Existence of traveling waves associated with lax shocks which violate oleiniks entropy criterion

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Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion Nguyen Huu Hiep, Mai Duc Thanh & Nguyen Dinh Huy

To cite this article: Nguyen Huu Hiep, Mai Duc Thanh & Nguyen Dinh Huy (2016): Existence of

traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion, Applicable Analysis, DOI: 10.1080/00036811.2016.1157864

To link to this article: http://dx.doi.org/10.1080/00036811.2016.1157864

Published online: 09 Mar 2016

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Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion

Nguyen Huu Hiepa,c, Mai Duc Thanhband Nguyen Dinh Huya

aFaculty of Applied Science, University of Technology, Ho Chi Minh City, Vietnam;bDepartment of Mathematics, International University, Vietnam National University-HCM, Ho Chi Minh City, Vietnam;cDepartment of Mathematics and Computer Science, University of Science, Vietnam National University-HCM, Ho Chi Minh City, Vietnam

ABSTRACT

This paper answers to the question whether a shock wave in conservation

laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion

is admissible under the vanishing viscosity-capillarity effects Such a shock

appears in van der Waals fluids when a secant line meets the graph of

the flux function at four distinct points, and the shock jumps between

the two farthest points The existence of the corresponding traveling

waves would justify the admissibility of the shock For this purpose, we

will first show that the corresponding traveling waves satisfy a system of

differential equations with two saddle points and two asymptotically stable

points Second, we estimate the domains of attraction of the asymptotically

stable equilibrium points, relying on Lyapunov’s stability theory Third,

we investigate the circumstances when an unstable trajectory leaving the

saddle point corresponding to the left-hand state of the shock will ever

enter the domain of attraction of each of the two asymptotically stable

equilibrium points Finally, we establish the existence of traveling waves

associated with a Lax shock but violating the Oleinik’s entropy criterion

ARTICLE HISTORY

Received 19 September 2015 Accepted 21 February 2016

COMMUNICATED BY

M Shearer

KEYWORDS

Shock wave; traveling wave; Lax shock inequalities; Oleinik’s entropy criterion; viscosity; capillarity; Lyapunov stability

AMS SUBJECT CLASSIFICATIONS

35L65; 74N20; 76N10; 76L05

1 Introduction

In this paper, we study the existence of traveling waves associated with a Lax shock but violating the Oleinik entropy criterion (see Oleinik [1], Lax [2], Liu [3]) of the following diffusive–dispersive conservation law

∂ t u + ∂ x f (u) = β(b(u)u x ) x + γ (c1(u)(c2(u)u x ) x ) x, x ∈ R I , t > 0, (1.1)

where the unknown u = u(x, t) > a0, a0is either a finite constant, or a0 = −∞, the positive constants

β and γ indicate small-scaled quantities, the function b(u) > 0, u > a0, represents the diffusion, and

the functions c1(u) > 0, c2(u) > 0, u > a0, represent the dispersion We assume that the functions

b(u), c1(u), and c2(u) are differentiable, and there is a positive constant c0so that

c1(u) ≥ c0, c2(u) ≥ c0,

for all u ∈ (a0,∞) The flux function f : (a0,∞) → R I is a twice differentiable function whose graph

changes the concavity twice and has two inflection points Precisely, we assume that there are two

CONTACT Mai Duc Thanh mdthanh@hcmiu.edu.vn

Figure 1.Flux function and slope of secant line determines the shock speeds.

constants a0 < a1< a2such that

f(u) > 0, for u ∈ (a0, a1) ∪ (a2,+∞)

f(u) < 0, for u ∈ (a1, a2)

lim

where(.) = d(.)/du, (.) = d2(.)/du2 The motivation for considering this kind of flux functions

comes from the shape of the pressure p = p(v, S0), v > a0> 0 of van der Waals fluids as a function of

the specific volume v for each fixed entropy S = S0 We note that the topic of shocks waves in van der Waals fluids is interesting and attracts the attention of many authors For simplicity, in the following

we always assume that

We note that the argument in this work may be applied for the case where a0is finite

Admissible shock waves of the conservation law

u t + f (u) x = 0. (1.4) can be obtained as the limit of traveling waves of (1.1) whenβ → 0+,γ → 0+.

Consider a straight line (d) passing through a given point(u−, f (u−)) on the graph of the flux

function z = f (u) with a slope s, see Figure1 Under the assumptions (1.2), the line (d) may also

meet the graph of the flux function f at three other points (u i , f (u i )), i = 1, 2, 3 Accordingly, we can

have three shock waves of (1.4) between the left-hand state u−and the right-hand state u+with the

same shock speed s, where u+can be u1, u2or u3 Recall that a weak solution of (1.4) of the form

u(x, t) =



u−, if x < st,

u+, if x > st, (1.5)

is called a shock wave between the left-hand state u−and the right-hand states u+with the shock

speed s As seen in van der Waals fluids, all of these three shock waves may be admissible in the

sense that they may all satisfy an entropy inequality and a kinetic relation The reader is referred

to Hayes and LeFloch [4], LeFloch [5] and LeFloch and Thanh [6,7] for nonclassical shock waves Furthermore, these three shocks can be classified into three kinds as follows

(i) The shock wave between u−and u+ = u3is called a classical shock, as it satisfies Oleinik’s

entropy criterion

f (u) − f (u−)

u − u− ≥

f (u+) − f (u−)

u+− u− for all u between u−and u+. (1.6) (ii) The shock wave between u−and u+ = u2 is called a nonclassical shock, because it violates

Oleinik’s entropy criterion (1.6) Note that this shock does not satisfy the Lax shock inequali-ties:

f(u+) < s < f(u−), (1.7) where(.)= d(.)/du.

(iii) The shock wave between u−and u+= u1is a nonclassical Lax shock, since it violates Oleinik’s

entropy criterion (1.6), but it satisfies the Lax shock inequalities (1.7)

The existence of traveling waves of (1.1) associated with a classical shock such as the one in (i) was studied in [8–12] The existence of traveling waves of (1.1) associated with a non-Lax, nonclassical shock such as the one in (ii) was studied in Bedjaoui and LeFloch [13–17], and Bedjaoui et al [18] See also the references therein Thus, admissible shock waves of the kinds (i) and (ii) (i.e they satisfy an entropy inequality as observed in [5,6]) have been shown to be also admissible under the vanishing viscosity-capillarity effects The question is whether the shock wave of the kind (iii) may also be admissible under the vanishing viscosity-capillarity effects This paper will give a positive answer to this question, by establishing the existence of traveling waves of (1.1) associated with a nonclassical Lax shock as in the case (iii) mentioned above

We note that traveling waves for diffusive–dispersive scalar equations were studied by Bona and Schonbek [19], Jacobs et al [20] Admissibility criteria by vanishing viscosity and capillarity effects using traveling waves were investigated by Slemrod [21,22] and Fan [23,24], Shearer and Yang [25] See also [26–28] for related works

The organization of this paper is as follows In Section2, we study the stability of the equilibria of

a system of ordinary differential equations associated with shock waves satisfying the Lax shock inequality, but violating Oleinik’s entropy criterion In Section 3, we investigate the attraction domains of the asymptotically stable equilibrium points, and we will provide estimates of these domains In Section4, we will establish the existence of traveling waves Finally, in Section5, we will present numerical simulations of these traveling waves

2 Traveling waves, equilibria and their stability

A traveling wave of (1.1) is a smooth solution of the form u = u(y), y = x − st, where s is a constant,

and satisfies the following boundary conditions

lim

y→±∞u(y) = u±, lim

y→±∞u(y) = lim

where(.)= d(.)/dy, (.)" = d2(.)/dy2 Substituting u = u(y), y = x − st into (1.1) and integrating over(−∞, y), using the boundary conditions (2.1), we obtain

−s(u − u−) + f (u) − f (u−) = βb(u)u+ γ c1(u)(c2(u)u). (2.2)

By letting y→ +∞ in (2.2), and using the boundary conditions (2.1), we have

−s(u+− u−) + f (u+) − f (u−) = 0, or s = f (u+) − f (u−)

u+− u− . (2.3)

This shows that s is the shock speed of the shock wave between the left-hand state u− and the

right-hand state u+

Setting v = c2(u)u, we can re-write the second-order differential Equation (2.2) as a system of

two first-order differential equations as follows

du

dy = v

c2(u)

dv

dy = 1

γ c1(u) h(u) −

β γ

b(u)

c1(u)c2(u) v,

(2.4)

where

h(u) = −s(u − u−) + f (u) − f (u−). (2.5) The boundary conditions (2.1) become

lim

y→±∞u(y) = u±, lim

y→±∞v(y) = lim

Thus, the problem (2.2)–(2.1) is reduced to (2.4)–(2.6) Let us study the behavior of trajectories of the autonomous system (2.4) Equilibrium points of (2.4) have the form(u, 0), where u satisfies

h(u) = 0,

which means that u, u−, s satisfy the Rankine–Hugoniot relation (2.3) for u = u+

Let us define

F(u, v) =  v

c2(u),

1

γ c1(u) h(u) −

β γ

b(u)

c1(u)c2(u) v



.

Then, the Jacobian matrix of (2.4) is given by

A(u, v) = DF(u, v) :=

⎛

⎜

⎝

−c2(u)

c22(u) v

1

c2(u)



h(u)

γ c1(u)



−



β γ

b(u)

c1(u)c2(u)



v −β

γ

b(u)

c1(u)c2(u)

⎞

⎟

⎠ , where(.)= d(.)/du So, the Jacobian matrix at any equilibrium point (u0, 0) of (2.4) is given by

A(u0) = A(u0, 0) :=

⎛

⎜

⎝

c2(u0)

−s + f(u0)

γ c1(u0) −

β γ

b(u0)

c1(u0)c2(u0)

⎞

⎟

⎠ , (2.7)

since h (u0) = 0, and h(u) = −s + f(u).

The boundary conditions (2.6) can be deduced using the stability of the equilibrium points(u±, 0).

The following proposition characterizes the stability of an equilibrium point(u0, 0) of (2.4)

Proposition 2.1: The point (u0, 0) is an equilibrium point of (2.4) iff h (u0) = 0, where h is defined

by (2.5) Moreover,

(i) If f(u0) > s, then the matrix A(u0) defined by (2.7) has two real eigenvalues of opposite signs,

so that (u0, 0) is a saddle point;

(ii) If f(u0) < s, then the matrix A(u0) defined by (2.7) has two eigenvalues of negative real parts,

so that (u0, 0) is asymptotically stable, either a stable node or a stable focus.

Proof: The characteristic equation of the matrix A(u0) defined by (2.7) is given by

λ2+β

γ

b(u0)

c1(u0)c2(u0) λ +

s − f(u0)

γ c1(u0)c2(u0) = 0. (2.8)

Since the left-hand side of (2.8) is a quadratic polynomial, the following conclusions hold

(i) Since s − f(u0)

γ c1(u0)c2(u0) < 0, the characteristic Equation (2.8) has two real roots of opposite signs.

(ii) Denote byλ1andλ2the two roots of the characteristic Equation (2.8) Then

λ1· λ2= γ c s − f(u0)

1(u0)c2(u0) > 0,

λ1+ λ2= −γ β b(u0)

c1(u0)c2(u0) < 0.

Thus, eitherλ1,λ2are two negative eigenvalues,

orλ1andλ2are complex and conjugate and have a negative real part Precisely, if

 :=



β γ

b(u0)

c1(u0)c2(u0)

2

− 4 s − f(u0)

γ c1(u0)c2(u0) ≥ 0,

then the two eigenvaluesλ1,λ2are the same real and:λ1 ≤ λ2 < 0 If  < 0, then λ1,λ2are complex and have a negative real part to be

−1 2

β γ

b(u0)

c1(u0)c2(u0) < 0.

This completes the proof of Proposition2.1 

As indicated by Proposition2.1, whenever f(u0) < s, the equilibrium point (u0, 0) of (2.4) is asymptotically stable So, it has the domain of attraction In the following we will find an estimate for this domain of attraction, which is a subset of it For this purpose, we will use a Lyapunov function, and an estimation of attraction domain of the asymptotically stable equilibrium point(u0, 0) will be

made through level sets of this Lyapunov function Precisely, we define

L(u, v) = u0

u

c2(ξ)

γ c1(ξ) h(ξ)dξ +

v2

It holds that

L(u0, 0) = 0, ∇L(u, v) =



− c2(u)

γ c1(u) h(u), v

.

The derivative of L along trajectories of (2.4) is given by

˙L(u, v) = − c2(u)

γ c1(u) h(u).u+ v.v

= − c2(u)

γ c1(u) h(u).

v

c2(u) + v.

 1

γ c1(u) h(u) −

β γ

b(u)

c1(u)c2(u) v

= −β

γ

b(u)

c1(u)c2(u) v2 < 0, for all v

On the other hand, since h(u0) = −s + f (u0) < 0, there exists a number θ = θ(u0) > 0 such that

h(u) < 0, u ∈ (u0− θ, u0+ θ),

then h (ξ) > h(u0) = 0, ξ ∈ [u0− θ, u0), and h(ξ) < h(u0) = 0, ξ ∈ (u0, u0+ θ] which yields

L(u, 0) =

u0

u

c2(ξ)

γ c1(ξ) h(ξ)dξ > 0, u ∈ [u0− θ, u0) ∪ (u0, u0+ θ].

This means that L is a Lyapunov function for the equilibrium point (u0, 0) in u ∈ (u0− θ, u0+ θ) Now, set p = u0− θ Sinceu0

p

c2(ξ)

γ c1(ξ) h(ξ)dξ > 0, and that the function u → h(u) is continuous,

there exists a number q > u0such that

u0

p

c2(ξ)

γ c1(ξ) h(ξ)dξ ≥

u0

q

c2(ξ)

γ c1(ξ) h(ξ)dξ > 0,

or

Let us choose a sufficiently large positive number M = M(p, q) such that

M2> |s| + max

u ∈[p,q]

c2(u)

c1(u) L f,

where L f is the Lipschitz constant of f over [p, q] Then, we define a region

G p,q (u0) =



(u, v) ∈ R I2|(u − u0)2+ 1

M2v2 ≤ |u0− q|2, u ≥ u0

 ,



(u, v) ∈ R I2|(u − u0)2+ |u0− p|2

(M|u0− q|)2v2≤ |u0− p|2, u ≤ u0

 , (2.11)

which is a compact set

The following proposition provides us with an estimate of the attraction domain of the asymptot-ically stable equilibrium point(u0, 0) of (2.4)

Lemma 2.2: Let (u0, 0) be an asymptotically stable equilibrium point of (2.4) satisfying f(u0) < s, and let G p,q (u0) be defined by (2.11) Then, for any positive number β < L(q, 0), the set

β = {(u, v) ∈ G p,q (u0)|L(u, v) ≤ β} (2.12)

is compact, positively invariant and lies entirely in G, and contains(u0, 0) as an interior point Moreover, every trajectory of (2.4) starting in

β∈(0,L(q,0))

must approach the set of equilibria in as y → +∞.

The proof of Lemma2.2is similar to the one in [11, Lem 3.2], so it is omitted

3 Attraction domain of the equilibria

Under the assumptions (1.2), we consider the case where the straight line (d) passing through a

given point(u−, f (u−)) with the slope s meets the graph at four distinct points at u−, u1, u2, u3 For simplicity we may assume in the sequel that

u1 < u2< u3 < u−, (3.1)

see Figure1 This intersection determines three shock waves, where u−is the left-hand state, s is the shock speed, and the right-hand state u+can be u1, u2or u3

It is not difficult to check that

h(u−) = h(u1) = h(u2) = h(u3) = 0,

where h is defined by (2.5) Moreover, it holds that

h(u) > 0, for u ∈ ( − ∞, u1) ∪ (u2, u3) ∪ (u−,+∞),

h(u) < 0, for u ∈ (u1, u2) ∪ (u3, u−),

h(u1) = −s + f(u1) < 0,

h(u3) = −s + f(u3) < 0,

h(u2) = −s + f(u2) > 0,

h(u−) = −s + f(u−) > 0.

(3.2)

It follows from Proposition2.1and (3.2) that(u−, 0) and (u2, 0) are saddle points, while (u1, 0) and (u3, 0) are asymptotically stable equilibrium points of (2.4) We define a Lyapunov function for each

of these asymptotically stable equilibrium points as follows For(u1, 0), we set

L1(u, v) =

u1

u

c2(ξ)

γ c1(ξ) h(ξ)dξ +

v2

and for(u3, 0), we set

L3(u, v) =

u3

u

c2(ξ)

γ c1(ξ) h(ξ)dξ +

v2

Lemma 3.1: Consider a Lax shock connecting the left-hand u−and the right-hand state u+ = u3

with the shock speed s If

L3(u2, 0) ≥ L3(u−, 0), (3.5)

then there exists a traveling wave connecting u−and u3.

Proof: Let us define an attraction domain of the asymptotically stable equilibrium point (u3, 0) by

3= {(u, v) ∈ G p,u−(u3)|L3(u, v) < L3(u−, 0)}, (3.6)

where G p,q (u0) is defined by (2.11) This attraction domain has the point(u−, 0) as a boundary point.

Letλ−> 0 be an eigenvalue of A(u−) corresponding to an eigenvector V−= (1, c2(u−)λ−) Then,

every trajectory of (2.3) starting at(u−, 0) has to approach the line through (u−, 0) with direction

V− So, this trajectory leaves(u−, 0) in one of the two quadrants

Q1(u−) = {(u, v) ∈ R I2|u < u−, v < 0},

Q2(u−) = {(u, v) ∈ R I2|u > u−, v > 0}. (3.7)

Let us consider the trajectory leaving(u−, 0) in the quadrant Q1

Since L (u, v) is smooth, the tangent of 3at(u−, 0) is vertical This shows that the line through (u−, 0) with direction V−intersects with 3 So, the trajectory leaving(u−, 0) at −∞ comes into the

attraction domain 3 According to Lemma2.2, this trajectory approaches the set of equilibria of (2.4) as y → +∞ in 3, which contains exactly one point(u3, 0). 

Now, we assume that L3(u2, 0) < L3(u−, 0) It holds that

L1(u−, 0) =

u1

u−

c2(ξ)

γ c1(ξ) h(ξ)dξ =

u1

u2

c2(ξ)

γ c1(ξ) h(ξ)dξ −

u3

u2

c2(ξ)

γ c1(ξ) h(ξ)dξ +

u3

u−

c2(ξ)

γ c1(ξ) h(ξ)dξ

= L1(u2, 0) − L3(u2.0) + L3(u−, 0) > L1(u2, 0) > 0.

By (1.2), L1(u, 0) =u1

u

c2(ξ)

γ c1(ξ) h(ξ)dξ

u→−∞

−−−−→ +∞ So, there exists ν < u1such that

L1(ν, 0) = L1(u−, 0) > 0. (3.8)

We can now define

= {(u, v) ∈ G ν,u−(u1) : L1(u, v) < L1(u−, 0)}. (3.9)

It holds that

L1(u i, 0) < L1(u−, 0), i = 1, 2, 3.

This implies that contains three equilibria (u i, 0), i = 1, 2, 3 It is easy to see that the points (u−, 0)

and(ν, 0) belong to the closure of

Lemma 3.2: Given an equilibrium point (u−, 0) of (2.4) and a constant s Assume that the straight

line through (u−, f (u−)) with the slope s cuts the graph of f at four distinct points:

u1< u2 < u3< u− Then, there is a trajectory of (2.4) leaving (u−, 0) at −∞ approaches one of the two asymptotically stable equilibrium points (u1, 0) and (u3, 0).

Proof: If L3(u2, 0) ≥ L3(u−, 0), the conclusion is established by Lemma3.1

Assume that L3(u2, 0) < L3(u−, 0) Arguing similarly as in the proof of Lemma3.1, we can show that there is a trajectory of (2.4) leaving(u−, 0) at −∞ goes to the attraction domain Thus, this

trajectory must approach the set of equilibria of (2.4) in Furthermore, contains three equilibria

{(u i, 0) : i = 1, 2, 3}, so this trajectory tends to one of these three points as y → +∞ It is now

sufficient to show that every trajectory of (2.4) leaving(u−, 0) cannot approach (u2, 0) Indeed, since

L3(u2, 0) < L3(u−, 0), there is a number ν3 ∈ (u3, u−) such that

L3(ν3, 0) = L3(u2, 0). (3.10) Then, we define a domain of attraction of(u3, 0) by

3= {(u, v) ∈ G u2 ,ν3(u3) : L3(u, v) < L3(u2, 0)}. (3.11)

It is easy to see that 

3has the point(u2, 0) on its boundary (Figure2) On the other hand, it holds

that L1(u2, 0) > 0 Thus, there exists a number ν1 < u1such that

L1(ν1, 0) = L1(u2, 0) > 0. (3.12) Then, we define a domain of attraction of(u1, 0) by

1= {(u, v) ∈ G ν1,u2(u1) : L1(u, v) < L1(u2, 0)}. (3.13) The point(u2, 0) also lie on the boundary of 1 If a trajectory of (2.4) tending to(u2, 0) as y → +∞,

it must approach(u2, 0) in the direction of the eigenvector V2 = (1, c2(u2)λ2) of A(u2), where λ2< 0

is the corresponding eigenvalue This implies that the trajectory would go to one of the attraction domains 1or 

3 This is a contradiction, since any trajectory going to an attraction set must remain

in that set Lemma2.2is completely proved  Next, we will study the properties of an arbitrarily trajectory of (2.4) leaving the saddle point

(u−, 0) at −∞ and going to the attraction domain of (u1, 0) Let y be the first time this trajectory cuts

the line v = 0 at u, that is

y = sup{α ∈ R I |v(y) < 0, ∀y ∈ ( − ∞, α)}, u = u(y). (3.14)

This also means that y = +∞ if (u(y), v(y)) does not meet the line v = 0 If a trajectory (u(y), v(y)) cuts the lines u = u i , i= 1, 2, 3, then we define

y i < y : u(y i ) = u i, i = 1, 2, 3. (3.15) Consider a trajectory of (2.4) leaving(u−, 0) in the quadrant Q1(u−) = {(u, v) ∈ R I2|u < u−, v < 0} For y < y, it holds that u(y) = v(y)

c2(u(y)) < 0 So, the trajectory part corresponding to y < y cuts the

lines u = u i , i = 1, 2, 3 at most once This shows that the definition of the points y i , i = 1, 2, 3 in (3.15) are reasonable

Proposition 3.3: Let (u(y), v(y)) be a trajectory of (2.3) which comes to the domain defined by

(3.9) Let y, u be given by (3.14) Then

u ∈ (ν, u1] ∪ (u2, u3]. (3.16)

Proof: Since is an attraction domain of (u3, 0), any trajectory comes into must stay in So,

ν < u < u−.

For every y < y, there exists ξ ∈ (y, y) such that

v(ξ) = v(y) − v(y)

y − y =

−v(y)

y − y > 0.

Letting y → y, we obtain

v(y) ≥ 0.

Substituting y = y into (2.4), we have

v(y) = h(u)

γ c1(u) ≥ 0.

This yields h (u) ≥ 0 This together with (3.2) imply that

u ∈ (ν, u1] ∪ [u2, u3] Finally, by Lemma3.2, any trajectory of (2.4) cannot approach(u2, 0 2 This completes the

4 Existence of traveling waves

In this section, we still assume the conditions (1.2), where u1< u2< u3 < u−, and

L3(u2, 0) < L3(u−, 0).