DSpace at VNU: Existence of traveling waves in elastodynamics with variable viscosity and capillarity

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Existence of traveling waves in elastodynamics with variable viscosity and capillarity

Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

Article history:

Received 17 May 2010

Accepted 4 June 2010

Keywords:

Conservation law

Traveling wave

Shock wave

Lax shock inequalities

Elastodynamics

Viscosity

Capillarity

Diffusion

Dispersion

Equilibria

Lyapunov function

Attraction domain

a b s t r a c t Motivated by our earlier works, Thanh (2010) [3,4], we study the global existence of travel-ing waves associate with a Lax shock of a model of elastodynamics where the viscosity and capillarity are functions of the strain The system is hyperbolic and may not be genuinely nonlinear The left-hand and right-hand states of a Lax shock correspond to a stable node and a saddle point By defining a Lyapunov-type function and using its level sets, we esti-mate the attraction domain of the stable node Then we show that the saddle point lies on the boundary of the attraction domain of the stable node Moreover, exactly one stable tra-jectory enters this attraction domain This gives a stable-to-saddle connection for 1-shocks (a saddle-to-stable connection for 2-shocks), and therefore defines exactly one traveling wave connecting the two states of the Lax shock

© 2010 Elsevier Ltd All rights reserved

1 Introduction

We are interested in the global existence of traveling waves associated with a Lax shock for the general model of nonlinear elastodynamics and phase transitions with a nonlinear viscosity and capillarity The model consists of the conservation law

of momentum and the continuity equation in elastodynamics describing the longitudinal deformations of an elastic body with negligible cross-section with variable nonlinear viscosityµ(w)and variable nonlinear capillarityλ(w):

∂tv − ∂xσ (w) =



λ′(w) wx2

2 − (λ(w)wx)x



x

+ (µ(w)vx)x,

∂tw − ∂xv =0.

(1.1)

Here, the unknownvandw > −1 represent the velocity and deformation gradient (the strain), respectively The constraint

w > −1 follows from the principle of impenetrability of matter, however, it is useless in this paper and we therefore do not impose this condition The stressσ = σ (w)is a function of the strainw The functionµ(w)characterizes the viscosity inducing diffusion effect and the functionλ(w)represents the positive capillarity inducing dispersion effect The reader is referred to LeFloch [1] for the derivation of the model(1.1) Throughout, the stress functionσis assumed to be differentiable and

such that the corresponding system of conservation laws without viscosity and capillarity

∗Tel.: +84 8 2211 6965; fax: +84 8 3724 4271.

E-mail addresses:mdthanh@hcmiu.edu.vn , hatothanh@yahoo.com

1468-1218/$ – see front matter © 2010 Elsevier Ltd All rights reserved.

∂t v − ∂xσ (w) =0,

is strictly hyperbolic, see [2], for example In addition, the continuous viscosity and smooth capillarity are required to satisfy the conditions

where m>0, κ >0 are constants

In our recent paper [3], the global existence of traveling waves of a single conservation law with constant viscosity and capillarity was established In this work, we propose a method of estimating attraction domain for the stable node of the resulted differential equations The argument of this method is completed by our second work [4], where by considering

an isothermal fluid with nonlinear diffusion and dispersion coefficients, we went further by proving that the saddle point

is in fact lying on the boundary of the stable node Moreover, we also pointed our that exactly one stable trajectory of the differential equations leaves the saddle point and enters the domain of attraction of the stable node This gives a complete description of a method of estimating the attraction domain for several systems of conservation laws with viscosity and capillarity effects Precisely, the method consists of several steps:

Step 1 Derive a system of nonlinear first-order differential equations corresponding to the given shock;

Step 2 Showing that one state of the shock corresponds to a stable node, the other state corresponds to a saddle point of

the system of differential equations given by Step 1;

Step 3 Define a suitable Lyapunov-type function for the stable node obtained in Step 2 Use the level sets of this function to

estimate the domain of attraction based on certain invariant properties of the level sets;

Step 4 Point out that there is one trajectory leaving the saddle and entering the attraction domain obtained in Step 3 This

gives a traveling wave connecting the two states of the given shock

Besides, the rate of change of a small quantity need not be small! In particular, if one can speak of small variable nonlinear viscosityµ(w)and capillarityλ(w), their effects may not be negligible when the rates of changeµ′(w), λ′(w)are quite large Our goal in this paper is to show that the above method still works even for the general model(1.1) Clearly, the arguments

in [3,4] have to be reconsidered or improved

Many important contributions for the study of traveling waves for viscous–capillary models such as(1.1)have been carried out by Hayes and LeFloch [5], then by Bedjaoui and LeFloch [6–8], and a recent joint work by Bedjaoui et al [9 Traveling waves for diffusive–dispersive scalar equations were earlier studied by Bona and Schonbek [10], Jacobs et al [11] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod and Fan [12–15], Shearer and Yang [16] Shock waves and entropy solutions of the hyperbolic system of conservation laws such as (1.3)were considered in [2,17–19] When the cross-section is taken into account, the Lax shocks and moreover the Riemann problem for the model of fluid flows in a nozzle with variable cross-section and the shallow water equations were considered

by LeFloch and Thanh [20,21], Kröner et al [22] and Thanh [23] See also the references therein for related works

This paper is organized as follows In Section2we recall the basic properties of Lax shocks of the system(1.3)and the derivation of a system of ordinary first-order differential equations obtained by substituting a traveling wave to the viscous–capillary model(1.1) We then point our that given a Lax shock of(1.3), there corresponds two equilibria of the system of differential equations in which one is a stable node, and the other is a saddle In Section3we estimate the domain

of attraction of the stable node which is large enough such that the saddle point belongs to the boundary of this domain In Section4we establish the existence of traveling waves by indicating that exactly one trajectory leaves the saddle point and enters the domain of attraction of the stable node Finally, we also include in Section4some numerical illustrations for the traveling waves

2 Basic concepts and results on Shock waves and traveling waves

2.1 Hyperbolicity and Shock waves of (1.3)

The Jacobian matrix of the system(1.3)is given by

A(v, w) =



0 − σ′(w)



which gives the characteristic equation

det(A− λI) = λ2− σ′(w) =0.

Sinceσ′(w) >0, the Jacobian matrix A admits two real and distinct eigenvalues

λ1(v, w) = −σ′(w) <0< λ2(v, w) = σ′(w).

A discontinuity of(1.3)connecting two given states u− = (v−, w−),u+ = (v+, w+)with the propagation speed of

discontinuity s is a weak solution of(1.3)of the form

u(x,t) =



(v−, w−) if x<st, (v , w ) if x>st,

and satisfies the Rankine–Hugoniot relations

s(v+− v−) + (σ (w+) − σ (w−)) =0,

The Eqs.(2.1)yield

s2= σ (w+) − σ (w−)

w+− w−

.

An admissible Lax shock, or a Lax shock for short, connecting the left-hand and the right-hand states u− = (v−, w−)

and u+ = (v+, w+), respectively, with the shock speed s = s(u−,u+)is a discontinuity of(1.3)satisfying the Lax shock inequalities

2.2 Traveling waves of (1.1)

Let us now turn to traveling waves We call a traveling wave of(1.1)connecting the left-hand state(v−, w−)and the right-hand state(v+, w+)a smooth solution of(1.1)of the form(v, w) = (v(y), w(y)),y=x−st where s is a constant,

and satisfies the boundary conditions

lim

y→±∞(v, w)(y) = (v±, w±)

lim

y→±∞

d

dy(v(y), w(y)) = lim

y→±∞

d2

Substituting(v, w) = (v, w)(y),y=x−st into(1.1), and re-arranging terms, we get

sv′+ (σ (w))′=



λ′(w) w′2

2 + λ(w)w′′

′

− (µ(w)v′)′,

sw′+ v′=0,

where(.)′=d(.)/dy Integrating the last equations on the interval(−∞,y), using the boundary conditions(2.1), we obtain

s(v − v−) + (σ (w) − σ (w−)) = λ′(w) w′2

2 + λ(w)w′′− µ(w)v′,

s(w − w−) + (v − v−) =0.

By letting y→ +∞, we can see that s and(v±, w±)satisfy the Rankine–Hugoniot relations(2.1) Substitutingv − v−=

−s(w − w−), v′ = −sw′

from the second equation in(2.1)into the second one, we obtain a second-order differential equation for the unknown functionw:

−s2(w − w−) + (σ (w) − σ(w−)) = λ′(w) w′2

2 + λ(w)w′′+sµ(w)w′

or

w′′= − λ′(w)

2λ(w) w

′ 2−sµ(w) λ(w) w

′+ σ (w) − σ (w−) −s2(w − w−)

Setting

z= w′, h(w) =s2(w − w−) − (σ (w) − σ(w−)),

we reduce the second-order differential equation(2.4)to the following 2×2 system of first-order differential equations

w′=z,

z′= − z

2λ(w) (λ

or, in a more compact form

dU

where

U= (w,z), F(U) =



z, − z

2λ(w) (λ

′(w)z+2sµ(w)) −hλ(w) (w)



The above argument reveals that a point U in the(w,z)-phase plane is an equilibrium point of the autonomous differential equations(2.6)if and only if U = (w ,0), wherew and the shock speed s are related by(2.1)

Since h(w±) =0, the Jacobian matrix of the system(2.6)is given by

DF(w±,0) =





σ′(w±) −s2

sµ(w±) λ(w±)



The characteristic equation of DF(v±,0)is then given by

|DF(w±,0) − β| =







σ′(w±) −s2

sµ(w±) λ(w±) − β







=0,

or

β2+sµ(w±)

λ(w±) β +

s2− σ (w±)

Assume that the jump satisfies the Lax shock inequalities

λ2(w−) >s> λ2(w+)

which yields

If s>0, we can see that the characteristic equation|DF(w−,0) − β|admits two real roots with opposite sign, and that the characteristic equation|DF(w+,0) − β|admits two roots with negative real parts There are similar arguments for the case

s<0 This leads us the the following conclusions

Proposition 2.1 (i) Given a Lax shock associate withλ1with the left-hand and right-hand states u− = (v−, w−),u+ = (v+, w+)and the shock speed s = s1(u+,u−) Then, the point (w−,0)is an asymptotically stable node, and the point

(w+,0)is a saddle of(2.5).

(ii) Given a Lax shock associate withλ2with the left-hand and right-hand states u−= (v−, w−),u+= (v+, w+)and the shock speed s=s2(u+,u−) Then, the point(w+,0)is an asymptotically stable node, and the point(w−,0)is a saddle of (2.5).

Proposition 2.1indicates that given a Lax shock, there is possibly a stable-to-saddle or saddle-to-stable connection Whenever such a connection is established, we obtain a traveling wave associated with the given Lax shock

3 Estimating the attraction domain

3.1 Assumptions and examples

Given a Lax 2-shock connecting the left-hand and right-hand states u− = (v−, w−)and u+ = (v+, w+), respectively,

with the shock speed s=s(u−,u+) Suppose for definitiveness that

w+< −w−.

Throughout, we assume the following hypotheses

(H1) The valuesw±satisfy

| w+− w−| ≤ 2κ

whereκ,m are positive constants as in(1.4)

(H2) There exists a valueν < w+such that

∫ w −

ν

h(ξ) λ(ξ)dξ <0, and

∫ w

w +

h(ξ)

where

h(w) =s2(w − w−) − (σ(w) − σ (w−)).

Example 3.1 We omit the conditionw > −1, and extend the functionσ = σ(w)to the whole−∞ < w < +∞ Let the functionσbe twice differentiable and strictly convex:

σ′′(w) >0, w ∈R.

Then the Lax shock inequalities

λ (w ) <s(u ,u ) < λ (w )

are equivalent to the condition

w−> w+.

And

∫ w −

−∞

h(ξ)

λ(ξ)dξ = −∞.

Thus, for any pair(w+, w−), there is always such aνsatisfying (H2)

Example 3.2 Let us take the model of elastodynamics described by the Eqs.(1.1), where the stressσis a twice differentiable function ofwsatisfying

σ′(0) >0, wσ′′(w) >0 forw ̸=0,

(which impliesσ′(w) >0 for allw > −1) and

lim

w→+∞σ′(w) = +∞.

(See [2]) Assumeλ(w) ≡ λ =constant Then, as argued similarly as in [2], we can see that for eachw−>0, there exists exactly one value denoted byϕ∞(w−) <0 such that

s2(w−, ϕ∞(w−))(ϕ∞(w−) − w−) − (σ(ϕ∞(w−)) − σ (w−)) =0

and that

s2(w−, w)(w − w−) − (σ (w) − σ (w−)) <0 ifw < ϕ∞(w−),

where

s2(w−, w) = σ (w) − σ(w−)

Moreover, for each pair(w0, w1), there is exactly one value denoted byϕ#(w0, w1)such that

s2(w0, w1) =s2(w0, ϕ#(w0, w1)).

Setting

w∗= ϕ#(w−, ϕ∞(w−)) := ζ (w−),

and takingw+such thatζ(w−) < w+< w−, and defining

ν = ϕ#(w−, w+),

we can see that the first condition of (H2) holds Moreover, since in this case the Lax shock inequalities are equivalent to the Liu entropy conditions, the second condition of (H2) also holds

3.2 Lyapunov-type function

Under the hypotheses (H1) and (H2), we now consider the autonomous system obtained from the previous section

dw

dy =z,

dz

dy = −

z

2λ(w) (λ

Let us define a Lyapunov-type function candidate

L(w,z) =

∫ w

w +

h(ξ) λ(ξ)dξ +

z2

The following lemma indicates that the function L defined by(3.7)is a Lyapunov-type function

Lemma 3.1 Setting D= (ν, w−) × {|z| < 2sκ

m} ∋ (w+,0) Under the hypotheses(H1),(H2), it holds that

L(w+,0) =0, L(w,z) >0, for(w,z) ∈D\ { (w+,0)},

˙

Proof First, we have immediately

L(w+,0) =0, L(w,z) ≥ ∫ w

w

h(ξ) λ(ξ)dξ >0, (w,z) ∈D, w ̸= w+,

which establishes the first line of(3.5) Second, the derivative of L along trajectories of(3.3)can be estimated as follows

˙

L(w,z) = ∇L(w,z) ·



dw

dy,dz

dy



=



h(w) λ(w) ,z

 

z, − z

2λ(w) (λ

′(w)z+2sµ(w)) − hλ(w) (w)



= − z2

2λ(w) (λ

′(w)z+2sµ(w))

< − z2

for(w,z)in D,z̸=0 This completes the proof ofLemma 3.1 

3.3 Estimation of attraction domain

It is derived from(3.1)that we can select a positive constant M such that

s≤M≤ 2sκ

Next, for each small numberεsuch thatw−− w+>2ε >0, we set

Gε =



(w,z) | (w − w+)2

(M(w+− (w−− ε)))2 ≤1, w ≥ w+



∪



(w,z) | (w − w+)2

(M| w+− (w−− ε)|)2 ≤1, w ≤ w+



where M is given by(3.7)andνis defined in(3.2) Then, it is not difficult to check that

Gε⊂ [ ν, w−) ×



|z| ≤2sκ

m

 Now, we claim that the function

∫ w

w +

h(ξ)

λ(ξ)dξ

is strictly increasing forwnearw−, w ≤ w− Indeed, the Lax shock inequalities(2.2)imply that there exists a positive number 0< θ < |w−− w+|such that

σ (w) − σ (w−)

w − w− >s2 for| w − w−| < θ.

This implies that for| w − w−| < θ,

h(w) >0,

which establishes the statement Fix thisθ >0, it then holds that

The following lemma provides us with properties of the sets Gε.

Lemma 3.2 For any positive numberεso that 0<2ε < θ < w−− w+, whereθ is given in(3.9), let Gεbe the set defined

by(3.8)and let∂Gεdenote its boundary It holds that

min

(w,z)∈∂Gε L(w,z) =L(w−− ε,0).

Moreover, the minimum value in(3.9)is strict, i.e.

Proof We need only to establish the second statement, i.e.,(3.10), since the first statement is a consequence of(3.10) On the semi-ellipse∂Gε, w ≤ w+, one has

z2=M2(|w − (w − ε)|2− (w − w )2).

Thus, along this left semi-ellipse, it holds that

L(w,z)|(w,z)∈∂Gε,w≤w + =

∫ w

w +

h(ξ) λ(ξ)dξ +

M2

2 (|w+− (w−− ε)|2− (w − w+)2) := g(w), w ∈ [w+, w−− ε].

Then, it holds for anyw ∈ (w+, w−− ε)that

dg(w)

h(w) λ(w) −M2(w − w+)

= (w − w+)

 1 λ(w)



s2− σ(w) − σ (w+)

w − w+



−M2



= (w − w+)



s2− σ′(ξ)

 , w+< ξ < w,

<0.

The function g is therefore strictly decreasing forw ∈ [w+, w−− ε]and attains its strict minimum on this interval at the end-pointw = w−− ε, i.e

L(w,z) >L(w−− ε,0), for all(w,z) ∈ ∂Gε\ { (w−− ε,0)}, w−− ε ≥ w ≥ w+.

Arguing similarly, we can see that

L(w,z) >L(ν,0), for all(w,z) ∈ ∂Gε\ { (ν,0)}, ν ≤ w ≤ w+.

The last two inequalities and(3.2)establish(3.10) The proof ofLemma 3.2is complete 

Properties of the level sets of the Lyapunov-type function(3.5)can be seen in the following lemma

Lemma 3.3 Under the assumptions and the notations of Lemma 3.2 , the set

is a compact set, lies entirely inside Gε, positively invariant with respect to(3.3), and has the point(w+,0)as an interior point.

As a consequence, the initial-value problem for(3.3)with initial condition(u(0), v(0)) = (w0, w0) ∈Ωεadmits a unique global solution(w(y),z(y))defined for all y≥0 Moreover, this trajectory converges to(w+,0)as y→ +∞, i.e.,

lim

y→+∞(w(y),z(y)) = (w+,0).

This means that the equilibrium point(w+,0)is asymptotically stable andΩεis a subset of the domain of attraction of(w+,0).

Proof Evidently,Ωεis a compact set We claim that the setΩεis in the interior of Gε Assume the contrary, then there is a

point U0∈Ωε∩ ∂Gε Then, as seen inLemma 3.2, the minimum of L over∂Gεis attained atw = w−− ε, so

L(U0) ≥L(w−− ε,0) >L(w−−2ε,0)

which is a contradiction, since U0∈Ωε,L(U0) ≤L(w−−2ε,0) Thus, the closed curve L(w,z) =L(w−−2ε,0)lies entirely

in the interior of Gε Moreover, it is derived fromLemma 3.1that

dL(w(y),z(y))

Thus,

L(w(y), w(y)) ≤L(w(0),z(0)) ≤L(w−−2ε,0), ∀y>0.

The last inequality means that any trajectory starting inΩεcannot cross the closed curve L(w,z) =L(w−−2ε,0) Therefore, the compact setΩεis positively invariant with respect to(4.3) As known in the standard existence theory of differential equations,(3.3)has a unique solution for y≥0 whenever U(0) ∈Ωε On the other hand, we set

E= { (w,z) ∈Ωε| ˙L(w,z) =0} = { (w,z) ∈Ωε|z=0}

It is derived from LaSalle’s invariance principle that every trajectory of(3.3)starting inΩεapproaches the largest invariant

set M of E as y→ ∞ Thus, to complete the proof, we need only to point out that

M= { (w+,0)}.

This can be done by proving that no solution can stay identically in E, except the trivial solution(w,z)(y) ≡ (w+,0) Indeed, let(w,z)be a solution that stays identically in E Then,

dw(y)

dy =z(y) ≡0,

which implies

w ≡ w−,

since(w−,0)is the unique equilibrium point inΩε Thus, every every trajectory of(2.6)starting any point inΩεmust approach(w−,0)as y→ ∞ The proof ofLemma 3.3is complete 

It follows fromLemma 3.3that the union

provides us with a sharp estimate for the attraction domain of the stable node(w+,0)

Clearly,

Ω = { (w,z) |L(w,z) <L(w−,0)}

=



(w,z) | ∫ w

w −

h(ξ) λ(ξ)dξ +

z2

2 <0



The following theorem is the main result of this section

Theorem 3.4 The set

Ω=



(w,z) |

∫ w

w −

h(ξ) λ(ξ)dξ +

z2

2 <0



is a subset of the domain of attraction of the stable node(w+,0): To every(w0, w0) ∈Ω, there exists a unique solution of the initial-value problem for(3.3)starting at(w0, w0)defined globally for all y≥0 which converges to(w+,0)as y→ +∞.

4 Existence of traveling waves and numerical illustration

4.1 The stable trajectory and existence of traveling waves

In this section we will establish the existence of traveling waves by finding out when the stable trajectory of a saddle point enters the attraction domain of the stable node For definitiveness, we still assume that we are still concerned with a 2-shock satisfying Lax shock inequalities and thatw+< w−, since the argument for the other cases are similar

Theorem 4.1 Under the assumptions(H1)and(H2)in the previous section, there exists a unique traveling wave of (1.1) con-necting the states(v−, w−)and(v+, w+).

Proof It follows fromProposition 2.1that the point(w−,0)is a saddle point Let us now consider the stable trajectories leaving the saddle point(w−,0) Since the stable trajectories are tangent to the eigenvector⟨1, β2(w−)⟩, whereβ2(w−) >0

is the positive root of the characteristic equation(2.8), one of them leaves the saddle point(w−,0)at y = −∞in the quadrant

and the other leaves the saddle point at y= −∞in the quadrant

If the straight line betweenw±does not intersect the graph ofσin the rangew > w−, then only the stable trajectory goes

into Q2may converge to the stable node And we will show that the stable trajectory goes into Q2in fact converges to the stable node(w+,0) Indeed, in a neighborhood of the saddle point(w−,0), says|z| ≤2sκ/m, it holds that

Multiplying it by the second equation of(3.3)by z=dw/dy, from(4.2)we get

z dz

dy = −

z2

2λ(w) (λ

′(w)z+2sµ(w)) − λ(w)h(w)dw

dy

< − λ(w)h(w)dw

Integrating the last inequality over(−∞,y), we get

∫ y

z dz

dy dy<

∫ y

−h(w) λ(w)

dw

dy dy

Fig 1 Numerical approximation of a traveling wave inExample 4.1

Fig 2 Numerical approximation of a traveling wave inExample 4.2

or

z2

2 < −

∫ w

w −

h(ξ) λ(ξ)dξ,

by using the boundary conditions(2.3) This yields

∫ w

w −

h(ξ)

λ(ξ)dξ +

z2

2 <0.

so that

whereΩ is given by(3.13) Thus, one stable trajectory leaving the saddle point(w−,0)at y = −∞enters the domain

of attractionΩ This establishes a saddle-to-stable connection between(w−,0)and(w+,0) The proof ofTheorem 4.1is complete 

4.2 Numerical illustration

We illustrate the existence of traveling waves by an approximation The trajectory of(2.5)leaving the saddle point (w−,0)at−∞and approaching the stable node(w+,0)at+∞is approximated by a trajectory starting near the saddle point converges to the stable node We use the solver ‘‘ode45’’ in MATLAB to generate approximate solutions of(3.3)

Example 4.1 We consider the system(1.1)where

σ(w) = w3+ w, λ(w) =2+sin(w), µ(w) =1+ | w|.

Let us take

w+=0.5, w−=1.5.

A trajectory of(3.3)starting at(w,z) = (w−−0.01; −0.001)converges to(w+,0) = (0.5,0)as y→ +∞, seeFig 1

Example 4.2 We take

σ(w) = w3+ w, λ(w) =2+cos(w), µ(w) =e−|w|.

Let us take

w+=1, w−=2.

A trajectory of(3.3)starting at(w,z) = (w−−0.01; −0.001)converges to(w+,0) = (1,0)as y→ +∞, seeFig 2

References

[1] P.G LeFloch, Hyperbolic Systems of Conservation Laws The Theory of Classical and Nonclassical Shock Waves, in: Lectures in Mathematics, ETH Zürich, Basel, 2002.

[2] P.G LeFloch, M.D Thanh, Nonclassical Riemann solvers and kinetic relations I An hyperbolic model of elastodynamics, Z Angew Math Phys 52 (2001) 597–619.

[3] M.D Thanh, Global existence of traveling wave for general flux functions, Nonlinear Anal.: TMA 72 (1) (2010) 231–239.

[4] M.D Thanh, Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion, Nonlinear Anal.: TMA 72 (6) (2010) 3136–3149 [5] B.T Hayes, P.G LeFloch, Non-classical shocks and kinetic relations: scalar conservation laws, Arch Ration Mech Anal 139 (1) (1997) 1–56 [6] N Bedjaoui, P.G LeFloch, Diffusive–dispersive traveling waves and kinetic relations I Non-convex hyperbolic conservation laws, J Differential Equations 178 (2002) 574–607.

[7] N Bedjaoui, P.G LeFloch, Diffusive–dispersive traveling waves and kinetic relations II A hyperbolic–elliptic model of phase-transition dynamics, Proc Roy Soc Edinburgh Sect A 132 (2002) 545–565.

[8] N Bedjaoui, P.G LeFloch, Diffusive–dispersive traveling waves and kinetic relations III An hyperbolic model from nonlinear elastodynamics, Ann Univ Ferra Sc Mat 44 (2001) 117–144.

[9] N Bedjaoui, C Chalons, F Coquel, P.G LeFloch, Non-monotone traveling waves in van der Waals fluids, Ann Appl 3 (2005) 419–446.

[10] J Bona, M.E Schonbek, Traveling-wave solutions to the Korteweg-de Vries–Burgers equation, Proc Roy Soc Edinburgh Sect A 101 (1985) 207–226 [11] D Jacobs, W McKinney, M Shearer, Travelling wave solutions of the modified Korteweg-deVries-Burgers equation, J Differential Equations 116 (2) (1995) 448–467.

[12] M Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch Ration Mech Anal 81 (1983) 301–315 [13] M Slemrod, The viscosity–capillarity criterion for shocks and phase transitions, Arch Ration Mech Anal 83 (1983) 333–361.

[14] H Fan, A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids, J Differential Equations 103 (1) (1993) 179–204 [15] H Fan, Traveling waves, Riemann problems and computations of a model of the dynamics of liquid/vapor phase transitions, J Differential Equations

150 (1) (1998) 385–437.

[16] M Shearer, Y Yang, The Riemann problem for a system of mixed type with a cubic nonlinearity, Proc Roy Soc Edinburgh Sect A 125 (1995) 675–699 [17] P.G LeFloch, M.D Thanh, Nonclassical Riemann solvers and kinetic relations II An hyperbolic–elliptic model of phase transition dynamics, Proc Roy Soc Edinburgh Sect A 132 (1) (2002) 181–219.

[18] P.G LeFloch, M.D Thanh, Nonclassical Riemann solvers and kinetic relations III A nonconvex hyperbolic model for van der Waals fluids, Electron J Differential Equations 72 (2000) 19 pp.

[19] P.G LeFloch, M.D Thanh, Properties of Rankine–Hugoniot curves for Van der Waals fluid flows, Japan J Indus Appl Math 20 (2) (2003) 211–238 [20] P.G LeFloch, M.D Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Comm Math Sci 1 (4) (2003) 763–797 [21] P.G LeFloch, M.D Thanh, The Riemann problem for shallow water equations with discontinuous topography, Comm Math Sci 5 (4) (2007) 865–885 [22] D Kröner, P.G LeFloch, M.D Thanh, The minimum entropy principle for fluid flows in a nozzle with discontinuous crosssection, ESAIM: Math Mod Numer Anal 42 (2008) 425–442.

[23] M.D Thanh, The Riemann problem for a non-isentropic fluid in a nozzle with discontinuous cross-sectional area, SIAM J Appl Math 69 (6) (2009) 1501–1519.