Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions

Digital Object Identifier (DOI) 10.1007/s00205-011-0450-y

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions

to a Riemann Problem

Feimin Huang, Yi Wang & Tong Yang

Communicated by A. Bressan

Abstract

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the super- position of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.

1. Introduction

Consider the compressible Navier–Stokes equations in the Lagrangian coordinates

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

vt −ux =0, ut +px =εux

v

x,

e+u²

2 _t+(pu)x =κ θx

v _x+εuux

v

x,

(1.1)

where the functionsv(x,t) >0,u(x,t), θ(t,x) >0 represent the specific volume, velocity and temperature of the gas, respectively. Here, we consider a perfect gas with pressure p= p(v, θ)and internal energy e=e(v, θ)given by

p=p(v, θ)= Rθ

v , e= R

γ−1θ,

respectively, withγ >1,R >0 being the gas constants. By using the entropy s, one can write

p= p(v,s)= Av^−γexp γ−1

R s for some constant A>0.

The main concern of this paper is the vanishing viscosity limit, which is an unsolved and still challenging problem with a long history. The main difficulty comes from the singularity, that is, shock waves, in the solutions to the inviscid compressible fluid, which has so far prevented solving the problem in the general setting by means of known analytic techniques and tools. Essential new ideas are needed to tackle this open problem; however, if the solution to the inviscid fluid is assumed to be smooth, then the problem is substantially easier and can be solved by a standard scaling method. Therefore, any attempt on this problem that involves the singularity in the inviscid solution can be viewed as progress to the solution for the general case.

Let us now review some related previous works along these lines. In fact, there are many results on the vanishing viscosity limit for a compressible fluid. First, for a system of hyperbolic conservation laws with artificial viscosity

ut+ f(u)x =εux x, (1.2)

Goodman and Xin [6] verified the limit for piecewise smooth solutions separated by non-interacting shock waves using a matched asymptotic expansion method.

Later, Yu [20] proved it for the hyperbolic conservation laws (1.2) with both shock and initial layers. In 2005, important progress made by Bianchini and Bressan [1] justified the vanishing viscosity limit in BV space, even though the problem is still unsolved for physical systems such as the Navier–Stokes equations.

For compressible isentropic Navier–Stokes equations in which the conserva- tion of energy in (1.1) is neglected in the isentropic regime, Hoff and Liu [7] first proved the vanishing viscosity limit for piecewise constant shock with an initial layer. Later Xin [18] justified the limit for rarefaction waves. Then Wang [16]

generalized the results of Goodmann and Xin [6] to the isentropic Navier–Stokes equations.

Recently, Chen and Perepelitsa [2] proved the vanishing viscosity to the com- pressible Euler equations for the isentropic compressible Navier–Stokes equations by using a compensated compactness method for the general initial data if the far field does not contain a vacuum. Note that this result is very general because it allows initial data containing a vacuum in the interior domain. However, the frame- work of compensated compactness is basically limited to 2×2 systems so far, so this result does not apply to the full compressible Navier–Stokes equations (1.1).

It is well known that the solution to the Riemann problem for the Euler equa- tions consists of three basic wave patterns: shock, rarefaction wave and contact discontinuity. Moreover, the Riemann solution is essential in the theory for Euler equations as it captures both the local and global behaviors of general solutions.

For the full Navier–Stokes equations, there are results on the limits to the Euler system for the basic wave patterns. We refer to Jiang et al. [10] and Xin and Zeng [19] for the rarefaction wave, Wang [17] for the shock wave, Ma [12] for the contact discontinuity and Huang et al. [9] for the superposition of two rarefaction

waves and a contact discontinuity. One can also refer to Huang et al. [8] for the hydrodynamic limit of Boltzmann equation to the compressible Euler system with contact discontinuity.

The limit of the full compressible Navier–Stokes equations with basic wave patterns is, in fact, closely related to the stability of viscous wave patterns. The strict monotonicity of the corresponding characteristic speed along the wave pro- file plays an important role in the studies on the shock wave and rarefaction wave.

Precisely, the shock wave is compressive so that the characteristic speed is mono- tone decreasing along the shock profile. To cope with this in the energy estimate, the anti-derivative variable of the perturbation should be used; see Goodman [5]

and Matsumura and Nishihara [13]. On the other hand, the rarefaction wave is expansive and the characteristic speed is monotone increasing along the profile, so the approach used for shock profile does not apply; see Matsumura and Nishihara [14]. Hence, the different frameworks for stability analyses on the viscous shock wave and rarefaction wave render the stability of the Riemann solution with both rarefaction and shock waves still unsolved.

In order to verify the vanishing viscosity limit to the compressible Euler equa- tions for the Riemann solution as a superposition of a rarefaction and a shock wave for any fixed time T , one main idea in this paper is to introduce “hyperbolic waves”

that capture the propagation of the extra mass created by the approximate hyper- bolic rarefaction wave profile in the viscous setting. With this tool, the vanishing viscosity limit can be formulated as a stability problem so that the energy method can be applied after some suitable scalings.

We now briefly explain why the hyperbolic waves is essential for the proof. First of all, in the regime of the vanishing viscosity limit of the compressible Navier–

Stokes equations (1.1) to the compressible Euler equations for the Riemann solution as a superposition of a rarefaction and a shock wave, we will carry out the analysis in the framework of the shock profile, that is, using anti-derivative variables of the perturbation and using the monotonicity of the characteristic speed along the shock profile. This is needed because of the compressibility of the shock profile. By doing so, the error coming from the inviscid approximate rarefaction wave in the setting of viscous system (1.1) is not good enough to get a decay rate with respect to the viscosity because we use the anti-derivative variables. Note that for the stability of the rarefaction wave for the Navier–Stokes equations, one cannot take the anti- derivative because the rarefaction wave is expansive, that is, opposite to the shock.

Therefore, to overcome this difficulty, we introduce “hyperbolic waves” to recover the dissipation terms for the inviscid rarefaction profile. We will also show that the “hyperbolic waves” decay like the first-order derivative of the rarefaction wave profile so that the decay properties given in statement (1) of Lemma2.3are good enough to carry out the analysis. In this way, we circumvent the difficulty mentioned above posed by the inviscid rarefaction wave. On the other hand, we also need to treat the interactions of the hyperbolic wave with both the shock and the rarefaction profiles. To this end, we observe that the hyperbolic wave has the same pointwise estimates as the 1-rarefaction wave away from the wave fan on the right side due to the underlying rarefaction wave structure. This also plays a very important role when dealing with the wave interactions. With these new estimates,

we can verify the vanishing viscosity limit of compressible Navier–Stokes equa- tions to the Riemann solutions of an Euler system containing a rarefaction wave and a shock wave.

We are now ready to formulate the problem. For the Navier–Stokes equations (1.1), formally, asεandκtend to zero, the limit system consists of the following compressible Euler equations

⎧⎪

⎪⎨

⎪⎪

⎩

vt−ux =0, ut+px =0,

e+u²

2 _t+(pu)x =0.

(1.3)

Keeping in mind that the Navier–Stokes equations (1.1) can be derived from the Boltzmann equation through the Chapman–Enskog expansion when the Knudsen number is close to zero, the following assumptions on the viscosity coefficientε and the heat conductivity coefficientκin system (1.1) are natural (see also [10]):

κ =O(ε) asε→0;

ν .= κ(ε)

ε c>0 for some positive constant c, asε→0. (1.4) Note that the eigenvalues of the Jacobi matrix of the Euler system (1.3) are

λ1(v, θ)= − γp

v , λ2≡0, λ3(v, θ)= γp

v . (1.5)

It is known that the first and third characteristic fields of (1.3) are genuinely nonlin- ear and the second characteristic field is linearly degenerate (see [3,15]). Consider the Riemann problem of (1.3) with Riemann initial data

(v,u, θ)(0,x)=

(v−,u₋, θ−),x<0,

(v₊,u₊, θ₊),x>0, (1.6) wherev±, θ±>0 and u_±are given constants. Then, in general, either a shock wave or a rarefaction wave is generated in the genuinely nonlinear fields, which contacts the discontinuity in the linearly degenerate field. In this paper, we consider the case when the Riemann solution of (1.3) contains a rarefaction wave and a shock wave, and without loss of generality, we assume the rarefaction wave is in the first family and the shock wave is in the third family.

Now recall the corresponding 1-rarefaction and 3-shock wave curves in the phase space, as follows.

• 1-Rarefaction wave curve:

R1(v₊,u₊, θ₊):=

(v,u, θ)

v < v₊,s(v, θ)=s₊, u = u₊+

_v+

v λ1(v,s₊)dv

, (1.7)

where s₊=s(v+, θ+), andλ1=λ1(v,s)=λ1(v,s(v, θ))is the first charac- teristic speed given in (1.5).

• 3-Shock wave curve:

It is well known that the Riemann problem (1.3), (1.6) admits a 3-shock wave if and only if the two states(v±,u_±, θ±)satisfy the Rankine–Hugoniot condition

⎧⎨

⎩

−s3(v₊−v₋)−(u₊−u₋)=0,

−s3(u₊−u₋)+(p₊−p₋)=0,

−s3(E₊−E₋)+(p₊u₊−p₋u₋)=0, (1.8) and Lax’s entropy condition

0< λ⁺₃ <s3< λ⁻₃, (1.9) where p_± = p(v_±, θ_±),E_± = _γ−^R₁θ_±+ ^u₂^2± andλ^±₃ = λ3(v_±, θ_±). The shock speed s3is uniquely determined by(v_±,u_±, θ_±)in (1.8). Denote the above 3-shock curve starting from(v+,u₊, θ+)by S3(v+,u₊, θ+).

Use (v−,u₋, θ−) ∈ R1(S3(v+,u₊, θ+)) to denote the case in which there exists a unique state (v∗,u_∗, θ∗) such that (v−,u₋, θ−) ∈ R1(v∗,u_∗, θ∗) and (v∗,u_∗, θ∗)∈S3(v+,u₊, θ+). Then in this case, the wave pattern(V¯,U¯,)(¯ t,x) consisting of a 1-rarefaction wave and a 3-shock wave that solves the corresponding Riemann problem of the Euler system (1.3) can be defined by

V¯ =v^r1 +v^s3−v∗, U¯ =u^r1+u^s3−u_∗, E¯=E^r1 +E^s3−E_∗, (1.10) where E^r1 = _γ−^R₁θ^r1 + ^(ur1₂⁾²,E^s3 = _γ^R₋₁θ^s3 +^(us3₂⁾²,E_∗ = _γ^R₋₁θ_∗+^u₂^2∗, and (v^r1,u^r1, θ^r1)(t,x)is the 1-rarefaction wave defined in (2.1) with the right state (v+,u₊, θ+)=(v∗,u_∗, θ∗), and(v^s3,u^s3, θ^s3)(t,x)is the 3-shock wave defined in (2.11) with the left state(v−,u₋, θ−)=(v∗,u_∗, θ∗).

Consequently, we can define ¯ = γ−1

R

E¯−U¯²

2 . (1.11)

Due to the singularity of the rarefaction wave at t =0, in this paper, we con- sider the problem on the time interval[h,T]for any small fixed h >0 up to any arbitrarily large but fixed time T >0. Investigation of the interaction between the waves and the initial layer is another interesting topic, but will not be discussed in this paper. The main theorem of this paper can be stated as follows.

Theorem 1. Suppose that the Riemann problem (1.3), (1.6) admits a solution (V¯,U¯,)(¯ t,x)defined in (1.10)–(1.11) which is a superposition of a 1-rarefaction wave and a 3-shock wave up to time T . Ifγ ∈ (1,3)and the viscosity and heat conductivity satisfy the relation (1.4), then there exist positive constantsε0andδ0

withδ0independent of h and T , such that ifε∈(0, ε0], and (γ −1)|v+−v∗|δ0,

then for any small h >0, the compressible Navier–Stokes system (1.1) admits a family of smooth solutions{(v^ε,h,u^ε,h, θ^ε,h)(t,x)}satisfying

sup

htT

sup

|x−s3t|h

|(v^ε,h,u^ε,h, θ^ε,h)(t,x)−(V¯,U,¯ )(t,¯ x)|Ch,T ε¹⁵|lnε|.

(1.12) Here, the constant Ch,T is independent ofε, but depends on h and T . Moreover,

(v^ε,h,u^ε,h, θ^ε,h)(t,x)→(V¯,U¯,)(t¯ ,x), a.e.in (0,T)×R, asε→0 and then h→0.

Remark 1. Note that the analysis in this paper can be applied to the case when the Riemann solution is a superposition of one shock and a contact discontinuity. Hence, together with our previous work on the superposition of two rarefaction waves and a contact discontinuity [9], the only Riemann solutions remaining unsolved are two cases: two shocks and one contact discontinuity, and the general case, that is, one shock, one rarefaction wave and a contact discontinuity. These two cases will be pursued by the authors in the future.

The rest of the paper is organized as follows. In Section2, we construct the approximate solution to the compressible Navier–Stokes equations (1.1) corre- sponding to the basic wave patterns to Euler system (1.3). Note that here we intro- duce the hyperbolic wave to recover the viscous terms to the inviscid approximate rarefaction wave. Thus we have detailed information of the difference between the Riemann solution to Euler system and the approximate solution to the Navier–

Stokes system by the construction. In Section3, we look for some exact solution to the compressible Navier–Stokes equations (1.1) around the approximate solu- tion by energy methods and give the proof of the main result. Finally, we give the detailed proof of the estimates of the hyperbolic waves in the Appendix.

Notations. In this paper, we will use the notations c,C,Ci(i = 1,2,3, . . .)to denote generic constants. We use · to denote the standard L2(R;d y)norm, and · Hⁱ (i =1,2,3, . . .)to denote the Sobolev Hⁱ(R)norm. Sometimes, we also use O(1)to denote a uniform bounded constant.

2. Construction of the Approximate Solution 2.1. Approximate Rarefaction Wave

Since there is no exact rarefaction wave profile for the compressible Navier–

Stokes equations, the following approximate rarefaction wave profile satisfying the Euler equations was motivated by Xin [18]. For completeness of presentation, we include its definition and properties in this subsection.

If (v−,u₋, θ−) ∈ R1(v+,u₊, θ+) as defined in (1.7), then there exists a 1-rarefaction wave (v^r1,u^r1, θ^r1)(x/t)which is a global solution of the follow- ing Riemann problem

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

vt−ux =0, ut+px =0,

e+u²

2 _t+(pu)x =0, (v,u, θ)(0,x)=

(v−,u₋, θ−),x<0, (v+,u₊, θ+),x>0.

(2.1)

Consider the following inviscid Burgers equation with Riemann data

⎧⎨

⎩

wt+wwx =0, w(t=0,x)=

w−, x<0, w+, x>0.

Ifw₋< w₊, then the above Riemann problem admits a rarefaction wave solution w^r(t,x)=w^r(x

t)=

⎧⎨

⎩

w₋, ^x_t w₋,

x

t, w₋ ^x_t w₊, w₊, ^x_t w₊.

As in [18], the approximate rarefaction wave(V^R1,U^R1, ^R1)(t,x)to the problem (1.1) can be constructed through the solution of the Burgers equation

wt +wwx =0,

w(0,x)=w_σ(x)=w(x

σ)= w++w−

2 +w+−w−

2 tanh x

σ, (2.2) whereσ >0 is a small parameter to be determined. In fact, we takeσ =ε¹⁵ in the following parts of this paper. Note that the solutionw^r_σ(t,x)of the problem (2.2) is given by

w^r_σ(t,x)=w_σ(x0(t,x)), x=x0(t,x)+w_σ(x0(t,x))t.

Note thatw_σ^r(t,x)has the following properties:

Lemma 2.1. Let w₋ < w₊, then (2.2) has a unique smooth solutionw_σ^r(t,x) satisfying

(1)w₋< w_σ^r(t,x) < w₊, (w^r_σ)x(t,x) >0;

(2) For any p(1p+∞), there exists a constant C such that _∂^∂_xw^r_σ(t,·)L^p(R)C min

(w+−w−)σ^−1+1/p, (w+−w−)^1/pt^−1+1/p , _∂^∂_x²2w^r_σ(t,·)L^p(R)C min

(w₊−w₋)σ^−2+1/p, σ^−1+1/pt⁻¹

; (3) If x−w₋t <0 andw₋>0, then

|w^r_σ(t,x)−w−|(w+−w−)e^−2|x−wσ^−t|,

|_∂^∂_x^kkw^r_σ(t,x)|^2(w+_σ^−wk ⁻⁾e^{−2|x−wσ−t|}, k=1,2;

If x−w₊t >0 andw₊<0, then

|w^r_σ(t,x)−w₊|(w₊−w₋)e^{−2|x−wσ+t|},

|_∂^∂_x^kkw^r_σ(t,x)|^2(w+_σ^−wk ⁻⁾e^{−2|x−w+σ t|}, k=1,2;

(4) sup

x∈R

|w_σ^r(t,x)−w^r(^x_t)|C min

(w₊−w₋),^σ_t[ln(1+t)+ |lnσ|]

.

The proof of statements (1), (2) and (4) can be found in [18], while the proof of statement (3) can be obtained similarly, as in [14].

Then the smooth approximate rarefaction wave profile denoted by(V^R1,U^R1, ^R1)(t,x)can be defined by

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

w±=λ1±:=λ1(v±, θ±),

w^r_σ(t,x)=λ1(V^R1(t,x), ^R1(t,x)), s(V^R1(t,x), ^R1(t,x))=s_±=s(v_±, θ_±), U^R1(t,x)=u₊−

_VR1(t,x) v+

λ1(v,s₊)dv.

(2.3)

Note that(V^R1,U^R1, ^R1)(t,x), defined above, satisfies

⎧⎪

⎨

⎪⎩

V_t^R1−U_x^R1 =0, U_t^R1+P_x^R1 =0, Et^R1 +(P^R1U^R1)x =0,

(2.4)

whereE^R1 =_γ^R₋₁^R1+^(UR1₂⁾² and P^R1 = p(V^R1, ^R1).

By Lemma2.1, the properties on the approximate rarefaction waves(V^R1,U^R1, ^R1)(t,x)can be summarized as follows.

Lemma 2.2. The approximate rarefaction waves (V^R1,U^R1, ^R1)(t,x) con- structed in (2.3) have the following properties:

(1) Ux^R1(t,x) >0 for x∈R,t >0;

(2) For any 1 p+∞,the following estimates hold, (V^R1,U^R1, ^R1)xL^p(d x)C min

δ^R1σ^−1+1/p, (δ^R1)^1/pt^−1+1/p (V^R1,U^R1, ^R1)x xL^p(d x)C min ,

δ^R1σ^−2+1/p, σ^−1+1/pt⁻¹ , whereδ^R1 = |v₊−v₋| is the rarefaction wave strength and the positive constant C depends only on p;

(3) If xλ1+t , then

|(V^R1,U^R1, ^R1)(t,x)−(v₊,u₊, θ₊)|Ce^{−2|x−λ1σ +t|},

|∂_x^k(V^R1,U^R1, ^R1)(t,x)| _σ^Cke^{−2|x−λ1σ+t|}, k=1,2;

(4) There exist positive constants C andσ0such that forσ ∈(0, σ0)and t >0, sup

x∈R

|(V^R1,U^R1, ^R1)(t,x)−(v^r1,u^r1, θ^r1)(x t)|C

t

σln(1+t)+σ|lnσ| .

2.2. Hyperbolic Waves

Since we consider the case of the combination of a rarefaction wave and a shock wave, it is suitable to introduce anti-derivative variables to the perturbations of the solutions near the wave profiles. From (2.4), we know that the approximate rarefaction wave(V^R1,U^R1, ^R1)(t,x)satisfies the compressible Euler equations exactly without viscous terms. Thus if we carry out the energy estimates to the anti-derivative variables, the error terms due to the viscous terms from the approx- imate rarefaction wave are not good enough to get the desired estimates. In order to overcome these difficulties, we introduce a new wave, called a hyperbolic wave.

These hyperbolic waves play a crucial role in our analysis. Now we give a detailed description of these hyperbolic waves. Let the hyperbolic waves(d1,d2,d3)(t,x) satisfy the linear system

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

d1t−d2x =0,

d2t+(p_v^R1d1+p_u^R1d2+p^R_E¹d3)x =ε

Ux^R1

V^R1

x

,

d3t+ [(pu)_v^R1d1+(pu)^Ru¹d2+(pu)^R_E¹d3]x =κ x^R1

V^R1

x

+ε

U^R1Ux^R1

V^R1

x

, (2.5) where p= ^R_v^θ = p(v,u,E)=^(γ−1)(₂^2E_v ^−u2) and p_v^R1 = p_v(V^R1,U^R1,E^R1), etc.

Now we want to solve this linear hyperbolic system (2.5) on the time interval[h,T]. First we diagonalize the above system. Rewrite the system (2.5) as

⎛

⎝d1

d2

d3

⎞

⎠

t

+

⎡

⎣A^R1

⎛

⎝d1

d2

d3

⎞

⎠

⎤

⎦

x

=

⎛

⎝ 0 B1

B2

⎞

⎠,

where B1=ε(^U_V^xR1_R1)x,B2=κ(_V^xR1_R1)x+ε(^UR1_VR1^UxR1)x and the matrix

A^R1 =

⎛

⎝ 0 −1 0 p_v^R1 pu^R1 p_E^R1 (pu)^R_v¹ (pu)u^R1 (pu)^R_E¹

⎞

⎠

with three distinct eigenvaluesλ^R₁¹:=λ1(V^R1,s_±) <0=λ₂^R1< λ3(V^R1,s_±):=λ₃^R1 and the corresponding left and right eigenvectors l^R_j¹,r^R_j¹ (j =1,2,3)satisfying

L^R1A^R1R^R1 =diag(λ^R₁¹,0, λ^R₃¹)≡^R1, L^R1R^R1 =Id.,

Here L^R1 =(l₁^R1,l₂^R1,l₃^R1)^t,R^R1 =(r₁^R1,r₂^R1,r₃^R1)with l_i^R1 =li(V^R1,U^R1,s_±) and r_i^R1 =ri(V^R1,U^R1,s_±) (i =1,2,3)and Id.is the 3×3 identity matrix. Now we set

(D1,D2,D3)^t =L^R1(d1,d2,d3)^t, (2.6)

then

(d1,d2,d3)^t =R^R1(D1,D2,D3)^t, (2.7) and(D1,D2,D3)satisfies the system

⎛

⎝D1

D2

D3

⎞

⎠

t

+

⎡

⎣^R1

⎛

⎝D1

D2

D3

⎞

⎠

⎤

⎦

x

=L^R1

⎛

⎝ 0 B1

B2

⎞

⎠ +L_t^R1R^R1

⎛

⎝D1

D2

D3

⎞

⎠+L_x^R1R^R1R1

⎛

⎝D1

D2

D3

⎞

⎠.

(2.8)

Because the 1-Riemann invariant is constant along the approximate rarefaction wave curve, we have that

L_t^R1 = −λ₁^R1L_x^R1.

Substituting the above equation into (2.8), we obtain the diagonalized system

⎧⎨

⎩

D1t+(λ₁^R1D1)x =b12B1+b13B2+a12V_x^R1D2+a13V_x^R1D3, D2t=b22B1+b23B2+a22V_x^R1D2+a23V_x^R1D3,

D3t+(λ₃^R1D3)x =b32B1+b33B2+a32V_x^R1D2+a33V_x^R1D3,

(2.9)

where ai j,bi j are given functions of V^R1,U^R1 and S^R1 =s₋=s₊. Note that in the diagonalized system (2.9), the equations of D2,D3are decoupled with D1due to the rarefaction wave structure of the system.

Now we impose the following conditions to the above linear hyperbolic system (2.9) on the domain(t,x)∈ [h,T] ×R:

D1(t=h,x)=0, D2(t =T,x)=D3(t =T,x)=0. (2.10) Now we can solve the linear diagonalized hyperbolic system (2.9) under the con- ditions (2.10). Moreover, we have the following estimates:

Lemma 2.3. There exist positive constant Ch,T independent ofε, such that (1)

∂^k

∂x^kdi(t,·) ²

L²(R)Ch,T ε²

σ^2k+1, i =1,2,3, k=0,1,2,3.

(2) If x> λ1+t , then we have

|di(x,t)|Ch,T

1

σe^{−|x−λ1σ+t|},

|di x(x,t)|Ch,T

1 σ²e^−|

x−λ1+t|

σ , i =1,2,3. The proof of Lemma2.3will be shown in the Appendix.

2.3. Viscous Shock Wave

If (v₋,u₋, θ₋) ∈ S3(v₊,u₊, θ₊), then the Riemann problem (1.3), (1.6) admits a 3-shock wave

(v^s3,u^s3, θ^s3)(t,x)=

(v₋,u₋, θ₋), x<s3t,

(v₊,u₊, θ₊), x>s3t. (2.11)

It is well known that the compressible Navier–Stokes system (1.1) admits a smooth traveling wave solution with shock profile (V^S3,U^S3, ^S3)(x −s3t) under the conditions (1.8) and (1.9) (see [15]).

We first recall some properties of the viscous 3-shock wave. The shock profile (V^S3,U^S3, ^S3)(ξ), ξ=x−s3t , is determined by

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

−s3(V^S3)−(U^S3)=0,

−s3(U^S3)+(P^S3)=ε((U^S3) V^S3 ),

−s3(E^S3)+(P^S3U^S3)=κ((^S3)

V^S3 )+ε(U^S3(U^S3) V^S3 ),

V^S3,U^S3, ^S3

(±∞)=(v±,u_±, θ±),

(2.12)

where = _d^d_ξ,P^S3 = p(V^S3, ^S3),E^S3 = _γ^R₋₁^S3 +^(US3₂⁾² and(v_±,u_±, θ_±) satisfy R-H condition (1.8) and Lax entropy condition (1.9), and s3is uniquely determined by (1.8). Integrating (2.12) on(±∞, ξ)gives

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩ s3εV_ξ^S3

V^S3 = −

P^S3+s₃²

V^S3−b1

s₃² , κ^S_ξ³

s3V^S3 = −

⎡

⎣E^S3−s²₃ 2

V^S3−b1

s₃² 2

+ b²₁ 2s²₃ −b2

⎤

⎦, U^S3 = −(s3V^S3 +a),

(2.13)

where a = −(s3v_±+u_±),b1 = p_± +s₃²v_±, b2 = e_±+ p_±v_±+s₃^2v

±2

2 and p_±= p(v_±, θ_±). From [4,11], we have the following Lemma:

Lemma 2.4. Assume that the two states (v±,u_±, θ±) satisfy R-H condition (1.8) and Lax entropy condition (1.9); then there exists a unique shock profile (V^S3,U^S3, ^S3)(ξ), up to a shift, of the ODE system (2.12). Moreover, there are positive constants C and c independent ofγ >1 such that

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

s3V_ξ^S3 = −U_ξ^S3 >0, _ξ^S3 <0,

|V^S3−v_±|,|U^S3 −u_±|,|^S3 −θ±|

γ−1 Cδ^S3e^{−cδS3ε|ξ|}, as ξ → ±∞,

V_ξ^S3,U_ξ^S3, _ξ^S3 γ−1

C(δ^S3)²

ε e^{−cδS3ε|ξ|}, as ξ → ±∞,

_ξ^S3 V_ξ^S3

C(γ −1), s₃²=γRθ₋

v+v−(1−q₊), θ₊=θ₋

1−v₊+v₋ v+ q₊ ,

(2.14)

whereδ^S3 = O(1)|v₊−v₋|is the strength of the 3-shock wave and q₊= ₁₊^m_m⁺₊ with m₊= ^(γ−_2v¹₊^)δS3.

2.4. Superposition of Rarefaction Wave and Shock Wave

The approximate wave profile(V,U, )(t,x)with the superposition of the 1-rarefaction and 3-shock waves to the compressible Navier–Stokes equations can be defined by

⎛

⎝V U E

⎞

⎠(t,x)=

⎛

⎝V^R1+d1+V^S3−v∗

U^R1+d2+U^S3−u_∗ E^R1+d3+E^S3−E_∗,

⎞

⎠(t,x) (2.15)

where(V^R1,U^R1,E^R1)(t,x)is the approximate 1-rarefaction wave defined in (2.3) with the right state(v+,u₊, θ+)replaced by(v∗,u_∗, θ∗), (d1,d2,d3)(t,x)are the hyperbolic waves defined in (2.5), and(V^S3,U^S3,E^S3)(t,x)is the viscous 3-shock wave defined in (2.14) with the left state(v₋,u₋, θ₋)replaced by(v_∗,u_∗, θ_∗).

From (2.15), we have

=^R1+^S3−θ_∗+γ−1

R d3−γ−1

2R [U²−(U^R1)²−(U^S3)²+u²_∗]. (2.16) Thus, from the construction of the approximate rarefaction wave and Lemmas2.2–2.4, we have the following relation between the approximate wave pattern (V,U,E, )(t,x)of the compressible Navier–Stokes equations and the exact inviscid wave pattern(V¯,U¯,E,¯ )(t,¯ x)to the Euler equations

|(V,U,E, )(t,x)−(V¯,U,¯ E,¯ )(t¯ ,x)|

C

!|(V^R1 −v^r1,U^R1−u^r1,E^R1−E^r1, ^R1−θ^r1)| + |(d1,d2,d3)|

+ |(V^S3 −v^s3,U^S3−u^s3,E^S3 −E^s3, ^S3−θ^s3)|"

C

!1 t

#σln(1+t)+σ|lnσ|$ + ε

σ +δ^S3e^−cδS3|

x−s3t|

ε

"

,

(2.17)

withσ =ε¹⁵.