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

Authors: Ruxu Lian; Guojing Zhang
Journal: Journal of Applied Mathematics
 (2013)

---

## 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.


---

## Body

## 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(1)ρt+(ρu)x=0,(ρu)t+(ρu2+P(ρ))x=(μ(ρ)ux)x,(x,t)∈R×[0,T],
where ρ>0 and u denote the flow density and velocity, respectively, the pressure-density function is taken as P(ρ)=ργ with γ>1, and the viscosity coefficient is μ(ρ)=ρα 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)=ργ(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 Rn [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 coefficientsμ(ρ)=ρα, 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. Asγ>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 Section2, 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:
(2)(ρ,u)(x,0)=(ρ0,u0),x∈[0,a0],u(0,t)=0,(ργ-ραux)(a(t),t)=0,t>0,
where x=a(t) is the free boundary defined by
(3)ddta(t)=u(a(t),t),a(0)=a0,t>0.Next, we give the definition of weak solution to the free boundary problem (1) and (2).Definition 1 (weak solution).
For anyT>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:
(4)0≤ρ∈L∞(0,T;L1([0,a(t)])∩Lγ([0,a(t)])),ρu∈L∞(0,T;L2([0,a(t)])),(ργ-ραux)∈L2(0,T;H1([0,a(t)])),
and (1) is satisfied in the sense of distributions. Namely, it holds for all φ∈C0∞([0,a(t)]×[0,T)) that
(5)∫0a0‍ρ0φ(x,0)dx+∫0T‍∫0a(t)‍ρφtdxdt+∫0T‍∫0a(t)‍ρρuφxdxdt=0
and for all ψ∈C0∞([0,a(t)]×[0,T)) that
(6)∫0a0‍ρ0u0ψ(x,0)dx+∫0T‍∫0a(t)‍(ρρuψt+(ρu)2ψx)dxdt+∫0T‍∫0a(t)‍(ργ-ραux)ψxdxdt=0.For simplicity, we consider the initial data in FBVP (1) and (2) with one discontinuous point y0∈(0,a0); namely, for some constant ρ->0(7)inf[0,y0)∪(y0,a0]ρ0≥ρ->0,(ρ0,u0)∈W1,∞([0,y0)∪(y0,a0]),(ρ0γ-ρ0αu0x)(a0)=0,ρ0(y0-0)>ρ0(y0+0),
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.
Letγ>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>0(8)ρ∈L∞(0,T;H1([0,y(t))∪(y(t),a(t)]))∩C0([0,y(t))∪(y(t),a(t)]×[0,T]),u∈L∞(0,T;H1([0,a(t)]))∩L2(0,T;H2([0,a(t)])),ut∈L2(0,T;L2([0,a(t)])),a(t)∈H1([0,T]),(ργ-ραux)∈L∞(0,T;L2([0,y(t))∪(y(t),a(t)])),
where x=y(t) is a curve defined by
(9)dy(t)dt=u(y(t),t),y(0)=y0,t>0,
along which the Rankine-Hugoniot conditions hold
(10)[u(y(t),t)]=0,[ργ(y(t),t)]=[ραux(y(t),t)],
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
(11)|[ρ0α(y0)]|e-C0t≤|[ρα(y(t),t)]|,
where C0 is a positive constant independent of time.
If it further holds thatu0∈H2([0,a0]), then (ρ,u,a,b) satisfies
(12)ρ∈L∞(0,T;H1([0,y(t))∪(y(t),a(t)])),ρt∈L∞(0,T;L2([0,y(t))∪(y(t),a(t)])),u∈L∞(0,T;H1([0,a(t)]))∩L2(0,T;H2([0,a(t)])),ut∈L∞(0,T;L2([0,a(t)]))∩L2(0,T;H1([0,a(t)])),a(t)∈H2([0,T]),(ργ-ραux)∈L∞(0,T;H1([0,y(t))∪(y(t),a(t)])).
The domain expands outwards at an algebraic rate in time as
(13)C(1+t)γ/(γ-α)≥a(t)≥{c(1+t),1<γ<2α,c(1+t)1-ν,γ=2α,c(1+t)α/(γ-α),γ>2α,
and the density decays pointwise to zero for any x∈[0,y(t))∪(y(t),a(t)] and t>0 as
(14)ρ(a(t),t)=((γ-α)t+ρ0(a0)α-γ)-1/(γ-α),(15)ρ(x,t)≤C(1+t)-1/γ-α+{C(1+t)-(γ-1)/(3γ+2α-1),kkkkkkkk.1<γ<2α,C(1+t)-((γ-1)/(3γ+2α-1))+ν,kkkkkkkkkkkkkkiγ=2α,C(1+t)-α(γ-1)/(3γ+2α-1)(γ-α),kkkkkkkkkkkkkkkkγ>2α,+{C(1+t)-2/(3γ+2α-1),kkkkkk1<γ<2α,C(1+t)-(2/(3γ+2α-1))+ν,kkkkkkkkkkki.γ=2α,C(1+t)-2α/(3γ+2α-1)(γ-α),kkkkkkkkkkkkk.γ>2α,
where C>0 and c>0 are positive constants independent of time, and ν∈(0,1) is a small constant.Remark 3.
Theorem2 holds for the Saint-Venant model for shallow water; that is, γ=2, α=1.Remark 4.
Fang-Zhang [10] obtained that
(16)|[ρ0α(y0)]|e-C0t≤|[ρα(y(t),t)]|,
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,
(17)|[ρα(y(t),t)]|≤C(1+t)-α/(γ-α)+{C(1+t)-α(γ-1)/(3γ+2α-1),kkkkkkkkk1<γ<2α,C(1+t)-(α(γ-1)/(3γ+2α-1))+αν,kkkkkkkkkkkkkkkkγ=2α,C(1+t)-α^2(γ-1)/(3γ+2α-1)(γ-α),kkkkk.kkkkkkkkkkkiγ>2α,+{C(1+t)-2α/(3γ+2α-1),.kkkkkki1<γ<2α,C(1+t)-(2α/(3γ+2α-1))+ν,kkkkkkkkkkkkiγ=2α,C(1+t)-2α^2/(3γ+2α-1)(γ-α),kkkkk.kkkkkkkkiγ>2α,
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
(18)dy(t)dt=u(y(t),t),y(0)=y0,t>0,
along which the Rankine-Hugoniot conditions hold
(19)[u(y(t),t)]=0,[ργ(y(t),t)]=[ραux(y(t),t)],
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
(20)ξ=∫0x‍ρ(z,t)dz,τ=t.
Since the conservation of total mass holds
(21)∫0a(t)‍ρ(z,t)dz=∫0a0‍ρ0(z)dz:=1>0,
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
(22)ξ0:=∫0y(t)‍ρ(z,t)dz=∫0y0‍ρ0(z)dz,
and the jump conditions become
(23)[u(ξ0,τ)]=0,[ργ(ξ0,τ)]=[ρ1+αuξ(ξ0,τ)].Meanwhile, the FBVP (1) and (2) is reformulated into
(24)ρτ+ρ2uξ=0,uτ+(ργ)ξ=(ρ1+αuξ)ξ,(ργ-ρ1+αuξ)(0,τ)=(ργ-ρ1+αuξ)(1,τ)=0,(ρ0,u0)=(ρ0,u0)(ξ),ξ∈[0,1],
where the initial data satisfies
(25)inf[0,ξ0)∪(ξ0,1]ρ0≥ρ->0,(ρ0,u0)∈W1,∞([0,ξ0)∪(ξ0,1]),(ρ0γ-ρ01+αu0x)(1)=0,ρ0(ξ0-0)>ρ0(ξ0+0),
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.
LetT>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (ρ,u) to the FBVP (24) that
(26)∫01‍(u22+1γ-1ργ-1)dξ+∫0τ‍∫01‍ρ1+αuξ2dξds=∫01‍(u022+1γ-1ρ0γ-1)dξ,τ∈[0,T],(27)ρ(ξ,τ)≤C,(ξ,τ)∈[0,ξ0)∪(ξ0,1]×[0,T],(28)∫01‍u2ndξ+n(2n-1)∫0τ‍∫01‍ρ1+αu2n-2uξ2dξds≤C(T),τ∈[0,T],(29)⨏01(ρα)ξ2ndξ≤C(T),τ∈[0,T],
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 ⨏01:=∫0ξ0‍+∫ξ01‍.Lemma 6.
LetT>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (ρ,u) to the FBVP (24) that
(30)ρ(0,τ)=((γ-α)τ+ρ0(0)α-γ)-1/(γ-α),(31)ρ(ξ0-0,τ)>ρ(ξ0+0,τ),τ∈[0,T].Proof.
From (24)1 and (24)3, we have
(32)ρτ(0,τ)+ργ-α+1(0,τ)=0,
which yields (30) similarly, because of (23) and (24)1, it holds that
(33)[ρα]τ+α[ργ]=0,
which together with (25) implies
(34)[ρα]=[ρ0α]exp{-α∫0τ‍[ργ][ρα]ds}<0.Lemma 7.
LetT>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (ρ,u) to the FBVP (24) that
(35)12⨏01(u+(ρα)ξα)2dξ+1γ-1∫01‍ργ-1dξ+γ∫0τ‍⨏01ργ+α-2ρξ2dξds+ργ(1,τ)a(τ)+γ∫0τ‍ρ2γ-α(1,s)a(s)ds-[ργ]y(τ)-γ∫0τ‍[ργ+1]uξy(s)ds=12⨏01(u0+(ρα)ξα)2dξ+1γ-1∫01‍ρ0γ-1dξ+ρ0γ(1)a0-[ρ0γ]y0kkkkkkkkkkkkkkkkkkkkkkkkkiτ∈[0,T],
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 ρα-1 gives
(36)(ρα)τα+ρ1+αuξ=0,
which leads to
(37)(ρα)τξα+(ρ1+αuξ)ξ=0.
Summing (24)2 and (37), we have
(38)(u+(ρα)ξα)τ+(ργ)ξ=0.
Multiplying (38) by (u+(ρα)_ξ/α) and integrating the result over [0,1]×[0,τ], we get
(39)12⨏01(u+(ρα)ξα)2dξ+1γ-1∫01‍ργ-1dξ+γ∫0τ‍⨏01ργ+α-2ρξ2dξds+∫0τ‍ργu|ξ=0ξ=ξ0-0ds+∫0τ‍ργu|ξ=ξ0+0ξ=1ds=12⨏01(u0+(ρ0α)ξα)2dξ+1γ-1∫01‍ρ0γ-1dξ,
which together with the fact that
(40)∫0τ‍ργu|ξ=0ξ=ξ0-0ds=∫0τ‍ργ(ξ0-0,s)y′(s)ds=ργ(ξ0-0,s)y(s)|0τ+γ∫0τ‍ργ-1ρ2(ξ0-0,s)uξ(ξ0,s)y(s)ds=ργ(ξ0-0,τ)y(τ)-ρ0γ(ξ0-0)y0+γ∫0τ‍ργ+1(ξ0-0,s)uξ(ξ0,s)y(s)ds,(41)∫0τ‍ργu|ξ=ξ0+0ξ=1ds=∫0τ‍ργ(1,τ)a′(s)ds-∫0τ‍ργ(ξ0+0,τ)y′(s)ds=ργ(1,s)a(s)|0τ+γ∫0τ‍ρ2γ-α(1,s)a(s)ds-ργ(ξ0+0,s)y(s)|0τ-γ∫0τ‍ργ+1(ξ0+0,s)uξ(ξ0,s)y(s)ds=ργ(1,τ)a(τ)-ρ0γ(1)a0+γ∫0τ‍b(s)ρ2γ-α(1,s)ds-ργ(ξ0+0,τ)y(τ)+ρ0γ(ξ0+0)y0-γ∫0τ‍ργ+1(ξ0+0,s)uξ(ξ0,s)y(s)ds,[ρ(ξ0,τ)]<0,uξ(ξ0,τ)=[ργ][ρ1+α]>0,a(τ)=∫01‍1ρ(ζ,τ)dζ>0,y(τ)=∫0ξ0‍1ρ(ζ,τ)dζ>0,
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],
(42)12⨏0a(t)ρ(u+ρ-1(ρα)x)2dx+1γ-1∫0a(t)‍ργdx+γ∫0t‍⨏0a(t)ργ+α-3ρx2dxds+ργ(a(t),t)a(t)+γ∫0t‍ρ2γ-α(a(s),s)a(s)ds-[ργ]y(t)-γ∫0t‍[ργ]uxy(s)ds=12⨏0a0ρ0(u0+ρ0-1(ρ0α)x)2dx+1γ-1∫0a0‍ρ0γdx+ρ0γ(a0)a0-[ρ0γ]y0.Lemma 9.
LetT>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
(43)∫0τ‍∥(ργ)ξ2n∥L∞([0,ξ0)∪(ξ0,1])ds≤C(T),τ∈[0,T].Proof.
It follows from (24)1,2 that
(44)(ρα)ξ(ξ,τ)=(ρ0α)ξ(ξ)-αu(ξ,τ)+αu0(ξ)-α∫0τ‍(ργ)ξ(ξ,s)ds.
By means of (25), (26), and (44), we have
(45)∫0τ‍∥(ργ)ξ2n∥L∞([0,ξ0)∪(ξ0,1])ds=γ2nα2n∫0τ‍∥ρ2n(γ-α)(ρα)ξ2n∥L∞([0,ξ0)∪(ξ0,1])ds≤C(T)+C(T)∫0τ‍∥ρ2n(γ-α)u2n∥L∞([0,ξ0)∪(ξ0,1])ds+C(T)∫0τ‍∫0s‍∥(ργ)ξ2n∥L∞([0,ξ0)∪(ξ0,1])dlds≤C(T)+C(T)∫0τ‍∫0s‍∥(ργ)ξ2n∥L∞([0,ξ0)∪(ξ0,1])dlds,
where we have used
(46)∫0τ‍∥ρ2n(γ-α)u2n∥L∞([0,ξ0)∪(ξ0,1])ds≤∫0τ‍⨏01ρ2n(γ-α)u2ndξds+∫0τ‍⨏01|(ρ2n(γ-α)u2n)ξ|dξds≤C(T)+C∫0τ‍⨏01(ρ2(2n(γ-α)-α)ρ2α-2ρξ2+u4nkkkkkkkkkkkk+ρ4n(γ-α)-(1+α)u2nkkkkkkkkkkki+ρ1+αu2n-2uξ2)dξds≤C(T),
which can be deduced from (26) and (27). Making use of Gronwall's inequality to (46), we obtain (43).Lemma 10.
LetT>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (ρ,u) to the FBVP (24) that
(47)ρ(ξ,τ)≥C(T),(ξ,τ)∈[0,ξ0)∪(ξ0,1]×[0,T].Proof.
Denote(48)v(ξ,τ)=1ρ(ξ,τ).
By (24)1, we have
(49)vτ=uξ.
Multiplying (49) by βvβ-1, integrating the result over [0,1]×[0,τ], and using (43), (44), we can obtain that for β∈(1,2)(50)∫01‍vβdξ+β(β-1)∫0τ‍∫01‍vα+β-1u2dξds=∫01‍v0βdξ+β∫0τ‍vβ-1uds|ξ=0ξ=ξ0-0+β∫0τ‍vβ-1uds|ξ0+0ξ=1+β(β-1)α∫0τ‍⨏01vα+β-1u(ρ0α)ξdξds+β(β-1)∫0τ‍∫01‍vα+β-1uu0dξds-β(β-1)×∫0τ‍∫01‍vα+β-1u∫0s‍(ργ)ξdldξds≤C(T)+C∫0τ‍vβ-1u(1,s)ds+β(β-1)2×∫0τ‍∫01‍vα+β-1u2dξds+C(T)∫0τ‍∫01‍vβdξds,
where we use the fact that
(51)∫0τ‍vβ-1(ξ0-0,s)u(ξ0,s)ds-∫0τ‍vβ-1(ξ0+0,s)u(ξ0,s)ds=∫0τ‍ρ1-β(ξ0-0,s)y′(s)ds-∫0τ‍ρ1-β(ξ0+0,s)y′(s)ds=ρ1-β(ξ0-0,s)y(s)|0τ-ρ1-β(ξ0+0,s)y(s)|0τ+(1-β)∫0τ‍ρ2-β(ξ0-0,s)uξ(ξ0,s)y(s)ds-(1-β)∫0τ‍ρ2-β(ξ0+0,s)uξ(ξ0,s)y(s)ds=-[ρ1-β]y(τ)+[ρ01-β]y0+(β-1)∫0τ‍[ρ2-β]uξ(ξ0,s)y(s)ds≤[ρ01-β]y0≤C.
Since it holds that
(52)ρ(1,τ)=((γ-α)τ+ρ0(1)α-γ)-1/(γ-α)≥C(T),
we have from (25) that
(53)∫0τ‍vβ-1u(1,s)dξ≤C(T)∫0τ‍((∫01‍u2dξ)1/2kkkkkkkkkkkk+(∫01‍ρ1+αuξ2dξ)1/2(∫01‍v1+αdξ)1/2)ds≤C(T)+C(T)∫0τ‍∫01‍vβdξds.
Substituting (53) in (50), we have
(54)∫01‍vβdξ+β(β-1)2∫0τ‍∫01‍vα+β-1u2dξds≤C(T)+C(T)∫0τ‍∫01‍vβdξds.
Using Gronwall's inequality, we get from (54) that
(55)∫01‍vβdξ≤C(T).
It follows from (29) and (55) that
(56)vβ(ξ,τ)≤∫01‍vβdξ+⨏01|vξβ|dξ≤C(T)+C(⨏01ρ-β-1|ρξ|dξ)≤C(T)+C(∫01‍ρ-2n(α+β)/(2n-1)dξ)(2n-1)/2nk×(⨏01(ρα)ξ2ndξ)1/2n≤C(T)+C(T)sup[0,ξ0)∪(ξ0,1]vα+β/2n(∫01‍vβdξ)(2n-1)/2n≤C(T)+C(T)sup[0,ξ0)∪(ξ0,1](vβ(ξ,τ))α/β+1/2n,
as α∈(0,1], β∈(1,2); for some n∈N large enough, we have
(57)αβ+12n<1,
which implies (47).We also have the regularity estimates for the solution(ρ,u) to the FBVP (24) as follows.Lemma 11.
LetT>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (ρ,u) to the FBVP (24) that
(58)ρ∈L∞(0,T;H1([0,ξ0)∪(ξ0,1]))∩C0([0,ξ0)∪(ξ0,1]×[0,T]),u∈L∞(0,T;H1([0,1]))∩L2(0,T;H2([0,1])),uτ∈L2(0,T;L2([0,1])),a(τ)∈H1([0,T]),(ργ-ρ1+αuξ)∈L∞(0,T;L2([0,ξ0)∪(ξ0,1])).
If it is also satisfied that(59)u0∈H2([0,1]);
then the piecewise regular solution (ρ,u) has the regularities
(60)ρ∈L∞(0,T;H1([0,ξ0)∪(ξ0,1])),ρτ∈L∞(0,T;L2([0,ξ0)∪(ξ0,1])),u∈L∞(0,T;H1([0,1]))∩L2(0,T;H2([0,1])),uτ∈L∞(0,T;L2([0,1]))∩L2(0,T;H1([0,1])),a(τ)∈H2([0,T]),(ργ-ρ1+αuξ)∈L∞(0,T;H1([0,ξ0)∪(ξ0,1])).Proof.
Multiplying (24)2 by uτ, integrating the result over [0,1], and making use of the boundary condition(24)3, after a direct computation and recombination, we deduce
(61)12∫01‍ρ1+αuξ2dξ+∫0τ‍∫01‍us2dξds=-∫01‍ρ0γu0ξdξ+12∫01‍ρ1+αu0ξ2dξ+∫01‍ργuξdξ+γ∫0τ‍∫01‍ρ1+γuξ2dξds-1+α2∫0τ‍∫01‍ρ2+αuξ3dξds≤C+14∫01‍ρ1+αuξ2dξ+∫0τ‍∫01‍|uξ|3dξds,
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
(62)ρ1+αuξ=ργ-∫ξ1‍uτdξ≤C+C(∫01‍uτ2dξ)1/2,
which implies
(63)sup[0,ξ0)∪(ξ0,1]uξ≤C(T)+C(T)(∫01‍uτ2dξ)1/2.
It holds from (61) and (63) that
(64)12∫01‍uξ2dξ+∫0τ‍∫01‍us2dξds≤C(T)+C(T)∫0τ‍sup[0,ξ0)∪(ξ0,1]|uξ|∫01‍uξ2dξds≤C(T)+C(T)∫0τ‍(1+∫01‍uτ2dξ)1/2∫01‍uξ2dξds≤12∫0τ‍∫01‍us2dξds+∫0τ‍∥uξ∥L22∫01‍uξ2dξds;
using Gronwall's inequality, (26), and (47), we have
(65)∫01‍uξ2dξ+∫0τ‍∫01‍us2dξds≤C(T),
where C(T)>0 denotes a constant dependent of time.
Differentiating (24)2 with respect to τ, we get
(66)uττ+(ρ)ξτγ=(ρ1+αuξ)ξτ.
Taking inner product between (66) and uτ, integrating the results over [0,1], and using the boundary conditions (24)3, it holds that
(67)12ddτ∫01‍uτ2dξ=∫01‍(ργ)τuξτdξ-∫01‍(ρ1+αuξ)τuξτdξ.
The terms on the right-hand side of (67) can be bounded, respectively,as described below:
(68)∫01‍(ργ)τuξτdξ=-∫01‍γργ+1uξuξτdξ≤-γ2ddτ∫01‍ργ+1uξ2dξ+C∫01‍(ρ1+αuξ2+ρ2γ-α+3uξ4)dξ,-∫01‍(ρ1+αuξ)τuξτdξ=-∫01‍((1+α)ραρτuξ+ρ1+αuξτ)uξτdξ≤-12∫01‍ρ1+αuξτ2dξ+C∫01‍ρ3+αuξ4dξ.
Summing (67) and (68) together and making use of (27) and (65), we obtain
(69)12ddτ∫01‍uτ2dξ+γ2ddτ∫01‍ργ+1uξ2dξ+12∫01‍ρ1+αuξτ2dξ≤C∫01‍(ρ1+αuξ2+ρ2γ-α+3uξ4)dξ+∫01‍ρ3+αuξ4dξ≤C(T)+C∥ρ1+αuξ∥L∞([0,1])2∫01‍ρ1-αuξ2dξ.
Substituting (62) into (69), it follows from (27), (47), and (59) that
(70)12∫01‍uτ2dξ+γ2∫01‍ργ+1uξ2dξ+12∫0τ‍∫01‍ρ1+αuξτ2dξds≤C(T)+C(T)∫0τ‍∫01‍us2dξds,
which together with Gronwall's inequality, (27), (47), and (65) yields
(71)∫01‍uξ2dξ+∫01‍uτ2dξ+∫0τ‍∫01‍us2dξds+∫0τ‍∫01‍uξs2dξds≤C(T),
which implies (ργ-ρ1+αuξ)∈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.
LetT>0. Under the assumptions of Theorem 2, it holds for any piecewise regular solution (ρ,u) to the FBVP (24) that
(72)|[ρ0α(ξ0)]|e-C0τ≤|[ρα(ξ0,τ)]|,τ∈[0,T],
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
(73)C(1+t)γ/(γ-α)≥a(t)≥{c(1+t),1<γ<2α,c(1+t)1-ν,γ=2α,c(1+t)α/(γ-α),γ>2α,
and the density decays pointwise to zero for any x∈[0,a(t)] and t>0 as
(74)ρ(a(t),t)=((γ-α)t+ρ0(a0)α-γ)-1/(γ-α),(75)ρ(x,t)≤C(1+t)-1/(γ-α)+{C(1+t)-(γ-1)/(3γ+2α-1),kkkkkknki1<γ<2α,C(1+t)-((γ-1)/(3γ+2α-1))+ν,kknnnnnnnnkkkiiiγ=2α,C(1+t)-α(γ-1)/(3γ+2α-1)(γ-α),kknnnnnnnnnn.kkkiγ>2α,+{C(1+t)-2/(3γ+2α-1),kkkk.ki1<γ<2α,C(1+t)-(2/(3γ+2α-1))+ν,kkkknnnnn.kkiγ=2α,C(1+t)-2α/(3γ+2α-1)(γ-α),kkkknnnnnnn.kkγ>2α,
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 functionalH(t) in the Eulerian form as [22, 28]:
(76)H(t)=∫0a(t)‍(x-(1+t)u(x,t))2ρ(x,t)dx+2γ-1(1+t)2∫0a(t)‍ργ(x,t)dx=∫0a(t)‍x2ρ(x,t)dx-2(1+t)×∫0a(t)‍xρudx+(1+t)2∫0a(t)‍(ρu2+2γ-1ργ)dx∶=I1+I2+I3.
Differentiating (76) with respect to t, using (1), (2), and a′(t)=u(a(t),t), we have
(77)I1′=∫0a(t)‍x2ρtdx+a2(t)ρ(a(t),t)a′(t)2.=2∫0a(t)‍xρudx+y2(t)y′(t)[ρ],I2′=-2∫0a(t)‍xρudx-2(1+t)×∫0a(t)‍(ρu2+ργ)dx+2(1+t)∫0a(t)‍ραuxdx-2(1+t)y(t)(y′(t))2[ρ],I3′=2(1+t)∫0a(t)‍(ρu2+2γ-1ργ)dx-2(1+t)2∫0a(t)‍ραux2dx+(1+t)2(y′(t))3[ρ]+2γ-1(1+t)2y′(t)[ργ].
Combining (77), we deduce
(78)H′(t)=2(3-γ)γ-1(1+t)∫0a(t)‍ργdx+2(1+t)×∫0a(t)‍ραuxdx-2(1+t)2∫0a(t)‍ραux2dx+(y(t)-(1+t)y′(t))2y′(t)[ρ]+2γ-1(1+t)2y′(t)[ργ]≤2(3-γ)γ-1(1+t)∫0a(t)‍ργdx+12∫0a(t)‍ραdx,
where we use the fact that as t>0 becomes large enough; it holds from (31) and
(79)y(t)=∫0ξ0‍1ρ(ζ,t)dζ=∫0t‍u(y(s),s)ds+y0>0,t∈[0,+∞),
that
(80)y′(t)=u(y(t),t)≥0.
Ifγ≥3, we have from (78) and the conservation of mass that
(81)H′(t)≤∫0a(t)‍ραdx≤(∫0a(t)‍ρdx)α(∫0a(t)‍1dx)1-α≤Ca(t)1-α.
Hence, it holds that
(82)H(t)≤H(0)+C∫0t‍a(s)1-αds≤C(1+∫0t‍a(s)1-αds),(83)∫0a(t)‍ργdx≤C(1+∫0t‍a(s)1-αds)(1+t)-2.
From (30) and (35), we obtain
(84)a(t)≤Cρ-γ(a(t),t)≤C(1+t)γ/(γ-α),
and then
(85)∫0t‍a(s)1-αds≤C(1+t)γ(1-α)/(γ-α)+1,
which with (83) implies
(86)∫0a(t)‍ργdx≤C(1+t)-α(γ-1)/(γ-α),γ≥3.
If1<γ<3, we deduce from (76), (78), and the conservation of mass that
(87)H′(t)=(3-γ)(1+t)-1H(t)+2∫0a(t)‍ραdxkkkk.≤(3-γ)(1+t)-1H(t)+Ca(t)1-α,
to which the application of Gronwall's inequality gives
(88)H(t)≤C(H(0)+∫0t‍a(s)1-α(1+s)γ-3ds)(1+t)3-γ,(89)∫0a(t)‍ργdx≤C(1+∫0t‍a(s)1-α(1+s)γ-3ds)(1+t)1-γ.
We get from (84) that
(90)∫0t‍a(s)1-α(1+s)γ-3ds≤{C(1+t)γ(1-α)/(γ-α)+γ-2,γ∈(1,3),γ≠2α,Cln(1+t),γ=2α,
which together with (89) yields
(91)∫0a(t)‍ργdx≤{C(1+t)-(γ-1),γ∈(1,2α),C(1+t)-(γ-1)ln(1+t),γ=2α,C(1+t)-α(γ-1)/(γ-α),γ∈(2α,3).
Also, it follows from the conservation of mass and Hölder's inequality that(92)0<(∫0a0‍ρ0(x)dx)γk=(∫0a(t)‍ρ(x,t)dx)γ≤a(t)γ-1∫0a(t)‍ργdx,
which together with (84), (86), and (91) gives (73). Similarly, we have
(93)0<(∫0y0‍ρ0(x)dx)γk=(∫0y(t)‍ρ(x,t)dx)γ≤y(t)γ-1∫0y(t)‍ργdx,
which together with (35) implies that
(94)y(t)≥{c(1+t),1<γ<2α,c(1+t)1-ν,γ=2α,c(1+t)α/(γ-α),γ>2α,(95)-[ργ]≤{c(1+t)-1,1<γ<2α,c(1+t)-1+ν,γ=2α,c(1+t)-α/(γ-α),γ>2α.
Finally, it follows from (30) that
(96)ρ(a(t),t)=((γ-α)t+ρ0α-γ(a0))-1/(γ-α),
and as t is large enough, it holds due to (35) that
(97)ρ(3γ+2α-1)/2(x,t)≤ρ(3γ+2α-1)/2(a(t),t)+∫0a(t)‍|(ρ(3γ+2α-1)/2)x|dx-[ρ(3γ+2α-1)/2]≤ρ(3γ+2α-1)/2(a(t),t)+C(∫0a(t)‍ργdx)1/2-C[ργ],
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.



---

*Source: 505108-2013-07-09.xml*