NAVIER-STOKES EQUATIONS IN THE PRESENCE OF THE SHOCK

http://actams.wipm.ac.cn

ZERO DISSIPATION LIMIT OF THE COMPRESSIBLE HEAT-CONDUCTING

NAVIER-STOKES EQUATIONS IN THE PRESENCE OF THE SHOCK

Wang Yi( )

Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China

E-mail: wangyi@amss.ac.cn

Abstract The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficientε satisfy κ =O(ε), ^κ_ε ≥ c >0, as ε→0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate ofε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].

Key words Zero dissipation limit, Navier–Stokes equations, shock waves 2000 MR Subject Classification 76N99, 35L67, 74J40

1 Introduction

We study the zero dissipation limit of the solution of the full Navier–Stokes system which reads in the Lagrangian coordinates

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

vt−ux= 0, ut+px=ε

ux

v

x,

e+u² 2

t+ (pu)_x=κ θx

v

x+ε uux

v

x,

(1.1)

where the functions v(x, t) > 0, u(x, t), and θ(x, t) > 0 represent the specific volume, the velocity, and the absolute temperature of the gas, respectively. Andp=p(v, θ) is the pressure,

∗Received December 30, 2006. Supported by the Knowledge Innovation Program of the Chinese Academy of Sciences

e=e(v, θ) is the internal energy,ε >0 is the viscosity constant andκ >0 is the coefficient of the heat conduction. Here we consider the perfect gas case, that is,

p= Rθ

v , e= Rθ

γ−1 + const, (1.2)

whereR >0 is the gas constant andγ >1 is the adiabatic exponent.

The study of the vanishing viscosity limit of viscous flow is an important problem in the theory of the compressible fluid flow. When the solution of the inviscid flow is smooth, the zero dissipation limit can be investigated through classical scaling method. However, the inviscid compressible solution is, in general, discontinuous like shock wave and its investigation is much more complicated. For the hyperbolic conservation laws with artificial viscosity

ut+f(u)_x=εuxx,

Goodman and Xin [3] proposed a matched asymptotic expansion method to study the zero dissipation limit in the presence of shock wave. For the vanishing viscosity limit of shock wave of the isentropic Navier–Stokes equation, whose viscosity matrix is only semi-positive definite, Hoff and Liu [4] first considered the case of single shock wave, where the initial data are discontinuous so that both initial layer and shock layer are included in the solution. Recently, Wang [15] applied the idea of [3] to study the isentropic Navier–Stokes equations in the case of a finite number of shock waves, whereas the initial value of the Navier–Stokes equation is well chosen so that there is no initial layer.

In this article, we consider the vanishing viscosity limit of shock wave for the full Navier–

Stokes equations (1.1). Motivated by [3], we first construct an approximate solution to the inviscid shock wave by matching the inner and outer expansion solutions and the higher order correction, then we use the scaling method to transform the zero dissipation limiting to the time asymptotic stability of the approximate solution. Note that the leading order of the inner solution in the approximate solution is the shock profile for the Navier–Stokes equations (1.1).

So we can use the energy method in [7] to show that, for a given piecewise smooth shock solution of the Euler system, there exists a smooth approximate solution for the Navier–Stokes system (1.1) such that it converges to the piecewise smooth shock solution away from the shock discontinuity at a rate ofεas εtends to zero. The precise statement of our main result is the following.

As in [6], where the vanishing viscosity limit of rarefaction wave was investigated, we

assume ⎧

⎨

⎩

κ=O(ε) as ε→0;

μ=. κ

ε ≥c >0 for some positive constantc, as ε→0. (1.3) The assumption (1.3) is reasonable because, when we consider the compressible Navier–Stokes system (1.1) as a derivation from the Bolzmann equation through Chapman–Enskog expansion, the heat conducting coefficientκand the viscosity coefficientεsatisfy (1.3).

Denote the total energy functionE byE= _γ−1^Rθ +^u₂², we have θ=γ−1

R

E−u² 2

, (1.4)

and

p=p(v, u, E) = (γ−1)E

v −(γ−1)u²

2v . (1.5)

Then, the system (1.1) can be written as

⎛

⎜⎜

⎝ v u E

⎞

⎟⎟

⎠

t

+A

⎛

⎜⎜

⎝ v u E

⎞

⎟⎟

⎠

x

=ε

⎡

⎢⎢

⎣B

⎛

⎜⎜

⎝ v u E

⎞

⎟⎟

⎠

x

⎤

⎥⎥

⎦

x

, (1.1)

where

A=

⎛

⎜⎜

⎜⎜

⎝

0 −1 0

−p

v −(γ−1)u v

γ−1 v

−pu

v p−(γ−1)u² v

(γ−1)u v

⎞

⎟⎟

⎟⎟

⎠, B =

⎛

⎜⎜

⎜⎜

⎝

0 0 0

0 1

v 0

0 (1−ν)u v

ν v

⎞

⎟⎟

⎟⎟

⎠,

andν =^κ(γ−1)_εR =^(γ−1)μ_R , whereμis defined in (1.3).

By a direct computation, the eigenvalues of the matrixAare λ1=−

γp

v , λ2= 0, λ3= γp

v . (1.6)

The corresponding Euler equation of the system (1.1) or (1.1) is

⎧⎪

⎪⎨

⎪⎪

⎩

vt−ux= 0, u_t+p_x= 0, Et+ (pu)_x= 0.

(1.7)

It is well known that the first and the third characteristic field of (1.7) are genuinely nonlinear and the second field is linearly degenerate (see [13]).

When we consider the initial shock discontinuity, i.e.,

(v, u, E)(x, t= 0) = (v⁰, u⁰, E⁰)(x), (1.8) the system (1.7) admits a single shock solution (v0, u0, E0)(x, t) (we assume it is 3-shock, without loss of generality) and the following four items are satisfied:

(1) (v0, u0, E0)(x, t) is a distribution solution of (1.7), (1.8).

(2) There exists a smooth shock curve x=s(t), 0 ≤t ≤T, such that (v0, u0, E0)(x, t) is smooth up to the curve x = s(t), and the left and right limits of (v0, u0, E0)(x, t) and its derivatives atx=s(t) exist.

(3) Across the shockx=s(t), the Rankine-Hugoniot condition holds:

⎧⎪

⎪⎨

⎪⎪

⎩

s˙(v₀⁻−v⁺₀) =u⁺₀ −u⁻₀, s˙(u⁻₀ −u⁺₀) =p⁻₀ −p⁺₀, s˙(E₀⁻−E₀⁺) =p⁻₀u⁻₀ −p⁺₀u⁺₀.

(1.9)

(4) Lax entropy condition holds:

0< λ⁺₃ <s < λ˙ ⁻₃. (1.10) In (1.9) and (1.10), ˙s=_dt^ds(t) andv⁻₀ =v0(s(t)−0, t), v₀⁺=v0(s(t) + 0, t), etc.

We also assume that there exist two positive constantsc∗andc^∗such that the initial data (1.8) satisfy

0< c∗< v⁰(x), θ⁰(x)< c^∗, (1.11) whereθ⁰(x) =^γ−1_R (E⁰−^(u0₂⁾²). Then we have

Theorem 1.1 Suppose that (1.7), (1.8) admit a 3-shock solution (v0, u0, E0)(x, t) up to timeT satisfying

7 k=1

_T

0

x=s(t)|∂_x^k(v0, u0, E0)(x, t)|²dxdt <∞. (1.12) Letγ∈(1,2], then there exist positive constantsε0 andε1 such that, ifε∈(0, ε0] and

(γ−1)|v⁺₀(t)−v⁻₀(t)|< ε1, ∀t∈[0, T], (1.13) the system (1.1) or (1.1) under the assumption (1.3) admits a unique smooth solution (v^ε, u^ε, E^ε) (x, t) satisfying

sup

0≤t≤T

|(v^ε, u^ε, E^ε)(x, t)−(v0, u0, E0)(x, t)|²dx≤Cε^α, ∀ α∈(2

3,1), (1.14) and

0≤t≤Tsup sup

|x−s(t)|≥h|(v^ε, u^ε, E^ε)(x, t)−(v0, u0, E0)(x, t)| ≤Chε, ∀h >0, (1.15) where the constantsC andC_hare independent ofε, butC_h depends onh.

Remark 1.1 The convergence rateεin (1.15) is optimal and Theorem 1 is also valid for the case that the system (1.7) has a finite number of noninteracting piecewise smooth shock waves.

Notations In this article, we use the notationsc, C to represent the generic constants which are independent of the variablesε, x, andt, except for additional explanations. And we use · to denote the usual L2(R) norm, and · _Hⁱ to denote the norm of Sobolev space Hⁱ(R) (i = 1,2,3,· · ·). We use O(1) to denote the uniform bounded constant and o(1) to denote the infinitesimal small quantity in some limit process.

2 Approximate Solutions

Motivated by [3], we use the matched asymptotic expansion method to construct the approximate solutions to the inviscid piecewise smooth solution (v0, u0, E0)(x, t).

2.1 Outer and inner expansion and the matching condition Away from the 3-shock locationx=s(t), we give an outer expansion

(v^ε, u^ε, E^ε)(x, t)∼(v0, u0, E0)(x, t) +ε(v1, u1, E1)(x, t) +ε²(v2, u2, E2)(x, t) +· · ·. (2.1)

Substituting (2.1) into (1.1) and equating the coefficients of powerεyield

O(1) :

⎧⎪

⎪⎨

⎪⎪

⎩

v0t−u0x= 0, u0t+p0x= 0, E0t+ (p0u0)_x= 0;

(2.2)

O(ε) :

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

v1t−u1x= 0,

u1t+ [∇p0·(v1, u1, E1)]_x= u0x

v0

x, E1t+ [∇(p0u0)·(v1, u1, E1)]_x=ν

E0x

v0

x+ (1−ν) u0u0x

v0

x;

(2.3)

O(ε²) :

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

v2t−u2x= 0,

u2t+ [∇p0·(v2, u2, E2)]_x= u1x

v0 −u0x

v²₀ v1

x−1

2[∇²p0·(v1, u1, E1)²]_x, E2t+ [∇(p0u0)·(v2, u2, E2)]_x=ν

E1x

v0 −E0x

v₀² v1

x

+ (1−ν)

(u0u1)_x

v0 −u0u0x

v²₀ v1

x−1

2[∇²(p0u0)·(v1, u1, E1)²]_x;

(2.4)

· · ·.

In the above expressions, we have used the notations p0=p(v0, u0, E0),∇p0=∇p(v0, u0, E0), etc.

Near the 3-shock locationx=s(t), we give an inner expansion by

(v^ε, u^ε, E^ε)(x, t)∼(V0, U0,E0)(ξ, t) +ε(V1, U1,E1)(ξ, t) +ε²(V2, U2,E2)(ξ, t) +· · ·, (2.5) whereξis the scaled variable defined by

ξ=x−s(t)

ε +δ(t, ε), (2.6)

whereδ(t, ε) has an expansion

δ(t, ε) =δ0(t) +εδ1(t) +ε²δ2(t) +· · ·. (2.7) Substituting (2.5)–(2.7) into (1.1), we have

O(ε⁻¹) :

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

−sV˙ 0ξ−U0ξ= 0,

−sU˙ 0ξ+P0ξ= U0ξ

V0

ξ,

−sE˙ _0ξ+ (P0U0)_ξ=ν E_0ξ

V0

ξ+ (1−ν) U0U0ξ

V0

ξ;

(2.8)

O(1) :

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

−sV˙ 1ξ−U1ξ =−V_0t−δ˙0V0ξ,

−sU˙ 1ξ+ [∇P0·(V1, U1,E1)]_ξ = U1ξ

V0 −U0ξ

V₀²V1

ξ−U0t−δ˙0U0ξ,

−sE˙ _1ξ+ [∇(P0U0)·(V1, U1,E1)]_ξ =ν E_1ξ

V0 −E_0ξ V₀²V1

ξ

+(1−ν)

(U0U1)_ξ

V0 −U0U0ξ

V₀² V1

ξ− E0t−δ˙0E0ξ;

(2.9)

O(ε) :

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

−sV˙ 2ξ−U2ξ=−V_1t−δ˙1V0ξ−δ˙0V1ξ,

−sU˙ 2ξ+ [∇P0·(V2, U2,E2)]_ξ = U2ξ

V0

+U0ξ

V₀³V₁²−U0ξ

V₀²V2−U1ξ

V₀²V1

ξ

−U_1t−δ˙0U1ξ−δ˙1U0ξ−1

2[∇²P0·(V1, U1,E1))²]_ξ,

−sE˙ 2ξ+ [∇(P0U0)·(V2, U2,E2)]_ξ=ν E2ξ

V0

+E0ξ

V₀³V₁²−E0ξ

V₀²V2−E1ξ

V₀²V1

ξ

+(1−ν)

U1U1ξ+ (U0U2)_ξ

V0 −(U0U1)_ξ

V₀² V1−U0U0ξ

V₀² V2+U0U0ξ

V₀³ V₁²

ξ

−E1t−δ˙0E1ξ−δ˙1E0ξ−1

2[∇²(P0U0)·(V1, U1,E1)²]_ξ;

(2.10)

· · ·,

where we have used the notationsP0=p(V0, U0,E0),∇P0= (∇p)(V0, U0,E0), etc.

The above inner expansion is supposed to be valid in a small zone of size O(ε) around x=s(t).

The outer expansion and the inner expansion are expected to be valid in the matching zone, where both|ξ| → ∞and|x−s(t)| small are true. Therefore, they must agree with each other there. Expressing the outer expansion solution in terms of ξ and using Taylor series to equate the coefficients of powerεin outer and inner expansions, we get the following matching condition asξ→ ±∞,

(V0, U0,E0)(ξ, t) = (v0, u0, E0)(s(t)±0, t) +o(1),

(V1, U1,E1)(ξ, t) = (v1, u1, E1)(s(t)±0, t) + (ξ−δ0)(v0x, u0x, E0x)(s(t)±0, t) +o(1), (V2, U2,E2)(ξ, t) = (v0, u0, E0)(s(t)±0, t) + (ξ−δ0)(v1x, u1x, E1x)(s(t)±0, t)

−δ1(v0x, u0x, E0x)(s(t)±0, t) +1

2(ξ−δ0)²(v0xx, u0xx, E0xx)(s(t)±0, t) +o(1),

(2.11)

etc.

We note that the matching condition (2.11) requires that the inner functions (Vi, Ui,E_i)(ξ, t) have algebraic growth rate|ξ|ⁱ (i= 0,1,2,· · ·) asξ→ ±∞.

2.2 Viscous shock profiles

This section is devoted to the traveling wave solution of (1.1), which has the form (v, u, E)(x, t) = (V, U,E)(ξ, t), ξ= x−s(t)

ε ,

which satisfies ⎧

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎩

−sV˙ −U= 0,

−sU˙ +P = (U V ),

−sE˙ + (P U) =ν E

V

+ (1−ν) U U

V

,

(2.12)

and

ξ→±∞lim (V, U,E)(ξ, t) = (v_±, u_±, E_±)(t), (2.13) where ˙s= _dt^ds(t), = _dξ^d, P =p(V, U,E) andv±(t)>0, u±(t), θ±(t) = ^γ−1_R (E±−^u₂^2±)(t)>0,

s(t) are the functions oft and satisfy the Rankine–Hugoniot condition

⎧⎪

⎪⎨

⎪⎪

⎩

s˙(v+−v−) =−(u+−u−), s˙(u+−u−) =p+−p−, s˙(E+−E−) =p+u+−p−u−.

(2.14)

For 3-shock wave, Lax entropy condition holds:

0< λ⁺₃ <s < λ˙ ⁻₃, or equivalently,

s˙(t)>0, v+(t)> v−(t), u+(t)< u−(t), ∀ t∈[0, T]. (2.15) Integrating (2.12) over [ξ,±∞) gives

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

−sV˙ −U =−( ˙sv±+u±),

−sU˙ +P = U

V + (−su˙ ±+p±),

−sE˙ +P U =νE

V + (1−ν)U U

V + (−sE˙ ±+p±u±).

(2.16)

Thus, we have ⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩ sV˙

V =−

P+ ˙s²(V −b1

s˙²)

, μΘ

sV˙ =− R

γ−1Θ−s˙²

2(V −b1

s˙²)²+ b²₁ 2 ˙s² −b2

, U =−sV˙ + ( ˙sv±+u±),

(2.17)

where Θ = _γ−1^R (E − ^U₂²), b1=p±+ ˙s²v±, and b2=_γ−1^R θ±+p±v±+^s˙2₂^v±2.

Regarding (v−, u−, E−) and ˙sas parameters, we denote the traveling wave solution (V, U,E) (ξ, t) and (v+, u+, E+) by

(V, U,E)(ξ, t) = (V, U,E)(ξ;v−, u−, E−,s˙), (v+, u+, E+)(t) = (v+, u+, E+)(v−, u−, E−,s˙). Similar to [7], the following lemma holds.

Lemma 2.1 There exists a traveling wave solution (V, U,E)(ξ, t), unique up to a shift, solving (2.12), (2.13) with the properties:

s >˙ 0, Vξ>0, Uξ =−sV˙ ξ <0, Θ_ξ<0, Θ_ξ Vξ

≤C(γ−1);

|(V, U,Θ)−(v±, u±, θ±)| ≤Ce^−c|ξ|,

|Vξ|,|Vξξ|,|Θ_ξξ| ≤C|v+−v−|e^−c|ξ|, |Θ_ξ| ≤C(γ−1)|v+−v−|e^−c|ξ|, as ξ→ ±∞;

∂(V, U,E)

∂(v−, u−, E−)−I3×3=O(1)e^−c|ξ|, ∂(V, U,E)

∂s˙ =O(1)e^−c|ξ|, asξ→ −∞;

∂(V, U,E)

∂(v−, u−, E−)−∂(v+, u+, E+)

∂(v−, u−, E−) =O(1)e^−c|ξ|, ∂(V, U,E)

∂s˙ −∂(v+, u+, E+)

∂s˙ =O(1)e^−c|ξ|,

asξ→+∞;

whereI3×3 represents the identity matrix of the third order.

2.3 The outer and the inner expansion solutions

Let (v0, u0, E0)(x, t) be the 3-shock solution described in Theorem 1 and (V0, U0,E0)(ξ, t) be the traveling wave solution (V, U,E)(ξ, t) in Lemma 2.1 with the end states

(v±, u±, E±)(t) = (v0, u0, E0)(s(t)±0, t).

Because the shift in Lemma 2.1 can be absorbed inδ(t, ε), we let it be zero.

We shall construct the next order terms (v1, u1, E1)(x, t), (V1, U1,E1)(ξ, t) and δ0(t) by studying the linear hyperbolic system (2.3), where the coefficients have jump across the shock curve x= s(t), and the ODE system (2.9) of ξ. The boundary values of (2.3) on the shock location x=s(t) and (2.9) on ξ = ±∞are related to the matching condition (2.11)₂. Once the boundary values for the systems (2.3) and (2.9) are known and the initial value for (2.3) is given, the corresponding solutions are immediately obtained. So the remaining problem is how to determine the boundary values for (2.3) and (2.9) by the previous terms (v0, u0, E0)(x, t), (V0, U0,E0)(ξ, t) and the matching condition (2.11)₂.

We first consider the system (2.9). Since (V1, U1,E1)(ξ, t) tends to infinity with order of O(|ξ|) as|ξ| → ∞due to (2.11)₂, we introduce a new variable ( ˜V1,U˜1,E˜1)(ξ, t) by

(V1, U1,E1)(ξ, t) = ( ˜V1,U˜1,E˜1)(ξ, t) + (D1, D2, D3)(ξ, t), where

(D1, D2, D3)(ξ, t) =ξ(v0x, u0x, E0x)(s(t)±0, t), if ±ξ≥1, so that

( ˜V1,U˜1,E˜1)(ξ, t) = (v1, u1, E1)(s(t)±0, t)−δ0(t)(v0x, u0x, E0x)(s(t)±0, t)

+o(1) as ξ→ ±∞, (2.18)

and ⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

−s˙V˜_1ξ−U˜1ξ=−δ˙0V0ξ+h1(ξ, t),

−s˙U˜1ξ+ [∇P0·( ˜V1,U˜1,E˜1)]_ξ= U˜1ξ

V0 −U0ξ

V₀²V˜1

ξ−δ˙0U0ξ+h2(ξ, t),

−s˙E˜1ξ+ [∇(P0U0)·( ˜V1,U˜1,E˜1)]_ξ =ν E˜1ξ

V0 −E0ξ

V₀²V˜1

ξ

+(1−ν)

(U0U˜1)_ξ

V0 −U0U0ξ

V₀² V˜1

ξ−δ˙0E0ξ+h3(ξ, t),

(2.19)

where

h1(ξ, t) = ˙sD1ξ+D2ξ−V0t

= ˙sv0x(s(t)±0, t) +u0x(s(t)±0, t)−[v0(s(t)±0, t)]_t

−[V0−v0(s(t)±0, t)]_t=O(1)e^−c|ξ|, as|ξ| → ∞, h2(ξ, t) = ˙sD2ξ−[∇P0·(D1, D2, D3)]_ξ−U0t+

D2ξ

V0 −U0ξ

V₀²D1

ξ

= ˙su0x(s(t)±0, t)− ∇p0·(v0x, u0x, E0x)(s(t)±0, t)

−[u0(s(t)±0, t)]_t+O(1)e^−c|ξ|

=O(1)e^−c|ξ|, as|ξ| → ∞,

h3(ξ, t) = ˙sD3ξ−[∇(P0U0)·(D1, D2, D3)]_ξ− E0t+ν D3ξ

V0 −U0ξ

V₀²D1

ξ

+(1−ν)(U0D2)_ξ

V0 −U0U0ξ

V₀² D1

ξ =O(1)e^−c|ξ|, as|ξ| → ∞.

Here we have used (2.17) and Lemma 2.1.

Integrating the system (2.19) over [0, ξ] yields

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

−s˙V˜1−U˜1=−δ˙0V0+H1(ξ, t) +C1(t),

−s˙U˜1+∇P0·( ˜V1,U˜1,E˜1) =U˜1ξ

V0 −U0ξ

V₀²V˜1−δ˙0U0+H2(ξ, t) +C2(t),

−s˙E˜1+∇(P0U0)·( ˜V1,U˜1,E˜1) =ν E˜1ξ

V0 −E0ξ

V₀²V˜1

+(1−ν)

(U0U˜1)_ξ

V0 −U0U0ξ

V₀² V˜1

−δ˙0E0+H3(ξ, t) +C3(t),

(2.20)

where Hi(ξ, t) = _ξ

0 hi(ζ, t)dζ, i = 1,2,3, and the integral constants Ci(t) (i = 1,2,3) will be determined later.

Let (˜v₁^±,u˜^±₁,E˜₁^±) = lim

ξ→±∞( ˜V1,U˜1,E˜1)(ξ, t). From (2.18), lim

ξ→±∞( ˜V1ξ,U˜1ξ,E˜_1ξ)(ξ, t) = (0,0,0).

Then, letting ξ→ ±∞in (2.20) implies

M^±

⎛

⎜⎜

⎝

˜v^±₁ u˜^±₁ E˜₁^±

⎞

⎟⎟

⎠=

⎛

⎜⎜

⎝

−δ˙0v₀^±+H₁^±(t) +C1(t)

−δ˙0u^±₀ +H₂^±(t) +C2(t)

−δ˙0E₀^±+H₃^±(t) +C3(t)

⎞

⎟⎟

⎠, (2.21)

where

M^±=

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

−s˙ −1 0

−p^±₀

v^±₀ −(γ−1)u^±₀

v₀^± −s˙ γ−1 v^±₀

−p^±₀u^±₀

v^±₀ p^±₀ −(γ−1)(u^±₀)² v₀^±

(γ−1)u^±₀ v₀^± −s˙

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

, (2.22)

H₁^±(t) =_±∞

0 h1(ζ, t)dζ, and v^±₀ = lim

ξ→±∞V0(ξ, t) =v0(s(t)±0, t)), etc. It is straight forward to compute

detM^±= ˙s(−s˙²+γp^±₀ v^±₀ )= 0,

due to the Lax entropy condition (1.10). There are four unknown quantities ˙δ0 and Ci(t), i= 1,2,3, in the formula (2.21).

On the other hand, from (2.18), (˜v^±₁,u˜^±₁,E˜^±₁) can be expressed by (v^±₁, u^±₁, E₁^±) = (v1, u1, E1) (s(t)±0, t), i.e., ⎧

⎪⎪

⎨

⎪⎪

⎩

v˜₁^±=v₁^±−δ0v_0x^±, u˜^±₁ =u^±₁ −δ0u^±_0x, E˜₁^±=E₁^±−δ0E_0x^±,

(2.23)

where, and in the sequel,v_0x^± =v0x(s(t)±0, t), etc. We turn to the linear hyperbolic system (2.3) for (v1, u1, E1). Because the coefficients of the system (2.3) have jump across the shock curve

x=s(t), we have to study it in Ω₊={(x, t);x > s(t), t≥0}and Ω₋={(x, t);x < s(t), t≥0}, respectively.

Let’s compute the Jacobi matrixA0 of (2.3). We have

A0=

⎛

⎜⎜

⎜⎜

⎜⎝

0 −1 0

−p0

v0 −(γ−1)u0

v0

γ−1 v0

−p0u0

v0 p0−(γ−1)u²₀ v0

(γ−1)u0

v0

⎞

⎟⎟

⎟⎟

⎟⎠. (2.24)

The eigenvalues ofA0 onx=s(t)±0 are λ^±₁ =−

γp⁺₀

v^±₀ , λ^±₂ = 0, λ^±₃ =

γp^±₀

v₀^± . (2.25)

When we consider (2.3) in the region Ω₊. From the Lax entropy condition (1.10), all eigenvalues λ⁺_i , i = 1,2,3 are less than the slope ˙s of the boundary x = s(t). That is all characteristics are incoming into the boundary and it is not necessary to impose the boundary values onx=s(t). So (v₁⁺, u⁺₁, E₁⁺) can be uniquely determined by the initial data.

When we consider the region Ω₋, the situation is subtle. The third wave is incoming on the boundary since λ⁻₃ > s˙. However, the eigenvalues λ⁻₁, λ⁻₂ are less than ˙s so that the corresponding waves are outgoing on the boundary. Let β_j⁻ = l⁻_j(v₁⁻, u⁻₁, E₁⁻)^t, i = 1,2,3, wherel⁻_j are the left eigenvectors ofA0(v₀⁻, u⁻₀, E₀⁻).From (2.21) and (2.23), we have

(β₁⁻, β₂⁻) = (G1, G2)(t)β₃⁻+ (G3, G4)(t), (2.26) and

δ˙0+G5(t)δ0=G6(t)β₃⁻+G7(t), (2.27) whereGi(t), i= 1,· · ·,7, are smooth known functions. The theory of linear hyperbolic system [9], [12] implies that the problem (2.3) and (2.26) in the domain Ω₋has a solution smooth up to the shock locationx=s(t) if the initial data are chosen to satisfy the compatibility conditions atx=s(0). Thus, we solve the hyperbolic equation (2.3) for (v1, u1, E1)(x, t). Then, we solve the ordinary differential equation (2.27) with δ0(0) = 0 to get δ0(t) in terms of the incoming waveβ₃⁻. Again using (2.21), we obtain the integral constantsCi(t), i= 1,2,3. Therefore, we get ( ˜V1,U˜1,E˜1)(ξ, t) from (2.20) and finally obtain (V1, U1,E1)(ξ, t). In summary, we have

Lemma 2.2 The functionsδ0(t),(v1, u1, E1)(x, t),(V1, U1,E1)(ξ, t) constructed above sat- isfy

(1) (v1, u1, E1)(x, t) and its derivatives are uniformly continuous up to the shockx=s(t) and

5 k=0

_T

0

x=s(t)|∂_x^k(v1, u1, E1)(x, t)|²dxdt <∞, (2.28) (2) (V1, U1,E1)(ξ, t) andδ0(t) are smooth functions, and there is a positive constantc >0 such that

(V1, U1,E1)(ξ, t) = (v1, u1, E1)(s(t)±0, t) + (ξ−δ0))(v0x, u0x, E0x)(s(t)±0, t) +O(1)e^−c|ξ|,

as ξ→ ±∞.

Similarly, the above procedure can be carried out to any higher order terms, in particu- larδ1(t),(v2, u2, E2)(x, t),(V2, U2,E2)(ξ, t) andδ2(t),(v3, u3, E3)(x, t),(V3, U3,E3)(ξ, t), with the similar estimates as in Lemma 2.2.

2.4 Construction of approximate solutions

With the outer and inner solutions determined in the previous section, we now construct the approximate solutions. Letm(z)∈C₀^∞(R) satisfying 0≤m(z)≤1 and

m(z) =

⎧⎨

⎩

1 |y| ≤1, 0 |y| ≥2. Choosingα∈(²₃,1), we construct the approximate solutions as

⎛

⎜⎜

⎝

¯v u¯ E¯

⎞

⎟⎟

⎠(x, t) =m(z)

⎛

⎜⎜

⎝ Vin

Uin

Ein

⎞

⎟⎟

⎠(x, t) + (1−m(z))

⎛

⎜⎜

⎝ vout

uout

Eout

⎞

⎟⎟

⎠(x, t) +

⎛

⎜⎜

⎝ d1

d2

d3

⎞

⎟⎟

⎠(x, t), (2.29)

where

z=x−s(t) ε^α ,

⎛

⎜⎜

⎝ Vin

Uin

Ein

⎞

⎟⎟

⎠(x, t) =

⎛

⎜⎜

⎝

V0+εV1+ε²V2+ε³V3

U0+εU1+ε²U2+ε³U3

E0+εE1+ε²E2+ε³E3

⎞

⎟⎟

⎠

x−s(t)

ε +δ0+εδ1+ε²δ2, t

, (2.30)

⎛

⎜⎜

⎝ vout

uout

Eout

⎞

⎟⎟

⎠(x, t) =

⎛

⎜⎜

⎝

v0+εv1+ε²v2+ε³v3

u0+εu1+ε²u2+ε³u3

E0+εE1+ε²E2+ε³E3

⎞

⎟⎟

⎠(x, t), (2.31)

anddi(x, t) (i= 1,2,3) are the higher order correction terms to be determined later.

The approximate solution (¯v,u,¯ E¯)(x, t) defined in (2.29) satisfies the following equations:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

v¯t−u¯x=q1(x, t) +d1t−d2x, u¯t+ ¯px=ε(u¯x

v¯ )_x+ 5

i=2

qi(x, t) +d2t−ε( d2x

¯v−d1

)_x, E¯t+ (¯pu¯)_x=εν(E¯x

v¯ )_x+ε(1−ν)(u¯u¯x

¯v )_x+ 9 i=6

qi(x, t) +d3t−εν( d3x

¯v−d1

)_x,

(2.32)

where

q1(x, t) =mt(Vin−vout)−mx(Uin−uout) +m((Vin)_t−(Uin)_x), (2.33) q2(x, t) =mt(Uin−uout)−εmx

(Uin)_x

Vin −(uout)_x vout

+{[p(¯v−d1,u¯−d2,E¯−d3)]_x−m(Pin)_x−(1−m)(pout)_x}, (2.34) q3(x, t) =m

(Pin−χ(Pin))_x−ε((Uin)_x

Vin −χ((Uin)_x Vin

))_x

+mε³(U3t+ ˙δ0U3ξ+ ˙δ1U2ξ+ ˙δ2U1ξ+εδ˙1U3ξ+εδ˙2U2ξ+ε²δ˙2U3ξ), (2.35)

q4(x, t) = (1−m)

(pout−χ(pout))_x−ε((uout)_x

vout −χ((uout)_x vout

))_x

, (2.36) q5(x, t) =−εu¯x

¯v −m(Uin)_x

Vin −(1−m)(uout)_x vout −d2x

v¯

x−ε d2x

¯v − d2x

v¯−d1

x

+[¯p−p(¯v−d1,u¯−d2,E¯−d3)]_x, (2.37) q6(x, t) =mt(Ein−Eout)−ενmx

(Ein)_x

Vin −(Eout)_x vout

−ε(1−ν)mx

Uin(Uin)_x

Vin −uout(uout)_x vout

+{[p(¯v−d1,u¯−d2,E¯−d3)·(¯u−d2)]_x−m(PinUin)_x−(1−m)(poutuout)_x}, (2.38) q7(x, t) =m

(PinUin−χ(PinUin))_x−εν((Ein)_x

Vin −χ((Ein)_x Vin

))_x

−ε(1−ν)(Uin(Uin)_x

Vin −χ(Uin(Uin)_x Vin

))_x

+mε³(E3t+ ˙δ0E3ξ+ ˙δ1E2ξ+ ˙δ2E1ξ+εδ˙1E3ξ+εδ˙2E2ξ+ε²δ˙2E3ξ), (2.39) q8(x, t) = (1−m)

(poutuout−χ(poutuout))_x−εν((Eout)_x

vout −χ((Eout)_x vout

))_x

−ε(1−ν)(uout(uout)_x

vout −χ(uout(uout)_x vout

))_x

, (2.40)

q9(x, t) =−ενE¯x

v¯ −m(Ein)_x

Vin −(1−m)(Eout)_x vout −d3x

¯v

x

−ε(1−ν)u¯u¯x

v¯ −mUin(Uin)_x

Vin −(1−m)uout(uout)_x vout

x

−ενd3x

v¯ − d3x

v¯−d1

x+ [¯pu¯−p(¯v−d1,u¯−d2,E¯−d3)·(¯u−d2)]_x, (2.41) wherePin=p(Vin, Uin,Ein),pout=p(vout, uout, Eout), and

χ(Pin) =P0+ε∇P0·(V1, U1,E1) +ε²∇P0·(V2, U2,E2) +ε³∇P0·(V3, U3,E3) +1

2ε²∇²P0·(V1, U1,E1)²+1

2ε³∇²P0·[(V1, U1,E1),(V2, U2,E2)], χ(pout) =p0+ε∇p0·(v1, u1, E1) +ε²∇p0·(v2, u2, E2) +ε³∇p0·(v3, u3, E3)

+1

2ε²∇²p0·(v1, u1, E1)²+1

2ε³∇²p0·[(v1, u1, E1),(v2, u2, E2)],

represent the expansions of the Taylor series of Pin and pout with respect to ε at the base state (V0, U0,E0), (v0, u0, E0) to the orderε³, respectively. Similar notations are used in (2.35), (2.36), (2.40) and (2.41).

Because qi(x, t), i = 5,9,are in conservative form, they are good terms for the stability analysis when introducing anti-derivative variables. The other terms qi(x, t), i= 5,9, though a little more complicated, can be estimated in the following Lemma 2.3 in the different zones (inner, outer and matching zones) by a tedious calculation. First, we compute

∂_x^k(Vin−vout, Uin−uout,Ein−Eout) =O(1)ε^(4−k)α, k= 0,1,2,3, (2.42) on the matching zone{(x, t) :ε^α≤ |x−s(t)| ≤2ε^α, t∈[0, T]}. We have

Lemma 2.3 qi(x, t), i= 5,9 satisfy the following properties (k= 0,1,2,3):

(1) suppq1, suppq3, suppq7 ⊆Inner zone: {(x, t) :|x−s(t)| ≤2ε^α, t∈[0, T]},

∂_x^k(q1, q3, q7)(x, t) =O(1)ε^(3−k)α. (2.43) (2) suppq2, suppq6 ⊆Matching zone: {(x, t) :ε^α≤ |x−s(t)| ≤2ε^α, t∈[0, T]},

∂_x^k(q2, q6)(x, t) =O(1)ε^(3−k)α. (2.44) (3) suppq4, suppq8 ⊆Outer zone: {(x, t) :|x−s(t)| ≥2ε^α, t∈[0, T]},

∂_x^k(q4, q8)(x, t) =O(1)ε^4−kα, (2.45)

T 0

∂_x^k(q4, q8) ²dt ¹₂

=O(1)ε^4−kα. (2.46)

Choose the higher order termsdi(x, t), i= 1,2,3,by

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

d1t−d2x=−q1(x, t), d2t−ε

d2x

v¯−d1

x=− 4 i=2

qi(x, t), d3t−εν

d3x

v¯−d1

x=− 8 i=6

qi(x, t),

d1(x, t= 0) =d2(x, t= 0) =d3(x, t= 0) = 0.

(2.47)

Then, the approximate solutions (¯v,u,¯ E¯)(x, t) satisfy the following equations:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

¯vt−u¯x= 0, u¯t+ ¯px=ε(¯ux

¯v )_x+Q1x, E¯t+ (¯pu¯)_x=εν(E¯x

¯v )_x+ε(1−ν)(u¯u¯x

v¯ )_x+Q2x,

(2.48)

where

Q1=−εu¯x

¯v −m(Uin)_x Vin

−(1−m)(uout)_x vout

−d2x

v¯

−εd2x

v¯ − d2x

¯v−d1

+ [¯p−p(¯v−d1,u¯−d2,E¯−d3)]

=O(1)ε|m_x(Uin−uout)|+O(1)m(1−m)ε|(Vin−vout)|

+O(1)ε|d1||d2x|+O(1)|(d1, d2, d3)|, (2.49) Q2=−ενE¯x

¯v −m(Ein)_x

Vin −(1−m)(Eout)_x vout −d3x

¯v −εν

d3x

v¯ − d3x

v¯−d1

−ε(1−ν) u¯u¯x

v¯ −mUin(Uin)_x

Vin −(1−m)uout(uout)_x vout

+[¯pu¯−p(¯v−d1,u¯−d2,E¯−d3)·(¯u−d2)]

=O(1)ε|m_x(Ein−Eout)|+O(1)m(1−m)ε|(Vin−vout, Uin−uout)|

+O(1)ε|d1||d_3x|+O(1)|(d1, d2, d3)|. (2.50)

It is convenient to use a variable ¯θ(x, t) = ^γ−1_R ( ¯E−^u¯₂²). With the approximate solutions defined in (2.29), we have

θ¯(x, t) =m

Θ₀+εγ−1

R (E1−U0U1) +ε²γ−1

R (E2−U₁²

2 −U0U2) +ε³γ−1

R (E3−U0U3−U1U2)

+ (1−m)

θ0+εγ−1

R (E1−u0u1) +ε²γ−1

R (E2−u²₁

2 −u0u2) +ε³γ−1

R (E3−u0u3−u1u2) +γ−1

R (d²₃−d²₁

2 ) +O(1)ε^4α−12, (2.51)

where Θ₀=^γ−1_R (E0−^U₂⁰²) andθ0= ^γ−1_R (E0−^u₂²⁰). Thus the approximate system (2.48) is equivalent to

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

v¯t−u¯x= 0, u¯t+ ¯px=ε(u¯x

¯v )_x+Q1x, E¯t+ (¯pu¯)_x=κ(θ¯x

v¯)_x+ε(u¯u¯x

¯v )_x+Q2x,

(2.48).

From (2.47), we have the following lemma about the properties of the higher order terms di(x, t), i= 1,2,3.

Lemma 2.4 The following estimates are valid fordi(x, t), i= 2,3:

∂_x^kdi(·, t) _L∞ ≤Cε^{(4−k)α−12} (k= 0,1,2,3), d_i(·, t) _L2 ≤Cε^4α−12,

∂_x^kdi(·, t) _L2≤Cε^{(4−k+12)α−12}, k= 1,2,3. The proof can be found in [3] and [15].

From (2.47)₁, we get

d1(x, t) = _t

0

d2x(x, τ)dτ− _t

0

q1(x, τ)dτ.

It follows that

∂_x^kd1(·, t) _L∞ ≤Cε^{(3−k)α−12} (k= 0,1,2,3), ∂_x^kd1(·, t) _L2 ≤Cε^{(3−k+12)α−12} (k= 0,1,2,3). From (2.49) and (2.50), we have, for i= 1,2,

∂_x^kQi(·, t) _L∞≤Cε^{(3−k)α−12} (k= 0,1,2), ∂_x^kQi(·, t) _L2≤Cε^{(3−k+12)α−12} (k= 0,1,2), ∂tQi(·, t) _L2≤Cε^52α−12.

(2.52)

Lemma 2.5 The approximate solutions satisfy (1)

(¯v,u,¯ E,¯ θ¯)(x, t) =

⎧⎨

⎩

(v0, u0, E0, θ0)(x, t) +O(1)ε if |x−s(t)| ≥ε^α, (V0, U0,E0,Θ₀)(x, t) +O(1)ε^α if |x−s(t)| ≤2ε^α.

(2.53)

(2) There exist positive constants v∗∗, v^∗∗ andθ∗∗, θ^∗∗ such that

0< v∗∗<¯v(x, t)< v^∗∗, 0< θ∗∗<θ¯(x, t)< θ^∗∗, if ε1. (2.54) (3) Introduce the scaled variables

y= x−s(t)

ε , τ = t

ε, (2.55)

then

∂(¯v,u,¯ E,¯ θ¯)

∂y =m(Voy, U0y,E0y) +O(1)ε, ∂(¯v,u,¯ E,¯ θ¯)

∂τ =O(1)ε. (2.56) The proof of Lemma 2.5 is similar in [3].

3 Stability Analysis

Let

(v, u, θ)(x, t) = (¯v,u,¯ θ¯)(x, t) + (φ, ψ, ω)(y, τ), (3.1) wherey, τ are the scaled variables defined in (2.55). Then we have

E−E¯ = R

γ−1ω+ψ²

2 + ¯uψ. (3.2)

Thus, (φ, ψ, ω)(y, τ) satisfies the system

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

φτ−sφ˙ y−ψy= 0,

ψ_τ−sψ˙ _y+ (p−p¯)_y =uy

v −u¯y

¯v

y−Q_1y, R

γ−1ω+ψ² 2 + ¯uψ

τ−s˙ R

γ−1ω+ψ² 2 + ¯uψ

y+ (pu−p¯u¯)_y

=μ θy

v −θ¯y

v¯

y+ uuy

v −¯uu¯y

¯v

y−Q2y, (φ, ψ, ω)(y, τ = 0) = (0,0,0).

(3.3)

Introduce the anti-derivative variables (Φ,Ψ,W˜)(y, τ) =

_y

−∞

(φ, ψ, E−E¯)(y, τ)dy. (3.4) Let

W(y, τ) =γ−1

R ( ˜W −¯uΨ). (3.5)

Then, we have

ω=Wy+γ−1 R

u¯yΨ−1 2Ψ²_y

. (3.6)

Integrating (3.3) over (−∞, ξ] and linearizing the resulted system yield

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Φ_τ−s˙Φ_y−Ψ_y= 0, Ψ_τ−s˙Ψ_y−η

¯vΦ_y+R

v¯Wy+γ−1

v¯ u¯yΨ−Ψ_yy

v¯ =J1−Q1, R

γ−1(Wτ−sW˙ y) +ηΨ_y−μ

¯v

Wy+γ−1 R u¯yΨ

y−s˙u¯yΨ +μ v¯θ¯yΦ_y

=J2−u¯τΨ + ¯uQ1−Q2, (Φ,Ψ, W)(y, τ = 0) = (0,0,0),

(3.7)