# Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with a Nonconstant Exterior Pressure

Authors: Ruxu Lian; Liping Hu
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We consider the free boundary value problem (FBVP) for one-dimensional isentropic compressible Navier-Stokes (CNS) equations with density-dependent viscosity coefficient in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure. Under certain assumptions imposed on the initial data and exterior pressure, we prove that there exists a unique global strong solution which is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate.


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## Body

## 1. Introduction

We will investigate the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient for regular initial data in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure in the present paper. In general, the one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient read as(1)ρt+(ρu)x=0,(ρu)t+(ρu2+P(ρ))x=(μ(ρ)ux)x,(x,t)∈R×[0,T],
where ρ>0, u, and P(ρ)=ργ(γ>1) stand for the flow density, velocity, and pressure, respectively, and the viscosity coefficient is μ(ρ)=ρα with α>0. We note here that as γ=2 and α=1 in (1), the system corresponds to the viscous Saint-Venant system for shallow water.There is huge literature on the studies of the compressible Navier-Stokes equations with density-dependent viscosity coefficients. For example, the mathematical derivations are achieved in the simulation of flow surface in shallow region [1, 2]. Bresch and Desjardins have investigated the existence of solutions to the 2D shallow water equations in [3, 4]. The global existence of classical solutions is proven by Mellet and Vasseur [5]. The qualitative patterns of behavior of global solutions and dynamical asymptotics of vacuum states are also shown, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behavior of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength; refer to [6–11] and references therein.Recently, there is much significant progress achieved on the free boundary value problems; for instance, 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 considered by many authors; refer to [12–24] and references therein. In addition, the free boundary value problems for multidimensional compressible viscous Navier-Stokes equations with constant viscosity coefficients for either barotropic or heat-conducive fluids are investigated by many authors, such as in the case that across the free surface stress tensor is balanced by a constant exterior pressure and/or the surface tension; classical solutions with strictly positive densities in the fluid regions to FBVP for CNS (1) with constant viscosity coefficients are shown locally in time for either barotropic flows [25–27] or heat-conductive flows [28–30]. In the case that across the free surface the stress tensor is balanced by exterior pressure [27], surface tension [31], or both exterior pressure and surface tension [32], respectively, as the initial data is assumed to be near to a nonvacuum equilibrium state, the global existence of classical solutions with small amplitude and positive densities in fluid region to the FBVP for CNS (1) with constant viscosity coefficients is proved. Global existence of classical solutions to FBVP for compressible viscous and heat-conductive fluids is also obtained with the stress tensor balanced by the exterior pressure and/or surface tension across the free surface; refer to [33, 34] and references therein.In the present paper, we focus on the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and a nonconstant exterior pressure, and the existence, regularities, and dynamical behavior of global strong solution will be addressed, and so forth. Asγ>1, 0<α≤1, we show that the free boundary value problem with regular initial data admits a unique global strong solution which is strictly positive from blow for any finite time and decays pointwise to zero at an algebraic time-rate (refer to Theorem 1 for details).The rest of the paper is arranged as follows. In Section2, the main results about the existence and dynamical behavior of global strong solution to FBVP for compressible Navier-Stokes equations are stated. Then, some important a priori estimates will be given in Section 3 and the theorem is proven in Section 4.


## 2. Main Results

We will investigate the global existence and dynamics of the free boundary value problem for (1) with the following initial data and boundary conditions:
(2)(ρ,u)(x,0)=(ρ0,u0),x∈[a0,b0],(ργ-ραux)(a(t),t)=pe(t),(ργ-ραux)(b(t),t)=pe(t),aaaaaaaaaaaaaaaaaaaat>0,
where x=a(t) and x=b(t) are the free boundaries defined by
(3)ddta(t)=u(a(t),t),a(0)=a0,ddtb(t)=u(b(t),t),b(0)=b0,t>0,
and the function pe(t)>0 is the nonconstant exterior pressure.Without the loss of generality, the total initial mass is renormalized to be one; that is,(4)∫a(t)b(t)‍ρ(x,t)dx=∫a0b0‍ρ0(x)dx≔1.
And we consider that the initial data satisfies
(5)inf⁡[a0,b0]ρ0≥ρ_>0,(ρ0,u0)∈W1,∞([a0,b0]),(ρ0γ-ρ0αu0x)(a0)=pe0,(ρ0γ-ρ0αu0x)(b0)=pe0,ρ0γ(b0)≥ρ0γ(a0)≥pe0,
where ρ_ is a positive constant and pe0≔pe(0); note that the compatibility conditions between initial data and boundary conditions hold. Then, we have the global existence and time-asymptotical behavior of strong solution as follows.Theorem 1 (FBVP).
Letγ>1 and 0<α≤1. Assume that the initial data satisfies (5); pe(t)=o((1+t)ν); ν>0 is a constant; pe(t)∈L1([0,T]) and pe′(t)∈L2([0,T]) uniformly for T>0. Then, there exists a unique global strong solution (ρ,u,a,b) to the FBVP (1) and (2) satisfying
(6)c≤ρ∈L∞(0,T;H1([a(t),b(t)]))∩C0([0,T]×[a(t),b(t)]),u∈L∞(0,T;H1([a(t),b(t)]))∩L2(0,T;H2([a(t),b(t)])),a(t),b(t)∈H1([0,T]),(ργ-ραux)∈L∞(0,T;L2([a(t),b(t)])),
with c>0 being a constant independent of time.
If it further holds thatu0∈H2([a0,b0]), then (ρ,u,a,b) satisfies
(7)(ρ,u)∈C0([0,T]×[a(t),b(t)]),ρ∈L∞(0,T;H1([a(t),b(t)])),ρt∈L∞(0,T;L2([a(t),b(t)])),u∈L∞(0,T;H2([a(t),b(t)]))∩L2(0,T;H3([a(t),b(t)])),ut∈L∞(0,T;L2([a(t),b(t)]))∩L2(0,T;H1([a(t),b(t)])),a(t),b(t)∈H2([0,T]),(ργ-ραux)∈C0([0,T]×([a(t),b(t)])).
The domain expands outwards in time as
(8)pe(t)-1≥b(t)-a(t)≥{c(1+t)λ/(γ-1),0<α<1,c(1+t)1/(γ-1),α=1,
where 0<λ≤γ-1 denotes a positive constant, and the density decays pointwise to zero for any x∈[a(t),b(t)] and t>0 as
(9)ρ(x,t)≤{c(1+t)-ηλ/(γ-1),0<α<1,c(1+t)-η/(γ-1),α=1,
where η>0 is a positive constant.Remark 2.
Theorem1 holds for one-dimensional Saint-Venant model for shallow water; that is, γ=2, α=1.Remark 3.
Equation (8) implies that as time goes to infinity, both the lower bound rate and the upper bound rate go to infinity.Remark 4.
In fact, we can choose the nonconstant exterior pressurepe(t) like these
(10)pe(t)=Ctae-bt,a∈R+,b∈R+,pe(t)=C(1+t)a(1+ebt)-1,a∈R,b∈R+,
or the linear combinations of these functions, and so forth.Remark 5.
In particular, letpe(t)=e-t; from (8), we have
(11)b(t)-a(t)≤et,
which implies that the upper bound rate of (b(t)-a(t)) expands at an exponential rate; however, we proved that the upper bound rate of (b(t)-a(t)) expands at an algebraic rate in [16] where we consider the free boundary value problem without the nonconstant exterior pressure.


## 3. The A Priori Estimates

Making use of the Lagrange coordinates, we can establish some a priori estimates. Define the Lagrange coordinates transform(12)ξ=∫a(t)x‍ρ(y,t)dy,τ=t.
Since the conservation of total mass holds, the boundaries x=a(t) and x=b(t) are transformed into ξ=0 and ξ=1, respectively, and the domain [a(t),b(t)] is transformed into [0,1]. The FBVP (1) and (2) is reformulated into
(13)ρτ+ρ2uξ=0,uτ+(ργ)ξ=(ρ1+αuξ)ξ,(ρ0,u0)=(ρ0,u0)(ξ),ξ∈[0,1],(ργ-ρ1+αuξ)(0,τ)=pe(τ),(ργ-ρ1+αuξ)(1,τ)=pe(τ),aaaaaaaaaaaaaaiτ∈[0,T],
where the initial data satisfies
(14)inf⁡[0,1]ρ0≥ρ_>0,(ρ0,u0)∈W1,∞([0,1]),(ρ0γ-ρ01+αu0x)(0)=pe0,(ρ0γ-ρ01+αu0x)(1)=pe0,ρ0γ(1)≥ρ0γ(0)≥pe0,
and the consistencies between initial data and boundary conditions hold.Next, we will deduce the a priori estimates for the solution(ρ,u) to the FBVP (13).Lemma 6.
LetT>0. Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(15)ργ(1,τ)≥ργ(0,τ)≥pe(τ).Proof.
From (13), we can find
(16)1α(ρα(0,τ)-peα/γ(τ))τ+ργ(0,τ)-pe(τ)+1γpeα/γ-1(τ)p′(τ)=0,1α(ρα(1,τ)-peα/γ(τ))τ+ργ(1,τ)-pe(τ)+1γpeα/γ-1(τ)p′(τ)=0,1α(ρα(1,τ)-ρα(0,τ))τ+ργ(1,τ)-ργ(0,τ)=0,
which imply that
(17)ρα(0,τ)-peα/γ(τ)=(ρ0α(0)-pe0α/γ)×exp⁡{-α∫0τ‍ργ(0,s)-pe(s)+(1/γ)peα/γ-1(s)p′(s)ρα(0,s)-peα/γ(s)ds}≥0,(18)ρα(1,τ)-peα/γ(τ)=(ρ0α(1)-pe0α/γ)×exp⁡{-α∫0τ‍ργ(1,s)-pe(s)+(1/γ)peα/γ-1(s)p′(s)ρα(1,s)-peα/γ(s)ds}≥0,(19)ρα(1,τ)-ρα(0,τ)=(ρ0α(1)-ρ0α(0))×exp⁡{-α∫0τ‍ργ(1,s)-ργ(0,s)ρα(1,s)-ρα(0,s)ds}≥0.Lemma 7.
LetT>0. Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(20)∫01‍(12u2+1γ-1ργ-1)dξ+∫0τ‍∫01‍ρ1+αuξ2dξds+∫0τ‍pe(s)(b(s)-a(s))′ds=∫01‍(12u02+1γ-1ρ0γ-1)dξ,τ∈[0,T],
where a(τ) satisfies a′(τ)=u(0,τ) and a(0)=a0 and b(τ) satisfies b′(τ)=u(1,τ) and b(0)=b0.Proof.
Taking the product of (13)2 with u, integrating on [0,1], and using boundary conditions, we have
(21)12ddτ∫01‍u2dξ+∫01‍ρ1+αuξ2dξ=∫01‍uξ(ργ-pe(τ))dξ=-∫01‍ργ-2ρτdξ-pe(τ)(u(1,τ)-u(0,τ))=-1γ-1ddτ∫01‍ργ-1dξ-pe(τ)(b(τ)-a(τ))′,
which leads to (20) after the integration with respect to τ∈[0,T], where we use the fact that, from (15), the domain (b(τ)-a(τ)) expands as the time grows up, and it holds that
(22)(b(τ)-a(τ))′=b′(τ)-a′(τ)=u(1,τ)-u(0,τ)≥0.Lemma 8.
LetT>0. Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(23)∫01‍(12(u+1α(ρα)ξ)2+1γ-1ργ-1)dξ+γ∫0τ‍∫01‍ργ+2α-1ρξ2dξds+ργ(1,τ)b(τ)-ργ(0,τ)a(τ)+γα∫0τ‍ργ-α(1,s)(ργ(1,s)-pe(s))b(s)ds-γα∫0τ‍ργ-α(0,s)(ργ(0,s)-pe(s))a(s)ds=∫01‍(12(u0+1α(ρ0α)ξ)2+1γ-1ρ0γ-1)dξ+ρ0γ(1)b0-ρ0γ(0)a0,τ∈[0,T].Proof.
Multiplying (13)1 by ρα-1 gives
(24)(ρα)τα+ρ1+αuξ=0,
which leads to
(25)(ρα)τξα+(ρ1+αuξ)ξ=0.
Summing (13)1 and (25), we deduce
(26)(u+(ρα)ξα)τ+(ργ)ξ=0.
Multiplying (26) by (u+(ρα)ξ/α) and integrating the result over [0,1]×[0,τ], we obtain
(27)∫01‍(12(u+1α(ρα)ξ)2+1γ-1ργ-1)dξ+γ∫0τ‍∫01‍ργ+2α-1ρξ2dξds+∫0τ‍(ργ(1,s)u(1,s)-ργ(0,s)u(0,s))ds=0,
where we have the fact that
(28)∫0τ‍(ργ(1,s)u(1,s)-ργ(0,s)u(0,s))ds=∫0τ‍ργ(1,s)b′(s)ds-∫0τ‍ργ(0,s)a′(s)ds=ργ(1,τ)b(τ)-ρ0γ(1)b0-γα∫0τ‍b(s)ργ-α(1,s)(ρα(1,s))sds-ργ(0,τ)a(τ)+ρ0γ(0)a0+γα∫0τ‍a(s)ργ-α(0,s)(ρα(0,s))sds=ργ(1,τ)b(τ)-ργ(0,τ)a(τ)-ρ0γ(1)b0+ρ0γ(0)a0+γα∫0τ‍ργ-α(1,s)(ργ(1,s)-pe(s))b(s)ds-γα∫0τ‍ργ-α(0,s)(ργ(0,s)-pe(s))a(s)ds,
which together with (15) and
(29)b(τ)-a(τ)=∫01‍1ρ(ζ,τ)dζ>0
gives rise to (23).Lemma 9.
LetT>0. Under the assumptions of Theorem 1, it holds that
(30)ρ(ξ,τ)≤C,(ξ,τ)∈[0,1]×[0,T],T>0,
where C is the positive constant independent of time.Proof.
Integrating (24) with respect to τ over [0,τ], we know
(31)ρα(ξ,τ)=ρ0α(ξ)-α∫0τ‍(ρ1+α)uξ(ξ,s)ds;
then integrating (13)2 over [0,ξ]×[0,τ] and using the boundary conditions, we have
(32)∫0ξ‍u(ζ,τ)dζ-∫0ξ‍u0(ζ)dζ+∫0τ‍ργ(s)ds-∫0τ‍pe(s)ds=∫0τ‍(ρ1+α)uξ(ξ,s)ds.
It holds from (31) and (32) that
(33)ρα(ξ,τ)+α∫0τ‍ργ(ξ,s)ds=ρ0α(ξ)-α∫0ξ‍u(ζ,τ)dζ+α∫0ξ‍u0(ζ)dζ+α∫0τ‍pe(s)ds≤C+C(∫01‍u2dξ)1/2+α∫0τ‍pe(s)ds≤C,
where C denotes the positive constant independent of time.Lemma 10.
LetT>0. Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(34)∫01‍u2ndξ+n(2n-1)∫0τ‍∫01‍ρ1+αu2n-2uξ2dξds+∫0τ‍pe(s)((b′(s))2n-1-(a′(s))2n-1)ds≤C(T),τ∈[0,T],
for any positive integer n∈N, and C(T)>0 denotes a constant dependent on time.Proof.
Multiplying (13)2 by 2nu2n-1 and integrating the result over [0,1]×[0,τ], we obtain
(35)∫01‍u2ndξ+2n(2n-1)∫0τ‍∫01‍ρ1+αu2n-2uξ2dξds+2n∫0τ‍p(s)(u2n-1(1,s)-u2n-1(0,s))ds=∫01‍u02ndξ+2n(2n-1)∫0τ‍∫01‍ργu2n-2uξ2dξds;
then it holds from Young’s inequality and (22) that
(36)∫01‍u2ndξ+n(2n-1)∫0τ‍∫01‍ρ1+αu2n-2uξ2dξds+2n∫0τ‍p(s)((b′(s))2n-1-(a′(s))2n-1)ds=∫01‍u02ndξ+C∫0τ‍∫01‍u2ndξds,
which together with Gronwall’s inequality gives (34).Lemma 11.
LetT>0, for n∈N, and n>(1+α)/4(γ-α). Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(37)∫0τ‍∥(ργ)ξ2n∥L∞([0,1])ds≤C(T),τ∈[0,T].Proof.
Integrating (26) with respect to τ over [0,τ], we have
(38)(ρα)ξ=(ρ0α)ξ+αu0(ξ)-αu(ξ,τ)-α∫0τ‍(ργ)ξ(ξ,s)ds,
which together with (20) gives
(39)∫0τ‍∥(ργ)ξ2n∥L∞([0,1])ds=γ2nα2n∫0τ‍∥ρ2n(γ-α)(ρα)ξ2n∥L∞([0,1])ds≤C(T)+C(T)∫0τ‍∥ρ2n(γ-α)u2n∥L∞([0,1])ds+C(T)∫0τ‍∫0s‍∥(ργ)ξ2n∥L∞([0,1])dlds≤C(T)+C(T)∫0τ‍∫0s‍∥(ργ)ξ2n∥L∞([0,1])dlds,
where we have used
(40)∫0τ‍∥ρ2n(γ-α)u2n∥L∞([0,1])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+u4n+ρ4n(γ-α)-(1+α)u2n+ρ1+αu2n-2uξ2)dξds≤C(T),
which can be deduced from (20), (23), (30), and (34). Making use of Gronwall’s inequality to (40), we obtain (37).Lemma 12.
LetT>0. Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(41)ρ(ξ,τ)≥C(T),(ξ,τ)∈[0,1]×[0,T].Proof.
Define that(42)v(ξ,τ)=1ρ(ξ,τ).
By (13)1, we have
(43)vτ=uξ.
Multiplying (43) by βvβ-1, where β=2(1+α), integrating the result over [0,1]×[0,τ], and using (37) and (38), we can find that
(44)∫01‍vβdξ+β(β-1)∫0τ‍∫01‍vα+β-1u2dξds=∫01‍v0βdξ+β∫0τ‍vβ-1uds|ξ=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)+β(∫0τ‍vβ-1(1,s)u(1,s)ds-∫0τ‍vβ-1(0,s)u(0,s)ds)+β(β-1)2∫0τ‍∫01‍vα+β-1u2dξds+C(T)∫0τ‍∫01‍vβdξds.
Since it holds that
(45)ρα(0,τ)=peα/γ(τ)+(ρ0α(0)-pe0α/γ)×exp⁡{-α∫0τ‍ργ(0,τ)-pe(τ)+(1/γ)peα/γ-1p′(τ)ρα(0,τ)-peα/γ(τ)}≥peα/γ(τ),
which implies
(46)ρ(0,τ)≥pe1/γ(τ)≥inf⁡τ∈[0,T]pe1/γ(τ)≔C(T),
we have from (20) and (23) that
(47)∫0τ‍vβ-1u(0,s)dξ≤C(T)∫0τ‍((∫01‍u2dξ)1/2+(∫01‍ρ1+αuξ2dξ)1/2+(∫01‍(ρα)ξ2dξ)1/2(∫01‍u2dξ)1/2)ds≤C(T).
Using the same method, we also have
(48)∫0τ‍vβ-1u(1,s)dξ≤C(T).
Substituting (47) and (48) into (44), we get
(49)∫01‍vβdξ+β(β-1)2∫0τ‍∫01‍vα+β-1u2dξds≤C(T)+C(T)∫0τ‍∫01‍vβdξds,
which together with Gronwall’s inequality yields
(50)∫01‍v2(1+α)dξ≤C(T).
We have from (23), (46), and (50) that
(51)v(ξ,τ)≤v(0,τ)+∫01‍|vξ|dξ≤C(T)+C(∫01‍v2(1+α)dξ)1/2(∫01‍|(ρα)ξ|2dξ)1/2≤C(T),
which implies
(52)ρ(ξ,τ)≥C(T).We also have the regularity estimates for the solution(ρ,u) to the FBVP (13) as follows.Lemma 13.
LetT>0. Under the assumptions of Theorem 1, it holds for any strong solution (ρ,u) to the FBVP (13) that
(53)ρ∈L∞(0,T;H1([0,1]))∩C0([0,T]×[0,1]),u∈L∞(0,T;H1([0,1]))∩L2(0,T;H2([0,1])),a(τ),b(τ)∈H1([0,T]),(ργ-ρ1+αuξ)∈L∞(0,T;L2([0,1])).
If it is also satisfied that(54)u0∈H2([0,1]),
then the strong solution (ρ,u) has the regularities
(55)(ρ,u)∈C0([0,T]×[0,1]),ρ∈L∞(0,T;H1([0,1])),ρτ∈L∞(0,T;L2([0,1])),u∈L∞(0,T;H2([0,1]))∩L2(0,T;H3([0,1])),uτ∈L∞(0,T;L2([0,1]))∩L2(0,T;H1([0,1])),a(τ),b(τ)∈H2([0,T]),(ργ-ρ1+αuξ)∈C0([0,T]×([0,1])).Proof.
Multiplying (13)2 by ρ-(1+α)uτ, integrating the result over [0,1], and making use of the boundary conditions, after a direct computation and recombination, we find
(56)ddτ∫01‍(12uξ2-(ργ-pe(τ))ρ-(1+α)uξ)dξ+∫01‍ρ-(1+α)uτ2dξ=[γ-(1+α)]∫01‍ργ-αuξ2dξ+(1+α)∫01‍pe(τ)ρ-αuξ2dξ+∫01‍pe′(τ)ρ-(1+α)uξdξ-(1+α)∫01‍(ργ-pe(τ))ρ-(2+α)ρξuτdξ+(1+α)∫01‍ρ-1ρξuξuτdξ.
Integrating (56) over [0,τ], from (20), (23), (30), (41), pe(τ)∈L1([0,T]), and pe′(τ)∈L2([0,T]), it is easily verified that
(57)∫01‍uξ2dξ+∫0τ‍∫01‍us2dξds≤C(T)+C(T)∫0τ‍∫01‍uξ2dξds+C(T)∫0τ‍∫01‍ρξ2dξds+C(T)∫0τ‍∫01‍ρξ2uξ2dξds≤C(T)+C(T)∫0τ‍∫01‍ρξ2uξ2dξds,
where C(T) denotes a constant dependent on time. From (13)2, (20), (23), (30), and (41), it holds that
(58)C(T)∫0τ‍∫01‍ρξ2uξ2dξds≤C(T)2∫0τ‍∫01‍ρξ2uξ2dξds+14∫0τ‍∫01‍us2dξds+C(T)∫0τ‍∫01‍ρξ2dξds+C(T)∫0τ‍∫01‍uξ2dξds.
Using (58), we can obtain that
(59)∫01‍uξ2dξ+∫0τ‍∫01‍us2dξds≤C(T),
which together with (13)2 implies
(60)∫01‍uξ2dξ+∫0τ‍∫01‍us2dξds+∫0τ‍∫01‍uξξ2dξds≤C(T).
Differentiating (13)2 with respect to τ, we get
(61)uττ+(ργ-pe(τ))ξτ=(ρ1+αuξ)ξτ.
Taking product between (61) and uτ, integrating the results over [0,1], and using the boundary conditions (13)4,5, we have
(62)12ddτ∫01‍uτ2dξ=∫01‍(ργ-pe(τ))τuξτdξ-∫01‍(ρ1+αuξ)τuξτdξ.
The terms on the right-hand side of (62) can be bounded, respectively, as described below:
(63)∫01‍(ργ-pe(τ))τuξτdξ=-∫01‍γργ+1uξuξτdξ-∫01‍pe′(τ)uξτdξ≤-γ2ddτ∫01‍ργ+1uξ2dξ+C∫01‍(ρ1+αuξ2+ρ2γ-α+3uξ4)dξ-∫01‍pe′(τ)uξτdξ,-∫01‍(ρ1+αuξ)τuξτdξ=-∫01‍((1+α)ραρτuξ+ρ1+αuξτ)uξτdξ≤-12∫01‍ρ1+αuξτ2dξ+C∫01‍ρ3+αuξ4dξ.
Summing (62)-(63) together and making use of (30) and (60), we have
(64)12ddτ∫01‍uτ2dξ+γ2ddτ∫01‍ργ+1uξ2dξ+12∫01‍ρ1+αuξτ2dξ≤C(T)∫01‍uξ2dξ+C(T)∥uξ∥L∞([0,1])2∫01‍uξ2dξ+14∫01‍ρ1+αuξτ2dξ+C(T)∫0τ‍(pe′(s))2ds≤C(T)+C(T)∥uξ∥L∞([0,1])2∫01‍uξ2dξ+14∫01‍ρ1+αuξτ2dξ.
Integrating (64) over [0,τ], it holds from (13)2, (20), and (60) that
(65)∫01‍uτ2dξ+∫01‍uξ2dξ+∫0τ‍∫01‍uξs2dξds≤C(T)+C(T)sup⁡τ∈[0,T]∥uξ∥L∞2≤C(T)+C(T)sup⁡τ∈[0,T](∫01‍uξ2dξ)1/2(∫01‍uξξ2dξ)1/2≤C(T)+12∫01‍uτ2dξ,
which gives
(66)∫01‍uτ2dξ+∫01‍uξ2dξ+∫0τ‍∫01‍uξτ2dξds≤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) and b′(τ)=u(1,τ) that a(τ), b(τ)∈H2([0,T]). The proof of this lemma is completed.Finally, we will give the large time behavior of the strong solution as follows.Lemma 14.
LetT>0. Under the assumptions of Theorem 1, it holds for α∈(0,1] and time t large enough that
(67)pe(t)-1≥b(t)-a(t)≥{c(1+t)λ/(γ-1),0<α<1,c(1+t)1/(γ-1),α=1,
where 0<λ≤γ-1 denotes a positive constant, and the density decays pointwise to zero for any x∈[a(t),b(t)] and t>0 as
(68)ρ(x,t)≤{c(1+t)-ηλ/(γ-1),0<α<1,c(1+t)-η/(γ-1),α=1,
where η>0 is a positive constant.Proof.
From (13) we can find that
(69)ddτ∫01‍u(ξ,τ)dξ=0,
and, without loss of generality, we can renormalize ∫01‍u0(ξ)dξ to be zero; then, we denote
(70)w=u-11+τ∫0ξ‍1ρdζ+11+τ∫01‍∫0ξ‍1ρdζdξ.
Applying (69), we can obtain
(71)wξ=uξ-1(1+τ)ρ=(1ρ)τ-1(1+τ)ρ,(72)wτ+w1+τ=uτ;
then the system (13) becomes
(73)ρτ+ρ2wξ+ρ1+τ=0,wτ+w1+τ+(ργ)ξ=(ρ1+αwξ+ρα1+τ)ξ,(ρ0,w0)=(ρ0,u0-11+τ∫0ξ‍1ρ0dζ+11+τ∫01‍∫0ξ‍1ρ0dζdξ)(ξ),ξ∈[0,1],(ργ-ρ1+α(wξ+1(1+τ)ρ))(0,τ)=pe(τ),(ργ-ρ1+α(wξ+1(1+τ)ρ))(1,τ)=pe(τ),τ∈[0,T].
Multiplying (73)2 by w and integrating the result over [0,1], after a straightforward calculation, we have
(74)12ddτ∫01‍w2dξ+11+τ∫01‍w2dξ=∫01‍(ρ1+αwξ)ξwdξ+11+τ∫01‍(ρα)ξwdξ-∫01‍(ργ-pe(τ))ξwdξ=-∫01‍ρ1+αwξ2dξ-11+τ∫01‍ραwξdξ+∫01‍(ργ-pe(τ))wξdξ,
which implies
(75)12ddτ∫01‍w2dξ+11+τ∫01‍w2dξ+∫01‍ρ1+αwξ2dξ=-11+τ∫01‍ραwξdξ+∫01‍ργwξdξ-∫01‍pe(τ)wξdξ,
and as 0<α<1 it holds from (71) that
(76)-11+τ∫01‍ραwξdξ=-11+τ∫01‍ρα{(1ρ)τ-1(1+τ)ρ}dξ=1(α-1)(1+τ)∫01‍(ρα-1)τdξ+1(1+τ)2∫01‍ρα-1dξ,∫01‍ργwξdξ=∫01‍ργ{(1ρ)τ-1(1+τ)ρ}dξ=11-γ∫01‍(ργ-1)τdξ-11+τ∫01‍ργ-1dξ,-∫01‍pe(τ)wξdξ=-∫01‍pe(τ){(1ρ)τ-1(1+τ)ρ}dξ=-∫01‍(pe(τ)ρ)τdξ+∫01‍pe′(τ)ρdξ+∫01‍pe(τ)(1+τ)ρdξ,
which together with (75) leads to
(77)ddτ∫01‍12w2dξ+1γ-1ddτ∫01‍ργ-1dξ+ddτ∫01‍pe(τ)ρdξ+11+τ∫01‍w2dξ+∫01‍ρ1+αwξ2dξ+11+τ∫01‍ργ-1dξ=1(α-1)(1+τ)∫01‍(ρα-1)τdξ+1(1+τ)2∫01‍ρα-1dξ+∫01‍pe′(τ)ρdξ+∫01‍pe(τ)(1+τ)ρdξ;
multiplying (77) by (1+t)λ for some 0<λ<1 to be determined later, we have
(78)ddτ∫01‍((1+τ)λ2w2+(1+τ)λγ-1ργ-1+(1+τ)λ-11-αρα-1+(1+τ)λpe(τ)ρ)dξ+(1-λ2)(1+τ)λ-1∫01‍w2dξ+(1+τ)λ∫01‍ρ1+αwξ2dξ+γ-1-λγ-1(1+τ)γ-1∫01‍ργ-1dξ+α-λ1-α(1+τ)λ-2∫01‍ρα-1dξ+∫01‍(1+τ)λ-1ρ(-(1+τ)pe′(τ)-(1+λ)pe(τ))dξ=0,
and we will prove the fact that as time τ is large enough, it holds that
(79)-(1+τ)pe′(τ)≥(1+λ)pe(τ),
which needs
(80)ddτ(ln⁡pe-1(τ))≥1+λ1+τ>0;
that is,
(81)pe(τ)≤C(1+τ)1+λ,
where C>0 is a positive constant independent of time and we can find that (81) is true as we assume pe(τ)=o((1+τ)ν), where ν is some positive constant. Then, as 0<λ≤min⁡{γ-1,α}, integrating (78) over [0,τ], we have
(82)∫01‍ργ-1dξ≤(1+τ)-λ.
Asα=1 and 0<λ≤γ-1, it holds from (75) that
(83)ddτ∫01‍((1+τ)λ2w2+(1+τ)λγ-1ργ-1+(1+τ)λpe(τ)ρ)dξ+(1-λ2)(1+τ)λ-1∫01‍w2dξ+(1+τ)λ∫01‍ρ1+αwξ2dξ+γ-1-λγ-1(1+τ)γ-1∫01‍ργ-1dξ+∫01‍(1+τ)λ-1ρ(-(1+τ)pe′(τ)-(1+λ)pe(τ))dξ=ddτ((1+τ)λ-1∫01‍ln⁡ρdξ)+(1-λ)(1+τ)λ-2∫01‍ln⁡ρdξ+(1+τ)λ-2;
integrating (83) over [0,τ] and using
(84)∫01‍ln⁡ρdξ≤∫01‍ρdξ≤C,
we have
(85)∫01‍ργ-1dξ≤(1+τ)-1.
From (82) and (85), we know
(86)∫a(t)b(t)‍ργdx≤{c(1+t)-λ,0<α<1,c(1+t)-1,α=1,
and it holds from the conservation of the mass that
(87)1=∫a0b0‍ρ0dx=∫a(t)b(t)‍ρdx≤(∫a(t)b(t)‍ργdx)1/γ(b(t)-a(t))(γ-1)/γ,
which gives
(88)b(t)-a(t)≥{c(1+t)1/(γ-1),α=1,c(1+t)λ/(γ-1),0<α<1.
Using (15), (17), and (23), we know
(89)b(t)-a(t)≤Cρ-γ(a(t),t)≤Cpe-1(t),
and we can choose the positive constant ν such that as time is large enough, it holds that
(90)pe(t)=(o(1+t)ν)≤{c(1+t)-1/(γ-1),α=1,c(1+t)-λ/(γ-1),0<α<1.
Finally, we will prove the decay rate of the density, and it holds from the Gagliardo-Nirenberg-Sobolev inequality, (23), and (30) that
(91)ρ(γ+2α-1)/2(x,t)=ρ(γ+2α-1)/2(ξ,τ)≤C∥ρ(γ+2α-1)/2(ξ,τ)∥L1([0,1])1/2∥(ρ(γ+2α-1)/2)ξ(ξ,τ)∥L1([0,1])1/2≤C(∫01‍ργ-1dξ)1/4(∫01‍(ρα)ξ2dξ)1/4≤C(∫a(t)b(t)‍ργdx)1/4,
which together with (86) gives (68).Remark 15.
We note that Lemmas7–14 are proved by means of the methods used in [9, 13, 16, 24], and Lemma 6 is new.


## 4. Proof of the Main Results

Proof.
The global existence of unique strong solution to the FBVP (1) and (2) can be established in terms of the short time existence carried out as in [15], the uniform a priori estimates, and the analysis of regularities, which indeed follow from Lemmas 6–13. We omit the details. The large time behavior follows from Lemma 14 directly. The proof of Theorem 1 is completed.



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*Source: 961014-2014-07-15.xml*