DSpace at VNU: Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Contents lists available atSciVerse ScienceDirect Nonlinear Analysis journal homepage:www.elsevier.com/locate/na Existence of traveling waves in compressible Euler equations with viscosi

Contents lists available atSciVerse ScienceDirect Nonlinear Analysis

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

Existence of traveling waves in compressible Euler equations with

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

a r t i c l e i n f o

Article history:

Received 6 September 2011

Accepted 2 April 2012

Communicated by Enzo Mitidieri

Keywords:

Compressible Euler equations

Traveling wave

Shock

Viscosity

Capillarity

Equilibria

Asymptotical stability

Lyapunov function

LaSalle’s invariance principle

Attraction domain

a b s t r a c t Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point

in fact lies on the boundary of this set Then, we establish a saddle-to-stable connection

by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point This gives us a traveling wave of the viscous–capillary compressible Euler equations

© 2012 Elsevier Ltd All rights reserved

1 Introduction

Naturally, it is interesting to see whether a shock wave admissible under a certain criterion can be approximated by some kind of smooth solutions when allowing viscosity and capillarity to the system This leads to the study of the existence

of traveling waves, which has attracted many authors for many years over the past In this paper, we are interested

in the global existence of traveling waves in compressible Euler equations when viscosity and capillarity are present Precisely, let us consider the following compressible Euler equations with viscosity and capillarity in Lagrange coordinates, see [1]:

vt −u x=0,

u t+p x= α(νu x)x− β(µvx)xx + β

2(µvv2

x)x,

E t+ (up)x = α(νuu x)x+ β  µv

2 uv2

x −u(µvx)x 

x

+ β (µu xvx)x, x∈R, t>0,

(1.1)

wherev,S,p, εand T denote the specific volume, entropy, pressure, internal energy, and temperature, respectively; u is the velocity, and E is the total energy The non-negative functionsµ = µ(v,S)andν = ν(v,S)represent the viscosity and capillarity coefficients of the fluid, respectively The positive numbersαandβmeasure the scale of the viscosity and the

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

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

0362-546X/$ – see front matter © 2012 Elsevier Ltd All rights reserved.

capillarity, respectively We note that the Lagrangian coordinates are chosen so that the calculations are simple only, since similar results hold for the Eulerian coordinates

Without viscosity and capillarity, the system(1.1)is the usual gas dynamics equations in Lagrange coordinates

vt −u x=0,

u t+p x=0,

E t+ (up)x =0, x∈R, t>0.

(1.2)

Solutions of the compressible Euler equations(1.2), and more general, hyperbolic systems of conservation laws, are found

in the weak forms In general, weak solutions are not unique due to the presence of discontinuities, or shock waves To

select a physical, unique solution, one implements the system with a certain admissibility criterion for shock waves

Various admissibility criteria have been known The concept of Lax shocks refers to shock waves that satisfy the Lax shock

inequalities; see [2] This criterion is widely used when the characteristic field of the system is genuinely nonlinear For systems of conservation laws where the characteristic fields are not genuinely nonlinear, one uses a more strict Liu’s entropy condition, see [3], to define classical shocks The concept of nonclassical shocks refers to the shock waves that

violate Liu’s entropy condition, but satisfy a single entropy inequality and a kinetic relation; see [4] Nonclassical shock may appear only when the characteristic field of the system fails to be genuinely nonlinear; see [5] and the references therein

Traveling waves are of a special kind of smooth solutions when viscosity and capillarity are added to the system, such

as(1.1) If a traveling wave connecting a left-hand state U−and a right-hand state U+exists, then its point-wise limit by vanishing viscosity and capillarity is a shock wave connecting these two states; see [5] Therefore, an admissibility criterion for shock waves could be justified physically in establishing the existence of the traveling waves

To simplify the calculations, from now on, we assume that the viscosity and capillarity coefficientsνandµare constants Furthermore, we still use the symbolsαandβto denote the quantitiesανandβµin(1.1), respectively We observe that the analysis in this paper can be extended to more general classes of viscosity and capillarity in which these quantities may depend on the specific volume and the entropy as in [1] The system(1.1)is therefore reduced to the following system

vt −u x=0,

u t+p x= αu xx− βvxxx,

E t+ (up)x = α(uu x)x+ β (u xvx−uvxx)x , x∈R, t>0.

(1.3)

In our earlier work [6], a method of estimating attraction domain of an asymptotically stable equilibrium point for traveling waves of a single conservation law with viscosity and capillarity was presented This method is relied on the Lyapunov stability techniques, in particular LaSalle’s invariance principle, and the characterization of admissible shock

waves of conservation laws under consideration In this paper, we will develop this method to the case of a3×3 system(1.3) Note that a complete set of compressible Euler equations with viscosity and capillarity effects, though has attracted attention

of many authors, but has rarely been fully investigated due to the complexity of the system As seen in the review below, most works in the literature have reduced to either the case of a single equation, or the isothermal case of a 2×2 system where

the equation of conservation of energy is ignored Our main result is the following: given any Lax shock of the compressible Euler equations(1.2), assuming that the fluid is polytropic and ideal, we will show that there exists a corresponding traveling wave

of (1.3)with a suitable choice of the viscosity and the capillarity.

The existence of traveling waves corresponding to a given nonclassical shock for viscous–capillary models was established by LeFloch and his collaborators and students; see [4,7–9,1,10,11] The existence of traveling waves corresponding to a given Lax shock,or a classical shock for viscous–capillary models was obtained in [6,12–14] These two approaches could provide the reader with a larger vision and better understanding of traveling waves for the same

or analogous viscous–capillary models Recall that traveling waves for diffusive–dispersive scalar equations were earlier studied by Bona and Schonbek [15], and Jacobs et al [16] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod [17,18] and Fan [19,20], and Shearer and Yang [21] Traveling waves

in Korteweg models in the isothermal case, Eulerian and Lagrangian capillarity models, and related topics were studied by Benzoni-Gavage and her collaborators; see [22–24] We note that a pioneering work on the related shock layers of the gas dynamics equations with viscosity and heat conduction effects (with zero capillarity) was presented by Gilbarg [25] See also the references therein

The organization of this paper is as follows In Section2, we first recall the basic concepts of hyperbolicity, jump relations, and Lax shock inequalities of the Euler equations(1.2) Second, we present the concept of traveling waves and derive a system of differential equations for these traveling waves Third, we present a result on the stability characteristics of the two equilibria of the differential equations satisfied by the traveling waves corresponding to a given Lax shock, where one equilibrium point is shown to be asymptotically stable, and the other one is shown to be a saddle point In Section3, we first define a Lyapunov function corresponding to the asymptotically stable equilibrium point This function is then shown

to possess a very desirable property: its level sets could provide us with a sharp estimation of the attraction domain of the asymptotically stable equilibrium point Finally, we establish the existence of a traveling wave for any given Lax shock, providing that the viscosity and capillarity are suitably chosen

2 Preliminaries

2.1 Hyperbolicity, jump relations and Lax shocks

Throughout, the fluid is assumed to be polytropic and ideal so that the equation of state is given by

ε = γ −pv

RT

where R > 0, 1 < γ < 5/3 are constant Let us choose the independent thermodynamic variablesvand S From the

thermodynamical identity

one has

εv= −p, εS =T

and one can express the pressure p as a function of the specific volumevand the entropy S:

p=p(v,S) = (γ −1)v− γexpS−S0

Cv



γ −1.

For smooth solutions U= (v,u,S), it is easy to see that the system(1.3)can be rewritten as

vt −u x=0,

u t+p x= αu xx− βvxxx,

TS t= αu2x.

(2.3)

In a similar way, Euler equations(1.2)can be written as

vt −u x=0,

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

S t =0.

(2.4)

Let us recall the basic concepts for the system(1.2) The Jacobian matrix of the system(2.4)and therefore of(1.2)is given by

A(v) =

pv 0 p S

 , which admits three distinct real eigenvalues

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

Thus, the system(1.2)is strictly hyperbolic Moreover, it has been known that the first and the third characteristic fields are

genuinely nonlinear, while the second characteristic field is linearly degenerate

Consider a Lax i-shock wave of the hyperbolic system(1.2), 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 i( U−,U+), i=1,3 These states and s satisfy the Rankine–Hugoniot jump relations

−s[ v] − [u] =0,

−s[u] + [p] =0,

−s[E] + [pu] =0,

(2.5)

where[ v] = v+− v−, [u] =u+−u−, etc., and the Lax shock inequalities

For polytropic ideal gas, it is known that the Lax shock inequalities, for 3-shock for example, are equivalent to each of the following conditions

(i) v−≤ v+,

(ii) p−≥p+,

(iii) S−≥S+,

(iv) u−≥u+

For a 1-shock, the inequalities in (i)–(iii) are reversed, while the inequality in (iv) is the same

2.2 Traveling waves and properties

Let us now turn to traveling waves We call a traveling wave of(1.3)connecting the left-hand state U−and the right-hand

state U+a smooth solution of(1.3)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

d

dy U(y) = lim

d2

Substituting U=U(y),y= (x−st), into(2.3), we get

−sv′−u′=0,

−su′+p′= αu′′+ βv′′′,

−sTS′= α(u′)2,

(2.8)

where(·)′=d(·)/dy Substituting u′from the first equation to the remaining two equations of(2.8)we can reduce the size

of the system as, assuming s̸=0,

−s2v′−p(v,S)′=sαv′′+ βv′′′,

Integrating the first equation on the interval(−∞,y), using the boundary conditions(2.7), we obtain

Thus, by setting

w = v′,

and using(2.10), we can rewrite the system(2.9)as a system of first-order differential equations

v′= w,

w′= −sα

1 β



p(v,S) −p(v−,S−) +s2(v − v−) ,

S′= −sα

T w2.

(2.11)

We will simplify further the system(2.11)as follows Multiplying(2.10)byv′, and using(2.2)and the third equation in

(2.11), we obtain

β

Integrating(2.12)over(−∞,y)and using the first equation of(2.11), we get

β

2w2= ε − ε−+p(v−,S−)(v − v−) −s2

Using equation of state(2.1), we can resolve for the pressure p as a function of(v, w)from(2.13)as

p= (γ −1)βw2

where

f(v) = γ −1

v



ε−−p−(v − v−) +s2

2(v − v−)2



Observe that

where the second identity follows from(2.5) Eq.(2.14)also determines the entropy as a function of the two variablesv andw:

S=S(v, w).

Therefore, the system(2.11)is reduced to the following system of two differential equations of first-order

v′= w,

w′= −sα

1

βh(v, w),

(2.17)

where

h(v, w) = (γ −1)βw2

2.3 Equilibria and their stability properties

Let(v, w)be an equilibrium point of the system(2.17) Then, by definition, it holds that

w =0,

where f is defined by(2.15) It is derived from(2.5),(2.16)and(2.19)that the points(v±,0)are equilibria of the system

(2.17)so that

g(v±) =0.

The function g is strictly convex, since it can be expressed as the sum of strictly convex functions and linear functions:

g(v) = (γ −1)

 ε−

v −p−+p−v−

s2

2



v −2v−+ v2

−

v



−p−+s2(v − v−).

Thus, the fact that g(v±) =0 implies that the strictly convex function g has exactly two zerosv± The system(2.17)therefore has exactly two equilibria(v±,0)

The following lemma provides us with the stability properties of the equilibria

Lemma 2.1 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(2.6) The following conclusions hold.

(a) The equilibrium point(v−,0)is a saddle point of the system of differential equations(2.17).

(b) The point(v+,0)is an asymptotically stable equilibrium point of (2.17).

Similar results hold for 1-shock waves

Proof First, it follows from the equation of state

p=p(v,S) = (γ −1)v− γexpS−S0

Cv



that

The Jacobian matrix of the system(2.17)at these equilibria is given by

B±=



− (f′(v±) +s2)/β −sα/β



which has the characteristic equation as

ξ2+sα

β ξ +

f′(v±) +s2

We have

f′(v) = 1− γ

v2



ε−−p−(v − v−) +s2

2(v − v−)2

 + γ −1

v −p−+s2(v − v−)

= 1− γ

v2



ε−−p−(v − v−) +s2

2(v − v−)2+p−v −s2v(v − v−)



Substitutingv = v−into(2.23), using(2.20), we obtain

f′(v−) = 1− γ

v2

−

(ε−+p−v−)

= − γp−

v−

Now, the Lax shock inequalities(2.6)and(2.24)imply that

−pv(v−,S−) = λ2>s2.

Thus,

f′(v−) +s2=pv(v−,S−) +s2<0,

so that the characteristic equation(2.22)admits two real roots having opposite signs:

ξ1<0< ξ2.

This establishes (a)

To prove (b), we estimate f′(v+)as follows Using the jump relations(2.5), we get

The relation(2.25)yields

[ ε] +p−[ v] −s2

2[ v]2= [ ε] +p−+p+

2 [ v] =0. Thus,

f′(v+) = 1v −2γ

+

ε+γ −1

v+

−p−+s2[ v]

= −p+

v+

− γ −1

v+

p+

= − γp+

v+

It is derived from the Lax shock inequalities(2.6)and(2.26)that

f′(v+) +s2=pv(v+,S+) +s2>0.

The last inequality implies that the characteristic equation(2.22)either has two real negative roots, or has two complex roots with the real part negative This establishes (b) 

3 Existence of traveling waves

As above, we consider an i-shock wave solution of the hyperbolic system(1.2), 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 For definitiveness, we consider a 3-shock such that conditions (i)–(iv) are fulfilled Similar argument can be made for 1-shocks Let us recall fromLemma 2.1that the point(v−,0)is a saddle point, and the point(v+,0)is an asymptotically stable equilibrium point of the autonomous system(2.17) Our purpose, however, is to estimate the attraction domain of the asymptotically stable node(v+,0) This will be done in this section using Lyapunov stability techniques

3.1 Lyapunov function

Let us define a Lyapunov function candidate

L(v, w) = β1

 v

v +

g(z)dz+ w2

where

g(v) =f(v) −p−+s2(v − v−)

= γ −1

v



ε−−p−(v − v−) +s2

2(v − v−)2



We now investigate properties of the function L as follows First, we have

L(v ,0) =0.

Thus, to show that L is positive definite, we will point out that

Indeed, let us show that

g(v) <0, v−< v < v+,

The function g is strictly convex, since it can be expressed as the sum of strictly convex functions and linear functions:

g(v) = (γ −1)

 ε−

v −p−+p−v−

s2

2



v −2v−+ v2

−

v



−p−+s2(v − v−).

Moreover,

g(v−) =g(v+) =0, v−< v+,

where the second identity follows from(2.16)and the jump relations(2.5) This establishes(3.4) Therefore,

 v

v +

g(z)dz>0, v−< v < v+.

Ifv > v+then g(v) >0 and therefore

 v

v +

g(z)dz>0.

This establishes(3.3) Moreover, it holds that

˙

L(v, w) = ∇L(v, w) · ⟨v′, w′⟩

= −

 αs

(γ −1)w

2v



whenever

αs

(γ −1)w

or

w ≥ (γ − −2αs

The above argument yields the following lemma

Lemma 3.1 The function L defined by(3.1)is a Lyapunov function in the domain

D=



(v, w) | v > v−, w > (γ − −2αs

1)β v



3.2 Estimating attraction domain

It is not difficult to check that

lim

v→∞

 v

v +

g(z)dz= ∞

Thus, one can always select a valueν > w+such that

 ν

v +

g(z)dz >

 v −

v +

For example, one can take

ν = w∗+1,

wherew∗> w+is the (unique) value such that

 w ∗

v +

g(z)dz=

 v −

v +

g(z)dz. The inequality(3.8)yields

Fig 1 The sets Gεdefined by (3.11) andΩδ defined by (3.14).

for any 0< ε < (w+− w−)/2 On the other hand, since the function f is strictly convex, f′is strictly increasing Thus, max

v∈[v − ,ν]|f

′(v)| =max{|f′(v−)|, |f′(ν)|}.

By(2.24), f′(v−) = −γp−/v− Define a number M as follows:

M := max{ γp−/v−, |f′(ν)|} +s2+11/ 2

>



max

v∈[v − ,ν]|f

′(v)| +s2

1 / 2

Now, we fix an arbitrary value 0< ε < (v+− v−)/2, and set

Gε =



(v, w) ∈R2| (v − v+)2+ 1

M2w2≤ | v+− (v−+ ε)|2, v ≤ v+







(v, w) ∈R2| (v − v+)2+ | v+− ν|2

(M| v+− (v−+ ε)|)2w2≤ | v+− ν|2, v ≥ v+



(seeFig 1)

One needs to implement a condition to make sure that these regions Gεare included in the domain of the Lyapunov function(3.1) In fact, one has

Gε⊂D,

providing that the tangent line(∆)passing through the origin to the curveC:

(v − v+)2+ 1

M2w2= | v+− v−|2, v ≤ v+, w <0, lies above the ‘‘boundary line’’(Γ)that determines the boundary of D:

w = (γ − −2αs

1)β v, v >0. SeeFig 2 This can be done by requiring that the slope of the tangent line(∆)is larger than the one of the boundary line (Γ), i.e.,

− M(v+− v−)



v2

+− (v+− v−)2 > (γ − −2αs

1)β , or

α

β >

M(γ −1)(v+− v−)

2s



v2

+− (v+− v−)2

Lemma 3.2 Let Gεbe the set defined by(3.11)and let∂Gεdenote its boundary It holds that

min

(v,w)∈∂Gε L(v, w) =L(v−+ ε,0).

Moreover, the minimum value is achieved at the unique point(v−+ ε,0), i.e.

Fig 2 In order that the set Gεdefined by (3.11) is always a subset of the domain of the Lyapunov function (3.1) given by (3.7), the tangent line(∆)must lie above the ‘‘boundary line’’Γforv ≥ v−

Proof We need only to establish(3.13) On the semi-ellipse∂Gε, v ≤ v+, one has

w2=M2(|v+− (v−+ ε)|2− (v − v+)2).

Thus, along this semi-ellipse, it holds that

L(v, w)|(v,w)∈∂Gε,v≤v + =

 v

v +

g(z)dz+M2

2 (|v+− (v−+ ε)|2− (v − v+)2) := ϕ(v), v ∈ [v−+ ε, v+]

Besides, it is derived from(2.16)and(2.25)that

g(v) =f(v) −f(v+) +s2(v − v+).

Therefore, one can estimate the derivative of the functionϕas follows

dϕ(v)

dv =g(v) −M2(v − v+)

= − (v − v+)



M2−



f(v) −f(v+)

v − v+

+s2



= (v+− v) M2− f′(ξ) −s2 , v < ξ < v+,

>0, v ∈ (v−+ ε, v+)

where the last inequality follows from(3.10) The function g is therefore strictly increasing forv ∈ [v−+ ε, v+]and attains its strict minimum on this interval at the end-pointv = v−+ ε, i.e

ϕ(v) > ϕ(v−+ ε), v ∈ (v−+ ε, v+]

This yields

L(v, w) >L(v−+ ε,0), for all(v, w) ∈ ∂Gε\ { (v−+ ε,0)}, v ≤ v+.

Arguing similarly, we can see that

L(v, w) >L(ν,0), for all(v, w) ∈ ∂Gε\ { (ν,0)}, v ≥ v+.

The last two inequalities and(3.9)establish(3.13) The proof ofLemma 3.2is complete 

The following lemma provides us with properties of the level sets of the Lyapunov function(3.1)

Lemma 3.3 Fix an arbitrary 0 < ε < (w+ − w−)/2, and let Gε be defined by(3.11) Then, for any positive number

0< δ <L(v−+ ε,0), the set

is a compact set, lies entirely inside Gε, positively invariant with respect to(2.17), and has the point(v+,0)as an interior point; see Fig 1 In addition, the initial-value problem for(2.17)with initial condition(u(0), v(0)) = (v0, v0) ∈Ωδadmits a unique

global solution(v(y), w(y))for all y≥0, which converges to(v+,0)as y→ +∞ Consequently, the setΩδis an attraction set

of the asymptotically stable equilibrium point(v ,0).

Proof The proof is similar to the one of Lemma 3.2, [6] However, for the sake of completeness, we will present the proof First, it is clear thatΩδis a compact set Next, we will show that the setΩδis in the interior of Gε Assume the contrary, then

there is a point U0∈Ωδwhich lies on the boundary of Gε Then, it follows from(3.13)that

L(U0) ≥L(v−+ ε,0) > δ

which is a contradiction, since U0 ∈ Ωδ,L(U0) ≤ δ Thus, the closed curve L(u, v) = δlies entirely in the interior of Gε Moreover, it is derived fromLemma 3.1that

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

dy ≤0.

This yields

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

The last inequality means that any trajectory starting inΩδ cannot cross the closed curve L(v, w) = δ Therefore, the compact setΩδis positively invariant with respect to(2.17) Therefore, a standard theory of differential equations implies that the system(2.17)has a unique global solution for y≥0 whenever U(0) ∈Ωδ Next, let us define a set

E= { (v, w) ∈Ωδ| ˙L(v, w) =0}

= { (v,0)| v > v−}

We will show that the set

M= { (v+,0)}

is the largest invariant set in E It is sufficient to show that no solution can stay identically in E, except the constant solution

v ≡ v+, w ≡0 Indeed, let(v, w)be a solution that stays identically in E Then,

v′= w ≡0, f(v) −p−+s2(v − v−) ≡0

which implies

(v, w) ≡ (v+,0).

Applying LaSalle’s invariance principle, we can see that any trajectory U starting inΩδconverges to(u+,0)as y→ ∞ The proof ofLemma 3.3is complete 

It follows fromLemma 3.3that we can express the setΩδas

And therefore, we can deduce that the set

0 <ε<(v + − v − )/ 2

is an attraction set of the asymptotically stable equilibrium point(v+,0) This set provides us with a convenient estimation

of attraction domain It is easy to see that

Ω = { (v, w) ∈D|L(v, w) <L(v−,0)}

=



(v, w) ∈R2, | v > v−, 1

β

 v

v −

g(z)dz+ w2

2 <0



whenever the condition(3.12)is fulfilled Therefore, one can see immediately from(3.17)that the saddle point(v−,0)lies

on the boundary of the setΩ:

3.3 Existence of traveling waves

We will show that one stable trajectory issuing from the saddle point enters the attraction domainΩ This establishes the existence of a traveling wave, as seen in the following theorem

Theorem 3.4 For any Lax 3-shock of (1.2)connecting the left-hand state U− = (v−,u−,S−) and the right-hand state

U+ = (v+,u+,S+), let the viscosityαand capillarityβbe chosen so that the condition(3.12)is fulfilled Then, there exists a traveling wave of(1.3)connecting these states.

Proof The eigenvectors of the matrix B−defined by(2.21)corresponding to the eigenvaluesξ1 = ξ1(v−,0)andξ2 =

ξ2(v−,0)can be chosen as