Boundary layers for compressible Navier–Stokes equations with density-dependent viscosity and cylindrical symmetry

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www.elsevier.com/locate/anihpc

Boundary layers for compressible Navier–Stokes equations with density-dependent viscosity and cylindrical symmetry

Lei Yao

^a

, Ting Zhang

^b

, Changjiang Zhu

^c,^∗

aDepartment of Mathematics, Northwest University, Xi’an 710127, China bDepartment of Mathematics, Zhejiang University, Hangzhou 310027, China

cThe Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received 22 September 2010; accepted 19 April 2011 Available online 7 May 2011

Abstract

In this paper, we consider the zero shear viscosity limit for the Navier–Stokes equations of compressible flows with density- dependent viscosity coefficient and cylindrical symmetry. The boundary layer effect as the shear viscosityμ=ερ^θgoes to zero (in fact,ε→0 in this paper, which impliesμ→0) is studied. We prove that the boundary layer thickness is of the orderO(ε^α), where 0< α <¹₂for the constant initial data and 0< α <¹₄for the general initial data, which extend the result in Frid and Shelukhin (1999) [4] to the case of density-dependent viscosity coefficient.

MSC:76N20; 35B40; 35Q30; 76N10; 76N17

Keywords:Navier–Stokes equations; Density-dependent viscosity; Cylindrical symmetry; Zero shear viscosity limit; Boundary layers;

BL-thickness

1. Introduction

In this paper, we shall study the effect of boundary layers as the shear viscosityμgoes to zero (in fact, ε→0 in this paper, which impliesμ→0) for an initial–boundary value problem to the Navier–Stokes equations of compressible flows with density-dependent viscosity coefficient and cylindrical symmetry. We restrict ourselves to the flows between two circular coaxial cylinders and assume that the corresponding solutions depend only on the radial variablex inΩ:= {x|0< a < x < b}and the time variablet ∈ [0, T]. Precisely, under the cylindrical symmetric transformation:

u=

ux₁

x −vx₂ x , ux₂

x +vx₁ x , w

, x=

x₁²+x₂², u=u(x, t ), v=v(x, t ), w=w(x, t ),

* Corresponding author.

E-mail addresses:yaolei1056@hotmail.com (L. Yao), zhangting79@zju.edu.cn (T. Zhang), cjzhu@mail.ccnu.edu.cn (C. Zhu).

doi:10.1016/j.anihpc.2011.04.006

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we consider the following three-dimensional compressible Navier–Stokes equations ∂_tρ+ ∇ ·(ρu) =0,

∂_t(ρu) + ∇ ·(ρu⊗ u)+ ∇p=div μ

∇ u+ ∇ u^T

+ ∇(λdivu).

Hereρis the density,p=Aρ^γ is the pressure, whereγ >1 is the gas constant andAis a positive constant, which will be normalized to be 1;uis the component of the velocity vectorualong the radial variablex(x∈Ω), vis the angular component of u, w is the axial component ofu; λandμare the bulk and shear viscosity coefficients, respectively.

λ >0 is a positive constant andμ=ερ^θwith 0θγ andεbeing a positive constant.

The reduced system is now of the form (see [8] for the case of constant viscosity coefficient):

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ρ_t+(ρu)_x+ρu x =0, ρ

u_t+uu_x−v² x

+p_x−(2μ+λ)

u_x+u x

x

−2μxu_x=0, ρ

v_t+uv_x+uv x

−μ

v_x+v x

x

−μ_xv_x+μ_xv x =0, ρ(w_t+uw_x)−μ

w_xx+w_x x

−μ_xw_x=0.

(1.1)

In the domainQT :=Ω×(0, T ), we consider the initial–boundary value problem given by (1.1) and

(ρ, u, v, w)|t=0=(ρ0, u0, v0, w0)(x), x∈Ω:=(a, b), (1.2) u|x=a,b=0, (v, w)|x=a=(v1, w1)(t ), (v, w)|x=b=(v2, w2)(t ). (1.3) The viscosity coefficient is often assumed to be a positive constant. However, it is well known that the viscosity coefficient of the flow is not constant. It is motivated by the physical consideration that in the derivation of the Navier–

Stokes equations from the Boltzmann equation through the Chapman–Enskog expansion to the second order, cf. [1, 6], the viscosity coefficient is a function of the temperature. Especially, for isentropic flow, this dependence of the viscosity is translated into the dependence on the density. For more physical background, please refer to [22,23,29]

and references therein.

The asymptotic behavior of viscous flows, as the viscosity vanishes, is one of the important topics in the theory of compressible flows, and the problem of small viscosity finds many applications, for example, in the boundary layer theory [14].

Shelukhin studied the zero shear viscosity limit for flows with heat conductivity between two parallel plates in [19,20] and the passage to the limit for a free-boundary problem of describing a joint motion of two compressible fluids with different viscosities, as the shear viscosity of the fluids vanishes in [21].

Frid and Shelukhin in [4] investigated the cylinder isentropic problem (1.1)–(1.3) with constant shear viscosity coefficientμ(i.e.θ=0). They proved that the problem possessed a unique global strong solution whenρ0∈H¹(Ω), infΩρ₀>0,u₀, v₀, w₀∈H₀¹(Ω), and also investigated the boundary layer effect and proved the existence of a bound- ary layer thickness (BL-thickness; see Definition 1.1 below) which is close to the valueO(μ¹²). For the non-isentropic flows with constant shear viscosity coefficient μand constant heat conductivity, Frid and Shelukhin in [5] justified the vanishing shear viscosity limit. For the non-isentropic flows with constant shear viscosity coefficientμand non- constant heat conductivity, Jiang and Zhang in [7] studied the boundary layer effect and convergence rates as the shear viscosityμgoes to zero. Their results showed that the BL-thickness and a convergence rate were of the ordersO(μ^α) with 0< α <¹₂andO(√

μ ), respectively.

But there are no relevant results for the problem with density-dependent viscosity coefficient. The main aim of this paper is to extend Frid and Shelukhin’s result in [4] to the case of density-dependent viscosity coefficient. We establish the convergence result asε→0 and also give BL-thickness of the orderO(ε^α), where 0< α <¹₂ for the constant initial data and 0< α <¹₄for the general initial data.

The boundary layers problem has been one of the fundamental and important issues in fluid dynamics. The Prandtl boundary layer (characteristic boundaries) are studied for the linearized case in [26–28] by using asymptotic analysis, while the boundary layer stability in the case of non-characteristic boundaries and one spatial dimension is discussed in [11,16]. For incompressible Navier–Stokes equations, there are a number of papers dedicated to the questions of

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the boundary layers, see [3,9,12,13,18,24,25] for example. We also mention that the fluids withμ=0 andλ >0 were discussed in [10,15,17].

To formulate the main results of this paper, we need the following notations which will be used.

Notation. From now on, we use the notations (f₁, f₂, . . . , f_n) ²= f₁ ²+ f₂ ² + · · · + f_n ² for functions f₁, f₂, . . . , f_n belonging to the same functional space equipped with a norm · . W^m,p(O)and W₀^m,p(O) (W^m,2(O)=H^m(O),W₀^m,2(O)=H₀^m(O))withm∈Z₊andp1 denote the usual Sobolev spaces defined overO, W^0,p(O)≡L^p(O). The symbol C^k(O)(resp.C^k(O)) with non-negative integerk represents the set of functions having continuous derivatives up to orderkinO(resp. inO). For simplicity, we will use the abbreviations:

· W^m,p:= · W^m,p(Ω), · L^p:= · L^p(Ω).

L^p(I, B)(resp. · L^p(I,B)) denotes the space of all strongly measurable,pth-power integrable (essentially bounded ifp= ∞) functions fromI toB(resp. norms), whereI⊂RandBis a Banach space.

Then, our main results can be stated as follows.

Theorem 1.1.Assume that the initial and boundary data satisfy the conditions:

(ρ0, v0, w0)∈W^1,2(Ω), u0∈W₀^1,2∩W^2,2(Ω), ρ0>0, ρ₀⁻¹

C(Ω)¯ <∞, (v_i, w_i)

C¹([0,T])<∞ (i=1,2),

u0(a)=u0(b)=0, (v0, w0)(a)=(v1, w1)(0), (v0, w0)(b)=(v2, w2)(0).

Then there exists a positive constantε₀,such that for allεε₀, we have:

(i) For anyT >0,there exists a unique solution(ρ, u, v, w)of the problem(1.1)–(1.3)satisfying ρ∈L^∞

0, T;W^1,2

, ρ_t ∈L^∞

0, T;L²

, ρ >0, u∈L^∞

0, T;W₀^1,2∩W^2,2

, ut∈L^∞

0, T;L²

∩L²

0, T;W^1,2 ,

where the norms are all uniform in ε. There also exists a positive constant C independent of ε, such that

|(v, w)(x, t )|Cfor all(x, t )∈ [a, b] × [0, T], and ε¹²(vx, wx)²

L^∞(0,T;L²)+ε³²(vxx, wxx)²

L²(0,T;L²)C.

(ii) There exist functions(ρ,¯ u,¯ v,¯ w)¯ such that asε→0, (ρ, u)→(ρ,¯ u)¯ strongly inC^α(Q¯_T)for anyα∈

0,1

2

,

(u_x, v, w)→(u¯_x,v,¯ w)¯ strongly inL^p¹(Q_T)for anyp₁∈ [1,∞), (ρ_x, ρ_t, u_xx, u_t) (ρ¯_x,ρ¯_t,u¯_xx,u¯_t) weakly-∗inL^∞

0, T;L² ,

εv_x², εw_x²

→(0,0) strongly inL^p²(Q_T)for anyp₂∈ [1,2], whereQT =(a, b)×(0, T ).

Furthermore, the limit functions(ρ,¯ u,¯ v,¯ w)¯ solve the problem(1.1)–(1.3)withε=0in the following sense:

Q_T

xρ(ψ¯ t+ ¯uψx) dx dt+

Ω

xρ0ψ (x,0) dx=0, (1.4)

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QT

x

¯ ρuφ¯ _t+

¯

ρu¯²+ ¯ρ^γ−λu¯_x−λu¯ x

φ_x

dx dt

+

Q_T

¯

ρv¯²+ ¯ρ^γ−λu¯x−λu¯ x

φ dx dt+

Ω

xρ0u0φ(x,0) dx=0, (1.5)

QT

xρ¯v¯

φ_t+ ¯uφ_x−u¯ xφ

dx dt+

Ω

xρ₀v₀φ(x,0) dx=0, (1.6)

Q_T

xρ¯w(φ¯ t+ ¯uφx) dx dt+

Ω

xρ0w0φ(x,0) dx=0, (1.7)

for any text functionsφ, ψ∈C¹(Q¯_T)withφ(x, T )=ψ (x, T )=0for allx∈Ω, andφ(a, t)=φ(b, t)=0for all t∈(0, T ).

Next, we state the result about the boundary layer effect. At first, let us give the definition of BL-thickness, which mainly comes from [4].

Definition 1.1.A functionδ(ε)is called a BL-thickness for the problem (1.1)–(1.3) with vanishing shear viscosityμ (in fact, vanishingεimplies vanishing shear viscosityμ), ifδ(ε)↓0 asε↓0, and

(ρ, u)→(ρ,¯ u)¯ strongly inC^α(Q¯T)for anyα∈

0,1 2

,asε→0, (1.8)

εlim→0

(v− ¯v, w− ¯w)

L^∞(0,T;L^∞(Ω_δ(ε)))=0, (1.9)

lim inf

ε→0

(v− ¯v, w− ¯w)

L^∞(0,T;L^∞(Ω))>0, (1.10)

whereΩ_δ:= {x∈Ω|a+δ < x < b−δ}, and(ρ, u, v, w)(resp.(ρ,¯ u,¯ v,¯ w))¯ is the solution to the problem (1.1)–(1.3) (resp. to (1.4)–(1.7)).

Clearly, this definition does not determine the BL-thickness uniquely, since any functionδ_∗(ε) satisfying the in- equalityδ_∗(ε)δ(ε)for smallεis also a BL-thickness.

Theorem 1.2(BL-thickness for the constant initial data). Under the conditions of Theorem1.1, we assume in addition that the initial data satisfy(ρ, u, v, w)|t=0=(ρ₀,0,0,0)where the initial densityρ₀is positive constant. Then the limit problem(1.4)–(1.7)has only trivial solution(ρ,¯ u,¯ v,¯ w)¯ =(ρ₀,0,0,0), and any function δ(ε), satisfying the conditionsδ(ε)→0andε¹²/δ(ε)→0asε→0,is a BL-thickness such that

(ρ, u)→(ρ₀,0) strongly inC^α(Q¯_T)for anyα∈

0,1 2

, asε→0,

εlim→0

(v, w)

L^∞(0,T;L^∞(Ωδ(ε)))=0, and

lim inf

ε→0

(v, w)

L^∞(0,T;L^∞(Ω))>0,

whenever the boundary conditionsvi andwi (i=1,2)are not identical zero.

Motivated by the ideas of [7], we can give some results about the BL-thickness for the general initial data under some additional initial assumptions. The main result is as follows:

Theorem 1.3 (BL-thickness for the general initial data). Let the assumptions given in Theorem1.1 hold. Assume in addition that (v₀, w₀)∈H³(Ω) and ρ₀∈H²(Ω). Then, any function δ(ε), satisfying the conditionsδ(ε)→0

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and ε¹⁴/δ(ε)→0, as ε→0, is a BL-thickness such that (1.8)–(1.10) hold, whenever the boundary conditions (v₁, w₁, v₂, w₂)are not identical(v(a, t ),¯ w(a, t ),¯ v(b, t ),¯ w(b, t )).¯

Remark 1.1.Whenθ=0, all the results in Theorems 1.1, 1.2 and 1.3 hold without the restrictionεε₀(we can choose the constantε₀as in (2.21)), which can be proved by using the ideas in [4,7]. So, in this paper, we only consider the case when 0< θγ.

Remark 1.2.The above results also hold for the case whenγ=1, where_γ^ρ₋^γ₁in (2.2) is replaced by(1−ρ+ρlnρ), which is positive forρ >0. We don’t refer this case in the present paper, see [4] for details.

The difficulty of this problem is to obtain the upper and lower bounds of the densityρ, especially for the lower bound. Whenεis small enough in some sense, using some newa prioriestimates for the solution, we can obtain the bounds of the density. The key ideas are using the classical continuity method and the followinga prioriassumption:

sup

t∈[0,T]ε² b

a

ρ^θ2

xdx1, for anyT >0.

In Lemmas 2.2–2.4, we get some uniforma prioriestimates (with respect toε) for the solution. Then, using these estimates and the smallness restriction onε, we can obtain

sup

t∈[0,T]

ε² b

a

ρ^θ2 xdx1

2, for anyT >0.

Using the classical continuity method, we can close this estimate.

Compared to [4,7], there are some other difficulties in our paper. Firstly, we have to seek a new method to estimate the bound ofρ because of the effect of density-dependent shear viscosityμ(see Lemma 2.2). Secondly, in order to obtain the bound ofv, we must deal withb

a v^2N

x^2N dxinstead ofb

av^2Ndx as in [7], whereN∈N. Finally, when we want to get the BL-thickness results, we have to obtain the estimates:

δ

b−δ

a+δ

v x

x

dxCε¹²

see (3.16) for the constant initial data(ρ₀,0,0,0)

, (1.11)

or δ

b−δ

a+δ

v− ¯v

x

dxCε¹⁴

see (3.40) for the general initial data

, (1.12)

instead of estimating in [7]

δ

b−δ

a+δ

|v_x|dxCε¹², (1.13)

or δ

b−δ

a+δ

(v− ¯v)_xdxCε¹⁴. (1.14)

In fact, as in [7], when we estimate (1.13) and (1.14) directly, we will encounter the terms like t

0

b

a

μxv

ρx z_xϕ_νξ_δ(x) dx dτ, z=v_x, (1.15)

because of the effect of density-dependent shear viscosityμ. However, we have no method to estimate (1.15).

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The paper is organized as follows. We first derive somea prioriestimates in Section 2. Then, at the beginning of Section 3, we illustrate the proof of Theorem 1.1 concisely, since the process of it is similar to that in [2,4,5,19,20].

Next, in Section 3.1, we apply uniform estimates in Section 2 to prove Theorem 1.2 by adapting and modifying the techniques used in [4,7]. Finally, in Section 3.2, we give some results on the convergence rate (w.r.t.ε) and the BL- thickness for the case of general initial data under certain additional initial conditions and this completes the proof of Theorem 1.3.

We should mention that the methods introduced by Frid and Shelukhin in [4,5] and Jiang and Zhang in [7] will play a crucial role in our proof here.

2. A prioriestimates

This section is devoted to the derivation ofa prioriestimates uniform inεfor the solution to the problem (1.1)–

(1.3). For simplicity, throughout the rest of this paper we shall denote byC the various positive constants dependent on the initial data,γ,λandT, but independent ofε.

At first, we give the basic energy estimates.

Lemma 2.1.For anyt∈(0, T ), we have b

a

xρ(x, t ) dx= b

a

xρ0(x) dx >0, (2.1)

and b

a

x

ρ

u²+v²+w² + ρ^γ

γ−1

dx

+ t

0

b

a

x

(2μ+λ) u²

x²+u²_x

+μ

vx−v x

2

+μw²_x

dx dτC. (2.2)

Proof. Firstly, it follows that (2.1) holds from (1.1)1and boundary conditions (1.3).

Secondly, rewrite (1.1) as 1

2ρ

u²+v²+w²

x+ ρ^γ γ−1x

t

+ 1

2xρu

u²+v²+w² + γ

γ−1ρ^γux

x

+

(2μ+λ)

u²_x+u² x²

+μ

v_x−v

x 2

+μw²_x

x

=

(2μ+λ)u_xux

x+ {μvv_xx}x− μv²

x+ {μww_xx}x. (2.3)

Integrating (2.3) with respect to(x, t )over(a, b)×(0, t )and using the boundary conditionsu(a, t )=u(b, t )=0, we have

b

a

x 1

2ρ

u²+v²+w² + ρ^γ

γ−1

dx+ t

0

b

a

x

(2μ+λ)

u²_x+u² x²

+μ

vx−v x

2

+μw_x²

dx dτ

= b

a

x 1

2ρ₀

u²₀+v²₀+w₀² + ρ₀^γ

γ−1

dx+ t

0

μvv_xx|^bx=adτ− t

0

μv²^b

x=adτ+ t

0

μww_xx|^bx=adτ.

(2.4)

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On the other hand, integrating Eqs. (1.1)3–(1.1)4over(a, x)and(x, b), respectively, we have by using the boundary conditions (1.3) that

μvvxx|x=a= a b−av1(t )

b

a

μ

vx−v x

dx− a

b−av1(t )d dt

b

a

x

a

ρv dy dx

−2av1(t ) b−a

b

a

x

a

1

yρuv dy dx−av₁(t ) b−a

b

a

ρuv dx

+ 2a b−av₁(t )

b

a

x

a

μ

v_x−v y

1

ydy dx+ μv²

x=a, (2.5)

μvv_xx|x=b= b b−av₂(t )

b

a

μ

v_x−v x

dx+ b

b−av₂(t )d dt

b

a

b

x

ρv dy dx

+2bv2(t ) b−a

b

a

b

x

1

yρuv dy dx−bv₂(t ) b−a

b

a

ρuv dx

− 2b b−av₂(t )

b

a

b

x

μ

v_x−v y

1

ydy dx+ μv²

x=b, (2.6)

μww_xx|x=a= 1 b−a

aw₁(t )

b

a

μw_xdx−aw₁(t )d dt

b

a

x

a

ρw dy dx−aw₁(t ) b

a

x

a

1

yρuw dy dx

−aw1(t ) b

a

ρuw dx+aw1(t ) b

a

x

a

1

yμwxdy dx

, (2.7)

and

μww_xx|x=b= 1 b−a

bw₂(t )

b

a

μw_xdx+bw₂(t )d dt

b

a

b

x

ρw dy dx+bw₂(t ) b

a

b

x

1

yρuw dy dx

−bw₂(t ) b

a

ρuw dx−bw₂(t ) b

a

b

x

1

yμw_xdy dx

. (2.8)

Substituting (2.5)–(2.8) into (2.4), we have b

a

x 1

2ρ

u²+v²+w² + ρ^γ

γ−1

dx

+ t

0

b

a

x

(2μ+λ)

u²_x+u² x²

+μ

v_x−v x

2

+μw_x²

dx dτ

C+C

t

0

b

a

xμ

v₁²(τ )+v₂²(τ )

dx dτ+1 2 t

0

b

a

μ

v_x−v x

2

x dx dτ

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+C t

0

b

a

ρv²dx dτ+C t

0

b

a

ρ

v_1t²(τ )+v_2t²(τ ) x dx dτ

+1 4

b

a

ρ

u²+v²+w²

x dx dτ+C b

a

ρ

v²₁(t )+v₂²(t )+w₁²(t )+w₂²(t ) x dx

+C t

0

b

a

v₁(τ )+v2(τ )ρ

u²+v²

x dx dτ+1 2 t

0

b

a

μw²_xx dx dτ

+C t

0

b

a

xμ

w₁²(τ )+w²₂(τ )

dx dτ+C t

0

b

a

ρw²x dx dτ

+C t

0

b

a

ρ

w²_1t(τ )+w²_2t(τ )

x dx dτ+C t

0

b

a

w₁(τ )+w2(τ )ρ

u²+w² x dx dτ

C+1

2 t

0

b

a

μ

v_x−v x

2

x dx dτ+1 2 t

0

b

a

μw²_xx dx dτ+1 4

b

a

ρ

u²+v²+w² x dx dτ

+C t

0

b

a

ρ

u²+v²+w²

x dx dτ+C t

0

b

a

xρ^γdx dτ.

Here we have used the Cauchy–Schwartz inequality, the Young inequality and 0< θγ. By Gronwall’s inequality, we deduce (2.2). This completes the proof of Lemma 2.1. 2

In order to estimate the upper and lower bounds ofρ, we suppose the followinga prioriassumption holds:

sup

t∈[0,T]ε² b

a

ρ^θ2

xdx1. (2.9)

Lemma 2.2.Under the assumptions of Theorem1.1and the a priori assumption(2.9), ifεε₀<1, then there exists a positive constantC, such that

C⁻¹ρ(x, t )C, for all(x, t )∈QT. (2.10)

Proof. For any fixed point(x₀, t₀)∈Q_T,we want to obtain the bound ofρat a fixed point(x₀, t₀). First, by (2.1) and the assumptions in Theorem 1.1, there existsx₁=x₁(t₀)∈(a, b)such that(b−a)x₁ρ(x₁, t₀)=_b

a xρ(x, t₀) dx= _b

a xρ₀(x) dx, which implies that there exists a positive constantCsuch that

C⁻¹ρ(x₁, t₀)C. (2.11)

LetL(x, t )=^2ε_θ ρ^θ+λlnρ.We shall examine the differenceL(x₁, t )−L(x₀, t )by considering its evolution along particle trajectories. DefineX_j(t )by

d

dtX_j=u(X_j, t ), X_j(t₀)=x_j, j =0,1,

and letL(t )=L(X₁(t ), t )−L(X₀(t ), t ). We then obtain from (1.1) that

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d

dtL(t )= −(2μ+λ) u

x +ux

^X¹

X0

= −

X₁

X₀

ρut+ρuux−ρv²

x +px+2μx

u x

dx

= −d

dtI (t )−p−

X₁

X₀

ρ x

u²−v² dx−

X₁

X₀

2μx

u

xdx, (2.12)

whereI (t )=_X₁

X₀ ρu dx,andp=p(ρ(·, t ))|^X_X¹₀. Now defineα(t )= ^{p(t )}_{L(t )}. It is easy to see thatα(t )0 by the representations ofLandp.

Rewrite (2.12) as the forms of a linear ODE d

dtL(t )+α(t )L(t )= −d dtI (t )−

X₁

X₀

ρ x

u²−v² dx−

X₁

X₀

2μx

u x dx, whose solution is

L(t )=e⁻

t 0α(τ ) dτ

L(0)+I (0)

−I (t ) +

t

0

e⁻

t sα(τ ) dτ

α(s)I (s)−

X₁

X₀

ρ x

u²−v² dx−2

X₁

X₀

μx

u x dx

ds, (2.13)

which implies

L(t )L(0)+I (0)+I (t ) +

t

0

e⁻^s^t^{α(τ ) dτ}

α(s)I (s)+

X1

X₀

ρ x

u²−v² +2μx

u x

dx

ds. (2.14)

Now we estimate the third term on the right-hand side of (2.14) as follows:

I (t )=

X₁

X₀

ρu dx

X₁

X₀

|ρu|dx

X₁

X₀

ρu²dx+C

X₁

X₀

xρ dxC, where we have used Lemma 2.1.

Substituting the above estimate into (2.14), we have L(t )C+C

t

0

e⁻

t

sα(τ ) dτα(s) ds+ t

0

X1

X0

ρ x

u²−v² +2μx

u x

dx

dτ. (2.15)

By Lemma 2.1 and thea prioriassumption (2.9), we have t

0

e⁻

t

sα(τ ) dτα(s) ds=1−e⁻

t

0α(τ ) dτ1, and

t

0

X₁

X0

ρ x

u²−v² +2μx

u x

dx dτ

C

t

0 X1

X0

ρu²+ρv²

dx dτ+C t

0 X1

X0

ε² ρ^θ2

xdx dτ+C t

0 X1

X0

u²

x²dx dτC.

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Thus (2.15) shows L(t₀)=

2ε

θ ρ^θ(x₁, t₀)+λlnρ(x₁, t₀)−2ε

θ ρ^θ(x₀, t₀)−λlnρ(x₀, t₀) C,

which implies the pointwise bound (2.10) follows from this and (2.11) under the assumptionεε₀<1. 2 Lemma 2.3.There is a positive constantCindependent ofε, such that

−Cv(x, t ), w(x, t )C for any(x, t )∈ ¯Q_T. (2.16)

Proof. Multiplying (1.1)4by 2N xw^2N⁻¹withN∈N, we have xρw^2N

t+

xρuw^2N

x=2N x(μwx)_xw^2N⁻¹+2N μw^2N⁻¹w_x. Integrating the above equality over(a, b)×(0, t ), we get

b

a

xρw^2Ndx= b

a

xρ₀w₀^2Ndx−2N (2N−1) t

0

b

a

xμw²_xw^2N⁻²dx dτ

+2N t

0

xμwxw^2N⁻¹^x⁼^b

x=a. (2.17)

We can use Lemmas 2.1, 2.2, (2.7) and (2.8) to deal with the boundary terms in (2.17), which can be controlled by CN²C^2Nthrough simple calculation. Then we get

b

a

xρw^2Ndx+2N (2N−1) t

0

b

a

xμw²_xw^2N⁻²dx dτ b

a

xρ₀w^2N₀ dx+CN²C^2NCN²C^2N,

where C is a positive constant, independent of N,ε. Now, we raise to the power _2N¹ for both sides of the above inequality and letN→ ∞to deduce that w(t ) _L∞C.

Next, we prove v(t ) _L∞C.Multiplying (1.1)3by 2N x⁻^2Nv^2N⁻¹withN∈N, we have x⁻^2Nρv^2N

t+

x⁻^2Nρuv^2N

x+(4N+1)ρuv^2Nx⁻^2N⁻¹

=2N x⁻^2N

μ

v_x+v x

x

v^2N⁻¹−4N x⁻^2Nv^2N⁻¹μ_xv x.

Integrating the above equality over(a, b)×(0, t )and integrating by parts, we get b

a

x⁻^2Nρv^2Ndx= b

a

x⁻^2Nρ₀v₀^2Ndx−(4N+1) t

0

b

a

x⁻^2N⁻¹ρuv^2Ndx dτ

−2N (2N−1) t

0

b

a

x⁻^2Nμv^2N⁻²

v_x−4N+1 4N−2 v x

2

dx dτ

+ 9N 4N−2

t

0

b

a

x⁻^2N⁻²μv^2Ndx dτ

+2N t

0

x⁻^2Nv^2N⁻¹μ

v_x−v x

^x⁼^b

x=a

dτ. (2.18)

We can use Lemmas 2.1, 2.2, (2.5) and (2.6) to deal with the boundary terms in (2.18), which also can be controlled byCN²C^2N.

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By (2.18) and Lemma 2.2, we get b

a

v^2N

x^2NdxCN²C^2N+CN t

0

1+u(τ )

L^∞

^b

a

v^2N x^2Ndx dτ

CN²C^2N+CN t

0

1+u_x(τ )

L²

^b

a

v^2N x^2Ndx dτ

,

where we have used the Cauchy–Schwartz inequality and Sobolev’s inequality u(t ) _L∞ u_x(t ) _L2 ∈L¹(0, T ).

Therefore, Gronwall’s inequality yields b

a

v^2N

x^2NdxCN²C^2Nexp{CN},

whereCis independent ofN,ε. Now, we raise to the power_2N¹ for both sides of the above inequality and letN→ ∞, then we get v(t ) _L∞C.

This proves Lemma 2.3. 2

Lemma 2.4.For anyt∈(0, T ), the following estimate holds b

a

ρ_x²(x, t ) dxC. (2.19)

Proof. From (1.1)1and (1.1)2, we have u_x+u

x = −ρ_t+uρ_x

ρ ,

and

ρ

2μ+λ ρ² ρ_x+u

t

+

ρu

2μ+λ ρ² ρ_x+u

x

+p_x+2μx

u x +ρ

x

u²−v²

=0.

Multiplying the above equation by(^2μ⁺^λ

ρ² ρ_x+u)and integrating the resulting equation over(a, b), we can get 1

2 d dt

b

a

ρ

2μ+λ ρ² ρ_x+u

2

dx+γ b

a

(2μ+λ)ρ^γ⁻³ρ_x²dx

= −γ b

a

ρ^γ⁻¹ρ_xu dx+1 2 b

a

ρu x

2μ+λ ρ² ρ_x+u

2

dx

− b

a

ρ x

u²−v²2μ+λ ρ² ρ_x+u

dx−2ε b

a

ρ^θ

x

u x

2μ+λ ρ² ρ_x+u

dx. (2.20)

The first three terms on the right-hand side of (2.20) can be estimated by Lemmas 2.1–2.3 and the Cauchy–Schwartz inequality as follows:

γ

b

a

ρ^γ⁻¹ρxu dx 1

2γ b

a

λρ^γ⁻³ρ_x²dx+C b

a

ρu²dx1 2γ

b

a

λρ^γ⁻³ρ_x²dx+C, and

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1 2 b

a

ρu x

2μ+λ ρ² ρ_x+u

2

dx +

b

a

ρ x

u²−v²2μ+λ ρ² ρ_x+u

dx

Cu(t )

L^∞

b

a

ρ

2μ+λ ρ² ρx+u

2

dx+C b

a

ρ

2μ+λ ρ² ρx+u

2

dx+C b

a

ρ

u²−v²2

dx

C

1+

b

a

u²_xdx b

a

ρ

2μ+λ ρ² ρ_x+u

2

dx+C+C b

a

u²_xdx, where we have also used Sobolev’s inequality u(t ) _L∞ u_x(t ) _L2∈L¹(0, T ).

And we can bound the last term on the right-hand side of (2.20) by using thea prioriassumption (2.9):

2ε

b

a

ρ^θ

x

u x

2μ+λ ρ² ρ_x+u

dx

C

b

a

ρu²

2μ+λ ρ² ρ_x+u

2

dx+Cε² b

a

ρ^θ2 xdx

C+C

b

a

u²_xdx b

a

ρ

2μ+λ ρ² ρx+u

2

dx.

Hence, inserting the above three estimates into (2.20) and using Lemmas 2.1–2.3, and applying Gronwall’s inequality, we obtain (2.19).

The proof of Lemma 2.4 is completed. 2

Now we prove that thea prioriassumption (2.9) is closed. From Lemmas 2.2 and 2.4, if we choose Cε₀²1

2, ε₀<1, (2.21)

then we have ε²

b

a

ρ^θ2

xdx=θ²ε² b

a

ρ^2θ⁻²ρ_x²dxCε²Cε²₀1 2, and this closes thea prioriassumption (2.9).

Lemma 2.5.For anyt∈(0, T ), we have u _L∞(0,T;L^∞(Ω))C,

and b

a

u²_xdx+ t

0

b

a

u²_t +u²_xx

dx dτC. (2.22)

Proof. Rewrite (1.1)2as

ρut+ρuux−(2μ+λ)uxx=(2μ+λ) u_x

x − u x²

+ρv²

x −px+2μxux, and

ρu_t−(2μ+λ)u_xx2

ρ⁻¹=ρ⁻¹

(2μ+λ) ux

x − u x²

−ρuu_x+ρv²

x −γρ^γ⁻¹ρ_x+2μxu_x 2

, which implies