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

Nonlinear Analysis: Real World Applications 12 (2011) 236–245

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Nonlinear Analysis: Real World Applications
journal homepage: www.elsevier.com/locate/nonrwa

Existence of traveling waves in elastodynamics with variable viscosity
and capillarity
Mai Duc Thanh ∗
Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam

article

abstract

info

Article history:
Received 17 May 2010
Accepted 4 June 2010

Motivated by our earlier works, Thanh (2010) [3,4], we study the global existence of traveling 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 estimate 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 trajectory 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.

Keywords:
Conservation law
Traveling wave
Shock wave
Lax shock inequalities
Elastodynamics
Viscosity
Capillarity
Diffusion
Dispersion
Equilibria
Lyapunov function
Attraction domain

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 viscosity µ(w) and variable nonlinear capillarity λ(w):



w2
∂t v − ∂x σ (w) = λ′ (w) x − (λ(w)wx )x + (µ(w)vx )x ,
2

x

(1.1)

∂t w − ∂x v = 0.
Here, the unknown v and w > −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 stress σ = σ (w) is a function of the strain w . The function µ(w) characterizes the viscosity
inducing diffusion effect and the function λ(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 function σ is assumed to be differentiable
and

σ′ > 0
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: ,

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.nonrwa.2010.06.010

(1.2)


M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

∂t v − ∂x σ (w) = 0,
∂t w − ∂x v = 0,

237

(1.3)

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

µ(w) > κ,
λ(w) > 0,
|λ′ (w)| < m for all w,
where m > 0, κ > 0 are constants.

(1.4)

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
viscosity µ(w) and capillarity λ(w), their effects may not be negligible when the rates of change µ′ (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 Section 2 we 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 Section 3 we 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
Section 4 we 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 Section 4 some 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
−1


−σ ′ (w)
0

which gives the characteristic equation
det(A − λI ) = λ2 − σ ′ (w) = 0.
Since σ ′ (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− )
(v+ , w+ )

if x < st ,
if x > st ,


238

M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

and satisfies the Rankine–Hugoniot relations
s(v+ − v− ) + (σ (w+ ) − σ (w− )) = 0,
s(w+ − w− ) + (v+ − v− ) = 0.

(2.1)

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

λi (u+ ) < s(u− , u+ ) < λi (u− ),

i = 1, 2.

(2.2)

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 (v, w)(y) = (v± , w± )

y→±∞

lim

y→±∞

d
dy

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

y→±∞

d2
dy2

(2.3)

(v(y), w(y)) = (0, 0).

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

w
s(v − v− ) + (σ (w) − σ (w− )) = λ′ (w)
+ λ(w)w′′ − µ(w)v ′ ,
2
s(w − w− ) + (v − v− ) = 0.
′2

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

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

w ′2
2

+ λ(w)w ′′ + sµ(w)w ′

or

w′′ = −

λ′ (w) ′2 sµ(w) ′ σ (w) − σ (w− ) − s2 (w − w− )
w −
w +
.
2λ(w)
λ(w)
λ(w)

(2.4)

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′ (w)z + 2sµ(w)) −

h(w)

λ(w)

(2.5)

,

or, in a more compact form
dU
dy

= F (U ),

−∞ < y < +∞

(2.6)

where
U = (w, z ),

F (U ) =



z, −

z
2λ(w)

(λ′ (w)z + 2sµ(w)) −

h(w)

λ(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), where w± and the shock speed s are related by (2.1).

M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

239

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



0



1

′
2
DF (w± , 0) =  σ (w± ) − s

λ(w± )

−

sµ(w± )  .

(2.7)

λ(w± )

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




 = 0,
sµ(w± )
−
− β 
λ(w± )



−β

|DF (w± , 0) − β| =  σ ′ (w± ) − s2
 λ(w )

1

±

or
sµ(w± )

β2 +

λ(w± )

β+

s2 − σ (w± )

λ(w± )

= 0.

(2.8)

Assume that the jump satisfies the Lax shock inequalities

λ2 (w− ) > s > λ2 (w+ )
which yields

σ (w− ) > s2 > σ (w+ ).

(2.9)

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 with λ1 with 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 with λ2 with 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 2.1 indicates 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 values w± satisfy

|w+ − w− | ≤

2κ

,

(3.1)

m
where κ, m are positive constants as in (1.4).
(H2) There exists a value ν < w+ such that
w−

∫
ν

h(ξ )

λ(ξ )

dξ < 0,

∫

w

and
w+

h(ξ )
dξ > 0,
λ(ξ )

w ∈ [ν, w− ],

(3.2)

where
h(w) = s2 (w − w− ) − (σ (w) − σ (w− )).
Example 3.1. We omit the condition w > −1, and extend the function σ = σ (w) to the whole −∞ < w < +∞. Let the
function σ be twice differentiable and strictly convex:

σ ′′ (w) > 0,

w ∈ R.

Then the Lax shock inequalities

λ2 (w+ ) < s(u+ , u− ) < λ2 (w− )


240

M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

are equivalent to the condition

w− > w+ .
And
w−

∫

−∞

h(ξ )
dξ = −∞.
λ(ξ )

Thus, for any pair (w+ , w− ), there is always such a ν satisfying (H2).
Example 3.2. Let us take the model of elastodynamics described by the Eqs. (1.1), where the stress σ is a twice differentiable
function of w satisfying

σ ′ (0) > 0,

wσ ′′ (w) > 0 for w ̸= 0,

(which implies σ ′ (w) > 0 for all w > −1) and
lim σ (w) = −∞,

lim σ ′ (w) = +∞.

w→+∞

w→−1

(See [2]). Assume λ(w) ≡ λ = constant. Then, as argued similarly as in [2], we can see that for each w− > 0, there exists
exactly one value denoted by ϕ∞ (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

if w < ϕ∞ (w− ),

where
s2 (w− , w) =

σ (w) − σ (w− )
.
w − w−

Moreover, for each pair (w0 , w1 ), there is exactly one value denoted by ϕ # (w0 , w1 ) such that
s2 (w0 , w1 ) = s2 (w0 , ϕ # (w0 , w1 )).
Setting

w∗ = ϕ # (w− , ϕ∞ (w− )) := ζ (w− ),
and taking w+ such that ζ (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
dz

= z,
=−

z

(λ′ (w)z + 2sµ(w)) −

h(w)

,

dy
2λ(w)
λ(w)
Let us define a Lyapunov-type function candidate
L(w, z ) =

∫

w

h(ξ )

w+

λ(ξ )

dξ +

z2
2

−∞ < y < +∞.

.

(3.3)

(3.4)

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)},
L˙ (w, z ) < 0 in D \ {z = 0},
L˙ (w, z ) = 0 on D ∩ {z = 0}.
Proof. First, we have immediately
L(w+ , 0) = 0,

L(w, z ) ≥

∫

w

w+

h(ξ )
dξ > 0,
λ(ξ )

(w, z ) ∈ D, w ̸= w+ ,

(3.5)


M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

241

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 , dz
dy dy



h(w)

z
h(w)
=
, z z, −
(λ′ (w)z + 2sµ(w)) −
λ(w)
2λ(w)
λ(w)
=−
<−

z2
2λ(w)
z2
2λ(w)

(λ′ (w)z + 2sµ(w))
(−m|z | + 2sκ) < 0,

(3.6)

for (w, z ) in D, z ̸= 0. This completes the proof of Lemma 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
m|w+ − w− |

.

(3.7)

Next, for each small number ε such that w− − w+ > 2ε > 0, we set

(w − w+ )2
z2
+
≤ 1, w ≥ w+
(w+ − (w− − ε))2
(M (w+ − (w− − ε)))2


z2
(w − w+ )2
+
≤
1
,
w
≤
w
∪ (w, z ) |
+ ,
(w+ − ν)2
(M |w+ − (w− − ε)|)2


Gε =

(w, z ) |



(3.8)

where M is given by (3.7) and ν is 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

h(ξ )

w+

λ(ξ )

dξ

is strictly increasing for w near w− , 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− )
> s2 for |w − w− | < θ .
w − w−
This implies that for |w − w− | < θ ,
h(w) > 0,
which establishes the statement. Fix this θ > 0, it then holds that
L(w− , 0) > L(w− − ε, 0),

0 < ε < θ.

(3.9)

The following lemma provides us with properties of the sets Gε .
Lemma 3.2. For any positive number ε so that 0 < 2ε < θ < w− − w+ , where θ is given in (3.9), let Gε be the set defined
by (3.8) and let ∂ Gε denote 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.
L(w, z ) > L(w− − ε, 0),

for all (w, z ) ∈ ∂ Gε \ {(w− , 0)}.

(3.10)

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
z 2 = M 2 (|w+ − (w− − ε)|2 − (w − w+ )2 ).


242

M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

Thus, along this left semi-ellipse, it holds that
L(w, z )|(w,z )∈∂ Gε ,w≤w+ =

∫

w

h(ξ )

M2

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

dξ +

Then, it holds for any w ∈ (w+ , w− − ε) that
dg (w)
dw

h(w)

− M 2 (w − w+ )




σ (w) − σ (w+ )
1
2
2
s −
−M
= (w − w+ )
λ(w)
w − w+
 2

s − σ ′ (ξ )
= (w − w+ )
− M 2 , w+ < ξ < w,
λ(w)
< 0.

=

λ(w)

The function g is therefore strictly decreasing for w ∈ [w+ , w− − ε] and attains its strict minimum on this interval at the
end-point w = 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 of Lemma 3.2 is 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

Ωε := {(w, z ) | L(w, z ) ≤ L(w− − 2ε, 0)}

(3.11)

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 ) ∈ Ωε admits 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 (w(y), z (y)) = (w+ , 0).

y→+∞

This means that the equilibrium point (w+ , 0) is asymptotically stable and Ωε is a subset of the domain of attraction of (w+ , 0).
Proof. Evidently, Ωε is a compact set. We claim that the set Ωε is in the interior of Gε . Assume the contrary, then there is a
point U0 ∈ Ωε ∩ ∂ Gε . Then, as seen in Lemma 3.2, the minimum of L over ∂ Gε is attained at w = 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 from Lemma 3.1 that
dL(w(y), z (y))
dy

≤ 0.

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 in Ωε cannot cross the closed curve L(w, z ) = L(w− − 2ε, 0). Therefore,
the compact set Ωε is 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 in Ωε approaches 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,

M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

243

which implies

w ≡ w− ,
since (w− , 0) is the unique equilibrium point in Ωε . Thus, every every trajectory of (2.6) starting any point in Ωε must
approach (w− , 0) as y → ∞. The proof of Lemma 3.3 is complete.
It follows from Lemma 3.3 that the union

Ω = ∪0<ε<(w− −w+ )/2 Ωε

(3.12)

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
h(ξ )
z2
= (w, z ) |
dξ +
<0 .
2
w− λ(ξ )

(3.13)

The following theorem is the main result of this section.
Theorem 3.4. The set


∫
Ω = (w, z ) |

w

h(ξ )

w−

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 that w+ < 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) connecting the states (v− , w− ) and (v+ , w+ ).
Proof. It follows from Proposition 2.1 that 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− )⟩, where β2 (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
Q1 = {(w, z )|w > w− , z > 0},

(4.1)

and the other leaves the saddle point at y = −∞ in the quadrant
Q2 = {(w, z )|w < w− , z < 0}.

(4.2)

If the straight line between w± does not intersect the graph of σ in the range w > w− , then only the stable trajectory goes
into Q2 may converge to the stable node. And we will show that the stable trajectory goes into Q2 in 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

λ′ (w)z + 2sµ(w) > 0.

(4.3)

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)) −

h(w) dw

λ(w) dy

,

h(w) dw

λ(w) dy

in Q2 ∩ {|z | ≤ 2sκ/m}.

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

∫

y

z
−∞

dz
dy

dy <

∫

y

−
−∞

h(w) dw

λ(w) dy

dy

(4.4)


244

M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

Fig. 1. Numerical approximation of a traveling wave in Example 4.1.

Fig. 2. Numerical approximation of a traveling wave in Example 4.2.

or

z2
2

<−

∫

w
w−

h(ξ )
dξ ,
λ(ξ )

by using the boundary conditions (2.3). This yields

∫

w

h(ξ )

w−

λ(ξ )

dξ +

z2
2

< 0.

so that

(w, z ) = (w(y), z (y)) ∈ Ω ,

y < 0,

(4.5)

where Ω is given by (3.13). Thus, one stable trajectory leaving the saddle point (w− , 0) at y = −∞ enters the domain
of attraction Ω . This establishes a saddle-to-stable connection between (w− , 0) and (w+ , 0). The proof of Theorem 4.1 is
complete.


M.D. Thanh / Nonlinear Analysis: Real World Applications 12 (2011) 236–245

245

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) = w 3 + 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 → +∞, see Fig. 1.
Example 4.2. We take

σ (w) = w 3 + 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 → +∞, see Fig. 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.