Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity

(1)

Volume 2012, Article ID 395209,22pages doi:10.1155/2012/395209

Research Article

Global Solutions to the Spherically

Symmetric Compressible Navier-Stokes Equations with Density-Dependent Viscosity

Ruxu Lian,

¹

Lan Huang,

¹

and Jian Liu

²

1College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

2Department of Mathematics, Capital Normal University, Beijing 100048, China

Correspondence should be addressed to Ruxu Lian,[email protected] Received 20 December 2011; Revised 21 March 2012; Accepted 31 March 2012 Academic Editor: Nazim I. Mahmudov

Copyrightq2012 Ruxu Lian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coeﬃcient in this paper. For regular initial density, we show that there exists a unique global strong solution to the exterior problem or the initial boundary value problem, respectively. In particular, the strong solution tends to the equilibrium state ast → ∞.

1. Introduction

The isentropic compressible Navier-Stokes equations with density-dependent viscosity coeﬃcients read as follows:

ρtdiv ρU

0, ρU

tdiv

ρU⊗U ∇P

ρ

−div μ

ρ

DU

− ∇ λ

ρ divU

0, 1.1

wheret∈0,∞is the time andx∈R^N,Nis the spatial coordinate,ρ >0 andudenote the density and velocity, respectively. Pressure function is taken asPρ ρ^γwithγ >1, and

DU ∇U^t∇U

2 1.2

(2)

is the strain tensor andμρ,λρare the Lam´e viscosity coeﬃcients satisfying μ

ρ

>0, μ ρ

Nλ ρ

≥0. 1.3

There are many important progress achieved recently on the compressible Navier- Stokes equations with density-dependent viscosity coeﬃcient. For instance, the mathematical derivations are derived in the simulation of flow surface in shallow region 1–4. The prototype model is the viscous Saint-Venant. The well posedness of solutions to the free boundary value problem with initial finite mass and the flow density being connected with the infinite vacuum either continuously or via jump discontinuity is investigated by many authors, refer to5–12and references therein. Mellet and Vasseur showed the global existence of strong solutions forα∈0,1/2 13. The qualitative behaviors of global solutions and dynamical asymptotics of vacuum states are also made, such as the finite time vanishing of finite vacuum or asymptotical formation of vacuum in large time, the dynamical behaviors of vacuum boundary, the large time convergence to rarefaction wave with vacuum, and the stability of shock profile with large shock strength, refer to14–17and references therein.

In this present paper, we consider the exterior problem and the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coeﬃcient and focus on the regularities and dynamical behaviors of global strong solution, and so forth. Asγ >1, we show that the exterior problem and the initial boundary value problem with regular initial data both admit the unique global strong solution. In particular, the strong solution tends to the equilibrium state ast → ∞.

The rest of the paper is arranged as follows. InSection 2, the main results about the dynamical behaviors of global strong solution for compressible Navier-Stokes equations are stated. Then, the theorems of the exterior problem and the initial boundary value problem are proved in Sections3and4, respectively.

2. Notations and Main Results

For simplicity, we will takeμρ ρandλρ 0 andDU ∇U in 1.1. The isentropic compressible Navier-Stokes equations become

ρtdiv ρU

0, ρU

tdiv

ρU⊗U ∇P

ρ

−div ρ∇U

0. 2.1

Firstly, we consider the exterior problem, namely, the initial data and boundary conditions of 2.1are imposed as follows:

ρ,U

x,0 ρ0,U0

x, x∈Ω, U0, on ∂Ω, lim

|x| →∞

ρ, u

x, t ρ,0

, t∈0, T, 2.2

where Ω : R³/Ωr₋,Ωr₋ is a ball of radiusr− centered at the origin in R³, and ρ > 0 is a constant.

(3)

We are concerned with the spherically symmetric solutions of system 2.1 in an spherically symmetric exterior domainΩ. To this end, we denote that

|x|r, ρx, t ρr, t, Ux, t ur, tx

r, 2.3

which leads to the following system of equations forr >0, ρt

ρu

r2ρu r 0, ρu

t

ρu²ρ^γ

r2ρu² r −

ρur

r−ρ 2u

r

0,

2.4

with the initial data and boundary conditions ρ, u

r,0 ρ0, u0

r, r ∈r₋,∞, ur−, t 0, lim

r→∞

ρ, u

r, t ρ,0

t∈0, T, 2.5

and the initial data satisfies for some constantsρandρ >0 ρ0−ρ∈L¹r−,∞∩L²r−,∞, inf

x∈r−,∞ρ0> ρ >0,

rρ0r ∈L²r−,∞, r²u0∈H²r−,∞. 2.6 Then, we define that

E0 : 1 2

_∞

r₋

ρ0u²₀r²dr _∞

r₋

ρ0

1 γ−1

ρ₀^γ−1−ρ^γ−1 ρ^γ

ρ₀⁻¹−ρ⁻¹ r²dr, E1 : 1

2 _∞

r−

ρ0

u0ρ⁻¹₀ ρ0

r

₂ r²dr

_∞

r−

ρ0

1 γ−1

ρ^γ−1₀ −ρ^γ−1 ρ^γ

ρ⁻¹₀ −ρ⁻¹ r²dr,

2.7

and give the main results as follows.

Theorem 2.1. Letγ >1. Assume that the initial data satisfies2.6andE0E1 < νr²₋ρ^γ/21/2,ν is a positive constant. Then, there exist two positive constantsρ∗ andρ^∗and a unique global strong solutionρ, uto the exterior problem2.4and2.5, namely, satisfying

0< ρ∗≤ρr, t≤ρ^∗, r, t∈r−,∞×0, T, ρ−ρ∈L^∞

0, T;L²r₋,∞

, ρr ∈L^∞

0, T;L²r₋,∞ , u∈L^∞

0, T;L²r−,∞

, ur ∈L^∞

0, T;L²r−,∞ .

2.8

Furthermore, the solution tends to the equilibrium stateρ,0

(4)

ρ−ρ, u

·, t

L^∞r−,∞−→0, t−→∞. 2.9

Then investigates the initial boundary value problem, and the initial data and boundary conditions of2.1are assumed as follow:

ρ,U

x,0 ρ0,U0

x, x∈Ω,

Ux, t 0, x∈∂Ω, t∈0, T. 2.10

By2.3, one considers2.4with the initial data and boundary conditions ρ, u

r,0 ρ0, u0

r, r ∈r−, r,

ur−, t ur, t 0, t∈0, T, 2.11

and the initial data satisfies for some constantρ >0

ρ0∈L¹r−, r∩W^1,∞r−, r, inf

x∈r−,rρ0> ρ >0,

r²u0∈H²r₋, r. 2.12

Then, can give the main results as follows.

Theorem 2.2. Let γ > 1. Assume that the initial data satisfies 2.12, there exist two positive constantsρ∗andρ^∗and a unique global strong solutionρ, uto the initial boundary value problem 2.4and2.11, namely, satisfying

0< ρ∗≤ρr, t≤ρ^∗, r, t∈r₋, r×0, T, ρ∈L^∞

0, T;L²r−, r

, ρr ∈L^∞

0, T;L²r−, r , u∈L^∞

0, T;L²r₋, r

, ur ∈L^∞

0, T;L²r₋, r .

2.13

Furthermore, the solutionρ, utends to the equilibrium state exponentially ρ−ρ, u

·, t

L^∞r−,r≤C0e^−C¹^t, 2.14

whereC0andC1are positive constants independent of time andρ_r

r−ρr, tr²dr.

(5)

Remark 2.3. Theorems2.1and2.2hold for one-dimensional Saint-Venant’s model for shallow water, that is,γ2,α1.

3. Proof of the Exterior Problem

3.1. The A Priori Estimates

It is convenient to make use of the Lagrangian coordinates so as to establish the uniformly a priori estimates. Take the Lagrange coordinates transform

x _r

r₋

ρr, tr²dr, τ t, 3.1

which mapr, t∈r₋,∞×Rintox, τ∈0,∞×R. The relation between Lagrangian and Eulerian coordinates are satisfied as

∂x

∂r ρr², ∂x

∂t −ρur². 3.2

The exterior problem2.4and2.5is reformulated to

ρτρ² r²u

x 0, r⁻²uτ

ρ^γ

x

ρ² r²u

x

x−2ρxu r , ρ, u

x,0 ρ0, u0

x, x∈0,∞, u0, t 0, lim

x→∞

ρ, u

ρ,0

, τ∈0,∞,

3.3

where the initial data satisfies

ρ0−ρ∈L¹0,∞∩L²0,∞, inf

x∈0,∞ρ0> ρ−>0, r²ρ0x∈L²0,∞, 1

r²ρ0

r²u0

∈L²0,∞,

r²ρ0

r²u0

x∈L²0,∞, 1

r²ρ0

r²ρ

r²ρ r²u0

x

∈L²0,∞.

3.4

First, we will establish the a-priori estimates for the solutionρ, uto the exterior problem 3.3.

(6)

Lemma 3.1. LetT >0. Under the conditions inTheorem 2.1, it holds for any solutionρ, uto the exterior problem3.3that

1 2

_∞

0

u²dx _∞

0

1 γ−1

ρ^γ−1−ρ^γ−1 ρ^γ

ρ⁻¹−ρ⁻¹ dx

_τ

0

_∞

0

2u²

r² ρ²u²_xr⁴

dx ds≤E0, τ ∈0, T.

3.5

Proof. Multiplying 3.32 byr²u and integrating the result with respect to x over0,∞, making use of3.31and3.4, we have

1 2

d dτ

_∞

0

u²dx d dτ

_∞

0

1 γ−1

ρ^γ−1−ρ^γ−1 ρ^γ

_∞

0

ρ² r²u₂

xdx 2

_∞

0

ρ ru²

xdx,

3.6

integrating3.6with respect toτ, we obtain 1

2 _∞

0

u²dx _∞

0

1 γ−1

ρ^γ−1−ρ^γ−1 ρ^γ

_τ

0

_∞

0

2u²

r² ρ²u²_xr⁴

dx ds 1

2 _∞

0

u²₀dx _∞

0

1 γ−1

ρ^γ−1₀ −ρ^γ−1 ρ^γ

ρ⁻¹₀ −ρ⁻¹ dx.

3.7

Lemma 3.1can be obtained.

1 2

_∞

0

ur²ρx

2

dx _∞

0

1 γ−1

ρ^γ−1−ρ^γ−1 ρ^γ

ρ⁻¹−ρ⁻¹ dx γ

_τ

0

_∞

0

ρ^γ−1ρ²_xr⁴dx ds≤E1, τ ∈0, T,

3.8

and there exist two constants 0< ρ_∗< ρ^∗such that

0< ρ_∗≤ρx, τ≤ρ^∗, x, τ∈0,∞×0, T. 3.9 Proof. Diﬀerentiating3.31with respect tox, we have

ρxτ ρ²

r²u

x

x0. 3.10

(7)

Summing3.10and3.32, we get

r⁻²uρx

τ

ρ^γ

x

r⁻²

τu−2ρxu

r . 3.11

Note that

r³x, τ r₋³3 _x

0

1

ρz, τdz, 3.12

and so

∂r

∂τ 1 r²

_x

0

1 ρ

t

z, tdz 1 r²

_x

0

r²u

zz, tdzux, τ, 3.13 which together with3.11yields

r⁻²uρx

τ

ρ^γ

x−2r⁻³u²−2ρxu

r . 3.14

Multiplying3.14byur²ρxr²and integrating the result with respect toxandτ, we have 1

2 _∞

0

ur²ρx

₂ dx

_∞

0

1 γ−1

ρ^γ−1−ρ^γ−1 ρ^γ

ρ⁻¹−ρ⁻¹ dx γ

_τ

0

_∞

0

ρ^γ−1ρ²_xr⁴dx ds 1

2 _∞

0

u0r²ρ0x

₂ dx

_∞

0

1 γ−1

ρ^γ−1₀ −ρ^γ−1 ρ^γ

ρ₀⁻¹−ρ⁻¹ dx.

3.15

Let

ϕ ρ

: 1

γ−1

ρ^γ−1−ρ^γ−1 ρ^γ

ρ⁻¹−ρ⁻¹ , ψ

ρ :

_ρ

ρ

ϕ η_1/2

dη.

3.16

It follows from3.6and3.13that ψ

ρ≤

_∞

y

∂xψ ρ

dx ≤r₋⁻²

_∞

y

ϕ ρ1/2

ρxr²dx

≤r⁻²₋ _∞

0

ϕ ρ

dy _∞

0

r²ρx

2

dx

^1/2≤r₋⁻²E0E1.

3.17

We can verify that

(8)

1Asρ → ∞, it holds forξθρ 1−θρ, whereθ∈0,1

ρlim→∞ψ ρ

lim

ρ→∞

_ρ

ρ

γ−2

ξ^γ−32ρ^γξ⁻³1/2η−ρdη

≥ lim

ρ→∞

γ−2

ξ^γ−32ρ^γξ⁻³_1/2_ρ

ρ

η−ρ dη lim

ρ→∞

1 2

γ−2

θρ 1−θρ_γ−3 2ρ^γ

θρ 1−θρ₋₃1/2 ρ−ρ2

−→∞.

3.18

2Asρ → 0, it holds forξθρ 1−θρ, whereθ∈0,1

ρlim→0ψ ρ

lim

ρ→0

_ρ

ρ

γ−2

ξ^γ−32ρ^γξ⁻³1/2η−ρdη

≤ −lim

ρ→0

γ−2

ξ^γ−32ρ^γξ⁻³_1/2_ρ

ρ

η−ρ dη −lim

ρ→0

1 2

γ−2

θρ 1−θρ_γ−3 2ρ^γ

θρ 1−θρ₋₃_1/2 ρ−ρ2

:−νρ^γ/21/2.

3.19

Applying3.17–3.19andE0E1 < νr₋²ρ^γ/21/2, whereνis a positive constant, we can prove3.9.

_∞

0

r²u₂

xdx _∞

0

r²u₂

τr⁻⁴dx _τ

0

_∞

0

r²u₂

sr⁻⁴dx ds

_τ

0

_∞

0

ρ² r²u₂

xsdx ds _τ

0

_∞

0

r²u₂

xxdx ds≤C, τ ∈0, T,

3.20

whereC >0 denotes a constant independent of time.

(9)

Proof. Multiplying 3.32 by ρ⁻²r²u_τ and integrating the result with respect to x over 0,∞, making use of3.4, we obtain

d dτ

_∞

0

1 2

r²u₂

x−ρ^γ−2 r²u

x

dx

_∞

0

ρ⁻² r²u₂

τr⁻⁴dx

γ−2

∞ 0

ρ^γ−1 r²u2

xdx−2 _∞

0

ρ^γ−3ρx

r²u

τdx2 _∞

0

ρ⁻¹ρx

r²u

x

r²u

τdx 2

_∞

0

ρ⁻²u² r²u

τr⁻³dx−2 _∞

0

ρ⁻²ρxu r²u

τr⁻¹dx,

3.21

which implies _∞

0

r²u2

xdx _τ

0

_∞

0

r²u2

sr⁻⁴dx ds

≤CC _∞

0

ρ^γ−2−ρ^γ−22

dxC _τ

0

_∞

0

u² r² u²_xr⁴

dx dsC _τ

0

_∞

0

ρ²_xr⁴dx ds C

_τ

0

_∞

0

ρ_x² r²u₂

xr⁴dx dsC _τ

0

_∞

0

u⁴r⁻²dx dsC _τ

0

_∞

0

ρ²_xu²r²dx ds

≤CC _τ

0

_∞

0

ρ²_x r²u2

xr⁴dx dsCsup

τ∈0,Tu²_L∞.

3.22

From3.32,3.5,3.8, and3.9, we can deduce that for some small∈0,1 _τ

0

_∞

0

ρ²_x r²u2

xr⁴dx ds≤ _τ

0

_∞

0

ρ_x² r²u2

xr⁴dx ds _τ

0

_∞

0

r²u2

xsr⁻⁴dx ds

_τ

0

_∞

0

ρ²_xdx dsC _τ

0

_∞

0

u² r² u²_xr⁴

dx ds,

3.23

sup

τ∈0,Tu²_L^∞ ≤ sup

τ∈0,T

_∞

0

r²u2

xdxCsup

τ∈0,T

_∞

0

u²dx, 3.24

using3.22–3.24, we can obtain that _∞

0

r²u2 xdx

_τ

0

_∞

0

r²u2

sr⁻⁴dx ds≤CC _τ

0

_∞

0

r²u2

xsr⁻⁴dx ds. 3.25

(10)

Diﬀerentiating 3.32 with respect to τ, multiplying the result by r²u_τ, and integrating the result with respect toxover0,∞, we have

1 2

d dτ

_∞

0

r²u₂

τr⁻⁴dx _∞

0

ρ² r²u₂

xτdx 2

_∞

0

uuτ

r²u

τr⁻³dx−1 2

_∞

0

r⁻⁴

τ

r²u₂

τdx2 _∞

0

u r⁻¹u

τ

r²u

τr⁻²dx

_∞

0

ρ^γ

τ

r²u

xτdx− _∞

0

ρ²

τ

r²u

x

r²u

xτdx− _∞

0

2ρxu r

τ

r²u

τdx 2

_∞

0

r⁻²

τr⁻¹u² r²u

τdx.

3.26

A complicated computation gives

d dτ

_∞

0

r²u₂

τr⁻⁴dx _∞

0

ρ² r²u₂

xτdx

≤C _∞

0

r²u₂

τr⁻⁴dxCsup

τ∈0,T

r²u₂

x

L^∞

_∞

0

r²u₂

xdx C

_∞

0

u² r² u²_xr⁴

dx

1 sup

τ∈0,T

_∞

0

r²u₂

xdx

,

3.27

integrating3.27with respect toτ, by means of3.32,3.5,3.8,3.9, and3.25, it holds that

_∞

0

r²u₂

τr⁻⁴dx _τ

0

_∞

0

ρ² r²u₂

xsdx ds

≤CC _τ

0

_∞

0

r²u₂

sr⁻⁴dx dsCsup

τ∈0,T

r²u₂

x

L^∞

Csup

τ∈0,T

_∞

0

r²u₂

xdx

≤CC _τ

0

_∞

0

r²u₂

sr⁻⁴dx dsCsup

τ∈0,T

_∞

0

r²u₂

xdx

1/2_∞

0

r²u₂

xxdx 1/2

Csup

τ∈0,T

_∞

0

r²u₂

xdx

≤CC _τ

0

_∞

0

ρ² r²u2

xsdx dsC _∞

0

r²u2 τr⁻⁴dx,

3.28

choosing the constantsmall suﬃciently, we can complete the proof ofLemma 3.3.

(11)

Remark 3.4. By Lemmas3.1–3.3, the following inequality holds:

_∞

0

u²dx _∞

0

ρ−ρ₂ dx

_∞

0

u²_xdx _∞

0

u²_τdx _∞

0

ρ²_xdx

_τ

0

_∞

0

ρ²_xdx ds _τ

0

_∞

0

u²_xdx ds _τ

0

_∞

0

u²_sdx ds

_τ

0

_∞

0

u²_xxdx ds _τ

0

_∞

0

u²_xτdx ds≤C.

3.29

Lemma 3.5. Under the conditions inTheorem 2.1, it holds for any solutionρ, u to the exterior problem3.3that

ρ−ρ, u

·, τ

L^∞0,∞−→0, τ−→∞, 3.30

whereC >0 denotes a constant independent of time.

Proof. From Lemmas3.1–3.3, we can obtain _∞

0

ρ−ρ, u

x ²

L²0,∞dτ≤r⁻⁴₋ _∞

0

ρ−ρ, u

xr² ²

L²0,∞dτ ≤C, 3.31

_∞

0

d

dτ ρ−ρ, u

x ²

L²0,∞

dτ

_∞

0

_∞

0

−4ρρ²_x r²u

x−2ρ²ρx

r²u

xx

dx2

_∞

0

uxuxτdx dτ

≤C _∞

0

_∞

0

ρ²_xdx dτC _∞

0

_∞

0

u²

r² u²_xr⁴ r²u2

xxu²_xτ

dx dτ≤C,

3.32

which together with3.31implies ρ−ρ, u

x ²

L²0,∞∈W^1,1R. 3.33

It holds from Gagliardo-Nirenberg-Sobolev inequality that ρ−ρ, u

L^∞0,∞≤ ρ−ρ, u ^1/2

L²0,∞ ρ−ρ, u

x ^1/2

L²0,∞, 3.34

which together with3.5,3.9, and3.33implies this lemma.

3.2. Proof ofTheorem 2.1

Proof. The global existence of unique strong solution to the exterior problem as 2.4and 2.5can be established in terms of the short-time existence carried out as in6, the uniform

(12)

a-priori estimates and the analysis of regularities, which indeed follow from Lemmas3.1–3.3.

We omit the details. The large time behaviors follow fromLemma 3.5directly. The proof of Theorem 2.1is completed.

4. Proof of the Initial Boundary Value Problem

4.1. The A-Priori Estimates

Take the Lagrange coordinates transform

x _r

r−

ρr, tr²dr, τ t. 4.1

By4.1and the conservation of mass forρ, u _r

r−

ρr, tr²dr _r

r−

ρ0rr²dr : 1, 4.2

the Lagrange coordinates transform4.1mapr, t∈r₋, r×Rintox, τ∈0,1×R. The relation between Lagrangian and Eulerian coordinates are satisfied as

∂x

∂r ρr², ∂x

∂t −ρur², 4.3

and the initial boundary value problem’s2.4and2.11are reformulated to ρτρ²

r²u

x 0, r⁻²uτ

ρ^γ

x

ρ² r²u

x

x−2ρxu r , ρ, u

x,0 ρ0, u0

x, x∈0,1, u0, t u1, t 0, τ ∈0,∞,

4.4

where the initial data satisfies

ρ0∈L¹0,1∩W^1,∞0,1, inf

x∈0,1ρ0> ρ >0, 1

r²ρ0

r²u0

∈L²0,1, r²ρ0

r²u0

x∈L²0,1, 1

r²ρ0

r²ρ

r²ρ r²u0

x

∈L²0,1.

4.5

Then, we will establish the a-priori estimates for the solutionρ, uto the initial boundary value problem4.4.

(13)

Lemma 4.1. LetT >0. Under the conditions inTheorem 2.2, it holds for any solutionρ, uto the initial boundary value problem4.4that

₁

0

1

2u² 1 γ−1ρ^γ−1

dx

_τ

0

₁

0

2u²

r² ρ²u²_xr⁴

dx ds

₁

0

1

2u²₀ 1 γ−1ρ^γ−1₀

dx, τ∈0, T.

4.6

Proof. Multiplying4.42byr²uand integrating the result with respect toxover0,1, using 4.41and4.5, we obtain

d dτ

₁

0

1

2u² 1 γ−1ρ^γ−1

dx

₁

0

ρ² r²u₂

xdx2 ₁

0

ρ ru²

xdx, 4.7

and integrating4.7with respect toτ, we obtain ₁

0

1

2u² 1 γ−1ρ^γ−1

dx

_τ

0

₁

0

2u²

r² ρ²u²_xr⁴

dx ds ₁

0

1

2u²₀ 1 γ−1ρ^γ−1₀

dx. 4.8

Lemma 4.1can be obtained.

1 2

₁

0

ur²ρx

₂

dx 1 γ−1

₁

0

ρ^γ−1dxγ _τ

0

₁

0

2 ₁

0

u0r²ρ0x

₂

dx 1 γ−1

₁

0

ρ^γ−1₀ dx, τ ∈0, T,

4.9

whereCis a positive constant independent of time.

Proof. Diﬀerentiating4.41with respect tox, we have

ρxτ ρ²

r²u

x

x0, 4.10

which together with4.42and∂r/∂τugives

r⁻²uρx

τ

ρ^γ

x−2r⁻³u²−2ρxu

r . 4.11

(14)

Multiplying4.11byur²ρxr², and integrating the result with respect toxandτ, it holds that

1 2

₁

0

ur²ρx

2

dx 1 γ−1

₁

0

ρ^γ−1dxγ _τ

0

₁

0

2 ₁

0

u0r²ρ0x

₂

dx 1 γ−1

₁

0

ρ^γ−1₀ dx.

4.12

The proof of4.9is completed.

Lemma 4.3. LetT >0. Under the conditions inTheorem 2.2, there exists a constantρ^∗>0 such that

0< ρx, τ≤ρ^∗, x, τ∈0,1×0, T. 4.13

Proof. It follows from4.6and4.9that

ρx, τ ρr, t≤ _r

r₋

ρr, tdr _r

r₋

ρrr, tdr

≤r₋⁻² _r

r−

ρr, tr²drr₋⁻² _r

r−

ρρrr, t

√ρ r²dr

≤CC _r

r−

1

ρρrr, t²r²dr

≤CC ₁

0

ρxx, τ²r⁴dx≤C:ρ^∗.

4.14

₁

0

u²ⁿdx _τ

0

₁

0

u²ⁿ

r² ρ²u²ⁿ⁻²u²_xr⁴

dx ds≤CT, 4.15

for any positive integern∈N, whereCTis a positive constant dependent of time.

Proof. Multiplying4.42withu²ⁿ⁻¹, integrating by parts over0,1, we have

1 2n

d dτ

₁

0

u²ⁿdx ₁

0

ρ² r²u

x

r²u²ⁿ⁻¹

xdx ₁

0

ρ^γ

r²u²ⁿ⁻¹

xρ 2ru²ⁿ

x

dx. 4.16

(15)

Since it holds that

r²u

x

r²u²ⁿ⁻¹

x 2u

ρr r²ux

2u²ⁿ⁻¹

ρr 2n−1r²u²ⁿ⁻²ux

4u²ⁿ

ρ²r² 2n−1u²ⁿ⁻²u²_xr⁴4nu²ⁿ⁻¹uxr

ρ ,

4.17

it follows from4.16that

d dτ

₁

0

u²ⁿ 2ndx2

₁

0

u²ⁿ

r² dx 2n−1 ₁

0

ρ²u²ⁿ⁻²u²_xr⁴dx 2

₁

0

ρ^γ−1u²ⁿ⁻¹

r dx 2n−1

₁

0

ρ^γu²ⁿ⁻²uxr²dx

≤ ₁

0

u²ⁿ r² dx

₁

0

ρ²u²ⁿ⁻²u²_xr⁴dxC ρ ^2γ−1

L^∞

₁

0

u²ⁿ⁻²dx,

4.18

which together with4.13and Young’s inequality yields

d dτ

₁

0

u²ⁿdx ₁

0

u²ⁿ r² dx

₁

0

ρ²u²ⁿ⁻²u²_xr⁴dx≤C ₁

0

u²ⁿdx, 4.19

and by applying the Gronwall’s inequality to4.19, we can obtain4.15.

Lemma 4.5. LetT >0, forn∈N, andn >1/2γ−1. Under the conditions inTheorem 2.2, it holds for any solutionρ, uto the initial boundary value problem4.4that

_τ

0

ρ^2nγ−1u²ⁿ

L^∞0,1ds≤CT, 4.20

_τ

0

ρ^γ

xr²²ⁿ

L^∞0,1ds≤CT, τ ∈0, T, 4.21 whereCTis a positive constant dependent of time.

(16)

Proof. By means of Sobolev imbedding theorem and Cauchy-Schwarz inequality, applying 4.6,4.13, and4.15, we get

_τ

0

ρ^2nγ−1u²ⁿ

L^∞0,1ds

≤ _τ

0

₁

0

ρ^2nγ−1u²ⁿdx ds _τ

0

₁

0

ρ^2nγ−1u²ⁿ

x

dx ds

≤CT C _τ

0

₁

0

ρ^2nγ−1−1ρxu²ⁿdx dsC _τ

0

₁

0

ρ^2nγ−1u²ⁿ⁻¹uxdx ds

≤CT C _τ

0

₁

0

ρx²r⁴dx dsC _τ

0

₁

0

ρ^{22nγ−1−1}u⁴ⁿr⁻⁴dx ds

_τ

0

₁

0

ρ^{22nγ−1−1}u²ⁿr⁻⁴dx dsC _τ

0

₁

0

ρ²u²ⁿ⁻²u²_xr⁴dx ds≤CT.

4.22

Next, we find that

ρxr²ρ0xr₀²u0−u− _τ

0

ρ^γ

xr²ds, 4.23

which together with4.13,4.15, and4.22gives _τ

0

ρ^γ

xr²²ⁿ L^∞

ds γ²ⁿ

_τ

0

ρ^2nγ−1ρxr²²ⁿ L^∞

ds γ²ⁿ

_τ

0

ρ^2nγ−1

ρ0xr₀²u0−u− _s

0

ρ^γ

xr²dl ²ⁿ

L^∞

ds

≤C _τ

0

ρ^2nγ−1

ρ²ⁿ_0xu²ⁿ₀ u²ⁿ

L^∞dsC _τ

0

ρ^2nγ−1 _s

0

ρ^γ

xr²dl 2n

L^∞

ds

≤CT C _τ

0

ρ^2nγ−1u²ⁿ

L^∞dsCT _τ

0

_s

0

ρ^γ

xr²²ⁿ L^∞

dl ds,

4.24

applying the Gronwall’s inequality to4.24, we obtain4.21.

Lemma 4.6. LetT >0. Under the conditions inTheorem 2.2, there exists a constantρ∗>0 such that

ρx, τ≥ρ∗>0, x, τ∈0,1×0, T. 4.25