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Publisher: Taylor & Francis

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Applicable Analysis: An International Journal

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Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws

Mai Duc Thanha & Nguyen Huu Hiepb a

Department of Mathematics, International University, Vietnam National University-Ho Chi Minh City, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam

b Faculty of Applied Science, University of Technology, 268 Ly Thuong Kiet str., District 10, Ho Chi Minh City, Vietnam

Published online: 30 Jul 2014

To cite this article: Mai Duc Thanh & Nguyen Huu Hiep (2014): Existence of traveling waves to any

Lax shock satisfying Oleinik’s criterion in conservation laws, Applicable Analysis: An International Journal, DOI: 10.1080/00036811.2014.915520

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Existence of traveling waves to any Lax shock satisfying Oleinik’s

criterion in conservation laws

Mai Duc Thanha ∗and Nguyen Huu Hiepb

a Department of Mathematics, International University, Vietnam National University-Ho Chi Minh City, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam; b Faculty of Applied Science, University of Technology, 268 Ly Thuong Kiet str., District 10, Ho Chi Minh City,

Vietnam

Communicated by M Shearer

(Received 9 March 2013; accepted 13 April 2014)

Given any shock wave of a conservation law where the flux function may not

be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock The paper develops the method of estimating attraction domain for traveling waves we have studied before

Keywords: Conservation laws; traveling wave; shock; diffusion; dispersion;

equilibria; asymptotical stability; Lyapunov function; LaSalle’s invariance principle; attraction domain

AMS Subject Classifications: 35L65; 74N20; 76N10; 76L05

1 Introduction

Let us consider the scalar conservation law

∂ t u (x, t) + ∂ x f (u(x, t)) = 0, x ∈ R I , t > 0, (1.1)

where the flux function f is assumed to be merely differentiable, and may not be convex

or concave, see Figure1

Solutions of (1.1) are understood in the sense of distributions and so are called weak

solutions Weak solutions are in general discontinuous as they may contain shock waves,

which are discontinuous jumps As well known, weak solutions are not unique Moreover, there are several kinds of shock waves, depending on the corresponding admissibility criterion, such as classical shocks and nonclassical shocks Each of these kinds of shock waves can be suitable for a particular application A reasonable mathematical modeling of the application can be made through the inclusion of some diffusion and dispersion terms

in the Equation (1.1) to get a diffusive-dispersive models

∗Corresponding author Email: mdthanh@hcmiu.edu.vn

Figure 1 A nonconvex flux function and a shock.

∂ t u (x, t) + ∂ x f (u(x, t)) = β(b(u)u x ) x + γ (c1(u)(c2(u)u x ) x ) x , x ∈ R I , t > 0, (1.2)

where β > 0, γ > 0 are small scales, the functions b(u), c1(u), c2(u), andu ∈ R I are

assumed to be differentiable and positive, and c2(u), u ∈ R I , is twice differentiable The first

and the second terms on the right-hand side of (1.2) represent the diffusion and dispersion coefficients, respectively It has been known that traveling waves of (1.2) connecting a

given left-hand state u−and a right-hand state u+tends to the shock wave connecting these left-hand and right-hand states whenβ and γ tend to zero Thus, the existence of traveling

wave solutions of (1.2) corresponding to a shock wave can justify the admissibility criterion used to select this shock wave

In [1], given a diffusive-dispersive model with constant viscosity and capillarity coeffi-cients, the author proposed a method of estimating attraction domain of an asymptotically stable equilibrium point to establish the existence of a traveling wave associated with a given Lax shock In this paper, we will develop the method and improve the argument in [1] to show that for any given Lax shock of (1.1) satisfying the Oleinik’s condition, there are corresponding traveling waves of (1.2), providing that the diffusion and dispersion coefficients are chosen in a convenient way

Traveling waves for diffusive and/or dispersive terms have attracted many authors Traveling waves were considered earlier for diffusive-dispersive scalar equations by Bona and Schonbek [2], Jacobs, McKinney, and Shearer [3] Traveling waves and admissibility criteria of the hyperbolic-elliptic model of phase transition dynamics were also studied by Slemrod [4,5] and Fan [6,7] Traveling waves corresponding to nonclassical shocks were studied by LeFloch and his collaborators and students, see [8 16] The developments of the method of estimating attraction domain of an asymptotically stable equilibrium point to establish the existence of a traveling wave for various models were carried out in [17–21] See also [21–23] for related works

2 Background on shock waves and traveling waves

First, let us recall that a shock wave of (1.1) is a weak solution of the form

u (x, t) =



u−, x < st,

where u−, u+are the left-hand and right-hand states, respectively, and s is the shock speed.

A function of the form (2.1) is a weak solution of the conservation law (1.1), the Rankine– Hugoniot relation

−s(u+− u−) + f (u+) − f (u−) = 0 (2.2) holds This relationship (2.2) means that the shock speed s can be evaluated by

s = s(u−, u+) = f (u+) − f (u−)

u+− u− .

Often, weak solutions are not unique Admissibility criteria have been set to select a unique solution For a single conservation law (1.1), one may use Oleinik’s criterion:

f (u) − f (u−)

u − u− ≥

f (u+) − f (u−)

u+− u− , for any u between u+and u−. (2.3)

The condition (2.3) is also stated as

f (u) − f (u+)

u − u+ ≤

f (u+) − f (u−)

u+− u− , for any u between u+and u−.

Geometrically, the inequality (2.3) means that if u+ < u−, the graph of f is lying below

the straight line() connecting the two points (u±, f (u±)) in the interval [u+, u−]

To deal with simultaneous conservation laws, one can make use of the Lax shock

inequalities, see [25] A shock wave of (1.1) is called a Lax shock if it satisfies the Lax

shock inequalities

f(u−) > s(u−, u+) > f(u+), u−= u+. (2.4)

Next, a traveling wave of (1.2) connecting a left-hand state u−to a right-hand state u+

is a smooth solution of the form u = u(y), y = x − st, where s is a constant, and satisfies

the boundary conditions

lim

y→±∞u (y) = u±,

lim

y→±∞

du

d y = lim

y→±∞

d2u

Substituting the traveling wave u into (1.2), we can see that the traveling wave u satisfies

the ordinary differential equation

−su+ ( f (u))= β(b(u)u)+ γ (c1(u)(c2(u)u)), (2.6) where(.)= d(.)/dy Integrating (2.6) and using the boundary condition (2.5), we obtain

βb(u)u+ γ c1(u)(c2(u)u)= −s(u − u−) + f (u) − f (u−). (2.7) Furthermore, the last equation and (2.5) give us

s= f (u+) − f (u−)

u+− u− ,

which means that u−, u+, and s satisfy the Rankine–Hugoniot relation (2.2) So, these quantities are, respectively, the left-hand state, right-hand state, and shock speed of a shock wave of (2.1)

v(y) = c2(u)u.

Then, forγ = 0, the second-order differential equation (2.7) can be re-written as a system

of two differential equations of first order

u= v

c2(u) ,

v= −γ c βb(u)v

1(u)c2(u) +

1

γ c1(u) (−s(u − u−) + f (u) − f (u−)), (2.8)

where u = u(y), v = v(y), y ∈ R I satisfying

lim

y→±∞u (y) = u±, lim

y→±∞v(y) = 0.

Setting

h (u) = −s(u − u−) + f (u) − f (u−),

U = (u, v) T , F(U) =



v

c2(u) , −

βb(u)v

γ c1(u)c2(u)+

1

γ c1(u) h (u)

T

, (2.9)

we can re-write the system (2.8) in the form

dU

It is easy to check that a point U in the (u, v)-phase plane is an equilibrium point of the

autonomous differential equations (2.8) if and only if U has the form U = (u+, 0) for some

constant u+so that the states u±and the shock speed s satisfy the Rankine–Hugoniot relation

(2.2) Consequently, u = u(x, t) defined by (2.1) is a weak solution of the conservation law (1.1) Conversely, a jump of (1.1) of the form (2.1) gives equilibria(u−, 0), (u+, 0) of the

differential equation (2.8)

The Jacobian matrix D F (U) of the system (2.10) at U = (u, v) is given by

D F (U) =

⎛

⎜

⎝

−c2(u)v

c22(u)

1

c2(u)

−βdγ (u)v −c1(u)h(u)

γ c2

1(u) +

f(u) − s

γ c1(u) −

βb(u)

γ c1(u)c2(u)

⎞

⎟

⎠ ,

where d (u) = b (u)

c1(u)c2(u) At u±, using the condition that h (u±) = 0, we have

D F (u±, 0) =

⎛

⎜

⎝

c2(u±)

f(u±) − s

γ c1(u±) −

βb(u±)

γ c1(u±)c2(u±)

⎞

⎟

⎠

The characteristic equation of D F (u±, 0) is then given by

λ2+ a1(u±)λ − a2(u±) = 0,

where

a1(u±) = βb(u±)

γ c1(u±)c2(u±) > 0, a2(u±) =

f(u±) − s

γ c1(u±)c2(u±) .

Assume that the shock satisfies the Lax shock inequalities

f(u−) > s(u−, u+) > f(u+), u−= u+.

Then,

a2(u+) < 0, a2(u−) > 0.

Since a2(u−) > 0, the Jacobian matrix at (u−, 0) admits two real eigenvalues having

opposite signs

λ1(u−, 0) = −a1(u−) −

a21(u−) + 4a2(u−)

λ2(u−, 0) = −a1(u−) +

a21(u−) + 4a2(u−)

The point(u−, 0) is thus a saddle point.

As seen above, a2(u+) < 0 So, the Jacobian matrix at (u+, 0) admits two eigenvalues

with negative real parts

λ1,2 (u+, 0) = −a1(u+) ±

a12(u+) + 4a2(u+)

Precisely, if a12(u+) + 4a2(u+) ≥ 0, then λ1,2 (u+, 0) are real and negative Otherwise,

λ1,2 (u+, 0) are complex, conjugate, and have the real negative part −a1(u+)/2 Thus, the

point(u+, 0) is asymptotically stable.

3 Existence of traveling waves

3.1 Estimate of attraction domain of the attracting equilibrium

Let us re-write the system (2.8) in the form

u= v

c2(u) ,

v= − βb(u)v

γ c1(u)c2(u)+

1

where

h (u) = −s(u − u−) + f (u) − f (u−).

We define a Lyapunov function candidate corresponding to the equilibrium point(u+, 0):

L (u, v) = γ1

u+

u

c2(ξ)

c1(ξ) h (ξ)dξ +

v2

Let the shock wave connecting the left-hand state u−with the right-hand state u+with

the shock speed s = s(u−, u+) satisfy the Oleinik’s criterion and the Lax shock inequalities.

For definitiveness, we assume that

u+< u−,

without restriction

Th e o r e m 3.1 There always exists a value u∗< u+such that

h (u) > 0, u∗< u < u+,

h (u) < 0, u+< u < u−. (3.3)

Consequently, the function L defined by (3.2) satisfies

L (u+, 0) = 0, L(u, v) > 0, , u∗< u < u−, u = u+,

˙L(u, v) = − β γ c b (u)v2

1(u)c2(u) < 0, for v = 0. (3.4) This means that L is a Lyapunov function on the set

D : u∗≤ u ≤ u− Proof Since h(u+) = −s + f(u+) < 0, by the Lax shock inequalities, and h(u+) = 0,

the continuity implies that h (u) > 0 for u ∈ (u+− ε, u+), for some ε > 0 This establishes

the first statement in (3.3) The second statement of (3.3) follows from the Oleinik criterion:

h (u) = −s(u − u−) + f (u) − f (u−) = (u − u−)



−s + f (u) − f (u−)

u − u−



< 0,

for u+< u < u−

Next, we have

L (u+, 0) = 0,

and

L (u, v) ≥ γ1

u+

u

c2(ξ)

c1(ξ) h (ξ)dξ > 0

by using (3.3) This establishes the statements in the first line of (3.4) Next, the derivative

of L along trajectories of (3.1) is given by

˙L(u, v) = ∇L(u, v)· < du

d y , d v

d y >

= −c2(u)h(u)

γ c1(u)

v

c2(u) +



− βb(u)v

γ c1(u)c2(u) +

h (u)

γ c1(u)



v

= − βb(u)v2

γ c1(u)c2(u) < 0, for v = 0,

Le m m a 3.2

(a) Let u m = (u∗+ u+)/2, where u∗is given by (3.3) Given any continuous function

c1(u) > 0, u ∈ R I , there are always infinitely many choices of a C∞ function

c2(u) > 0, u ∈ R I such that

1

c2

c1 L∞[u+ ,u−]

u m

u∗

c2(u)

c1(u) du >

||h|| L∞[u+ ,u−]

min[u∗ ,u m]h (u) |u+− u−|. (3.5)

Figure 2 An estimate β of the attraction domain of the asymptotically stable equilibrium point

(u+, 0).

Conversely, given any positive continuous function c2(u), u ∈ R I , there are always infinitely many choices of a C∞function c

1(u), u ∈ R I satisfying (3.5).

(b) Let c1, c2 be a pair of positive continuous functions satisfy (3.5) Then, for any

positive number 0 < β < L(u−, 0), the set

β := {(u, v) ∈ D| L(u, v) ≤ β} (3.6)

(see Figure2) is a compact set, positively invariant with respect to (3.1), and has

the point (u+, 0) as an interior point.

Proof

(a) Observe that a C∞function has the derivative up to any order Setα to be the value

of the right-hand side of the inequality (3.5):

α = ||h|| L∞[u+ ,u−]

min[u∗ ,u m]h (u) |u+− u−|. (3.7)

Obviously,α > 0 Let us take an arbitrary positive continuous function ω(u), u ∈ R I ,

and define a function

c (u) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

2α

u m − u∗max{c1(u), ω(u)}, if u ∈ [u∗, u m ],

1

2min{c1(u), ω(u)}, if u ∈ [u+, u−],

c (u m ) + c (u+) − c(u m )

u+− u m (u − u m ), if u ∈ [u m , u+],

2α

u m − u∗max{c1(u∗), ω(u∗)} = c(u∗), if u ≤ u∗,

1

2min{c1(u−), ω(u−)} = c(u−), if u ≥ u−.

(3.8)

It is easy to see that the function c defined by (3.8) is positive and continuous on R I ,

and

c (u) ≥ 2α

u m − u∗c1(u), u ∈ [u∗, u m ],

c (u) ≤ 1

2c1(u), u ∈ [u+, u−]. (3.9)

Then, we will consider the mollification c ε in the interval I = (u∗− 1, u−+ 1) of

c defined by (3.8) Recall that a standard mollifier is defined by

η(u) =



C exp



1

|u|2 −1



, if |u| < 1,

where

1

C =

1

−1exp

 1

|u|2− 1



du

Then, for eachε > 0, the function

η ε (u) = 1

ε η(u/ε), u ∈ R I ,

is C∞, has the support in(−ε, ε), and satisfies

R I

η ε (u)du = 1.

A mollification of c on I is defined by

c ε (u) = (c ∗ η ε )(u) =

u−+1

u∗−1 η ε (u − v)c(v)dv =

ε

−ε η ε (v)c(u − v)dv (3.10)

As well known, c ε ∈ C∞(I ε ), where I ε = (u∗−1+ε, u−+1−ε) It is not difficult

to check that

c ε (u) = c(u∗), u ∈ (u∗− 1 + ε, u∗− ε),

c ε (u) = c(u−), u ∈ (u−+ ε, u−+ 1 − ε),

for 0< ε < 1/2 Thus, a natural extension of c ε to be constant outside I εmakes sure

that it is of C∞(R I ) Moreover, since c is continuous, c ε → c as ε → 0 uniformly

on compact subsets of the interval I = (u∗− 1, u−+ 1) Thus, since c(u), u ∈ R I ,

is positive, there existsε0> 0 such that

2c (u) > c ε (u) > c(u)/2, u ∈ [u+, u−] ∪ [u∗, u m ], 0 < ε < ε0.

From (3.9) and the last inequalities, considering c ε as c2, we can estimate the left-hand side of (3.5) by

1

c ε

c1 L∞[u+ ,u−]

u m

u∗

c ε (u)

c1(u) du >

1

2c

c1 L∞[u+,u−]

u m

u∗

c (u)

2c1(u) du

2(1/2)

u m

u∗

2α

2(u m − u∗) du

= α, 0 < ε < ε0,

... (ξ)dξ > 0

by using (3.3) This establishes the statements in the first line of (3.4) Next, the derivative

of L along trajectories of (3.1) is given by

˙L(u,... 9

Figure An estimate β of the attraction domain of the asymptotically stable equilibrium point

(u+,... s(u−, u+) satisfy the Oleinik’s criterion and the Lax shock inequalities.

For definitiveness, we assume that

u+< u−,