Global Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Abstract

We consider the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data in this paper. For piecewise regular initial density with bounded jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially as 𝑡→+∞.

1. Introduction

The isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients read as follows: 𝜌_𝑡+div𝜌𝐔=0,𝜌𝐔_𝑡𝜌ℎ𝜌𝐷(𝐔)𝑔𝜌+div𝜌𝐔⊗𝐔+∇𝑃−div−∇div𝐔=0, (1) where 𝑡∈(0,+∞) is the time and 𝐱∈𝑅^𝑁, 𝑁 is the spatial coordinate, and 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 (2) is the strain tensor and ℎ(𝜌), 𝑔(𝜌) are the Lamé viscosity coefficients satisfying ℎ𝜌>0,ℎ𝜌+𝑁𝑔𝜌≥0. (3)

There is huge literature on the studies of the global existence of weak solutions and dynamical behaviors of jump discontinuity for the compressible Navier-Stokes equations with discontinuous initial data; for instance, if the viscosity coefficients ℎ(𝜌) and 𝑔(𝜌) are both constants, for the case of one space dimension, Hoff investigated the global existence of discontinuous solutions of the Navier-Stokes equations [1–3]. Hoff derived the construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data [4]; therein, Hoff also showed that the discontinuities persist for all time, convecting along particle trajectories, and decaying at a rate inversely proportional to the viscosity coefficient. Hoff also obtained the global existence theorems for the multidimensional Navier-Stokes equations of isothermal compressible flows with the polytropic equation of state 𝑝(𝜌)=𝜌^𝛾(𝛾≥1) [5, 6]. The global existence of weak solutions was proved for the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data by Chen et al. in [7]. Hoff showed the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting fluids in two and three space dimensions when the initial data may be discontinuous across a hypersurface of 𝑅^𝑛 [8]. The global existence of solutions of the Navier-Stokes equations for compressible, barotropic flow in two space dimensions which exhibit convecting singularity curves, was proved by Hoff in [9].

If the viscosity coefficients ℎ(𝜌)=𝜌^𝛼, 𝑔(𝜌)=0, for the case of one space dimension, Fang and Zhang proved the global existence of unique piecewise smooth solution to the free boundary value problem for (1) with 0<𝛼<1, where the initial density is piecewise smooth with possibly large jump discontinuities [10]. Lian et al. addressed the initial boundary value problem for (1) with 0<𝛼≤1 subject to piecewise regular initial data with initial vacuum state included [11], where they obtained the global existence of unique piecewise regular solution and the finite time vanishing of vacuum state. In particular, they got that the jump discontinuity of density decays exponentially but never vanishes in any finite time and the piecewise regular solution tends to the equilibrium state exponentially as 𝑡→+∞.

In this present paper, we consider the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data and focus on the regularities and dynamical behaviors of global weak solution and so forth. As 𝛾>1, we show that the initial boundary value problem with piecewise regular initial data admits a unique global piecewise regular solution, where the discontinuity in piecewise regular initial density is bounded. In particular, the jump discontinuity of density decays exponentially and the piecewise regular solution tends to the equilibrium state exponentially as 𝑡→+∞.

There are also many significant progresses achieved recently on the compressible Navier-Stokes equations with density-dependent viscosity coefficients. For instance, the mathematical derivations are obtained in the simulation of flow surface in shallow region [12, 13]. The prototype model is the viscous Saint-Venant equation (corresponding to (1) with 𝑃(𝜌)=𝜌², ℎ(𝜌)=𝜌, and 𝑔(𝜌)=0). Many authors considered 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; refer to [14–22] and references therein. The global existence of classical solutions is shown by Mellet and Vasseur in [23]. 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 long time, the dynamical behaviors of vacuum boundary, the long time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [24–26] and references therein.

The rest part of the paper is arranged as follows. In Section 2, the main results about the existence and dynamical behaviors of global piecewise regular solution for compressible Navier-Stokes equations are stated. Then, some important a priori estimates will be given in Section 3. Finally, the theorem is proved in Section 4.

2. Notations and Main Results

We will take ℎ(𝜌)=𝜌 and 𝑔(𝜌)=0 and 𝐷(𝐔)=∇𝐔 in (1) for simplicity. The isentropic compressible Navier-Stokes equations read as 𝜌_𝑡+div𝜌𝐔=0,𝜌𝐔_𝑡+div𝜌𝐔⊗𝐔+∇𝜌^𝛾−div𝜌∇𝐔=0, (4) with the initial data and boundary condition 𝜌,𝐔(𝐱,0)=𝜌₀,𝐔₀(𝐱)𝐔()[],,𝐱∈Ω,𝐱,𝑡=0,𝐱∈𝜕Ω,𝑡∈0,𝑇 (5) where Ω∶={𝐱∣0<𝑟₋≤|𝐱|≤𝑟₊} and 𝑟₋ and 𝑟₊ are constants.

We will investigate the spherically symmetric solutions of the system (4) in a spherically symmetric domain Ω and denote that |𝐱|()()()()𝐱=𝑟,𝜌𝐱,𝑡=𝜌𝑟,𝑡,𝐔𝐱,𝑡=𝑢𝑟,𝑡𝑟, (6) which gives the following system for 𝑟>0: 𝜌_𝑡+𝜌𝑢_𝑟+2𝜌𝑢𝑟=0,𝜌𝑢_𝑡+𝜌𝑢²+𝜌^𝛾_𝑟+2𝜌𝑢²𝑟−𝜌𝑢_𝑟𝑟−𝜌2𝑢𝑟_𝑟=0, (7) and the initial data and boundary condition become 𝜌,𝑢(𝑟,0)=𝜌₀,𝑢₀(𝑟),𝑟∈𝑟₋,𝑟₊,𝑢𝑟₋,𝑡=𝑢𝑟₊,𝑡=0,𝑡∈[0,𝑇]. (8)

For simplicity, we consider the initial data with one discontinuous point 𝜉₀∈(𝑟₋,𝑟₊); namely, 𝑟²𝜌₀∈𝐿¹∩𝑊^1,∞𝑟₋,𝜉₀∪𝜉₀,𝑟₊,inf_{𝑟𝐱∈−,𝜉0∪𝜉0,𝑟+}𝜌₀>𝜌₋𝜌>0,₀𝜉₀−0≠𝜌₀𝜉₀,𝜌+0₀𝜉₀+0−𝜌₀𝜉₀𝑟−0<𝛿,₂𝑢₀∈𝐻²𝑟₋,𝜉₀∪𝜉₀,𝑟₊, (9) where 𝜌₋ and 𝛿>0 are positive constants, and 𝛿>0 is bounded.

Then, we can give the main results as follows.

Theorem 1. Let 𝛾>1. Assume that the initial data satisfies (9). Then, there exist two positive constants 𝜌_∗, 𝜌^∗ and a unique global piecewise regular solution (𝜌,𝑢) to the initial boundary value problem (7)-(8), namely, satisfying 0<𝜌_∗≤𝜌(𝑟,𝑡)≤𝜌^∗,(𝑟,𝑡)∈𝑟₋,𝜉(𝑡)∪𝜉(𝑡),𝑟₊×[0,𝑇],𝜌∈𝐿^∞[0,𝑇];𝐻¹𝑟₋,𝜉(𝑡)∪𝜉(𝑡),𝑟₊,𝑢∈𝐿^∞[0,𝑇];𝐻¹𝑟₋,𝜉(𝑡)∪𝜉(𝑡),𝑟₊,𝑢_𝑟∈𝐿²[0,𝑇];𝐻¹𝑟₋,𝜉(𝑡)∪𝜉(𝑡),𝑟₊, (10) where 𝑟=𝜉(𝑡) is a curve defined by 𝑑𝜉(𝑡)(𝜉(𝑡))(0)𝑑𝑡=𝑢,𝑡,𝜉=𝜉₀,𝑡>0, (11) along which the Rankine-Hugoniot conditions hold 𝑢(𝜉(𝑡)𝜌,𝑡)=0,_𝛾(𝜉(𝑡),𝑡)=𝜌𝑢_𝑟+2𝜌𝑢𝑟(𝜉(𝑡),𝑡), (12) where [𝑓(𝜉(𝑡),𝑡)]∶=𝑓(𝜉(𝑡)+0,𝑡)−𝑓(𝜉(𝑡)−0,𝑡), and along the discontinuity 𝑟=𝜉(𝑡) the jump decays exponentially 𝜌₀𝜉₀𝑒^−𝐶0𝑡≤𝜌(𝜉(𝑡),𝑡)≤𝜌₀𝜉₀𝑒^−𝐶1𝑡, (13) and furthermore, the solution (𝜌,𝑢) tends to the equilibrium state exponentially 𝜌−𝜌,𝑢(⋅,𝑡)_{𝐿∞([𝑟−,𝜉(𝑡))∪(𝜉(𝑡),𝑟+])}≤𝐶₂𝑒^−𝐶3𝑡, (14) where 𝐶₀, 𝐶₁, 𝐶₂, and 𝐶₃ are positive constants independent of time and 𝜌=1/(𝑟₊−𝑟₋)⨍^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟, where ⨍^𝑟+_𝑟−∶=∫_𝑟^𝜉(𝑡)₋+∫^𝑟+_𝜉(𝑡).

Remark 2. Theorem 1 holds for the Saint-Venant model for shallow water; that is, 𝛾=2.

Remark 3. For the piecewise regular initial data, Hoff [4] has proved that there is a (piecewise regular) weak solution to the initial boundary value problem (7)-(8) in terms of the difference scheme. In addition, there is a curve 𝑟=𝜉(𝑡) starting from 𝜉(0)=𝜉₀, such that the density is discontinuous cross the curve 𝑟=𝜉(𝑡), and the Rankine-Hugoniot conditions hold 𝑢(𝜉(𝑡)𝜌,𝑡)=0,_𝛾(𝜉(𝑡),𝑡)=𝜌𝑢_𝑟+2𝜌𝑢𝑟(𝜉(𝑡),𝑡). (15)
To extend the local solution globally in time, we have to prove the uniformly a priori estimates.

3. The a Priori Estimates

According to the analysis made in [27], there is a curve 𝑟=𝜉(𝑡) defined by 𝑑𝜉(𝑡)(𝜉(𝑡))(0)𝑑𝑡=𝑢,𝑡,𝜉=𝜉₀,𝑡>0, (16) along which the Rankine-Hugoniot conditions hold 𝑢(𝜉(𝑡)𝜌,𝑡)=0,_𝛾(𝜉(𝑡),𝑡)=𝜌𝑢_𝑟+2𝜌𝑢𝑟(𝜉(𝑡),𝑡), (17) where [𝑓(𝜉(𝑡),𝑡)]∶=𝑓(𝜉(𝑡)+0,𝑡)−𝑓(𝜉(𝑡)−0,𝑡). It is convenient to make use of the Lagrange coordinates so as to establish the uniformly a priori estimates and take the Lagrange coordinates transform 𝑥=^𝑟_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟,𝜏=𝑡. (18)

By (18) and the conservation of mass for (𝜌,𝑢)^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟=^𝑟+_𝑟−𝜌₀(𝑟)𝑟²𝑑𝑟∶=1, (19) the Lagrange coordinates transform (18) maps (𝑟,𝑡)∈[𝑟₋,𝑟₊]×𝑅⁺ into (𝑥,𝜏)∈[0,1]×𝑅⁺. The curve 𝑟=𝜉(𝑡) in Eulerian coordinates is changed to a line 𝑥=𝑥₀ in Lagrangian coordinates, where 𝑥₀=_𝑟^𝜉(𝑡)₋𝜌(𝑟,𝑡)𝑟²𝑑𝑟=^𝜉0_𝑟−𝜌₀(𝑟)𝑟²𝑑𝑟∈(0,1), (20) and the jump conditions become 𝑢𝑥₀,𝜏=0,𝜌^𝛾𝑥₀,𝜏=𝜌²𝑟²𝑢_𝑥𝑥₀,𝜏. (21)

The relations between Lagrangian and Eulerian coordinates are satisfied as 𝜕𝑥𝜕𝑟=𝜌𝑟²,𝜕𝑥𝜕𝑡=−𝜌𝑢𝑟², (22) and the initial boundary value problem (7)-(8) is reformulated to 𝜌_𝜏+𝜌²𝑟²𝑢_𝑥𝑟=0,₋₂𝑢_𝜏+𝜌^𝛾_𝑥=𝜌²𝑟²𝑢_𝑥𝑥−2𝜌_𝑥𝑢𝑟,𝜌,𝑢(𝑥,0)=𝜌₀,𝑢₀(𝑥)[],𝑢()()[),,𝑥∈0,10,𝑡=𝑢1,𝑡=0,𝜏∈0,+∞ (23) where the initial data satisfies 𝜌₀∈𝐿¹([0,1])∩𝑊^1,∞0,𝑥₀∪𝑥₀,1,inf_{𝑥∈0,𝑥0∪𝑥0,1}𝜌₀>𝜌₋𝜌>0,₀𝑥₀−0≠𝜌₀𝑥₀,𝜌+0₀𝑥₀+0−𝜌₀𝑥₀1−0<𝛿,𝑟²𝜌₀𝑟²𝑢₀∈𝐿²0,𝑥₀∪𝑥₀,1,𝑟²𝜌₀𝑟²𝑢_0𝑥∈𝐿²0,𝑥₀∪𝑥₀,1,1𝑟²𝜌₀𝑟²𝜌₀𝑟²𝜌₀𝑟²𝑢_0𝑥𝑥∈𝐿²0,𝑥₀∪𝑥₀,1. (24)

Next, we will deduce the a priori estimates for the solution (𝜌,𝑢) to the initial boundary value problem (23). To prove the a priori estimates, we assume a priori that there are constants 𝜌_±>0 such that 0<𝜌₋≤𝜌(𝑥,𝜏)≤𝜌₊,(𝑥,𝜏)∈0,𝑥₀∪𝑥₀,1×[0,𝑇],𝑇>0. (25)

Lemma 4. Let 𝑇>0. Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (23) that ¹₀14𝑢²+1𝜌𝛾−1^𝛾−1𝑑𝑥+^𝜏₀¹₀2𝑢²𝑟²+12𝜌²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠≤¹₀12𝑢²₀+1𝜌𝛾−1^0𝛾−1𝑑𝑥,𝜏∈[0,𝑇], (26) where ⨍¹₀∶=∫^𝑥0₀+∫¹_𝑥0.

Proof. Multiplying (23)₂ by 𝑟²𝑢 and integrating the result with respect to 𝑥 over [0,1], it holds from (21) and (23)₁ that 𝑑𝑑𝜏¹₀12𝑢²+1𝜌𝛾−1^𝛾−1𝑑𝑥+¹₀𝜌²𝑟²𝑢²_𝑥𝑑𝑥=2¹₀𝜌𝑟𝑢²_𝑥𝑑𝑥+2𝜌𝑟𝑢²_𝑥=𝑥0, (27) and integrating (27) with respect to 𝜏, we have ¹₀12𝑢²+1𝜌𝛾−1^𝛾−1𝑑𝑥+^𝜏₀¹₀2𝑢²𝑟²+𝜌²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠=¹₀12𝑢²₀+1𝜌𝛾−1^0𝛾−1𝑑𝑥+^𝜏₀2𝜌𝑟𝑢²_𝑥=𝑥0𝑑𝑠. (28)
Using (21) and (23)₁, we deduce 𝜌_𝜏+𝜌^𝛾=0, (29) which gives 𝜌=𝜌₀exp−^𝜏₀𝜌^𝛾𝜌𝑑𝑠. (30)
It holds from (25) that 𝛾𝜌₋^𝛾−1≤𝜌^𝛾𝜌≤𝛾𝜌₊^𝛾−1. (31)
From (24), (30), and (31), we have ^𝜏₀2𝜌𝑟𝑢²_𝑥=𝑥0≤𝑑𝑠𝐶1+𝜌₋^−1𝜏₀𝜌₀𝑒^{−𝛾𝜌−𝛾−1𝑠1}₀𝑢²𝑑𝑥+¹₀𝑢²_𝑥≤𝑑𝑥𝑑𝑠𝐶1+𝜌₋⁻¹𝜌₀sup_{[]𝜏∈0,𝑇}¹₀𝑢²𝑑𝑥^𝜏₀𝑒^{−𝛾𝜌−𝛾−1𝑠}+𝑑𝑠𝐶1+𝜌₋⁻¹𝜌₀𝜌₋⁻²𝑟₋^−4𝜏₀¹₀𝜌²𝑢²_𝑥𝑟⁴≤1𝑑𝑥𝑑𝑠4sup_{[]𝜏∈0,𝑇}¹₀𝑢²1𝑑𝑥+2^𝜏₀¹₀𝜌²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠, (32) where 𝐶 is a positive constant independent of time and we assume that 0<𝜌₀𝛾≤min4𝐶1+𝜌₋⁻¹𝜌₋^𝛾−1,12𝐶1+𝜌₋⁻¹𝜌²₋𝑟⁴₋∶=𝑀₁. (33)
From (28) and (32), Lemma 4 can be proved.

Lemma 5. Let 𝑇>0. Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (23) that 12¹₀𝑢+𝑟²𝜌_𝑥²1𝑑𝑥+𝛾−1¹₀𝜌^𝛾−1𝑑𝑥+𝛾^𝜏₀¹₀𝜌^𝛾−1𝜌²_𝑥𝑟⁴≤1𝑑𝑥𝑑𝑠2¹₀𝑢₀+𝑟²𝜌_0𝑥²+1𝑑𝑥𝛾−1¹₀𝜌₀^𝛾−1𝑑𝑥+𝐶,𝜏∈[0,𝑇], (34) where 𝐶 is a positive constant independent of time.

Proof. Differentiating (23)₁ with respect to 𝑥, we get 𝜌_𝑥𝜏+𝜌²𝑟²𝑢_𝑥𝑥=0. (35)
Summing (35) and (23)₂, it holds 𝑟⁻²𝑢+𝜌_𝑥𝜏+𝜌^𝛾_𝑥=𝑟⁻²_𝜏𝑢−2𝜌_𝑥𝑢𝑟. (36)
Note that 𝑟³(𝑥,𝜏)=𝑟³₋+3^𝑥₀1𝜌(𝑧,𝜏)𝑑𝑧, (37) and so 𝜕𝑟=1𝜕𝜏𝑟^2𝑥₀1𝜌_𝑡()=1𝑧,𝑡𝑑𝑧𝑟^2𝑥₀𝑟²𝑢_𝑧(𝑧,𝑡)𝑑𝑧=𝑢(𝑥,𝜏), (38) which together with (36) implies 𝑟⁻²𝑢+𝜌_𝑥𝜏+𝜌^𝛾_𝑥=−2𝑟⁻³𝑢²−2𝜌_𝑥𝑢𝑟. (39)
Multiplying (39) by (𝑢+𝑟²𝜌_𝑥)𝑟² and integrating the result with respect to 𝑥 and 𝜏, it holds that 12¹₀𝑢+𝑟²𝜌_𝑥²1𝑑𝑥+𝛾−1¹₀𝜌^𝛾−1𝑑𝑥+𝛾^𝜏₀¹₀𝜌^𝛾−1𝜌²_𝑥𝑟⁴=1𝑑𝑥𝑑𝑠2¹₀𝑢₀+𝑟²𝜌_0𝑥²1𝑑𝑥+𝛾−1¹₀𝜌₀^𝛾−1+𝑑𝑥_𝜏0𝜌^𝛾𝑟²𝑢_𝑥=𝑥0𝑑𝑠. (40)
From (24), (26), (30), and (31), we have that ^𝜏₀𝜌^𝛾𝑟²𝑢_𝑥=𝑥0≤𝐶𝑑𝑠1+𝜌₋^−2𝜏₀𝜌₊^𝛾−1𝜌¹₀𝑢²𝑑𝑥+¹₀𝑢²_𝑥𝑑𝑥^1/2≤𝐶𝑑𝑠1+𝜌₋^−2𝜏₀𝜌₀𝑒^{−𝛾𝜌−𝛾−1𝑠}𝜌₊^2(𝛾−1)+¹₀𝑢²𝑑𝑥+𝜌₋⁻²𝑟₋⁻⁴¹₀𝜌²𝑢²_𝑥𝑟⁴≤𝐶𝑑𝑥𝑑𝑠1+𝜌₋⁻²𝜌₀𝜌₋^1−𝛾𝜌₊^2(𝛾−1)+𝜌₋^1−𝛾+𝜌₋⁻²𝑟₋⁻⁴≤𝐶, (41) where 𝐶 is a positive constant independent of time and we assume that 0<𝜌₀𝜌<min^−𝛾−1𝜌₊^2(1−𝛾)1+𝜌₋⁻²,𝜌₋^𝛾−11+𝜌₋⁻²,𝜌²₋𝑟⁴₋1+𝜌₋⁻²∶=𝑀₂. (42)
The proof of (34) is completed.

Lemma 6. Let 𝑇>0. Under the conditions in Theorem 1, there exists a constant 𝜌^∗>0 such that 0<𝜌(𝑥,𝜏)≤𝜌^∗,(𝑥,𝜏)∈[0,1]×[0,𝑇]. (43)

Proof. It follows from (19) and (34) that 𝜌()()≤𝑥,𝜏=𝜌𝑟,𝑡_𝑟+𝑟−𝜌(𝑟,𝑡)𝑑𝑟+^𝑟+_𝑟−𝜌_𝑟(𝑟,𝑡)𝑑𝑟≤𝑟₋^−2𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟+𝑟₋^−2𝑟+_𝑟−𝜌^1/2𝜌^1/2_𝑟(𝑟,𝑡)𝑟²𝑑𝑟≤𝐶+𝐶^𝑟+_𝑟−𝜌𝑟²𝑑𝑟^1/2𝑟+_𝑟−𝜌^1/2_𝑟(𝑟,𝑡)²𝑟²𝑑𝑟^1/2≤𝐶+𝐶max¹₀𝜌_𝑥(𝑥,𝜏)²𝑟⁴𝑑𝑥^1/2≤𝐶, (44) which yields 𝜌(𝑥,𝜏)≤𝐶∶=𝜌^∗. (45)
Thus, we can choose 𝜌₊=𝜌^∗+1 to have 0<𝜌(𝑥,𝜏)≤𝜌^∗≤𝜌₊,(𝑥,𝜏)∈0,𝑥₀∩𝑥₀,1×[0,𝑇]. (46)

Lemma 7. Let 𝑇>0. Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (23) that ¹₀𝑢^2𝑛𝑑𝑥+^𝜏₀¹₀𝑢^2𝑛𝑟²+𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶(𝑇), (47) for any positive integer 𝑛∈𝑁, where 𝐶(𝑇) is a positive constant dependent of time.

Proof. Multiplying (23)₂ with 𝑟²𝑢^2𝑛−1, integrating by parts over [0,1], we have 𝑑𝑑𝜏¹₀𝑢^2𝑛2𝑛𝑑𝑥+¹₀𝜌²𝑟²𝑢_𝑥𝑟²𝑢^2𝑛−1_𝑥=𝑑𝑥₁₀𝜌^𝛾𝑟²𝑢^2𝑛−1_𝑥+𝜌2𝑟𝑢^2𝑛_𝑥𝜌𝑑𝑥+2𝑟𝑢^2𝑛_𝑥=𝑥0. (48)
Since it holds that 𝑟²𝑢_𝑥𝑟²𝑢^2𝑛−1_𝑥=2𝑢𝜌𝑟+𝑟²𝑢_𝑥2𝑢^2𝑛−1+()𝑟𝜌𝑟2𝑛−1²𝑢^2𝑛−2𝑢_𝑥=4𝑢^2𝑛𝜌²𝑟²+(2𝑛−1)𝑢^2𝑛−2𝑢²_𝑥𝑟⁴+4𝑛𝑢^2𝑛−1𝑢_𝑥𝑟𝜌, (49) it follows from (48) that 𝑑𝑑𝜏¹₀𝑢^2𝑛2𝑛𝑑𝑥+2¹₀𝑢^2𝑛𝑟²𝑑𝑥+(2𝑛−1)¹₀𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴𝑑𝑥=2¹₀𝜌^𝛾−1𝑢^2𝑛−1𝑟𝑑𝑥+(2𝑛−1)¹₀𝜌^𝛾𝑢^2𝑛−2𝑢_𝑥𝑟²𝜌𝑑𝑥+2𝑟𝑢^2𝑛_𝑥=𝑥0≤¹₀𝑢^2𝑛𝑟²𝑑𝑥+¹₀𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴𝑑𝑥+𝐶𝜌_𝐿^2(𝛾−1)_∞¹₀𝑢^2𝑛−2𝜌𝑑𝑥+2𝑟𝑢^2𝑛_𝑥=𝑥0, (50) which together with (43) and Young’s inequality yields 𝑑𝑑𝜏¹₀𝑢^2𝑛𝑑𝑥+¹₀𝑢^2𝑛𝑟²𝑑𝑥+2(𝑛−1)¹₀𝑢^2𝑛−2𝑢²_𝑥𝑟⁴𝑑𝑥≤𝐶+𝐶¹₀𝑢^2𝑛𝑑𝑥+2𝜌𝑟𝑢^2𝑛_𝑥=𝑥0, (51) and applying the Gronwall’s inequality to (51), we can obtain ¹₀𝑢^2𝑛𝑑𝑥+^𝜏₀¹₀𝑢^2𝑛𝑟²()𝑑𝑥𝑑𝑠+2𝑛−1^𝜏₀¹₀𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴(𝑇)𝑑𝑥𝑑𝑠≤𝐶+2^𝜏₀𝜌𝑟𝑢^2𝑛_𝑥=𝑥0𝑑𝑠, (52) where 𝐶(𝑇) is a positive constant dependent of time. It holds from (24), (26), (30), and (31) that ^𝜏₀𝜌𝑟𝑢^2𝑛_𝑥=𝑥0≤𝐶𝑑𝑠1+𝜌₋^−1𝜏₀𝜌₀𝑒^{−𝛾𝜌−𝛾−1𝑠}×¹₀𝑢^2𝑛𝑑𝑥+¹₀𝑢^2𝑛−2𝑢²_𝑥≤𝐶𝑑𝑥𝑑𝑠1+𝜌₋⁻¹𝜌₀sup_{[]𝜏∈0,𝑇}¹₀𝑢^2𝑛𝑑𝑥^𝜏₀𝑒^{−𝛾𝜌−𝛾−1𝑠}+𝐶𝑑𝑠1+𝜌₋⁻¹𝜌₀𝜌₋⁻²𝑟₋^−4𝜏₀¹₀𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴≤1𝑑𝑥𝑑𝑠2sup_{[]𝜏∈0,𝑇}¹₀𝑢^2𝑛𝑑𝑥+𝑛−12^𝜏₀¹₀𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠, (53) where 𝐶 is a positive constant independent of time and we assume that 0<𝜌₀𝛾≤min2𝐶1+𝜌₋⁻¹𝜌₋^𝛾−1,(𝑛−1)2𝐶1+𝜌₋⁻¹𝜌²₋𝑟⁴₋∶=𝑀₃. (54)
From (52) and (53), we can deduce (47).

Lemma 8. Let 𝑇>0, for 𝑛∈𝑁, and 𝑛>1/2(𝛾−1). Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (23) that ^𝜏₀𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛_{𝐿∞([0,𝑥0)∪(𝑥0,1])}𝑑𝑠≤𝐶_𝜏0(𝑇),|(𝜌^𝛾)_𝑥𝑟²|^2𝑛_{𝐿∞([0,𝑥0)∪(𝑥0,1])}𝑑𝑠≤𝐶(𝑇),𝜏∈[0,𝑇], (55)(56) where 𝐶(𝑇) is a positive constant dependent of time.

Proof. By means of Sobolev imbedding theorem and Cauchy-Schwartz inequality, applying (34), (43), and (47), we get ^𝜏₀𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛_{𝐿∞([0,𝑥0)∪(𝑥0,1])}𝑑𝑠≤^𝜏₀¹₀𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛𝑑𝑥𝑑𝑠+^𝜏₀¹₀𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛_𝑥(𝑇)𝑑𝑥𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝜌^{2𝑛(𝛾−1)−1}𝜌_𝑥𝑢^2𝑛𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛−1𝑢_𝑥(𝑇)𝑑𝑥𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝜌²_𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌^{2(2𝑛(𝛾−1)−1)}𝑢^4𝑛𝑟⁻⁴+𝑑𝑥𝑑𝑠_𝜏010𝜌^{2(2𝑛(𝛾−1)−1)}𝑢^2𝑛𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌²𝑢^2𝑛−2𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶(𝑇). (57)
Next, we find that 𝜌_𝑥𝑟²=𝜌_0𝑥𝑟²+𝑢₀−𝑢−^𝜏₀𝜌^𝛾_𝑥𝑟²𝑑𝑠, (58) which together with (43), (47), and (57) gives ^𝜏₀𝜌^𝛾_𝑥𝑟^22𝑛_𝐿∞𝑑𝑠=𝛾^2𝑛𝜏₀𝜌^{2𝑛(𝛾−1)}𝜌_𝑥𝑟^22𝑛_𝐿∞𝑑𝑠=𝛾^2𝑛𝜏₀𝜌^{2𝑛(𝛾−1)}𝜌_0𝑥𝑟²+𝑢₀−𝑢−^𝑠₀𝜌^𝛾_𝑥𝑟²𝑑𝑙^2𝑛_𝐿∞𝑑𝑠≤𝐶^𝜏₀𝜌^{2𝑛(𝛾−1)}𝜌^2𝑛_0𝑥+𝑢₀^2𝑛+𝑢^2𝑛_𝐿∞𝑑𝑠+𝐶^𝜏₀𝜌^{2𝑛(𝛾−1)𝑠}₀𝜌^𝛾_𝑥𝑟²𝑑𝑙^2𝑛_𝐿∞(𝑇)𝑑𝑠≤𝐶+𝐶^𝜏₀𝜌^{2𝑛(𝛾−1)}𝑢^2𝑛_𝐿∞(𝑇)𝑑𝑠+𝐶^𝜏₀^𝑠₀𝜌^𝛾_𝑥𝑟^22𝑛_𝐿∞𝑑𝑙𝑑𝑠, (59) and applying the Gronwall’s inequality to (59), we obtain (56).

Lemma 9. Let 𝑇>0. Under the conditions in Theorem 1, there exists a constant 𝜌_∗>0 such that 𝜌(𝑥,𝜏)≥𝜌_∗>0,(𝑥,𝜏)∈[0,1]×[0,𝑇]. (60)

Proof. It is easy to verify 𝜌_𝐿^(𝛾+1)/2(𝜏)1∶=𝑥₀^𝑥0₀𝜌^(𝛾+1)/2()≥𝑟𝑥,𝜏𝑑𝑥₂₋𝑟²₊𝑥_0𝑟^𝜉(𝑡)₋𝜌(𝑟,𝑡)𝑟²𝑑𝑟^(𝛾+3)/2=𝑟²₋𝑟²₊𝑥₀^𝜉0_𝑟−𝜌₀(𝑟)𝑟²𝑑𝑟^(𝛾+3)/2>0,𝜌^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝜏)²_{𝐿2([0,𝑥0))}∈𝑊^1,1([0,𝑇]). (61)(62)
Indeed, it holds that ^𝜏₀𝜌^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝑠)²_𝐿20,𝑥0𝑑𝑠≤𝐶^𝜏₀𝜌^(𝛾+1)/2_𝑥𝑟²²_𝐿20,𝑥0𝑑𝑠≤𝐶, (63) which together with (23), (34), and (43) gives ^𝜏₀𝑑𝜌𝑑𝑠^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝑠)²_𝐿20,𝑥0≤𝑑𝑠𝛾+1^𝜏₀^𝑥0₀𝜌^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝑠)×𝜌^(𝛾−1)/2𝜌_𝑠𝑑𝑥𝑑𝑠+2^𝜏₀^𝑥0₀𝜌^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝑠)×𝜌_𝐿^(𝛾+1)/2(𝑠)_𝑠𝑑𝑥𝑑𝑠≤𝐶^𝜏₀^𝑥0₀𝜌^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝑠)²𝑑𝑥𝑑𝑠+𝐶^𝜏₀^𝑥0₀𝑢²𝑟²+𝜌²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶. (64)
Similarly, we can obtain 𝜌^(𝛾+1)/2−𝜌_𝑅^(𝛾+1)/2(𝜏)²_𝐿2𝑥0,1∈𝑊^1,1([0,𝑇]), (65) where we have used 𝜌_𝑅^(𝛾+1)/2(𝜏)1∶=1−𝑥₀¹_𝑥0𝜌^(𝛾+1)/2()≥𝑟𝑥,𝜏𝑑𝑥₂₋𝑟²₊1−𝑥₀^𝑟+_𝜉(𝑡)𝜌(𝑟,𝑡)𝑟²𝑑𝑟^(𝛾+3)/2=𝑟²₋𝑟²₊𝑥₀^𝑟+_𝜉0𝜌₀(𝑟)𝑟²𝑑𝑟^(𝛾+3)/2>0. (66)
By Gagliardo-Nirenberg-Sobolev inequality, (62), and (65), we have 𝜌^(𝛾+1)/2−𝜌_𝐿^(𝛾+1)/2(𝜏)²_{𝐿∞0,𝑥0}+𝜌^(𝛾+1)/2−𝜌_𝑅^(𝛾+1)/2(𝜏)²_{𝐿∞𝑥0,1}⟶0as𝜏⟶+∞. (67)
Thus, there is a 𝑇₀>0 and a constant 𝜌₁>0 such that 𝜌(𝑥,𝜏)≥𝜌₁,𝑥∈0,𝑥₀∪𝑥₀,1,𝜏∈𝑇₀,+∞. (68)
For 𝜏∈[0,𝑇₀], denote 𝑣()=1𝑥,𝜏𝜌(),𝑥,𝜏 (69) and then from (23), we can obtain 𝑣_𝑡=𝑟²𝑢_𝑥, (70) and multiplying (70) by 4𝑣³ and integrating the result over ([0,𝑥₀)∪(𝑥₀,1])×[0,𝑇₀], we have ¹₀𝑣⁴𝑑𝑥=¹₀𝑣⁴₀𝑑𝑥+4^𝜏₀¹₀𝑣³𝑟²𝑢_𝑥=𝑑𝑥𝑑𝑠₁₀𝑣⁴₀𝑑𝑥−4^𝜏₀𝑣³𝑟²𝑢_𝑥=𝑥0𝑑𝑠+12^𝜏₀¹₀𝑣⁴𝑢𝜌_𝑥𝑟²=𝑑𝑥𝑑𝑠₁₀𝑣⁴₀𝑑𝑥−4^𝜏₀𝑣³𝑟²𝑢_𝑥=𝑥0𝑑𝑠+12^𝜏₀¹₀𝑣⁴𝑢𝜌_0𝑥𝑟²+𝑢₀−−𝑢_𝑠0𝜌^𝛾_𝑥𝑟²𝑑𝑙𝑑𝑥𝑑𝑠. (71)
From (56), it holds that ¹₀𝑣⁴𝑑𝑥+12^𝜏₀¹₀𝑣⁴𝑢²=𝑑𝑥𝑑𝑠₁₀𝑣⁴₀𝑑𝑥−4^𝜏₀𝑣³𝑟²𝑢_𝑥=𝑥0𝑑𝑠+12^𝜏₀¹₀𝑣⁴𝑢𝜌_0𝑥𝑟²𝑑𝑥𝑑𝑠+12^𝜏₀¹₀𝑣⁴𝑢𝑢₀𝑑𝑥𝑑𝑠−12^𝜏₀¹₀𝑣⁴𝑢^𝑠₀𝜌^𝛾_𝑥𝑟²𝑑𝑙𝑑𝑥𝑑𝑠≤6^𝜏₀¹₀𝑣⁴𝑢²𝑑𝑥𝑑𝑠+𝐶𝑇₀^𝜏₀¹₀𝑣⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀𝑣³𝑟²𝑢_𝑥=𝑥0𝑑𝑠. (72)
It holds from (24), (26), (30), and (31) that ^𝜏₀𝑣³𝑟²𝑢_𝑥=𝑥0𝑑𝑠≤𝐶1+𝜌₋^−2𝜏₀𝜌₊⁻⁴𝜌¹₀𝑢²𝑑𝑥+¹₀𝑢²_𝑥𝑑𝑥^1/2𝑑𝑠≤𝐶1+𝜌₋^−2𝜏₀𝜌₀𝑒^{−𝛾𝜌−𝛾−1𝑠}𝜌₊⁻⁸+¹₀𝑢²𝑑𝑥+𝜌₋⁻²𝑟₋⁻⁴¹₀𝜌²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠≤𝐶1+𝜌₋⁻²𝜌₀𝜌₋^1−𝛾𝜌₊⁻⁸+𝜌₋^1−𝛾+𝜌₋⁻²𝑟₋⁻⁴≤𝐶, (73) where 𝐶 is a positive constant independent of time and we assume that 0<𝜌₀<min𝑀₁,𝑀₂,𝑀₃,𝜌₋^𝛾−1𝜌⁸₊1+𝜌₋⁻²,𝜌₋^𝛾−11+𝜌₋⁻²,𝜌²₋𝑟⁴₋1+𝜌₋⁻²∶=𝛿. (74)
Then, it follows from (72) and (73) that ¹₀𝑣⁴𝑑𝑥≤𝐶+𝐶𝑇₀^𝜏₀¹₀𝑣⁴𝑑𝑥𝑑𝑠, (75) where 𝐶(𝑇₀) is a positive constant independent of time 𝑇₀. By Gronwall’s inequality, (75) leads to ¹₀𝑣⁴𝑑𝑥=¹₀1𝜌⁴𝑑𝑥≤𝐶𝑇₀. (76)
It holds for (𝑥,𝜏)∈([0,𝑥₀)∪(𝑥₀,1])×[0,𝑇₀] that 1𝜌=¹₀1𝜌𝑑𝑥+¹₀1𝜌_𝑥𝑑𝑥≤𝐶+𝐶¹₀1𝜌⁴𝑑𝑥^1/21₀𝜌²_𝑥𝑟⁴𝑑𝑥^1/2≤𝐶𝑇₀; (77) namely, 𝜌(𝑥,𝜏)≥𝐶𝑇₀∶=𝜌₂,(𝑥,𝜏)∈0,𝑥₀∪𝑥₀,1×0,𝑇₀. (78)
Therefore, we can choose 𝜌_∗=min𝜌₁,𝜌₂,𝜌₋=12𝜌_∗, (79) to get 𝜌≥𝜌_∗>𝜌₋,(𝑥,𝜏)∈0,𝑥₀∪𝑥₀,1×[0,+∞). (80)

Lemma 10. Let 𝑇>0. Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (23) that 𝜌₀𝑥₀𝑒^−𝐶0𝜏≤𝜌𝑥₀,𝜏≤𝜌₀𝑥₀𝑒^−𝐶1𝜏,𝜏∈[0,𝑇], (81) where 𝐶₀, 𝐶₁ are positive constants independent of time.

Proof. From (30), (31), (43), and (60), we can get (81).

Lemma 11. Let 𝑇>0. Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (23) that ¹₀𝑟²𝑢²_𝑥𝑑𝑥+¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥+^𝜏₀¹₀𝑟²𝑢²_𝑠𝑟⁻⁴+𝑑𝑥𝑑𝑠_𝜏010𝜌²𝑟²𝑢²_𝑥𝑠𝑑𝑥𝑑𝑠+^𝜏₀¹₀𝑟²𝑢²_𝑥𝑥[],𝑑𝑥𝑑𝑠≤𝐶,𝜏∈0,𝑇 (82) where 𝐶>0 denotes a constant independent of time.

Proof. Multiplying (23)₂ by 𝜌⁻²(𝑟²𝑢)_𝜏 and integrating the result with respect to 𝑥 over [0,1], making use of (21) and (24), it holds that 𝑑𝑑𝜏¹₀12𝑟²𝑢²_𝑥−𝜌^𝛾−2𝑟²𝑢_𝑥𝑑𝑥+¹₀𝜌⁻²𝑟²𝑢²_𝜏𝑟⁻⁴=𝑑𝑥𝛾−2¹₀𝜌^𝛾−1𝑟²𝑢²_𝑥𝑑𝑥−2¹₀𝜌^𝛾−3𝜌_𝑥𝑟²𝑢_𝜏𝑑𝑥+2¹₀𝜌⁻¹𝜌_𝑥𝑟²𝑢_𝑥𝑟²𝑢_𝜏𝑑𝑥+2¹₀𝜌⁻²𝑢²𝑟²𝑢_𝜏𝑟⁻³𝑑𝑥−2¹₀𝜌⁻²𝜌_𝑥𝑢𝑟²𝑢_𝜏𝑟⁻¹+𝜌𝑑𝑥₋₂𝜌^𝛾−𝜌²𝑟²𝑢_𝑥𝑟²𝑢_{𝜏𝑥=𝑥+0}, (83) which gives ¹₀𝑟²𝑢²_𝑥𝑑𝑥+^𝜏₀¹₀𝑟²𝑢²_𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠≤𝐶+𝐶¹₀𝜌^2(𝛾−2)𝑑𝑥+𝐶^𝜏₀¹₀𝑢²𝑟²+𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝑥𝑟²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝑢⁴𝑟⁻²𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝑥𝑢²𝑟²+𝑑𝑥𝑑𝑠_𝜏0𝜌⁻²𝜌^𝛾−𝜌²𝑟²𝑢_𝑥𝑟²𝑢_{𝑠𝑥=𝑥+0}𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝜌²_𝑥𝑟²𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠+𝐶sup_{[]𝜏∈0,𝑇}‖𝑢‖²_𝐿^∞+^𝜏₀𝜌⁻²𝜌^𝛾−𝜌²𝑟²𝑢_𝑥𝑟²𝑢_{𝑠𝑥=𝑥+0}𝑑𝑠. (84)
It holds from (23)₂, (26), (34), and (43) that 𝐶^𝜏₀¹₀𝜌²_𝑥𝑟²𝑢²_𝑥𝑟⁴≤𝐶𝑑𝑥𝑑𝑠3^𝜏₀¹₀𝜌²_𝑥𝑟²𝑢²_𝑥𝑟⁴1𝑑𝑥𝑑𝑠+6^𝜏₀¹₀𝑟²𝑢²_𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝑥𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝑢²𝑟²+𝑢²_𝑥𝑟⁴𝑑𝑥𝑑𝑠,𝐶sup_{[]𝜏∈0,𝑇}‖𝑢‖²_𝐿^∞≤14sup_{[]𝜏∈0,𝑇}¹₀𝑟²𝑢²_𝑥𝑑𝑥+𝐶sup_{[]𝜏∈0,𝑇}¹₀𝑢²𝑑𝑥,_𝜏0𝜌⁻²𝜌^𝛾−𝜌²𝑟²𝑢_𝑥𝑟²𝑢_{𝑠𝑥=𝑥+0}𝑑𝑠≤𝐶^𝜏₀𝜌₀𝑒^−𝐶𝑠𝜌^𝛾−𝜌²𝑟²𝑢_𝑥𝐿∞𝑟²𝑢_𝑠𝐿∞𝑑𝑠≤𝐶^𝜏₀𝑒^−𝐶𝑠𝜌^𝛾2_𝐿∞𝑑𝑠+𝐶^𝜏₀¹₀𝜌²_𝑥𝑑𝑥𝑑𝑠+𝐶^𝜏₀¹₀𝑢²𝑟²+𝑢²_𝑥𝑟⁴+1𝑑𝑥𝑑𝑠4^𝜏₀¹₀𝑟²𝑢²_𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠+𝐶𝜀^𝜏₀¹₀𝑟²𝑢²_𝑥𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠, (85)(86)(87) where 𝐶 denotes a positive constant independent of time and 𝜀∈(0,1) is a small constant which will be chosen later. It holds from (84)–(87) that ¹₀𝑟²𝑢²_𝑥𝑑𝑥+^𝜏₀¹₀𝑟²𝑢²_𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠≤𝐶+𝐶𝜀^𝜏₀¹₀𝑟²𝑢²_𝑥𝑠𝑟⁻⁴𝑑𝑥𝑑𝑠. (88)
Differentiating (23)₂ with respect to 𝜏, multiplying the result by (𝑟²𝑢)_𝜏, and integrating the result with respect to 𝑥 over [0,1], we deduce 12𝑑𝑑𝜏¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥+¹₀𝜌²𝑟²𝑢²_𝑥𝜏𝑑𝑥=2¹₀𝑢𝑢_𝜏𝑟²𝑢_𝜏𝑟⁻³1𝑑𝑥−2¹₀𝑟⁻⁴_𝜏𝑟²𝑢²_𝜏𝑑𝑥+2¹₀𝑢𝑟⁻¹𝑢_𝜏𝑟²𝑢_𝜏𝑟⁻²+𝑑𝑥₁₀𝜌^𝛾_𝜏𝑟²𝑢_𝑥𝜏𝑑𝑥−¹₀𝜌²_𝜏𝑟²𝑢_𝑥𝑟²𝑢_𝑥𝜏−𝑑𝑥₁₀2𝜌_𝑥𝑢𝑟_𝜏𝑟²𝑢_𝜏𝑑𝑥+2¹₀𝑟⁻²_𝜏𝑟⁻¹𝑢²𝑟²𝑢_𝜏𝑑𝑥. (89)
A complicated computation implies 𝑑𝑑𝜏¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥+¹₀𝜌²𝑟²𝑢²_𝑥𝜏𝑑𝑥≤𝐶¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥+𝐶¹₀𝑢²𝑟²+𝑢²_𝑥𝑟⁴𝑑𝑥¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥+𝐶sup_{[]𝜏∈0,𝑇}𝑟²𝑢²_𝑥𝐿∞¹₀𝑟²𝑢²_𝑥𝑑𝑥+𝐶¹₀𝑢²𝑟²+𝑢²_𝑥𝑟⁴𝑑𝑥, (90) by means of Gronwall’s inequality, (23)₂, (26), (34), (43), and (88), we have ¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥+^𝜏₀¹₀𝜌²𝑟²𝑢²_𝑥𝑠𝑑𝑥𝑑𝑠≤𝐶+𝐶𝜀^𝜏₀¹₀𝜌²𝑟²𝑢²_𝑥𝑠𝑑𝑥𝑑𝑠+𝐶sup_{[]𝜏∈0,𝑇}𝑟²𝑢²_𝑥𝐿∞≤𝐶+𝐶𝜀^𝜏₀¹₀𝜌²𝑟²𝑢²_𝑥𝑠𝑑𝑥𝑑𝑠+𝐶sup_{[]𝜏∈0,𝑇}¹₀𝑟²𝑢²_𝑥𝑑𝑥^1/21₀𝑟²𝑢²_𝑥𝑥𝑑𝑥^1/2≤𝐶+𝐶𝜀^𝜏₀¹₀𝜌²𝑟²𝑢²_𝑥𝑠1𝑑𝑥𝑑𝑠+2¹₀𝑟²𝑢²_𝜏𝑟⁻⁴𝑑𝑥, (91) where 𝐶 denotes a positive constant independent of time; choosing the constant 𝜀 small sufficiently, we complete the proof of Lemma 11.

Remark 12. By Lemmas 4–11, the following inequality holds: ¹₀𝑢²𝑑𝑥+¹₀𝑢²_𝑥𝑑𝑥+¹₀𝑢²_𝜏𝑑𝑥+¹₀𝜌²_𝑥𝑑𝑥+^𝜏₀¹₀𝜌²_𝑥+𝑑𝑥𝑑𝑠_𝜏010𝑢²_𝑥𝑑𝑥𝑑𝑠+^𝜏₀¹₀𝑢²_𝑠+𝑑𝑥𝑑𝑠_𝜏010𝑢²_𝑥𝑥𝑑𝑥𝑑𝑠+^𝜏₀¹₀𝑢²_𝑥𝜏𝑑𝑥𝑑𝑠≤𝐶. (92)

Remark 13. In fact, there are several fundamental estimates, such as (26) in Lemma 4, (34) in Lemma 5, (43) in Lemma 6, (47) in Lemma 7, (60) in Lemma 9, and (82) in Lemma 11, which can be also found in [15, 20, 22, 26] and so forth.

Lemma 14. Under the conditions in Theorem 1, it holds for any solution (𝜌,𝑢) to the initial boundary value problem (7)-(8) that 𝜌−𝜌,𝑢(⋅,𝑡)_{𝐿∞([𝑟−,𝜉(𝑡))∪(𝜉(𝑡),𝑟+])}≤𝐶₂𝑒^−𝐶3𝑡,𝑡>0, (93) where 𝐶₂ and 𝐶₃ denote the constants independent of time and 𝜌=1/(𝑟₊−𝑟₋)⨍^𝑟+_𝑟−𝜌(𝑟,𝑡)𝑟²𝑑𝑟.

Proof. In a similar argument to show (28) and (40) with modification, we can obtain (94) and (95) 𝑑𝑑𝑡^𝑟+_𝑟−12𝜌𝑢²+1𝜌𝛾−1^𝛾−1−𝜌^𝛾−1−𝛾𝜌^𝛾−1𝜌−𝜌𝑟²+𝑑𝑟_𝑟+𝑟−2𝜌𝑢²+𝜌𝑢²_𝑟𝑟²𝑑𝑟=2𝜉(𝑡)𝜌𝑢²_{𝑟=𝜉(𝑡)},12𝑑𝑑𝑡^𝑟+_𝑟−𝜌𝑢+𝜌⁻¹𝜌_𝑟²𝑟²+1𝑑𝑟𝑑𝛾−1𝑑𝑡^𝑟+_𝑟−𝜌^𝛾−1−𝜌^𝛾−1−𝛾𝜌^𝛾−1𝜌−𝜌𝑟²𝑑𝑟+𝛾^𝑟+_𝑟−𝜌^𝛾−2𝜌²_𝑟𝑟²𝑑𝑟=𝜉(𝑡)²𝜌^𝛾𝑢_{𝑟=𝜉(𝑡)}𝑑𝑠. (94)(95)
It holds from (34), (43), and (60) that ^𝑟+_𝑟−𝜌−𝜌²𝑟²𝑑𝑟≤𝐶^𝑟+_𝑟−𝜌^𝛾−2𝜌²_𝑟𝑟²𝑑𝑟. (96)
Denote 𝐸(𝑡)∶=^𝑟+_𝑟−𝜌𝑢²+𝑢+𝜌⁻¹𝜌_𝑟²𝑟²+𝑑𝑟_𝑟+𝑟−𝜌^𝛾−1−𝜌^𝛾−1−𝛾𝜌^𝛾−1𝜌−𝜌𝑟²𝑑𝑟. (97)
By (94)–(97), a complicated computation gives rise to 𝑑𝐸(𝑡)+(𝑡)𝜌𝑢𝑑𝑡𝐶𝐸≤𝐶²+𝜌^𝛾𝑢, (98) where 𝐶<𝐶₁ is a positive constant and 𝐶₁ is the constant obtained in Lemma 10. From (97), we have 𝐸(𝑡)≤𝐸(0)𝑒⁻𝐶_𝑡+𝐶𝑒⁻𝐶_𝑡^𝑡₀𝑒𝐶_𝑠𝜌𝑢²+𝜌^𝛾𝑢𝑑𝑠≤𝐶₂𝑒^−𝐶3𝑡, (99) where we have used (26), (30), and (31). By the fact 𝐸(𝑡)≥𝑐‖𝑢‖²_{𝐿2𝑟−(𝑡∪𝜉(𝑡),𝜉),𝑟+}+𝜌−𝜌²_{𝐿2𝑟−(𝑡∪𝜉(𝑡),𝜉),𝑟+}+𝜌_𝑟²_{𝐿2𝑟−(𝑡)∪𝜉(𝑡),𝜉,𝑟+}, (100) where 𝑐>0 is a constant independent of time and Gagliardo-Nirenberg-Sobolev inequality 𝜌−𝜌,𝑢_{𝐿∞([𝑟−,𝜉(𝑡))∪(𝜉(𝑡),𝑟+])}≤𝐶𝜌−𝜌,𝑢_𝐿^1/2_{2𝑟−(𝑡∪𝜉(𝑡),𝜉),𝑟+}×𝜌−𝜌,𝑢_𝑟𝐿^1/2_{2𝑟−(𝑡∪𝜉(𝑡),𝜉),𝑟+}, (101) we obtain (93).

4. Proof of the Main Results

Proof. The global existence of unique piecewise regular solution to the initial boundary value problem (7)-(8) can be established in terms of the short time existence carried out as in [2, 10], the uniform a priori estimates, and the analysis of regularities, which indeed follow from Lemmas 4–11. In addition, one can show that (𝜌,𝑢) is the global weak solution to the initial boundary value problem (7)-(8) with the initial data satisfying (9). We omit the details. The large time behaviors follow from Lemma 14 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.

Acknowledgments

The research of R. X. Lian is supported by NNSFC no. 11101145, China Postdoctoral Science Foundation no. 2012M520360, Doctoral Foundation of North China University of Water Sources and Electric Power no. 201032, and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.