DSpace at VNU: Global existence of diffusive-dispersive traveling waves for general flux functions

Using Lyapunov stability techniques, we reduce the global problem of finding traveling waves to considering local behaviors of a stable trajectory of the saddle point.. This is a certain

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Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Global existence of diffusive–dispersive traveling waves for general

flux functions

Mai Duc Thanh∗

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 28 April 2008

Accepted 12 June 2009

MSC:

35L65

76N10

76L05

Keywords:

Traveling wave

Conservation law

Diffusive

Dispersive

Shock wave

a b s t r a c t

We establish a global existence of traveling waves for diffusive–dispersive conservation laws for locally Lipschitz flux functions Using Lyapunov stability techniques, we reduce the global problem of finding traveling waves to considering local behaviors of a stable trajectory of the saddle point

© 2009 Elsevier Ltd All rights reserved

1 Introduction

We consider in this paper the existence of a certain kind of smooth solutions, called the traveling waves, of the following

third-order partial differential equation

∂t u(x,t) + ∂x f(u(x,t)) =a∂xx u(x,t) +b∂xxx u(x,t), x∈R,t >0, (1.1)

where, a,b represent the diffusion and dispersion coefficients, respectively Here, we assume that a and b are positive

constants

When traveling waves of(1.1)exist, one is interested in their limit when a,b→0+ This is a certain kind of admissibility criteria for shock waves of the conservation law

Conversely, when a shock wave of(1.2)exists, it has been shown that the corresponding traveling waves also exist, under certain circumstances, see [1

Diffusive–dispersive traveling waves have been studied by many authors, see [2–8], etc In [1], the relationship between the existence of traveling waves of(1.1)and the existence of classical and nonclassical shock waves was considered A

geometrical distinction between the classical shocks and nonclassical shocks is that in the case of classical shocks, the line connecting the two left-hand and right-hand states does not cross the graph of the flux function in the interval between these two states, while it is the case for nonclassical shocks The reader is referred to [9–15] for classical shocks, to [4,16,17,

16,1,18–20] for nonclassical shock waves Recently, non-monotone traveling waves for van der Waals fluids with diffusion and dispersion terms were obtained in [21]

∗Tel.: +84 8 37242181; fax: +84 8 37242195.

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

0362-546X/$ – see front matter © 2009 Elsevier Ltd All rights reserved.

The present paper devotes to establishing a global existence of traveling waves of(1.1), where the flux function f is

solely locally Lipschitz Our strategy is as follows First, we transform the problem of finding a traveling wave connecting

a left-hand state u−to a right-hand state u+to a 2×2 system of ordinary differential equations Second, we consider the asymptotical behavior of trajectories of the two equilibria(u±,0)of the system, which turn out to be a stable node and a saddle point Third, we define a Lyapunov function in such a way that this function enables us to estimate the domain of attraction of the stable node We then show that the saddle point is in fact on the boundary of the attraction domain of the node Since a saddle point always admits stable trajectories, this raises the hope that a stable trajectory from the saddle would eventually enter the domain of attraction of the node Whenever this happens, a connection between the stable node

and the saddle is established This also gives us a traveling wave connecting the states u± Finally, a sharp estimation of the domain of attraction of the node using Lyapunov function yields the existence result

The organization of the paper is as follows In Section2, we will provide basic concepts and properties of traveling waves

of(1.1)and the stability of equilibria of the associated differential equation Furthermore, we will establish an invariance result concerning traveling waves of(1.1), relying on LaSalle’s invariance principle In Section3we will demonstrate that traveling waves of(1.1)exist whenever there is a Lax shock of the associate conservation law(1.2)satisfying Oleinik’s entropy condition

2 Traveling waves and stability of equilibria

Let us consider traveling waves of(1.1)i.e., smooth solution u=u(y)depending on the re-scaled variable

y:= αx− λt

a =

x− λt

√

for some constant speedλand

α =a/ √b.

Substituting u=u(y)to(1.1), after re-scaling, the traveling wave u connecting a left-hand state u−to a right-hand state

u+satisfies the ordinary differential equation

− λdu

dy+

df(u)

dy = αd2u

dy2 +d3u

and the boundary conditions

lim

y→±∞u(y) =u±,

lim

y→±∞

du

dy =y→±∞lim

d2u

Integrating(2.2)and using the boundary condition(2.3), we find u such that

d2u

dy2 + αdu

Using(2.3)again, we deduce from(2.4)

λ = f(u+) −f(u−)

Setting

v = du

dy

we can re-write the second-order differential equation(2.4)to the following second-order system

du(y)

dy = v(y),

dv(y)

dy = − αv(y) − λ(u(y) −u−) +f(u(y)) −f(u−).

(2.6)

The system(2.6)can be written in a more compact of autonomous differential equations

dU(y)

where U= (u, v) ∈R2and

F(U) = (v, −αv +h(u)), h(u) = −λ(u−u−) +f(u) −f(u−).

We observe that the function F is locally Lipschitz in R2if f is locally Lipschitz in R From the local existence theory, it is

not difficult to check the following result, which provides us with the global existence for(2.7)

Lemma 2.1 Let f be locally Lipschitz in R Suppose that there exists a compact set W⊂R2such that any solution of

dU(y)

dy =F(U), y>0,U(0) =U0

lies entirely in W Then, there is a unique solution passing through U0defined for all y≥0.

Next, we want to study the asymptotic behavior of trajectories of(2.7) For this purpose, we consider the stability of equilibria of(2.7) It is derived from(2.5)that

F(u+,0) =0, F(u−,0) =0,

which means that(u+,0)and(u−,0)are equilibrium points of the differential equation(2.7) By definition, a point U0is

called an equilibrium point of(2.7)if

F(U0) =0.

Thus, any equilibrium point of(2.7)has the form U= (u,0), where u satisfies

h(u) = −λ(u−u−) +f(u) −f(u−) =0.

The last equality means that u,u−, λsatisfy the Rankine–Hugoniot relation for the associate conservation law(1.2) We therefore conclude that

Proposition 2.2 A point U is an equilibrium point of the autonomous differential equation(2.7)if and only if U has the form

U = (u+,0)for some constant u+so that the states u±and the shock speedλsatisfy the Rankine–Hugoniot relation for the associate conservation law(1.2).

Consequently, when U= (u+,0)is an equilibrium point of(2.7), the function

u(x,t) =



u−, x< λt,

is a weak solution of the conservation law(1.2) Conversely, a jump of(1.2)of the form(2.8)gives equilibria(u−,0), (u+,0)

of the differential equation(2.7)

Now, let us study the boundary conditions(2.3) We recall some basic concepts The reader is referred to [22] for more

details An equilibrium point U0= (u0,0)of(2.7)is positively (negatively) stable if for eachε >0, there existsδ = δ(ε) >0 such that

kU(0) −U0k < δ ⇒ kU(y) −U0k < ε, ∀y≥0,

(∀y≤0, respectively) The equilibrium point U0is positively (negatively) asymptotically stable if it is positively (negatively)

stable andδcan be chosen such that

kU(0) −U0k < δ ⇒ lim

y→∞U(y) =U0, ( lim

y→−∞U(y) =U0respectively).

Whenever a Lyapunov function defined on a domain D containing the equilibrium point U0is found, the equilibrium point

is stable A Lyapunov function for(2.7)is a continuously differentiable function L which is positive definite:

L(U0) =0, L(U) >0, U∈D\ {U0}

and such that its derivative along trajectories of(2.7)is non-positive:

˙

L(U) := ∇L(U) ·F(U) ≤0, U∈D.

Lyapunov’s stability theorem says that if such a function exists, then the equilibrium point U0is stable in D Moreover, if

˙

L(U) = ∇L(U) ·F(U) <0, U∈D\ {U0}

then the equilibrium point U0is asymptotically stable

When an equilibrium point is asymptotically stable, it is interesting to see how far from U0, the trajectories of(2.7)still

converges to U0as y approaches infinity The domain of attraction of an asymptotically stable equilibrium point U0is the

set of all points in D such that any solution of(2.7)starting from such a point exists for all y≥0 and converges to U0as y

approaches infinity

Let us consider the stability of equilibria of(2.6), or(2.7) It follows fromProposition 2.2that the setΓ of equilibria has the form(u i,0),i∈I and that

λ = f(u i) −f(u−)

This yields

h(u) = −λ(u−u−) +f(u) −f(u−)

Geometrically,Γis the intersection of the straight line connecting(u±,0)and the graph of h.

The Jacobian matrix DF(U)is given by

DF(U) =



(f0(u) − λ) −α



The characteristic equation of DF(U)is

|DF(U) − βI| = β2+ αβ − (f0(u) − λ) =0

which admits two roots as

β1= − α

2 +

r

 α 2

2

+f0(u) − λ, β2= − α

2 −

r

 α 2

2

Since we consider the asymptotic behaviors u→u+as y→ ∞and u→u−as y→ −∞, we have

Proposition 2.3. (i) If f0(u+) < λthenβ2< β1<0 The point(u+,0)is asymptotically stable.

(ii) If f0(u+) > λthenβ2<0< β1 The point(u+,0)is a saddle.

(iii) If f0(u−) < λthenβ2< β1<0 The point(u−,0)is unstable.

(iv) If f0(u−) > λthenβ2<0< β1 The point(u−,0)is a saddle.

Thus, traveling waves would exist in the cases of stable-to-saddle connection and saddle-to-saddle connection, only However, generally, it is difficult to establish saddle-to-saddle connection In what follows, we will study only the case of stable-to-saddle connection

In the rest of this section, we will establish the asymptotical behaviors of trajectories of(2.7)relying on LaSalle’s invariance

principle We first provide some more definitions.

A set M⊂D is said to be an invariant set with respect to(2.7)if

A set M⊂D is said to be a positively invariant set with respect to(2.7)if

And similarly for a negatively invariant set.

Therefore, a set M is invariant if and only if it is both positively and negatively invariant.

We also say that U(y)approaches a set M as y approaches infinity, if for everyε >0, there is Y >0 such that the distance

from a point p to a set M is less thanε:

dist(U(y),M) := inf

U∈M

Suppose that there exists a continuous differentiable function L:D→R such that

˙

We defined

LaSalle’s invariance principle states that: If Ωis a compact set that is positively invariant with respect to(2.7), and M is the

largest invariant set in E, then every solution starting inΩapproaches M as y→ ∞

Proposition 2.4 Let f be locally Lipschitz Suppose that there exists a compact setΩthat is positively invariant with respect to

(2.7) Then, every trajectory of (2.7)starting inΩapproaches the set M of equilibria of(2.7)as y→ ∞.

Proof First, it is derived fromLemma 2.1that any solution U of(2.7)starting inΩexists globally for y≥0.

Next, we will establish the asymptotic behavior of trajectories of(2.7)starting inΩ For this purpose, we will find a

function L, and a set E satisfying(2.15)and(2.16) Set

L(U) =L(u, v) =

Z u+

u

h(v)dv +1

where

h(u) := −λ(u−u−) +f(u) −f(u−).

The function L(u, v)is continuously differentiable and satisfies

˙

L(u, v) = −h(u)v + v(−αv +h(u)) = −αv2≤0, ∀(u, v) ∈R,

so that˙L(u, v)is semi-negative definite Thus, we define

E= { (u, v) ∈R2| ˙L(u, v) =0}

The set E can be simplified as follows Suppose

˙

L(u, v) =0.

Then

αv2=0

or

v =0.

Thus,

Applying LaSalle’s invariance principle, we conclude that: every trajectory of(2.7)approaches the largest invariant set

M of E as y→ ∞

Let us next show that the largest invariant set M in E coincides with the set of equilibria This can be done by proving that no solution can stay identically in E, except constant solutions u(y) ≡u i,i∈I Indeed, let(u, v)be a solution that stays

identically in E Then,

du(y)

dy = v(y) ≡0,

which implies

u≡u0=constant.

Thus,(u(y), v(y)) ≡ (u0,0)and coincides with some equilibrium point We can therefore deduce that no solution can stay

identically in E, except constant solutions This implies that the largest invariant set M in E is the set of equilibria

Thus, every trajectory of(2.7)starting any point inΩmust approach M as y→ ∞ 

3 Existence of traveling waves

In this section, we will show that the existence of a Lax shock of(1.2)between a left-hand u−and a right-hand state u+

implies the existence of a traveling wave of(1.1)connecting(u−,0)and(u+,0)

First, let us provide a brief introduction to the concept of shock waves A discontinuity of the form

u(x,t) =



u−, x< λt,

where u−,u+are relatively the left-hand and right-hand states andλis the speed of discontinuity propagation, is a weak

solution of the conservation law(1.2)in the sense of distributions iff it satisfies the Rankine–Hugoniot relation

Eq.(3.2)implies that the speed of discontinuity propagation is given by

λ =f(u+) −f(u−)

u −u .

It is known that weak solutions are not unique To select a unique solution, one constraints weak solutions to admissibility

entropy conditions In the case of scalar conservation laws, one often uses the Oleinik entropy criterion, which requires

f(u) −f(u−)

u−u−

> f(u+) −f(u−)

u+−u−

The condition(3.3)is equivalent to

f(u) −f(u+)

u−u+ < f(u+) −f(u−)

u+−u− , for any u between u+and u−.

A shock wave of(1.2)is a weak solution of the form(3.1)and satisfies the Oleinik entropy criterion(3.3) In brief, a shock

wave connecting a left-hand state u−to a right-hand state u+with shock speedλis given by(3.1), where u±andλare such that the Rankine–Hugoniot relation(3.2)and the Oleinik criterion(3.3)hold

Assume for definitiveness that u+<u− Geometrically, the inequality(3.2)means that in the interval[u+,u−], the graph

of f is lying below the straight line(∆)connecting the two points(u±,f(u±))

We now study trajectories approaching the equilibrium point(u+,0)as y tends to+∞ So we consider the differential equation

dU(y)

where

U= (u, v), F(U) = (v, −αv +h(u)),

h(u) = −λ(u−u−) +f(u) −f(u−).

Let Lip(f|[2u+−u−,u−])be the Lipschitz constant of the flux function f on the interval[2u+−u−,u−] Take an arbitrary constantγsuch that

0< γ < 1

We define

D:= { (u, v) ∈R2| (u−u+)2+ γ v2≤ (u−−u+)2} ,

L(u, v) = Z u+

u

h(v)dv +1

We claim that

Lemma 3.1 The function L1(u, v) :=L(u+u+, v), (u, v) ∈D, is positive definite in D:

L(u+,0) =0, L(u, v) >0, (u, v) ∈D

and the derivative along trajectories˙L(u, v) = ∇L(u, v) · (u0, v0)of (3.4)is semi-negative definite:

˙

L(u, v) = −αv2≤0.

Proof First, let us check thatL˙ (u, v)is semi-negative definite Indeed, it holds that

˙

L(u, v) = ∇L(u, v) · (u0, v0)

= −h(u)v + v(−αv +h(u))

= − αv2≤0, ∀(u, v) ∈D,

which means that dL(u, v)/dy is semi-negative definite.

Next, we prove that L is positive definite Obviously, L(u+,0) =0 Take any number 0<r≤ |u−−u+|and set

E r = { (u, v) ∈R2| (u−u+)2+ γ v2≤r2} ⊂D. (3.7) Let∂E r be the boundary of E r In fact,∂E ris the ellipse where the two principal axes are the line segments between(u+−r,0) and(u++r,0), and(u+, −r/√γ)and(u+,r/√γ) To prove that L is positive definite, it is sufficient to show that L attains

its positive minimum on each ellipse∂E r In other words, we will check that for any positive number r≤ |u−−u+|, it holds that

m= min

Indeed, on∂E r, one has

v2= 1

γ (r2− (u−u+)2).

Substitutingvfrom the last equation to the expression of L, we have

m= min

(u,v)∈ ∂E r

L(u, v)

u∈[u+−r,u++r]

Z u+

u

h(v)dv + 1

2γ (r2− (u−u+)2)

 Setting

φ(u) := Z u+

u

h(v)dv + 1

2γ (r2− (u−u+)2), u∈ [u+−r,u++r] ,

we reduce the above extremum problem of L to finding the minimum value of the function of one variableφ A straightfor-ward calculation gives

dφ(u)

du = −h(u) − γ1(u−u+)

= −

 1

γ + λ

 (u−u+) +f(u) −f(u+)

= (u−u+)



f(u) −f(u+)

u−u+

−

 1

γ + λ



<0,

for u∈ [u+−r,u++r], where the last inequality is derived from(3.5) Thus, the functionφis decreasing and therefore

attains at u=u++r its minimum value

m=

Z u+

u++r

h(v)dv

=

Z u++r

u+

(u−u+)



f(u) −f(u+)

u−u+

− λ



where the Oleinik criterion is used Since L is positive on any ellipse centered at(u+,0)with the two principal axes to be the line segments between(u+−r,0)and(u++r,0), and(u+, −r/√γ)and(u+,r/√γ), where 0<r <u−−u+, L is positive on D\ { (u+,0)}.Lemma 3.1is completely proved 

Let us now try to find an estimate of the domain of attraction of the equilibrium point(u+,0) Set

We will show thatΩcis a subset of the attraction domain of the equilibrium point(u+,0)

Lemma 3.2 The setΩc is a compact set and is positively invariant with respect to(3.4) Consequently, the system(3.4)has a unique global solution for y≥0 whenever U(0) ∈Ωc Moreover, this trajectory converges to(u+,0)as y→ ∞.

Proof Obviously,Ωcis a compact set We claim that the setΩc is in the interior of E r Assume the contrary, then there is a

point U0∈Ωc which lies on the boundary of E r Then, by definition of minimum

L(U0) ≥m>c

which is a contradiction, since U0 ∈ Ωc,L(U0) ≤ c Thus, the closed curve L(u, v) = c lies entirely in the interior of E r Moreover, it is derived fromLemma 3.1that

dL(u(y), v(y))

Thus,

L(u(y), v(y)) ≤L(u(0), v(0)) ≤c, ∀y>0.

The last inequality means that any trajectory starting inΩc cannot cross the closed curve L(u, v) =c Therefore, the compact

setΩcis positively invariant with respect to(3.4) According toLemma 2.1,(3.4)has a unique solution for y≥0 whenever

U(0) ∈Ωc It is derived fromProposition 2.4that any trajectory U starting inΩcconverges to(u+,0)as y→ ∞ The proof

ofLemma 3.2is complete 

In the following, we will show that the attraction domain, or even the its subsetΩc can be large enough such that stable trajectories corresponding of the saddle(u−,0)eventually enters Whenever this happens, there is a traveling wave connecting the two points(u±,0) The following proposition implies that the point(u−,0)is in fact a limit point of the attraction domain of the equilibrium point(u+,0)

Proposition 3.3 The domain of attraction of the stable node(u+,0)includes the setΩ, whereΩis defined by



(u, v) ∈D|L(u, v) +

Z u−

u+

h(w)dw <0



The setΩis open, connected and contains the line segment[u+,u−)×{0} Consequently, the saddle(u−,0)lies on the boundary

∂Ω:

(u−,0) ∈ ∂Ω.

Proof Taking r =u−−u+− δ/2 for an arbitrary smallδ >0, we get u++r− δ/2=u−− δ Then, choosing a constant

c as

c:=

Z u++r− δ/ 2

u+

−h(v)dv =

Z u−− δ

u+

−h(v)dv,

since h(u) <0,u∈ (u+,u−), we have

0<c<m.

From(3.10)it holds that

Ωc = { (u, v) ∈D|L(u, v) ≤c}

=



(u, v) ∈D|

Z u−− δ

u

h(w)dw + v2

2 ≤0



Lettingδ →0, we can see that the domain of attraction of the equilibrium point(u+,0)includes the setΩ, where



(u, v) ∈D|

Z u−

u

h(w)dw + v2

2 <0



=



(u, v) ∈D|L(u, v) + Z u−

u+

h(w)dw <0



Due to the continuity, the setΩis open, connected and contains the line segment[u+,u−)×{0} This implies that the saddle point(u−,0)lies on the boundary∂Ω The proof ofProposition 3.3is complete 

The above argument leads to the following theorem which provides us with the existence of traveling waves associate with shock waves

Theorem 3.4 Assume that the flux function f is locally Lipschitz, and that there is a shock wave of (1.2)connecting the left-hand state u−and the right-hand state u+with the shock speedλsatisfying the Oleinik entropy condition In addition, assume that there is a point(u0, v0) ∈ Ω, whereΩis defined by(3.11), such that the trajectory of (2.7)starting at(u0, v0)converges to

(u−,0)as y→ −∞ Then, there exists a traveling wave of (1.1)connecting the states u−,u+.

Proof Let(u1, v1) = (u1(y), v1(y)),y≤0 be the trajectory starting at(u0, v0)and converging to(u−,0)as y→ −∞ It follows fromProposition 3.3that there exists a trajectory(u2, v2) = (u2(z), v2(z))of(3.4)starting at(u0, v0), defined for

all z≥0, such that this trajectory tends to(u+,0)as z→ +∞ Therefore, the function u(y)defined by

u(y) =



u1(y), for y≤y0,

u2(y−y0), for y≥y0,

is the solution of(2.4)satisfying the boundary conditions(2.3) Consequently, u=u(y),y∈R is a traveling wave of(1.1)

connecting u±and passing through u0∈ (u−,u+).Theorem 3.4is completely proved 

Acknowledgements

The author would like to express his sincere thanks to the reviewers for their very constructive comments and helpful suggestions

[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] J Bona, M.E Schonbek, Traveling-wave solutions to the Korteweg–de Vries–Burgers equation, Proc Roy Soc Edinburgh, Sec A 101 (1985) 207–226 [3] D Jacobs, W McKinney, M Shearer, Travelling wave solutions of the modified Korteweg–deVries–Burgers equation, J Differential Equations 116 (2) (1995) 448–467.

[4] B.T Hayes, P.G LeFloch, Non-classical shocks and kinetic relations: Scalar conservation laws, Arch Ration Mech Anal 139 (1) (1997) 1–56 [5] B Hayes, M Shearer, Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes, Proc Roy Soc Edinb., Sect A, Math 129 (4) (1999) 733–754.

[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 132 A (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 Ferrara Sci Mat 44 (2001) 117–144.

[9] O.A Oleinik, Construction of a generalized solution of the cauchy problem for a quasi-linear equation of first order by the introduction of vanishing viscosity, Amer Math Soc Transl Ser 2 33 (1963) 277–283.

[10] P.D Lax, Shock waves and entropy, in: E.H Zarantonello (Ed.), Contributions to Nonlinear Functional Analysis, 1971, pp 603–634.

[11] T.P Liu, The Riemann problem for general 2×2 conservation laws, Trans Amer Math Soc 199 (1974) 89–112.

[12] D Kröner, P.G LeFloch, M.D Thanh, The minimum entropy principle for fluid flows in a nozzle with discontinuous crosssection, Math Mod Numer Anal 42 (2008) 425–442.

[13] 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 [14] P.G LeFloch, M.D Thanh, The Riemann problem for shallow water equations with discontinuous topography, Comm Math Sci 5 (4) (2007) 865–885 [15] 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.

[16] B.T Hayes, P.G LeFloch, Nonclassical shocks and kinetic relations: Strictly hyperbolic systems, SIAM J Math Anal 31 (2000) 941–991.

[17] P.G LeFloch, An introduction to nonclassical shocks of systems of conservation laws, in: D Kröner, M Ohlberger, C Rohde (Eds.), An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, in: Lect Notes Comput Sci Eng., vol 5, Springer, 1999, pp 28–72.

[18] 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.

[19] 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 132 A (1) (2002) 181–219.

[20] 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.

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

[22] H.K Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 2002.