Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

Abstract

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and discontinuous initial data in this paper. For piecewise regular initial density, we show that there exists a unique global piecewise regular solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate, and the jump discontinuity of density also decays at an algebraic time-rate as the time tends to infinity.

1. Introduction

In the present paper, we consider the free boundary value problem to one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient for piecewise regular initial data connected with the infinite vacuum via jump discontinuity. In general, the one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient read 𝜌_𝑡+𝜌𝑢_𝑥=0,𝜌𝑢_𝑡+𝜌𝑢²+𝑃𝜌_𝑥=𝜇𝜌𝑢_𝑥𝑥,(𝑥,𝑡)∈𝑅×[0,𝑇], (1) where 𝜌>0 and 𝑢 denote the flow density and velocity, respectively, the pressure-density function is taken as 𝑃(𝜌)=𝜌^𝛾 with 𝛾>1, and the viscosity coefficient is 𝜇(𝜌)=𝜌^𝛼 with 𝛼>0. Note here that the case 𝛾=2 and 𝛼=1 in (1) corresponds to the viscous Saint-Venant system.

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 example, as the viscosity coefficients are both constants, the global existence of discontinuous solutions of one-dimensional Navier-Stokes equations was derived by Hoff [1–3]. Hoff investigated the construction of global spherically symmetric weak solutions of compressible Navier-Stokes equations for isothermal flow with large and discontinuous initial data [4]; therein it is also proved that the discontinuities in the density and pressure persist for all time, convecting along particle trajectories and decaying at a rate inversely proportional to the viscosity coefficient. 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) were also showed by Hoff [5, 6]. Chen et al. obtained the global existence of weak solutions for the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data [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 also proved by Hoff [9].

If the viscosity coefficients 𝜇(𝜌)=𝜌^𝛼, 𝜆(𝜌)=0, for the case of one space dimension, the global existence of unique piecewise smooth solution to the free boundary value problem was obtained by Fang-Zhang for (1) with 0<𝛼<1, where the initial density is piecewise smooth with possibly large jump discontinuities [10]. Lian et al. considered the initial boundary value problem for (1) with 0<𝛼≤1 subject to piecewise regular initial data with initial vacuum state included in [11]. Lian et al. also addressed the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient [12]; in these two cases above, they proved the global existence of unique piecewise regular solution and the finite time vanishing of vacuum state was proved in [11]. 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 as 𝑡→+∞.

Recently, there are also many significant progresses achieved 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 [13, 14]. The good-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 was considered by many authors; refer to [15–22] and references therein. The global existence of classical solutions is shown by Mellet and Vasseur [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 [24–28] and references therein.

In this present paper, we consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes equations and focus on the existence, regularities, and dynamical behaviors of global piecewise regular solution, and so forth. As 𝛾>1, 0<𝛼≤1, we show that the free boundary value problem with piecewise regular initial data admits a unique global piecewise regular solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic, and the jump discontinuity of density also decays at an algebraic time-rate as 𝑡→+∞ (refer to Theorem 2 for details).

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. Main Results

We are interested in the global existence and dynamics of the free boundary value problem for (1) with following initial data and boundary conditions: 𝜌,𝑢(𝑥,0)=𝜌₀,𝑢₀,𝑥∈0,𝑎₀,𝑢(0,𝑡)=0,𝜌^𝛾−𝜌^𝛼𝑢_𝑥(𝑎(𝑡),𝑡)=0,𝑡>0, (2) where 𝑥=𝑎(𝑡) is the free boundary defined by 𝑑𝑎(𝑡)(𝑎(𝑡))(0)𝑑𝑡=𝑢,𝑡,𝑎=𝑎₀,𝑡>0. (3)

Next, we give the definition of weak solution to the free boundary problem (1) and (2).

Definition 1 (weak solution). For any 𝑇>0, (𝜌,𝑢) is said to be a weak solution of the free boundary problems (1) and (2), if (𝜌,√𝜌𝑢) has the following regularities: 0≤𝜌∈𝐿^∞0,𝑇;𝐿¹([0,𝑎(𝑡)])∩𝐿^𝛾(𝑡,√([0,𝑎)])𝜌𝑢∈𝐿^∞0,𝑇;𝐿²(𝑡,𝜌([0,𝑎)])_𝛾−𝜌^𝛼𝑢_𝑥∈𝐿²0,𝑇;𝐻¹([0,𝑎(𝑡)]), (4) and (1) is satisfied in the sense of distributions. Namely, it holds for all 𝜑∈𝐶^∞₀([0,𝑎(𝑡)]×[0,𝑇)) that ^𝑎0₀𝜌₀𝜑(𝑥,0)𝑑𝑥+^𝑇₀₀^𝑎(𝑡)𝜌𝜑_𝑡+𝑑𝑥𝑑𝑡_{𝑇00𝑎(𝑡)}√𝜌√𝜌𝑢𝜑_𝑥𝑑𝑥𝑑𝑡=0 (5) and for all 𝜓∈𝐶^∞₀([0,𝑎(𝑡)]×[0,𝑇)) that ^𝑎0₀𝜌₀𝑢₀𝜓(𝑥,0)𝑑𝑥+^𝑇₀₀^𝑎(𝑡)√𝜌√𝜌𝑢𝜓_𝑡+√𝜌𝑢²𝜓_𝑥+𝑑𝑥𝑑𝑡_{𝑇00𝑎(𝑡)}𝜌^𝛾−𝜌^𝛼𝑢_𝑥𝜓_𝑥𝑑𝑥𝑑𝑡=0. (6)

For simplicity, we consider the initial data in FBVP (1) and (2) with one discontinuous point 𝑦₀∈(0,𝑎₀); namely, for some constant 𝜌₋>0inf_{[0,𝑦0)∪(𝑦0,𝑎0]}𝜌₀≥𝜌₋𝜌>0,₀,𝑢₀∈𝑊^1,∞0,𝑦₀∪𝑦₀,𝑎₀,𝜌^𝛾₀−𝜌^𝛼₀𝑢_0𝑥𝑎₀𝜌=0,₀𝑦₀−0>𝜌₀𝑦₀+0, (7) and the compatibility conditions between initial data and boundary conditions hold.

We will give the global existence and time-asymptotic behavior of piecewise regular solution as follows.

Theorem 2. Let 𝛾>1 and 0<𝛼≤1. Assume that the initial data satisfies (7). Then, there exists a unique global piecewise regular solution (𝜌,𝑢,𝑎) to the FBVP (1) and (2) satisfying for 𝑇>0𝜌∈𝐿^∞0,𝑇;𝐻¹0,𝑦(𝑡)∪𝑦(𝑡),𝑎(𝑡)∩𝐶⁰0,𝑦(𝑡)∪𝑦(𝑡),𝑎(𝑡)×[0,𝑇],𝑢∈𝐿^∞0,𝑇;𝐻¹([0,𝑎(𝑡)])∩𝐿²0,𝑇;𝐻²(𝑡,𝑢([0,𝑎)])_𝑡∈𝐿²0,𝑇;𝐿²(𝑡,𝑎(𝑡)([0,𝑎)])∈𝐻¹,𝜌([0,𝑇])_𝛾−𝜌^𝛼𝑢_𝑥∈𝐿^∞0,𝑇;𝐿²0,𝑦(𝑡)∪𝑦(𝑡),𝑎(𝑡), (8) where 𝑥=𝑦(𝑡) is a curve defined by 𝑑𝑦(𝑡)𝑦(𝑡)(0)𝑑𝑡=𝑢,𝑡,𝑦=𝑦₀,𝑡>0, (9) along which the Rankine-Hugoniot conditions hold 𝑢𝑦(𝑡),𝑡=0,𝜌^𝛾𝑦(𝑡),𝑡=𝜌^𝛼𝑢_𝑥𝑦(𝑡),𝑡, (10) where [𝑓(𝑦(𝑡),𝑡)]∶=𝑓(𝑦(𝑡)+0,𝑡)−𝑓(𝑦(𝑡)−0,𝑡), and along the discontinuity 𝑟=𝜉(𝑡) the jump satisfies 𝜌^𝛼₀𝑦₀𝑒^−𝐶0𝑡≤𝜌^𝛼𝑦(𝑡),𝑡, (11) where 𝐶₀ is a positive constant independent of time.
If it further holds that 𝑢₀∈𝐻²([0,𝑎₀]), then (𝜌,𝑢,𝑎,𝑏) satisfies 𝜌∈𝐿^∞0,𝑇;𝐻¹(𝑡∪𝑦(𝑡)(𝑡,𝜌0,𝑦),𝑎)_𝑡∈𝐿^∞0,𝑇;𝐿²0,𝑦(𝑡)∪𝑦(𝑡),𝑎(𝑡),𝑢∈𝐿^∞0,𝑇;𝐻¹([0,𝑎(𝑡)])∩𝐿²0,𝑇;𝐻²(𝑡,𝑢([0,𝑎)])_𝑡∈𝐿^∞0,𝑇;𝐿²([0,𝑎(𝑡)])∩𝐿²0,𝑇;𝐻¹(𝑡,𝑎(𝑡)([0,𝑎)])∈𝐻²,𝜌([0,𝑇])_𝛾−𝜌^𝛼𝑢_𝑥∈𝐿^∞0,𝑇;𝐻¹0,𝑦(𝑡)∪𝑦(𝑡),𝑎(𝑡). (12) The domain expands outwards at an algebraic rate in time as 𝐶(1+𝑡)^{𝛾/(𝛾−𝛼)}(𝑡)≥𝑐()≥𝑎1+𝑡,1<𝛾<2𝛼,𝑐(1+𝑡)^1−𝜈,𝛾=2𝛼,𝑐(1+𝑡)^{𝛼/(𝛾−𝛼)},𝛾>2𝛼, (13) and the density decays pointwise to zero for any 𝑥∈[0,𝑦(𝑡))∪(𝑦(𝑡),𝑎(𝑡)] and 𝑡>0 as 𝜌(𝑎(𝑡),𝑡)=𝛾−𝛼𝑡+𝜌₀𝑎₀^{𝛼−𝛾−1/(𝛾−𝛼)},𝜌(𝑥,𝑡)≤𝐶(1+𝑡)^{−1/𝛾−𝛼}+𝐶(1+𝑡)^{−(𝛾−1)/(3𝛾+2𝛼−1)},𝐶()1<𝛾<2𝛼,1+𝑡^{−((𝛾−1)/(3𝛾+2𝛼−1))+𝜈},𝐶()𝛾=2𝛼,1+𝑡^{−𝛼(𝛾−1)/(3𝛾+2𝛼−1)(𝛾−𝛼)},+𝐶()𝛾>2𝛼,1+𝑡^{−2/(3𝛾+2𝛼−1)},𝐶()1<𝛾<2𝛼,1+𝑡^{−(2/(3𝛾+2𝛼−1))+𝜈},𝐶()𝛾=2𝛼,1+𝑡^{−2𝛼/(3𝛾+2𝛼−1)(𝛾−𝛼)},𝛾>2𝛼, (14)(15) where 𝐶>0 and 𝑐>0 are positive constants independent of time, and 𝜈∈(0,1) is a small constant.

Remark 3. Theorem 2 holds for the Saint-Venant model for shallow water; that is, 𝛾=2, 𝛼=1.

Remark 4. Fang-Zhang [10] obtained that 𝜌^𝛼₀𝑦₀𝑒^−𝐶0𝑡≤𝜌^𝛼𝑦(𝑡),𝑡, (16) which show that the discontinuity in the density persists for all time. However, in this paper, from (15) we can shows that the discontinuity in the density decays at an algebraic rate in time; namely, 𝜌^𝛼𝑦(𝑡),𝑡≤𝐶(1+𝑡)^{−𝛼/(𝛾−𝛼)}+𝐶(1+𝑡)^{−𝛼(𝛾−1)/(3𝛾+2𝛼−1)},𝐶()1<𝛾<2𝛼,1+𝑡^{−(𝛼(𝛾−1)/(3𝛾+2𝛼−1))+𝛼𝜈},𝐶()𝛾=2𝛼,1+𝑡^{−𝛼2(𝛾−1)/(3𝛾+2𝛼−1)(𝛾−𝛼)},+𝐶()𝛾>2𝛼,1+𝑡^{−2𝛼/(3𝛾+2𝛼−1)},𝐶()1<𝛾<2𝛼,1+𝑡^{−(2𝛼/(3𝛾+2𝛼−1))+𝜈},𝐶()𝛾=2𝛼,1+𝑡^{−2𝛼2/(3𝛾+2𝛼−1)(𝛾−𝛼)},𝛾>2𝛼, (17) where 𝐶>0 is a positive constant independent of time, and 𝜈∈(0,1) is a small constant.

3. The A Priori Estimates

According to the analysis made in [29], there is a curve 𝑥=𝑦(𝑡) defined by 𝑑𝑦(𝑡)𝑦(𝑡)(0)𝑑𝑡=𝑢,𝑡,𝑦=𝑦₀,𝑡>0, (18) along which the Rankine-Hugoniot conditions hold 𝑢𝑦(𝑡),𝑡=0,𝜌^𝛾𝑦(𝑡),𝑡=𝜌^𝛼𝑢_𝑥𝑦(𝑡),𝑡, (19) where [𝑓(𝑦(𝑡),𝑡)]∶=𝑓(𝑦(𝑡)+0,𝑡)−𝑓(𝑦(𝑡)−0,𝑡).

It is convenient to make use of the Lagrange coordinates in order to establish the uniformly a-priori estimates. Let (𝜌,𝑢,𝑎) be a piecewise regular solution to the FBVP (1) and (2), and take the Lagrange coordinates transform 𝜉=^𝑥₀𝜌(𝑧,𝑡)𝑑𝑧,𝜏=𝑡. (20) Since the conservation of total mass holds ₀^𝑎(𝑡)𝜌(𝑧,𝑡)𝑑𝑧=^𝑎0₀𝜌₀(𝑧)𝑑𝑧∶=1>0, (21) the boundaries 𝑥=0 and 𝑥=𝑎(𝑡) are transformed into 𝜉=0 and 𝜉=1, respectively, and the domain [0,𝑎(𝑡)] is transformed into [0,1], the curve 𝑥=𝑦(𝑡) in the Eulerian coordinates is changed to a line 𝜉=𝜉₀ in the Lagrangian coordinates, where 𝜉₀∶=₀^𝑦(𝑡)𝜌(𝑧,𝑡)𝑑𝑧=^𝑦0₀𝜌₀(𝑧)𝑑𝑧, (22) and the jump conditions become 𝑢𝜉₀,𝜏=0,𝜌^𝛾𝜉₀,𝜏=𝜌^1+𝛼𝑢_𝜉𝜉₀,𝜏. (23)

Meanwhile, the FBVP (1) and (2) is reformulated into 𝜌_𝜏+𝜌²𝑢_𝜉𝑢=0,_𝜏+𝜌^𝛾_𝜉=𝜌^1+𝛼𝑢_𝜉𝜉,𝜌^𝛾−𝜌^1+𝛼𝑢_𝜉(0,𝜏)=𝜌^𝛾−𝜌^1+𝛼𝑢_𝜉()𝜌1,𝜏=0,₀,𝑢₀=𝜌₀,𝑢₀(𝜉),𝜉∈[0,1], (24) where the initial data satisfies inf_{[0,𝜉0)∪(𝜉0,1]}𝜌₀≥𝜌₋𝜌>0,₀,𝑢₀∈𝑊^1,∞0,𝜉₀∪𝜉₀,𝜌,1_𝛾0−𝜌₀^1+𝛼𝑢_0𝑥1)𝜌(=0,₀𝜉₀−0>𝜌₀𝜉₀+0, (25) for some constant 𝜌₋>0, and the consistencies between initial data and boundary value hold.

Next, we will give the a-priori estimates for the solution (𝜌,𝑢) to the FBVP (24). Similarly to the arguments used in [10, 16, 20], we can establish the following a priori estimates and omit the details here.

Lemma 5. Let 𝑇>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that ¹₀𝑢²2+1𝜌𝛾−1^𝛾−1𝑑𝜉+^𝜏₀¹₀𝜌^1+𝛼𝑢²_𝜉=𝑑𝜉𝑑𝑠₁₀𝑢²₀2+1𝜌𝛾−1^0𝛾−1[],𝜌()()∈𝑑𝜉,𝜏∈0,𝑇𝜉,𝜏≤𝐶,𝜉,𝜏0,𝜉₀∪𝜉₀,1×[0,𝑇₁₀],𝑢^2𝑛𝑑𝜉+𝑛(2𝑛−1)^𝜏₀¹₀𝜌^1+𝛼𝑢^2𝑛−2𝑢²_𝜉(𝑇)[],𝑑𝜉𝑑𝑠≤𝐶,𝜏∈0,𝑇₁₀(𝜌^𝛼)_𝜉^2𝑛𝑑𝜉≤𝐶(𝑇),𝜏∈[0,𝑇], (26)(27)(28)(29) for any positive integer 𝑛∈𝑁, 𝐶>0 denotes a constant independent of time and 𝐶(𝑇)>0 denotes a constant dependent on time, where ⨏¹₀∶=∫^𝜉0₀+∫¹_𝜉0.

Lemma 6. Let 𝑇>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that 𝜌(0,𝜏)=𝛾−𝛼𝜏+𝜌₀(0)^{𝛼−𝛾−1/(𝛾−𝛼)},𝜌𝜉₀−0,𝜏>𝜌𝜉₀+0,𝜏,𝜏∈[0,𝑇]. (30)(31)

Proof. From (24)₁ and (24)₃, we have 𝜌_𝜏(0,𝜏)+𝜌^{𝛾−𝛼+1}(0,𝜏)=0, (32) which yields (30) similarly, because of (23) and (24)₁, it holds that [𝜌^𝛼]_𝜏+𝛼𝜌^𝛾=0, (33) which together with (25) implies 𝜌^𝛼=𝜌^𝛼₀exp−𝛼^𝜏₀𝜌^𝛾𝜌^𝛼𝑑𝑠<0. (34)

Lemma 7. Let 𝑇>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that 12¹₀𝜌𝑢+^𝛼𝜉𝛼²1𝑑𝜉+𝛾−1¹₀𝜌^𝛾−1𝑑𝜉+𝛾^𝜏₀¹₀𝜌^{𝛾+𝛼−2}𝜌²_𝜉𝑑𝜉𝑑𝑠+𝜌^𝛾(1,𝜏)𝑎(𝜏)+𝛾^𝜏₀𝜌^2𝛾−𝛼(1,𝑠)𝑎(𝑠)𝑑𝑠−𝜌^𝛾𝑦(𝜏)−𝛾^𝜏₀𝜌^𝛾+1𝑢_𝜉𝑦(𝑠)=1𝑑𝑠2¹₀𝑢₀+𝜌^𝛼_𝜉𝛼²+1𝑑𝜉𝛾−1¹₀𝜌₀^𝛾−1𝑑𝜉+𝜌^𝛾₀(1)𝑎₀−𝜌^𝛾₀𝑦₀𝜏∈[0,𝑇], (35) where 𝑎(𝜏) satisfies 𝑎^′(𝜏)=𝑢(0,𝜏) and 𝑎(0)=𝑎₀, 𝑦(𝜏) satisfies 𝑦^′(𝜏)=𝑢(𝜉₀,𝜏) and 𝑦(0)=𝑦₀.

Proof. Multiplying (24)₁ by 𝜌^𝛼−1 gives (𝜌^𝛼)_𝜏𝛼+𝜌^1+𝛼𝑢_𝜉=0, (36) which leads to (𝜌^𝛼)_𝜏𝜉𝛼+(𝜌^1+𝛼𝑢_𝜉)_𝜉=0. (37) Summing (24)₂ and (37), we have 𝜌𝑢+^𝛼𝜉𝛼_𝜏+(𝜌^𝛾)_𝜉=0. (38) Multiplying (38) by (𝑢+(𝜌^𝛼)_𝜉/𝛼) and integrating the result over [0,1]×[0,𝜏], we get 12¹₀𝜌𝑢+^𝛼𝜉𝛼²1𝑑𝜉+𝛾−1¹₀𝜌^𝛾−1𝑑𝜉+𝛾^𝜏₀¹₀𝜌^{𝛾+𝛼−2}𝜌²_𝜉𝑑𝜉𝑑𝑠+^𝜏₀𝜌^𝛾𝑢^{𝜉=𝜉0−0}_𝜉=0𝑑𝑠+^𝜏₀𝜌^𝛾𝑢^𝜉=1_𝜉=𝜉0+0=1𝑑𝑠2¹₀𝑢₀+𝜌^𝛼_0𝜉𝛼²1𝑑𝜉+𝛾−1¹₀𝜌₀^𝛾−1𝑑𝜉, (39) which together with the fact that ^𝜏₀𝜌^𝛾𝑢^{𝜉=𝜉0−0}_𝜉=0=𝑑𝑠_𝜏0𝜌^𝛾𝜉₀−0,𝑠𝑦^′(𝑠)𝑑𝑠=𝜌^𝛾𝜉₀−0,𝑠𝑦(𝑠)^𝜏₀+𝛾^𝜏₀𝜌^𝛾−1𝜌²𝜉₀−0,𝑠𝑢_𝜉𝜉₀,𝑠𝑦(𝑠)𝑑𝑠=𝜌^𝛾𝜉₀−0,𝜏𝑦(𝜏)−𝜌^𝛾₀𝜉₀−0𝑦₀+𝛾^𝜏₀𝜌^𝛾+1𝜉₀−0,𝑠𝑢_𝜉𝜉₀,𝑠𝑦(𝑠)𝑑𝑠,_𝜏0𝜌^𝛾𝑢^𝜉=1_𝜉=𝜉0+0=𝑑𝑠_𝜏0𝜌^𝛾(1,𝜏)𝑎^′(𝑠)−𝑑𝑠_𝜏0𝜌^𝛾𝜉₀+0,𝜏𝑦^′(𝑠)𝑑𝑠=𝜌^𝛾(1,𝑠)𝑎(𝑠)^𝜏₀+𝛾^𝜏₀𝜌^2𝛾−𝛼(1,𝑠)𝑎(𝑠)𝑑𝑠−𝜌^𝛾𝜉₀+0,𝑠𝑦(𝑠)|^𝜏₀−𝛾^𝜏₀𝜌^𝛾+1𝜉₀+0,𝑠𝑢_𝜉𝜉₀,𝑠𝑦(𝑠)𝑑𝑠=𝜌^𝛾(1,𝜏)𝑎(𝜏)−𝜌^𝛾₀(1)𝑎₀+𝛾^𝜏₀𝑏(𝑠)𝜌^2𝛾−𝛼(1,𝑠)𝑑𝑠−𝜌^𝛾𝜉₀+0,𝜏𝑦(𝜏)+𝜌^𝛾₀𝜉₀+0𝑦₀−𝛾^𝜏₀𝜌^𝛾+1𝜉₀+0,𝑠𝑢_𝜉𝜉₀𝑦(𝑠)𝜌𝜉,𝑠𝑑𝑠,₀𝑢,𝜏<0,_𝜉𝜉₀=𝜌,𝜏^𝛾𝜌^1+𝛼𝑎(𝜏)>0,=¹₀1𝜌()𝑦(𝜏)=𝜁,𝜏𝑑𝜁>0,_𝜉001𝜌(𝜁,𝜏)𝑑𝜁>0, (40)(41) gives rise to (35).

Remark 8. The estimate (35) can be written in the following form in the Eulerian coordinates; that is to say, for all 𝑡∈[0,𝑇], 12₀^𝑎(𝑡)𝜌𝑢+𝜌⁻¹𝜌^𝛼_𝑥²1𝑑𝑥+𝛾−1₀^𝑎(𝑡)𝜌^𝛾𝑑𝑥+𝛾^𝑡₀₀^𝑎(𝑡)𝜌^{𝛾+𝛼−3}𝜌²_𝑥𝑑𝑥𝑑𝑠+𝜌^𝛾(𝑎(𝑡),𝑡)𝑎(𝑡)+𝛾^𝑡₀𝜌^2𝛾−𝛼(𝑎(𝑠))𝑎(𝑠)−𝜌,𝑠𝑑𝑠_𝛾𝑦(𝑡)−𝛾^𝑡₀𝜌^𝛾𝑢_𝑥𝑦(𝑠)=1𝑑𝑠2^𝑎0₀𝜌₀𝑢₀+𝜌₀⁻¹𝜌^𝛼_0𝑥²+1𝑑𝑥𝛾−1^𝑎0₀𝜌^𝛾₀𝑑𝑥+𝜌^𝛾₀𝑎₀𝑎₀−𝜌^𝛾₀𝑦₀. (42)

Lemma 9. Let 𝑇>0, for 𝑛∈𝑁, and 𝑛>(1+𝛼)/4(𝛾−𝛼). Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that ^𝜏₀𝜌^𝛾_𝜉^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}𝑑𝑠≤𝐶(𝑇),𝜏∈[0,𝑇]. (43)

Proof. It follows from (24)_1,2 that (𝜌^𝛼)_𝜉(𝜉,𝜏)=𝜌^𝛼_0𝜉(𝜉)−𝛼𝑢(𝜉,𝜏)+𝛼𝑢₀(𝜉)−𝛼^𝜏₀𝜌^𝛾_𝜉(𝜉,𝑠)𝑑𝑠. (44) By means of (25), (26), and (44), we have ^𝜏₀𝜌^𝛾_𝜉^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}=𝛾𝑑𝑠_2𝑛𝛼^2𝑛𝜏₀𝜌^{2𝑛(𝛾−𝛼)}(𝜌^𝛼)_𝜉^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}(𝑇)(𝑇)𝑑𝑠≤𝐶+𝐶^𝜏₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}(𝑇)𝑑𝑠+𝐶^𝜏₀^𝑠₀(𝜌^𝛾)_𝜉^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}(𝑇)(𝑇)𝑑𝑙𝑑𝑠≤𝐶+𝐶^𝜏₀^𝑠₀(𝜌^𝛾)_𝜉^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}𝑑𝑙𝑑𝑠, (45) where we have used ^𝜏₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛_{𝐿∞([0,𝜉0)∪(𝜉0,1])}𝑑𝑠≤^𝜏₀¹₀𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛+𝑑𝜉𝑑𝑠_𝜏010(𝜌^{2𝑛(𝛾−𝛼)}𝑢^2𝑛)_𝜉(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝜌^{2(2𝑛(𝛾−𝛼)−𝛼)}𝜌^2𝛼−2𝜌²_𝜉+𝑢^4𝑛+𝜌^{4𝑛(𝛾−𝛼)−(1+𝛼)}𝑢^2𝑛+𝜌^1+𝛼𝑢^2𝑛−2𝑢²_𝜉(𝑇),𝑑𝜉𝑑𝑠≤𝐶 (46) which can be deduced from (26) and (27). Making use of Gronwall's inequality to (46), we obtain (43).

Lemma 10. Let 𝑇>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that 𝜌(𝜉,𝜏)≥𝐶(𝑇),(𝜉,𝜏)∈0,𝜉₀∪𝜉₀,1×[0,𝑇]. (47)

Proof. Denote 𝑣()=1𝜉,𝜏𝜌().𝜉,𝜏 (48) By (24)₁, we have 𝑣_𝜏=𝑢_𝜉. (49) Multiplying (49) by 𝛽𝑣^𝛽−1, integrating the result over [0,1]×[0,𝜏], and using (43), (44), we can obtain that for 𝛽∈(1,2)¹₀𝑣^𝛽𝑑𝜉+𝛽𝛽−1^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢²=𝑑𝜉𝑑𝑠₁₀𝑣^𝛽₀𝑑𝜉+𝛽^𝜏₀𝑣^𝛽−1𝑢𝑑𝑠^{𝜉=𝜉0−0}_𝜉=0+𝛽^𝜏₀𝑣^𝛽−1𝑢𝑑𝑠_𝜉^𝜉=1₀₊₀+𝛽𝛽−1𝛼^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢(𝜌^𝛼₀)_𝜉𝑑𝜉𝑑𝑠+𝛽𝛽−1^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢𝑢₀×𝑑𝜉𝑑𝑠−𝛽𝛽−1_𝜏010𝑣^{𝛼+𝛽−1}𝑢^𝑠₀(𝜌^𝛾)_𝜉(𝑇)𝑑𝑙𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀𝑣^𝛽−1𝑢()𝛽1,𝑠𝑑𝑠+𝛽−12×^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢²𝑑𝜉𝑑𝑠+𝐶(𝑇)^𝜏₀¹₀𝑣^𝛽𝑑𝜉𝑑𝑠, (50) where we use the fact that ^𝜏₀𝑣^𝛽−1𝜉₀−0,𝑠𝑢𝜉₀−,𝑠𝑑𝑠_𝜏0𝑣^𝛽−1𝜉₀+0,𝑠𝑢𝜉₀=,𝑠𝑑𝑠_𝜏0𝜌^1−𝛽𝜉₀−0,𝑠𝑦^′(𝑠)−𝑑𝑠_𝜏0𝜌^1−𝛽𝜉₀+0,𝑠𝑦^′(𝑠)𝑑𝑠=𝜌^1−𝛽𝜉₀−0,𝑠𝑦(𝑠)^𝜏₀−𝜌^1−𝛽𝜉₀+0,𝑠𝑦(𝑠)^𝜏₀+1−𝛽^𝜏₀𝜌^2−𝛽𝜉₀−0,𝑠𝑢_𝜉𝜉₀𝑦(𝑠)−,𝑠𝑑𝑠1−𝛽^𝜏₀𝜌^2−𝛽𝜉₀+0,𝑠𝑢_𝜉𝜉₀𝑦(𝑠)𝜌,𝑠𝑑𝑠=−^1−𝛽𝑦(𝜏)+𝜌₀^1−𝛽𝑦₀+𝛽−1^𝜏₀𝜌^2−𝛽𝑢_𝜉𝜉₀𝑦(𝑠)≤𝜌,𝑠𝑑𝑠_01−𝛽𝑦₀≤𝐶. (51) Since it holds that 𝜌(1,𝜏)=𝛾−𝛼𝜏+𝜌₀(1)^{𝛼−𝛾−1/(𝛾−𝛼)}≥𝐶(𝑇), (52) we have from (25) that ^𝜏₀𝑣^𝛽−1𝑢()(𝑇)1,𝑠𝑑𝜉≤𝐶^𝜏₀¹₀𝑢²𝑑𝜉^1/2+¹₀𝜌^1+𝛼𝑢²_𝜉𝑑𝜉^1/21₀𝑣^1+𝛼𝑑𝜉^1/2(𝑇)(𝑇)𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝑣^𝛽𝑑𝜉𝑑𝑠. (53) Substituting (53) in (50), we have ¹₀𝑣^𝛽𝛽𝑑𝜉+𝛽−12^𝜏₀¹₀𝑣^{𝛼+𝛽−1}𝑢²(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝑣^𝛽𝑑𝜉𝑑𝑠. (54) Using Gronwall's inequality, we get from (54) that ¹₀𝑣^𝛽𝑑𝜉≤𝐶(𝑇). (55) It follows from (29) and (55) that 𝑣^𝛽(𝜉,𝜏)≤¹₀𝑣^𝛽𝑑𝜉+¹₀𝑣^𝛽_𝜉(𝑇)𝑑𝜉≤𝐶+𝐶¹₀𝜌^−𝛽−1𝜌_𝜉(𝑇)𝑑𝜉≤𝐶+𝐶¹₀𝜌^{−2𝑛(𝛼+𝛽)/(2𝑛−1)}𝑑𝜉^{(2𝑛−1)/2𝑛}×¹₀𝜌^𝛼_𝜉^2𝑛𝑑𝜉^1/2𝑛≤𝐶(𝑇)+𝐶(𝑇)sup_{[0,𝜉0)∪(𝜉0,1]}𝑣^{𝛼+𝛽/2𝑛1}₀𝑣^𝛽𝑑𝜉^{(2𝑛−1)/2𝑛}≤𝐶(𝑇)+𝐶(𝑇)sup_{[0,𝜉0)∪(𝜉0,1]}𝑣^𝛽(𝜉,𝜏)^{𝛼/𝛽+1/2𝑛}, (56) as 𝛼∈(0,1], 𝛽∈(1,2); for some 𝑛∈𝑁 large enough, we have 𝛼𝛽+12𝑛<1, (57) which implies (47).

We also have the regularity estimates for the solution (𝜌,𝑢) to the FBVP (24) as follows.

Lemma 11. Let 𝑇>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that 𝜌∈𝐿^∞0,𝑇;𝐻¹0,𝜉₀∪𝜉₀,1∩𝐶⁰0,𝜉₀∪𝜉₀,1×[0,𝑇],𝑢∈𝐿^∞0,𝑇;𝐻¹([0,1])∩𝐿²0,𝑇;𝐻²,𝑢([0,1])_𝜏∈𝐿²0,𝑇;𝐿²([0,1]),𝑎(𝜏)∈𝐻¹,𝜌([0,𝑇])_𝛾−𝜌^1+𝛼𝑢_𝜉∈𝐿^∞0,𝑇;𝐿²0,𝜉₀∪𝜉₀,1. (58)
If it is also satisfied that 𝑢₀∈𝐻²([0,1]); (59) then the piecewise regular solution (𝜌,𝑢) has the regularities 𝜌∈𝐿^∞0,𝑇;𝐻¹0,𝜉₀∪𝜉₀,𝜌,1_𝜏∈𝐿^∞0,𝑇;𝐿²0,𝜉₀∪𝜉₀,1,𝑢∈𝐿^∞0,𝑇;𝐻¹([0,1])∩𝐿²0,𝑇;𝐻²,𝑢([0,1])_𝜏∈𝐿^∞0,𝑇;𝐿²([0,1])∩𝐿²0,𝑇;𝐻¹,𝑎(𝜏)([0,1])∈𝐻²,𝜌([0,𝑇])_𝛾−𝜌^1+𝛼𝑢_𝜉∈𝐿^∞0,𝑇;𝐻¹0,𝜉₀∪𝜉₀,1. (60)

Proof. Multiplying (24)₂ by 𝑢_𝜏, integrating the result over [0,1], and making use of the boundary condition(24)₃, after a direct computation and recombination, we deduce 12¹₀𝜌^1+𝛼𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠=−¹₀𝜌^𝛾₀𝑢_0𝜉1𝑑𝜉+2¹₀𝜌^1+𝛼𝑢²_0𝜉𝑑𝜉+¹₀𝜌^𝛾𝑢_𝜉𝑑𝜉+𝛾^𝜏₀¹₀𝜌^1+𝛾𝑢²_𝜉𝑑𝜉𝑑𝑠−1+𝛼2^𝜏₀¹₀𝜌^2+𝛼𝑢³_𝜉1𝑑𝜉𝑑𝑠≤𝐶+4¹₀𝜌^1+𝛼𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢_𝜉³𝑑𝜉𝑑𝑠, (61) where we have used (26) and (27). On the other hand, integrating (24)₂ over [𝜉,1] and making use of (23) and the boundary conditions (24)₃, it holds that 𝜌^1+𝛼𝑢_𝜉=𝜌^𝛾−¹_𝜉𝑢_𝜏𝑑𝜉≤𝐶+𝐶¹₀𝑢²_𝜏𝑑𝜉^1/2, (62) which implies sup_{[0,𝜉0)∪(𝜉0,1]}𝑢_𝜉≤𝐶(𝑇)+𝐶(𝑇)¹₀𝑢²_𝜏𝑑𝜉^1/2. (63) It holds from (61) and (63) that 12¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀sup_{0,𝜉0∪𝜉0,1}𝑢_𝜉¹₀𝑢²_𝜉(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀1+¹₀𝑢²_𝜏𝑑𝜉^1/21₀𝑢²_𝜉≤1𝑑𝜉𝑑𝑠2^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠+^𝜏₀𝑢_𝜉²_𝐿2¹₀𝑢²_𝜉𝑑𝜉𝑑𝑠; (64) using Gronwall's inequality, (26), and (47), we have ¹₀𝑢²_𝜉𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠≤𝐶(𝑇), (65) where 𝐶(𝑇)>0 denotes a constant dependent of time.
Differentiating (24)₂ with respect to 𝜏, we get 𝑢_𝜏𝜏+(𝜌)^𝛾_𝜉𝜏=(𝜌^1+𝛼𝑢_𝜉)_𝜉𝜏. (66) Taking inner product between (66) and 𝑢_𝜏, integrating the results over [0,1], and using the boundary conditions (24)₃, it holds that 12𝑑𝑑𝜏¹₀𝑢²_𝜏𝑑𝜉=¹₀(𝜌^𝛾)_𝜏𝑢_𝜉𝜏𝑑𝜉−¹₀(𝜌^1+𝛼𝑢_𝜉)_𝜏𝑢_𝜉𝜏𝑑𝜉. (67) The terms on the right-hand side of (67) can be bounded, respectively,as described below: ¹₀(𝜌^𝛾)_𝜏𝑢_𝜉𝜏𝑑𝜉=−¹₀𝛾𝜌^𝛾+1𝑢_𝜉𝑢_𝜉𝜏𝛾𝑑𝜉≤−2𝑑𝑑𝜏¹₀𝜌^𝛾+1𝑢²_𝜉𝑑𝜉+𝐶¹₀𝜌^1+𝛼𝑢²_𝜉+𝜌^{2𝛾−𝛼+3}𝑢⁴_𝜉−𝑑𝜉,₁₀𝜌^1+𝛼𝑢_𝜉𝜏𝑢_𝜉𝜏𝑑𝜉=−¹₀(1+𝛼)𝜌^𝛼𝜌_𝜏𝑢_𝜉+𝜌^1+𝛼𝑢_𝜉𝜏𝑢_𝜉𝜏1𝑑𝜉≤−2¹₀𝜌^1+𝛼𝑢²_𝜉𝜏𝑑𝜉+𝐶¹₀𝜌^3+𝛼𝑢⁴_𝜉𝑑𝜉. (68) Summing (67) and (68) together and making use of (27) and (65), we obtain 12𝑑𝑑𝜏¹₀𝑢²_𝜏𝛾𝑑𝜉+2𝑑𝑑𝜏¹₀𝜌^𝛾+1𝑢²_𝜉1𝑑𝜉+2¹₀𝜌^1+𝛼𝑢²_𝜉𝜏𝑑𝜉≤𝐶¹₀𝜌^1+𝛼𝑢²_𝜉+𝜌^{2𝛾−𝛼+3}𝑢⁴_𝜉𝑑𝜉+¹₀𝜌^3+𝛼𝑢⁴_𝜉(𝑇)𝜌𝑑𝜉≤𝐶+𝐶^1+𝛼𝑢_𝜉²_{𝐿∞([0,1])}¹₀𝜌^1−𝛼𝑢²_𝜉𝑑𝜉. (69) Substituting (62) into (69), it follows from (27), (47), and (59) that 12¹₀𝑢²_𝜏𝛾𝑑𝜉+2¹₀𝜌^𝛾+1𝑢²_𝜉1𝑑𝜉+2^𝜏₀¹₀𝜌^1+𝛼𝑢²_𝜉𝜏(𝑇)(𝑇)𝑑𝜉𝑑𝑠≤𝐶+𝐶^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠, (70) which together with Gronwall's inequality, (27), (47), and (65) yields ¹₀𝑢²_𝜉𝑑𝜉+¹₀𝑢²_𝜏𝑑𝜉+^𝜏₀¹₀𝑢²_𝑠𝑑𝜉𝑑𝑠+^𝜏₀¹₀𝑢²_𝜉𝑠𝑑𝜉𝑑𝑠≤𝐶(𝑇), (71) which implies (𝜌^𝛾−𝜌^1+𝛼𝑢_𝜉)∈𝐿^∞(0,𝑇;𝐻¹([0,1])), and it follows from the definition of 𝑎^′(𝜏)=𝑢(0,𝜏) that 𝑎(𝜏)∈𝐻²([0,𝑇]). The proof of this Lemma is completed.

Lemma 12. Let 𝑇>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (𝜌,𝑢) to the FBVP (24) that 𝜌^𝛼₀𝜉₀𝑒^−𝐶0𝜏≤𝜌^𝛼𝜉₀,𝜏,𝜏∈[0,𝑇], (72) where 𝐶₀ is a positive constant independent of time.

Proof. From (27) and (34), we can obtain (72).

Finally, we will give the large time behaviors of the interface and decay rate of the density as follows.

Lemma 13. Let (𝜌,𝑢,𝑎) be any piecewise regular solution to the FBVP (1) and (2). Under the assumptions of Theorem 2, it holds for 𝛼∈(0,1] and time 𝑡>0 large enough that 𝐶(1+𝑡)^{𝛾/(𝛾−𝛼)}(𝑡)≥𝑐()≥𝑎1+𝑡,1<𝛾<2𝛼,𝑐(1+𝑡)^1−𝜈,𝛾=2𝛼,𝑐(1+𝑡)^{𝛼/(𝛾−𝛼)},𝛾>2𝛼, (73) and the density decays pointwise to zero for any 𝑥∈[0,𝑎(𝑡)] and 𝑡>0 as 𝜌(𝑎(𝑡),𝑡)=𝛾−𝛼𝑡+𝜌₀𝑎₀^{𝛼−𝛾−1/(𝛾−𝛼)},𝜌(𝑥,𝑡)≤𝐶(1+𝑡)^{−1/(𝛾−𝛼)}+𝐶(1+𝑡)^{−(𝛾−1)/(3𝛾+2𝛼−1)},1<𝛾<2𝛼,𝐶(1+𝑡)^{−((𝛾−1)/(3𝛾+2𝛼−1))+𝜈},𝑖𝑖𝛾=2𝛼,𝐶(1+𝑡)^{−𝛼(𝛾−1)/(3𝛾+2𝛼−1)(𝛾−𝛼)},+𝛾>2𝛼,𝐶(1+𝑡)^{−2/(3𝛾+2𝛼−1)},1<𝛾<2𝛼,𝐶(1+𝑡)^{−(2/(3𝛾+2𝛼−1))+𝜈},𝛾=2𝛼,𝐶(1+𝑡)^{−2𝛼/(3𝛾+2𝛼−1)(𝛾−𝛼)},𝛾>2𝛼, (74)(75) where 𝐶>0 and 𝑐>0 are positive constants independent of time and 𝜈∈(0,1) is a small constant.

Proof. We introduce the following functional 𝐻(𝑡) in the Eulerian form as [22, 28]: 𝐻(𝑡)=₀^𝑎(𝑡)(𝑥−(1+𝑡)𝑢(𝑥,𝑡))²𝜌()+2𝑥,𝑡𝑑𝑥𝛾−1(1+𝑡)²₀^𝑎(𝑡)𝜌^𝛾(𝑥,𝑡)𝑑𝑥=₀^𝑎(𝑡)𝑥²𝜌()()×𝑥,𝑡𝑑𝑥−21+𝑡_0𝑎(𝑡)𝑥𝜌𝑢𝑑𝑥+(1+𝑡)²₀^𝑎(𝑡)𝜌𝑢²+2𝜌𝛾−1^𝛾𝑑𝑥∶=𝐼₁+𝐼₂+𝐼₃. (76) Differentiating (76) with respect to 𝑡, using (1), (2), and 𝑎^′(𝑡)=𝑢(𝑎(𝑡),𝑡), we have 𝐼^′₁=₀^𝑎(𝑡)𝑥²𝜌_𝑡𝑑𝑥+𝑎²(𝑡)𝜌(𝑎(𝑡),𝑡)𝑎^′(𝑡)=2₀^𝑎(𝑡)𝑥𝜌𝑢𝑑𝑥+𝑦²(𝑡)𝑦^′(𝑡𝜌,𝐼_′2)=−2₀^𝑎(𝑡)()×𝑥𝜌𝑢𝑑𝑥−21+𝑡_0𝑎(𝑡)𝜌𝑢²+𝜌^𝛾𝑑𝑥+2(1+𝑡)₀^𝑎(𝑡)𝜌^𝛼𝑢_𝑥()𝑦(𝑡)𝑑𝑥−21+𝑡(𝑦^′(𝑡))²𝜌,𝐼^′₃=2(1+𝑡)₀^𝑎(𝑡)𝜌𝑢²+2𝜌𝛾−1^𝛾𝑑𝑥−2(1+𝑡)²₀^𝑎(𝑡)𝜌^𝛼𝑢²_𝑥+()𝑑𝑥1+𝑡²(𝑦^′(𝑡))³𝜌+2()𝛾−11+𝑡²𝑦^′(𝑡)𝜌^𝛾. (77) Combining (77), we deduce 𝐻^′(𝑡)=23−𝛾()𝛾−11+𝑡₀^𝑎(𝑡)𝜌^𝛾()×𝑑𝑥+21+𝑡_0𝑎(𝑡)𝜌^𝛼𝑢_𝑥𝑑𝑥−2(1+𝑡)²₀^𝑎(𝑡)𝜌^𝛼𝑢²_𝑥(𝑡)−()𝑦𝑑𝑥+(𝑦1+𝑡^′(𝑡))²𝑦^′(𝑡𝜌+2)𝛾−1(1+𝑡)²𝑦^′(𝑡)𝜌^𝛾≤23−𝛾()𝛾−11+𝑡₀^𝑎(𝑡)𝜌^𝛾1𝑑𝑥+2₀^𝑎(𝑡)𝜌^𝛼𝑑𝑥, (78) where we use the fact that as 𝑡>0 becomes large enough; it holds from (31) and 𝑦(𝑡)=^𝜉0₀1𝜌()=𝜁,𝑡𝑑𝜁_𝑡0𝑢𝑦(𝑠),𝑠𝑑𝑠+𝑦₀>0,𝑡∈[0,+∞), (79) that 𝑦^′(𝑡)=𝑢𝑦(𝑡),𝑡≥0. (80)
If 𝛾≥3, we have from (78) and the conservation of mass that 𝐻^′(𝑡)≤₀^𝑎(𝑡)𝜌^𝛼≤𝑑𝑥_0𝑎(𝑡)𝜌𝑑𝑥^𝛼₀^𝑎(𝑡)1𝑑𝑥^1−𝛼≤𝐶𝑎(𝑡)^1−𝛼. (81) Hence, it holds that 𝐻(𝑡)≤𝐻(0)+𝐶^𝑡₀𝑎(𝑠)^1−𝛼𝑑𝑠≤𝐶1+^𝑡₀𝑎(𝑠)^1−𝛼𝑑𝑠_0𝑎(𝑡),𝜌^𝛾𝑑𝑥≤𝐶1+^𝑡₀𝑎(𝑠)^1−𝛼𝑑𝑠(1+𝑡)⁻². (82)(83) From (30) and (35), we obtain 𝑎(𝑡)≤𝐶𝜌^−𝛾(𝑎(𝑡),𝑡)≤𝐶(1+𝑡)^{𝛾/(𝛾−𝛼)}, (84) and then ^𝑡₀𝑎(𝑠)^1−𝛼𝑑𝑠≤𝐶(1+𝑡)^{𝛾(1−𝛼)/(𝛾−𝛼)+1}, (85) which with (83) implies ₀^𝑎(𝑡)𝜌^𝛾𝑑𝑥≤𝐶(1+𝑡)^{−𝛼(𝛾−1)/(𝛾−𝛼)},𝛾≥3. (86)
If 1<𝛾<3, we deduce from (76), (78), and the conservation of mass that 𝐻^′(𝑡)=3−𝛾(1+𝑡)⁻¹𝐻(𝑡)+2₀^𝑎(𝑡)𝜌^𝛼≤()𝑑𝑥3−𝛾1+𝑡⁻¹𝐻(𝑡)+𝐶𝑎(𝑡)^1−𝛼, (87) to which the application of Gronwall's inequality gives 𝐻(𝑡)≤𝐶𝐻(0)+^𝑡₀𝑎(𝑠)^1−𝛼(1+𝑠)^𝛾−3𝑑𝑠(1+𝑡)^3−𝛾,_0𝑎(𝑡)𝜌^𝛾𝑑𝑥≤𝐶1+^𝑡₀𝑎(𝑠)^1−𝛼(1+𝑠)^𝛾−3𝑑𝑠(1+𝑡)^1−𝛾. (88)(89) We get from (84) that ^𝑡₀𝑎(𝑠)^1−𝛼(1+𝑠)^𝛾−3≤𝑑𝑠𝐶(1+𝑡)^{𝛾(1−𝛼)/(𝛾−𝛼)+𝛾−2}()(),𝛾∈1,3,𝛾≠2𝛼,𝐶ln1+𝑡,𝛾=2𝛼, (90) which together with (89) yields ₀^𝑎(𝑡)𝜌^𝛾𝑑𝑥≤𝐶(1+𝑡)^{−(𝛾−1)}(),𝐶(),𝛾∈1,2𝛼1+𝑡^{−(𝛾−1)}ln(1+𝑡),𝛾=2𝛼,𝐶(1+𝑡)^{−𝛼(𝛾−1)/(𝛾−𝛼)},𝛾∈(2𝛼,3). (91)
Also, it follows from the conservation of mass and Hölder's inequality that 0<^𝑎0₀𝜌₀(𝑥)𝑑𝑥^𝛾=₀^𝑎(𝑡)𝜌(𝑥,𝑡)𝑑𝑥^𝛾≤𝑎(𝑡)^𝛾−1₀^𝑎(𝑡)𝜌^𝛾𝑑𝑥, (92) which together with (84), (86), and (91) gives (73). Similarly, we have 0<^𝑦0₀𝜌₀(𝑥)𝑑𝑥^𝛾=₀^𝑦(𝑡)𝜌(𝑥,𝑡)𝑑𝑥^𝛾≤𝑦(𝑡)^𝛾−1₀^𝑦(𝑡)𝜌^𝛾𝑑𝑥, (93) which together with (35) implies that 𝑦(𝑡)≥𝑐(1+𝑡),1<𝛾<2𝛼,𝑐(1+𝑡)^1−𝜈,𝛾=2𝛼,𝑐(1+𝑡)^{𝛼/(𝛾−𝛼)}−𝜌,𝛾>2𝛼,_𝛾≤𝑐(1+𝑡)⁻¹,1<𝛾<2𝛼,𝑐(1+𝑡)^−1+𝜈,𝛾=2𝛼,𝑐(1+𝑡)^{−𝛼/(𝛾−𝛼)},𝛾>2𝛼. (94)(95)
Finally, it follows from (30) that 𝜌(𝑎(𝑡),𝑡)=𝛾−𝛼𝑡+𝜌₀^𝛼−𝛾𝑎₀^{−1/(𝛾−𝛼)}, (96) and as 𝑡 is large enough, it holds due to (35) that 𝜌^{(3𝛾+2𝛼−1)/2}(𝑥,𝑡)≤𝜌^{(3𝛾+2𝛼−1)/2}(𝑎(𝑡))+,𝑡_0𝑎(𝑡)𝜌^{(3𝛾+2𝛼−1)/2}_𝑥𝑑𝑥−𝜌^{(3𝛾+2𝛼−1)/2}≤𝜌^{(3𝛾+2𝛼−1)/2}(𝑎(𝑡),𝑡)+𝐶₀^𝑎(𝑡)𝜌^𝛾𝑑𝑥^1/2−𝐶𝜌^𝛾, (97) which together with (86), (91), (95), and (96) gives (75).

4. Proof of the Main Results

Proof. The global existence of unique piecewise regular solution to the FBVP (1) and (2) can be established in terms of the short-time existence carried out as in [1, 2, 4], the uniform a-priori estimates, and the analysis of regularities, which indeed follow from Lemmas 5–11. We omit the details. The large time behaviors follow from Lemmas 12–13 directly. The proof of Theorem 2 is completed.

Acknowledgments

The authors thank the referee for the helpful comments and suggestions on the paper. 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. The research of G. J. Zhang is supported by NSF no. GFQQ2460101710.