Irrotational approximation to the quasi-1-d gas flow

Z. angew. Math. Phys. 60 (2009) 1053–10730044-2275/09/061053-21DOI 10.1007/s00033-008-8085-zc© 2009 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Irrotational approximation to the quasi-1-d gas flow

Jinbo Geng and Yongqian Zhang

Abstract. Under the assumptions that both initial data and the cross-section have sufficientlysmall total variation and that the initial data are supersonic (or are subsonic respectively),we prove that in any bounded domain the L1 norm of the difference between solutions of thehyperbolic system of balance laws and the potential flow system of balance laws with the sameinitial data can be bounded by the cube of the sum over total variations of the initial data andthe cross-section.

Mathematics Subject Classification (2000). Primary: 76N15, 76L05; Secondary: 35L65,35L67.

Keywords. Irrotational approximation, Riemann solver, h-Riemann solver, stability, semi-group.

1. Introduction

The isentropic gas flow through a duct with slowly varying cross section a(x) isgoverned by the system

{∂t[a(x)ρ] + ∂x[a(x)ρu] = 0,

∂t[a(x)ρu] + ∂x[a(x)ρu2 + a(x)P (ρ)] = a′(x)P (ρ),

where u is the velocity of fluid, ρ is the density, the pressure P (ρ) = ργ with γ > 1,and inf

x∈Ra(x) > 0, a(x) ∈ C2, sup

x∈R|a′(x)| < +∞.

This system can be written equivalently as:{

∂tρ + ∂x(ρu) = Bρu,

∂t(ρu) + ∂x(ρu2 + ργ) = Bρu2,(1.1)

where B(x) = −a′(x)a(x) .

System (1.1) has a good approximation:{

∂tρ + ∂x(ρu) = Bρu,

∂tu + ∂x(u2

2 + γγ−1ργ−1) = 0,

(1.2)

1054 J. Geng and Y. Zhang ZAMP

which is called 1-D potential flow equation, see Majda [12]. Equation (1.2) is calledthe irrotational approximation to (1.1) here.

The multi-D potential flow equation is

∂tρ +N∑

k=1

∂xk(ρuk) = 0,

uk = φxk,

(1.3)

where ρ is the density, v = (u1, ..., uN ) is the fluid velocity vector, and φ is thevelocity potential satisfying

φt +|∇φ|2

2+

γργ−1

γ − 1= 0.

Equations (1.3) are an approximation to

∂tρ +N∑

k=1

∂xk(ρuk) = 0,

∂t(ρv) + div(ρv ⊗ v + P I) = 0(1.4)

(see Majda [12]). Here the second equation in (1.3) implies that the flow is ir-rotational. These equations have been widely studied for the transonic flow andmulti-dimension shock waves, for instance, see [6, 9, 10, 11, 12, 13] and referencestherein. As is well known, the potential flow equation is a good approximation tothe compressible isentropic Euler equations, since it approximates to the isentropicEuler equations up to third-order in shock strength for the flow containing weakshocks. When N = 1, Majda used weakly nonlinear geometric optics expansionto derive the L1-difference between the solutions of systems (1.3) and (1.4) withsame initial data in [12], that is, to derive L1-difference between the solutions ofsystems (1.1) and (1.2) with same initial data for the case that a(x) ≡ constant.But for N > 1, it is still an open problem to estimate the L1-difference betweenthe solutions of systems (1.3) and (1.4).

The main purpose of this paper is to deal with the quasi-1-D gas flow, thatis to estimate the L1-difference between the weak solutions of systems (1.1) and(1.2) with same initial data.

DenoteD1 =

{(ρ, u)| ρ > 0, |u| >

√γργ−1

},

andD2 =

{(ρ, u)| ρ > 0, |u| <

√γργ−1

}.

Theorem 1.1. Let U0(x) = (ρ0(x), u0(x))> a bounded measurable function withbounded total variation such that lim

x→±∞U0(x) = U

(0)0 for some constant state

U(0)0 = (ρ(0)

0 , u(0)0 )> with ρ

(0)0 > 0 and |u(0)

0 | 6=√

γ(ρ(0)0 )γ−1. Let U1 = (ρ1(t, x),

u1(t, x))> and U2 = (ρ2(t, x), u2(t, x))> be respective solutions on R+ × R withrespect to systems (1.1) and (1.2) both taking U0 = (ρ0(x), u0(x))> as initial data.

Vol. 60 (2009) Approximation to hyperbolic systems of the balance laws 1055

Assume that T.V.(U0) + T.V.(a) is sufficiently small such that U1 and U2 are welldefined globally with U1, U2 ∈ Dn for all t > 0 and x ∈ R1. Here n ∈ {1, 2}. Thenthere exist constants δ > 0 and K > 0 such that if T.V.(U0) + T.V.(a) < δ and ifU0 − U

(0)0 ∈ L1, then there holds the following

‖U1(t, ·)− U2(t, ·)‖L1 ≤ K(T.V.(U0) + T.V.(a)

)3

t (1.5)

for all t > 0. Here and in the sequel T.V.(U0) and T.V.(a) stand for the totalvariation of U0(x) and a(x) respectively. The superscript > denotes the transpose,while ‖ · ‖L1 the L1 norm.

The proof of Theorem 1.1 will be given in section 5. We follow the idea fromBressan [3] to compare the solutions in L1 norm. Such approach has been used byBianchini and Colombo in [2] to derive the L1 stability of the Riemann semigroupswith respect to the flux functions, and by Saint-Raymond in [14] to compare thedifference in L1 of the solutions for 1-D isentropic and non-isentropic Euler equa-tions, and by Chen–Christofolou–Zhang in [4] and [5] to treat the non-relativitylimits and zero-Mach limits in 1-D, and by Zhang in [16] to deal with the irro-tational approximations to the 2-D steady supersonic flow. In the homogeneouscases as in [14], [4], [5] and [16], the standard Riemann semigroups generated bythe homogeneous systems were used. When the initial data is a piecewise constantwith finite jumps, then for a short time, the trajectory of the standard Riemannsemigroup generated by a homogeneous system can be obtained by piecing togetherthe homogeneous Riemann solvers determined by the jumps. Then it enables oneto compare the difference in L1 of the entropy solutions by comparing the differ-ence between the Riemann solutions for different systems. But the trajectoriesof the Lipschitz semigroups generated by the non-homogeneous systems in thispaper are more complex even for the initial data that are piecewise-constant-statefunctions with finite jumps(see [1]). To overcome these difficulties, we use theglobal ε, h−approximate solutions Uε,h and and the approximate semigroups Ph

for systems (1.1) and (1.2) constructed in [1]; the definitions of these approximatesolutions and semigroups are given in section 4. Then, we obtain the difference inL1 of the approximate solutions for (1.1) and (1.2) by inequality (5.8). Finally, wecomplete the proof of the main result, Theorem 1.1, by letting both ε and h go tozero.

The remaining is organized as follows. In sections 2, we study the homoge-neous Riemann solvers and h-Riemann solvers for systems (1.1) and (1.2), andanalyze some of their properties. In section 3, we compare the homogeneous Rie-mann solvers and h-Riemann solvers for these systems. In section 4, we presentthe existence and the stability results from [1]. In section 5, we estimate thedifference between the Riemann solutions for systems (1.1) and (1.2) with thesame initial data at first, which gives the estimates on the difference between theε, h-approximate solutions. Finally by letting ε, h go to zero, we prove the mainresult.


2. The Riemann solver and h-Riemann solver

2.1. Homogeneous Riemann solver

The homogeneous systems,{

∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + ργ) = 0,(2.1)

and {∂tρ + ∂x(ρu) = 0,

∂tu + ∂x(u2

2 + γγ−1ργ−1) = 0,

(2.2)

are strictly hyperbolic and genuinely nonlinear for ρ > 0.These systems have the same set of eigenvalues:

λ(1)1 = λ

(2)1 = u−

√γργ−1 ,

λ(1)2 = λ

(2)2 = u +

√γργ−1 ,

with the same set of eigenvectors:

r(1)1 = r

(2)1 =

(1,−

√γργ−1

ρ

)>, (2.3)

r(1)2 = r

(2)2 =

(1,

√γργ−1

ρ

)>, (2.4)

where λ(1)k and λ

(2)k are the eigenvalues of systems (2.1) and (2.2) respectively,

and both r(1)k and r

(2)k are the corresponding eigenvectors. Here and in sequel,

the superscript (i) stands for system (2.i), while the subscript k stands for the kth

characteristic family.Direct computation shows that r

(1)k · 5λ

(1)k = r

(2)k · 5λ

(2)k 6= 0, k = 1, 2, for

ρ > 0 (see also Smoller [15]).Now consider systems (2.1) and (2.2), endowed with the same Riemann initial

datum:

U(0, x) =

{UL = (ρL, uL)> if x < x0 ,

UR = (ρR, uR)> if x > x0 .(2.5)

We assume that the constant states UL, UR are sufficiently close. As in [8] and[15], using Lax parameters we parameterize the elementary wave curves by C2

functions ψ(1)k (σk, UL) and ψ

(2)k (σk, UL) for systems (2.1) and (2.2) respectively.

When σk ≥ 0, ψ(i)k (σk, UL) is the rarefaction wave; when σk < 0, ψ

(i)k (σk, UL) is

the shock wave.Moreover,

∂ψ(i)k

∂σk

∣∣∣σk=0

= r(i)k (UL), (2.6)


ψ(i)k (σk = 0, UL) = UL, (2.7)

d

dσkψ

(i)k (σk, UL) = r

(i)k (ψ(i)

k (σk, UL)) for σk ≥ 0, (2.8)

∂2ψ(i)k

∂σ2k

(0, UL) = ∇r(i)k (UL) · r(i)

k (UL), (2.9)

where r(i)k = r

(i)k

r(i)k ·5λ

(i)k

.

2.2. Non-homogeneous h-Riemann solver

Systems (1.1) and (1.2) can be written as the following compact forms respectively:

Ut + f1(U)x = g1, (2.10)

Ut + f2(U)x = g2, (2.11)

whereU = (ρ, ρu)> or U = (ρ, u)>,

f1(U) = (ρu, ρu2 + ργ)>, f2(U) =(ρu,

u2

2+

γ

γ − 1ργ−1

)>,

g1 = −a′(x)a(x)

(ρu, ρu2)>, g2 = −a′(x)a(x)

(ρu, 0)>.

For small h > 0, since

det(

∂f1(U)∂(ρ, u)

) ∣∣∣∣∣U=U

(0)0

=∣∣∣∣

u0 ρ0

u20 + γργ−1

0 2ρ0u0

∣∣∣∣ = ρ0(u20 − γργ−1

0 ) 6= 0,

and

det(

∂f2(U)∂(ρ, u)

) ∣∣∣∣∣U=U

(0)0

=∣∣∣∣

u0 ρ0

γργ−20 u0

∣∣∣∣ = u20 − γργ−1

0 6= 0,

for U(0)0 ∈ Dn, n ∈ {1, 2}, then the map U 7→ fi(U) is invertible in some neigh-

borhood of U = U(0)0 . Then, as in [1], we can define

Φ(i)h (x0, U) = f−1

i

(fi(U) +

∫ h

0

gi(x0 + s, U)ds

), i ∈ {1, 2}. (2.12)

Definition 2.1. Given h > 0 suitably small, x0 ∈ R, we say that U(t, x) is anh-Riemann solver for system (1.1) (or (1.2)), if the following conditions hold

(a) There exist two states U−, U+ ∈ Dn which satisfy U+ = Φ(i)h (x0, U

−);


(b) U(t, x) coincides, on the set {(t, x)| t ≥ 0, x < x0}, with the solution to(2.1)(or (2.2)) with the Riemann initial values UL, U−, and on the set {(t, x)|t ≥ 0, x > x0}, with the solution to (2.1)(or (2.2)) with the Riemann initialvalues U+, UR;

(c) The Riemann problem between UL and U− is solved only by waves with negativespeeds;

(d) The Riemann problem between U+ and UR is solved only by waves with positivespeeds.

Here and in the sequel, the front

{U+, x > x0

U−, x < x0

is called zero-wave.

Due to [1], the Riemann solvers for the homogeneous systems and h-Riemannsolvers for the non-homogeneous systems are the building blocks in constructingapproximate solutions to systems (1.1) and (1.2).

3. Comparison of the Riemann solvers

In this section we will compare the Riemann solvers to systems (2.1) and (2.2),and the h-Riemann solvers to systems (1.1) and (1.2) with the same initial data.

3.1. Comparison of Riemann solvers for homogeneous systems

Lemma 3.1. Suppose (ρL, uL)> ∈ Oδ1(U(0)0 ) ⊂ (D1 ∪D2) for some δ1 > 0, then

for αj ≥ 0, j = 1, 2,

ψ(1)j (αj , UL) = ψ

(2)j (αj , UL), (3.1)

Here and in the sequel Oδ1(U(0)0 ) is the open ball in R2 centered at U

(0)0 with radius

δ1.

Proof. Since r(1)j (U) = r

(2)j (U) for U ∈ D1 ∪ D2, j = 1, 2, then by (2.8), for

αj ≥ 0, j = 1, 2, we can see that ψ(1)j (αj , UL) and ψ

(2)j (αj , UL) satisfy the same

ordinary differential equation. Note that

ψ(1)j (0, UL) = ψ

(2)j (0, UL) = UL,

then, we have (3.1) for αj ≥ 0 by the uniqueness of solutions for ODEs. The proofis complete.

Lemma 3.2. Suppose that (ρL, uL)> ∈ Oδ1(U(0)0 ) ⊂ (D1 ∪D2) for some δ1 > 0.

Then for k = 1 or 2, the equation

ψ(1)2

(β2, ψ

(1)1 (β1, UL)

)= ψ

(2)k (αk, UL) (3.2)


has a unique solution (β1, β2) in some neighborhood of β2 = β1 = αk = 0 withβ2, β1 ∈ C2. Moreover,

βk = αk + O(1)|α−k |3, (3.3)βj = O(1)|α−k |3 for j 6= k. (3.4)

Here UL = (ρL, uL)>, a− = min(a, 0), the bound of O(1) is independent of αk andUL.

Proof. We prove the result for the case that k = 1. Since

det

(∂ψ

(1)2

(β2, ψ

(1)1 (β1, UL)

)

∂(β1, β2)

) ∣∣∣∣∣β1=β2=0

= det(r(1)1 (UL), r(1)

2 (UL))

6= 0, (3.5)

then applying the implicit function theorem to (3.2) gives the existence of theunique solution, (β1, β2), with βj = βj(α1, UL) ∈ C2 and βj(α1 = 0, UL) = 0(j = 1, 2).

To get the estimate of βj , we differentiate (3.2) with respect to α1 and takeα1 = 0, then

2∑

j=1

∂βj

∂α1

∣∣∣α1=0

r(1)j (UL) = r

(2)1 (UL).

Since r(1)1 (UL) and r

(1)2 (UL) are linearly independent, we have

∂βj

∂α1

∣∣∣α1=0

= δj1 =

{1 j = 1,

0 j 6= 1.(3.6)

Now we differentiate (3.2) twice, and let α1 = 0, then

2∑

j=1

∂2βj

∂α21

∣∣∣α1=0

r(1)j (UL) +

∂2ψ(1)1 (β1, UL)∂β2

1

∣∣∣β1=0

=∂2ψ

(2)1 (α1, UL)

∂α21

∣∣∣α1=0

.

By using (2.3) and (2.9), we have

∂2ψ(1)1 (β1, UL)∂β2

1

∣∣∣β1=0

=∂2ψ

(2)1 (α1, UL)

∂α21

∣∣∣α1=0

.

Therefore,∂2βj

∂α21

∣∣∣α1=0

= 0. (3.7)

Then, by (3.6) and (3.7), we can get (3.3) and (3.4) for α1 < 0.When α1 ≥ 0, due to Lemma 3.1, equation (3.2) becomes

ψ(1)2

(β2, ψ

(1)1 (β1, UL)

)= ψ

(1)1 (α1, UL).


Then, with (3.5), applying the implicit function theorem to the above equations,we get

β1 = α1, β2 = 0,

which concludes the proof of the case that α1 ≥ 0. For the case k = 2, the proofcan be carried out in the same way. The proof is complete.

3.2. Comparison of h-Riemann solvers for non-homogeneous systems

Consider the following equations:{

ρR′uR

′ = ρL′uL

′(1 + µ),ρR

′(uR′)2 + (ρR

′)γ = ρL′(uL

′)2 + (ρL′)γ + µρL

′(uL′)2,

(3.8)

and {ρRuR = ρLuL(1 + µ),12 (uR)2 + γ

γ−1 (ρR)γ−1 = 12 (uL)2 + γ

γ−1 (ρL)γ−1,(3.9)

for the given constant µ and constant states UL = (ρL, uL)> ∈ Dn, U ′L =

(ρL′, u′L)> ∈ Dn for some n ∈ {1, 2}.

Lemma 3.3. Suppose that (ρL, uL)>, (ρL′, uL

′)> ∈ Oδ1(U(0)0 ) ⊂ (D1 ∪ D2) for

some δ1 > 0. Then systems (3.8) and (3.9) have unique solutions respectively

(ρR′, uR

′)> = H(1)(ρL′, uL

′, µ) ∈ C2,

and(ρR, uR)> = H(2)(ρL, uL, µ) ∈ C2

in neighborhoods {|µ| < δ′, |U ′L − U

(0)0 | < δ1, |U ′

R − U(0)0 | < δ1} and {|µ| <

δ′, |UL − U(0)0 | < δ1, |UR − U

(0)0 | < δ1} with H(1)(ρL

′, uL′, 0) = (ρL

′, uL′)> and

H(2)(ρL, uL, 0) = (ρL, uL)>.

Proof. We consider system (3.8) at first. Set

F1(UR′, UL

′, µ) = ρR′uR

′ − ρL′uL

′(1 + µ),

F2(UR′, UL

′, µ) = ρR′(uR

′)2 + (ρR′)γ − ρL

′(uL′)2 − (ρL

′)γ − µρL′(uL

′)2,

andF(UR

′, UL′, µ) =

(F1(UR

′, UL′, µ), F2(UR

′, UL′, µ)

).

Since

det(

∂F(UR′, UL

′, µ)∂(ρR

′, uR′)

) ∣∣∣∣∣UR

′=UL′

= ρR′ ((uR

′)2 − γ(ρR′)γ−1

)∣∣∣∣∣UR

′=UL′

6= 0


for UL′ ∈ Oδ1(U

(0)0 ) ⊂ (D1 ∪D2), then applying the implicit function theorem to

(3.8) yields the existence of the unique solution

(ρR′, uR

′)> = H(1)(ρL′, uL

′, µ) ∈ C2

with H(1)(ρL′, uL

′, µ = 0) = (ρL′, uL

′)>.System (3.9) can be dealt with in the same way.Thus, the proof is complete.

Remark 3.1. If µ =∫ h

0B(x0 + s)ds, then

H(1)(ρL′, uL

′, µ) = Φ(1)h (x0, ρL

′, uL′),

H(2)(ρL, uL, µ) = Φ(2)h (x0, ρL, uL).

Lemma 3.4. Suppose that (ρL, uL) ∈ Oδ1(U(0)0 ) ⊂ (D1 ∪ D2) for some δ1 > 0.

Then there holds the following:

H(1)(ρL, uL, µ)−H(2)(ρL, uL, µ) = O(1)µ2, (3.10)

the bound of O(1) is independent of µ and (ρL, uL).

Proof. Using the notations in the proof of Lemma 3.3, and differentiating (3.8)with respect to µ, we get

∂ρR′

∂µ=

∣∣∣∣∣−∂F1

∂µ∂F1∂uR

′

−∂F2∂µ

∂F2∂uR

′

∣∣∣∣∣∣∣∣∣∣

∂F1∂ρR

′∂F1∂uR

′

∂F2∂ρR

′∂F2∂uR

′

∣∣∣∣∣

=−ρL

′(uL′)2 + 2ρL

′uL′uR

′

(uR′)2 − γ(ρR

′)γ−1, (3.11)

and

∂uR′

∂µ=

∣∣∣∣∣∂F1∂ρR

′ −∂F1∂µ

∂F2∂ρR

′ −∂F2∂µ

∣∣∣∣∣∣∣∣∣∣

∂F1∂ρR

′∂F1∂uR

′

∂F2∂ρR

′∂F2∂uR

′

∣∣∣∣∣

=ρL′(uL

′)2uR − ρL′uL

′(uR′)2 − γ(ρR

′)γ−1ρL′uL

′

ρR′ ((uR

′)2 − γ(ρR′)γ−1)

.

(3.12)Similarly, differentiating (3.9) with respect to µ, we have

∂ρR

∂µ=

ρLuLuR

uR2 − γρR

γ−1, (3.13)

∂uR

∂µ=−γρR

γ−2ρLuL

uR2 − γρR

γ−1. (3.14)

Now let ρL′ = ρL, uL

′ = uL in (3.11)-(3.14), then

∂ρR′

∂µ

∣∣∣∣∣µ=0

=∂ρR

∂µ

∣∣∣∣∣µ=0

,


∂uR′

∂µ

∣∣∣∣∣µ=0

=∂uR

∂µ

∣∣∣∣∣µ=0

,

which leads toH(1)(ρL, uL, µ)−H(2)(ρL, uL, µ) = O(1)µ2.

Therefore, the proof is complete.

Now we have the following result:

Lemma 3.5. Suppose that (ρL, uL) ∈ Oδ1(U(0)0 ) ⊂ D2 for some δ1 > 0. Then the

equationψ

(1)2

(ε2,H

(1)(ψ

(1)1 (ε1, ρL, uL), µ

))= H(2)(ρL, uL, µ) (3.15)

has a unique solution (ε1, ε2) in some neighborhood {|µ| < δ′, |ε1| < δ′, |ε2| <δ′} ×Oδ(U0) with εj = εj(µ, ρL, uL) ∈ C2, j = 1, 2. Moreover,

εj(µ, ρL, uL) = O(1)µ2, (3.16)

where the bound of O(1) is independent of µ and (ρL, uL).

Proof. Set

G(ε1, ε2, µ) = ψ(1)2

(ε2,H

(1)(ψ

(1)1 (ε1, ρL, uL), µ

))−H(2)(ρL, uL, µ). (3.17)

Then,

det(

∂G(ε1, ε2, µ)∂(ε1, ε2)

) ∣∣∣∣∣ε1=ε2=µ=0

= det(r(1)1 (ρL, uL), r(1)

2 (ρL, uL))

6= 0,

therefore applying the implicit function theorem to (3.15) gives the existence ofthe unique solution, (ε1, ε2), with εj = εj(µ, ρL, uL) ∈ C2 and εj(0, ρL, uL) =0, j = 1, 2.

Now, with εj = εj(µ, ρL, uL), j = 1, 2, we differentiate (3.15) with respect toµ and take µ = 0, then

∂ψ(1)2

∂H(1)

(∂H(1)

∂ψ(1)1

∂ψ(1)1

∂ε1

∂ε1

∂µ+

∂H(1)

∂µ

) ∣∣∣∣∣µ=0

+∂ψ

(1)2

∂ε2

∂ε2

∂µ

∣∣∣∣∣µ=0

− ∂H(2)

∂µ

∣∣∣∣∣µ=0

= 0,

which is equivalent to

∂H(1)

∂µ

∣∣∣∣∣µ=0

+2∑

j=1

∂εj

∂µ

∣∣∣∣∣µ=0

r(1)j (ρL, uL)− ∂H(2)

∂µ

∣∣∣∣∣µ=0

= 0. (3.18)

Here, to derive (3.18), we use the following

∂ψ(1)2

∂H(1)

∣∣∣∣∣µ=ε1=ε2=0

= I2×2 for (ρL, uL) ∈ (D1 ∪D2),


where I2×2 is the identity matrix.Then, applying Lemma 3.4 to (3.18) yields

2∑

j=1

∂εj

∂µ

∣∣∣∣∣µ=0

r(1)j (ρL, uL) = 0.

Note that r(1)j (ρL, uL) are linearly independent, then we have

∂εj

∂µ

∣∣∣∣∣µ=0

= 0 for j = 1, 2,

which prove (3.16). Thus, the proof is complete.

In the same way, we have the following:

Lemma 3.6. Suppose that (ρL, uL) ∈ Oδ1(U(0)0 ) ⊂ D1 with uL > 0, Then the

equationψ

(1)2

(ε2, ψ

(1)1

(ε1,H

(1)(ρL, uL, µ)))

= H(2)(ρL, uL, µ) (3.19)

has a unique solution (ε1, ε2) in some neighborhood {|µ| < δ′, |ε1| < δ′, |ε2| <δ′} ×Oδ(U0) with εj = εj(µ, ρL, uL) ∈ C2, j = 1, 2. Moreover,

εj(µ, ρL, uL) = O(1)µ2, (3.20)

where the bound of O(1) is independent of µ and (ρL, uL).

Remark 3.2. For case that uL < −√

γργ−1L , the same estimates as (3.16) hold

for the solution, (ε1, ε2), to

ψ(1)2

(ε2, ψ

(1)1

(ε1,H

(1)(ρL, uL, µ)))

= H(2)(ρL, uL, µ). (3.21)

Its proof could be carried out in the same way.

4. The existence and stability of the weak solutions

We introduce the definition of ε, h-approximate solutions of system (1.1) (or system(1.2)).

Definition 4.1. Given ε, h > 0, we say that a continuous map

Uε,h : [0, +∞) → L1loc(R,R2)

is an ε, h-approximate solution of system (1.1) (or (1.2)) if the following conditionshold:(1) As a function of two variables, Uε,h is piecewise constant with discontinuitiesoccurring along finitely many straight lines in x, t plane. Only finitely many wave-front interactions occur, each involving exactly two wave-fronts, and jumps can be


of four types: shocks fronts, rarefaction fronts, non-physical waves and zero-waves,denote as J = S ∪R ∪NP ∪ Z.(2) Along each shock xα = xα(t), α ∈ S, the values on both sides U− = Uε,h(t, xα−)and U+ = Uε,h(t, xα+) are related by

U+ = ψ(i)kα

(σα, U−)

for some kα, i ∈ {1, 2} and some wave size σα < 0. Moreover, the errors betweenthe shock front speeds xα and the true shock speed λ

(i)kα

(U−, U+) satisfy

|xα − λ(i)kα

(U−, U+)| ≤ ε,

where λ(i)kα

(U−, U+) is the speed of the shock front prescribed by the classical Rankine–Hugoniot conditions.(3) Along each rarefaction front xα = xα(t), α ∈ R, one has

U+ = ψ(i)kα

(σα, U−)

for some kα ∈ {1, 2} and some wave size σα ∈ (0, ε). Moreover,

|xα − λ(i)kα

(U+)| ≤ ε.

(4) All non-physical fronts xα = xα(t), α ∈ NP have the same speed:

xα ≡ λ,

where λ is a fixed constant greater than all characteristic speeds. The total strengthof all non-physical fronts remains uniformly small, i.e.

∑

α∈NP|Uε,h(t, xα+)− Uε,h(t, xα−)| ≤ ε, for all t ≥ 0.

(5) The zero-waves are located at every point x = jh, j ∈ (− 1εh , 1

εh ) ∩ Z. Along azero-wave located at xα = jαh, α ∈ Z, the value U− = Uε,h(t, xα−) and U+ =Uε,h(t, xα+) satisfy U+ = Φ(i)

h (xα, U−) for all t > 0 except at the interactionpoints.(6) The total variation in space T.V.

(Uε,h(t, ·)) is uniformly bounded for all t ≥ 0

and ε, h > 0. The total variation in time T.V.{Uε,h(·, x); [0,+∞)} is uniformlybounded for x 6= jh, j ∈ Z.(7) The initial data Uε,h(0, x) satisfy ‖Uε,h(0, ·)− U0‖L1(R) ≤ ε.

Due to [1], we have the following for system (1.2):

Proposition 4.1. Under the basic assumptions in Theorem 1.1, there exist strictlypositive constants δ0 , such that the following holds. If T.V.(U0) + T.V.(a) ≤ δ0

and U0 − U(0)0 ∈ L1(R,R2), then the Cauchy problem of system (1.2) admits ε, h-

approximate solutions Uε,h2 = Uε,h

2 (t, x) for ε, h > 0 sufficiently small and all t > 0such that

‖Uε,h2 (t1, ·)− Uε,h

2 (t2, ·)‖L1(R) ≤ M |t1 − t2|,


for all t1, t2 ∈ [0, +∞). Moreover, there exists a unique U2(t, x) ∈ L1∩BV (R;R2)s.t.

‖Uε,h2 (t, x)− U2(t, x)‖L1(R) → 0 as ε, h → 0.

Here M is a positive constant independent of ε, h and Uε,h2 (t, x).

For system (1.1), we have the following, which is a result in [1].

Proposition 4.2. For system (1.1), there exist constants δ > 0 and C0 > 0,such that if T.V.(a) is sufficiently small, then for any small h > 0, there exist anon-empty closed domain Dh(δ) and a unique uniformly Lipschitz semigroup Ph :[0,+∞) × Dh(δ) → Dh(δ) whose trajectories U(t, ·) = Ph

t U0 solve the followingequation,

∫ ∞

0

∫

R[Uϕt + f1(U)ϕx]dxdt

+∫ ∞

0

∑

j∈Z

ϕ(t, jh)

(∫ h

0

g1(jh + s, U(t, jh−))ds

)dt = 0,

for any ϕ ∈ C1c ((0, +∞)×R), and are obtained as the limit of any sequence of ε, h-

approximate solutions of system (1.1) as ε tends to zero with fixed h. In particularPh satisfies, for any V0, V1 ∈ Dh(δ), t, s ≥ 0,

Ph0 V0 = V0,

Pht ◦ Ph

sV0 = Phs+tV0,

‖Pht V0 − Ph

s V1‖L1 ≤ L(‖V0 − V1‖L1(R) + |t− s|),for some positive constant L independent of h. Here

Dh(δ) = {U = (ρ, u)> : U − U(0)0 ∈ D0

h(δ)},where D0

h(δ)=cl{W ∈ L1(R,R2) : W piecewise constant, Vh(W )+C0Qh(W ) < δ},and cl stands for closure in L1(R),

Vh(W ) =∑

{|σα| : α is a wave inW},

Qh(W ) =∑

{|σασβ | : α and β are approaching waves in W},and the subscript “h” means that all waves in W are generated by the h-Riemannproblems and Riemann problems for W as in [1].

Remark 4.1. Let V0 be a piecewise constant function with finitely many jumps.Due to [1], Ph

t V0 is the limit in L1 of the ε, h-approximate solutions for system(1.1). Then for 0 < t ¿ 1, Ph

t V0 is glued by the following:(1) shocks, rarefaction fronts and non-physical fronts for (2.1) issuing from thejump-point x0 in V0 for jh < x0 < (j + 1)h.(2) zero-waves and elementary waves for h-Riemann solver issuing from the point(0, jh), j ∈ Z.


Moreover, due to [1], we have the following:

Proposition 4.3. If T.V.(a) is sufficiently small, there exist constants L > 0 andδ∗ > 0, a non-empty closed domain D and a unique semigroup P : [0, +∞)×D →D with the following properties:(1) for any U ∈ D, U − U

(0)0 ∈ L1(R) and TV (U) < δ∗;

(2) for some δ > 0 and for all h > 0 small enough, D ⊂ Dh(δ);(3) P0U = U, ∀ U ∈ D; Pt ◦ PsU = Ps+tU, ∀ U ∈ D, t, s ≥ 0;(4) ‖PtU − PsV ‖L1 ≤ L(‖U − V ‖L1(R) + |t− s|), ∀U ∈ D, t, s ≥ 0;(5) for all U0 ∈ D, the function U(t, x) = PtU0 is a entropy solution of system(1.1) with initial data U0;(6) there exists a sequence of semigroups Phi such that Phi

t U converges in L1 toPtU as i → +∞ for any U ∈ D.

For the proof of Propositions 4.1–4.3, see [1].

5. Proof of the main results

To get the estimate on the difference of solutions to systems (1.1) and (1.2), wefirst consider two simple cases presented in the following lemmas, Lemmas 5.1and 5.2.

Lemma 5.1. Suppose that UL, UR ∈ Oδ1(U(0)0 ) ⊂ (D1 ∪ D2). Let λ be a fixed

constant speed such that |λ(i)k (U)| ≤ λ for U ∈ Oδ1(U

(0)0 ) and all k and i, and let

W = W (t, x) be a Riemann solver of homogeneous system (2.1) with the followinginitial data:

W (0, x) =

{UL = (ρL, uL)> if x < x0,

UR = (ρR, uR)> if x > x0.

Denote

V (t, x) =

{UL if x < x0 + λt,

UR if x ≥ x0 + λt,

where λ is a constant such that |λ| < λ Then, we have the following.

(i) In the general case, one has

1t‖V (t, ·)−W (t, ·)‖L1(R) = O(1) · |UR − UL|. (5.1)

(ii) If UL and UR have additional relations: UR = ψ(2)k (σk, UL) and λ = λ

(2)k (UR)

for some σk > 0, k = 1 or 2, then we have a sharper estimate:

1t‖V (t, ·)−W (t, ·)‖L1(R) = O(1) · σ2

k. (5.2)


(iii) If UL and UR are related by a shock-wave, i.e. UR = ψ(2)k (σk, UL) and λ =

s(2)k (UR, UL) for some σk < 0, k = 1 or 2, and s

(2)k (UR, UL) is the corresponding

shock speed of system (2.2), then we have

1t‖V (t, ·)−W (t, ·)‖L1(R) = O(1) · |UR − UL|3. (5.3)

Proof. (i) Estimate (5.1) comes from:

1t‖V (t, ·)−W (t, ·)‖L1(R) =

1t

∫ ∞

−∞|V (t, x)−W (t, x)|dx

=1t

∫ tλ

−tλ

|V (t, x)−W (t, x)|dx

= O(1) · |UR − UL|.(ii) We only prove the case of k = 1. For k = 2, the proof is quite the same here.If σ1 > 0, the Riemann solution of system (2.1) can be given by the followingequation:

ψ(1)2

(β2, ψ

(1)1 (β1, UL)

)= ψ

(2)1 (σ1, UL), (5.4)

By Lemma 3.2, equation (5.4) has a unique solution:

β1 = σ1, β2 = 0.

So we have:1t‖V (t, ·)−W (t, ·)‖L1(R) =

1t

∫ ∞

−∞|V (t, x)−W (t, x)|dx

=1t

∫ tλ(2)1 (UR)

tλ(2)1 (UL)

|V (t, x)−W (t, x)|dx

= O(1) · |UR − UL| ·(λ

(2)1 (UR)− λ

(2)1 (UL)

)

= O(1) · σ21 .

(iii) Similarly, to get the Riemann solution of system (2.1), we shall first solve thefollowing equation:

ψ(1)2

(β2, ψ

(1)1 (β1, UL)

)= ψ

(2)1 (σ1, UL). (5.5)

By Lemma 3.2, the unique solution to (5.5) is given by:

β1 = σ1 + O(1) · |σ1|3,β2 = O(1) · |σ1|3 .

If we denote WM the intermediate state occurring in the solution of Riemannproblem (2.1), then we can easily have:

|WM − UR| = O(1) · |β2| = O(1) · |UR − UL|3.


Note that when β1 < 0, then there is a shock corresponding to the 1-th character-istic field of system (2.1), call s

(1)1 the respective shock speed of system (2.1), then

we have:

s(2)1 (σ1) = λ

(2)1 (UL) +

12σ1 + O(1) · σ2

1 ,

s(1)1 (β1) = λ

(1)1 (UL) +

12β1 + O(1) · σ2

1

= λ(2)1 (UL) +

12σ1 + O(1) · σ2

1 .

So we haves(1)1 − s

(2)1 = O(1) · |σ1|2 = O(1) · |UR − UL|2.

Denote

q1M = max(s(2)

1 , s(1)1 ), q1

m = min(s(2)1 , s

(1)1 ),

q2M = max(λ(2)

2 (UR), λ(2)2 (WM )), q2

m = min(λ(2)2 (UR), λ(2)

2 (WM )).

Then,1t‖V (t, ·)−W (t, ·)‖L1(R)

=1t

∫ ∞

−∞|V (t, x)−W (t, x)|dx

=1t

( ∫ q1M t

q1mt

+∫ q2

M t

q1M t

)|V (t, x)−W (t, x)|dx

= O(1) · |UR − UL| · |s(2)1 − s

(1)1 |+ O(1) · |UR − UL|3

= O(1) · |UR − UL|3.The proof is complete.

Lemma 5.2. Suppose that UL, UR ∈ Oδ1(U(0)0 ) ⊂ (D1 ∪ D2) for some δ1 > 0,

and that UL and UR are related by a zero-wave, i.e. UR = Φ(2)h (jh, UL) for some

j ∈ (− 1εh , 1

εh ) ∩ Z. Let λ be a fixed constant speed such that |λ(i)k (U)| ≤ λ for

U ∈ Oδ1(U(0)0 ) and all k and i, and let W = W (t, x) be a h-Riemann solver of

system (1.1) with the following initial data:

W (0, x) =

{UL = (ρL, uL)> if x < x0,

UR = (ρR, uR)> if x > x0.

Denote

V (t, x) =

{UL if x < x0,

UR if x ≥ x0.

Then,1t‖V (t, ·)−W (t, ·)‖L1(R) = O(1)µ2

j , (5.6)


where µj =∫ h

0B(jh + s)ds and the bound of the O(1) is independent of µj and j.

Proof. For UL, UR ∈ Oδ1(U(0)0 ) ⊂ D2, to get the h-Riemann solution of (1.1), it

suffices to solve the following equation:

ψ(1)2

(ε2,H

(1)(ψ

(1)1 (ε1, ρL, uL), µj

))= H(2)(ρL, uL, µj). (5.7)

By Lemma 3.5, equation (5.7) has a unique solution (ε1, ε2) with

εi = O(1)µ2j for i = 1, 2.

Then,

1t‖V (t, ·)−W (t, ·)‖L1(R1) =

1t

∫ ∞

−∞|V (t, x)−W (t, x)|dx

=1t

∫ tλ

−tλ

|V (t, x)−W (t, x)|dx

= O(1) · (ε1 + ε2)= O(1) · µ2

j .

For UL, UR ∈ Oδ1(U(0)0 ) ⊂ D1, the result can be derived in the same way by

Lemma 3.6 and Remark 3.2.Thus, the proof is complete.

Lemmas 5.1 and 5.2 give the local estimates on the difference of ε, h-approxi-mate solutions. To get the global estimates on the difference of the approximatesolutions, we need the following lemma, which is a consequence of Theorem 2.9in [3].

Lemma 5.3. Assume that the map w : I = [0, +∞) → Dh(δ) satisfies w−U(0)0 ∈

Lip(I, L1(R)

). Then, for any t ≥ 0,

‖w(t, ·)− Pht w(0)‖L1(R)

≤ L

∫ t

0

(lim infs→0+

‖w(τ + s)− Phs w(τ)‖L1(R)

s

)dτ, (5.8)

where L is independent of h and t, and Dh(δ) is given by Propositions 4.2 and 4.3,and

Lip(I, L1(R)

)= {v|v(t) ∈ L1(R) ∀t ∈ I, sup

t 6=τ

∫R |v(t, x)− v(τ, x)|dx

|t− τ | < ∞}.

Now, we can carry out the proof of the main result.

Proof of Theorem 1.1. By Proposition 4.1, for every ε > 0 and h > 0, theCauchy problem for (1.2) with initial data U0 admits an ε, h-approximate fronttracking solution Uε,h

2 (t, x) =(ρε,h2 (t, x), uε,h

2 (t, x))

with Uε,h2 (t, x) = U

(0)0 outside

a compact interval depending on x for each t > 0.


Let Ph be the semigroup of weak solution of system (1.1) as in Proposition 4.2.Then, by Proposition 4.1 and Lemma 5.3, we have:

‖Uε,h2 (t)− Ph

t Uε,h2 (0)‖L1(R)

≤ L

∫ t

0

lim infs→0+

‖Uε,h2 (τ + s)− Ph

s Uε,h2 (τ)‖L1(R)

sdτ. (5.9)

To estimate the integrand on the right hand side of (5.9), we consider any timeτ ∈ [0, T ] where no interaction takes place. Let Uε,h

2 (τ, ·) have jumps at pointsx1 < x2 < · · · < xN . Let S be the set of indices α ∈ {1, 2, · · ·, N} such thatUε,h

2 (τ, xα−) and Uε,h2 (τ, xα+) are connected by shock fronts. Moreover, let R be

the set of indices α corresponding to a rarefaction front in Uε,h2 , and NP the set

of indices α corresponding to non-physical fronts in Uε,h2 , and Z the set of indices

α corresponding to zero-waves in Uε,h2 .

For each α, denote by Wα(t, x) the h-Riemann solution of system (1.1) with ini-tial data Wα(t = τ, x) = UL = Uε,h

2 (τ, xα−) for x < xα and Wα(t = τ, x) = UR =Uε,h

2 (τ, xα+) for x > xα. Then, for s small enough, the semigroup s 7→ Phs Uε,h

2 (τ)is obtained by piecing together the h-Riemann solvers (see also Remark 4.1).Therefore, by Lemmas 5.1 and 5.2, we have

lims→0+

‖Uε,h2 (τ + s)− Ph


s

=∑

α∈R∪S∪NP∪Z

(lim

s→0+

1s

∫ xα+ρ

xα−ρ

|Uε,h2 (τ + h, x)−Wα(h, x− xα)|dx

)

=∑

α∈SO(1) ·

(ε · |Uε,h

2 (τ, xα+)− Uε,h2 (τ, xα−)|+ |Uε,h

2 (τ, xα+)− Uε,h2 (τ, xα−)|3

)

+∑

α∈RO(1) · |σα|

(|σα|+ ε)

+∑

α∈NPO(1) ·

(|Uε,h

2 (τ, xα+)− Uε,h2 (τ, xα−)|

)

+∑

α∈ZO(1)µ2

j , (5.10)

where σα stands for the corresponding strength of the rarefaction front, and µj =∫ h

0B(jh+s)ds =

∫ (j+1)h

jhB(x)dx. The presence of ε in (5.10) comes from properties

(2) and (3) in Definition 4.1:

|xα − λ(i)kα

(U−, U+)| ≤ ε.

for the shock front x = xα with speed λ(i)kα

(U−, U+), and

|xα − λ(i)kα

(U+)| ≤ ε.

for the rarefaction front x = xα with speed λ(i)kα

(U+), where U+ = Uε,h2 (τ, xα+),

and xα denotes the slope of the front x = xα in Uε,h2 .


SinceT.V.(Uε,h

2 (τ, ·)) ≤ O(1)(T.V.(Uε,h

2 (0, ·)) + T.V.(a))

(see [1]), then at τ ∈ [0, t] where no interaction takes place, we have

lims→0+

‖Uε,h2 (τ + s)− Ph


s

= O(1) ·(

ε + ε · T.V.(Uε,h2 (τ, ·)) + T.V.(Uε,h

2 (τ, ·))3)

+∑

α∈ZO(1)µ2

j

= O(1) ·(

ε + ε ·(T.V.(Uε,h

2 (τ, ·)))

+(T.V.(Uε,h

2 (τ, ·)))3

)+

∑

α∈ZO(1)µ2

j

= O(1) ·(

ε + ε ·(T.V.(Uε,h

2 (0, ·)) + T.V.(a))

+(T.V.(Uε,h

2 (0, ·)) + T.V.(a))3

)

+∑

j∈Z

O(1)µ2j

= O(1)

(ε +

(T.V.(U0) + T.V.(a)

)3

)+

∑

j∈Z

O(1)h∫ (j+1)h

jh

|a′(x)|dx

= O(1)(ε +

(T.V.(U0) + T.V.(a)

)3 + h

∫

R

|a′(x)|dx). (5.11)

Therefore, using (5.9), we have

‖Uε,h2 (t, ·)− Ph

t Uε,h2 (0)‖L1(R)

≤O(1) ·(

ε + h +(T.V.(U0) + T.V.(a)

)3)

t. (5.12)

On the other hand,

‖Uε,h2 (t, ·)− U2(t, ·)‖L1(R) −−−−→

ε,h→00, (5.13)

and

‖Pht Uε,h

2 (0)− U1(t, ·)‖L1

= ‖Pht Uε,h

2 (0)− PtU0‖L1

≤ ‖Pht Uε,h

2 (0)− Pht U0‖L1 + ‖Ph

t U0 − PtU0‖L1

≤ L · ‖Uε,h2 (0)− U0‖L1 + ‖Ph

t U0 − PtU0‖L1

= O(1)ε + ‖Pht U0 − PtU0‖L1 . (5.14)

Therefore, letting ε → 0, h = hi → 0 in (5.12), we can get the desired result by(5.12), (5.13) and (5.14). Here {hi} is the sequence given in (5) of Proposition 4.3.

The proof of Theorem 1.1 is complete.


Acknowledgments

This work was supported by NSFC Project 10531020, by the 111 Project B08018and by Project STCSM(06JC14005). The authors would like to thank professorShuxing Chen for his helpful discussion.

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Jinbo GengSchool of Mathematical SciencesandLaboratory of Mathematics for Nonlinear ScienceFudan UniversityShanghai 200433P.R.Chinae-mail: [email protected]


Yongqian ZhangSchool of Mathematical SciencesandLaboratory of Mathematics for Nonlinear ScienceFudan UniversityShanghai 200433P.R.Chinae-mail: [email protected]

(Received: August 4, 2008; revised: November 11, 2008)

Published Online First: May 4, 2009

To access this journal online:www.birkhauser.ch/zamp

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