(𝜌)=lim_𝜌→+∞^𝜌_𝜌(𝛾−2)𝜉^𝛾−3≥lim_𝜌→+∞+2𝜌^𝛾𝜉⁻³^1/2||𝜂−𝜌||𝑑𝜂=lim_𝜌→+∞(𝛾−2)𝜉^𝛾−3+2𝜌^𝛾𝜉⁻³^1/2^𝜌_𝜌𝜂−𝜌𝑑𝜂12(𝛾−2)𝜃𝜌+(1−𝜃)𝜌^𝛾−3+2𝜌^𝛾𝜃𝜌+(1−𝜃)𝜌⁻³^1/2𝜌−𝜌²⟶+∞.(3.18)(2) As 𝜌→0, it holds for 𝜉=𝜃𝜌+(1−𝜃)𝜌, where 𝜃∈(0,1)lim_𝜌→0𝜓(𝜌)=lim_𝜌→0^𝜌_𝜌(𝛾−2)𝜉^𝛾−3≤−lim_𝜌→0+2𝜌^𝛾𝜉⁻³^1/2||𝜂−𝜌||𝑑𝜂=−lim_𝜌→0(𝛾−2)𝜉^𝛾−3+2𝜌^𝛾𝜉⁻³^1/2^𝜌_𝜌𝜂−𝜌𝑑𝜂12(𝛾−2)𝜃𝜌+(1−𝜃)𝜌^𝛾−3+2𝜌^𝛾𝜃𝜌+(1−𝜃)𝜌⁻³^1/2𝜌−𝜌²=∶−𝜈𝜌^𝛾/2+1/2.(3.19) Applying (3.17)–(3.19) and 𝐸₀+𝐸₁<𝜈𝑟²⁻𝜌^𝛾/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+∞𝑟²𝑢^2𝑥𝑑𝑥+^0+∞𝑟²𝑢^2𝜏𝑟⁻⁴𝑑𝑥+^𝜏0^0+∞𝑟²+𝑢^2𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠_𝜏0^0+∞𝜌²𝑟²𝑢^2𝑥𝑠𝑑𝑥𝑑𝑠+^𝜏0^0+∞𝑟²𝑢^2𝑥𝑥𝑑𝑥𝑑𝑠≤𝐶,𝜏∈[0,𝑇],(3.20) where 𝐶>0 denotes a constant independent of time.

Proof. Multiplying (3.3)₂ by 𝜌⁻²(𝑟²𝑢)_𝜏 and integrating the result with respect to 𝑥 over [0,+∞), making use of (3.4), we obtain 𝑑𝑑𝜏^0+∞21𝑟²=(𝛾−2)𝑢_0+∞^2𝑥−𝜌^𝛾−2𝑟²𝑢_𝑥𝑑𝑥+^0+∞𝜌⁻²𝑟²𝑢^2𝜏𝑟⁻⁴𝑑𝑥+2_0+∞𝜌^𝛾−1𝑟²𝑢^2𝑥𝑑𝑥−2^0+∞𝜌^𝛾−3𝜌_𝑥𝑟²𝑢_𝜏𝑑𝑥+2^0+∞𝜌⁻¹𝜌_𝑥𝑟²𝑢_𝑥𝑟²𝑢_𝜏𝑑𝑥𝜌⁻²𝑢²𝑟²𝑢_𝜏𝑟⁻³𝑑𝑥−2^0+∞𝜌⁻²𝜌_𝑥𝑢𝑟²𝑢_𝜏𝑟⁻¹𝑑𝑥,(3.21) which implies ^0+∞𝑟²𝑢^2𝑥𝑑𝑥+^𝜏0^0+∞𝑟²≤𝐶+𝐶_0+∞𝑢^2𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶_𝜏0𝜌^𝛾−2−𝜌^𝛾−2²𝑑𝑥+𝐶^𝜏0^0+∞𝑢𝑟²²+𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏0^0+∞𝜌^2𝑥𝑟⁴𝑑𝑥𝑑𝑠^0+∞𝜌^2𝑥𝑟²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏0^0+∞𝑢⁴𝑟⁻²𝑑𝑥𝑑𝑠+𝐶^𝜏0^0+∞≤𝐶+𝐶_𝜏0𝜌^2𝑥𝑢²𝑟²𝑑𝑥𝑑𝑠^0+∞𝜌^2𝑥𝑟²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶sup_{[]𝜏∈0,𝑇}‖𝑢‖^2𝐿_∞.(3.22) From (3.3) ₂, (3.5), (3.8), and (3.9), we can deduce that for some small 𝜖∈(0,1)^𝜏0^0+∞𝜌^2𝑥𝑟²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝜖^𝜏0^0+∞𝜌^2𝑥𝑟²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝜖^𝜏0^0+∞𝑟²+𝜖_𝜏0𝑢^2𝑥𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠^0+∞𝜌^2𝑥𝑑𝑥𝑑𝑠+𝐶(𝜖)^𝜏0^0+∞𝑢𝑟²²+𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠,(3.23)sup_{[]𝜏∈0,𝑇}‖𝑢‖^2𝐿_∞≤𝜖sup_{[]𝜏∈0,𝑇}^0+∞𝑟²𝑢^2𝑥𝑑𝑥+𝐶(𝜖)sup_{[]𝜏∈0,𝑇}^0+∞𝑢²𝑑𝑥,(3.24) using (3.22)–(3.24), we can obtain that ^0+∞𝑟²𝑢^2𝑥𝑑𝑥+^𝜏0^0+∞𝑟²𝑢^2𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠≤𝐶+𝐶𝜖^𝜏0^0+∞𝑟²𝑢^2𝑥𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠.(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 12𝑑𝑑𝜏^0+∞𝑟²=2_0+∞𝑢^2𝜏𝑟⁻⁴𝑑𝑥+^0+∞𝜌²𝑟²𝑢^2𝑥𝜏𝑑𝑥𝑢𝑢_𝜏𝑟²𝑢_𝜏𝑟⁻³𝑑𝑥−12^0+∞𝑟⁻⁴_𝜏𝑟²+𝑢^2𝜏𝑑𝑥+2^0+∞𝑢𝑟⁻¹𝑢_𝜏𝑟²𝑢_𝜏𝑟⁻²𝑑𝑥_0+∞(𝜌^𝛾)_𝜏𝑟²𝑢_𝑥𝜏𝑑𝑥−^0+∞𝜌²_𝜏𝑟²𝑢_𝑥𝑟²𝑢_𝑥𝜏𝑑𝑥−^0+∞2𝜌^𝑥𝑢𝑟_𝜏𝑟²+2_0+∞𝑢_𝜏𝑑𝑥𝑟⁻²_𝜏𝑟⁻¹𝑢²𝑟²𝑢_𝜏𝑑𝑥.(3.26) A complicated computation gives 𝑑𝑑𝜏^0+∞𝑟²≤𝐶_0+∞𝑢^2𝜏𝑟⁻⁴𝑑𝑥+^0+∞𝜌²𝑟²𝑢^2𝑥𝜏𝑑𝑥𝑟²𝑢^2𝜏𝑟⁻⁴𝑑𝑥+𝐶sup_{𝜏∈[0,𝑇]}‖‖𝑟²𝑢^2𝑥‖‖_𝐿∞^0+∞𝑟²+𝐶_0+∞𝑢^2𝑥𝑑𝑥𝑢𝑟²²+𝑢^2𝑥𝑟⁴𝑑𝑥1+sup_{[]𝜏∈0,𝑇}^0+∞𝑟²𝑢^2𝑥𝑑𝑥,(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+∞𝑟²𝑢^2𝜏𝑟⁻⁴𝑑𝑥+^𝜏0^0+∞≤𝐶+𝐶_𝜏0𝜌²𝑟²𝑢^2𝑥𝑠𝑑𝑥𝑑𝑠^0+∞𝑟²𝑢^2𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶sup_{[]𝜏∈0,𝑇}‖‖𝑟²𝑢^2𝑥‖‖_𝐿∞+𝐶sup_{𝜏∈[0,𝑇]}^0+∞𝑟²≤𝐶+𝐶_𝜏0𝑢^2𝑥𝑑𝑥^0+∞𝑟²𝑢^2𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶sup_{[]𝜏∈0,𝑇}^0+∞𝑟²𝑢^2𝑥𝑑𝑥^1/2^0+∞𝑟²𝑢^2𝑥𝑥𝑑𝑥^1/2+𝐶sup_{[]𝜏∈0,𝑇}^0+∞𝑟²≤𝐶+𝐶𝜖_𝜏0𝑢^2𝑥𝑑𝑥^0+∞𝜌²𝑟²𝑢^2𝑥𝑠𝑑𝑥𝑑𝑠+𝐶𝜖^0+∞𝑟²𝑢^2𝜏𝑟⁻⁴𝑑𝑥,(3.28) choosing the constant 𝜖 small sufficiently, we can complete the proof of Lemma 3.3.

Remark 3.4. By Lemmas 3.1–3.3, the following inequality holds: ^0+∞+𝑢²𝑑𝑥+^0+∞𝜌−𝜌²𝑑𝑥+^0+∞𝑢^2𝑥𝑑𝑥+^0+∞𝑢^2𝜏𝑑𝑥+^0+∞𝜌^2𝑥𝑑𝑥_𝜏0^0+∞𝜌^2𝑥𝑑𝑥𝑑𝑠+^𝜏0^0+∞𝑢^2𝑥𝑑𝑥𝑑𝑠+^𝜏0^0+∞+𝑢^2𝑠𝑑𝑥𝑑𝑠_𝜏0^0+∞𝑢^2𝑥𝑥𝑑𝑥𝑑𝑠+^𝜏0^0+∞𝑢^2𝑥𝜏𝑑𝑥𝑑𝑠≤𝐶.(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 ‖‖𝜌−𝜌,𝑢(⋅,𝜏)‖‖_{𝐿∞([0,+∞))}⟶0,𝜏⟶+∞,(3.30) where 𝐶>0 denotes a constant independent of time.

Proof. From Lemmas 3.1–3.3, we can obtain ^∞0‖‖𝜌−𝜌,𝑢_𝑥‖‖^2𝐿_2([0,+∞))𝑑𝜏≤𝑟⁻⁻⁴^∞0‖‖𝜌−𝜌,𝑢_𝑥𝑟²‖‖^2𝐿_2([0,+∞))𝑑𝜏≤𝐶,(3.31)^∞0|||𝑑𝑑𝜏‖‖𝜌−𝜌,𝑢_𝑥‖‖^2𝐿_2([0,+∞))=|||𝑑𝜏_∞0||||^0+∞≤𝐶_∞0−4𝜌𝜌^2𝑥𝑟²𝑢_𝑥−2𝜌²𝜌_𝑥𝑟²𝑢_𝑥𝑥𝑑𝑥+2^0+∞𝑢_𝑥𝑢_𝑥𝜏𝑑𝑥||||𝑑𝜏^0+∞𝜌^2𝑥𝑑𝑥𝑑𝜏+𝐶^∞0^0+∞𝑢𝑟²²+𝑢^2𝑥𝑟⁴+𝑟²𝑢^2𝑥𝑥+𝑢^2𝑥𝜏𝑑𝑥𝑑𝜏≤𝐶,(3.32) which together with (3.31) implies ‖‖𝜌−𝜌,𝑢_𝑥‖‖^2𝐿_2([0,+∞))∈𝑊^1,1𝑅⁺.(3.33) It holds from Gagliardo-Nirenberg-Sobolev inequality that ‖‖𝜌−𝜌,𝑢‖‖_{𝐿∞([0,+∞))}≤‖‖𝜌−𝜌,𝑢‖‖^𝐿1/2_2([0,+∞))‖‖𝜌−𝜌,𝑢_𝑥‖‖^𝐿1/2_2([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 3.1–3.3. 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=^𝑟𝑟₋𝜌(𝑟,𝑡)𝑟²𝑑𝑟,𝜏=𝑡.(4.1) By (4.1) and the conservation of mass for (𝜌,𝑢)^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟=^𝑟+_𝑟−𝜌₀(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 𝑟𝜌_𝜏+𝜌²𝑟²𝑢_𝑥=0,₋₂𝑢_𝜏+(𝜌^𝛾)_𝑥=𝜌²𝑟²𝑢_𝑥_𝑥−2𝜌^𝑥𝑢(𝜌,𝑢)(𝑥,0)=𝜌₀𝑟,𝑢(0,𝑡)=𝑢(1,𝑡),=𝑢₀0,(𝑥)𝜏,∈𝑥[0∈,+[∞0,)1,],(4.4) where the initial data satisfies 1𝜌₀∈𝐿¹([0,1])∩𝑊^1,∞([0,1]),inf_[]𝑥∈0,1𝜌₀>𝜌>0,√𝑟²𝜌₀𝑟²𝑢1₀∈𝐿²([0,1]),𝑟²𝜌₀𝑟²𝑢₀_𝑥∈𝐿²([0,1]),√𝑟²𝜌₀𝑟²𝜌𝑟²𝜌𝑟²𝑢₀_𝑥_𝑥∈𝐿²([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 ¹⁰12𝑢²+1𝛾−1𝜌^𝛾−1𝑑𝑥+^𝜏0¹⁰2𝑟𝑢²²+𝜌²𝑢^2𝑥𝑟⁴=𝑑𝑥𝑑𝑠₁₀12𝑢²⁰+1𝛾−1𝜌^0𝛾−1𝑑𝑥,𝜏∈[0,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 𝑑𝑑𝜏¹⁰12𝑢²+1𝛾−1𝜌^𝛾−1𝑑𝑥+¹⁰𝜌²𝑟²𝑢^2𝑥𝑑𝑥=2¹⁰𝜌𝑟𝑢²_𝑥𝑑𝑥,(4.7) and integrating (4.7) with respect to 𝜏, we obtain ¹⁰12𝑢²+1𝛾−1𝜌^𝛾−1𝑑𝑥+^𝜏0¹⁰2𝑟𝑢²²+𝜌²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠=¹⁰12𝑢²⁰+1𝛾−1𝜌^0𝛾−1𝑑𝑥.(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 12¹⁰𝑢+𝑟²𝜌_𝑥²𝑑𝑥+1𝛾−1¹⁰𝜌^𝛾−1𝑑𝑥+𝛾^𝜏0¹⁰=1𝜌^𝛾−1𝜌^2𝑥𝑟⁴𝑑𝑥𝑑𝑠2¹⁰𝑢₀+𝑟²𝜌_0𝑥²𝑑𝑥+1𝛾−1¹⁰𝜌^0𝛾−1𝑑𝑥,𝜏∈[0,𝑇],(4.9) where 𝐶 is a positive constant independent of time.

Proof. Differentiating (4.4)₁ with respect to 𝑥, we have 𝜌_𝑥𝜏+𝜌²𝑟²𝑢_𝑥_𝑥=0,(4.10) which together with (4.4)₂ and 𝜕𝑟/𝜕𝜏=𝑢 gives 𝑟⁻²𝑢+𝜌_𝑥_𝜏+(𝜌^𝛾)_𝑥=−2𝑟⁻³𝑢²−2𝜌^𝑥𝑢𝑟.(4.11) Multiplying (4.11) by (𝑢+𝑟²𝜌_𝑥)𝑟², and integrating the result with respect to 𝑥 and 𝜏, it holds that 12¹⁰𝑢+𝑟²𝜌_𝑥²𝑑𝑥+1𝛾−1¹⁰𝜌^𝛾−1𝑑𝑥+𝛾^𝜏0¹⁰=1𝜌^𝛾−1𝜌^2𝑥𝑟⁴𝑑𝑥𝑑𝑠2¹⁰𝑢₀+𝑟²𝜌_0𝑥²𝑑𝑥+1𝛾−1¹⁰𝜌^0𝛾−1𝑑𝑥.(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)∈[0,1]×[0,𝑇].(4.13)

Proof. It follows from (4.6) and (4.9) that 𝜌(𝑥,𝜏)=≤𝜌𝑟(₋₋₂𝑟,𝑡)≤^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑑𝑟+^𝑟+_𝑟−||𝜌_𝑟(𝑟,𝑡)||𝑑𝑟^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟+𝑟⁻⁻²^𝑟+_𝑟−√𝜌||𝜌^𝑟(𝑟,𝑡)||√𝜌𝑟²≤𝐶+𝐶_𝑟+𝑟−𝑑𝑟1𝜌||𝜌^𝑟≤𝐶+𝐶₁₀(𝑟,𝑡)||²𝑟²𝑑𝑟||𝜌_𝑥(𝑥,𝜏)||²𝑟⁴𝑑𝑥≤𝐶=∶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 ¹⁰𝑢^2𝑛𝑑𝑥+^𝜏0¹⁰𝑢𝑟^2𝑛2+𝜌²𝑢^2𝑛−2𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶(𝑇),(4.15) for any positive integer 𝑛∈𝑁, where 𝐶(𝑇) is a positive constant dependent of time.

Proof. Multiplying (4.4)₂ with 𝑢^2𝑛−1, integrating by parts over [0,1], we have 12𝑛𝑑𝑑𝜏¹⁰𝑢^2𝑛𝑑𝑥+¹⁰𝜌²𝑟²𝑢_𝑥𝑟²𝑢^2𝑛−1_𝑥𝑑𝑥=¹⁰𝜌^𝛾𝑟²𝑢^2𝑛−1_𝑥+𝜌2𝑟𝑢^2𝑛_𝑥𝑑𝑥.(4.16) Since it holds that 𝑟²𝑢_𝑥𝑟²𝑢^2𝑛−1_𝑥==4𝑢2𝜌𝑢𝑟_2𝑛+𝑟²𝑢_𝑥2𝑢𝜌𝑟^2𝑛−1+(2𝑛−1)𝑟²𝑢^2𝑛−2𝑢_𝑥𝜌²𝑟²+(2𝑛−1)𝑢^2𝑛−2𝑢^2𝑥𝑟⁴+4𝑛𝑢^2𝑛−1𝑢_𝑥𝑟𝜌,(4.17) it follows from (4.16) that 𝑑𝑑𝜏¹⁰𝑢^2𝑛2𝑛𝑑𝑥+2¹⁰𝑢^2𝑛𝑟²=2₁₀𝑑𝑥+(2𝑛−1)¹⁰𝜌²𝑢^2𝑛−2𝑢^2𝑥𝑟⁴𝑑𝑥𝜌^𝛾−1𝑢^2𝑛−1𝑟𝑑𝑥+(2𝑛−1)¹⁰≤𝜌^𝛾𝑢^2𝑛−2𝑢_𝑥𝑟²𝑑𝑥₁₀𝑢^2𝑛𝑟²𝑑𝑥+¹⁰𝜌²𝑢^2𝑛−2𝑢^2𝑥𝑟⁴𝑑𝑥+𝐶‖𝜌‖^{𝐿2(𝛾−1)}_∞¹⁰𝑢^2𝑛−2𝑑𝑥,(4.18) which together with (4.13) and Young’s inequality yields 𝑑𝑑𝜏¹⁰𝑢^2𝑛𝑑𝑥+¹⁰𝑢^2𝑛𝑟²𝑑𝑥+¹⁰𝜌²𝑢^2𝑛−2𝑢^2𝑥𝑟⁴𝑑𝑥≤𝐶+¹⁰𝑢^2𝑛𝑑𝑥,(4.19) and by applying the Gronwall’s inequality to (4.19), we can obtain (4.15).

Lemma 4.5. Let 𝑇>0, for 𝑛∈𝑁, and 𝑛>1/2(𝛾−1). Under the conditions in Theorem 2.2, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (4.4) that ^𝜏0‖‖𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛‖‖_{𝐿∞([0,1])}𝑑𝑠≤𝐶(𝑇),(4.20)_𝜏0‖‖||(𝜌^𝛾)_𝑥𝑟²||^2𝑛‖‖_{𝐿∞([0,1])}𝑑𝑠≤𝐶(𝑇),𝜏∈[0,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 ^𝜏0≤‖‖𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛‖‖_{𝐿∞([0,1])}𝑑𝑠_𝜏0¹⁰||𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛||𝑑𝑥𝑑𝑠+^𝜏0¹⁰≤𝐶(𝑇)+𝐶_𝜏0||𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛_𝑥||𝑑𝑥𝑑𝑠¹⁰𝜌^{2𝑛(𝛾−1)−1}||𝜌_𝑥||𝑢^2𝑛𝑑𝑥𝑑𝑠+𝐶^𝜏0¹⁰≤𝐶(𝑇)+𝐶_𝜏0𝜌^{2𝑛(𝛾−1)}||𝑢^2𝑛−1𝑢_𝑥||𝑑𝑥𝑑𝑠¹⁰||𝜌_𝑥||²𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏0¹⁰+𝜌^{2(2𝑛(𝛾−1)−1)}𝑢^4𝑛𝑟⁻⁴𝑑𝑥𝑑𝑠_𝜏0¹⁰𝜌^{2(2𝑛(𝛾−1)−1)}𝑢^2𝑛𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶^𝜏0¹⁰𝜌²𝑢^2𝑛−2𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶(𝑇).(4.22) Next, we find that 𝜌_𝑥𝑟²=𝜌_0𝑥𝑟²⁰+𝑢₀−𝑢−^𝜏0(𝜌^𝛾)_𝑥𝑟²𝑑𝑠,(4.23) which together with (4.13), (4.15), and (4.22) gives ^𝜏0=‖‖||(𝛾𝜌^𝛾_2𝑛)_𝑥𝑟²||^2𝑛‖‖_𝐿∞𝑑𝑠^𝜏0=𝛾_2𝑛‖‖𝜌^{2𝑛(𝛾−1)}||𝜌_𝑥𝑟²||^2𝑛‖‖_𝐿∞𝑑𝑠^𝜏0‖‖‖‖𝜌_{2𝑛(𝛾−1)}||||𝜌_0𝑥𝑟²⁰+𝑢₀−𝑢−^𝑠0(𝜌^𝛾)_𝑥𝑟²𝑑𝑙||||^2𝑛‖‖‖‖_𝐿∞≤𝐶_𝜏0𝑑𝑠‖‖𝜌^{2𝑛(𝛾−1)}𝜌^2𝑛0𝑥+𝑢^02𝑛+𝑢^2𝑛‖‖_𝐿^∞𝑑𝑠+𝐶^𝜏0‖‖‖‖𝜌_{2𝑛(𝛾−1)}^𝑠0(𝜌^𝛾)_𝑥𝑟²𝑑l^2𝑛‖‖‖‖_𝐿∞≤𝐶(𝑇)+𝐶_𝜏0𝑑𝑠‖‖𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛‖‖_𝐿∞𝑑𝑠+𝐶(𝑇)^𝜏0^𝑠0‖‖||(𝜌^𝛾)_𝑥𝑟²||^2𝑛‖‖_𝐿∞𝑑𝑙𝑑𝑠,(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)∈[0,1]×[0,𝑇].(4.25)

Proof. It is easy to verify that 𝜌^(𝛾+1)/2(𝜏)=∶¹⁰𝜌^(𝛾+1)/2(𝑥,𝜏)𝑑𝑥≥𝑟²⁻𝑟²⁺^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟^1+(𝛾+1)/2=𝑟²⁻𝑟²⁺^𝑟+_𝑟−𝜌₀(𝑟)𝑟²𝑑𝑟^1+(𝛾+1)/2>0,(4.26)‖‖𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝜏)‖‖^2𝐿_2([])0,1∈𝑊^1,1([0,𝑇]).(4.27) Indeed, it holds that ^𝜏0‖‖𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝑠)‖‖^2𝐿_2([])0,1𝑑𝑠≤𝐶^𝜏0‖‖𝜌^(𝛾+1)/2_𝑥𝑟²‖‖^2𝐿_2([])0,1𝑑𝑠≤𝐶,(4.28) which together with (4.4), (4.9), and (4.13) gives ^𝜏0|||𝑑𝑑𝑠‖‖𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝑠)‖‖^2𝐿_2([])0,1≤(𝛾+1)_𝜏0|||𝑑𝑠||||¹⁰+2_𝜏0𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝑠)𝜌^{(𝛾+1)/2−1}𝜌_𝑠𝑑𝑥||||𝑑𝑠||||¹⁰≤𝐶_𝜏0𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝑠)𝜌^(𝛾+1)/2(𝑠)_𝑠𝑑𝑥||||𝑑𝑠¹⁰𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝑠)²𝑑𝑥𝑑𝑠+^𝜏0¹⁰𝜌²𝑢^2𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶.(4.29) By Gagliardo-Nirenberg-Sobolev inequality and (4.27), it follows that ‖‖𝜌^(𝛾+1)/2−𝜌^(𝛾+1)/2(𝜏)‖‖^2𝐿_∞([])0,1⟶0as𝜏⟶+∞.(4.30) Thus, there is a 𝑇₀>0 and a constant 𝜌₁>0 such that 𝜌(𝑥,𝜏)≥𝜌₁,𝑥∈[0,1],𝜏∈𝑇₀,+∞.(4.31)
For 𝜏∈[0,𝑇₀], denote 𝑣(𝑥,𝜏)=∶1𝜌(𝑥,𝜏)𝑟²(𝑥,𝜏),(4.32) then from (4.4), we can obtain 𝑣_𝜏=𝑟²𝑢_𝑥𝑟²−2𝑣𝑢𝑟,(4.33) multiplying (4.33) by 4𝑣³ and integrating the result over [0,1]×[0,𝑇₀], we have ¹⁰𝑣⁴𝑑𝑥=¹⁰𝑣⁴⁰𝑑𝑥+4^𝜏0¹⁰𝑣³𝑟²𝑢_𝑥𝑟²𝑑𝑥𝑑𝑠−8^𝜏0¹⁰𝑣⁴𝑢=𝑟𝑑𝑥𝑑𝑠₁₀𝑣⁴⁰𝑑𝑥+12^𝜏0¹⁰𝑣⁴𝑢𝜌_𝑥𝑟²𝑑𝑥𝑑𝑠+24^𝜏0¹⁰𝑣⁴𝑢=𝑟𝑑𝑥𝑑𝑠₁₀𝑣⁴⁰𝑑𝑥+12^𝜏0¹⁰+24_𝜏0𝑣⁴𝑢𝜌_0𝑥𝑟²⁰+𝑢₀−𝑢−^𝑠0(𝜌^𝛾)_𝑥𝑟²𝑑𝑙𝑑𝑥𝑑𝑠¹⁰𝑣⁴𝑢𝑟𝑑𝑥𝑑𝑠.(4.34) From (4.21), it holds that ¹⁰𝑣⁴𝑑𝑥+12^𝜏0¹⁰=𝑣⁴𝑢²𝑑𝑥𝑑𝑠₁₀𝑣⁴⁰𝑑𝑥+12^𝜏0¹⁰𝑣⁴𝑢𝜌_0𝑥𝑟²⁰𝑑𝑥𝑑𝑠+12^𝜏0¹⁰−12_𝜏0𝑣⁴𝑢𝑢₀𝑑𝑥𝑑𝑠¹⁰𝑣⁴𝑢^𝑠0(𝜌^𝛾)_𝑥𝑟²𝑑𝑙𝑑𝑥𝑑𝑠+24^𝜏0¹⁰𝑣⁴𝑢≤6_𝜏0𝑟𝑑𝑥𝑑𝑠¹⁰𝑣⁴𝑢²𝑑𝑥𝑑𝑠+𝐶𝑇₀^𝜏0¹⁰𝑣⁴𝑑𝑥𝑑𝑠,(4.35) which yields to ¹⁰𝑣⁴𝑑𝑥≤𝐶+𝐶𝑇₀^𝜏0¹⁰𝑣⁴𝑑𝑥𝑑𝑠,(4.36) where 𝐶(𝑇₀) is a positive constant dependent of time 𝑇₀. By Gronwall’s inequality, (4.36) leads to ¹⁰𝑣⁴𝑑𝑥=¹⁰1𝜌⁴𝑟⁸𝑑𝑥≤𝐶𝑇₀.(4.37) It holds for (𝑥,𝜏)∈([0,1])×[0,𝑇₀] that 1𝜌=¹⁰1𝜌𝑑𝑥+¹⁰||||1𝜌_𝑥≤𝐶+𝐶₁₀||||𝑑𝑥1𝜌⁴𝑟⁸𝑑𝑥+𝐶¹⁰1𝜌⁴𝑟⁸𝑑𝑥^1/2¹⁰𝜌^2𝑥𝑟⁴𝑑𝑥^1/2≤𝐶𝑇₀,(4.38) namely, 𝜌(𝑥,𝜏)≥𝐶𝑇₀=∶𝜌₂,(𝑥,𝜏)∈([0,1])×0,𝑇₀.(4.39) Therefore, we can choose 𝜌_∗=min𝜌₁,𝜌₂,(4.40) to get 𝜌≥𝜌_∗,(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 ¹⁰𝑟²𝑢^2𝑥𝑑𝑥+¹⁰𝑟²𝑢^2𝜏𝑟⁻⁴𝑑𝑥+^𝜏0¹⁰𝑟²+𝑢^2𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠_𝜏0¹⁰𝜌²𝑟²𝑢^2𝑥𝑠𝑑𝑥𝑑𝑠+^𝜏0¹⁰𝑟²𝑢^2𝑥𝑥𝑑𝑥𝑑𝑠≤𝐶,𝜏∈[0,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 4.1–4.7, the following inequality holds: ¹⁰𝑢²+𝑑𝑥+¹⁰𝜌−𝜌²𝑑𝑥+¹⁰𝑢^2𝑥𝑑𝑥+¹⁰𝑢^2𝜏𝑑𝑥+¹⁰𝜌^2𝑥𝑑𝑥_𝜏0¹⁰𝜌^2𝑥𝑑𝑥𝑑𝑠+^𝜏0¹⁰𝑢^2𝑥𝑑𝑥𝑑𝑠+^𝜏0¹⁰+𝑢^2𝑠𝑑𝑥𝑑𝑠_𝜏0¹⁰𝑢^2𝑥𝑥𝑑𝑥𝑑𝑠+^𝜏0¹⁰𝑢^2𝑥𝜏𝑑𝑥𝑑𝑠≤𝐶.(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 ‖(𝜌−𝜌̃,𝑢)(⋅,𝑡)‖_𝐿^∞_{([𝑟−,𝑟+])}≤𝐶₀𝑒^−𝐶1𝑡,𝑡>0,(4.44) where 𝐶₀ and 𝐶₁ denote the constants independent of time and 𝜌̃=∫^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟.

Proof. In a similar argument to show (4.8) and (4.12) with modification, we can obtain the following 𝑑𝑑𝑡^𝑟+_𝑟−12𝜌𝑢²+1𝛾−1𝜌^𝛾−𝜌̃^𝛾−𝛾𝜌̃^𝛾−1(𝜌−𝜌̃)𝑟²𝑑𝑟+^𝑟+_𝑟−2𝜌𝑢²+𝜌𝑢^2𝑟𝑟²𝑑𝑟=0,(4.45)12𝑑𝑑𝑡^𝑟+_𝑟−𝜌𝑢+𝜌⁻¹𝜌_𝑟²𝑟²𝑑𝑟+1𝛾−1𝑑𝑑𝑡^𝑟+_𝑟−𝜌^𝛾+𝛾_𝑟+𝑟−−𝜌̃^𝛾−𝛾𝜌̃^𝛾−1(𝜌−𝜌̃)𝑟²𝑑𝑟𝜌^𝛾−2𝜌^2𝑟𝑟²𝑑𝑟=0.(4.46) It holds from (4.9), (4.13), and (4.25) that ^𝑟+_𝑟−(𝜌−𝜌̃)²𝑟²𝑑𝑟≤𝐶^𝑟+_𝑟−𝜌^𝛾−2𝜌^2𝑟𝑟²𝑑𝑟.(4.47) Denote 𝐸(𝑡)=∶^𝑟+_𝑟−𝜌𝑢²+𝑢+𝜌⁻¹𝜌_𝑟²𝑟²𝑑𝑟+^𝑟+_𝑟−𝜌^𝛾−𝜌̃^𝛾−𝛾𝜌̃^𝛾−1(𝜌−𝜌̃)𝑟²𝑑𝑟.(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)𝑒^−𝐶𝑡.(4.50) By the fact that 𝐸(𝑡)≥𝑐‖𝑢‖^2𝐿_{2𝑟−,𝑟+}+‖𝜌−𝜌̃‖^2𝐿_{2𝑟−,𝑟+}+‖‖𝜌_𝑟‖‖^2𝐿_{2𝑟−,𝑟+},(4.51) where 𝑐>0 is a constant independent of time and Gagliardo-Nirenberg-Sobolev inequality ‖(𝜌−𝜌̃,𝑢)‖_𝐿^∞_{([𝑟−,𝑟+])}≤𝐶‖(𝜌−𝜌̃,𝑢)‖^𝐿1/2_{2([𝑟−,𝑟+])}‖‖(𝜌−𝜌̃,𝑢)_𝑟‖‖^𝐿1/2_{2𝑟−,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.