DSpace at VNU: Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion

Contents lists available atScienceDirect Nonlinear Analysis journal homepage:www.elsevier.com/locate/na Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion D

Contents lists available atScienceDirect Nonlinear Analysis

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

Attractor and traveling waves of a fluid with nonlinear diffusion

and dispersion

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 4 September 2009

Accepted 2 December 2009

MSC:

35L65

74N20

76N10

76L05

Keywords:

Traveling wave

Fluid

Diffusion

Dispersion

Nonclassical shock

Equilibria

Lyapunov stability

Attractor

a b s t r a c t This work completes the description of the method of estimating attraction domain for traveling waves for several systems of conservation laws with viscosity and capillarity ef-fects proposed in our earlier work Thanh (2010) [26] Precisely, we establish the global existence of traveling waves for an isentropic fluid with nonlinear diffusion and dispersion coefficients The shock wave can be classical or nonclassical Interestingly, we in particular show the existence of a traveling wave for a given Lax shock but rather nonclassical when the straight line connecting the two left-hand and right-hand states crosses the graph of the pressure function two more times in the middle region Furthermore, we also discuss all the possibilities of saddle–stable, saddle–saddle, or stable–stable connections

© 2009 Elsevier Ltd All rights reserved

1 Introduction

The interest for the study of nonclassical traveling waves has been justified by the general theory of nonclassical solutions (Riemann problem, initial-value problem, numerical approximations) developed by LeFloch and his collaborators for many years (see [1] and the references therein) On the other hand, nonlinear diffusion and dispersion have been found useful in many applications of fluid dynamics and material sciences This paper addresses the global existence of traveling waves of

an isothermal fluid flow with nonlinear diffusion and dispersion coefficients

∂tv − ∂xu=0,

where u, v >0 and p denote the velocity, specific volume, and the pressure, respectively The system(1.1)are conserva-tion laws of mass and momentum of gas dynamics equaconserva-tions in Lagrange coordinates The system can be obtained from the

common gas dynamics equations in Lagrange coordinates by writing the equation of state of the form p= p(v,S), where

S is the entropy and assume that S is constant The diffusion and dispersion terms are similar to those in [2], except the involvement of the small positive constantsε >0, δ > 0 which mean that the sizes of the diffusion and dispersion are

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

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

0362-546X/$ – see front matter © 2009 Elsevier Ltd All rights reserved.

(v+,u+), if x>st, (see [1], for example) Furthermore, a shock wave(1.3)obtained in this way is admissible under the admissibility condition of viscosity–capillarity zero limit, according to Slemrod [3–5] Conversely, given a shock wave of the form(1.3), we would like

to know whether there is a traveling wave, and this is the goal of the current paper In the case where the pressure p is convex,

Slemrod [3] showed that given a Lax shock (i.e a shock satisfies the Lax shock inequalities), then there exists a corresponding traveling waves See [6–9] and the references therein for Lax shocks in gas dynamics equations and related topics LeFloch, Bedjaoui and their collaborators, see [10,1,11–14,2] have paid lots of contributions on the existence of traveling waves, focusing mainly on traveling waves associate with nonclassical shocks (the one violating Liu’s entropy condition) In these works, existence results basically come from the saddle-to-saddle connection between the two stable trajectories leaving a saddle point at−∞and approaching the other saddle point at+∞ See [10,15,16,1,17–20] for nonclassical shocks Traveling waves for diffusive–dispersive scalar equations were earlier studied by Bona and Schonbek [21], Jacobs, McKinney, and Shearer [22] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied by Slemrod [3,

4] and Fan [23,24], Shearer and Yang [25] See also the references therein

In exerting on the model(1.1), this paper will complete the description of the attractor method for estimating attraction

domain of a set of equilibria to construct traveling waves for hyperbolic conservation laws with viscosity and capillarity

coefficients The method was proposed in our earlier work [26] exploring the applications of LaSalle’s invariance principle

An additional argument taking a stable trajectory into this domain of attraction establishes a connection We observe that our analysis only requires the shock satisfies the one-sided relaxed Lax shock inequalities The relaxation means that the shock may emerge from or arrive at a state an elliptic region The shock can also jump across an elliptic region (phase transitions) Moreover, the shock may be nonclassical and violates Liu’s entropy condition In particular, the line between the left-hand and the right-hand states can cross the graph of the pressure function up to four times as in the case of van der Waals fluids,

or more times

The organization of this paper is as follows In Section2we provide basic facts concerning shock waves and traveling waves for(1.1)and(1.2)and we describe briefly the relation between these two kinds of waves Section3is devoted to the proof of the asymptotical stability of an equilibrium point and the estimation of its attraction domain In Section4we will establish the stable–saddle or saddle–saddle connection that gives a traveling wave We also provide numerical illustrations

of the traveling waves Finally, in Section5we show that there is no stable-to-stable connection in the nonlinear case of diffusion (as in the linear case) by pointing out that the corresponding equilibrium point does not admit any asymptotically stable trajectory

2 Preliminaries: Shock waves and traveling waves

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

A(v) =



p0(v) 0



(2.1) which has the characteristic equation

λ2+p0(v) =0.

Thus, if p0(v) ≤0, A(v)admits two real eigenvalues

λ+(v) = − p

−p0(v) ≤0≤ λ2(v) = p

−p0(v).

Otherwise, it has two distinct complex conjugate eigenvalues

λ+(v) = −ipp0(v), λ2(v) =ipp0(v), where i2= −1.

Consider a shock wave solution of the hyperbolic system(1.2), connecting a given left-hand state(u−, v−)to some right-hand state(u+, v+)and propagating with the speed s These states and s satisfy the Rankine–Hugoniot relations

s(v+− v−) +u+−u−=0,

Eqs.(2.2)determine the shock speed s as

s2= −p(v+) −p(v−)

and thus define the Hugoniot set of the form u+=u−−s(v−v−) Therefore, providing that(p(v+)−p(v−))/(v+− v−) ≤0, the shock speed

s

−p(v+) −p(v−)

v+− v−

is well defined and independent of u−and u, so we simply set s=s(v−, v+), where the 1- and 2-shocks correspond to the

minus and the plus sign, respectively The Hugoniot set is thus composed of two Hugoniot curves corresponding to s≤0

and s≥0

Let us recall standard entropy criteria for hyperbolic systems of conservation laws The Lax shock inequalities require that any discontinuity connecting the left-hand state(v−,u−)and the right-hand state(v+,u+)satisfies

where s i stands for the i-shock speed, i=1,2 Since we do not assume the hyperbolicity, a shock may start from or arrive

at a state where the system may fail to be hyperbolic Therefore, we require a relaxed version of the Lax shock inequalities for a shock connecting the left-hand state(v−,u−)and a right-hand state(v+,u+):

p0(v+) < −s2<p0(v−) for 1-shocks

For a hyperbolic system conservation laws with non-genuinely nonlinear characteristic fields, Lax shock inequalities are usually replaced by Liu’s entropy condition to ensure the uniqueness Liu’s entropy condition is the one that imposes along Hugoniot curves:

s(v−, v) ≥s(v+, v−) for anyvbetweenv+andv−,

where s(v+, v)denote the speed of the discontinuity connectingvandv+ Thus, Liu’s entropy condition means that any discontinuity connecting the left-hand state(v−,u−)and the right-hand state(v+,u+)fulfils

–for 1-shocks

p(v) −p(v−)

v − v−

≥p(v+) −p(v−)

v+− v− , for anyvbetweenv+andv−

–for 2-shocks

p(v) −p(v−)

v − v−

≤p(v+) −p(v−)

Observe that the Liu strict entropy condition (where the inequalities ‘‘≥’’ and ‘‘≤’’ above are replaced by the strict inequalities

‘‘>’’ and ‘‘<’’, respectively) implies the Lax shock inequalities(2.4)

Remark In the next section, we will use the relaxed Lax shock inequalities(2.5)as the sole admissibility condition for each admissible discontinuity of(1.2), and we refer to it as a shock wave Since we do not impose any condition on the convexity/concavity of the pressure function, the relaxed Lax shock inequalities are in fact weaker than the Liu (strict)

entropy condition Consequently, a shock in this sense may be classical or nonclassical SeeFig 1

Let us now turn to traveling waves We call a traveling wave of(1.1)connecting the left-hand state(v−,u−)and the right-hand state(v+,u+)a smooth solution of(1.1)depending on the re-scaled variable

(v,u) = (v(y),u(y)), y= x−st

√

δ ,

where s is a constant, and satisfying the boundary conditions

lim

y→±∞(v,u)(y) = (v±,u±)

lim

y→±∞

d

dy(v(y),u(y)) = lim

y→±∞

d2

Substituting(v,u) = (v,u)(y), y= (x−st)/ √ δinto(1.1), and re-arranging terms, we get

sv0+u0=0,

−su0+ (p(v))0= ε

δ(q+ 1 )/ 2(a(v)|v0|q u0)0− v000,

Fig 1 A discontinuity fulfils Lax shock inequalities but violates Liu’s entropy condition.

where(.)0=d(.)/dy Integrating the last equations on the interval(−∞,y), using the boundary conditions(2.7), we obtain

s(v − v−) +u−u−=0,

Substituting u=u−−s(v − v−), u0= −sv0

from the first equation in(2.8)into the second one, we obtain a second-order differential equation for the unknown functionv:

v00= − ε

where, by letting y→ +∞in(2.8), s and(v±,u±)satisfy the Rankine–Hugoniot relations(2.2) As usual, the second-order differential equation(2.9)is equivalent to the following 2×2 system of first-order differential equations

v0=z,

Setting

h(v) :=p(v) −p(v−) +s2(v − v−),

U= (v,z), F(U) = (z, −γsa(v)|z|q z−h(v)),

we can re-write the system(2.10)in the form

dU

The above argument ensures that a point U in the(v,z)-phase plane is an equilibrium point of the autonomous differential equations(2.11)if and only if U= (v+,0), wherev±and the shock speed s are related by(2.3) The Jacobian matrix of the system(2.11)is given by

DF(U) =



− γsa0(v)|z|q z− (p0(v) +s2) −γsa(v)(q+1)|z|q



Observe that h(v±) =0 Thus,(v±,0)are equilibria of(2.11) Moreover,

DF(v±,0) =



−p0(v±) −s2 0



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

λ2+p0(v±) +s2=0.

Thus, we arrive at the following conclusion

Proposition 2.1. (a)Given a 1-shock((v±,u±);s)of the system(1.2)satisfying the relaxed Lax shock inequalities(2.5) The following conclusions hold for the corresponding differential equation(2.11):

(i) The Jacobian matrix of (2.11)at(v+,0)admits two eigenvalues having opposite signs

¯

λ1(v+,0) = − p

−p0(v+) −s2<0< ¯λ2(v−,0) = p

The point(v+,0)is thus a saddle point of the differential equation(2.11).

(ii) The Jacobian matrix at(v−,0)admits two purely imaginary eigenvalues

¯

The point(v−,0)is thus a focus (of the linearized system).

(b)Given a 2-shock((v±,u±);s)of the system(1.2)satisfying the relaxed Lax shock inequalities(2.5) The following conclusions hold for the corresponding differential equation(2.11):

(iii) The Jacobian matrix of (2.11)at(v−,0)admits two eigenvalues having opposite signs

¯

λ1(v−,0) = − p

−p0(v−) −s2<0< ¯λ2(v−,0) = p

The point(v−,0)is thus a saddle point of the differential equation(2.11).

(iv) The Jacobian matrix at(v+,0)admits two purely imaginary eigenvalues

¯

The point(v+,0)is thus a focus (of the linearized system).

Later on, we will establish a traveling wave using a saddle-to-stable connection As seen fromProposition 2.1, linearization does not give us a stable node of the nonlinear system(2.11) Thus, we need another criterion to find out stable nodes

3 The stable node and the estimation of its attraction domain

In this section, we assume simply that the pressure p= p(v)is a regular function, says, p∈C1 For definitiveness, we assume that we are concerned with a 2-shock and that

v+> v−,

without restriction, since similar argument can be made for 1-shocks and/orv+< v− Given a 2-shock wave

(v,u)(x,t) =

 (v−,u−), if x<st,

satisfying a one-sided relaxed Lax shock inequality

In this case, as shown byProposition 2.1, the point(v+,0)is a focus for the linearized system of(2.11) We aim to prove that the point(v+,0)is actually a stable node of the differential equations

dv

dy =z,

dz

dy = − γsa(v)|z|q z−h(v), −∞ <y< +∞,

(3.3)

where

h(v) =p(v) −p(v−) +s2(v − v−), s2= −p(v+) −p(v−)

v+− v−

Moreover, we will use the level sets of a Lyapunov-type function to estimate its domain of attraction

Let us define a Lyapunov-type function

L(v,z) = Z v

v +

h(ξ)dξ +z2

The function in(3.5)satisfies

L(v ,0) =0, ∇L(v,z) = hh(v),zi

Fig 2 The sets G defined by(3.11) andΩβ defined by (3.14)

The derivative of L along trajectories of(3.3)can be estimated by

˙

L(v,z) = ∇L(v,z) ·

dv

dy,dz

dy

= hh(v),zihz, −γsa(v)|z|q z−h(v)i

sinceγ >0,s>0,a>0

The Rankine–Hugoniot relations(2.2)yield

h(v) =p(v) −p(v+) +s2(v − v+)

= (v − v+)



p(v) −p(v+)

v − v+

+s2

 Moreover, the relaxed Lax shock inequality(3.2)implies that there exists a valueθ >0 such that

p(v) −p(v+)

v − v+

+s2>0, |v − v+| ≤ θ.

Thus,

Z v

v +

h(ξ)dξ = Z v

v +

(ξ − v+)



p(ξ) −p(v+)

ξ − v+

+s2



Settingν1= v++ θ, from(3.7)we have

Z ν1

v +

h(ξ)dξ >0.

Then, by continuity, it is derived from(3.7)that we can always take a pointν2< v+such that

Z ν1

v +

h(ξ)dξ > Z ν2

v +

or

Fix these valuesν1, ν2 Let us take a sufficiently large number M so that

M2> max

v∈[ν2,ν1 ]

Define the set



(v,z) ∈R2| (v − v+)2+ 1

M2z2≤ | v+− ν2|2, v ≤ v+



∪



(v,z) ∈R2| (v − v+)2+ | v+− ν1|2

(M| v+− ν2| )2z2≤ | v+− ν1|2, v ≥ v+



(seeFig 2)

Lemma 3.1 Let G be the set defined by(3.11)and let∂G denote its boundary It holds that

min

Moreover, the minimum value in(3.12)is achieved at the unique point(ν2,0), i.e.

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

z2=M2(|v+− ν2|2− (v − v+)2).

Thus, along this semi-ellipse, it holds that

L(v,z)|(v,z)∈∂G,v≤v + =

Z v

v +

h(ξ)dξ +M2

2 (|v+− ν2|2− (v − v+)2) :=g(v), v ∈ [nu2, v+]

We have

dg(v)

dv = h(v) −M2(v − v+)

= − (v − v+)



M2−



p(v) −p(v+)

v − v+

+s2



= (v+− v) M2−p0(ξ) −s2

, v+< ξ < v,

> 0, v ∈ (ν2, v+)

where the last inequality follows from the definition of M in(3.10) The function g is therefore strictly increasing for

v ∈ [ν2, v+]and attains its strict minimum on this interval at the end-pointv = ν2, i.e

L(v,z) >L(ν2,0), for all(v,z) ∈ ∂G\ { (ν2,0)}, v ≤ v+.

Arguing similarly, we can see that

L(v,z) >L(ν1,0), for all(v,z) ∈ ∂G\ { (ν1,0)}, v ≥ v+.

The last two inequalities and(3.9)establish(3.13) The proof ofLemma 3.1is complete 

Properties of the level sets of the Lyapunov-type function(3.5)can be seen in the following lemma

Lemma 3.2 Letν1, ν2be defined as in(3.9)and G be defined by(3.11) For any positive number 0< β <L(ν2,0), the set

is a compact set, lies entirely inside G, positively invariant with respect to(3.3), and has the point(v+,0)as an interior point, (see

Figs 2and3).

In addition, assume that ν1, ν2 in(3.9)be chosen such that the point(v+,0)is the sole equilibrium point in the domain

ν2 < v < ν1 Then, the initial-value problem for(3.3)with initial condition(u(0), v(0)) = (v0, v0) ∈ Ωβadmits a unique global solution(v(y),z(y))for all y≥0 Moreover, this trajectory converges to(v+,0)as y→ +∞, i.e.,

lim

y→+∞(v(y),z(y)) = (v+,0).

This means that the equilibrium point(v+,0)is asymptotically stable andΩβis a subset of the domain of attraction of (v+,0).

We will omit the proof, since it is similar to the one of Lemma 3.2, [26]

4 The stable trajectory, multiple equilibria and 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 and thatv+> v−, since the argument for the other cases are similar

4.1 Traveling waves associate with classical shocks

First, let us establish the existence of traveling waves when the shock satisfying the relaxed Lax shock inequalities(2.5)

and Liu’s entropy condition(2.6) Recall that a shock of this kind is known as a classical shock

Fig 3 The level setΩβ of a van der Waals fluid.

Theorem 4.1. (i)Given a 2-shock wave(3.1)satisfying the relaxed Lax shock inequalities(3.2) Suppose that there exists a value

ν > v+such that

Z ν

v +

h(ξ)dξ >

Z v −

v +

and that(v+,0)is the unique equilibrium point of(3.3)in the rangev−< v < ν.

Then, there exists a unique traveling wave of (1.1)connecting the states(v−,u−)and(v+,u+).

(ii)Similar result holds for 1-shock waves.

Proof By applyingLemma 3.2whereν1= ν, ν2= v−, we can see that the level setsΩβ,0< β <L(v−,0)are subsets of the domain of attraction of the stable node(v+,0) Their union

therefore provides us with a sharp estimate for the attraction domain Clearly,

Ω =  (v,z) ∈R2, v ∈ (v−, ν) : L(v,z) −L(v−,0) <0

=



(v,z) ∈R2, v ∈ (v−, ν) :

Z v

v −

h(ξ)dξ +z2

2 <0



SeeFig 3

Since the shock satisfying the relaxed Lax shock inequalities, the point(v−,0)is a saddle point Let us now consider the stable trajectories leaving the saddle point(v−,0) Since the stable trajectories are tangent to the eigenvectorh1, λ2(v−,0)i, one of them leaves the saddle point(v−,0)in the quadrant

Q1= { (v,z)|v > v−,z>0}

and the other leaves the saddle point in the quadrant

Q2= { (v,z)|v < v−,z<0}

Only the stable trajectory goes into Q1may converge to the stable node

Reversing the sides of the first equation of(3.3)and multiplying it by the second equation of(3.3)side-by-side, and then integrating the resulting equation from(−∞,y), we get

Z y

−∞

z dz

dy dy=

Z y

−∞

dv

dy(−γsa(v)|z|q z−h(v))dy

or

z2

2 =

Z v

v (−γsa(ξ)|z|q z−h(ξ))dξ.

Fig 4 van der Waals fluid: Illustration of the right part of the traveling wave in(y, v)-plane (above) and the trajectory in the(v,z)-plane (below).

Fig 5 van der Waals fluid: Illustration of the right part of the traveling wave in(y, v)-plane (above) and the trajectory in the(v,z)-plane (below) for longer time.

Since(v,z)is Q1, z<0 Thus, we have

0≤ z2

2 <

Z v

v −

−h(ξ)dξ or

Z v

v −

h(ξ)dξ +z2

2 <0.

It follows from(4.3)and the last inequality that the stable trajectory leaving the saddle point enters the attraction domain

of the stable node:

(v(y),z(y)) ∈Ω, y<0,

which establishes a saddle-to-stable connection The proof ofTheorem 4.1is complete SeeFigs 4and5for the illustration

of a trajectory starting near the saddle point converges to the stable node 

Fig 6 Four equilibria.

4.2 Multiple equilibria and traveling waves for a given shock speed

In this subsection, given a left-hand state(v−,u−)and a 2-shock speed s>0, we will establish the existence of a unique traveling wave connecting to some state on the right(v+,u+)determined by the Rankine–Hugoniot relations(2.2) Although the argument can be applied for a general case, we only consider the case of a van der Waals fluid

p(v) = 8e(3(γ −1)S0/8

3v −1)γ −

3

where S0=0.1

We are interested in the situation where the straight line between(v±,p(v±))cuts the graph of p at exactly other two

points(vi,p(vi)), i=1,2 withv−< v1< v2< v+ Clearly, the two points(vi,0),i=1,2 are also the equilibria, together with(v±,0)of the differential equations(2.10) Let us denote by M the set of the three equilibria:

SeeFig 6

Theorem 4.2. (a)Given a left-hand state(v−,u−)and a 2-shock speed s >0 such that the straight line through(v−,p(v−))

with the slope−s2 cuts the graph of the pressure function at exactly four points(v±,p(v±)), (vi ,p(vi)), i = 1,2 where

v−< v1< v2< v+ Let(vi,u i), i=1,2 be the corresponding states in the(v,u)-phase domain.

There exists a unique traveling wave leaving(v−,u−)at−∞and converging to the state corresponding to one of the remaining three equilibria(v+,0), (vi,0), i=1,2 Thus, there are the following possibilities: the traveling wave

(i) Either associate with a classical shock between(v−,u−)and(v1,u1) This case exhibits a saddle-to-stable connection, (ii) or associate with a nonclassical shock between(v−,u−)and(v2,u2) This case exhibits a saddle-to-saddle connection, (iii) or associate with a nonclassical shock between(v−,u−)and(v+,u+) This case exhibits a saddle-to-stable connection.

(b)Similar result holds for 1-shock waves.

Proof It is easy to check that the pressure function is convex forv ≥ v+ Thus,

Z v

v +

and the condition(4.1)clearly holds Arguing similarly as in the proof ofTheorem 4.1, we can see that exactly one stable trajectory leaving the saddle(v−,0)at−∞enters the domain of attraction of equilibriaΩ, where

Ω =  (v,z) ∈R2, v ∈ (v−, ν) : L(v,z) −L(v−,0) <0

=



(v,z) ∈R2, v ∈ (v−, ν) :

Z v

v −

h(ξ)dξ +z2

2 <0



SeeFig 7

Thus, the stable trajectory enters a setΩβfor some 0< β <L(v−,0) As seen fromLemma 3.2, the setΩβdefined by

(3.14)is a compact set and is positively invariant with respect to(3.3) Thus, any solution of(3.3)starting inΩβlies entirely