DSpace at VNU: Existence of traveling waves in van der Waals fluids with viscosity and capillarity effects

Contents lists available atScienceDirect Nonlinear Analysis journal homepage:www.elsevier.com/locate/na Existence of traveling waves in van der Waals fluids with viscosity and capillarit

Contents lists available atScienceDirect Nonlinear Analysis

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

Existence of traveling waves in van der Waals fluids with

viscosity and capillarity effects

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

bFaculty of Applied Science, University of Technology, 268 Ly Thuong Kiet, str District 10, Ho Chi Minh City, Viet Nam

cNguyen Huu Cau High School, 07 Nguyen Anh Thu, Trung Chanh, Hoc Mon, Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

Article history:

Received 9 August 2012

Accepted 16 October 2013

Communicated by Enzo Mitidieri

Keywords:

van der Waals fluid

Traveling wave

Shock wave

Viscosity

Capillarity

Lyapunov stability

a b s t r a c t

We establish the existence of traveling waves of non-isentropic van der Waals fluids with viscosity and capillarity effects The method developed the one for simpler models The nonconvex equation of state of the fluid causes much difficulty in evaluating the related quantities, and so the argument and the analysis are much more involved than the convex equation of state The point is to estimate the pressure along the Hugoniot curves such that

a Lyapunov function can be defined in an appropriate way

© 2013 Elsevier Ltd All rights reserved

1 Introduction

In this paper we study the existence of traveling waves in van der Waals fluids with the effects of viscosity and capillarity The viscous–capillary model is given by

vt−u x=0,

u t+p x=

 λ

vu x



x

− (µvx)xx,

E t+ (up)x=

 λ

vuu x



x

− (u(µvx)x)x+ (µu xvx)x,

(1.1)

for x ∈ R and t > 0 As usual, the symbolsρ, v = 1/ρ,S,p, ε,T and u denote the density, specific volume, entropy,

pressure, internal energy, temperature, and velocity, respectively, and

E= ε + u2

µ

2v2

is the total energy The quantitiesλandµrepresent the viscosity and capillarity coefficients, respectively For simplicity, throughout we assume thatλandµare positive constants Besides, a van der Waals fluid is characterized by the equation

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

E-mail addresses:mdthanh@hcmiu.edu.vn, mdthanh1@gmail.com (M.D Thanh), dinhhuy56@hcmut.edu.vn (N.D Huy),

nguyenhuuhiep47@yahoo.com (N.H Hiep), cuongnhc82@gmail.com (D.H Cuong).

0362-546X/$ – see front matter © 2013 Elsevier Ltd All rights reserved.

http://dx.doi.org/10.1016/j.na.2013.10.017

of state of the form

p= RT

v −b−

a

where a>0,b>0 and R>0 are constants Nonclassical (non-Lax) shocks have been known to appear in van der Waals fluids, and they exist at the expenses of Lax shocks by the nucleation criterion, which prefers non-Lax shocks over Lax shocks Moreover, Riemann solvers with kinetics use both kinds of Lax shocks and of nonclassical (non-Lax) shocks, see [1–3], for example Traveling waves associated with nonclassical shocks in the Euler equations for van der Waals fluids with viscosity and capillarity effects were obtained by Bedjaoui–LeFloch [4] The question is whether or not the Euler equations for van der Waals fluids with viscosity and capillarity effects can also possess traveling waves associated with Lax shocks? Our aim

in this paper is to seek a positive answer to this question Therefore, our results indicate that the viscous–capillary models are appropriate for applications involving both kinds of Lax shocks and nonclassical shock waves

The existence of traveling waves has attracted many authors Recently, Thanh [5] studied the existence of traveling waves

in the Euler equations for polytropic ideal fluids with viscosity and capillarity So, the result is applied for convex equations of state In this work, the viscosity and capillarity coefficients are slightly different from the ones in [5] More interestingly, the fluid is of van der Waals type(1.3) This is a typical nonconvex equation of state Consequently, the system of Euler equations may be of mixed type as an elliptic–hyperbolic model, and the characteristic fields are not entirely genuinely nonlinear The analysis and the argument will be much more involved for van der Waals fluids, since they possess complicated features not only in the characteristic fields, but also in the Rankine–Hugoniot relations, and the admissibility criteria for shock waves, etc However, we can establish the existence of traveling waves of(1.1)for van der Waals fluids when the viscosity and capillarity are suitably chosen The point is that we can estimate the pressure along the Hugoniot curves such that we can suitably define a Lyapunov function We will show that the viscous–capillary model(1.1)for van der Waals fluids can still yield nice properties of the corresponding Lyapunov function Accordingly, the level sets near the saddle point of the Lyapunov function will be proved to provide sharp estimates for the attraction domain of the asymptotically stable equilibrium point

A stable trajectory from the saddle point will then be shown to enter the attraction domain of the asymptotically stable equilibrium point This saddle-to-stable connection gives us the traveling wave of(1.1)

We observe that traveling waves corresponding to a given non-Lax shock for viscous–capillary models were considered

by LeFloch and his collaborators and students, see [6–9,4,10,11] Traveling waves corresponding to a given Lax shock for vis-cous–capillary models were obtained by Thanh [12–15,5] Traveling waves were considered earlier for diffusive–dispersive scalar equations by Bona and Schonbek [16], Jacobs, McKinney, and Shearer [17] Traveling waves and admissibility criteria

of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod [18,19] and Fan [20,21], Shearer and Yang [22] See also [23–25] for related works

This paper is organized as follows In Section2we provide basic concepts and properties of the Euler equations for van der Waals fluids In Section3we study the system of ordinary differential equations which are derived from(1.1)for the traveling wave Its equilibria and the stability of equilibria using linearization will be presented In Section4, we define a Lyapunov function, estimate the attraction domain, and establish the existence of the traveling wave

2 Preliminaries

2.1 Hyperbolicity

Several equations of state of a van der Waals fluid other than(1.3)are given by

T = d

(v −b)2 / 3exp



2S 3R −

5 3

 ,

ε = 3R

2T −

a

v =



p+ a

v2

3(v −b)

a

v ,

where d>0 is a parameter, see [26] Substituting T from the last system into(1.3), one obtains an equation of state of the

form p=p(v,S), where

p(v,S) = Rd

(v −b)5 / 3exp



2S 3R −

5 3



v2. SeeFig 1

Consider the fluid dynamics equations in the Lagrangian coordinates

vt−u x=0,

u t+p x=0,

Fig 1 A typical isentrope p=p(v,S)of van der Waals fluids in the(v,p)-plane.

Choosing U= (v,u,S), we can re-write(2.1)as

vt−u x=0,

u t+pv(v,S)vx+p S(v,S)S x=0,

S t=0.

The Jacobian matrix of the last system is given by

A=

pv 0 p S

 Providing that

pv(v,S) = −5Rd

3(v −b)8 / 3exp



2S 3R −

5 3

 +2a

v3 <0,

the Jacobian matrix A admits three distinct real eigenvalues

λ1= −  −pv(v,S) < λ2=0< λ3=  −pv(v,S),

so that the system(1.2)is strictly hyperbolic

2.2 Shock waves and admissibility criteria

A shock wave of(2.1)is a weak solution U of the form

U(x,t) =



U−, if x<st,

where U−and U+are constant states, U−̸=U+, called the left-hand and right-hand states, respectively, and the constant s

is the shock speed These quantities must satisfy the Rankine–Hugoniot relations

s(v+− v−) + (u+−u−) =0,

−s(u+−u−) +p+−p−=0,

ε+− ε−+ p++p−

Often, one fixes a left-hand state U−and consider the Hugoniot setH(U−)consisting of all the right-hand states U =U+

that can be connected to U− by a shock wave(2.2) As well-known, the Hugoniot set contains the Hugoniot curves

Hi(U−),i= 1,3 associated with the first and the third characteristic fields, respectively, for van der Waals fluids can be parameterized by the specific volumev Precisely, the relations(2.3)for U=U+determine the Hugoniot curves

p=p(U−; v) := −p−v3+ (2ε−+p−v−)v2−av +3ab

v2(4v − (3b+ v−)) ,

u=u(U−; v) :=u−± (v − v−)



− (p(U−; v) −p−)

where the plus sign corresponds to the 1-Hugoniot curveH1(U−), and the minus sign corresponds to the 3-Hugoniot curve

H3(U−) The shock speed s along the Hugoniot curves is given by

s=s(U−,U) = ±



−p(U−; v) −p−

v − v−

provided

p(U−; v) −p−

v − v−

≤0.

In(2.4), the minus sign corresponds to 1-shocks, and the plus sign corresponds to 3-shocks Recall that for any given state

U−, the set of all states U+can be connected to U−by a shock wave satisfying the Rankine–Hugoniot relations, and is called

the Hugoniot set issuing from U−, and is denoted byH(U−)

The i-shock wave between the left-hand state U−and the right-hand state U+with shock speed s(U−,U+)is said to be

a Lax shock if it satisfies the Lax shock inequalities, see [27],

For example, for a 3-shock between a left-hand state U−= (v−,u−,S−)and a right-hand state U+= (v+,u+,S+)with the

shock speed s(U−,U+) = √ − (p+−p−)/(v+− v−), the Lax shock inequalities(2.5)read



−pv(v+,S+) <



−p+−p−

v+− v− < −pv(v−,S−), or,

pv(v+,S+) >p+−p−

v+− v−

= −s2>pv(v−,S−).

As well-known, the characteristic fields of the system(2.1)for a van der Waals fluid are not genuinely nonlinear For this kind of systems, one often uses Liu’s entropy condition, see [28], which imposes along Hugoniot curves that

s(U−,U) ≥s(U+,U−) for any U between U+and U−.

Thus, Liu’s entropy condition means that any discontinuity connecting the left-hand state U−and the right-hand state

U+satisfies

–for 1-shocks

p(U−, v) −p−

v − v−

≥ p+−p−

v+− v−

= −s2, for anyvbetweenv+andv−; –for 3-shocks

p(U−; v) −p−

v − v−

≤p+−p−

v+− v−

2.3 Equation of the entropy

Let us now find the equation of conservation of energy in terms of the specific entropy S The left-hand side of the equation

of conservation of energy of(1.1)can be re-written as

E t+ (up)x = εt+uu t+up x+pu x+

 µ

2v2

x



t

=TS t−pvt+uu t+up x+pu x+

 µ

2v2

x



t

=TS t−p(vt−u x) +u(u t+p x) +  µ

2v2

x



t. The 2nd term of the last equation vanishes due to the conservation of mass The 3rd term can be substituted by the equation

of conservation of momentum in(1.1) Thus, it follows from the last system that

E t+ (up)x =TS t+u

 λ

vu x



x

− (µvx)xx

 +

 µ

2v2

x



t

=TS t+u

 λ

vu x



− (µvx)xx



The right-hand side of the equation of conservation of energy of(1.1)is given by

 λ

vuu x



x

− (u(µvx)x)x=u

 λ

vu x



x

+ λ

vu

2

Equating both sides of(1.1), using(2.7),(2.8)andvt=u x, and simplifying the terms, we get

TS t = (µu xvx) − [µvxvxt+u x(µvx)x] + λ

vu

2

x. Sincevxt = vtx=u xx, it is derived from the last equation that

TS t = (µu xvx) − [µvx u xx+u x(µvx)x] + λ

vu2x,

or, equivalently,

TS t = λ

vu2x.

The above argument shows that the system(1.1)is equivalent to the following system

vt−u x=0,

u t+p x=

 λ

vu x



x

− (µvx)xx,

TS t = λ

vu2x.

(2.9)

3 Traveling waves and equilibria

3.1 Traveling waves

A traveling waves of(1.1)connecting the left-hand state U−and the right-hand state U+is a smooth solution of(1.1)

depending on the variable

U=U(y) = (v(y),u(y),S(y)), y=x−st,

where s is a constant, and satisfying the boundary conditions

lim

y→±∞U(y) =U±

lim

y→±∞

d

dy U(y) = lim

y→±∞

d2

Substituting U=U(y),y= (x−st), into(2.9), we get

−sv′−u′=0,

−su′+p′=

 λ

vu

′

′

− (µv′)′′,

−sTS′= λ

v (u

′)2,

(3.2)

where(.)′ =d(.)/dy Eliminating u′in the 2nd and the 3rd equations of(3.2)by substituting u′from the first equation of

(3.2), we obtain

s2v′+p′= −s

 λ

v v

′

′

− (µv′)′′,

−TS′=sλ

v (v

′)2.

(3.3)

Integrate the first equation of(3.3)on the interval(−∞,y)and use the boundary conditions(3.1)to get

s2(v − v−) +p(v,S) −p(v−,S−) = −sλ

v v

Re-arranging terms of the last equation to get

(µv′)′= −sλ

v v

We can also simplify the 2nd equation in(3.3) Indeed, multiplying both sides of Eq.(3.4)by sv′

one gets

s2(v − v−)v′+s(p−p−)v′= −sλ

v (v

Add(3.6)to the 2nd equation in(3.3)side-by-side to get



s2(v − v−)v′+s(p−p−)v′ −TS′= (µv′)′v′.

Simplifying terms in the last equations yields

s2(v − v−)v′+ (p−p−)v′−TS′= −1

22µv′v′′, or

s2(v − v−)v′−p−v′− (−pv′+TS′) = −1

2

 µ(v′)2′

. Sinceε(v,S)′= −pv′+TS′, the last equation gives

1

2



µ(v′)2′

= ε(v,S)′+p−v′−s2(v − v−)v′. Integrating the last equation in(−∞,y), we have

1

2µ(v′)2= (ε(v,S) − ε−) +p−(v − v−) −s2

It follows from(3.5)and(3.7)that by settingw = µv′, we can re-write the system(3.3)as

v′= w

µ ,

w′= −s λ

µv w −p+p−−s2(v − v−),

w2

2µ = (ε − ε−) +p−(v − v−) −s2

2(v − v−)2,

(3.8)

where p=p(v,S), ε = ε(v,S) The third equation in(3.8)determines the entropy as a function

S=S(v, w)

so that by substituting S=S(v, w)into p=p(v,S)in the second equation of(3.8), we obtain a simpler system However,

we can do this in a direct way by substituting

ε = p+ a

v2

3(v −b)

a

v , into the third equation of(3.8)to get

w2

2µ =



p+ a

v2

3(v −b)

a

v −



ε−−p−(v − v−) +s2

2(v − v−)2

 This yields

p=p(v,S(v, w)) = w2

where

˜

p(U−; v) := 2

3(v −b)



a

v +



ε−−p−(v − v−) +s2

2(v − v−)2



Substitute p from(3.9)into the second equation of(3.8), we obtain the following 2×2 ordinary differential equations of first order

v′= w

µ ,

w′= −s λ

µv w −

w2

where p(U−; v)is given by(3.10)

Lemma 3.1 The system(3.8)is equivalent to the following system

v′= w

µ ,

w′= −s λ

µv w −p+p+−s2(v − v+),

w2

2µ = (ε − ε+) +p+(v − v+) −s2

2(v − v+)2.

(3.12)

Consequently, the system(3.11)is equivalent to the system

v′= w

µ ,

w′= −s λ

µv w −

w2

3µ(v −b) −  ˜p(U+; v) −p++s2(v − v+) ,

where

˜

p(U+; v) := 2

3(v −b)



a

v +



ε+−p+(v − v+) +s2

2(v − v+)2



v2.

In addition, it holds that

˜

p(U−; v) = ˜p(U+; v).

Proof Equating the right-hand sides of the second equations of(3.8)and(3.12)to get

−p+p−−s2(v − v−) = −p+p+−s2(v − v+)

which gives

s2= −p+−p−

v+− v−.

This is exactly the definition of the shock speed Equating the right-hand sides of the third equations of(3.8)and(3.12)to get

(ε − ε−) +p−(v − v−) −s2

2(v − v−)2= (ε − ε+) +p+(v − v+) −s2

2(v − v+)2 which gives

(ε+− ε−) +p−(v − v−) −p+(v − v+) +s2

2

 (v − v+)2− (v − v−)2 =0. Simplifying the last equation gives us

ε+− ε−+ p++p−

2 (v+− v−) =0 which is exactly the third Rankine–Hugoniot relation in(2.3) The remaining conclusion immediately follows 

3.2 Equilibria and their stability

Lemma 3.2 Given a 3-shock between the left-hand state(v−,u−,S−)and the right-hand state(v+,u+,S+)with shock speed

s satisfying the Lax shock inequalities(2.5)and the Liu entropy condition(2.6) Then,(v±,0)are equilibria of the system of differential equations(3.11)and following conclusions hold

(a) The equilibrium point(v−,0)is a saddle point.

(b) The point(v+,0)is an asymptotically stable equilibrium point.

Proof Setting the right-hand side of(3.11), we get

w =0, p˜ (U−; v) −p−+s2(v − v−) =0.

Clearly,

p(U , v ) =p .

Moreover, it follows from(2.4)that

˜

p(U−; v+) =p−−s2(v+− v−) =p+.

Thus, the two points(v±,0)are equilibria of the system(3.11)

To investigate the stability of the equilibria(v±,0), we use the linearization method Precisely, we will evaluate the Jacobian matrix of the system(3.11)and its eigenvalues at these equilibria The Jacobian matrix of the system(3.11)at these equilibria is given by

B±=





− (˜p′(U−, v±) +s2) − µvsλ

±



 ,

which has the characteristic equation as

ξ2+ sλ

µv±ξ +p˜

′(U−, v±) +s2

Since sλ/µv±>0, the last equation admits two roots which have opposite signs or have two (complex) roots of negative

real parts depending on the sign of p′(U−, v±) +s2<0 It follows from the Liu entropy condition that

s2≤ −p(U−, v) −p−

v − v−

. This implies that

˜

p(U−; v) = 2

3(v −b)



a

v +



ε−−p−(v − v−) +s2

2(v − v−)2



v2

3(v −b)



a

v +



ε−−p−(v − v−) −p(U−; v) −p−



v2

3(v −b)



a

v +



ε−−p(U−; v) +p−



v2

3(v −b)

a

v + ε(U−; v)  − a

v2

Observe that along the Hugoniot curves parameterized by the specific volume it holds that

p′(U−; v−) =pv(v−,S−),

and that

p′(U−; v+) =p′(U+; v+) =pv(v+,S+).

Moreover, since p(U−, v±) =p±, it follows from(3.13)that

˜

p′(U−; v−) ≤p′(U−; v−) =pv(v−,S−), p˜′(U−; v+) ≥p′(U−; v+) =pv(v+,S+).

By the Lax shock inequalities, we deduce that

˜

p′(U−; v−) +s2≤pv(v−,S−) +s2<0,

and that

˜

which terminates the proof 

4 Existence of traveling waves

Given a shock wave of the form(2.2)of the hyperbolic system(2.1)connecting a given left-hand state U−= (v−,u−,S−)

to some right-hand state U+= (v+,u+,S+)and propagating with the speed s=s(U−,U+),i=1,3 We will assume that the shock satisfying the Lax shock inequalities(2.5)and the Liu entropy condition(2.6) For definitiveness, we consider a

3-shock between U−and U+, where s>0 and

v−< v+.

Note that similar arguments can be made for other cases

The system(3.11)can be re-written as

v′= w

µ ,

w′= −s λ

µv w −

w2

where

˜

p(U−; v) = 2

3(v −b)



a

v +



ε−−p−(v − v−) +s2

2(v − v−)2



v2,

q(U−; v) = ˜p(U−; v) −p−+s2(v − v−).

(4.2)

4.1 Lyapunov function

Let us define a Lyapunov function candidate

L(v, w) = µ  v

v +

q(U−;z)dz+ w2

where q(U−, v)is defined by(4.2)

Obviously,

L(v+,0) =0.

It follows from(3.13)that

q(U−; v) ≤ p(U−; v) −p−+s2(v − v−)

= (v − v−)



p(U−; v) −p−

v − v−

+s2



≤0, v ∈ (v−, v+).

Moreover, by the Lax shock inequalities, the last inequality is strict, at least forvnearv− Thus,

L(v, w) = µ

 v

v +

q(U−;z)dz+ w2

2 > w2

2 ≥0, v ∈ (v−, v+).

As seen in the proof ofLemma 3.2,p˜ (U−, v±) =p±, which yields

q(U−; v±) =0.

Moreover, from(3.14)we have

q′(U−; v+) = ˜p′(U−, v+) +s2>0.

Thus

q(U−; v) >0

for at leastv+< v < ˜v, for somev > v ˜ + Set

Assume in the following that

 ν

v −

It is easy to see that the valueνsatisfies

L(ν,0) = µ  ν

v +

q(U−;z)dz> max

v − ≤ v≤v +

L(v,0) =L(v−,0) >0.

The above argument shows that the Lyapunov function candidate L is positive definite forv ∈ (v−, ν) We will show that it

is in fact a Lyapunov function on the set

D:=



(v, w)|v−< v < ν, w >3sλ



b

v −1



Fig 2 The domain D of the Lyapunov function defined by(4.6) is above the curve C and on the right of the linev =b in the(v, w)-plane.

Lemma 4.1 Under the condition(4.5), the function L defined by(4.3)is a Lyapunov function in the set D given by(4.6) Precisely, the following conclusions hold

L(v+,0) =0, L(v, w) >0, forv ∈ (v−, ν), v ̸= v+,

˙

L(v, w) <0 forw ̸=0, w >3sλ



b

v −1

 ,

˙

L(w,z) =0 on{ w =0} ,

(4.7)

where L denotes the derivative of L along trajectories of˙ (4.1).

Proof Let us consider the derivative of L along the trajectories of(4.1):

˙

L(v, w) = ∇L(v, w) · ⟨v′, w′⟩

= ⟨ µ ˜p(U−; v) −p−+s2(v − v−) , w⟩ ·

 w

µ , −s

λ

µv w −

w2

3µ(v −b) −  ˜p(U−; v) −p−+s2(v − v−)



= −s λ

µv w2−

w3

3µ(v −b)

= − w2

µ



sλ

w

3(v −b)

 The last equality establishes the third line of(4.7)and implies that˙L(v, w) <0, w ̸=0 whenever

sλ

w

3(v −b) >0,

which holds if

w > −3sλ(1−b/v) =3sλ



b

v −1

 This completes the proof ofLemma 4.1 

The set D defined by(4.6)is the region above the curveC : w = 3sλ b

v−1

 and on the right of the linev =b in the

(v, w)-plane, seeFig 2

4.2 Estimating attraction domain

First, we will define the elementary sets that include the estimates of the attraction domain as follows Set

θ := 9b(1−b/v−)

(v+− v−)v2

+

Assume that