Scientific Research Publishing · 2013. 12. 24. · Title: Multidimensional Stability of Subsonic...

Applied Mathematics, 2012, 3, 673-684 http://dx.doi.org/10.4236/am.2012.37101 Published Online July 2012 (http://www.SciRP.org/journal/am)

Multidimensional Stability of Subsonic Phase Transitions in a Non-Isothermal Van Der Waals Fluid*

Shuyi Zhang School of Science, Shanghai Institute of Technology, Shanghai, China

Email: [email protected]

Received February 27, 2012; revised May 22, 2012; accepted May 29, 2012

ABSTRACT We show the multidimensional stability of subsonic phase transitions in a non-isothermal van der Waals fluid. Based on the existence result of planar waves in our previous work [1], a jump condition is posed on non-isothermal phase boundaries which makes the argument possible. Stability of planar waves both in one dimensional and multidimensional spaces are proved. Keywords: Non-Isothermal Phase Transitions; Euler Equations; Multidimensional Stability

1. Introduction The motion of a 2-dimensional non-isothermal van der Waals fluid is governed by the following Euler equations

2 2

00

1 1 02 2

t

t

t

p

e i

uu u u

u u u

(1)

where , , Tx y is the density, is the velocity with

, Tu vu2 2u vu 2 , p is the pressure satis-

fying the following state equation

2, ,R ap b

b

(2)

with 1 being the specific volume, being the temperature, R being the perfect gas constant and a, b being positive constants, e is the specific internal energy given by

, vae ,C

(3)

and i is the specific enthalpy given by

.i e p (4)

Otherwise, according to the second law of thermody- namics, the specific entropy s and the specific free energy f of the fluid is defined by

ln lnv vs R b C C (5) and

ln lnvaf R b C

(6)

respectively. Regarding , , ,u v s , , , TU u v s

as independent vari-

ables and denoting ,

02

,12

uF U v

q e

2

1

2

,12

uv p

F U uv

u q i

2221

2

vuv

F U v p

v q i

and

11 0 1

2

0 00 ,

,0 0 00 0 0

U U

s

A U F U F U

uc u p s

uu

*Project 10901107 supported by National Natural Science Foundation of China. Project 11101076 supported by National Natural Science Foundation of China. Supported by “The Shanghai Committee of Science and Technology(11ZR1400200)”.

Copyright © 2012 SciRes. AM

S. Y. ZHANG 674

12 0 2

2

0 00 0 0

,0 ,

0 0 0

U U

s

A U F U F U

vv

c v p sv

s

where is the sound speed, we can rewrite (1) as

2 ,c p

0 1 2 0t x yF U F U F U (7) or

1 2 0.t x yU A U U A U U (8)

When 4 , 27 8a bR a bR , the state Equation (2) is not monotonic with respect to , which means that there exist and such that

, 0 , ,

, 0 , .

p b

p

(9)

The fluid is in liquid phase in the region ,b ,

,

while it is in vapor phase in the region . The region , highly unstable region (spi- nodal region) where no state can be found in experiments

is a

onic phase transition is a discontinuous solution to

tt

[2]. Due to such monotonicity, subsonic phase transitions can be found in a van der Waals fluid, which is different from the well-known classical nonlinear waves such as shock waves, rarefaction waves and contact discontin- uities.

A subs the Euler Equation (1) with a single discontinuity,

which changes phases across the discontinuity and satisfies certain subsonic condition on both sides of the discontinuity. To explain the concept with more detail, let us consider the following planar subsonic phase transition

0

0

, , ,, ,

, , ,U u v s x

U t x yU u v s x

，

(10)

where 0, , ,u v s are constant states of the flow, is the c of the discontinuity onstant speed x t a nd belong to different phases. The soluti atis-

the Rankine-Hugoniot condition

on (10) sfies

0 1 0,F U F U (11)

and the subsonic condition

1,uM

c

(12)

where tinu

denotes the difference of a function across the discon ity x t , M 2 ,d p s are c and

onthe Mach number and the sound speed on each side of the disc tinuity x t respectively.

Due to the s pr ty (12) Lax ub er , the well-known

veral ad a were intro- du

wer papers available. Slemrod [11] an

w that the corresp

sonic opentropy inequality [3] is violated for subsonic phase transitions. Hence, se missibility criteri

ced to select the physical admissible subsonic phase transitions, among which the viscosity capillarity criterion proposed by Slemrod [4] is an important one. Ever since, for a long time, attention has been paid to isothermal phase transitions and related problems with numer- ous works devoted to such topics. For problems in one dimensional spaces, see [2,4-6] and references therein. For problems in multi-dimensional spaces, see [7-10] and references therein.

Compared with isothermal phase transitions, there is much less knowledge on non-isothermal phase transitions and there are fe

d Grinfeld [12] proved the existence of traveling waves in Lagrange coordinates by Conley index theory. Hattori [13] considered certain cases of the Riemann problem by the entropy rate criterion. Recently, the au- thor [1] proved the existence and structural stability of traveling waves by using the center manifold method, in light of which, we can expect to reveal more insights of multidimensional phase transitions.

The purpose of this paper is to study the multidimensional stability of non-isothermal phase transitions. With straightforward computation, we sho

onding linearized initial boundary problem for the planar phase transition satisfies the uniform Lopatinski condition [14,15]. Without giving much detail, here we briefly state the main result of this paper

Theorem 1.1 There exists 1 0 and K1 > 0 depending on the bounds of U and given in (10) and given in (18), such that for 10 and 0 < K < K1, the , K -admissible phase sition (10) is uniformly stable.

The definitions of the paramete

tran

K, , rs and , K - admissib ll be given in Section 2, and the uniform stability will be d

le wiescribed in detail in Section 4.

. o brie

es. In Se

The paper is arranged as follows Secti n 2 is a f recall of the viscosity capillarity criterion for phase transitions and related existence results of traveling wav

ction 3, we propose the main problem and prove the stability of phase transitions in one dimensional spaces. The multidimensional stability of phase transitions is presented and proved in Section 4.

For the simplicity of notations, we will need the following quantities in the coming arguments.

2, ,I IIRp p a

b


S. Y. ZHANG 675

, ,I II vae e C

ln( ), ln ,I II v vs R b s C C

ln , ln .I II vaf R b f C

Considering the planar subsonic phase transition (10), we denote by j u

2 the mass transfer flux,

and π p j and 2 2π 2e j . Then,

the R (11) and the s bsonic condition (12) can be rewritten as

ankine-Hugoniot condition u

0, 0j (13) and

π 0,

2j

1,,d p s

(14)

ively.

2. Viscosity Capillarity ProfilesAnalogue to the traveling wave method for viscous

sity capillarity criterion is applied to wave (10) which admits the existence of

respect

shocks, the viscofind the planar the following traveling wave

x tU U

(15)

satisfying U U and the Navier-Stokes equations

2 1

2 2

2 1

0

1 12 2

,

t

t

t

p

e q i

u

u u u u

u u

u u u

where

(16)

2 2x y is the Laplace operator, is the

viscosity coefficient, 2 is the capillarity c efficient and

o is the heat conductivity coefficient with 0 ,

0 the R

, 0 . Sunkine-Hugonng heterocli

bstitu g (15) into (16) andiot tions (11), we ca

tin condi

non deri

wn

ticing ve the a

followi nic problem for the unkno functions , ,

,

where the prime ' denotes the derivative of a function

with respect to

2

2 22

π ,

1 , π2 2

j p j

je

(17)

. In order to deal with the above problem by the center

manifold method, we proposed the following assuin [1],

mption

1 1 as ,vv

O CC

A

which was later simplified as 1 K (18)

with M 1

v

KC

being a positive constant and . Employ-

ing the Rankine-Hugoniot conditions (13), the hectero- clinic problem (17) becomes

2

2π2

a j j

2

22

π ,

π

, , .

j p j

a jjK

2

2 2

Therefore, the admissibility of subsonic phase transitions can be defined by

Definition 2.1 The planar phase transition (1missible if and only if the problem (19) has a solution. The solution

(19)

0) is ad-

, , or

is called the viscosity capillarity profile , K -profile for simplicity. The pair , , , is called , K -admissible.

To state the existence result of , K -profile, we will need the following quantities. As usual for fixed 4 ,8 27bR a bR , the Maxwell equilibrium a

,m M is defined by the equal area rule

, , d 0.M

mp

th ts a unique po

mp

Then ere exis int 1 ,m , which satisfies that the chord connecting the points

1 1, ,p and , ,m mp is t - of

an

gent to the graph ,p p at the point 1 1, ,p . Denote

12

11m

j, ,

.mp p

When

0K , the 0,0 -profile satisfies

π ,

, ,

p

2jjM

,

(20)

which implies . Setting , there exists


S. Y. ZHANG 676

; ,j generalized e

satisfying the first equation of (20) by the qual area rule as in [8], which means

Moreover, for every

d 0.p j 2π , 4 ,8 27a bR a bR and j

2 210 < j j , a unique pair , , ,j j can be found such that , and , can be connected by the -profile with the parameters j 0,0and

sed.

the existence of Ba on 0,0 -profile, the following theorem shows the existence of , K -profile for small and small K in [1].

Theorem 2.1 For every 4 ,8 27a bR a bR and 2 21< j j0 , there exist 0 0 , and n r

hoods , 0 0K eighbo -

0 0 , 0 of j , , , ,j j , respectively, such that, for 0 0 00, K , there are unique p 0, and 0

, , 0,j K air , for

which , and , are , K -admissible with the parameters j and .

Moreover, an additional jump c ndition can be derived for (10), which plays an essential role in the study of the

o

stability of phase transitions. In the isothermal casdue to t (12 he Ra

1) ent to guara ess of th bou problem for phase

ituation that we encounter ial case.

ultiplying the first n of

e [4], he subsonic condition ), t nkine-Hugoniot

is not suffic ntee the ell- condition (1 i wposedn e ndary value transitions, which is also the s n the study of non-isotherm

By m equatio (19) with and integrating from to with respect to , we get the following jump condition on the phase boundary

22π , ; ,

2

, ; ,

I jf a j K

Kb j K

(21)

where

2, ; , d ,a j K j

, ; , dIb j K s

with K , lnIs R b and , being the , K -profile with the parame-

ters j and Remark 2.1

. form

is a bounded function which also a uni limit as possesses 0K , 0

ve . Indeed,

from the second n of

,

where

equatio (19), we ha

, 0

j h

h

2 2 2

2 2π π2

a j a j2 2

h

.

Simple calculation yields

d 0jj

jh e h j

0

.

djjh e

0K , 0 , has a uniform limit When

00

d 0lim

d 0

j

jK

jh e j

jh e h j

,

where is the 0,0 -profile. Moreover, from wing jump conditions the follo

2

22

22

0

π 02

π , ; , , ; , .2

IK e

I

p j

j

jf a j K Kb j K

(22)

th

e functions , , of , , ,j K can be determined for by the implicity function theorem , K near 0,0 and every ,j

e satisfying

following identities can be the

Theoremeasily verified

conditions given in 2.1. Th

2 22

2 2 2

02

, ,

0 1

,,

0

I I I

I I I

j

K

j p s sC CC

j p s sC C C

a jC

02

,a jC

02

,b j

(23)

02

,,

0

C

b jC

where − denotes the value of a function for , K 0,0 and

0 , , ;0a j a j ,0 ,

0 , , ;b j b j , 0,0


S. Y. ZHANG


677

2 2,C p . j

3. Linearized Problems and One Dimensional Stability

In this section, we propose the nonlinear problem for a multidimensional subsonic phase transition and derive the corresponding linearized problem. Then we pro1-dimensional stability for the linear problem.

3.1. Linearized Problems Endow the Euler Equation (1) with the following initial data

(24)

00

0 0

,,

,U x y x y

U x yU x y

x y

0

where 0x y is the initial discontinuity and 0 belong to different phases. If the initial data (24) satisfies certain compatibility conditions, then we can expect to construct the following multidimensional subsonic phase transition ve the

, , ,

, ,U t x y x t y

U t x y (25) , , ,U t x y x t y

which satisfies the following nonlinear invalue problem

itial boundary

1 2

2

2

000 0

0,

, , ; , , , ,2 1

t x y

y t

y

t t

U A U U A U U

U

u vI s a j K K x t y

U U x

where the third equation is a reformulation of the jump conditi

0 1 2 0,t yF U F U F

, 0

,

, ;

x t y

x t y

Kb j

(26)

on (21) with , II s f p ,

21,

y t yx y t

j u v

,

2, ; , d ,a j K j , ; , d ,Ib j K s

and satisfying ; , ,j K , ; , ,j K

2

2 2 22 2

,

π ,

π π2 2 2

, ,x t y

j p j

a j a jjK j

.

Following [15], we introduce the following transformation to map the free boundary ,x t y into a fixed boundary 0x

= , , ,

, , , ,

x x t y x t y

y yt tU t x y U t x y

> 0

(27)

Then the problem (26) becomes

1 2 2

0 1 2 0,t yF U F U F U x

2

2

000 0

0, , 0

0

, , ; , , ; , , 02 1

,

t y t x y

y t

y

t t

U A U A U I U A U U x t

u vI s a j K Kb j K x

U U x y

(28)

S. Y. ZHANG 678

where we have dropped the tildes for simplicity of notations.

Consider the perturbation, , ,U U , satisfies the problem

of the planar phase transition (10), which (26) and

0, , , ,U U U U x

. Deno te

0

d, , , ,d

V V U U

earized problem for the unk

. Then, the following lin-

nowns , ,V V can be derived from (26),

1 2

0 1

0

( ( ) ) ( ) , , 00

, , vanish

t x y

t y

t

V A U I V A U V f x tb b V V g xV V

(29) where

200

, ,F UF U

b b

0 1

0

u vu

F U F Ul

l

1

1 0

,

F U F U

with

, ,0, ,I Il u

s

ˆ ,

ˆ,0,

Il u

Ius s

and .ˆ,j ja K b a K b

Remark 3.1 Simple calculation yields

2 2 , ,I I I II Ic K p s K p ss

2 , , and , .I IsK p K p s Ks

p

Noticing that the boundary conditions of (29) involve the quantities ja , jb , and , we will need the following lemma to deal with these quantities.

Lemma 3.1 For all

a b

0 0, 0, 0,K K , , ; ,b j K

er, their derivatives are c

the functions and are continuously

ov ontinuous , ; ,a j K tiable. More

to nded de-

pending on the bounds of and U given in (10) and the constant M given in s (18). There exist 0 such that for all j

differenwith respect , K at 0,0 and are bou

, 0lim

K

he estim we pr

Let us show the continuity

,0, 0.a j K

j

(30)

Proof. T a mm diate from ma 2 in [7], once ove the ntinuity

, ;

te (30) is i co

of

Lem, ;a j

e of ,j K .

ja and jb . Differ en- tiating t(21) with respect o j, we get

2 2π .I j js j a K bj j

By Tayl ula, we

(31)

or’s form have

1 ,I Is s o

1 ,2

K o rj j K

2 2π π π π 1 ,K o r

j j j j K

where 2 2r K and is an in as r goes to zero. Substituting the an

1o finitesimaabove identities into (31)

K b

l

d employing the calculations (23), we get

0 0 ,j j j ja a K b o r w

hich implies

0 0, 0,0 , 0,0. .j j j jK Ka a b b lim , lim

Similar argum ield ontinuity of a

ents y the c

and b as the following

,lim , lim .

Ka a b

0,0 , 0,0Kb 0 0

3.2. One Dimensional Stability Th

ely, e one dimensional stability concerns the stability of

the problem (29) without terms of y-derivatives, nam

1 , , 0t x

t

V A U I V f x

b V V g

0

t

x0

0, , vanish.

tV V

(32)

the stability of planar ph l spaces.

Theorem 3.2 There exists depe on the bounds of

nding

The following theorem shows ase transitions in one dimensiona

0K 0 and given and ant in (10)U the const

given ), suc hat fo din (18 h t r any fixe (00 0 is given in mma ) and Le 3.1. 00 K K

e with respect, the subsonic to pert

(3

show th

phase transition (10) is stabltions in the x-direction, which means the problem 2) is well-posed.

Proof. The main idea of the proof is to at the

urba-


S. Y. ZHANG 679

bounda utgoing characteristics and the free boundary can be determined by the boundary conditions, for which we need to investigate the eigenval

ry values of o

ues and the eigenvectors of the matrix 1A U I . The nval- ues of

eige 1A U I are

.1 3,u c u c

of multiplicity 1 and

2 u

of multiplicity 2. The corresponding right eigenve rsare

T

c

respectively. Denote by

osi f

cto

1, ,0,0 ,Tr c r 1 3

and

1, ,0,0c

221 220,0,1,0 , ,0,0,TT Ir r Kp

1 1 21 21 22 22 3 3V v r v r v r v r

the decomp tions o V on the bases 1 21 22 3, , ,r r r r respectively. Since the mass transfer flux j u is nonzero, we assume 0j . Then the subsonic condition (14) becomes

1 2 30 . (33)

Accordingly, we rewrite the boundary condition of (32) as

t

21

0 21 22 3 1 22

3

1

1

22

3

, , , ,v

b r r r r vvv

v

vv

211 21 22 3, , , vg r r r r

to sepafree bo

rate the outgoing characteristics together with the undary from the incoming characteristics. The

necessary and sufficient condition for the w dness of the problem (32) is that the determinant

ell-pose

1det , , , ,b r r r r0 21 22 3 (34)ompu

does not vanish. Direct c tation yields

2 23 1 3 10j c

a

1 1

1

jjj c c a

O K

(35)

where 1O of U

denotes a bounded term depending on the bounds ,

on tgiven in (10) and M given n (19).

The determ he right side of (35) takes the value i

inant

23 1 0j

c ca

for with 0 given in Lemma 3.1. 00 Therefore, we can find 0K depending on the bounds

of U , given in (10) and the constant given in (18) such that for 0 0K K and 0 0 , the problemcarried out

(32) is well-posed. Sim j

ilar arguments can be for the case 0 .

4. Multidimensional Stability First let us introduce the uniform stability in [15] and state the main result in detail. Denote by

and

, TV V V

2 01ˆ , , , , d d

2πst i yV s x e V t x y y t

the Laplace-Fourier transform of V in -variables with Then, from (29) we kn satisfies

,t yow that Re 0s . V̂

ˆ ˆˆ,V B s V fx

(36)

where

11 2

11 2

0,

0

A U I sI i A UB s

A U I sI i A U


S. Y. ZHANG 680

and

1 ˆ 1ˆ ˆ 1 1, Tf A U I f A U I f . Denote by

1

lj j

all the distinct eigenvalues of

,B s with multiplicity being mj. Obviously, we have

Introduce

81

, .jl m

jj

Ker I B s

,

, Re 0,1jm

j j j

E s

v Ker I B s j l

the space of boundary values of all bounded solutions of the special form

1

0 0

ˆ ,!

jj

j

m p px

as follows Theorem 4.1 There exist 1 0 and 1 0K de-

pending on the bounds of U , given in (10) and th econstant given in (1 at fo fixed 8), such th r any

10 and 10 K K the viscosity-capillarity admissible phase transition (10) is uniformly stable, i.e. there is 0 such that the estimate

2 2

20 1Re 0

12 22 2

infs

s

b s ib V V

V V

(37)

holds for all , ,V V V E s and .

4.1. The Space ,E s

Re, , j j

p

xV x e I B s vp

to (36) with ˆ 0f .

For simplicity, we shall only consider the c ase 0j u

and thes

Thus, we can state the uniform stability result in detail

0j other case can be studied similarly. Taking the Laplace-Fourier transform on the equation

of (29) with 0f yields

22

2 22

ˆ 10

Vi c d

x d

2 2

2

ˆ

0 0 0

I

I

I

s u s KspKsps u i c

Vsd i d Kpu u u

sdu

(38)

here

i u sc

w 220 and .s s i v d c u As in [15], if we introduce the transformation

0 0 1 0

02 ˆ ,

02

I

I

pc

1

12 20 0 0 1

c K2 2Ẑ V

Kp

cc

th ivalent to

(39)

en (38) is equ

ˆ

ˆ, ,Z N s Zx

(40)

where

02 2

0 02, .

0

0 0 0

iu u

i cu cu cN s

i c

su

0su c2 u c

c i csu

s


S. Y. ZHANG 681

The eigenvalues of ,N s with negative real part

for are Re 0s

1s

u

of multiplicity 2 and

2 2 22 2

s u c sd

d

iplicity 1, where the of mult denotes the sitive real pa onding eigenvectors are

(41)

port square root of a complex value. The corresp

11 0,0,0,1 ,Te

12 2 , , ,0 Te s i u i u (42) and

2 2 ,e i c

2

2

,

,0T

s u c

s u c

(43)

ly. The eigenvalue of

respective ,N s with a negative real part for is Re 0s

2 2 23 2

s u c s dd

and the corresponding eigenvector is

3 3

3

2 ,

,0 .T

e i c s u c

s u c

, (44)

ove eigenbe con i d to the

ation

Remark 4.1 The ab values and eigenvectors can t nuously extende case Re 0s . With a little abuse of the not , we still use it e those extensions of square roots a pearin case

. vectors, we have

are linearly independent for

to denotg in the p

Re 0s As in [15], for these Proposition 4.2 11 12 2 3, , ,e e e e 2 2 1 and Re ss except at 0

22 2, Re 0s s u s .

In the above cases, the following proposition help us

to find the bases of ,E s . Proposition 4.3 1) If s u and

then Re 0s ,

1 2

with the following eiand the vectors

genvectors (41), (44) together

12 2, , ,0 Te i i 2 ,0, 2 ,0 Te u c i u

are linearly independent.

2) If s u then 1 2 he following

and Re 0s , and the 44) er with teigenvectors

vectors (41), ( togeth

,0 Ti 12 2, ,e i

2 ,0, 2 ,0 Te u c i u are linearly independent.

As in [15], in the critical case , the bases of 0s ,E s

Proposiis given by tion 4.4 If and then 0s 0 ,

1 2 30, ,c c

d d

and the corresponding eigenvectors

11 0,0,0,1 ,Te

12 0,1, 1,0 ,Te

2 1, , ,0 ,2 2

Ti c u i c u

ed d

3 1, , ,02 2

Ti c u i c u

ed d

are linearly independent. Combining the above propositions, if we naturally

expand the eigenvectors as

,

then the bases of

11 12 2

3

0, , ,

0 0 0e e e

e

,E s are given for 2 2 1s and .

eterminNow we can show the uniform stability of the phase transition.

Proof of Theorem 4.1. Taking the Laplace-Fourier ition in (29) with

Re 0s

4.2. Lopatinski D ant

transformation on the boundary cond0g , multiplying it with the invertible matrix

02

0

0 1 0 0,

0 1 020 0 0 0 1

vq i v

1 0 0 0 01 0 0 0

with 2 2 20q u v and introducing the transformation


S. Y. ZHANG 682

(3

where

9), we get

0,c Z x ˆ ˆˆ 0 onZ

0

,i pclj j s

0

u s

s

with

3 12

2 2 2

02 2

I

I

Kj pcc c

Kj p

3 10 2

3 1

0,2 2

0 0 0

02 2

cc cj

u u j

3 12

2

2 cc

2 212

23 13 1

2

02

20 0 0

02 2 2 2

I

I

I

Kj pc

Kj pcc

jj j Kj pu u j

cc c

3

0 0 2 c

and

2 23 10, , ,

I Ip sl

2 ,2 2 2 2

I I I IIK K p s K p sK s

c

c c

2 2 23 3 1 1

2 2

ˆˆ0, , , .2 2 2 2 2 2

I I I I I I IIK p s K p s K p s j K pl K s

c c cc c c c

2

To achieve the result, we need to verify the determi-

nant (45)

being nonzero. Noticing that the eigenvector remains the same in

all the cases mentioned in n 4.1, the following simplification can be made to

K (46)

where is a bounded term depending on the

11 12 2 3det , , , ,c e e e e

11e

Sectio,

12 2 3ˆ ˆ ˆdet , , , 1 ,j c e e e O 1O

bounds of U , given in (10) and given in (19),

302 2

1

2 23 1

3 1

0

0 0ˆ ,2 20 0

0 02

c c

c c

0j

2

3 1

2 23 1

3 3 1 1

0 02 2

0 0ˆ .2 20 0

0 02 2 2 2

c c

c cj

c c

0

For sufficiently small , the determinant is nonzero as long as the determinant

0K

1 12 2ˆ ˆ ˆdet , , ,c e e e 3

doesn’t vanish. Considering , one can find that it is similar to the Lopatinski dete inant for the corresponding problem in the isothermal case [7,9]. Noticing Propo- sition 4.2-4.4, we need to consider the following three cases:

1)

1rm

22 and 0s u s . We obtain


S. Y. ZHANG 683

22 2

1

4 2,j j

j s uI a K b II

(47)

where

22 2 2 2 2 2 ,

jI s d s d c c

s

s (48)

and

2 2 2 2 2

2 2 2 2 2 2

2 2 2.

d j sc j s dc cII s d s ds sc j s d

(49)

Following [9], we claim that IIjI a is nonzero for sufficiently small 0 . In fact, when , if we have Re 0s 0 , thenI

2

2 2 2 2 2 2 4 ,c cs d s d s

u u

(50)

which implies

2 22 2 2 2 2 2

22

4 1

2 1

d d d d d d M Ms

M M

(51)

with uMc

being the Mach numbers. From the

subsonic property (12) of the phase transition, we have . Due to , we deduce that one

plus si , which is not the root of . Thus, I onzero, which gives that

0 1M should take theI o viouslythe

Re 0s gn in (51)

is always nbre exist constants 1 0M , and 1 0 , such that for

any 0 1 , we have

1 0.jI aII M

with

(52)

When Re s 0 22 2s u and 0s , we know that if 2s does not equ to the right hand side of (51) with the minus sign, then the inequality (52) holds for any 10

al

with sufficiently small 1 0 . If 2s vanish

satisfies (51) However,49) i

with the minus sign, the imaginary part of

n thes. the

(

e

term I erm II the t

given in s nonzero, which implies that for any fixed 0 , ther a c nstant 1 0M such that the inequal-

ity (52) hol e is ods as well.

Therefore, we can find 1 0 and depending on and

1 0K U give and nstant n in (10) the co

for given in 18) s ch th do nish (1

u at 1 es not va0 and 10 K K

2.

2) 2 2s u . In this case, we get

3 2 2 21 8 1

for 0.

ic j s d c O

(53)

which is nonzero for sufficiently small 0 and .

3) 0K

0s . In this case, we have

14

,d d j j

c c

(54)

which is also nonzero for sufficiently small 0 and .

Therefore, combining the above arguments, we draw the conclusion of the Theorem 4.1.

REFERENCES [1] S.-Y. Zhang, “Existence of Travelling Waves in Non-

Isothermal Phase Dynamics,” Journal of Hyperbolic Dif-.

0K

ferential Equations, Vol. 3, No. 4, 2007, pp. 391-400 doi:10.1142/S0219891607001197

[2] H. Fan and M. Slemrod, “Dynamic Flows with Liquid/ Vapor Phase Transitions,” In: H. Fan and . Slemrod, Eds., Handbook of Mathematical Fluid Dynamics, Vol. I,

M

North-Holland, Amsterdam, 2002, pp. 373-420. [3] P. D. Lax, “Hyperbolic Systems of Conservation Laws.

II,” Communications on Pure and Applied Mathematics, Vol. 10, 1957, pp. 537-467. doi:10.1002/cpa.3160100406

[4] M. Slemrod, “Admissibility Criteria for Propagating Phase Boundaries in a Van Der ,” Archive for

echanics and Anal , .

[5] M. Shearer, “Nonuniqueness of Admissible Solutions of Riemann Initial Value Problem for a System of Conser-

Waals FluidRational M ysis, Vol. 81, No. 4, 1983pp. 301-315


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vation Laws of Mixed Type,” Archive for Rational Me-chanics and Analysis, Vol. 93, No. 1, 1986, pp. 45-59. doi:10.1007/BF00250844

[6] P. G. LeFloch, “Propagating Phase Boundaries: Formula-tion of the Problem and Existence via the Glimm Method,”

ive for Rational Mechanics and Analysis, Vol. 123, 153-197. doi:10.1007/BF00695275

ArchNo. 2, 1993, pp.

[7] S. Benzoni-Gavage, “Stability of Multi-Dimensional Phase Transitions in a Van Der Waals Fluid,” Nonlinear Analy-

s, Vol. 31, No. 1-2, 1998, pp. 243-263. /S0362-546X(96)00309-4

sidoi:10.1016

[8] S. Benzoni-Gavage, “Stability of Subsonic Planar Phase Boundaries in a Van Der Waals Fluid,” Archive for Ra-tional Mechanics and Analysis, Vol. 150, No. 1, 1999, pp. 23-55. doi:10.1007/s002050050179

[9] ilitl Subsonic Phase Tr athe-

maticae Applicatae Sinica. English Series, Vol. 19, No. 4, 2003, pp. 529-558.

[10] S.-Y. Zhang, “Discontinuous Solutions to the Euler Equa-an Der Waals Fluid,” Applied Mathematics

Letters, Vol. 20, No. 2, 2007, pp. 170-176. doi:10.1016/j.aml.2006.03.010

[11] M. Slemrod, “Dynamic Phase Transitions in a Van Der Waals Fluid,” JournNo. 1, 1984, pp. 1-23. doi:10.1016/0022-0396(84)90130-X

Y.-G. Wang and Z. Xin, “Stab xistence of Mul-y and Etidimensiona ansitions,” Acta M

tions in a V

al of Differential Equations, Vol. 52,

[12] M. Grinfeld, “Nonisothermal Dynamic Phase Transi-

tions,” Quarterly of1989, pp. 71-84.

[13] H. Hattori, “The Riemann Problem for a Van Der Waals fluid with Entropy Rate Admissibility Criterion Noniso-thermal Case,” Journal of DiffereNo. 2, 1986, pp. 158-174. doi:10.1016/0022-0396(86)90031-8

Applied Mathematics, Vol. 47, No. 1,

ntial Equations, Vol. 65,

iss, “I Boundary Value Problems for Hy-stems mmunications on Pure and Applied

Mathematics, Vol. 23, 1970, pp. 227-298. doi:10.1002/cpa.3160230304

[14] H. O. Kre nitialperbolic Sy ,” Co

[15] A. Majda, “The St

Fronts,” Memoirs of thVol. 41, No. 275, 1983, pp. 1-95.

ability of Multi-Dimensional Shock e American Mathematical Society,


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