Viscous Singular Shock Structure for
a Nonhyperbolic Two-Fluid Model
Barbara Lee Key tz Michael Sever y Fu Zhang z
Abstract. We consider a system of two nonhyperbolic conservation laws modeling incom-
pressible two-phase
ow in one space dimension. The purpose of this paper is to justify the use
of singular shocks in the solution of Riemann problems. We prove that both strictly and weakly
overcompressive singular shocks are limits of viscous structures. Using Riemann solutions we
solve Cauchy problems with piecewise constant data for the nonhyperbolic two-
uid model.
1 Introduction
This paper examines a nonhyperbolic system arising in the modeling of incompressible
two-
uid
ows. The principal result we obtain here is that approximate
viscous pro les can be constructed for both strictly and weakly overcompressive
singular shocks that are important in solving Riemann problems. In addition,
we use this stability result for Riemann solutions to show that a class of Cauchy
problems for the nonhyperbolic two-
uid model can be solved.
We consider a system of two equations, wt+q(w)x = 0, for a state w = ( ; v)>,
which for the two-
uid model takes the form
t + (vB1( ) + K )x = 0
vt +
v2B2( ) + Kv
x = 0;
(1.1)
with
B1( ) =
( 1)( 2)
; B2( ) =
2 1 2
2 2 : (1.2)
This system is equivalent to the system of four equations modeling one-dimensional
unsteady isothermal incompressible two-phase
ow, [2, p 248]. The volume
fractions 1 and 2 = 1 1 have been replaced by a density-weighted volume
element
= 2 1 + 1 2;
Department of Mathematics, University of Houston, Houston, Texas 77204-3008. Research sup-
ported by the Department of Energy, grant DE-FG02-03ER25575, the National Science Foundation,
Grant DMS 03-06307, and the Texas Advanced Research Program, grant 003652-0076-2001.
y Department of Mathematics, The Hebrew University, Jerusalem, Israel.
z Department of Mathematics, University of Houston, Houston, Texas 77204-3008. Research sup-
ported by the Texas Advanced Research Program, grant 003652-0076-2001 and the Department of
Energy, grant DE-FG02-03ER25575.
Keyfitz, Sever and Zhang
and the momentum equations replaced by a single equation for the momentum
di erence,
v = 1u1 2u2 ( 1 2)K:
Here K(t) = 1u1 + 2u2 is constant in space as a consequence of conservation
of mass. In this paper we take K = 0, for convenience. The system (1.1) has
been nondimensionalized: the densities i and have been scaled by dividing
by 1 2, so we take 1 2 = 1 throughout and note that i and are
dimensionless. Hence v has the dimensions of velocity; we may replace v and K
by v=v0 and K=v0, where v0 is some characteristic velocity, and simultaneously
scale x=t by v0, making (1.1) dimensionless. Ordinarily, there is a balance term,
G = F1= 1 F2= 2 in the second equation of (1.1); G models buoyancy, drag
and other interfacial forces. Research on two-
uid models usually concentrates
on correct modeling of the forces Fi. However, the focus of this article is on the
mathematical structure of the di erential operator on the left side of the system
(1.1), and we set G to zero.
It is well known [2, 13] that the system is not hyperbolic, as the characteristics
at any mixed-phase state ( 2 < < 1), are complex. We de ne j(w), j = 1; 2,
to be the eigenvalues of dq(w), and note that 1 = 2 if 2 < < 1, while
1 = 2 if = i, i = 1 or 2. Key tz, Sanders and Sever have discussed this
equation in earlier papers [4, 6]. Two important facts are that the system, at
least by a formal asymptotic expansion, admits solutions of very low regularity,
called singular shocks, and that by means of singular shocks one can solve the
Riemann problem for (1.1) for all initial discontinuities.
In this paper, we justify the use of singular shocks by showing that they
appear as weak limits of smooth solutions to a regularized, viscous system,
wt + q(w)x = "wxx: (1.3)
More precisely, we construct a sequence of functions w"(x; t) such that, as " ! 0,
w"( ; t)t + q(w"( ; t))x "w"( ; t)xx * 0; (1.4)
weakly in the space of measures on R, pointwise in t. The sequence w" converges
to a singular shock, in the same sense.
Establishing a relation like (1.4) is a critical step in arguing that singular
shock solutions have a physical interpretation for the system modeled by (1.1).
As is well known, even for hyperbolic systems of conservation laws, which are
linearly well-posed (unlike our nonhyperbolic system), solutions exist in only
the weak sense, and establishing well-posedness requires admissibility conditions
for shocks. Admissibility conditions can be derived in a number of ways, as
described in Smoller's monograph [12], including perturbation of the system by
a second-order term representing viscosity, as in (1.3). Another motivation for
introducing a viscous approximation like (1.3) is that nite-di erence numerical
2
Viscous Singular Shocks
schemes for conservation laws achieve stability by introducing a small amount
of dissipation, which can be modeled at the continuous level with a system like
(1.3). In principle, one would like to prove admissibility via the use of a physically
realistic viscosity, or, equivalently, to prove that the precise form of viscosity
does not a ect important macroscopic properties of shocks such as their speed
and stability. This is important both for establishing physical relevance and
for studying the convergence of numerical approximations. The precise form
of physical viscosity is not known for the system (1.1). Modeling numerical
dissipation is also complicated. `Arti cial' viscosity, as in (1.3), is often chosen
in theoretical studies, for convenience. We choose it here, because it is the only
choice which leads to relatively simple viscous structures. Finding comparable
structures for other forms of viscous perturbation remains an open problem, as
we discuss in the concluding section.
In related work, Sanders and Sever construct approximate viscous pro les for
nonhyperbolic systems which exhibit a symmetry [9]. The method outlined in
[9] applies to the model system (1.1). However, that paper is limited to strictly
overcompressive singular shocks, and hence, for the system (1.1), does not include
all the shock connections needed to solve Riemann problems. In this paper, we
give a complete proof of the construction for (1.1), in both the overcompressive
and weakly overcompressive cases.
For a model strictly hyperbolic system originally studied by Key tz and
Kranzer [5], existence of self-similar shock pro les for singular shocks when selfsimilar
viscosity of the form "twxx is added to the system has recently been
proved by Schecter [10]. Schecter uses geometric singular perturbation theory to
nd smooth solutions for su ciently small, positive ". It is possible that a geometric
perturbation result can be found for nonhyperbolic systems such as (1.1),
perhaps using some new scalings as suggested by this paper. The construction in
our paper provides viscous structure for a regularized system with viscosity "wxx,
and also gives self-similar viscous structure with viscosity "twxx in some cases,
but not in the important case of weakly overcompressive shocks.
Recent work of Kreiss and Ystrom [7], on the relation of viscous regularizations
to their limits, considers a number of model equations, not including (1.1) but
with the nonhyperbolic equations of the two-
uid model very much in mind.
Their work shows that the existence of certain kinds of energy bounds can prevent
unbounded amplitude growth in the parabolic approximations, and suggests that
weak convergence to a limit of bounded but high-frequency oscillations may occur
instead. Amplitude bounds do not exist for (1.1), and the type of convergence we
report here is di erent from the nonlinear examples in [7]. Numerical experiments
by Dinh, Nourgaliev and Theofanous [1] on the model (1.1) con rm the singular
behavior predicted by our analysis.
In the next section, we give the de nition of a singular shock solution, and
also review the singular shock solutions of (1.1), described in detail in [6]. Section
3 states the main theorem, along with the theorem of Sever from [11] which we
3
Keyfitz, Sever and Zhang
use to prove it. The construction of fw"g is carried out in Section 4, while the
proof that this construction provides an approximation as required by the theory
in [11] occupies the three sections which follow. As an application, we solve
some Cauchy problems for (1.1) in Section 8. The signi cance of our results is
explained in Section 9.
2 Definitions
A singular shock solution to a system wt + q(w)x = 0 is of the form w = ! +
,
where !, the regular part of the solution, is a discontinuous function and
, the
singular part, is a measure supported on the discontinuity set of !. A self-similar
singular shock has the following structure:
!(x; t) =
w; x < st
w+; x > st
; (2.1)
(x; t) = t e(w;w+; s) (x st); (2.2)
where
e(w;w+; s) = s(w+ w) q(w+) + q(w): (2.3)
A measure-valued function such as w does not satisfy the conservation law in the
usual weak sense. Hence, it can be di cult to assign a signi cance to singular
shocks. Based on a standard approach to analysing conservation laws by studying
limits of regularized approximations, we attempt to justify singular shocks by
nding them as limits of solutions of approximating systems. This might mean
strong limits in an appropriate measure space, as was carried out by Key tz and
Kranzer for a hyperbolic model system [5]. However, one can work directly with
weak limits [11], and that is the approach we adopt here. In [11, Chapter 3, x1],
Sever gives a de nition of singular shock solutions, which we use in this paper.
Definition 2.1 A singular shock solution of a system wt + q(w)x = 0 is a
measure of the form
w(x; t) = !(x; t) +
X
i
Mi(t) i(t) (x xi(t));
where ! is a classical weak solution away from the singularities, i is the charac-
teristic function of an interval [Ai;Bi); Mi 2 L1, and xi 2 W1;1. The function
w is the weak limit of a sequence w" with w"( ; t) 2 L1
loc uniformly with respect to
", pointwise in t; satisfying
w"( ; t) * w( ; t) and w"( ; t)t + q(w"( ; t))x "(Aw"( ; t)x)x * 0; (2.4)
as " ! 0, weakly in the space of measures on R, pointwise with respect to t, for
some positive de nite matrix A.
4
Viscous Singular Shocks
Remark 2.2 A function f( ; ") is said to be in L1
loc uniformly with respect to
" if, for any set J R of nite measure,
R
J jf( ; ")j M = M(J), where
M is independent of ". In general, a property holds uniformly with respect to
a parameter if the bound which establishes the property is independent of the
parameter.
Remark 2.3 The matrix A need not be constant in this de nition. Two important
cases are the scalar matrices A = I and A = It, corresponding to the
standard concept of `arti cial' viscosity. These are the only choices which appear
to give simple viscous structures for singular shocks. The signi cance of this is
discussed in the concluding section of the paper.
The measures de ned this way generalize the self-similar singular shocks described
by (2.1), (2.2) and (2.3); however, for any candidate satisfying (2.1){(2.3), one
must verify (2.4) to establish that the object is indeed a singular shock. In
[11], Sever established some important properties of singular shocks. First, the
singular mass M(t) evolves according to the Rankine-Hugoniot de cit, de ned in
equation (2.3) above. Along the i-th singular shock,
dMi(t)
dt
= e
w(xi(t) 0; t);w(xi(t) + 0; t); x0
i(t)
: (2.5)
Second, singular shocks are generally overcompressive discontinuities; for equations
with real characteristics this means j(w) s j(w+), where j = 1; 2,
s = x0
i, and w are the limits of w from the right and left sides of the discontinuity.
Furthermore, for systems with real characteristics, the existence of
convex entropies implies a direction for the vector e. In a number of hyperbolic
model problems this direction picks out a coordinate system such that the rst
component of e is identically zero, which means that the rst Rankine-Hugoniot
condition holds.
The last two points do not immediately generalize to our system (1.1) with
complex characteristic speeds. However, in [6] it was shown that if we use a
generalization of the overcompressibility condition, obtained by examining the
real part of the characteristic speeds,
Re
j(w)
s Re
j(w+)
; j = 1; 2; (2.6)
and if we satisfy the rst Rankine-Hugoniot condition at each singular shock
jump, then there exist solutions of the form (2.1), (2.2), (2.3) by means of which
one can solve all Riemann problems. The case was also made in [6], based on
the derivation of (1.1), that in the rst equation is genuinely conserved, while
the rationale for the form of the second equation is weaker. Hence it seems
appropriate to satisfy the rst Rankine-Hugoniot relation.
In the remainder of this paper, we shall assume condition (2.6), and also take
A = I or It in De nition 2.1. An explanation of this choice is given in Section 9.
5
Keyfitz, Sever and Zhang
w−
w0
Q(w o.c. shock −)
H ®
¬ H
H¯
¬ s=Re(l−)
¬ s=Re(l+)
b
v
¬ r2 r1 ®
Figure 2.1: The Region Q of Overcompressive Shocks
The Riemann solution of (1.1) is not unique [6]. We assume that the rst
Rankine-Hugoniot condition is satis ed, so that
s =
[vB1]
[ ]
v+B1( +) vB1( )
+
(2.7)
is determined by w and w+. For any w, we de ne Q(w) to be the set of
points w+ in phase space such that (2.6), which is now a condition on w+, holds.
Then, two solutions exist for each w+ 2 Q(w). One solution consists of a single,
singular shock, while the other exhibits a complicated wave pattern consisting of
singular shocks, rarefactions, and a contact discontinuity [6]. When w+ is not
in Q(w), then only the second type of solution can be constructed. The wave
in this case is always composite, although if w and w+ are in di erent vertical
half-planes there is only a single rarefaction and no contact discontinuity. Furthermore,
in a composite wave, the singular shock connects w with a particular
state, w0, the point where @Q, the boundary of Q, intersects H, which we de ne
to be the set of points in the physical region with real characteristics. See Figure
2.1. A shock between w and a point on @Q is only weakly overcompressive.
One detail of the composite Riemann solutions is that two points on the
boundary of Q are needed to construct a composite wave: w0(w) and the analogous
point, which we called w1(w+) in [6], in H, which can be joined to w+ on
the right with a weakly overcompressive singular shock with 1(w1) = 2(w1) =
s > Re( j(w+)), j = 1; 2. In this paper, we show that these points in @Q \ H
admit viscous pro le connections.
Without loss of generality, we concentrate on the right half of the pro le; in
particular, we consider only shocks where strict overcompressibility fails at the
state on the right. That is,
Re
j(w)
s = Re
j(w+)
; j = 1; 2: (2.8)
The conclusions of the paper also apply if strict overcompressibility fails on the
left instead, or even if it fails on both right and left, as the convergence arguments
at the right and left endpoints proceed independently of each other.
6
Viscous Singular Shocks
There are three ways that singular shocks which are weakly overcompressive
on the right might arise in (1.1). In examining Riemann problems, one nds that
states w with = 1 and v > 0 can be connected to w+ = ( 2; 0) via a
weakly overcompressive discontinuity satisfying (2.7) and (2.8) with speed s = 0.
The most important case we consider, however, is that with w in the interior
of the physical region and w+ 2 @Q(w) \ H, so w+ = w0(w), as in Figure 2.1:
these are the singular shocks which appear in composite waves. Finally, it is in
principle possible that a state w+ 2 @Q(w) \ f 2 < < 1g could be joined to
w via a singular shock. Such connections are not necessary to solve Riemann
problems, since composite waves are also available. Singular shocks of this form
do not appear to have viscous structure.
A principal di erence between strictly and weakly overcompressive shocks, as
pointed out in [6], is the rate of convergence of viscous trajectories to their end
states, which is exponential in the case of strictly overcompressive shocks, but
not in the weakly overcompressive case. In this paper, we show that convergence
holds when w+ 2 H, and it is strong enough to prove the existence of singular
shocks in the sense of De nition 2.1.
3 Viscous Profiles
We review the construction in [11], which generates a sequence of functions satisfying
(1.4). To motivate this, recall that the classical traveling wave construction
for conservation laws uses the similarity variable = (x st)=" to nd solutions
of wt + q(w)x = "wxx in the form
w = y( ): (3.1)
This leads to an ordinary di erential equation for y:
y00 = (q(y) sy)0; (3.2)
and if we now assume that s is the shock speed corresponding to a pair of states
w which satisfy the Rankine-Hugoniot relation s(w+ w) = q(w+) q(w),
then we can integrate (3.2) to obtain y0 = (q(y)sy)+C, where C is the common
value of sw q(w ). For given data w and s, we can assume C = 0. Thus the
fundamental dynamical system in the study of classical shock pro les is
y0 = q(y) sy: (3.3)
However, singular shocks connect states which do not satisfy the Rankine-Hugoniot
relation, and so the premise underlying (3.1) is
awed. We instead let
w"(x; t) = y
x st
"
;
t
"
y( ; ); (3.4)
7
Keyfitz, Sever and Zhang
and de ne
z( ; ) = y q(y) + sy; (3.5)
the function z measures, in some sense, the extent to which the problem fails to
have a classical viscous pro le solution. Then the theorem proved in [11] is
Theorem 3.1 (Theorem 3.3.1 of [11]) Suppose that there exist functions y
and z, related by (3.5), such that
(1) z( ; ) is constant for j j su ciently large;
(2) lim
! 1
y( ; ) = w , uniformly in ;
(3) y ( ; ) 2 L1(R) uniformly in ;
(4)
Z
R
y ( ; ) d = e(w;w+; s) + o(1) as ! 1;
(5) y( ; 1) 2 BV and z( ; ) 2 BV uniformly in .
Then the sequence fw"g de ned by (3.4) satis es De nition 2.1 with A = I.
(See Remark 2.2 for the interpretation of uniformity in .) Sever [11] also proves
that if an additional constraint holds:
Z
R j j
jy ( ; 1)j + jy ( ; )j
d < 1; (3.6)
then we can de ne
w"(x; t) = y
x st
"t
;
1
"
; (3.7)
which satis es De nition 2.1 with A = tI. (In this sense, there is not much
di erence between the standard and the self-similar viscosity criteria.)
Using this theorem, the question of nding singular shocks can be reduced to
nding a pair of functions y and z which satisfy the hypotheses of Theorem 3.1.
The theorem we prove in this paper is
Theorem 3.2 For initial conditions w, w+ 2 Q(w), and s = [vB1]=[ ] satis-
fying either
Re( j(w)) > s > Re( j(w+)); j = 1; 2; (3.8)
or w+ 2 H \ @Q(w) and
Re( j(w)) s = 1(w+) = 2(w+); j = 1; 2; (3.9)
there exists a sequence fw"g of approximate viscous pro le connections satisfying
(2.4). Hence singular shock connections exist.
8
Viscous Singular Shocks
4 The Construction of Approximate Viscous Profiles
If one applies a formal asymptotic expansion to a solution concentrated at a
singular shock (with remaining bounded, as is physically reasonable, but with
no constraint on v), as was done in [6], then the problem reduces, to dominant
order, to equation (3.3) with s = 0 (equation (4.2) below). We now show that,
as suggested by the formal treatment, there are viscous structures satisfying
De nition 2.1.
The construction is based on the ansatz that near the singular shock locus
the viscous solution w" behaves like a solution to the simple system (3.3), w0 =
q(w) sw, while far away, as ! 1 it tends to solutions of di erent systems
w0 = q(w) sw C , where C = q(w ) sw . Furthermore, in the central
interval, an approximate solution can be found by a simple scale change.
The point of departure of the construction is the observation that the system
(1.1) possesses a scaling invariance: If
t 7! t0 = t= ; v 7! v0 = v; (4.1)
then solutions of wt+q(w)x = 0 are mapped to solutions of ( ; v0)>
t0+q( ; v0)>
x = 0.
Also to the point is the observation that the
ux vector eld, q(w), admits a oneparameter
family of heteroclinic connections. The dynamical system
w0 = q(w) (4.2)
is just (3.3) with s = 0; in Section 4.1, we explain the role of s in comparing (4.2)
to (3.3).
The heteroclinic orbits of (4.2) connect points ( 1; 0) and ( 2; 0); we shall
assume v 0 and then W = ( 1; 0), W+ = ( 2; 0). The entire physical region
with v > 0 and 2 < 1 is in the stable set for W+. There is a similar
construction when v 0, with ( 1; 0) the attractor.
The heteroclinic connections can be found explicitly.
Proposition 4.1 For the functions B1 and B2 given by equation (1.2), solutions
to the system (4.2) in the strip ( 2; 1) \ fv > 0g form a one-parameter family
with asymptotic behavior
2 +
4 3
2
V 2 2 and v
2 2
=
1
B2( 2)
; as ! 1; (4.3)
1
4 3
1
V 2 2 and v
2 1
=
1
B2( 1)
; as ! 1; (4.4)
where the parameter V is a positive constant.
Proof: We write
d
dv
=
1
v
B1
B2
;
9
Keyfitz, Sever and Zhang
and solve by separating the variables, recalling that B2 = B0
1 =2, to obtain the
family of solutions
v = V
p
jB1j = V
( 1)( 2)
1
2
; (4.5)
for any positive constant V . Thus the equation for ( ) is
d
d
= V
( 1 )( 2)
3=2
; (4.6)
and ( ) is given by quadrature; since ! 2 as ! 1, then from (4.6),
d
d V
2
2
3
2
as ! 1. Integrating, we get the rst limit in (4.3), and the second comes from
(4.5). By observing that ! 1 as ! 1, we obtain (4.4) in the same way.
The solutions are parameterized by V .
The system also admits a translation symmetry, ! 0. A convenient
normalization for is (0) = p 1 2 where B1 attains its minimum. Then
v(0) = V (p 1 p 2)
is the maximum of v along the trajectory. In particular, we can identify the
nontrivial parameter V with max v. A scaling symmetry like (4.1) also holds
for the dynamical system: If = = , and v0 = v, then solutions in are
mapped into solutions in . We use this scaling in de ning the second variable
in Theorem 3.1.
As a consequence of the scaling, we can express all solutions of (4.2) in terms
of a single function,
W0( ) = ( 0; v0) with v0(0) = 1: (4.7)
For any other solution, we have
( ); v( )
=
0( ); v0( )
; (4.8)
for some ; we take = ( ) to handle the dependence of solutions on .
4.1 The Role of s
In typical constructions of viscous pro les for conservation laws arising in
uid
dynamics, one makes a Galilean change of variables x 7! x st to reduce to the
case s = 0. However, we have already used the change of variables x 7! xKt in
10
Viscous Singular Shocks
yx=q(y)−sy−C− yx=q(y)−sy−C+
yx~ q(y)−sy
2x− x− 0 x+ 2x+
Figure 4.1: Asymptotic Dynamics of the Approximate Pro les
setting K = 0 in (1.1). Thus, we must consider both the cases s = 0 and s 6= 0.
We nd the di erence in the technical detail that (3.3) does not possess the useful
heteroclinic connection of (4.2). However, we can use the system (4.2) to replace
(3.3) in the neighborhood of = 0 by introducing a stretched variable ( ), with
(0) = 0 and d =d > 0. With respect to a xed heteroclinic connection, for
example W0 = ( 0; v0), the function
y( ; ) =
0( );
d
d
v0( )
>
; (4.9)
satis es (3.3) approximately. In particular, y = q(y) + (0; v0d2 =d 2)>. If the
last term were sy, we would have (3.3) and then z = 0 in (3.5). However, even
if ! 1 as ! 1 (as we nd to be the case), only the second component of
y becomes unbounded as ! 1, and so z is a function with bounded variation
if we choose the second component of y in (4.9) to satisfy the second equation of
(3.3). That is, we set
v0
d2
d 2 = e2(sy) = s
d
d
v0 or
d2
d 2 = s
d
d
:
(Here e2(f) refers to the second component of vector f.) Thus, d =d = es ;
and now, imposing (0) = 0, we get ( ) = (1 es )=s. This explains the form
of the function we use below in equation (4.12).
4.2 The Five Zones of the Construction
Since the component v of the solution grows unboundedly at the shock as " !
0, we use the scaling of v(0) to represent the " dependence, and, recalling the
ansatz at the beginning of Section 4, we match solutions of the three di erent
dynamical systems near = 1, = 0 and = 1. See Figure 4.1. On the
central interval [ ; +], singular behavior occurs and the dynamics are governed
by (3.3); the two asymptotic systems hold on (1; 2 ] and [2 +;1), and in
transition regions, (2 ; ) and ( +; 2 +), we go from one system to another.
The choice of transition points is essentially arbitrary: analytic and numerical
evidence, presented in [6], suggests a rapid transition from the singular shock to
the asymptotic regime. We designate by y the candidate for w" when scaled as
11
Keyfitz, Sever and Zhang
in (3.4) or (3.7), while z, related to y by (3.5), will be shown to be small in the
sense of Theorem 3.1.
If y is to approach the limit states w at = 1, then, from (3.5), z( 1; )
takes the values C = q(w ) sw . We choose z to take these constant values
outside (2 ; 2 +), thus satisfying (1) of Theorem 3.1. Then, for large j j, the
dynamics of y, by (3.5), are the asymptotic dynamics of
y = q(y) sy C : (4.10)
The dependence on (or ") is driven by the initial condition v(0) in the
central dynamical system in the central interval. It is noteworthy that di erent
systems with singular shocks show quite di erent scalings; in the system studied
in this paper, the scaling is exponential in . (This exponential scaling, which
di ers from the polynomial scaling of the hyperbolic system in [5], seems to be
responsible for the failure of Schecter's geometric construction in [10] to carry
over immediately to (1.1).) Thus, in the middle interval [ ; +], beginning with
the heteroclinic connection W0 = ( 0; v0) de ned in (4.7), we choose the scaling
parameter in (4.8) to be an increasing function ( ) with the properties that
(1) = 1, (1) = 1 and
lim
!1
0( )
( )
= L; (4.11)
where L will be determined in the proof. (It is through the choice of L that
condition (4) of Theorem 3.1| see (2.3) | is satis ed.) Nothing is lost if we
simply assume ( ) = eL( 1). Finally, we introduce to obtain the form (4.9),
and we have, for 2 [ ; +],
y = yV ( ; )
0
( )
1 es
s
; ( )es v0
( )
1 es
s
>
: (4.12)
Then (3.5) implies
z = zV ( ; ) s
0
( )
1 es
s
; 0
>
: (4.13)
The case s = 0 is included in the formulas above, since lims!0(1 es )=s = ,
but there are simpler expressions in this case:
yV ( ; ) =
0
( )
; ( )v0
( )
> and zV ( ; ) = (0; 0)>: (4.14)
We de ne
y = lim
!1
yV ( ; ): (4.15)
From (4.3) and (4.4), explicit formulas for the limits y give the nite value
y+ =
2;
es +
B2( 2) ( +)
>
: (4.16)
12
Viscous Singular Shocks
From now on, we focus on the right hand intervals only.
According to our ansatz, between + and 2 +, the function y moves from the
point yV ( +; ), which tends to y+ in equation (4.16) as ! 1, to a point p1
in the stable set
0(w+) of w+. We choose p1 to be independent of and de ne
y( ; ) for 2 [ +; 2 +] as a straight line:
y( ; ) =
2 +
+
yV ( +; ) +
+
+
p1: (4.17)
Finally, y( ; ) in (2 +;1) is determined from the asymptotic dynamics, equation
(4.10), and continuity at +, as the solution of
y = q(y) q(w+) s(y w+); for > 2 +; y(2 +) = p1: (4.18)
Consistent with (4.10), (4.18) is an autonomous system for > 2 +, with
equilibrium w+; note that y is independent of in this interval.
The auxiliary function z is now determined for 2 ( +;1) by
z( ; ) = y q(y) + sy; (4.19)
consistent with (3.5). Note that z = sw+ q(w+) is constant for > 2 +. In
addition, the behavior of z in ( +; 2 +) is as required for Theorem 3.1, (5).
Proposition 4.2 In the interval [ +; 2 +], z 2 BV, uniformly in .
Proof: The function z is given by (4.19), and since y is the straight line (4.17),
y takes the constant value
y =
1
+
p1 yV ( +; )
:
Since yV ( +; ) tends to the limit y+ (see equation (4.16)), the value y is uniformly
bounded in ; hence the total variation of y, and thence q(y) and thence
z, is uniformly bounded on this interval.
We have now de ned the functions y and z for all ( ; ). Of the ve conditions
in Theorem 3.1, the rst, (1), on asymptotic behavior of z, holds by construction.
We next establish (2), uniform convergence of y to the equilibrium w+ of (4.10),
in Section 5. Then in Section 6 we prove that y ( ; ) is uniformly integrable for
all 1, establishing (3). Finally in Section 7 we complete the proof of Theorem
3.2 by showing that conditions (4) and (5) hold in Theorem 3.1.
13
Keyfitz, Sever and Zhang
r2
v+
O b
W0
v
G
Figure 5.1: The Nullcline v = 0
5 Convergence of the Approximate Solutions at Infinity
For strictly overcompressive connections with s > Re( j(w+)), j = 1; 2, the
point w+ is a spiral sink for (4.10), and convergence of y to w+ takes place at an
exponential rate, provided only that p1 be chosen in the open basin of attraction
of w+.
There are three weakly overcompressive cases:
(1) w+ = ( 2; 0)> and s = 0;
(2) w+ = w0(w) = ( 2; v+)> 2 @Q \ f = 2g, and s = 2B2( 2)v+ = v+= 2;
(3) w+ = ( +; +)> 2 @Q \ f 2 < < g, and s = 2B2( +)v+;
and we prove
Proposition 5.1 In cases (1) and (2) above, y ! w+.
Proof: In case (1), when v+ = 0, then s = 0. Since w+ = ( 2; 0)> = W+, then
equation (4.18) is the same as (4.2) for large . The open set
0(w+) = f( ; v)j 2 < < 1; v > 0g;
is in the basin of attraction of w+ = ( 2; 0), and y is the solution yV of (4.2)
given by (4.12) for all > and all . This implies the uniform approach of
y to w+, as can be seen from (4.3). Thus, condition (2) of Theorem 3.1 holds.
To establish the integrability of y (item (3) of Theorem 3.1), we choose y to be
independent of for > 2 +. (See the proof of Proposition 6.5.) For this we
choose p1 to be any point, independent of , in
0(w+).
Case (2) requires a more careful analysis of equation (4.18), an autonomous
system with equilibrium ( 2; v+), for large . We write
= vB1( ) +
v+
2
( 2) =
1
( 2)(v v+) +
1v+
2
( 2)2;
v = v2B2( ) v2
+B2( 2) +
v+
2
(v v+) = v2B2( ) +
v+
2
v
v+
2
= B2( )(v v+)2 +
1v+
2 2
v
v+
2
( + 2)( 2):
(5.1)
14
Viscous Singular Shocks
We note that w+ has an open basin of attraction. A nullcline fv = 0g of equation
(5.1) is given by
=
B2( ) =
v+
2v2
v
v+
2
; (5.2)
which can be solved for (v), v v+. On , we have
d
dv
=
3(v v+)
v3 1 2
;
which is zero at ( 2; v+); so in the phase plane ( ; v), is tangent to the line
= 2. The open set
0(w+) = f( ; v)j 2 < < 1(v); v > v+g;
between and the line = 2, is in the stable set of w+. See Figure 5.1. By
choosing p1 independent of in the interior of
0 we ensure that we have y(2 +; )
in
0. Then solutions of (5.1) for > 2 + have the property that y( ; ) ! w+
as ! 1.
For the third type of potential connection, points in the interior, 2 < + <
, on @Q(w), the construction of viscous pro les does not seem possible. To
study the type of the equilibrium w+ 2 @Q(w) of the system, we can x w
and consider a family of states w+ crossing @Q. Then when w+ is on @Q(w),
the Hopf bifurcation theorem [3, Theorem 3.4.2] implies that one of three things
must happen: either w+ is a center, or supercritical or subcritical Hopf bifurcation
takes place at w+. Calculation of the normal form of the equilibrium up to third
order is inconclusive. Numerical simulations indicate that w+ is a center.
6 Integrals of the Approximate Solutions
An important technical feature of the construction is that the L1 integral of
y (:; ) is bounded uniformly in . Absolute integrability of y may not be an
intrinsic requirement of viscous pro les. However, we need this condition in order
to apply Theorem 3.1. Potential sources of di culty occur near the origin, where
y ! 1 with , and near = 1, where in the weakly overcompressive case y
does not tend exponentially fast to w+. For the system studied in this paper,
we have eliminated the di culty near in nity by choosing approximate solutions
which are independent of outside a nite interval.
In this section we prove integrability near = 0, in Theorem 6.1. Part (iii) of
Theorem 6.1 will be used to obtain condition (4) of Theorem 3.1.
In addition, in Proposition 6.5, we establish the uniform integrability of y on
the remaining interval, [ +; 2 +].
We rst prove the result when s = 0; then as Corollary 6.4 we show it follows
for any s.
15
Keyfitz, Sever and Zhang
Theorem 6.1 Let y be the function yV given by (4.14). For given < 0 < +,
and with lim !1 0= = L,
(i)
R +
jy ( ; )j d is bounded uniformly in ;
(ii) For any 2 ( ; +) with 6= 0, lim !1 y ( ; ) = 0;
(iii) lim !1
R +
( ; ) d = 0, and lim !1
R +
v ( ; ) d = 2L( 1 + 2).
We rst prove some lemmas about the functions ( ; v) de ned by (4.14).
Lemma 6.2 For any xed > 0, v ( ; ) changes sign at most once when < 0
and once when > 0.
Proof: We may assume v (0; ) > 0, since v(0; ) = ( )v0(0) ! 1 as ! 1.
Di erentiating v( ; ) with respect to , we have
v ( ; ) = 0( )v0( )
1 + B2( 0( ))v0( )
: (6.1)
The factor 1 + B2( 0)v0 can be rewritten T( ) = 1 + B2( 0( ))v0( ), whose
behavior depends only on the xed solution ( 0; v0). Let = be a point where
T( ) = 0. Then at = (using B2v0 = 1 and the di erential equation (4.2))
we have
T0( ) = B2v0 + B0
2 0
0v0 + B2v0
0 = v2
0B1B0
2;
and B1 < 0 and B0
2 > 0 imply that sgn T 0 = sgn . In other words, as a function
of , T can go only from positive to negative when > 0 and the other way when
< 0. Thus, either T > 0 everywhere or there are unique values < 0 < + at
which T changes sign.
The function T can be computed explicitly using a standard solver, and changes
sign twice. We de ne ( ) = = ( ) as the points where v changes sign. Now
we prove
Lemma 6.3 The integral
R +( )
( ) jv ( ; )j d is bounded uniformly in .
Proof: From Lemma 6.2, we may drop the absolute value sign, and from (6.1),
for a given > 0, we write
Z +( )
( )
v ( ; ) d =
0( )
( )
Z +( )
( )
v0( ) + v0
0( ) d( ):
Carrying out the integration gives
Z +
v d =
0
v0( )
+
=
0
+ v0( +) v0( )
:
16
Viscous Singular Shocks
Now, from T( ) = 0 the last expression becomes
0
1
B2( 0( +))
+
1
B2( 0( ))
;
which is uniformly bounded in .
We can now prove Theorem 6.1.
Proof: We begin with (ii), pointwise bounds for and v . For any > 0,
0( ) ! 2 and v0( ) ! 1=B2( 2) as ! 1, by Proposition 4.1; so we
have
( ; ) =
0( )
( )
v0( )B1( 0( )) ! 0
as ! 1. We note also that, since v0( ) = v0( ) is bounded, is in
fact uniformly bounded. For v , we use (6.1), multiplying and dividing by and
inserting limits from Proposition 4.1:
v ( ; ) ! L
1
B2( 2)
1 B2( 2)
1
B2( 2)
= 0;
as ! 1. The proof for < 0 is the same.
Now we establish (iii). The formula ( ; ) = 0( ) 0
0 ( ( ) ) = 0 v0B1
gives
sgn ( ; ) = sgn : (6.2)
For any 0 > 0 with 0 < minf ; +g, using (6.2) we have
Z 0
0
d =
0( )
( )
Z 0
0
0
0( ) d
0
0
Z 0
0
0
0( ) d +
Z 0
0
0
0( ) d
=
0
0 [ 0( 0 ) 0( 0 )]
0
0( 2 1):
We may assume that 0= 2L for all . For any > 0, since the integral is
negative, take 0 small enough that
Z 0
0
d
<
2
:
By (ii), given > 0 and 0 > 0 we can nd T > 0 such that, if > T,
Z 0
d
<
4
and
Z +
0
d
<
4
:
Therefore if > T,
Z +
( ; ) d
< :
17
Keyfitz, Sever and Zhang
Now we check the second component of y . As in the proof of Lemma 6.3 we
integrate v :
Z +
v ( ; ) d =
0( )
( )
+v0( +) v0( )
:
By (4.3) and (4.4),
+v0( +) ! 2 2 and v0( ) ! 2 1; (6.3)
as ! 1, while 0= ! L. The second relation in (iii) follows.
Finally, we prove (i). For any we have
Z +
j ( ; )j d =
Z 0
( ; ) d
Z +
0
( ; ) d
=
0( )
( )
Z 0
0
0( ( ) ) d( )
Z +
0
0
0( ( ) ) d( )
2L( 1 2) maxf ; +g:
This is the desired result for . We separate the v integral into its positive and
negative parts:
Z +
v ( ; ) d =
Z
v ( ; ) d +
Z +
v ( ; ) d +
Z +
+
v ( ; ) d :
As we have shown in (iii) that the left side tends to 2L( 1+ 2) while from Lemma
6.3 the middle integral is uniformly bounded, we have uniform bounds on the two
outer integrals for large , and
Z +
jv ( ; )j d =
Z
v ( ; ) d +
Z +
+
v ( ; ) d
!
+
Z +
v ( ; ) d
is also uniformly bounded.
This completes the proof of the theorem.
Corollary 6.4 Let y be the solution yV given by (4.12) with s 6= 0. Then the
relations in Theorem 6.1 hold.
Proof: When s 6= 0, we have
( ; ) = 0
( )
1 es
s
; v( ; ) = ( )es v0
( )
1 es
s
:
As in Section 4.1, let = (1 es )=s. Then
( ; ) = 0
) = ( ; ); v( ; ) = (1 s ) v0
) = (1 s ) v( ; );
18
Viscous Singular Shocks
where ( ; v) are given by the formulas in (4.14). Hence the estimates in Theorem
6.1 apply to ( ; v). Since any points are just mapped to points ( ), while
the factor 1 s is bounded on [ ( ); ( +)], these bounds hold also for ( ; v).
Finally, the nonzero estimate in (iii) is also valid since
R
v ! 0 as ! 1.
Proposition 6.5 When w+ = W+ = ( 2; 0), or w+ = ( 2; v+), and if p1 is
chosen in the stable set
0(w+), independent of , then
R 1
+ jy j d ! 0 as ! 1.
Proof: The only contribution to this integral is from the interval + < < 2 +,
since y = 0 for > 2 +. Since y and y are linear in , from (4.17) we have
Z 2 +
+ jy j d =
+
2
@
@
yV ( +; )
;
and this expression tends to zero as ! 1, by Theorem 6.1, (ii).
We have now established condition (3) of Theorem 3.1.
7 Completion of the Proof of Theorem 3.2
It remains to show that conditions (4) and (5) of Theorem 3.1 hold. Using
Proposition 6.5 and Theorem 6.1, (iii), we have
lim
!1
Z
R
y ( ; ) d = lim
!1
Z +
y ( ; ) d =
0
2L( 1 + 2)
;
and letting L be the nonzero component of e(w;w+; s)=2( 1 + 2) (see equation
(2.3)) yields (4).
Finally, from the construction of y and z, y( ; ) is Lipschitz continuous for
2 R, 1 and approaches constants at 1, so y( ; 1) is of bounded variation.
To show that z( ; ) is of bounded variation uniformly in , we compute (noting
0
0 < 0) the rst component z1 from equation (4.13):
Z +
jz1; j d = s
Z +
0
0( )es d s
Z 1
1
0
0( )
d
;
which tends to zero as ! 1. Proposition 4.2 gives uniformly bounded variation
for z 2 [ +; 2 +], and z is constant for > 2 +. Thus, (5) holds.
Thus we can apply Theorem 3.1 to obtain Theorem 3.2.
In the case of strictly overcompressive shocks, the condition (3.6) holds whenever
the conditions of Theorem 3.1 are met, and we can conclude that both ordinary
(viscosity "wxx) and self-similar (viscosity "twxx) viscous structures exist.
However, in the weakly overcompressive cases of Theorem 3.2, a rough calculation
of the solutions near the attractor w+ indicates that convergence of the integral
of j y j is not rapid enough to give (3.6).
19
Keyfitz, Sever and Zhang
8 The Cauchy Problem with Piecewise Constant Data
Here we take K = 0, G = 0, and use the dependent variable Z given by
Z =
( 2)( 1 )v
; (8.1)
noting that Z uniquely determines the volume fraction and the velocity of any
phase present [6]. The region H (in which all phases present have zero velocity)
corresponds to Z2 = 0.
The entropy Riemann solutions obtained in [6] have the following properties.
(i) All of the intermediate states created are in H.
(ii) An admissible discontinuity separating a region where the state is ZH 2 H
from a region where the state is Z 62 H moves towards the region where
the state is Z. (Points in space change only from state Z to ZH.)
(iii) For a given Z 62 H, there is a positive such that all admissible discontinuities
connecting Z to any point ZH 2 H have speed greater than .
(iv) An admissible discontinuity connecting two states in H has speed zero.
Now consider the Cauchy problem with piecewise constant initial data Z( ; 0),
assuming only a nite number of di erent values and with a locally nite set of
points of discontinuity. Solutions are easily constructed by wave front tracking.
Indeed, in general a plethora of corresponding solutions exists, as at any point
(x; t) where Z is continuous and Z(x; t) 62 H, one may choose the nontrivial
Riemann solution of the trivial Riemann problem.
However, all such solutions have a very simple form.
From (i), it follows that for any t > 0
fZ( ; t)g fZ( ; 0)g [ H: (8.2)
Denote by S(t) the interior points of the set fx j Z(x; t) 2 Hg. Then from (ii)
and (iv), for any x 2 S(t0) it follows that
Z(x; t) = Z(x; t0); 8t > t0: (8.3)
From (8.2), the assumption that Z(:; 0) assumes only a nite number of di erent
values, and (iii), there is a positive such that no admissible discontinuities
of speed less than connect states ZH in H to states not in H.
Assume now that the closure of S(t) is not empty and is not all of R. Then the
boundary of S(t) necessarily contains a point at which an admissible discontinuity
is moving out of S(t) at a speed of at least . So if the complement of S(0) has
nite measure, then the closure of S(t) will be all of R in nite time, independent
20
Viscous Singular Shocks
of which solution is chosen. After this nite time, from (8.3) the solution is
independent of t; from (1.1), recalling that K = 0, we deduce that v vanishes
identically. Thus, this model predicts that if there is a nite amount of mixed
phase
uid initially, it will separate into single phase states in nite time. In
other words, absent any interfacial momentum transfer terms (the balance terms
we have omitted), the mathematical solution correctly predicts that the Bernoulli
e ect will dominate [8].
9 Conclusions
Singular shocks represent a novel type of solution to conservation law systems,
and may provide solutions in situations, such as large data for some hyperbolic
systems as well as the present context, in which classical shock solutions do
not su ce to solve all initial value problems. Because they are not yet well
understood, it is important to apply to them the same tests that have been
used, classically, to establish the admissibility and stability of weak solutions of
conservation law systems.
In this paper we have established viscous structure for singular shocks which
can be used to solve Riemann problems for a nonhyperbolic model system. This
exercise is not based on a physically realistic form of
uid-dynamic viscosity.
Nonetheless, it is an important step in justifying the use of singular shocks to
solve the model problem. In particular, it shows that there is a sense in which
solutions to the ill-posed system (1.1) can be recovered as limits of solutions of
well-posed systems of the form (1.3). Since (1.1) is equivalent to a standard
two-
uid model in widespread use, a better understanding of the mathematical
structure of its solutions will have signi cant application.
Many questions remain. It would be valuable to understand how singular
shocks can be approximated when di erent forms of viscous regularization are
used. This question has not been answered even in the simpler case that the
underlying equation is hyperbolic. In a second direction, critically important to
the application to two-phase
ow, we would like to understand what happens
to singular shock solutions of (1.1) when balance terms are added to the second
equation of (1.1). The technique of this paper suggests adding both viscous perturbations
and balance terms, and seeking scale-invariant or asymptotic solutions.
We are currently studying this problem.
Acknowledgements
It is a pleasure to acknowledge useful conversations with Richard Sanders, and
help from Marty Golubitsky and Steve Schecter on dynamics and viscous pro les.
Finally, we thank Nam Dinh of the Center for Risk Studies and Safety at the
University of California, Santa Barbara, for his encouragement of our pursuit of
nonhyperbolic models.
21
Keyfitz, Sever and Zhang
References
[1] T. N. Dinh, R. R. Nourgaliev, and T. G. Theofanous. Understanding
the ill-posed two-
uid model. In The 10th International Topical Meeting
on Nuclear Reactor Thermal Hydraulics (NURETH-10). Seoul, Korea, 2003.
[2] D. A. Drew and S. L. Passman. Theory of Multicomponent Fluids.
Springer, New York, 1999.
[3] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical
Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1986.
[4] B. L. Keyfitz. Mathematical properties of nonhyperbolic models for incompressible
two-phase
ow. In E. E. Michaelides, (ed), Proceedings of
the Fourth International Conference on Multiphase Flow, New Orleans (CD
ROM). Tulane University, 2001.
[5] B. L. Keyfitz and H. C. Kranzer. Spaces of weighted measures for conservation
laws with singular shock solutions. Journal of Di erential Equa-
tions, 118:420{451, 1995.
[6] B. L. Keyfitz, R. Sanders, and M. Sever. Lack of hyperbolicity in the
two-
uid model for two-phase incompressible
ow. Discrete and Continuous
Dynamical Systems, 3:541{563, 2003.
[7] H.-O. Kreiss and J. Ystrom. Parabolic problems which are ill-posed in
the zero dissipation limit. Mathematical and Computer Modelling, 35:1271{
1295, 2002.
[8] A. Prosperetti. Private communication.
[9] R. Sanders and M. Sever. Computations with singular shocks. In
preparation, 2003.
[10] S. Schecter. Existence of Dafermos pro les for singular shocks. Preprint,
2003.
[11] M. Sever. Distribution solutions of nonlinear systems of conservation laws.
Preprint, 2003.
[12] J. A. Smoller. Shock Waves and Reaction-Di usion Equations. Springer-
Verlag, New York, 1983.
[13] H. B. Stewart and B. Wendroff. Two-phase
ow: Models and methods.
Journal of Computational Physics, 56:363{409, 1984.
22
a Nonhyperbolic Two-Fluid Model
Barbara Lee Key tz Michael Sever y Fu Zhang z
Abstract. We consider a system of two nonhyperbolic conservation laws modeling incom-
pressible two-phase
ow in one space dimension. The purpose of this paper is to justify the use
of singular shocks in the solution of Riemann problems. We prove that both strictly and weakly
overcompressive singular shocks are limits of viscous structures. Using Riemann solutions we
solve Cauchy problems with piecewise constant data for the nonhyperbolic two-
uid model.
1 Introduction
This paper examines a nonhyperbolic system arising in the modeling of incompressible
two-
uid
ows. The principal result we obtain here is that approximate
viscous pro les can be constructed for both strictly and weakly overcompressive
singular shocks that are important in solving Riemann problems. In addition,
we use this stability result for Riemann solutions to show that a class of Cauchy
problems for the nonhyperbolic two-
uid model can be solved.
We consider a system of two equations, wt+q(w)x = 0, for a state w = ( ; v)>,
which for the two-
uid model takes the form
t + (vB1( ) + K )x = 0
vt +
v2B2( ) + Kv
x = 0;
(1.1)
with
B1( ) =
( 1)( 2)
; B2( ) =
2 1 2
2 2 : (1.2)
This system is equivalent to the system of four equations modeling one-dimensional
unsteady isothermal incompressible two-phase
ow, [2, p 248]. The volume
fractions 1 and 2 = 1 1 have been replaced by a density-weighted volume
element
= 2 1 + 1 2;
Department of Mathematics, University of Houston, Houston, Texas 77204-3008. Research sup-
ported by the Department of Energy, grant DE-FG02-03ER25575, the National Science Foundation,
Grant DMS 03-06307, and the Texas Advanced Research Program, grant 003652-0076-2001.
y Department of Mathematics, The Hebrew University, Jerusalem, Israel.
z Department of Mathematics, University of Houston, Houston, Texas 77204-3008. Research sup-
ported by the Texas Advanced Research Program, grant 003652-0076-2001 and the Department of
Energy, grant DE-FG02-03ER25575.
Keyfitz, Sever and Zhang
and the momentum equations replaced by a single equation for the momentum
di erence,
v = 1u1 2u2 ( 1 2)K:
Here K(t) = 1u1 + 2u2 is constant in space as a consequence of conservation
of mass. In this paper we take K = 0, for convenience. The system (1.1) has
been nondimensionalized: the densities i and have been scaled by dividing
by 1 2, so we take 1 2 = 1 throughout and note that i and are
dimensionless. Hence v has the dimensions of velocity; we may replace v and K
by v=v0 and K=v0, where v0 is some characteristic velocity, and simultaneously
scale x=t by v0, making (1.1) dimensionless. Ordinarily, there is a balance term,
G = F1= 1 F2= 2 in the second equation of (1.1); G models buoyancy, drag
and other interfacial forces. Research on two-
uid models usually concentrates
on correct modeling of the forces Fi. However, the focus of this article is on the
mathematical structure of the di erential operator on the left side of the system
(1.1), and we set G to zero.
It is well known [2, 13] that the system is not hyperbolic, as the characteristics
at any mixed-phase state ( 2 < < 1), are complex. We de ne j(w), j = 1; 2,
to be the eigenvalues of dq(w), and note that 1 = 2 if 2 < < 1, while
1 = 2 if = i, i = 1 or 2. Key tz, Sanders and Sever have discussed this
equation in earlier papers [4, 6]. Two important facts are that the system, at
least by a formal asymptotic expansion, admits solutions of very low regularity,
called singular shocks, and that by means of singular shocks one can solve the
Riemann problem for (1.1) for all initial discontinuities.
In this paper, we justify the use of singular shocks by showing that they
appear as weak limits of smooth solutions to a regularized, viscous system,
wt + q(w)x = "wxx: (1.3)
More precisely, we construct a sequence of functions w"(x; t) such that, as " ! 0,
w"( ; t)t + q(w"( ; t))x "w"( ; t)xx * 0; (1.4)
weakly in the space of measures on R, pointwise in t. The sequence w" converges
to a singular shock, in the same sense.
Establishing a relation like (1.4) is a critical step in arguing that singular
shock solutions have a physical interpretation for the system modeled by (1.1).
As is well known, even for hyperbolic systems of conservation laws, which are
linearly well-posed (unlike our nonhyperbolic system), solutions exist in only
the weak sense, and establishing well-posedness requires admissibility conditions
for shocks. Admissibility conditions can be derived in a number of ways, as
described in Smoller's monograph [12], including perturbation of the system by
a second-order term representing viscosity, as in (1.3). Another motivation for
introducing a viscous approximation like (1.3) is that nite-di erence numerical
2
Viscous Singular Shocks
schemes for conservation laws achieve stability by introducing a small amount
of dissipation, which can be modeled at the continuous level with a system like
(1.3). In principle, one would like to prove admissibility via the use of a physically
realistic viscosity, or, equivalently, to prove that the precise form of viscosity
does not a ect important macroscopic properties of shocks such as their speed
and stability. This is important both for establishing physical relevance and
for studying the convergence of numerical approximations. The precise form
of physical viscosity is not known for the system (1.1). Modeling numerical
dissipation is also complicated. `Arti cial' viscosity, as in (1.3), is often chosen
in theoretical studies, for convenience. We choose it here, because it is the only
choice which leads to relatively simple viscous structures. Finding comparable
structures for other forms of viscous perturbation remains an open problem, as
we discuss in the concluding section.
In related work, Sanders and Sever construct approximate viscous pro les for
nonhyperbolic systems which exhibit a symmetry [9]. The method outlined in
[9] applies to the model system (1.1). However, that paper is limited to strictly
overcompressive singular shocks, and hence, for the system (1.1), does not include
all the shock connections needed to solve Riemann problems. In this paper, we
give a complete proof of the construction for (1.1), in both the overcompressive
and weakly overcompressive cases.
For a model strictly hyperbolic system originally studied by Key tz and
Kranzer [5], existence of self-similar shock pro les for singular shocks when selfsimilar
viscosity of the form "twxx is added to the system has recently been
proved by Schecter [10]. Schecter uses geometric singular perturbation theory to
nd smooth solutions for su ciently small, positive ". It is possible that a geometric
perturbation result can be found for nonhyperbolic systems such as (1.1),
perhaps using some new scalings as suggested by this paper. The construction in
our paper provides viscous structure for a regularized system with viscosity "wxx,
and also gives self-similar viscous structure with viscosity "twxx in some cases,
but not in the important case of weakly overcompressive shocks.
Recent work of Kreiss and Ystrom [7], on the relation of viscous regularizations
to their limits, considers a number of model equations, not including (1.1) but
with the nonhyperbolic equations of the two-
uid model very much in mind.
Their work shows that the existence of certain kinds of energy bounds can prevent
unbounded amplitude growth in the parabolic approximations, and suggests that
weak convergence to a limit of bounded but high-frequency oscillations may occur
instead. Amplitude bounds do not exist for (1.1), and the type of convergence we
report here is di erent from the nonlinear examples in [7]. Numerical experiments
by Dinh, Nourgaliev and Theofanous [1] on the model (1.1) con rm the singular
behavior predicted by our analysis.
In the next section, we give the de nition of a singular shock solution, and
also review the singular shock solutions of (1.1), described in detail in [6]. Section
3 states the main theorem, along with the theorem of Sever from [11] which we
3
Keyfitz, Sever and Zhang
use to prove it. The construction of fw"g is carried out in Section 4, while the
proof that this construction provides an approximation as required by the theory
in [11] occupies the three sections which follow. As an application, we solve
some Cauchy problems for (1.1) in Section 8. The signi cance of our results is
explained in Section 9.
2 Definitions
A singular shock solution to a system wt + q(w)x = 0 is of the form w = ! +
,
where !, the regular part of the solution, is a discontinuous function and
, the
singular part, is a measure supported on the discontinuity set of !. A self-similar
singular shock has the following structure:
!(x; t) =
w; x < st
w+; x > st
; (2.1)
(x; t) = t e(w;w+; s) (x st); (2.2)
where
e(w;w+; s) = s(w+ w) q(w+) + q(w): (2.3)
A measure-valued function such as w does not satisfy the conservation law in the
usual weak sense. Hence, it can be di cult to assign a signi cance to singular
shocks. Based on a standard approach to analysing conservation laws by studying
limits of regularized approximations, we attempt to justify singular shocks by
nding them as limits of solutions of approximating systems. This might mean
strong limits in an appropriate measure space, as was carried out by Key tz and
Kranzer for a hyperbolic model system [5]. However, one can work directly with
weak limits [11], and that is the approach we adopt here. In [11, Chapter 3, x1],
Sever gives a de nition of singular shock solutions, which we use in this paper.
Definition 2.1 A singular shock solution of a system wt + q(w)x = 0 is a
measure of the form
w(x; t) = !(x; t) +
X
i
Mi(t) i(t) (x xi(t));
where ! is a classical weak solution away from the singularities, i is the charac-
teristic function of an interval [Ai;Bi); Mi 2 L1, and xi 2 W1;1. The function
w is the weak limit of a sequence w" with w"( ; t) 2 L1
loc uniformly with respect to
", pointwise in t; satisfying
w"( ; t) * w( ; t) and w"( ; t)t + q(w"( ; t))x "(Aw"( ; t)x)x * 0; (2.4)
as " ! 0, weakly in the space of measures on R, pointwise with respect to t, for
some positive de nite matrix A.
4
Viscous Singular Shocks
Remark 2.2 A function f( ; ") is said to be in L1
loc uniformly with respect to
" if, for any set J R of nite measure,
R
J jf( ; ")j M = M(J), where
M is independent of ". In general, a property holds uniformly with respect to
a parameter if the bound which establishes the property is independent of the
parameter.
Remark 2.3 The matrix A need not be constant in this de nition. Two important
cases are the scalar matrices A = I and A = It, corresponding to the
standard concept of `arti cial' viscosity. These are the only choices which appear
to give simple viscous structures for singular shocks. The signi cance of this is
discussed in the concluding section of the paper.
The measures de ned this way generalize the self-similar singular shocks described
by (2.1), (2.2) and (2.3); however, for any candidate satisfying (2.1){(2.3), one
must verify (2.4) to establish that the object is indeed a singular shock. In
[11], Sever established some important properties of singular shocks. First, the
singular mass M(t) evolves according to the Rankine-Hugoniot de cit, de ned in
equation (2.3) above. Along the i-th singular shock,
dMi(t)
dt
= e
w(xi(t) 0; t);w(xi(t) + 0; t); x0
i(t)
: (2.5)
Second, singular shocks are generally overcompressive discontinuities; for equations
with real characteristics this means j(w) s j(w+), where j = 1; 2,
s = x0
i, and w are the limits of w from the right and left sides of the discontinuity.
Furthermore, for systems with real characteristics, the existence of
convex entropies implies a direction for the vector e. In a number of hyperbolic
model problems this direction picks out a coordinate system such that the rst
component of e is identically zero, which means that the rst Rankine-Hugoniot
condition holds.
The last two points do not immediately generalize to our system (1.1) with
complex characteristic speeds. However, in [6] it was shown that if we use a
generalization of the overcompressibility condition, obtained by examining the
real part of the characteristic speeds,
Re
j(w)
s Re
j(w+)
; j = 1; 2; (2.6)
and if we satisfy the rst Rankine-Hugoniot condition at each singular shock
jump, then there exist solutions of the form (2.1), (2.2), (2.3) by means of which
one can solve all Riemann problems. The case was also made in [6], based on
the derivation of (1.1), that in the rst equation is genuinely conserved, while
the rationale for the form of the second equation is weaker. Hence it seems
appropriate to satisfy the rst Rankine-Hugoniot relation.
In the remainder of this paper, we shall assume condition (2.6), and also take
A = I or It in De nition 2.1. An explanation of this choice is given in Section 9.
5
Keyfitz, Sever and Zhang
w−
w0
Q(w o.c. shock −)
H ®
¬ H
H¯
¬ s=Re(l−)
¬ s=Re(l+)
b
v
¬ r2 r1 ®
Figure 2.1: The Region Q of Overcompressive Shocks
The Riemann solution of (1.1) is not unique [6]. We assume that the rst
Rankine-Hugoniot condition is satis ed, so that
s =
[vB1]
[ ]
v+B1( +) vB1( )
+
(2.7)
is determined by w and w+. For any w, we de ne Q(w) to be the set of
points w+ in phase space such that (2.6), which is now a condition on w+, holds.
Then, two solutions exist for each w+ 2 Q(w). One solution consists of a single,
singular shock, while the other exhibits a complicated wave pattern consisting of
singular shocks, rarefactions, and a contact discontinuity [6]. When w+ is not
in Q(w), then only the second type of solution can be constructed. The wave
in this case is always composite, although if w and w+ are in di erent vertical
half-planes there is only a single rarefaction and no contact discontinuity. Furthermore,
in a composite wave, the singular shock connects w with a particular
state, w0, the point where @Q, the boundary of Q, intersects H, which we de ne
to be the set of points in the physical region with real characteristics. See Figure
2.1. A shock between w and a point on @Q is only weakly overcompressive.
One detail of the composite Riemann solutions is that two points on the
boundary of Q are needed to construct a composite wave: w0(w) and the analogous
point, which we called w1(w+) in [6], in H, which can be joined to w+ on
the right with a weakly overcompressive singular shock with 1(w1) = 2(w1) =
s > Re( j(w+)), j = 1; 2. In this paper, we show that these points in @Q \ H
admit viscous pro le connections.
Without loss of generality, we concentrate on the right half of the pro le; in
particular, we consider only shocks where strict overcompressibility fails at the
state on the right. That is,
Re
j(w)
s = Re
j(w+)
; j = 1; 2: (2.8)
The conclusions of the paper also apply if strict overcompressibility fails on the
left instead, or even if it fails on both right and left, as the convergence arguments
at the right and left endpoints proceed independently of each other.
6
Viscous Singular Shocks
There are three ways that singular shocks which are weakly overcompressive
on the right might arise in (1.1). In examining Riemann problems, one nds that
states w with = 1 and v > 0 can be connected to w+ = ( 2; 0) via a
weakly overcompressive discontinuity satisfying (2.7) and (2.8) with speed s = 0.
The most important case we consider, however, is that with w in the interior
of the physical region and w+ 2 @Q(w) \ H, so w+ = w0(w), as in Figure 2.1:
these are the singular shocks which appear in composite waves. Finally, it is in
principle possible that a state w+ 2 @Q(w) \ f 2 < < 1g could be joined to
w via a singular shock. Such connections are not necessary to solve Riemann
problems, since composite waves are also available. Singular shocks of this form
do not appear to have viscous structure.
A principal di erence between strictly and weakly overcompressive shocks, as
pointed out in [6], is the rate of convergence of viscous trajectories to their end
states, which is exponential in the case of strictly overcompressive shocks, but
not in the weakly overcompressive case. In this paper, we show that convergence
holds when w+ 2 H, and it is strong enough to prove the existence of singular
shocks in the sense of De nition 2.1.
3 Viscous Profiles
We review the construction in [11], which generates a sequence of functions satisfying
(1.4). To motivate this, recall that the classical traveling wave construction
for conservation laws uses the similarity variable = (x st)=" to nd solutions
of wt + q(w)x = "wxx in the form
w = y( ): (3.1)
This leads to an ordinary di erential equation for y:
y00 = (q(y) sy)0; (3.2)
and if we now assume that s is the shock speed corresponding to a pair of states
w which satisfy the Rankine-Hugoniot relation s(w+ w) = q(w+) q(w),
then we can integrate (3.2) to obtain y0 = (q(y)sy)+C, where C is the common
value of sw q(w ). For given data w and s, we can assume C = 0. Thus the
fundamental dynamical system in the study of classical shock pro les is
y0 = q(y) sy: (3.3)
However, singular shocks connect states which do not satisfy the Rankine-Hugoniot
relation, and so the premise underlying (3.1) is
awed. We instead let
w"(x; t) = y
x st
"
;
t
"
y( ; ); (3.4)
7
Keyfitz, Sever and Zhang
and de ne
z( ; ) = y q(y) + sy; (3.5)
the function z measures, in some sense, the extent to which the problem fails to
have a classical viscous pro le solution. Then the theorem proved in [11] is
Theorem 3.1 (Theorem 3.3.1 of [11]) Suppose that there exist functions y
and z, related by (3.5), such that
(1) z( ; ) is constant for j j su ciently large;
(2) lim
! 1
y( ; ) = w , uniformly in ;
(3) y ( ; ) 2 L1(R) uniformly in ;
(4)
Z
R
y ( ; ) d = e(w;w+; s) + o(1) as ! 1;
(5) y( ; 1) 2 BV and z( ; ) 2 BV uniformly in .
Then the sequence fw"g de ned by (3.4) satis es De nition 2.1 with A = I.
(See Remark 2.2 for the interpretation of uniformity in .) Sever [11] also proves
that if an additional constraint holds:
Z
R j j
jy ( ; 1)j + jy ( ; )j
d < 1; (3.6)
then we can de ne
w"(x; t) = y
x st
"t
;
1
"
; (3.7)
which satis es De nition 2.1 with A = tI. (In this sense, there is not much
di erence between the standard and the self-similar viscosity criteria.)
Using this theorem, the question of nding singular shocks can be reduced to
nding a pair of functions y and z which satisfy the hypotheses of Theorem 3.1.
The theorem we prove in this paper is
Theorem 3.2 For initial conditions w, w+ 2 Q(w), and s = [vB1]=[ ] satis-
fying either
Re( j(w)) > s > Re( j(w+)); j = 1; 2; (3.8)
or w+ 2 H \ @Q(w) and
Re( j(w)) s = 1(w+) = 2(w+); j = 1; 2; (3.9)
there exists a sequence fw"g of approximate viscous pro le connections satisfying
(2.4). Hence singular shock connections exist.
8
Viscous Singular Shocks
4 The Construction of Approximate Viscous Profiles
If one applies a formal asymptotic expansion to a solution concentrated at a
singular shock (with remaining bounded, as is physically reasonable, but with
no constraint on v), as was done in [6], then the problem reduces, to dominant
order, to equation (3.3) with s = 0 (equation (4.2) below). We now show that,
as suggested by the formal treatment, there are viscous structures satisfying
De nition 2.1.
The construction is based on the ansatz that near the singular shock locus
the viscous solution w" behaves like a solution to the simple system (3.3), w0 =
q(w) sw, while far away, as ! 1 it tends to solutions of di erent systems
w0 = q(w) sw C , where C = q(w ) sw . Furthermore, in the central
interval, an approximate solution can be found by a simple scale change.
The point of departure of the construction is the observation that the system
(1.1) possesses a scaling invariance: If
t 7! t0 = t= ; v 7! v0 = v; (4.1)
then solutions of wt+q(w)x = 0 are mapped to solutions of ( ; v0)>
t0+q( ; v0)>
x = 0.
Also to the point is the observation that the
ux vector eld, q(w), admits a oneparameter
family of heteroclinic connections. The dynamical system
w0 = q(w) (4.2)
is just (3.3) with s = 0; in Section 4.1, we explain the role of s in comparing (4.2)
to (3.3).
The heteroclinic orbits of (4.2) connect points ( 1; 0) and ( 2; 0); we shall
assume v 0 and then W = ( 1; 0), W+ = ( 2; 0). The entire physical region
with v > 0 and 2 < 1 is in the stable set for W+. There is a similar
construction when v 0, with ( 1; 0) the attractor.
The heteroclinic connections can be found explicitly.
Proposition 4.1 For the functions B1 and B2 given by equation (1.2), solutions
to the system (4.2) in the strip ( 2; 1) \ fv > 0g form a one-parameter family
with asymptotic behavior
2 +
4 3
2
V 2 2 and v
2 2
=
1
B2( 2)
; as ! 1; (4.3)
1
4 3
1
V 2 2 and v
2 1
=
1
B2( 1)
; as ! 1; (4.4)
where the parameter V is a positive constant.
Proof: We write
d
dv
=
1
v
B1
B2
;
9
Keyfitz, Sever and Zhang
and solve by separating the variables, recalling that B2 = B0
1 =2, to obtain the
family of solutions
v = V
p
jB1j = V
( 1)( 2)
1
2
; (4.5)
for any positive constant V . Thus the equation for ( ) is
d
d
= V
( 1 )( 2)
3=2
; (4.6)
and ( ) is given by quadrature; since ! 2 as ! 1, then from (4.6),
d
d V
2
2
3
2
as ! 1. Integrating, we get the rst limit in (4.3), and the second comes from
(4.5). By observing that ! 1 as ! 1, we obtain (4.4) in the same way.
The solutions are parameterized by V .
The system also admits a translation symmetry, ! 0. A convenient
normalization for is (0) = p 1 2 where B1 attains its minimum. Then
v(0) = V (p 1 p 2)
is the maximum of v along the trajectory. In particular, we can identify the
nontrivial parameter V with max v. A scaling symmetry like (4.1) also holds
for the dynamical system: If = = , and v0 = v, then solutions in are
mapped into solutions in . We use this scaling in de ning the second variable
in Theorem 3.1.
As a consequence of the scaling, we can express all solutions of (4.2) in terms
of a single function,
W0( ) = ( 0; v0) with v0(0) = 1: (4.7)
For any other solution, we have
( ); v( )
=
0( ); v0( )
; (4.8)
for some ; we take = ( ) to handle the dependence of solutions on .
4.1 The Role of s
In typical constructions of viscous pro les for conservation laws arising in
uid
dynamics, one makes a Galilean change of variables x 7! x st to reduce to the
case s = 0. However, we have already used the change of variables x 7! xKt in
10
Viscous Singular Shocks
yx=q(y)−sy−C− yx=q(y)−sy−C+
yx~ q(y)−sy
2x− x− 0 x+ 2x+
Figure 4.1: Asymptotic Dynamics of the Approximate Pro les
setting K = 0 in (1.1). Thus, we must consider both the cases s = 0 and s 6= 0.
We nd the di erence in the technical detail that (3.3) does not possess the useful
heteroclinic connection of (4.2). However, we can use the system (4.2) to replace
(3.3) in the neighborhood of = 0 by introducing a stretched variable ( ), with
(0) = 0 and d =d > 0. With respect to a xed heteroclinic connection, for
example W0 = ( 0; v0), the function
y( ; ) =
0( );
d
d
v0( )
>
; (4.9)
satis es (3.3) approximately. In particular, y = q(y) + (0; v0d2 =d 2)>. If the
last term were sy, we would have (3.3) and then z = 0 in (3.5). However, even
if ! 1 as ! 1 (as we nd to be the case), only the second component of
y becomes unbounded as ! 1, and so z is a function with bounded variation
if we choose the second component of y in (4.9) to satisfy the second equation of
(3.3). That is, we set
v0
d2
d 2 = e2(sy) = s
d
d
v0 or
d2
d 2 = s
d
d
:
(Here e2(f) refers to the second component of vector f.) Thus, d =d = es ;
and now, imposing (0) = 0, we get ( ) = (1 es )=s. This explains the form
of the function we use below in equation (4.12).
4.2 The Five Zones of the Construction
Since the component v of the solution grows unboundedly at the shock as " !
0, we use the scaling of v(0) to represent the " dependence, and, recalling the
ansatz at the beginning of Section 4, we match solutions of the three di erent
dynamical systems near = 1, = 0 and = 1. See Figure 4.1. On the
central interval [ ; +], singular behavior occurs and the dynamics are governed
by (3.3); the two asymptotic systems hold on (1; 2 ] and [2 +;1), and in
transition regions, (2 ; ) and ( +; 2 +), we go from one system to another.
The choice of transition points is essentially arbitrary: analytic and numerical
evidence, presented in [6], suggests a rapid transition from the singular shock to
the asymptotic regime. We designate by y the candidate for w" when scaled as
11
Keyfitz, Sever and Zhang
in (3.4) or (3.7), while z, related to y by (3.5), will be shown to be small in the
sense of Theorem 3.1.
If y is to approach the limit states w at = 1, then, from (3.5), z( 1; )
takes the values C = q(w ) sw . We choose z to take these constant values
outside (2 ; 2 +), thus satisfying (1) of Theorem 3.1. Then, for large j j, the
dynamics of y, by (3.5), are the asymptotic dynamics of
y = q(y) sy C : (4.10)
The dependence on (or ") is driven by the initial condition v(0) in the
central dynamical system in the central interval. It is noteworthy that di erent
systems with singular shocks show quite di erent scalings; in the system studied
in this paper, the scaling is exponential in . (This exponential scaling, which
di ers from the polynomial scaling of the hyperbolic system in [5], seems to be
responsible for the failure of Schecter's geometric construction in [10] to carry
over immediately to (1.1).) Thus, in the middle interval [ ; +], beginning with
the heteroclinic connection W0 = ( 0; v0) de ned in (4.7), we choose the scaling
parameter in (4.8) to be an increasing function ( ) with the properties that
(1) = 1, (1) = 1 and
lim
!1
0( )
( )
= L; (4.11)
where L will be determined in the proof. (It is through the choice of L that
condition (4) of Theorem 3.1| see (2.3) | is satis ed.) Nothing is lost if we
simply assume ( ) = eL( 1). Finally, we introduce to obtain the form (4.9),
and we have, for 2 [ ; +],
y = yV ( ; )
0
( )
1 es
s
; ( )es v0
( )
1 es
s
>
: (4.12)
Then (3.5) implies
z = zV ( ; ) s
0
( )
1 es
s
; 0
>
: (4.13)
The case s = 0 is included in the formulas above, since lims!0(1 es )=s = ,
but there are simpler expressions in this case:
yV ( ; ) =
0
( )
; ( )v0
( )
> and zV ( ; ) = (0; 0)>: (4.14)
We de ne
y = lim
!1
yV ( ; ): (4.15)
From (4.3) and (4.4), explicit formulas for the limits y give the nite value
y+ =
2;
es +
B2( 2) ( +)
>
: (4.16)
12
Viscous Singular Shocks
From now on, we focus on the right hand intervals only.
According to our ansatz, between + and 2 +, the function y moves from the
point yV ( +; ), which tends to y+ in equation (4.16) as ! 1, to a point p1
in the stable set
0(w+) of w+. We choose p1 to be independent of and de ne
y( ; ) for 2 [ +; 2 +] as a straight line:
y( ; ) =
2 +
+
yV ( +; ) +
+
+
p1: (4.17)
Finally, y( ; ) in (2 +;1) is determined from the asymptotic dynamics, equation
(4.10), and continuity at +, as the solution of
y = q(y) q(w+) s(y w+); for > 2 +; y(2 +) = p1: (4.18)
Consistent with (4.10), (4.18) is an autonomous system for > 2 +, with
equilibrium w+; note that y is independent of in this interval.
The auxiliary function z is now determined for 2 ( +;1) by
z( ; ) = y q(y) + sy; (4.19)
consistent with (3.5). Note that z = sw+ q(w+) is constant for > 2 +. In
addition, the behavior of z in ( +; 2 +) is as required for Theorem 3.1, (5).
Proposition 4.2 In the interval [ +; 2 +], z 2 BV, uniformly in .
Proof: The function z is given by (4.19), and since y is the straight line (4.17),
y takes the constant value
y =
1
+
p1 yV ( +; )
:
Since yV ( +; ) tends to the limit y+ (see equation (4.16)), the value y is uniformly
bounded in ; hence the total variation of y, and thence q(y) and thence
z, is uniformly bounded on this interval.
We have now de ned the functions y and z for all ( ; ). Of the ve conditions
in Theorem 3.1, the rst, (1), on asymptotic behavior of z, holds by construction.
We next establish (2), uniform convergence of y to the equilibrium w+ of (4.10),
in Section 5. Then in Section 6 we prove that y ( ; ) is uniformly integrable for
all 1, establishing (3). Finally in Section 7 we complete the proof of Theorem
3.2 by showing that conditions (4) and (5) hold in Theorem 3.1.
13
Keyfitz, Sever and Zhang
r2
v+
O b
W0
v
G
Figure 5.1: The Nullcline v = 0
5 Convergence of the Approximate Solutions at Infinity
For strictly overcompressive connections with s > Re( j(w+)), j = 1; 2, the
point w+ is a spiral sink for (4.10), and convergence of y to w+ takes place at an
exponential rate, provided only that p1 be chosen in the open basin of attraction
of w+.
There are three weakly overcompressive cases:
(1) w+ = ( 2; 0)> and s = 0;
(2) w+ = w0(w) = ( 2; v+)> 2 @Q \ f = 2g, and s = 2B2( 2)v+ = v+= 2;
(3) w+ = ( +; +)> 2 @Q \ f 2 < < g, and s = 2B2( +)v+;
and we prove
Proposition 5.1 In cases (1) and (2) above, y ! w+.
Proof: In case (1), when v+ = 0, then s = 0. Since w+ = ( 2; 0)> = W+, then
equation (4.18) is the same as (4.2) for large . The open set
0(w+) = f( ; v)j 2 < < 1; v > 0g;
is in the basin of attraction of w+ = ( 2; 0), and y is the solution yV of (4.2)
given by (4.12) for all > and all . This implies the uniform approach of
y to w+, as can be seen from (4.3). Thus, condition (2) of Theorem 3.1 holds.
To establish the integrability of y (item (3) of Theorem 3.1), we choose y to be
independent of for > 2 +. (See the proof of Proposition 6.5.) For this we
choose p1 to be any point, independent of , in
0(w+).
Case (2) requires a more careful analysis of equation (4.18), an autonomous
system with equilibrium ( 2; v+), for large . We write
= vB1( ) +
v+
2
( 2) =
1
( 2)(v v+) +
1v+
2
( 2)2;
v = v2B2( ) v2
+B2( 2) +
v+
2
(v v+) = v2B2( ) +
v+
2
v
v+
2
= B2( )(v v+)2 +
1v+
2 2
v
v+
2
( + 2)( 2):
(5.1)
14
Viscous Singular Shocks
We note that w+ has an open basin of attraction. A nullcline fv = 0g of equation
(5.1) is given by
=
B2( ) =
v+
2v2
v
v+
2
; (5.2)
which can be solved for (v), v v+. On , we have
d
dv
=
3(v v+)
v3 1 2
;
which is zero at ( 2; v+); so in the phase plane ( ; v), is tangent to the line
= 2. The open set
0(w+) = f( ; v)j 2 < < 1(v); v > v+g;
between and the line = 2, is in the stable set of w+. See Figure 5.1. By
choosing p1 independent of in the interior of
0 we ensure that we have y(2 +; )
in
0. Then solutions of (5.1) for > 2 + have the property that y( ; ) ! w+
as ! 1.
For the third type of potential connection, points in the interior, 2 < + <
, on @Q(w), the construction of viscous pro les does not seem possible. To
study the type of the equilibrium w+ 2 @Q(w) of the system, we can x w
and consider a family of states w+ crossing @Q. Then when w+ is on @Q(w),
the Hopf bifurcation theorem [3, Theorem 3.4.2] implies that one of three things
must happen: either w+ is a center, or supercritical or subcritical Hopf bifurcation
takes place at w+. Calculation of the normal form of the equilibrium up to third
order is inconclusive. Numerical simulations indicate that w+ is a center.
6 Integrals of the Approximate Solutions
An important technical feature of the construction is that the L1 integral of
y (:; ) is bounded uniformly in . Absolute integrability of y may not be an
intrinsic requirement of viscous pro les. However, we need this condition in order
to apply Theorem 3.1. Potential sources of di culty occur near the origin, where
y ! 1 with , and near = 1, where in the weakly overcompressive case y
does not tend exponentially fast to w+. For the system studied in this paper,
we have eliminated the di culty near in nity by choosing approximate solutions
which are independent of outside a nite interval.
In this section we prove integrability near = 0, in Theorem 6.1. Part (iii) of
Theorem 6.1 will be used to obtain condition (4) of Theorem 3.1.
In addition, in Proposition 6.5, we establish the uniform integrability of y on
the remaining interval, [ +; 2 +].
We rst prove the result when s = 0; then as Corollary 6.4 we show it follows
for any s.
15
Keyfitz, Sever and Zhang
Theorem 6.1 Let y be the function yV given by (4.14). For given < 0 < +,
and with lim !1 0= = L,
(i)
R +
jy ( ; )j d is bounded uniformly in ;
(ii) For any 2 ( ; +) with 6= 0, lim !1 y ( ; ) = 0;
(iii) lim !1
R +
( ; ) d = 0, and lim !1
R +
v ( ; ) d = 2L( 1 + 2).
We rst prove some lemmas about the functions ( ; v) de ned by (4.14).
Lemma 6.2 For any xed > 0, v ( ; ) changes sign at most once when < 0
and once when > 0.
Proof: We may assume v (0; ) > 0, since v(0; ) = ( )v0(0) ! 1 as ! 1.
Di erentiating v( ; ) with respect to , we have
v ( ; ) = 0( )v0( )
1 + B2( 0( ))v0( )
: (6.1)
The factor 1 + B2( 0)v0 can be rewritten T( ) = 1 + B2( 0( ))v0( ), whose
behavior depends only on the xed solution ( 0; v0). Let = be a point where
T( ) = 0. Then at = (using B2v0 = 1 and the di erential equation (4.2))
we have
T0( ) = B2v0 + B0
2 0
0v0 + B2v0
0 = v2
0B1B0
2;
and B1 < 0 and B0
2 > 0 imply that sgn T 0 = sgn . In other words, as a function
of , T can go only from positive to negative when > 0 and the other way when
< 0. Thus, either T > 0 everywhere or there are unique values < 0 < + at
which T changes sign.
The function T can be computed explicitly using a standard solver, and changes
sign twice. We de ne ( ) = = ( ) as the points where v changes sign. Now
we prove
Lemma 6.3 The integral
R +( )
( ) jv ( ; )j d is bounded uniformly in .
Proof: From Lemma 6.2, we may drop the absolute value sign, and from (6.1),
for a given > 0, we write
Z +( )
( )
v ( ; ) d =
0( )
( )
Z +( )
( )
v0( ) + v0
0( ) d( ):
Carrying out the integration gives
Z +
v d =
0
v0( )
+
=
0
+ v0( +) v0( )
:
16
Viscous Singular Shocks
Now, from T( ) = 0 the last expression becomes
0
1
B2( 0( +))
+
1
B2( 0( ))
;
which is uniformly bounded in .
We can now prove Theorem 6.1.
Proof: We begin with (ii), pointwise bounds for and v . For any > 0,
0( ) ! 2 and v0( ) ! 1=B2( 2) as ! 1, by Proposition 4.1; so we
have
( ; ) =
0( )
( )
v0( )B1( 0( )) ! 0
as ! 1. We note also that, since v0( ) = v0( ) is bounded, is in
fact uniformly bounded. For v , we use (6.1), multiplying and dividing by and
inserting limits from Proposition 4.1:
v ( ; ) ! L
1
B2( 2)
1 B2( 2)
1
B2( 2)
= 0;
as ! 1. The proof for < 0 is the same.
Now we establish (iii). The formula ( ; ) = 0( ) 0
0 ( ( ) ) = 0 v0B1
gives
sgn ( ; ) = sgn : (6.2)
For any 0 > 0 with 0 < minf ; +g, using (6.2) we have
Z 0
0
d =
0( )
( )
Z 0
0
0
0( ) d
0
0
Z 0
0
0
0( ) d +
Z 0
0
0
0( ) d
=
0
0 [ 0( 0 ) 0( 0 )]
0
0( 2 1):
We may assume that 0= 2L for all . For any > 0, since the integral is
negative, take 0 small enough that
Z 0
0
d
<
2
:
By (ii), given > 0 and 0 > 0 we can nd T > 0 such that, if > T,
Z 0
d
<
4
and
Z +
0
d
<
4
:
Therefore if > T,
Z +
( ; ) d
< :
17
Keyfitz, Sever and Zhang
Now we check the second component of y . As in the proof of Lemma 6.3 we
integrate v :
Z +
v ( ; ) d =
0( )
( )
+v0( +) v0( )
:
By (4.3) and (4.4),
+v0( +) ! 2 2 and v0( ) ! 2 1; (6.3)
as ! 1, while 0= ! L. The second relation in (iii) follows.
Finally, we prove (i). For any we have
Z +
j ( ; )j d =
Z 0
( ; ) d
Z +
0
( ; ) d
=
0( )
( )
Z 0
0
0( ( ) ) d( )
Z +
0
0
0( ( ) ) d( )
2L( 1 2) maxf ; +g:
This is the desired result for . We separate the v integral into its positive and
negative parts:
Z +
v ( ; ) d =
Z
v ( ; ) d +
Z +
v ( ; ) d +
Z +
+
v ( ; ) d :
As we have shown in (iii) that the left side tends to 2L( 1+ 2) while from Lemma
6.3 the middle integral is uniformly bounded, we have uniform bounds on the two
outer integrals for large , and
Z +
jv ( ; )j d =
Z
v ( ; ) d +
Z +
+
v ( ; ) d
!
+
Z +
v ( ; ) d
is also uniformly bounded.
This completes the proof of the theorem.
Corollary 6.4 Let y be the solution yV given by (4.12) with s 6= 0. Then the
relations in Theorem 6.1 hold.
Proof: When s 6= 0, we have
( ; ) = 0
( )
1 es
s
; v( ; ) = ( )es v0
( )
1 es
s
:
As in Section 4.1, let = (1 es )=s. Then
( ; ) = 0
) = ( ; ); v( ; ) = (1 s ) v0
) = (1 s ) v( ; );
18
Viscous Singular Shocks
where ( ; v) are given by the formulas in (4.14). Hence the estimates in Theorem
6.1 apply to ( ; v). Since any points are just mapped to points ( ), while
the factor 1 s is bounded on [ ( ); ( +)], these bounds hold also for ( ; v).
Finally, the nonzero estimate in (iii) is also valid since
R
v ! 0 as ! 1.
Proposition 6.5 When w+ = W+ = ( 2; 0), or w+ = ( 2; v+), and if p1 is
chosen in the stable set
0(w+), independent of , then
R 1
+ jy j d ! 0 as ! 1.
Proof: The only contribution to this integral is from the interval + < < 2 +,
since y = 0 for > 2 +. Since y and y are linear in , from (4.17) we have
Z 2 +
+ jy j d =
+
2
@
@
yV ( +; )
;
and this expression tends to zero as ! 1, by Theorem 6.1, (ii).
We have now established condition (3) of Theorem 3.1.
7 Completion of the Proof of Theorem 3.2
It remains to show that conditions (4) and (5) of Theorem 3.1 hold. Using
Proposition 6.5 and Theorem 6.1, (iii), we have
lim
!1
Z
R
y ( ; ) d = lim
!1
Z +
y ( ; ) d =
0
2L( 1 + 2)
;
and letting L be the nonzero component of e(w;w+; s)=2( 1 + 2) (see equation
(2.3)) yields (4).
Finally, from the construction of y and z, y( ; ) is Lipschitz continuous for
2 R, 1 and approaches constants at 1, so y( ; 1) is of bounded variation.
To show that z( ; ) is of bounded variation uniformly in , we compute (noting
0
0 < 0) the rst component z1 from equation (4.13):
Z +
jz1; j d = s
Z +
0
0( )es d s
Z 1
1
0
0( )
d
;
which tends to zero as ! 1. Proposition 4.2 gives uniformly bounded variation
for z 2 [ +; 2 +], and z is constant for > 2 +. Thus, (5) holds.
Thus we can apply Theorem 3.1 to obtain Theorem 3.2.
In the case of strictly overcompressive shocks, the condition (3.6) holds whenever
the conditions of Theorem 3.1 are met, and we can conclude that both ordinary
(viscosity "wxx) and self-similar (viscosity "twxx) viscous structures exist.
However, in the weakly overcompressive cases of Theorem 3.2, a rough calculation
of the solutions near the attractor w+ indicates that convergence of the integral
of j y j is not rapid enough to give (3.6).
19
Keyfitz, Sever and Zhang
8 The Cauchy Problem with Piecewise Constant Data
Here we take K = 0, G = 0, and use the dependent variable Z given by
Z =
( 2)( 1 )v
; (8.1)
noting that Z uniquely determines the volume fraction and the velocity of any
phase present [6]. The region H (in which all phases present have zero velocity)
corresponds to Z2 = 0.
The entropy Riemann solutions obtained in [6] have the following properties.
(i) All of the intermediate states created are in H.
(ii) An admissible discontinuity separating a region where the state is ZH 2 H
from a region where the state is Z 62 H moves towards the region where
the state is Z. (Points in space change only from state Z to ZH.)
(iii) For a given Z 62 H, there is a positive such that all admissible discontinuities
connecting Z to any point ZH 2 H have speed greater than .
(iv) An admissible discontinuity connecting two states in H has speed zero.
Now consider the Cauchy problem with piecewise constant initial data Z( ; 0),
assuming only a nite number of di erent values and with a locally nite set of
points of discontinuity. Solutions are easily constructed by wave front tracking.
Indeed, in general a plethora of corresponding solutions exists, as at any point
(x; t) where Z is continuous and Z(x; t) 62 H, one may choose the nontrivial
Riemann solution of the trivial Riemann problem.
However, all such solutions have a very simple form.
From (i), it follows that for any t > 0
fZ( ; t)g fZ( ; 0)g [ H: (8.2)
Denote by S(t) the interior points of the set fx j Z(x; t) 2 Hg. Then from (ii)
and (iv), for any x 2 S(t0) it follows that
Z(x; t) = Z(x; t0); 8t > t0: (8.3)
From (8.2), the assumption that Z(:; 0) assumes only a nite number of di erent
values, and (iii), there is a positive such that no admissible discontinuities
of speed less than connect states ZH in H to states not in H.
Assume now that the closure of S(t) is not empty and is not all of R. Then the
boundary of S(t) necessarily contains a point at which an admissible discontinuity
is moving out of S(t) at a speed of at least . So if the complement of S(0) has
nite measure, then the closure of S(t) will be all of R in nite time, independent
20
Viscous Singular Shocks
of which solution is chosen. After this nite time, from (8.3) the solution is
independent of t; from (1.1), recalling that K = 0, we deduce that v vanishes
identically. Thus, this model predicts that if there is a nite amount of mixed
phase
uid initially, it will separate into single phase states in nite time. In
other words, absent any interfacial momentum transfer terms (the balance terms
we have omitted), the mathematical solution correctly predicts that the Bernoulli
e ect will dominate [8].
9 Conclusions
Singular shocks represent a novel type of solution to conservation law systems,
and may provide solutions in situations, such as large data for some hyperbolic
systems as well as the present context, in which classical shock solutions do
not su ce to solve all initial value problems. Because they are not yet well
understood, it is important to apply to them the same tests that have been
used, classically, to establish the admissibility and stability of weak solutions of
conservation law systems.
In this paper we have established viscous structure for singular shocks which
can be used to solve Riemann problems for a nonhyperbolic model system. This
exercise is not based on a physically realistic form of
uid-dynamic viscosity.
Nonetheless, it is an important step in justifying the use of singular shocks to
solve the model problem. In particular, it shows that there is a sense in which
solutions to the ill-posed system (1.1) can be recovered as limits of solutions of
well-posed systems of the form (1.3). Since (1.1) is equivalent to a standard
two-
uid model in widespread use, a better understanding of the mathematical
structure of its solutions will have signi cant application.
Many questions remain. It would be valuable to understand how singular
shocks can be approximated when di erent forms of viscous regularization are
used. This question has not been answered even in the simpler case that the
underlying equation is hyperbolic. In a second direction, critically important to
the application to two-phase
ow, we would like to understand what happens
to singular shock solutions of (1.1) when balance terms are added to the second
equation of (1.1). The technique of this paper suggests adding both viscous perturbations
and balance terms, and seeking scale-invariant or asymptotic solutions.
We are currently studying this problem.
Acknowledgements
It is a pleasure to acknowledge useful conversations with Richard Sanders, and
help from Marty Golubitsky and Steve Schecter on dynamics and viscous pro les.
Finally, we thank Nam Dinh of the Center for Risk Studies and Safety at the
University of California, Santa Barbara, for his encouragement of our pursuit of
nonhyperbolic models.
21
Keyfitz, Sever and Zhang
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