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Phase boundary solutions to model kinetic equations via the Conley index theory. Part II
MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical
and Computer
Modelling
36 (2002)
139331408 www.elsevier.com/locate/mcm
Phase Boundary Solutions Model Kinetic Equations via to the Conley Index Theory. Part II B. KA~MIERCZAK Polish Academy of Sciences, IPPT PAN, Swietokrzyska 21 PL 00-049 Warszawa, Poland K. PIECH~R Department of Mathematics, Technology, and Natural Sciences University of Bydgoszcz, 85..064 Bydgoszcz, Poland and Polish Academy of Sciences, IPPT PAN, Swietokrzyska 21 PL 00-049 Warszawa, Poland
(Received
March
2002;
accepted
April
2002)
Abstract-we consider phase boundary solutions to a four-velocity kinetic model of a kinetrc equation governing the motion of van der Waals fluids. These solutions connect such equrlibrrum states, which are saddle critical points of the related dynamic system. Solutrons of thrs type can be interpreted as dynamic phase transition. The mathematical apparatus is that of the Conley index theory. @ 2002 Elsevier Science Ltd. All rights reserved. Keywords--van
der Waals fluids, Model discrete ling waves: Conley connection index.
velocity
kinetic
theory.
Phase boundary
travel-
1. INTRODUCTION This paper
deals with the phase
governing
the
motion
consisting
of a discretization
As was already number
of velocities,
already
attent,ion
points of the dynamic
mentioning
supported
0895-7177/02/$ - see front matter PII: SOS95-7177(02)00296-O
to the Enskog-Vlasov phenomena
wave solutions
analysis by Grant
schemes
as documented in this paper
[a].
to a small occurring
the saddle critical
Boltzmann
problems. equation
in some recent
is also related
papers,
can be generalised
analysis It is worth e.g., [4-71.
also to models
KBN 7T07A00919.
@ 2002 Elsevier
Science
Ltd.
in
Hence, this paper deals with
of fluids and with the qualitative
(or discretized)
proposed
limited
develops the considera-
connecting
in this case, wave propagation
in the discrete
in il].
equation
of some interesting This paper further
equation
proposed
being
properties
problems;
of computational
the qualitative
This paper was partially
of travelling
of a kinetic
model,
the discretization
related to the above class of equations.
phenomenological
the interest
to the development In principle,
of existence
model
mathematical
corresponds despite
provides the description
of the mathematical
that
The
to the van der Waals ones.
system
of qualitative
to a four-velocity
fluids.
by four velocities,
tions of [3] with the analysis the analysis
solutions
shown in [1,3], the model,
fluids with special
of the solution
state
of the van der Waals
,211 rights reserved.
Typeset
by dn/ls-‘Q$
B. KA~MIERCZAK AND K. PIECH~R
1394
which discretize energy
the original
in the collision,
The contents
equation
by conservative
schemes
preserving
mass, momentum,
and
see [4,6].
of this paper
are organised
into four sections.
Section
2 deals with
a concise
description of the model and the statement of the mathematical problem. Some preliminar? technical results are given in Section 3, while the last section deals with the mathematical problem related to wave phenomena.
2. STATEMENT This paper
is a continuation
sake of completeness results
of Part
The present system
of partial
of our previous
paper
[3], referred to as Part I. However, for the convenience, we quote here the most important
of theses and for the reader’s
I, revising, paper
OF THE PROBLEM
at this opportunity,
concerns
differential
some misprints
the proof of existence
contained
of travelling
there.
wave solutions
to the following
equations:
a
a ax
-w--u=o, at
(2.1)
-gu+&[~(l+$p)-~+~]=E~[~~u] a ++
5 p
a
2
EW
i()
and
a
tuatq
a
- 2uaxq
l-U2
a
w
ax
+ ----_u
(2.2)
2a2 ---_w W5dX2
I
’
4P = --q,
(2.3)
E
where p= p(w) = h
1 0 W
Here t is the time, X is the Lagrangian coordinate, w is the specific volume, ‘~1is the velocity of the fluid in the X-direction, q is an extra quantity which has no direct physical interpretation. and a, b, CY, E are positive constants characterising the fluid. This system of equations is an approximation to a kinetic four-velocity model of the Enskog-Vlasov equation. Since its derivation is described in detail in [l], and briefly, it is also presented in Part I, we omit a closer discussion of their structure. REMARK
1. The specific volume
the following
constraints
w, the mean velocity
w > b,
Throughout
this paper,
U, as well as the parameters
a, b satisfy
[I]:
the above conditions
and
u2 < 1 - ;T.
will be used without
any further
reference.
I
Similarly, as in Part I, we are interested in the travelling wave solutions to equations (2.1). (2.3). Prom the mathematical point of view, the existence analysis splits naturally into two parts: 0 > 0 and 0 < 0, where g is the wave speed. Though the proofs in both of these cases proceed along the same lines, there are some essential differences. Moreover, the physical significance of the case 0 < 0 has encouraged us to treat it separately. A travelling wave solution to the kinetic is a so1utio.n of the form equations (2.1)-(2.3)
(‘w,u, q)(t,X) = (WIu, n)(E)1
x - fft
(C--..--
&
E
BP,
(2.4)
Phase Boundary
Solutions
1395
where 0 = const is the wave speed, such that (2.5) (2.6) (2.7)
,Jym(w’,u’,Q’)(0 = (O,O,O), ,!jf& (w”,u”‘) (0 = (0, 0,O),
(2.8)
where the dash by a character means differentiation with respect to [. Now, we act in a very standard way. Namely, we substitute (2.4) into equations perform one integration with respect to E, and use the limit conditions (2.5)-(2.8). that, we find the left and right states of rest related algebraically
(2.1)-(2.3), Having done
(2.9) These relations are called the Rankine-Hugoniot is given by
conditions.
Next, we find that the velocity u
u = Ul - a(w - Wl),
(2.10)
whereas w = w(E) and q = q(r) are coupled through the following system of ordinary differential equations: w’ = z, “=-A(w)
1
LA’ (w)z’ 2 UJ
11
+(w) + 4~3
1-(u1-c+wl))2Z+
Q(w)
q’ = -cr w (2 (U1 + OWL)- aw)
(2.11)
q,
2(ul+awl)-aw
(2.12)
f(W;w~,21~r~)=~2(W-w’W1)+p(W,‘1L1--(~--W1))-~(W1,~1)~ where P(W,U) =
1 - u2
(2.13)
2(w - b)
is the pressure, and A(w) = 2cy/ws. In our considerations, we do not make use of the specific form of the function A(w). need is just that this function is positive for all w > b, and that its derivative A’(w) for w > b. Prom (2.5)-(2.8), we obtain the following limit conditions:
w(E)= WI,
,
(2.14)
;;iJ& w(J) = W,
/?im_qt<) = 0,
(2.15)
Elim -0c)
pya 40
What we is negative
= 0,
E lir z’(t) = 0, -+ca
,liI=,
These conditions must be supplemented by the Rankine-Hugoniot
f(.Wv;w7w,~) = 0.
q’(E) = 0.
(2.16)
condition (2.9), which implies (2.17)
In this part of the paper, we consider only the case of left-going waves with a negative wave speed. The idea of the proof of existence of such waves is the same as in Part I and consists of the use of the Conley index theory. However, in the case of the negative wave speed, some details
1396
B. KA~MIERCZAK AND K. PIECH~R
and estimates the Conley system
are different
connection
into a simpler
following system
from the previously
index consists
considered
of the possibility
one, which can be effectively
of ordinary
differential
case of positive
of making
analysed
wave speeds.
a homotopy
(see [8,9]).
Use of
of the considered
So, we will analyse
the
equations:
w’ = z,
- rj + VA(W)]-
z’ = -[I
’ ;A;(w)z'
1 - (“1 - cr (w - wL))2z + q’ = -al7 w (2 (u1 + awl) - aw) where
71 E [O,l],
(2.18)
coincides
correspond
subject
to the limit
to the well-analysed
4w3
1
4P(W) 2(Ul +awl) -0w
conditions
with the initial system
The basic ingredient
ab2d4
+
system
of the existence
(2.14)-(2.16).
whereas
(2.11),
+ (1 - rl)
describing
Note
(2.18)
1
1 4,
that,
for q = 1, system
for 7 = 0, the first two equations
the
isothermal
proof is constructing
phase
transitions
a family of compact
of (2.18) (see
[9]).
neighbourhoods
N(g, 77)>rl E [O, 11and o from a suitable closed interval in IR1, such that for every 7) and 0: the set N(a, q) has the maximal invariant set with respect to system (2.18) in its interior.
3. BASIC The-critical equation
ASSUMPTIONS
points
(2.17). o.2
for system
Explicitly,
2b)
2 In general,
due to the complexity
in detail the dependence
(2.18)
either
w:(wo+ Wl -
AND
PRELIMINARY
RESULTS
are all of the form (~0~0, U), where wo is a solution
wg = WL, or wg satisfies
+azl&
-
of the coefficients
of its solutions
of
the cubic equation
a (w - b)
ab
Wl”
Wl I
in equation
on the wave speed 0.
wo+3
(3.1),
=o.
we are unable
However,
(3.1) to discuss
from the analysis
done
in (lo], one can deduce the following. PROPOSITION
the algebraic
1.
There
equation
exists an open set Z such that 0 = 0 is contained (2.17)
has exactly
four distinct
b < ‘~1 < WI(~)
in it, or if 0 # 0. the11
roots satisfying
< w,(a)
< wz(c),
with
We assume ASSUMPTION ASSUMPTION
[wl, w,(a)]
the following. 1.
The infimum
2.
There
of Z is negative,
is a compact,
connected
whereas
the supremum
interval
P c
(b, oo) such that
for any c E 2:
C P, 21~+ owl > 0, 2(u~ + GW~) > gw, for any w E P.
Similarly,
as in Part
PROPOSITION
2.
I
I, we can prove the following.
If (w(e), z(J),q(t))
and q(E) are bounded ~(6; 0, rl) I ~(0,
functions
IS an y continuous
of <, i.e., there are ~(a,
V) and SUP< MC; 0, q)l =
solution
of system
77) and $a,~)
(2.18)
such that w(t)
such that b < ~(a.
&(a, rl) < co; then supE Iz(c; a,~)1 = Z(a,q)
LEMMA 1. Let (w(0,z(<),q(E)) b e a continuous solution of system (2.18) such that is bounded and w(E;a,q) is uniformly bounded function of < E R1, o E 1, and q E SUP< Id<;
such
that
or ~11 =
b < g
u
of Z is nonnegative.
Q(a, 7) < 00, and there are constants 5 w([,a,q)
w, G independent
< cc.
‘7) 5 1
q(<; 0: q)
[0, 11,i.e..
of 0 E Z and q c [0, l]
5 G for any [ E JR-l,CYE Z, and 77 E [0, 11. Then
there
are
Phase Boundary
Solutions
1397
constants Q > 0 and Z > 0 independent of u E Z and r] E [0,11 such that, for any < E IR’, CJ E 1, and q E [0, l], the following estimates hold true: sup< /Q(<; o,~)]
= Q((T,~) < Q and
supE MC; o, ~)l = z(o, 7) < 2, i.e., q(t; ot V) and z(E; o, 7) are also uniformly
bounded
functions
of < E IR’, CTE I, and 77E [O,l].
I
LEMMA 2. Let (w(<),z(G,q(E)) b e a continuous solution of system (2.18) satisfying the limit (2.14)-(2.16) such that w(J;a,q) is a uniformly bounded function of 5 E RI. CTE 1.
conditions
and 17E [0, 11, then there is a positive such /q([;a,q)]
constant
Q* independent
of < E EC’, 0 E 2, and 77E [O. 11.
5 q]o]Q*, for any E E R’, (T E 2, and fl E [0,11.
I
PROOF. Let (w(e), z(E), q(5)) b e a continuous solution of system (2.18). equation of this system, we obtain
Then, from the third
50 1 f E xic,-a?) J 4dNC)Ml) [i 4([; 0~77)= exp
71 - 77+ /J(W(T)))
d7’
I
z(c) 6 (w CC)) (1 - ~2(w))
exp
-
50
(3.3)
(1 - rl+ Pd(W(T)))dT & * Eo I 1
where cc is an arbitrarily fixed value of the independent variable, Cc is a constant of integration, and b(w) =
4P(W) 2u+ffw’
(3.4)
This function satisfies boundary condition (2.15) only if
J
b(w(C))(1- u2W>))
O3
co = UT)
z(s)
4P(W(GMC)
Eo
exp
c
[J
(1 - q + &W(T))) 50
d7
1
d<.
Due to our assumptions on z(c; or q), this integral is convergent, so the constant Ce is well defined. Using it in (3.3), we obtain
Jw S(w(C))(1- 2 WI))) m z(p))(1-+JK))) d(’ = uJ’ b(z) (1-~“(4) dx Jw(E) JE 4P(W(<)M~) z(()
X
(1 - rl+ &W(T)))
4P(W(0)W(C)
F
dr
d<
Since the integral
4PCX)X
is convergent for any finite <, we can use the Bonnet formula (the second mean value theorem) and conclude that there is <* > c such that
J
E’
4(<;g>77)=
b(d<)) (1- ~“bJK))>
z(c)
dC,
4P(W(l))W(<)
E
Now, recalling that z(f) = ~‘(0,
we can rewrite the last formula as follows:
4(C;fl> .rl)=
JTn.b(x)(1- u”(x)> dx
w
4P(X)X
’
where w = w(t), w* = w(<*) 2 b > 0. Using here (2.10) and (3.4), we can perform explicitly the integration to get
4(J;o,77) = orl
1
o(w* -
w)
2V + -log 1 1u;
[
1
+
w2Vl (2Vl (w*-
cw*) w)
I)’
B. KA~MIERCZAKAND K. PIECH~R
1398
where Q = ~1 + gwl is positive
by Assumption
2. For cs < 0, we h&e
the following
estimate
for
C7(<;0,77):
Since
wlul is independent
)fl( f (1 - $)/2
of q and a continuous
function
of Ir E Z, then taking
we prove the thesis.
I
The linearisation
of system
(2.18)
around
any critical
state
(wo,O, 0) gives us the system
(;)=K(‘). where
0
1
0
ab’po
_.f&o;w~+~,~) K=
(3.5)
rl
-4~,3(1-77+~Ao)
1 -V+VAO
-
arl(1- 4)
do
point,
and A0 = A(wo),
rl+rlbo
l-
4wo PO where wg is a critical
(3.6)
-2(1-q+qAo)
/
po = p(wo), UO= ul - a(wo - ~1). In addition,
4:‘%wo.
60 E 6 (wo) = 221 0 REMARK
2.
We have 6(wl) > 0, 6(wl(a))
This remark
follows from Proposition
The eigenvalues
of K are solutions (,J, - po)
> 0, b(w,(a))
> 0.
1 and Assumption
I
2.
of the following third-order
[A” + Au
(Go +
Doq’) -
equation:
a;]=aDo17’A2.
(3.7)
where
Do = Do(a) =
Go = Here PO(~), DO(~), and Remark
Go(o) =
2, whereas
3.
Let
(1 - v + rlAo) (1 -
rl + r160)’
b’po 4wo3(1 -rl+qAo)’
and Go are positive
due to the assumption
the sign of cy2(o) coincides
In the case of 7 = 1, equation PROPOSITION
60(1 - 4) 8~0~0
(3.7)
CJ be negative,
reduces
imposed
on the flow velocity
u,
with that of fh(.wo, wl, ul, 0).
to equation
with 10) sufficiently
(4.7) of our earlier
small.
If fh(wo,
paper wl,ul,g)
[lo]. < 0. i.e..
and if ~0 < ,&, or explicitly,
wg = WI or w. = w,(a), f;
(wo, ~1, UL,~) > (-1
- rl + q-40) (1 - rl + $0)’
3
(3.8)
1399
PhaseBoundarySolutions
then equation (3.7) has three real roots such that one of them is negative and the two others are positive. They can be arranged as follows:
(3.9)
A@) < 0 < (Yg < A@) < xC3) < PO. to X(j), j = 1,2,3,
Moreover, the right eigenvectors r (j) of the matrix K corresponding chosen so that
x(3) - p.
x(3) - p.
&I
r(3) = (
aTjX(3) ’
“7j
(1 - u;) 4WoPo
’-
can be
(3.10)
\ )
Also the following asymptotic formula holds:
x(3) - po a77
= O(rl)>
as ffq + 0.
PROOF. For small values of 101, we can treat the coefficients CY,/3, C, and G in equation (3.7) as if they were independent of e. In this case, o:(o) is positive due to (3.2). By means of the implicit function theorem, the following expressions for the roots of equation (3.7) hold:
xC3) -- PO + CT=
+ 0 (2)
.
Having these formulae, we check immediately that inequalities (3.9) hold true; also, by a direct I check, that the vectors r(j) given by (3.10) are the right eigenvectors of the matrix K. PROPOSITION 4. For 0 < 0 with 1~1 sufficiently small, all the characteristic exponents correI
sponding to the critical point (WI(~), 0,O) have their real part positive. PROOF. In this case, o;(a)
is negative due to (3.2).So we set a;(a)
= -o:(a),
where
By means of the implicit function theorem, the following expressions: X(l)(a)7
= 1 - n + $1 + O(a),
for the roots of equation (3.7) can be proved to be correct as e --+ 0. proposition is an immediate conclusion following from these formulae.
The assertion of the I
In this paper, we consider the case of cr < 0 with 101sufficiently small. Also, we are interested in such solutions to (2.11) along with (2.14)-(2.17) CJ= 0 can be excluded due to the following. ASSUMPTION 3. Let
where 201 5 w(J;a,n)
< w,(a).
The case
1400
B. KA~MIERCZAK AND
K. PIECH~R
Also, let 1. F(w,(O);wl,U2,0) 2. there exist 6,~
E Z such that infZ
(3.12)
> 0,
< 6 < a < 0 and such that
F(w,(~);wl,w,a)
the inequality (3.13)
5 0
holds for every g E [&, ~1. REMARK
3.
For comparison,
I
in Part I, for the case g > 0, we assumed that
and that F(W”(a);
for 0 E [a, 6.1, where 0 5 ?? < 6 were some real numbers, In Part
I, assumption
(4.17)
derived from the present We discuss
of that
solution.
0 = 0. Setting
instead
of (3.12)
paper was a little different
and (3.13),
from (3.14),
respectively.
but it can be easily
formulation.
in more detail,
of the Maxwell
(3.15)
W> W, 0) L 0,
I
assumptions
This
(3.12),(3.13).
type of solution
0 = 0 in equations
(2.10),
To this end, we will need the notion
describes
a particular
phase transition
in which
we obtain (3.16)
whereas
the only bounded
solution
of equations
(2.11)
is q(c)
= 0, < E IR1, and the equation
for w(E) takes the form (3.17) Using
the facts
conditions
that
w(+oo)
for the existence
= w, and w/(+00)
of solutions
of equation
= 0, we deduce (3.15)
easily
that
the necessary
are
WA4 F(luM;%,u1,0)
=
P(WM,%) Condition
(3.18)
is the well-known
is only one solution that ul is suitably
because
w,(a)
pair w,, chosen.
satisfies
equation
(g[ sufficiently
vicinity
of w,.
small.
(3.19)
Maxwell equal area rule of equilibrium
WM with b < w, They
< wM satisfying
are called the Maxwellian
(2.17).
Hence,
(3.18)
P(wrL>Q)l 4 = 0,
=P(Wm,W)
If ul > 0, then we have f~(w;w~,u~,~) with
s
w”~ [P(<, UI) -
But,
from (2.12),
equations
states.
It is easy to note that F(w,(O);
There provided
that
we have
FL(w,(O); wl,ul,O)
(3.18),(3.19),
Let us notice
> 0, for all w E (wl,w,(O)]
in particular,
phase transition.
wl,ul,O)
and all o > 0 or all 0 5 0 > 0, for all wl from some
> 0, if only wl > w,,
but sufficiently
Phase Boundary Solutions
1401
close to it, and F(wr(O);wl,ut,O) > 0 tends to zero as wl tends to w,~. Thus, by using the implicit function theorem, we conclude that, if only wz - W m is positive and sufficiently close to zero, then there exists a locally unique
a
such t h a t
F(wr
;wl, ut, ~) = 0. Moreover, ~ < 0~
l
F(w~(o);wl,ut,a) > 0 for all o - o > 0 sufficiently small, and for all wl - W m > 0 sufficiently / l small, we have F£(w~(a);wt,uz,~) > 0. As a consequence, F(w~(o);wl,ut,a) < 0 for all a < sufficiently close to it. F r o m the above, we can conclude the following. REMARK 4. T h e case a > 0 has to be accompanied by the condition wl < win, but sufficiently close to it (this was tacitly assumed in P a r t I), and the case a < 0 has to be accompanied by the condition wl > win, but also sufficiently close to it (this is the case of the present paper). | LEMMA 3. Let (3.12) hold. Then there exists ~ E (~_,O) such that for all rl E [0, 1], and all a E (-g,O) sut~ciently small, there is no trajectory connecting (wl, 0, 0) and (w~(~), O, O) with the nonnegative derivative w' (~). | PROOF. In the case of assumption (3.14), the proof m a y be found in P a r t I. So, let us assume t h a t (3.12) holds. T h e above l e m m a is implied by the fact t h a t we are able to find a priori estimates for z(~) g o o d for all admissible values of o (see L e m m a 1 in P a r t I). Now, we write the second equation of s y s t e m (2.18) in the form [1 - ~ +
~A(w)]z' + ~-A~(w)z 22 '
crb2p(w) 4w 3 z - f (w; wl, ul, o) - ~q. r]
Multiplying by z, integrating from - o 0 to ~, and making use of the t h e o r e m proved in [1t, C h a p t e r 1], we obtain
[w(~) (°b2p(u)
~
JW
q(.)
)
a.,
(3.20)
l
where n ( w , . )
=
1 [1
-
+ r/A(w)],
(3.21)
and the functions z and q are treated as functions of w. Let the derivative w'(~) = z(~) be nonnegative. As z is, under our conditions, b o u n d e d uniformly with respect to cr and r], then, as follows from L e m m a s 1 and 2, respectively, the first term on the right-hand side of (3.20) is of order of a as o ~ 0, whereas for the third one, we have the estimate q(~, a) = O i a ) as o ~ 0. Therefore, for small values of o, the right-hand side of (3.20) can be m a d e arbitrarily close to
fj
~'(~) f (u; wl, ul, 0) du. I
Thus, taking ~ = oe, we arrive at a contradiction. T h e l e m m a is proved. REMARK 5. This l e m m a is a c o u n t e r p a r t of L e m m a 2 of P a r t I, which used (3.14) and o belonging to the interval [0, a], with _a positive and sufficiently small.
Assmnption 3.2, there is no trajectory connecting the points (wt,O,O) and (wr(o), O, O) with nonnegative derivative w' ({) and nonpositive q( ~) for o E [d, a_] and r? c [0, 1]. | PROOF. From relation (3.20), we obtain LEMMA 4. Under
so, the facts t h a t z is nonnegative and q is nonpositive negative lead to the contradiction.
|
REMARK 6. This l e m m a is a c o u n t e r p a r t of L e m m a 3 of P a r t I, in the proof of which we used (3.15) instead of (3.13). |
B. KA~MIERCZAK AND K. PIECH~R
1402
4. THE LEFT-GOING Isolating
neighbourhoods
for system
(2.18)
WAVES
will be defined in a similar way as in the case of
0 > 0, however, now q will be assumed to be nonpositive, case under consideration
which is the main difference
between the
and the case treated in Part I. Below, we take n E: [0, l] and (T E ia, a].
where a and ZYare such as above (see Assumption
3 and Lemma
3).
1. Let N;(a,rl)
= {(‘w, z,q)
Let AL(E) = B((w~,O),E) denotes
: ‘w E [w~,w-(~)], .z E K’, 2 (a,rl)l>4 E i-Q (0,~) ,011. x [-E,E]
and AT(u,c)
a closed ball in (w, z)-plane
= B((w,(~),~),E)
x [--E,E], where B(,,E)
with the centre at the point (.) and the radius equal
to &. Let Al(a,y)
= B((~~(cr),O)~y)
in (w,z)-plane tion (2.17),
x (-E,E),
where B((wi(a),O),y)
with the centre at (wi(o),O),
denotes
where WI(C) is the ‘second’
and the radius y with y sufficiently
an open
ball
root of equa-
small.
Let Nl(a,%E)
{( w,z,q)
=
:w
E [W,WT
(Q)l,25E
Finally, let
N(o,rl,~) = Nl(a,rl,~) u AL(E)u M~‘,E) \ Al(a,y). The set N(a! n, E) is shown in Figure 1.
Figure 1. The candidate for the isolating neighbourhood We prove the following. LEMMA 5. For isolating
E >
0 sufficiently
neighbourhood,
small, every n E [0, l] and cr E [a,~],
i.e., its maximal
First, we prove some propositions, PROPOSITION 5. If I(fi(a,
in its interior.
making the proof of the lemma easier.
q, E)) d enotes ~~(vLE)
invariant set is contained
the set N(a! ‘q, E) is a good
the maximal
invariant set contained
=N~(~,~~,&)UA~(E)UA,(~,&),
in the set
I
Phase Boundary Solutions
1403
where E > 0 is taken sufficiently small, then the set of points in I(fi(o,
7, E)) lying on the
trajectories tending to (WI(~), 0,O) for [ --+ --co is empty.
I
The assertion is a simple consequence of Proposition 3.
PROOF.
PROPOSITION
I
For every n E [0, l] and g E [g, ??I, with )o( sufficiently small, the following
6.
equality holds: I(Nr(o,n,s))
n dNl(o,n,s)
= {critical points
2 Nr(o,n,E)}.
Moreover, for all E E IR1, we have q(J) < 0 and q(e) = 0, if q = 0, for any trajectory to
belonging I
I(Nl(O,%&)).
We must show that if the trajectory touches the boundary of Ni (a, 7, E) for, say, < = &, unless it is one of the critical points. (The then this trajectory cannot belong to I(Ni(a,v,~)) dependence of the trajectory on the parameters g and n will be omitted below in notation.) PROOF.
POINT 1. w([c) = wl or ~( 0, then the trajectory leaves Ni(a,n,e) in the appropriate time direction. So, let us suppose that ~(6s) = 0. So that the trajectory stays inside Ni (a, n, E), we should have ~‘((0) = 0, as otherwise, it leaves Ni (a, n, E) in the appropriate time direction. But this implies also q = 0, so at 6 = 0, then the trajectory leaves Ni(o, 7, E) in the appropriate time direction. On the other hand, if q’(Jo) = 0, then p = 0, and also, ~~.z(~s) = 0, as in Point 1. This implies that for q > 0, z’(&,) = 0, and hence, w(Ec) is equal to one of the roots to equation (2.17), so at t = [a, we have one of the critical points. However, if 07 = 0, then we must have q(J) = 0 for all t E IR’. POINT
3.
z(&)
= 0.
As before, this implies that z’(&,) = 0, hence, (4.1)
If n = 0, this implies f(w(&,); wl, UL,D) = 0, so at [ = ta, If 77 > 0, then relation (4.1) takes place only if q(co) = 0. point. To see this, let us suppose that q([o) # 0, i.e., q(ca) w,(o) > w(E) > (to) = wc. Integrating the second equation and making use of the theorem proved in [ll, Chapter l], we w
R(w)z - ;
J W”
R’(C) d( =
lJJb2PK) dC
-0
J wg 4c3
_
Ef(w(T);
J Eo
we have one of the critical points. Hence, again we are at a singular < 0. Let us consider [ such that of system (2.18) with respect to 5 obtain
1 ‘w1,u1>0) dr - 2’1
4(+k JEa E
(4.2)
where R(w) = R(w; n) is given by (3.21), and the function z is treated as a function of w. This is possible as long as z(E) 2 0. From the third equation of system (2.18), we get
s[ co E
4P(W(7)) rl2(Ul + owl) - aw(r)
1
+ (1 - 77) q(W(r))dr=q(<)-q(
to 1-(Ul -4-wH2 &, J ?JJgC(2(7Q + OWL)- CC)
We define G=
4P(W) inf + (1-q) rl zuE[wo,w,(a)l 2(Ul + awl) - crw
1
> 0.
Let us note that, according to the previous point, for any solution staying in Ni (a, n, E), q(E) 5 0, for all <. Hence,
(4.3)
B. KA~MIERCZAK AND
1404
K. PIECH~R
Using (4.3) in (4.2), we get the inequality
_-77q(t) 2
4
s
((0) -- av2
G
2G
lD 1 - (Ul - u(C -
w1))2
‘wg < (2 (u1 + owl) - OC)
dC
At the critical point, we have
due to equation (4.1). The last term of the right-hand side of the above inequality is positive, due to relations (3.2) and the fact that z’(E) 2 0, for all < E R1, so ignoring it, we get
In the last inequality, l l l
the left-hand side is negative, the sum of the first two terms on the right-hand side is negative as well, the third term on the right-hand side of this inequality is positive.
Hence, for sufficiently small values of 1~71,this inequality cannot be satisfied. So we have to have q(&,) = 0. We cannot have w(
w(Eo) = w,(a).
According to the saddle-point theorem, for E sufficiently small, the sets Al(&) and AT(a, E) are isolating neighbourhoods with the maximal invariant sets equal to the critical points (wl,O, 0) and (w,(a), 0, O), respectively. Now, we prove the following. PROPOSITION
7. A trajectory from I(Nl(a,
TJ,E)) cannot enter any of the sets A,(a,
or the set AI(E) \ Nl(a,q,&). PROOF.
Suppose, first, that a trajectory
E) \NI (CT!77,E) I
belonging to I(Nl(a,
77,E)) leaves NI(CT, q, E) and enters
the set A,(a,
77,E) \ Nl(a, 7, E). So that the considered trajectory could reach the singular point (w,(a), O,O), it should coincide with the stable one-dimensional manifold S of this point. The eigenvector tangent asymptotically to S has the form given by (3.10), where X* is the unique negative eigenvalue of the linearisation matrix K : E} at a point where z > 0 and q 5 0, and the set {(w, z, q) : w = w, + E} (also w = w,transversally) at a point where z < 0 and q 2 0. It is obvious that the trajectory cannot reach \ Nl(o, 7,~) as it should leave this set first (and the point (w,(a), 0,O) inside the set Ar(o,q,&) consequently, the set N(a, 77,E)). So it must enter the set NI(~,
However, the considered trajectory cannot enter the set that this trajectory reaches this set at some time [ = to to the region w < w, - E. Then we must have z(e) < But the points (w, z, Q) such that w < w, - E and z < 0 contradiction.
77,E) n {(w, z, q) : w = W- - E}.
{(w, z, q) : w = w, - E}. For, suppose and passes from the region w > W, - E 0 for all < < &J sufficiently close to it. are not in N1 (a, 77,E), so we arrive at a
1405
Phase Boundary Solutions
Now, let us consider
matrix
K around
eigenvectors.
Al(&).
the singular
The vectors
which is tangent
Let XI’) < 0 < Xi2) < X3 be the eigenvalues (~1, 0,0), next
point
let rl(i), i = 1,2,3,
rji), a = 2,3, span the unstable
to the unstable
manifold
manifold
Ul of the nonlinear
of the linearisation be the corresponding
of the linearized
system
system
(2.18) at the singular
(3.5) point
(wl,O, 0). Let Nz*,Ns* denote the trajectories of the nonlinear system corresponding, due to the Hartman-Grobman theorem [ll, Chapter IX; 121, in a one-to-one way, in some vicinity of the singular point (~1, 0,O) (independent of D and q), to the half lines
rl) = {(w, *) : (w, ~~4) = *b-j”‘,
k(c, (see Figure
2). Let Ns*
denote
the upper
i = 2,3,
C I 0) ,
or lower branch
2). We set Al =
of N3 (see Figure
{(w, z, q) : (w - ~1)’ + z2 _< E, q E [-q*, q*]} and Bl = Al f~Ul, where q* = -oQ* and 0 < E < q* 2), so AL(E) c Al. If P E 1 is sufhciently small then, obviously, in the set Bl. the following conditicns are satisfied. (see Lemma
1. Ns-
intersects
the plane
q = -q*
and Ns+ intersects
the plane q = q*.
2. f~(w;w~,~~,~) < 0 for (w,z,q) E Bl = Al nUl. 3. Every trajectory of the nonlinear system (2.18) lying in Ul leaves the set Bl for sufficiently large 5. Now,
we will use the fact
that
Ul is two dimensional.
For 77 = 0, the
last
equation
of
system (2.18) separates from the first two, so it is easy to see that all the trajectories from In this case, the monotonicity of w(l), i.e., the I(Ni(a, Q, E)) must lie in the plane q = 0. positives of z(E), is obvious. So, let us consider singular
point
the case of 7 > 0. In this case, in the assumed
(WI, 0,0), we can treat N3-
w and z as functions
= @J* (0 32, (0 1q* (0)
(2< 0)
I
and
N3+
small
of the variable = b*(<),
t*iE)>
(2 = 0)
7---___-___-____
N3+ {q=q’)
- (4’0)
Figure 2. The set Bl.
vicinity
q. Let
q*(O).
of Bl, the
1406
B. KA~MIERCZAK
We claim that,
for all CJ E 2, r) E (0,11,
~(6) Indeed,
AND K. PIECH~R
> 0
and
asymptotically,
z*(c)
as E +
as NY_ stays in Bl, because
-00,
as long as Ns-
< 0, Ns-
is tangent
the third component
and Ns+ stay in Bl.
to rl(3). Therefore,
we have q*(t) < 0 as long
of this vector is strictly
small values of cr E Z and r) E [0, l] (see (2.10)),
that is why we conclude
large, we have z*(t)
> 0 for 6 E (-co,&)
> 0.
from the second equation
Suppose of system
z’ (to) = - P -
rl +
This leads to a contradiction.
z*(E)
(2.18),
we have
that,
independently
for all sufficiently
for (-<)
and z,(&)
sufficient11
= 0.
But
then.
+;ari(50) ;Wl,W,~‘) 1>0.
VA(w (EON-~ [lf (w (to)
The same proof may be done for Ns+.
Owing to the construction to parts,
that
negative
of the vector rl(3) is positive
of 0 E Z and 77 E [0,11.Next, the second component
(4.4)
of the set Bl and (4.4),
say, the left and the right ones.
I
we infer that the line Ns divides this set into
Any trajectory
starting
from the singular
point (~1~ 0,O)
and lying in the left part of BL leaves this set either from within the region z < 0 or through
one
of the planes q = +q*.
the
Any such trajectory
cannot
right part of Bl. In this case, any trajectory from the point the second
(wl,O, 0) has to be asymptotically,
component
of cr, we infer that
of this vector
asymptotically
is positive
positive
Na-,
as < +
7, E). So let us consider
and different
-00,
tangent
from Ns*,
to the vector
and the third one is negative
we have w > WI, z > 0, and q < 0. Suppose
leaves the region z > 0 (for the first time) if q becomes
belong to Ni(a,
T, including
for some < = [i.
for some < = .$ <
This is possible
starting
rj2).
for negative
Since values
that the trajectory
only if q([l) > 0, i.e..
we would have z’(ci)
> 0 which follows
from the relation
but this is a contradiction. consequently, trajectory
leaving
The trajectory
r has to return
second time.
can either
N(a, n, E) or it can attempt
leave the set Bl from the region z < 0. a~nd
to get into N~(Q, 7, E). In the latter
to the region z > 0 crossing
Let it take place for 6 = &. First,
case, the
the line Ul n {(w, z, q) : z = 0) for the
we will prove that
we cannot
have (3.5)
4(<3) > 4((l)'
We have z’(&)
The
last
maximum, from (4.5)
2 0 and
inequality at [
=
results
from the fact
and the fact that
minimum,
z’([i)
that
z’(&)
> 0, we can obtain,
f(w
Subtracting
(4.8)
from (4.7),
w(E)
attains
function
at < =
of [
E (
Now.
< 0, it follows that
f (w(b);W1rwr~) Using the fact that
the function
and is a decreasing
+ ;w(tl)
in a similar
(<3) ;w,w,a)
we obtain
+ $?u(G)
(4.7)
> 0.
fashion,
< 0.
the following
inequality: (4.8)
Phase Boundary Solutions
1407
qi = q(Ji), and i = 1,2. Since &(w; W~,UL,g) < 0 in Bl (see the definition of this set), then, owing to (4.6), we have
where wi = w(rt),
0.
(4.10)
Now, from (4.5) and (4.10), we infer that the left-hand side of (4.9) is negative. Consequently, If q(&) < q(&), th en either 7 would leave Bl from within the region z < 0 or there would exist a segment of this trajectory lying entirely in the region z I 0
we arrived at a contradiction.
joining the points (wl, 0, ql) and (w”, 0, q2) such that q1 < q2. But this is impossible according to the considerations above. Hence, any trajectory tangent asymptotically to the vector rl(2) having once entered the region of z < 0 cannot come back to the region of z > 0. Hence, any such trajectory
will either leave the set Nl(o, 77,E) from the region z < 0 or will stay in the region
z > 0. If it leaves Bl crossing the line Ul n {q = q*} or the line Vl n {q = -q*}, then it leaves the set N1 (a, 77,E). Hence, the only opportunity is that the trajectory leaves Bl from the region z > 0 and -q* < q < q*, thus, staying in the set Nl(a, q, E). REMARK
7.
I
Let us note that, for g = 0, 7 E [0,11,the set Nl(O,
77,E)u
B ((w~,o,o),~)u~((~,(~),O,O),~)\Al(~,~),
where B(P,G) denotes the three-dimensional ball with the centre P and radius Lz,is a good isolating neighbourhood, only if 3 is taken sufficiently small. This follows from the fact that for u = 0, any invariant trajectory lies in the plane q = 0, thus, the system behaves as two dimensional. Due to the robustness of the Conley index, the same isolating neighbourhood is good for the system with 1~1sufficiently small. However, proceeding in this way, one does not prove the w-monotonicity of the invariant trajectories near the points (~1, 0,0) and (w,(c), 0,O). This information is important from the physical point of view. Also, the necessary smallness of 1~1 is not controlled, which is very significant in view of Assumption 3. In Part I, the monotonicity property was implied by the results of paper [lo]. I Let Assumptions 1-3 be satisfied. Then, there is at least one CTE (a,??), such that there exists a heteroclinic solution to system (2.11) connecting critical points (~1, 0,O) and (w,(o), 0,O) such that w’(t) > 0 for all [ E (--00, co). I
MAIN
THEOREM.
PROOF.
For q = 0, system (2.18) takes the form
wi=
2,
ab2dw)
z’ = -
[
-z+f(W;w,w7~) 4w3
q’ = q.
1 7
(4.11)
Any compact invariant set for this system must be contained in the plane q E 0. Thus, for every g E [a, YY],we can find a family of isolating neighbourhoods Nx (a, E), X E [0,11,such that N1(a, E) = N(a, 0, E) and No(cr, E) = Nzuz(gr E) x [-E, E], where N,,(o, E) denotes the projection of the set N(g, 0, E) onto the (w, z)-plane. We use the Conley connection index theory [8]. To do this, we append to system (2.18) the equation
0’ =
pqw, z, q)
( I7 - F),
(4.12)
where @ is a sufficiently small positive number. Let U’ and U” denote open neighbourhoods
of
(w~rO,O)~{a}U(w~,0,O)x{~}and(w,(a),O,O)x{a}U(w,(~),O,O)x{~}inR3x(a-~,~+~), respectively, having disjoint closures. The real valued continuous function @ is arbitrary except for the fact that it is positive on U’ and negative on U” (see [S]). Then we must compute the
1408
B. KA~MIERCZAK AND K. PIECH~
Conley index he of the maximal
invariant
by Sc.
Since the third
of system
system,
we infer that
the invariant equation) with
equation
set in Ne = &ek,?l (4.11)
this index is homotopic
set comprised
in the set [-E,E]
to the
flow generated
index h is computed to 0. Thus,
(with respect
in the paper by Grinfeld (wr(g),O,O).
and Sl: = IL,,,,] spheres X2. Therefore, In general,
S, =
h” ”
set comprised
equations
[9] or by Conley
and Gardner
of the hyperbolic
of this index of
by the third
in &k,al
of system
hand, let S& = &k,zl(~~,
The indices
C1 is the Conley
to the flow generated
invariant
by the remaining
ho 2 C’ A 0 g 0. On the other
from the other two equations
to h” P, h, where
and h is the Conley index of the maximal
respect
Ne(o, E). This set will be denoted
uncouples
NW2 (0. E)
(4.11),(4.12).
The
[8]. It is homotopic
0,O) = (wl, 0,O) x [(T,CT]
critical
points
are both
pointed
the “wedge” sum of the indices of Sl, and S$’ is equal to ( C2 A C’) VC2 y 6.
we can define for 17 E [0,11:
maximal
invariant
set in
n
N(~,v,E)
with respect
to system
(2.18),
(4.12)
OE!g,F] For 17 E [O,l], the triple triple
(SL, Si, Se).
Also, the Conley respect
indices
points with exactly
of the points
two positive
from the critical
f orms a connection
points.
77) n N(~,v,E).
(wi, 0,O) and (~,(a*),
triple,
which is a continuation
index of the first one, for q E [0, 11, is homotopic
S’(a),
by system
a value of D* E (a,??)
in the set Ni(a, connect
(Sk, St, S,)
the connection
to the flow generated
there exists different
Hence,
S;(g),
(2.18)
characteristic
0 E [a,~],
are homotopic
exponents).
(as isolated
invariant
to C2 (these
points
According
such that for 0 = g*, there exists In fact, according As it cannot
0,O).
to Propositions
to the results
an invariant
of the to 0.
sets with are saddle
given in [8],
set in N(a*,
1, E)
6 and 7, this set is contained
be comprised
in the plane {w = 0, z = 0}, it must
The proof is complete.
I
REFERENCES A four-velocity model for vanderwaals fluids, Arch. Mech. 47. 1089-1111 (1995). Cinetique des Gaz a RCpartition Discrete des Vitesses, Lect. Notes Physzcs. Volume 36 Springer-Verlag, (1975). B. Kaimierczak and K. Piechor, Phase boundary solutions to model kinetic equat,ions via the Conley index theory. Part I, Mathl. Comput. Modellzng 31 (13), 77-92 (2000). L. Preziosi and E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. fnd. Appl. Math. 14, 3999415 (1997). C. Ringhofer, Dissipative discretization methods for approximations, Math. Models Meth. Appl. Ser. 11, 1333148 (2001). D. Goersch, Boltzmann operators in unbounded domains, Math. Models Meth. Appl. Scz. 12 (1). 49-76 (2002). H. Babovsky, A kinetic multiscale model, Math. Models Meth. Appl. Sci. 12 (2), 309-332 (2002) C. Conley and R. Gardner, An application of the generalised Morse index to travelling wave solutions of a competitive reaction diffusion model, Indiana Unzverszty Mathematzcal Journal 33, 319-343 (1984). M. Grinfeld, Nonisothermal dynamic phase transitions, Quarterly of Applied Mathematics XLVII (l), 71-84 (1989). B. Kaimierczak and K. Piechor, Shock waves by a kinetic model of vanderwaals fluids, ARI 51, 203-215 (1999). P. Hartman, Ordinary Difjerential Equations, Wiley, (1964). S. Sastry, No&near Systems: Analyszs, Stability, and Control, Springer, (1999).
1. K. Piechor,
2. R. Gatignol, 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Mathematical
and Computer
Modelling
36 (2002)
139331408 www.elsevier.com/locate/mcm
Phase Boundary Solutions Model Kinetic Equations via to the Conley Index Theory. Part II B. KA~MIERCZAK Polish Academy of Sciences, IPPT PAN, Swietokrzyska 21 PL 00-049 Warszawa, Poland K. PIECH~R Department of Mathematics, Technology, and Natural Sciences University of Bydgoszcz, 85..064 Bydgoszcz, Poland and Polish Academy of Sciences, IPPT PAN, Swietokrzyska 21 PL 00-049 Warszawa, Poland
(Received
March
2002;
accepted
April
2002)
Abstract-we consider phase boundary solutions to a four-velocity kinetic model of a kinetrc equation governing the motion of van der Waals fluids. These solutions connect such equrlibrrum states, which are saddle critical points of the related dynamic system. Solutrons of thrs type can be interpreted as dynamic phase transition. The mathematical apparatus is that of the Conley index theory. @ 2002 Elsevier Science Ltd. All rights reserved. Keywords--van
der Waals fluids, Model discrete ling waves: Conley connection index.
velocity
kinetic
theory.
Phase boundary
travel-
1. INTRODUCTION This paper
deals with the phase
governing
the
motion
consisting
of a discretization
As was already number
of velocities,
already
attent,ion
points of the dynamic
mentioning
supported
0895-7177/02/$ - see front matter PII: SOS95-7177(02)00296-O
to the Enskog-Vlasov phenomena
wave solutions
analysis by Grant
schemes
as documented in this paper
[a].
to a small occurring
the saddle critical
Boltzmann
problems. equation
in some recent
is also related
papers,
can be generalised
analysis It is worth e.g., [4-71.
also to models
KBN 7T07A00919.
@ 2002 Elsevier
Science
Ltd.
in
Hence, this paper deals with
of fluids and with the qualitative
(or discretized)
proposed
limited
develops the considera-
connecting
in this case, wave propagation
in the discrete
in il].
equation
of some interesting This paper further
equation
proposed
being
properties
problems;
of computational
the qualitative
This paper was partially
of travelling
of a kinetic
model,
the discretization
related to the above class of equations.
phenomenological
the interest
to the development In principle,
of existence
model
mathematical
corresponds despite
provides the description
of the mathematical
that
The
to the van der Waals ones.
system
of qualitative
to a four-velocity
fluids.
by four velocities,
tions of [3] with the analysis the analysis
solutions
shown in [1,3], the model,
fluids with special
of the solution
state
of the van der Waals
,211 rights reserved.
Typeset
by dn/ls-‘Q$
B. KA~MIERCZAK AND K. PIECH~R
1394
which discretize energy
the original
in the collision,
The contents
equation
by conservative
schemes
preserving
mass, momentum,
and
see [4,6].
of this paper
are organised
into four sections.
Section
2 deals with
a concise
description of the model and the statement of the mathematical problem. Some preliminar? technical results are given in Section 3, while the last section deals with the mathematical problem related to wave phenomena.
2. STATEMENT This paper
is a continuation
sake of completeness results
of Part
The present system
of partial
of our previous
paper
[3], referred to as Part I. However, for the convenience, we quote here the most important
of theses and for the reader’s
I, revising, paper
OF THE PROBLEM
at this opportunity,
concerns
differential
some misprints
the proof of existence
contained
of travelling
there.
wave solutions
to the following
equations:
a
a ax
-w--u=o, at
(2.1)
-gu+&[~(l+$p)-~+~]=E~[~~u] a ++
5 p
a
2
EW
i()
and
a
tuatq
a
- 2uaxq
l-U2
a
w
ax
+ ----_u
(2.2)
2a2 ---_w W5dX2
I
’
4P = --q,
(2.3)
E
where p= p(w) = h
1 0 W
Here t is the time, X is the Lagrangian coordinate, w is the specific volume, ‘~1is the velocity of the fluid in the X-direction, q is an extra quantity which has no direct physical interpretation. and a, b, CY, E are positive constants characterising the fluid. This system of equations is an approximation to a kinetic four-velocity model of the Enskog-Vlasov equation. Since its derivation is described in detail in [l], and briefly, it is also presented in Part I, we omit a closer discussion of their structure. REMARK
1. The specific volume
the following
constraints
w, the mean velocity
w > b,
Throughout
this paper,
U, as well as the parameters
a, b satisfy
[I]:
the above conditions
and
u2 < 1 - ;T.
will be used without
any further
reference.
I
Similarly, as in Part I, we are interested in the travelling wave solutions to equations (2.1). (2.3). Prom the mathematical point of view, the existence analysis splits naturally into two parts: 0 > 0 and 0 < 0, where g is the wave speed. Though the proofs in both of these cases proceed along the same lines, there are some essential differences. Moreover, the physical significance of the case 0 < 0 has encouraged us to treat it separately. A travelling wave solution to the kinetic is a so1utio.n of the form equations (2.1)-(2.3)
(‘w,u, q)(t,X) = (WIu, n)(E)1
x - fft
(C--..--
&
E
BP,
(2.4)
Phase Boundary
Solutions
1395
where 0 = const is the wave speed, such that (2.5) (2.6) (2.7)
,Jym(w’,u’,Q’)(0 = (O,O,O), ,!jf& (w”,u”‘) (0 = (0, 0,O),
(2.8)
where the dash by a character means differentiation with respect to [. Now, we act in a very standard way. Namely, we substitute (2.4) into equations perform one integration with respect to E, and use the limit conditions (2.5)-(2.8). that, we find the left and right states of rest related algebraically
(2.1)-(2.3), Having done
(2.9) These relations are called the Rankine-Hugoniot is given by
conditions.
Next, we find that the velocity u
u = Ul - a(w - Wl),
(2.10)
whereas w = w(E) and q = q(r) are coupled through the following system of ordinary differential equations: w’ = z, “=-A(w)
1
LA’ (w)z’ 2 UJ
11
+(w) + 4~3
1-(u1-c+wl))2Z+
Q(w)
q’ = -cr w (2 (U1 + OWL)- aw)
(2.11)
q,
2(ul+awl)-aw
(2.12)
f(W;w~,21~r~)=~2(W-w’W1)+p(W,‘1L1--(~--W1))-~(W1,~1)~ where P(W,U) =
1 - u2
(2.13)
2(w - b)
is the pressure, and A(w) = 2cy/ws. In our considerations, we do not make use of the specific form of the function A(w). need is just that this function is positive for all w > b, and that its derivative A’(w) for w > b. Prom (2.5)-(2.8), we obtain the following limit conditions:
w(E)= WI,
,
(2.14)
;;iJ& w(J) = W,
/?im_qt<) = 0,
(2.15)
Elim -0c)
pya 40
What we is negative
= 0,
E lir z’(t) = 0, -+ca
,liI=,
These conditions must be supplemented by the Rankine-Hugoniot
f(.Wv;w7w,~) = 0.
q’(E) = 0.
(2.16)
condition (2.9), which implies (2.17)
In this part of the paper, we consider only the case of left-going waves with a negative wave speed. The idea of the proof of existence of such waves is the same as in Part I and consists of the use of the Conley index theory. However, in the case of the negative wave speed, some details
1396
B. KA~MIERCZAK AND K. PIECH~R
and estimates the Conley system
are different
connection
into a simpler
following system
from the previously
index consists
considered
of the possibility
one, which can be effectively
of ordinary
differential
case of positive
of making
analysed
wave speeds.
a homotopy
(see [8,9]).
Use of
of the considered
So, we will analyse
the
equations:
w’ = z,
- rj + VA(W)]-
z’ = -[I
’ ;A;(w)z'
1 - (“1 - cr (w - wL))2z + q’ = -al7 w (2 (u1 + awl) - aw) where
71 E [O,l],
(2.18)
coincides
correspond
subject
to the limit
to the well-analysed
4w3
1
4P(W) 2(Ul +awl) -0w
conditions
with the initial system
The basic ingredient
ab2d4
+
system
of the existence
(2.14)-(2.16).
whereas
(2.11),
+ (1 - rl)
describing
Note
(2.18)
1
1 4,
that,
for q = 1, system
for 7 = 0, the first two equations
the
isothermal
proof is constructing
phase
transitions
a family of compact
of (2.18) (see
[9]).
neighbourhoods
N(g, 77)>rl E [O, 11and o from a suitable closed interval in IR1, such that for every 7) and 0: the set N(a, q) has the maximal invariant set with respect to system (2.18) in its interior.
3. BASIC The-critical equation
ASSUMPTIONS
points
(2.17). o.2
for system
Explicitly,
2b)
2 In general,
due to the complexity
in detail the dependence
(2.18)
either
w:(wo+ Wl -
AND
PRELIMINARY
RESULTS
are all of the form (~0~0, U), where wo is a solution
wg = WL, or wg satisfies
+azl&
-
of the coefficients
of its solutions
of
the cubic equation
a (w - b)
ab
Wl”
Wl I
in equation
on the wave speed 0.
wo+3
(3.1),
=o.
we are unable
However,
(3.1) to discuss
from the analysis
done
in (lo], one can deduce the following. PROPOSITION
the algebraic
1.
There
equation
exists an open set Z such that 0 = 0 is contained (2.17)
has exactly
four distinct
b < ‘~1 < WI(~)
in it, or if 0 # 0. the11
roots satisfying
< w,(a)
< wz(c),
with
We assume ASSUMPTION ASSUMPTION
[wl, w,(a)]
the following. 1.
The infimum
2.
There
of Z is negative,
is a compact,
connected
whereas
the supremum
interval
P c
(b, oo) such that
for any c E 2:
C P, 21~+ owl > 0, 2(u~ + GW~) > gw, for any w E P.
Similarly,
as in Part
PROPOSITION
2.
I
I, we can prove the following.
If (w(e), z(J),q(t))
and q(E) are bounded ~(6; 0, rl) I ~(0,
functions
IS an y continuous
of <, i.e., there are ~(a,
V) and SUP< MC; 0, q)l =
solution
of system
77) and $a,~)
(2.18)
such that w(t)
such that b < ~(a.
&(a, rl) < co; then supE Iz(c; a,~)1 = Z(a,q)
LEMMA 1. Let (w(0,z(<),q(E)) b e a continuous solution of system (2.18) such that is bounded and w(E;a,q) is uniformly bounded function of < E R1, o E 1, and q E SUP< Id<;
such
that
or ~11 =
b < g
u
of Z is nonnegative.
Q(a, 7) < 00, and there are constants 5 w([,a,q)
w, G independent
< cc.
‘7) 5 1
q(<; 0: q)
[0, 11,i.e..
of 0 E Z and q c [0, l]
5 G for any [ E JR-l,CYE Z, and 77 E [0, 11. Then
there
are
Phase Boundary
Solutions
1397
constants Q > 0 and Z > 0 independent of u E Z and r] E [0,11 such that, for any < E IR’, CJ E 1, and q E [0, l], the following estimates hold true: sup< /Q(<; o,~)]
= Q((T,~) < Q and
supE MC; o, ~)l = z(o, 7) < 2, i.e., q(t; ot V) and z(E; o, 7) are also uniformly
bounded
functions
of < E IR’, CTE I, and 77E [O,l].
I
LEMMA 2. Let (w(<),z(G,q(E)) b e a continuous solution of system (2.18) satisfying the limit (2.14)-(2.16) such that w(J;a,q) is a uniformly bounded function of 5 E RI. CTE 1.
conditions
and 17E [0, 11, then there is a positive such /q([;a,q)]
constant
Q* independent
of < E EC’, 0 E 2, and 77E [O. 11.
5 q]o]Q*, for any E E R’, (T E 2, and fl E [0,11.
I
PROOF. Let (w(e), z(E), q(5)) b e a continuous solution of system (2.18). equation of this system, we obtain
Then, from the third
50 1 f E xic,-a?) J 4dNC)Ml) [i 4([; 0~77)= exp
71 - 77+ /J(W(T)))
d7’
I
z(c) 6 (w CC)) (1 - ~2(w))
exp
-
50
(3.3)
(1 - rl+ Pd(W(T)))dT & * Eo I 1
where cc is an arbitrarily fixed value of the independent variable, Cc is a constant of integration, and b(w) =
4P(W) 2u+ffw’
(3.4)
This function satisfies boundary condition (2.15) only if
J
b(w(C))(1- u2W>))
O3
co = UT)
z(s)
4P(W(GMC)
Eo
exp
c
[J
(1 - q + &W(T))) 50
d7
1
d<.
Due to our assumptions on z(c; or q), this integral is convergent, so the constant Ce is well defined. Using it in (3.3), we obtain
Jw S(w(C))(1- 2 WI))) m z(p))(1-+JK))) d(’ = uJ’ b(z) (1-~“(4) dx Jw(E) JE 4P(W(<)M~) z(()
X
(1 - rl+ &W(T)))
4P(W(0)W(C)
F
dr
d<
Since the integral
4PCX)X
is convergent for any finite <, we can use the Bonnet formula (the second mean value theorem) and conclude that there is <* > c such that
J
E’
4(<;g>77)=
b(d<)) (1- ~“bJK))>
z(c)
dC,
4P(W(l))W(<)
E
Now, recalling that z(f) = ~‘(0,
we can rewrite the last formula as follows:
4(C;fl> .rl)=
JTn.b(x)(1- u”(x)> dx
w
4P(X)X
’
where w = w(t), w* = w(<*) 2 b > 0. Using here (2.10) and (3.4), we can perform explicitly the integration to get
4(J;o,77) = orl
1
o(w* -
w)
2V + -log 1 1u;
[
1
+
w2Vl (2Vl (w*-
cw*) w)
I)’
B. KA~MIERCZAKAND K. PIECH~R
1398
where Q = ~1 + gwl is positive
by Assumption
2. For cs < 0, we h&e
the following
estimate
for
C7(<;0,77):
Since
wlul is independent
)fl( f (1 - $)/2
of q and a continuous
function
of Ir E Z, then taking
we prove the thesis.
I
The linearisation
of system
(2.18)
around
any critical
state
(wo,O, 0) gives us the system
(;)=K(‘). where
0
1
0
ab’po
_.f&o;w~+~,~) K=
(3.5)
rl
-4~,3(1-77+~Ao)
1 -V+VAO
-
arl(1- 4)
do
point,
and A0 = A(wo),
rl+rlbo
l-
4wo PO where wg is a critical
(3.6)
-2(1-q+qAo)
/
po = p(wo), UO= ul - a(wo - ~1). In addition,
4:‘%wo.
60 E 6 (wo) = 221 0 REMARK
2.
We have 6(wl) > 0, 6(wl(a))
This remark
follows from Proposition
The eigenvalues
of K are solutions (,J, - po)
> 0, b(w,(a))
> 0.
1 and Assumption
I
2.
of the following third-order
[A” + Au
(Go +
Doq’) -
equation:
a;]=aDo17’A2.
(3.7)
where
Do = Do(a) =
Go = Here PO(~), DO(~), and Remark
Go(o) =
2, whereas
3.
Let
(1 - v + rlAo) (1 -
rl + r160)’
b’po 4wo3(1 -rl+qAo)’
and Go are positive
due to the assumption
the sign of cy2(o) coincides
In the case of 7 = 1, equation PROPOSITION
60(1 - 4) 8~0~0
(3.7)
CJ be negative,
reduces
imposed
on the flow velocity
u,
with that of fh(.wo, wl, ul, 0).
to equation
with 10) sufficiently
(4.7) of our earlier
small.
If fh(wo,
paper wl,ul,g)
[lo]. < 0. i.e..
and if ~0 < ,&, or explicitly,
wg = WI or w. = w,(a), f;
(wo, ~1, UL,~) > (-1
- rl + q-40) (1 - rl + $0)’
3
(3.8)
1399
PhaseBoundarySolutions
then equation (3.7) has three real roots such that one of them is negative and the two others are positive. They can be arranged as follows:
(3.9)
A@) < 0 < (Yg < A@) < xC3) < PO. to X(j), j = 1,2,3,
Moreover, the right eigenvectors r (j) of the matrix K corresponding chosen so that
x(3) - p.
x(3) - p.
&I
r(3) = (
aTjX(3) ’
“7j
(1 - u;) 4WoPo
’-
can be
(3.10)
\ )
Also the following asymptotic formula holds:
x(3) - po a77
= O(rl)>
as ffq + 0.
PROOF. For small values of 101, we can treat the coefficients CY,/3, C, and G in equation (3.7) as if they were independent of e. In this case, o:(o) is positive due to (3.2). By means of the implicit function theorem, the following expressions for the roots of equation (3.7) hold:
xC3) -- PO + CT=
+ 0 (2)
.
Having these formulae, we check immediately that inequalities (3.9) hold true; also, by a direct I check, that the vectors r(j) given by (3.10) are the right eigenvectors of the matrix K. PROPOSITION 4. For 0 < 0 with 1~1 sufficiently small, all the characteristic exponents correI
sponding to the critical point (WI(~), 0,O) have their real part positive. PROOF. In this case, o;(a)
is negative due to (3.2).So we set a;(a)
= -o:(a),
where
By means of the implicit function theorem, the following expressions: X(l)(a)7
= 1 - n + $1 + O(a),
for the roots of equation (3.7) can be proved to be correct as e --+ 0. proposition is an immediate conclusion following from these formulae.
The assertion of the I
In this paper, we consider the case of cr < 0 with 101sufficiently small. Also, we are interested in such solutions to (2.11) along with (2.14)-(2.17) CJ= 0 can be excluded due to the following. ASSUMPTION 3. Let
where 201 5 w(J;a,n)
< w,(a).
The case
1400
B. KA~MIERCZAK AND
K. PIECH~R
Also, let 1. F(w,(O);wl,U2,0) 2. there exist 6,~
E Z such that infZ
(3.12)
> 0,
< 6 < a < 0 and such that
F(w,(~);wl,w,a)
the inequality (3.13)
5 0
holds for every g E [&, ~1. REMARK
3.
For comparison,
I
in Part I, for the case g > 0, we assumed that
and that F(W”(a);
for 0 E [a, 6.1, where 0 5 ?? < 6 were some real numbers, In Part
I, assumption
(4.17)
derived from the present We discuss
of that
solution.
0 = 0. Setting
instead
of (3.12)
paper was a little different
and (3.13),
from (3.14),
respectively.
but it can be easily
formulation.
in more detail,
of the Maxwell
(3.15)
W> W, 0) L 0,
I
assumptions
This
(3.12),(3.13).
type of solution
0 = 0 in equations
(2.10),
To this end, we will need the notion
describes
a particular
phase transition
in which
we obtain (3.16)
whereas
the only bounded
solution
of equations
(2.11)
is q(c)
= 0, < E IR1, and the equation
for w(E) takes the form (3.17) Using
the facts
conditions
that
w(+oo)
for the existence
= w, and w/(+00)
of solutions
of equation
= 0, we deduce (3.15)
easily
that
the necessary
are
WA4 F(luM;%,u1,0)
=
P(WM,%) Condition
(3.18)
is the well-known
is only one solution that ul is suitably
because
w,(a)
pair w,, chosen.
satisfies
equation
(g[ sufficiently
vicinity
of w,.
small.
(3.19)
Maxwell equal area rule of equilibrium
WM with b < w, They
< wM satisfying
are called the Maxwellian
(2.17).
Hence,
(3.18)
P(wrL>Q)l 4 = 0,
=P(Wm,W)
If ul > 0, then we have f~(w;w~,u~,~) with
s
w”~ [P(<, UI) -
But,
from (2.12),
equations
states.
It is easy to note that F(w,(O);
There provided
that
we have
FL(w,(O); wl,ul,O)
(3.18),(3.19),
Let us notice
> 0, for all w E (wl,w,(O)]
in particular,
phase transition.
wl,ul,O)
and all o > 0 or all 0 5 0 > 0, for all wl from some
> 0, if only wl > w,,
but sufficiently
Phase Boundary Solutions
1401
close to it, and F(wr(O);wl,ut,O) > 0 tends to zero as wl tends to w,~. Thus, by using the implicit function theorem, we conclude that, if only wz - W m is positive and sufficiently close to zero, then there exists a locally unique
a
such t h a t
F(wr
;wl, ut, ~) = 0. Moreover, ~ < 0~
l
F(w~(o);wl,ut,a) > 0 for all o - o > 0 sufficiently small, and for all wl - W m > 0 sufficiently / l small, we have F£(w~(a);wt,uz,~) > 0. As a consequence, F(w~(o);wl,ut,a) < 0 for all a < sufficiently close to it. F r o m the above, we can conclude the following. REMARK 4. T h e case a > 0 has to be accompanied by the condition wl < win, but sufficiently close to it (this was tacitly assumed in P a r t I), and the case a < 0 has to be accompanied by the condition wl > win, but also sufficiently close to it (this is the case of the present paper). | LEMMA 3. Let (3.12) hold. Then there exists ~ E (~_,O) such that for all rl E [0, 1], and all a E (-g,O) sut~ciently small, there is no trajectory connecting (wl, 0, 0) and (w~(~), O, O) with the nonnegative derivative w' (~). | PROOF. In the case of assumption (3.14), the proof m a y be found in P a r t I. So, let us assume t h a t (3.12) holds. T h e above l e m m a is implied by the fact t h a t we are able to find a priori estimates for z(~) g o o d for all admissible values of o (see L e m m a 1 in P a r t I). Now, we write the second equation of s y s t e m (2.18) in the form [1 - ~ +
~A(w)]z' + ~-A~(w)z 22 '
crb2p(w) 4w 3 z - f (w; wl, ul, o) - ~q. r]
Multiplying by z, integrating from - o 0 to ~, and making use of the t h e o r e m proved in [1t, C h a p t e r 1], we obtain
[w(~) (°b2p(u)
~
JW
q(.)
)
a.,
(3.20)
l
where n ( w , . )
=
1 [1
-
+ r/A(w)],
(3.21)
and the functions z and q are treated as functions of w. Let the derivative w'(~) = z(~) be nonnegative. As z is, under our conditions, b o u n d e d uniformly with respect to cr and r], then, as follows from L e m m a s 1 and 2, respectively, the first term on the right-hand side of (3.20) is of order of a as o ~ 0, whereas for the third one, we have the estimate q(~, a) = O i a ) as o ~ 0. Therefore, for small values of o, the right-hand side of (3.20) can be m a d e arbitrarily close to
fj
~'(~) f (u; wl, ul, 0) du. I
Thus, taking ~ = oe, we arrive at a contradiction. T h e l e m m a is proved. REMARK 5. This l e m m a is a c o u n t e r p a r t of L e m m a 2 of P a r t I, which used (3.14) and o belonging to the interval [0, a], with _a positive and sufficiently small.
Assmnption 3.2, there is no trajectory connecting the points (wt,O,O) and (wr(o), O, O) with nonnegative derivative w' ({) and nonpositive q( ~) for o E [d, a_] and r? c [0, 1]. | PROOF. From relation (3.20), we obtain LEMMA 4. Under
so, the facts t h a t z is nonnegative and q is nonpositive negative lead to the contradiction.
|
REMARK 6. This l e m m a is a c o u n t e r p a r t of L e m m a 3 of P a r t I, in the proof of which we used (3.15) instead of (3.13). |
B. KA~MIERCZAK AND K. PIECH~R
1402
4. THE LEFT-GOING Isolating
neighbourhoods
for system
(2.18)
WAVES
will be defined in a similar way as in the case of
0 > 0, however, now q will be assumed to be nonpositive, case under consideration
which is the main difference
between the
and the case treated in Part I. Below, we take n E: [0, l] and (T E ia, a].
where a and ZYare such as above (see Assumption
3 and Lemma
3).
1. Let N;(a,rl)
= {(‘w, z,q)
Let AL(E) = B((w~,O),E) denotes
: ‘w E [w~,w-(~)], .z E K’, 2 (a,rl)l>4 E i-Q (0,~) ,011. x [-E,E]
and AT(u,c)
a closed ball in (w, z)-plane
= B((w,(~),~),E)
x [--E,E], where B(,,E)
with the centre at the point (.) and the radius equal
to &. Let Al(a,y)
= B((~~(cr),O)~y)
in (w,z)-plane tion (2.17),
x (-E,E),
where B((wi(a),O),y)
with the centre at (wi(o),O),
denotes
where WI(C) is the ‘second’
and the radius y with y sufficiently
an open
ball
root of equa-
small.
Let Nl(a,%E)
{( w,z,q)
=
:w
E [W,WT
(Q)l,25E
Finally, let
N(o,rl,~) = Nl(a,rl,~) u AL(E)u M~‘,E) \ Al(a,y). The set N(a! n, E) is shown in Figure 1.
Figure 1. The candidate for the isolating neighbourhood We prove the following. LEMMA 5. For isolating
E >
0 sufficiently
neighbourhood,
small, every n E [0, l] and cr E [a,~],
i.e., its maximal
First, we prove some propositions, PROPOSITION 5. If I(fi(a,
in its interior.
making the proof of the lemma easier.
q, E)) d enotes ~~(vLE)
invariant set is contained
the set N(a! ‘q, E) is a good
the maximal
invariant set contained
=N~(~,~~,&)UA~(E)UA,(~,&),
in the set
I
Phase Boundary Solutions
1403
where E > 0 is taken sufficiently small, then the set of points in I(fi(o,
7, E)) lying on the
trajectories tending to (WI(~), 0,O) for [ --+ --co is empty.
I
The assertion is a simple consequence of Proposition 3.
PROOF.
PROPOSITION
I
For every n E [0, l] and g E [g, ??I, with )o( sufficiently small, the following
6.
equality holds: I(Nr(o,n,s))
n dNl(o,n,s)
= {critical points
2 Nr(o,n,E)}.
Moreover, for all E E IR1, we have q(J) < 0 and q(e) = 0, if q = 0, for any trajectory to
belonging I
I(Nl(O,%&)).
We must show that if the trajectory touches the boundary of Ni (a, 7, E) for, say, < = &, unless it is one of the critical points. (The then this trajectory cannot belong to I(Ni(a,v,~)) dependence of the trajectory on the parameters g and n will be omitted below in notation.) PROOF.
POINT 1. w([c) = wl or ~(
3.
z(&)
= 0.
As before, this implies that z’(&,) = 0, hence, (4.1)
If n = 0, this implies f(w(&,); wl, UL,D) = 0, so at [ = ta, If 77 > 0, then relation (4.1) takes place only if q(co) = 0. point. To see this, let us suppose that q([o) # 0, i.e., q(ca) w,(o) > w(E) > (to) = wc. Integrating the second equation and making use of the theorem proved in [ll, Chapter l], we w
R(w)z - ;
J W”
R’(C) d( =
lJJb2PK) dC
-0
J wg 4c3
_
Ef(w(T);
J Eo
we have one of the critical points. Hence, again we are at a singular < 0. Let us consider [ such that of system (2.18) with respect to 5 obtain
1 ‘w1,u1>0) dr - 2’1
4(+k JEa E
(4.2)
where R(w) = R(w; n) is given by (3.21), and the function z is treated as a function of w. This is possible as long as z(E) 2 0. From the third equation of system (2.18), we get
s[ co E
4P(W(7)) rl2(Ul + owl) - aw(r)
1
+ (1 - 77) q(W(r))dr=q(<)-q(
to 1-(Ul -4-wH2 &, J ?JJgC(2(7Q + OWL)- CC)
We define G=
4P(W) inf + (1-q) rl zuE[wo,w,(a)l 2(Ul + awl) - crw
1
> 0.
Let us note that, according to the previous point, for any solution staying in Ni (a, n, E), q(E) 5 0, for all <. Hence,
(4.3)
B. KA~MIERCZAK AND
1404
K. PIECH~R
Using (4.3) in (4.2), we get the inequality
_-77q(t) 2
4
s
((0) -- av2
G
2G
lD 1 - (Ul - u(C -
w1))2
‘wg < (2 (u1 + owl) - OC)
dC
At the critical point, we have
due to equation (4.1). The last term of the right-hand side of the above inequality is positive, due to relations (3.2) and the fact that z’(E) 2 0, for all < E R1, so ignoring it, we get
In the last inequality, l l l
the left-hand side is negative, the sum of the first two terms on the right-hand side is negative as well, the third term on the right-hand side of this inequality is positive.
Hence, for sufficiently small values of 1~71,this inequality cannot be satisfied. So we have to have q(&,) = 0. We cannot have w(
w(Eo) = w,(a).
According to the saddle-point theorem, for E sufficiently small, the sets Al(&) and AT(a, E) are isolating neighbourhoods with the maximal invariant sets equal to the critical points (wl,O, 0) and (w,(a), 0, O), respectively. Now, we prove the following. PROPOSITION
7. A trajectory from I(Nl(a,
TJ,E)) cannot enter any of the sets A,(a,
or the set AI(E) \ Nl(a,q,&). PROOF.
Suppose, first, that a trajectory
E) \NI (CT!77,E) I
belonging to I(Nl(a,
77,E)) leaves NI(CT, q, E) and enters
the set A,(a,
77,E) \ Nl(a, 7, E). So that the considered trajectory could reach the singular point (w,(a), O,O), it should coincide with the stable one-dimensional manifold S of this point. The eigenvector tangent asymptotically to S has the form given by (3.10), where X* is the unique negative eigenvalue of the linearisation matrix K : E} at a point where z > 0 and q 5 0, and the set {(w, z, q) : w = w, + E} (also w = w,transversally) at a point where z < 0 and q 2 0. It is obvious that the trajectory cannot reach \ Nl(o, 7,~) as it should leave this set first (and the point (w,(a), 0,O) inside the set Ar(o,q,&) consequently, the set N(a, 77,E)). So it must enter the set NI(~,
However, the considered trajectory cannot enter the set that this trajectory reaches this set at some time [ = to to the region w < w, - E. Then we must have z(e) < But the points (w, z, Q) such that w < w, - E and z < 0 contradiction.
77,E) n {(w, z, q) : w = W- - E}.
{(w, z, q) : w = w, - E}. For, suppose and passes from the region w > W, - E 0 for all < < &J sufficiently close to it. are not in N1 (a, 77,E), so we arrive at a
1405
Phase Boundary Solutions
Now, let us consider
matrix
K around
eigenvectors.
Al(&).
the singular
The vectors
which is tangent
Let XI’) < 0 < Xi2) < X3 be the eigenvalues (~1, 0,0), next
point
let rl(i), i = 1,2,3,
rji), a = 2,3, span the unstable
to the unstable
manifold
manifold
Ul of the nonlinear
of the linearisation be the corresponding
of the linearized
system
system
(2.18) at the singular
(3.5) point
(wl,O, 0). Let Nz*,Ns* denote the trajectories of the nonlinear system corresponding, due to the Hartman-Grobman theorem [ll, Chapter IX; 121, in a one-to-one way, in some vicinity of the singular point (~1, 0,O) (independent of D and q), to the half lines
rl) = {(w, *) : (w, ~~4) = *b-j”‘,
k(c, (see Figure
2). Let Ns*
denote
the upper
i = 2,3,
C I 0) ,
or lower branch
2). We set Al =
of N3 (see Figure
{(w, z, q) : (w - ~1)’ + z2 _< E, q E [-q*, q*]} and Bl = Al f~Ul, where q* = -oQ* and 0 < E < q* 2), so AL(E) c Al. If P E 1 is sufhciently small then, obviously, in the set Bl. the following conditicns are satisfied. (see Lemma
1. Ns-
intersects
the plane
q = -q*
and Ns+ intersects
the plane q = q*.
2. f~(w;w~,~~,~) < 0 for (w,z,q) E Bl = Al nUl. 3. Every trajectory of the nonlinear system (2.18) lying in Ul leaves the set Bl for sufficiently large 5. Now,
we will use the fact
that
Ul is two dimensional.
For 77 = 0, the
last
equation
of
system (2.18) separates from the first two, so it is easy to see that all the trajectories from In this case, the monotonicity of w(l), i.e., the I(Ni(a, Q, E)) must lie in the plane q = 0. positives of z(E), is obvious. So, let us consider singular
point
the case of 7 > 0. In this case, in the assumed
(WI, 0,0), we can treat N3-
w and z as functions
= @J* (0 32, (0 1q* (0)
(2< 0)
I
and
N3+
small
of the variable = b*(<),
t*iE)>
(2 = 0)
7---___-___-____
N3+ {q=q’)
- (4’0)
Figure 2. The set Bl.
vicinity
q. Let
q*(O).
of Bl, the
1406
B. KA~MIERCZAK
We claim that,
for all CJ E 2, r) E (0,11,
~(6) Indeed,
AND K. PIECH~R
> 0
and
asymptotically,
z*(c)
as E +
as NY_ stays in Bl, because
-00,
as long as Ns-
< 0, Ns-
is tangent
the third component
and Ns+ stay in Bl.
to rl(3). Therefore,
we have q*(t) < 0 as long
of this vector is strictly
small values of cr E Z and r) E [0, l] (see (2.10)),
that is why we conclude
large, we have z*(t)
> 0 for 6 E (-co,&)
> 0.
from the second equation
Suppose of system
z’ (to) = - P -
rl +
This leads to a contradiction.
z*(E)
(2.18),
we have
that,
independently
for all sufficiently
for (-<)
and z,(&)
sufficient11
= 0.
But
then.
+;ari(50) ;Wl,W,~‘) 1>0.
VA(w (EON-~ [lf (w (to)
The same proof may be done for Ns+.
Owing to the construction to parts,
that
negative
of the vector rl(3) is positive
of 0 E Z and 77 E [0,11.Next, the second component
(4.4)
of the set Bl and (4.4),
say, the left and the right ones.
I
we infer that the line Ns divides this set into
Any trajectory
starting
from the singular
point (~1~ 0,O)
and lying in the left part of BL leaves this set either from within the region z < 0 or through
one
of the planes q = +q*.
the
Any such trajectory
cannot
right part of Bl. In this case, any trajectory from the point the second
(wl,O, 0) has to be asymptotically,
component
of cr, we infer that
of this vector
asymptotically
is positive
positive
Na-,
as < +
7, E). So let us consider
and different
-00,
tangent
from Ns*,
to the vector
and the third one is negative
we have w > WI, z > 0, and q < 0. Suppose
leaves the region z > 0 (for the first time) if q becomes
belong to Ni(a,
T, including
for some < = [i.
for some < = .$ <
This is possible
starting
rj2).
for negative
Since values
that the trajectory
only if q([l) > 0, i.e..
we would have z’(ci)
> 0 which follows
from the relation
but this is a contradiction. consequently, trajectory
leaving
The trajectory
r has to return
second time.
can either
N(a, n, E) or it can attempt
leave the set Bl from the region z < 0. a~nd
to get into N~(Q, 7, E). In the latter
to the region z > 0 crossing
Let it take place for 6 = &. First,
case, the
the line Ul n {(w, z, q) : z = 0) for the
we will prove that
we cannot
have (3.5)
4(<3) > 4((l)'
We have z’(&)
The
last
maximum, from (4.5)
2 0 and
inequality at [
=
results
from the fact
and the fact that
minimum,
z’([i)
that
z’(&)
> 0, we can obtain,
f(w
Subtracting
(4.8)
from (4.7),
w(E)
attains
function
at < =
of [
E (
Now.
< 0, it follows that
f (w(b);W1rwr~) Using the fact that
the function
and is a decreasing
+ ;w(tl)
in a similar
(<3) ;w,w,a)
we obtain
+ $?u(G)
(4.7)
> 0.
fashion,
< 0.
the following
inequality: (4.8)
Phase Boundary Solutions
1407
qi = q(Ji), and i = 1,2. Since &(w; W~,UL,g) < 0 in Bl (see the definition of this set), then, owing to (4.6), we have
where wi = w(rt),
0.
(4.10)
Now, from (4.5) and (4.10), we infer that the left-hand side of (4.9) is negative. Consequently, If q(&) < q(&), th en either 7 would leave Bl from within the region z < 0 or there would exist a segment of this trajectory lying entirely in the region z I 0
we arrived at a contradiction.
joining the points (wl, 0, ql) and (w”, 0, q2) such that q1 < q2. But this is impossible according to the considerations above. Hence, any trajectory tangent asymptotically to the vector rl(2) having once entered the region of z < 0 cannot come back to the region of z > 0. Hence, any such trajectory
will either leave the set Nl(o, 77,E) from the region z < 0 or will stay in the region
z > 0. If it leaves Bl crossing the line Ul n {q = q*} or the line Vl n {q = -q*}, then it leaves the set N1 (a, 77,E). Hence, the only opportunity is that the trajectory leaves Bl from the region z > 0 and -q* < q < q*, thus, staying in the set Nl(a, q, E). REMARK
7.
I
Let us note that, for g = 0, 7 E [0,11,the set Nl(O,
77,E)u
B ((w~,o,o),~)u~((~,(~),O,O),~)\Al(~,~),
where B(P,G) denotes the three-dimensional ball with the centre P and radius Lz,is a good isolating neighbourhood, only if 3 is taken sufficiently small. This follows from the fact that for u = 0, any invariant trajectory lies in the plane q = 0, thus, the system behaves as two dimensional. Due to the robustness of the Conley index, the same isolating neighbourhood is good for the system with 1~1sufficiently small. However, proceeding in this way, one does not prove the w-monotonicity of the invariant trajectories near the points (~1, 0,0) and (w,(c), 0,O). This information is important from the physical point of view. Also, the necessary smallness of 1~1 is not controlled, which is very significant in view of Assumption 3. In Part I, the monotonicity property was implied by the results of paper [lo]. I Let Assumptions 1-3 be satisfied. Then, there is at least one CTE (a,??), such that there exists a heteroclinic solution to system (2.11) connecting critical points (~1, 0,O) and (w,(o), 0,O) such that w’(t) > 0 for all [ E (--00, co). I
MAIN
THEOREM.
PROOF.
For q = 0, system (2.18) takes the form
wi=
2,
ab2dw)
z’ = -
[
-z+f(W;w,w7~) 4w3
q’ = q.
1 7
(4.11)
Any compact invariant set for this system must be contained in the plane q E 0. Thus, for every g E [a, YY],we can find a family of isolating neighbourhoods Nx (a, E), X E [0,11,such that N1(a, E) = N(a, 0, E) and No(cr, E) = Nzuz(gr E) x [-E, E], where N,,(o, E) denotes the projection of the set N(g, 0, E) onto the (w, z)-plane. We use the Conley connection index theory [8]. To do this, we append to system (2.18) the equation
0’ =
pqw, z, q)
( I7 - F),
(4.12)
where @ is a sufficiently small positive number. Let U’ and U” denote open neighbourhoods
of
(w~rO,O)~{a}U(w~,0,O)x{~}and(w,(a),O,O)x{a}U(w,(~),O,O)x{~}inR3x(a-~,~+~), respectively, having disjoint closures. The real valued continuous function @ is arbitrary except for the fact that it is positive on U’ and negative on U” (see [S]). Then we must compute the
1408
B. KA~MIERCZAK AND K. PIECH~
Conley index he of the maximal
invariant
by Sc.
Since the third
of system
system,
we infer that
the invariant equation) with
equation
set in Ne = &ek,?l (4.11)
this index is homotopic
set comprised
in the set [-E,E]
to the
flow generated
index h is computed to 0. Thus,
(with respect
in the paper by Grinfeld (wr(g),O,O).
and Sl: = IL,,,,] spheres X2. Therefore, In general,
S, =
h” ”
set comprised
equations
[9] or by Conley
and Gardner
of the hyperbolic
of this index of
by the third
in &k,al
of system
hand, let S& = &k,zl(~~,
The indices
C1 is the Conley
to the flow generated
invariant
by the remaining
ho 2 C’ A 0 g 0. On the other
from the other two equations
to h” P, h, where
and h is the Conley index of the maximal
respect
Ne(o, E). This set will be denoted
uncouples
NW2 (0. E)
(4.11),(4.12).
The
[8]. It is homotopic
0,O) = (wl, 0,O) x [(T,CT]
critical
points
are both
pointed
the “wedge” sum of the indices of Sl, and S$’ is equal to ( C2 A C’) VC2 y 6.
we can define for 17 E [0,11:
maximal
invariant
set in
n
N(~,v,E)
with respect
to system
(2.18),
(4.12)
OE!g,F] For 17 E [O,l], the triple triple
(SL, Si, Se).
Also, the Conley respect
indices
points with exactly
of the points
two positive
from the critical
f orms a connection
points.
77) n N(~,v,E).
(wi, 0,O) and (~,(a*),
triple,
which is a continuation
index of the first one, for q E [0, 11, is homotopic
S’(a),
by system
a value of D* E (a,??)
in the set Ni(a, connect
(Sk, St, S,)
the connection
to the flow generated
there exists different
Hence,
S;(g),
(2.18)
characteristic
0 E [a,~],
are homotopic
exponents).
(as isolated
invariant
to C2 (these
points
According
such that for 0 = g*, there exists In fact, according As it cannot
0,O).
to Propositions
to the results
an invariant
of the to 0.
sets with are saddle
given in [8],
set in N(a*,
1, E)
6 and 7, this set is contained
be comprised
in the plane {w = 0, z = 0}, it must
The proof is complete.
I
REFERENCES A four-velocity model for vanderwaals fluids, Arch. Mech. 47. 1089-1111 (1995). Cinetique des Gaz a RCpartition Discrete des Vitesses, Lect. Notes Physzcs. Volume 36 Springer-Verlag, (1975). B. Kaimierczak and K. Piechor, Phase boundary solutions to model kinetic equat,ions via the Conley index theory. Part I, Mathl. Comput. Modellzng 31 (13), 77-92 (2000). L. Preziosi and E. Longo, On a conservative polar discretization of the Boltzmann equation, Japan J. fnd. Appl. Math. 14, 3999415 (1997). C. Ringhofer, Dissipative discretization methods for approximations, Math. Models Meth. Appl. Ser. 11, 1333148 (2001). D. Goersch, Boltzmann operators in unbounded domains, Math. Models Meth. Appl. Scz. 12 (1). 49-76 (2002). H. Babovsky, A kinetic multiscale model, Math. Models Meth. Appl. Sci. 12 (2), 309-332 (2002) C. Conley and R. Gardner, An application of the generalised Morse index to travelling wave solutions of a competitive reaction diffusion model, Indiana Unzverszty Mathematzcal Journal 33, 319-343 (1984). M. Grinfeld, Nonisothermal dynamic phase transitions, Quarterly of Applied Mathematics XLVII (l), 71-84 (1989). B. Kaimierczak and K. Piechor, Shock waves by a kinetic model of vanderwaals fluids, ARI 51, 203-215 (1999). P. Hartman, Ordinary Difjerential Equations, Wiley, (1964). S. Sastry, No&near Systems: Analyszs, Stability, and Control, Springer, (1999).
1. K. Piechor,
2. R. Gatignol, 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.