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Some remarks on the Stefan problem with surface structure

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Abstract

This paper discusses a generalized Stefan problem which allows for supercooling and superheating and for capillarity in the interface between phases. Simple solutions are obtained indicating the chief differences between this problem and the classical Stefan problem. A weak formulation of the general problem is given.
QUARTERLY OF APPLIED MATHEMATICS
VOLUME L, NUMBER 2
JUNE 1992, PAGES 291-303
SOME REMARKS ON THE STEFAN PROBLEM
WITH SURFACE STRUCTURE
By
MORTON E. GURTIN and H. METE SONER
Carnegie-Mellon University, Pittsburgh, Pennsylvania
Abstract. This paper discusses a generalized Stefan problem which allows for su-
percooling and superheating and for capillarity in the interface between phases. Sim-
ple solutions are obtained indicating the chief differences between this problem and
the classical Stefan problem. A weak formulation of the general problem is given.
1. Introduction. In this paper we discuss a generalized Stefan problem which allows
for capillarity in the interface between phases. We use the following notation: £2 is
a region in E6' occupied by a two-phase continuum; £2,-(/) is the subregion of £2
occupied by phase i (i = 1,2); j(t) is the interface1 between £2,(/) and Q2(?);
n(x, t) is the unit normal to j(t) directed outward from £2,(0; v(x, t) is the
normal velocity of j(t) in the direction of n(x , t);
L(x, t) = -V^x, t) e lin(n(x, t)±)
is the curvature tensor and k(x , t) — {d - l)_l trace L(x, t) the mean curvature of
j(t); and
f 0 if x e £2. (/),
-f cn , U
I 1 if X G £1,(0 ,
is the phase distribution. Here lin(^) is the set of linear transformations of A into
itself, while is the surface gradient on j(t).
The problem under consideration consists of: the partial differential equations
c.u =-divq, q = -K.V« in phase 1 ,
1 ' (1.2)
c2ut - divq, q =-K2Vw in phase 2;
the free-boundary conditions3
u = B(n) L fi{n)v , Iv = [q] • n on the interface; (1.3)
1 To avoid discussions of contact conditions, we assume throughout that the interface does not touch the
container walls: n{t) n dQ = 0 .
2 V and div will denote the gradient and divergence in . The subscripts t and r denote partial
derivatives.
3 T
For linear transformations A, B of an inner-product space into itself, we write A B = trace(AB )
with Bt the transpose of B .
291
292 M. E. GURTIN and H. M. SONER
the initial conditions
u(x, 0) = U0(x), x(x,0) = x0(x) (1-4)
for x G Q; and suitable boundary conditions and/or conditions at infinity. Here
"in phase i " means in Qt(t) for t > 0 , while "on the interface" means on a(t) for
/ > 0. Modulo appropriate scalings, u is the temperature; q is the heat flux-, [q]
is the jump in q across the interface (phase 2 minus phase 1); ci and K( e lin(R )
are the specific heat and conductivity tensor for phase i (i = 1, 2); / is the latent
heat; B(n) e lin(nx) is the capillarity tensor, /i(n) is the kinetic coefficient, uQ is the
initial temperature-distribution; and Xo is the initial phase-distribution.
In the classical Stefan problem the temperature is strictly negative in the solid and
strictly positive in the liquid, and is generally used to characterize the individual phase
regions. Here we allow positive temperatures in the solid (superheating), negative
temperatures in the liquid (supercooling), and therefore use the phase distribution to
characterize the individual regions.
We assume that
/> 0 (1.5)
which is consistent with
phase 1 = solid, phase 2 = liquid.
When the two phases and the interface are isotropic, (1.2) and (1.3) have the simple
forms c.u, = -divq, q =-k.Vu in the solid,
1 ' (1.6)
c2u( = - divq, q -—k-,Vu in the liquid,
and
u = aK — fiv , Iv = [q]n on the interface, (1-7)
with Kia , p scalar constants.
Important special cases of the general problem are based on the following assump-
tions:
(i) cl c2 0 {fast-diffusion);
(ii) C[=0, K, = 0 (one active phase)-,
(iii) /?(n) = 0 (no surface kinetics)-,
(iv) B(n) = 0 (no capillarity)-,
(v) cx = c2 = kx k2 I = 0 (no bulk structure)-,
with the initial condition for u in phase i omitted when ci 0 . The isotropic prob-
lem without surface kinetics and without capillarity is the classical Stefan problem;
the isotropic one-phase problem with capillarity, but without surface kinetics and
under the assumption of fast diffusion, 4 was introduced by Mullins [3] to explain
the formation of grooves on an interface separating a solid phase from a saturated5
4 These equations also describe the motion of an interface separating immiscible viscous fluids, when
the fluids lie in the narrow gap between parallel plates (Hele-Shae cell) (cf. Saffman and Taylor [2] and
McLean and Saffman [5]).
5 In the papers by Mullins [3] and Yokoyama and Kuroda [18] the underlying transport mechanism is
mass diffusion rather than heat conduction.
THE STEFAN PROBLEM WITH SURFACE STRUCTURE 293
fluid phase; the isotropic problem with two active phases but without surface kinetics
was introduced by Mullins and Sekerka [4] to study the stability of crystal growth; the
anisotropic one-phase problem with fast diffusion and no capillarity was introduced
by Yokoyama and Kuroda [18] to study pattern information in the growth of snow
crystals; the isotropic 6 problem with no bulk structure and u 0 was introduced by
Mullins [ 1 ] to study the motion of grain boundaries; the general problem was derived
within a continuum-thermodynamical framework by Gurtin [10].
In this paper we consider the equations described above in conjunction with the
boundary conditions
u = U on a portion of dQ, q v = 0 on the remainder of dQ, (1.8)
with U constant and u normal to <9Q, so that a portion of the boundary is isother-
mal, the remainder insulated. In addition, we allow Q to be unbounded, but only
for the special case in which
u(x, t) —> U as |x| —► oo (1.9)
in a sufficiently regular manner.
In Sec. 2 we discuss an energy equation associated with the problems described
above. We show that, because of the instabilities associated with supercooling and
superheating, it is plausible to expect solutions in which one phase grows from a seed
of zero volume.
In Sec. 3 we discuss the spherically symmetric problem, without surface kinetics,
under the assumption of fast diffusion, with the liquid supercooled at infinity. We
consider some simple problems that indicate the chief differences between the prob-
lems studied here and the classical Stefan problem. In particular, we show that:7
(i) for Q = R3, a ball of the solid phase of sufficiently small size disappears in
finite time, but a sufficiently large ball grows without bound;
•5
(ii) for Q = M and the solid phase initially situated in a spherical shell of thickness
e , the thickness of the solid shell initially increases, but the inner radius of this region
decreases to zero in finite time T; the solid ball remaining at time T disappears at a
later time or grows without bound according as e is less than or greater than a critical
value; in the limit e 0 the region occupied by the solid disappears infinitely fast;
the problem has no solution for £ = 0 ;8
(iii) when Q is the region exterior to a sphere of radius R, with the boundary
r = R insulated and with the solid phase initially in a spherical shell of zero thickness
at r = R, the solid phase grows without bound provided R is sufficiently large.
6 The anisotropic problem without bulk structure is discussed by Angenent and Gurtin [12].
7 While (ii) and (iii) are probably of little practical interest, they do demonstrate—within a manageable
geometry—the possibility of growth from a seed of zero volume.
gThe one-dimensional theory of a flat interface with and without fast diffusion, but without surface
kinetics, is discussed by Gurtin, Kossioris, and Soner [15]. There, because there is supercooling but no
capillarity, there are solutions in which the solid grows from a set of zero one-dimensional measure. Here
the curvature of the interface brings capillary forces into play, and the solid region must have nonzero
volume to nucleate.
294 M. E. GURTIN and H. M. SONER
In Sec. 4 we use recent ideas9 of Sethian [9] and Osher and Sethian [11] to note
a weak formulation of the system (1.2), (1.3).
2. The free-energy inequality. For a large class of problems of physical interest the
capillarity tensor B(n) is derivable from an interfacial free-energy fin) > 0 in the
sense that (cf. [10])
®(n) = /(n)l(n) + Z)2/(n), (2.1)
2 j_ j_ 2
where l(n),Z) /(n) e lin(n ) with l(n) the identity on n andZ)*"/(n) the second
derivative of /(n) on the surface of the unit ball. In this section we assume that
such a free-energy exists. We assume in addition that both conductivity tensors are
positive definite, while p(n) > 0.
For convenience, we introduce functions c(x , t) and K(x , t) defined by
c = rlc2x + rlci( \-x), K = r1K2/ + r1K1(i-z), (2.2)
and write
vol = tZ-dimensional Lebesque measure in M ,
area = (d - l)-dimensional Hausdorff measure on surfaces in M0'.
Given any temperature field u(x) and any phase distribution x(x) > we define the
free-energy e{u, x) by
e{u, x} = [f{n) + Uvol(Q,) + j [ c(u - U)2. (2.3)
Then any sufficiently regular solution u(x, t), x{x, t) of (1.2), (1.3), (1.8), and (1.9)
(for Q unbounded) satisfies the free-energy inequality (cf. [10, Eq. (7.9)]):
> X} + j(c2 - c,) J{u - U)2v = - J Vm KVu - J/l(n)v2 < 0. (2.4)
The right side of this equation represents energy dissipated by heat conduction and
in the exchange of atoms between phases.
We remark, for future use, that (2.4) is (at least formally) valid when any of
cl,c2, /(n), and (l(n) vanish. Also, when the material is isotropic, /(n) = / =
a/(d - 1) and
/(n) = / area(^). (2.5)
By (2.4), if c, = c2 (which holds trivially in the case of fast diffusion), then
the free-energy (2.3) is nonincreasing. (2.6)
Supercooled liquids are inherently unstable; as we shall see, this instability allows
for the spontaneous formation of the solid 10 phase from a seed of zero volume. To
study this phenomenon, we consider problems with Q initially almost all liquid in
the sense that
/(x,0) = l for a.e jc e O. (2.7)
9 Cf. also Barles [8], Chen, Giga, and Goto [13], Evans and Spruck [14], and Soner [17],
10 The problem for a liquid seed is strictly analogous: in the ensuing discussions simply replace the word
solid by liquid, the boundary temperature U by U , and the word supercooled by superheated.
THE STEFAN PROBLEM WITH SURFACE STRUCTURE 295
Assume that the diffusion is fast, so that c} = c2 — 0. Then, as an immedi-
ate consequence of the free-energy inequality (2.4), a solid seed cannot grow when
U > 0, but such growth is plausible when U < 0. In this latter case, when the bound-
ary of the liquid phase intersects the portion of the boundary on which u = U, at
least some of the liquid must be supercooled; thus, in general, supercooling of the
liquid is at least formally necessary for the spontaneous growth of a solid seed from
a set of zero volume.
Thus suppose that
U < 0. (2.8)
Then the liquid equilibrium
u(x, t) = U, x(x, t) = 1 (2.9)
is a solution of the basic equations and boundary conditions, but, since U < 0, the
solid equilibrium
u(x, t) = U, x(x,t) = 0 (2.10)
has lower free-energy (2.3): the presence of supercooling renders the solid phase more
stable than the supercooled liquid phase. In fact, the solid equilibrium globally min-
imizes the free-energy. In the absence of interfacial free-energy, the free-energy is
—\U\ vol(Qj) and is hence lowered by any solution in which the solid phase grows in
volume from its initial distribution with zero volume. Interfacial free-energy is sta-
bilizing: in its presence the solid phase cannot grow in volume from zero volume and
zero surface area; such solutions violate (2.4), since they have area(j(t))/vol(fi,(?))
unbounded as t -+ 0+ . On the other hand, it is possible to construct sets which
grow from zero volume but nonzero surface area and which initially lower the free-
energy. Thus, granted a supercooled boundary, spontaneous growth of a solid seed of
zero volume but suitable shape is to be expected. In the next section we will exhibit
solutions which display this phenomenon.
3. The spherically symmetric problem with fast diffusion and without surface kinet-
ics, with the liquid supercooled. The effects of capillarity are most easily discussed
under the assumptions of isotropy, fast diffusion, negligible surface kinetics, and
equal conductivities for the two phases:
c, = c2 = /? = 0, k] = k2. (3.1)
As a further simplification, we restrict our attention to problems involving spherical
symmetry with the underlying space M3, and therefore seek solutions of the form
u(r, t), xXr, t) with r \x\. The partial differential equations (1.6) then reduce to
un + 2r lur = 0 in the solid and in the liquid, (3.2)
while the free-boundary conditions (1.7) at a spherical surface of radius C(0 take
the form
«(f(o,o = ±"C(o-1, c'(t) = urm],t)-urm2,t), (3.3)
where, for convenience, we have chosen a length scale with kjl = k2/l = 1 . Here
C{t) i indicates the limit as C(0 is approached from the solid phase, and similarly
296 M. E. GURTIN and H. M. SONER
for C,{t)2 , and the plus or minus sign is chosen according as the material is liquid or
solid for r close to CM with r < £(0 . We will discuss solutions of (3.2), (3.3) for
Q all of R3 with
u{r,t)-*U<0 as r —► oo; (3.4)
or for Q the region exterior to a sphere with
ur = 0 ondQ, u(r, t) —» U < 0 as r —> oc, (3.5)
so that the inner spherical boundary is insulated. In addition, we will always have
the exterior phase region (the phase region containing the point at infinity) liquid, so
that, by (3.4) or (3.5), the liquid is supercooled for all sufficiently large r.
Remarks. (3a) The general solution of (3.2) is B(t) + G(t)r~l ; thus the tem-
perature must be spatially constant in any phase region that contains the origin, or,
when Q is the region exterior to a sphere, in any phase region that contains the inner
boundary. On the other hand, the temperature must have the form U + G(t)r~x in
the exterior phase region.
(3b) Since ut(x, t) does not enter (3.2)-(3.5), solutions u(x, t) need not be
continuous in t, although the continuity of £(/) does impose some regularity.
3.1. Growth of the solid phase from a ball. Let Q = M3, and suppose that initially
the solid occupies a ball of radius R , with the region exterior to R liquid. Then, by
(3.3) and Remark (3a), the interfacial radius £(0 satisfies the initial-value problem:
C'= -(UC + a)/t:2, t(0) = R. (3.6)
Thus U < 0 is necessary for the growth of the solid phase. Granted this,
£(t) —* 0 in finite time if R < a/\U\,
C(?) —* oo as/—»oo if/?> a/\U\,
C{t) = R if R = a/\U\. (3.7)
Therefore, for growth of a ball of the solid phase, it is necessary and sufficient that
the liquid be supercooled and that the initial radius of the ball be larger than a/\U\.
Let R < a/\U\, and let T denote the time at which the solid phase disappears.
Then for t > T the solution consists of the liquid equilibrium (2.9) (cf. Remark
(3b)).
Remark (3c). This discussion applies almost without change for Q a ball of radius
R0, u — U on dQ, and R < R0, the only difference occurring when R > a/\U\.
In this case £(t) —► R0 in finite time T(j (cf. (3.7)2), so that for t > T0 the solution
is the solid equilibrium (2.10).
3.2. Growth from a set of arbitrarily small volume. Again let £2 = K* . We choose
an arbitrary point (0 e (0,1) and seek a solution in which the solid phase is a
spherical shell p(t) < r < £(t) with
p(0) = pQ, C(0) = />0 + £, (3.8)
with e > 0 small. In view of (3.3) and Remark (3a), finding such a solution reduces
THE STEFAN PROBLEM WITH SURFACE STRUCTURE 297
quid
Fig. 1. The phase portrait for a solid shell of inner radius p and
outer radius £ growing from the shell p(0) = p0 , f(0) = p0 + e .
The inner ball of liquid disappears in finite time leaving a solid ball
surrounded by liquid. This solid ball disappears in finite time or grows
without bound according as e is less than or greater than £0 .
to solving the differential equations
P =-a(P + C)/p2(C- P)> (3 9)
C, = -{2a + U(C-p)}/C(C-p)
subject to (3.8) and the constraint
0 < p{t) < C(0- (3.10)
Equations (3.9) are not defined at f = p. In fact, we will show at the end of the
section that there is no solution of (3.9), (3.10) for t > 0 that satisfies the initial
condition (3.8) with e = 0.
The phase diagram for (3.9) is shown in Fig. 1. The system (3.8)—(3.10) has a
solution for each e > 0, and each such solution has p(t) —> 0 in a finite time T(e).
At t = T(e) the inner liquid-phase disappears, and we are left with a solid of radius
R = C(T(e)) surrounded by liquid. Thus for t > T(e) the solution is given by the
solution of (3.6) with the initial condition applied at time T(e) (cf. Remark (3b)).
Therefore, by (3.7) and Fig. 1, for t > T(e) the solid ball disappears in finite time
T*(e) or grows without bound according as s is less than or greater than a critical
value. For e small, the solution for t > T*(e) is the liquid equilibrium (2.9).
Consider the initial time-interval t < T(e). Letting 8 = C, - p, we may use (3.9)
to conclude that
6' = {a(C2 + pC~ 2P2) - Up2S}/(di;p2), (3.11)
and therefore, even though £ decreases, the distance S between the interfaces in-
creases.
298 M. E. GURTIN and H. M. SONER
The limit e —» 0 is interesting. We now show that, in this limit, the region occupied
by the solid disappears infinitely fast-, precisely,
r*(e)-0 as e —> 0. (3.12)
To prove (3.12), choose p0 > 0 and e > 0, and let (pg, C£) denote the solu-
tion of (3.8)—(3.10) on [0, r(e)], with extended continuously via (3.6), to
[T(e), r*(e)]. Further, for R e (0, p0), let
T(s, R) = inf{* > 0 : ps(t) = R}
denote the time at which the inner radius reaches R. Our first step will be to show
that for pQ > 0 and R e (0, p0) fixed,
T(s,R)-> 0 as e —► 0. (3.13)
Fix e > 0 and suppress the subscript e . To verify (3.13), let Q = 5_1 ; then (3.9)
has the form
p' = -a{l+2pQ)/p2, (3H)
Q' = -Q2{a( 1 + 3 pQ) + \U\p2Q}/p\l + pQ).
Let t be confined to the interval [0, T(e, /?)]. Then, since (1 + 3pQ)/( \ + pQ) < 3
for all pQ > 0, and since p(t) > R,
Q'>~CQ2, C = {3a + \U\R)/R2.
Further, Q(0) = e 1 ; thus
Q(t)>(e + Ct) (3.15)
On the other hand, (3.14), yields
p\t) < -2aQ(t)lp(t) < -2a/[p0{e + Ct)];
hence
Pit) < Pq~ (2a/PqC) ln[l + (Ct/e)].
Let S{e, R) denote the time at which the right side of this inequality reaches the
value R . Then S{e, R) —* 0 as £ —> 0 . But T{e, R) < S(e, R); thus (3.13) is
satisfied.
Next, (3.14), yields
(p2)' - —4aQ 2a/p > —4aQ 2a/R.
Thus, since R = p{T(e, R)),
•T(s,R)
2 2 f M£'K)
R > P0 - (2a/R)T(e, R) - 4a / Q(s)ds.
J o
Similarly, by (3.9)2,
(3.16)
(C2)' = —4aQ + 2\U\,
so that
t2it) = ip0 + e)2 -4a I Q(s)ds + 2\U\t.
f
Jo
THE STEFAN PROBLEM WITH SURFACE STRUCTURE 299
Thus, by (3.16),
f2(J,
which allows us to choose e0 = e0(R) small enough that
C2(T(e, R)) < (pQ + s)2 - pi + R2 + 2R~\a + \U\R)T(e, R)
C(T(e,R))<2R (3.17)
for e e (0, e0).
We now consider times t e [T(e, R), T*(e)]. The function £(t) satisfies (3.9)2
on [T(s, R), r(e)] and (3.6), on [7(e), r*(e)];thus
(C2)'<2|t/|-(2Q/C)
on [r(e, R), T*(e)]. We now require that R be small enough that
R<a/A\U\\ (3.18)
then, by (3.17) the right side of the above inequality is negative. Hence £ is nonin-
creasing on [T(e, R), T*(e)] and, by (3.17) and (3.18), (C2)' < -a/2R, so that
C2(t)<4R2 ~(a/2R)[t-T(e,R)]
for t £ [T(e, R), r*(e)]. Consequently
T*(e) < T(e, R) + 8R*/a.
Letting e -+ 0 and then R —> 0, we arrive at (3.12).
We conclude this section by showing that the problem (3.8)—(3.10) with e = 0 has
no solution. Suppose there were a solution. Choose a time tQ > 0 and an R e (0, p0)
such that the solution exists on [0, ;0] and p(t) e (R, p0] on [0, ?0]. Then the steps
leading to (3.15) here imply the existence of constants C0 and C such that
Q(t)>(C0 + Ct)-1.
Further, since <5(0) = 0, we must have C0 = 0 . Thus (3.14), yields
p\t) < -2aQ{t)/p{t) < -2a/pQCt,
and, for t (0, /0),
P{t0) < p(t) - (2a/p0C) In(t0/t).
Letting t 0 yields a contradiction, since /?(0+) = p0 Thus the problem with
£ = 0 has no solution.
3.3. Region exterior to a sphere with inner boundary insulated: growth from a set
of zero volume. This problem furnishes an example of growth from a set of zero
volume. Here Q is the region in M3 exterior to a sphere of radius R, and the
"boundary conditions" are (3.5), so that the inner boundary is insulated, while the
boundary at infinity is supercooled. We seek a solution in which the solid phase is a
spherical shell R < r < £(t) for t > 0 with
m = R, (3.19)
so that initially the solid phase has zero volume.
300 M. E. GURTIN and H. M. SONER
In view of Remark (3a), the current problem—(3.2), (3.3), (3.5), and (3.19)—
leads to (3.6), and we may therefore conclude that for R > a/\U\ there is a solution
in which the solid phase grows without bound. For R < a/\U\ there is no solution
with the solid phase a spherical shell R < r < £(t) for t > 0; for this range of R
the solid phase disappears immediately, and the solution is the liquid equilibrium
(2.9). For R = a/\U\ there are two possibilities: the liquid equilibrium; the formal
solution in which the solid phase remains in a thin spherical shell of zero thickness
at r — R. The second solution has larger free-energy (2.3) than the first, since the
solid-liquid interface has interfacial free-energy (2.5).
3.4. Some general remarks. In the problems of the last three subsections, the
volume vol(Q,(/)) of the solid phase is continuous, and so is the interfacial area,
area(^(f)), except when the interface reaches the boundary, as in Remark (3c). (We
define area(^(?)) = 0 for the solid and liquid equilibria.)
The problem discussed in Sec. 3.3 consists in solving (3.2)-(3.4) for t > 0 in con-
junction with the initial condition: /{r, 0) = 1 , r / £0 , x(C0, 0) = 0, a condition
which represents an infinitesimally thin spherical shell of the solid phase situated at
r = C0, so that, formally, area(<*(0)) ^ 0. There are at least two solutions of this
problem: the solution given in Sec. 3.2 and the liquid equilibrium (2.10). For the
latter, area(a(t)) is discontinuous at t - 0, but area(<*(f)) is discontinuous also in
the solution discussed in Remark (3c). In addition, the liquid equilibrium lowers the
free-energy as the two interfaces at r = £0 disappear.
Given a solution, a time tQ , and a radius £0 in the liquid region at tQ , one might
ask whether it is reasonable to consider a second solution in which a solid region of
zero thickness at r = C0 forms spontaneously at t0 . Such a solution is not acceptable
physically, since it raises the total free-energy at time tQ by an amount equal to the
interfacial free-energy of the two interfaces at r = f0 . In fact, a physical requirement
for any solution is that areanot suffer a positive jump discontinuity.
3.5 Reduction to a Stefan problem. Ostensibly, one expects the problem with cap-
illarity to be more stable than the Stefan problem, but, interestingly, the former may
be reduced to the latter by a change in dependent variable, at least when the interface
consists of a single spherical surface of radius C(t).
Assume first that Q does not contain the origin and define a new dependent
variable
w(r, /) = u(r, t) ± ar ', (3.20)
where the plus or minus sign is chosen according as the material is solid or liquid for
r close to C(0 with r < C(0 . Then w(r, t) satisfies
wrr + 2r~lwr = 0 in the solid and in the liquid, (3.21)
and
w(C(t),t) = 0, C'(0 = "r(C(01,0-"r(C(02,0, (3.22)
which are exactly the free-boundary conditions of the Stefan theory.
When Q contains the origin, we replace (4.15) by
w(r, t) = u(r, t) ± f(r), (3.23)
(4.1a)
THE STEFAN PROBLEM WITH SURFACE STRUCTURE 301
where f(r) is smooth for all r and equal to ar~x outside a ball T of sufficiently
small radius. Then—provided the interface lies outside of T—the free-boundary
conditions (3.22) remain unchanged; on the other hand, the differential equation
(3.21) will have a nonzero right-hand side within T.
The transformations (3.20) and (3.23) apply also to the problem without the as-
sumption of fast diffusion, in which case (3.22) remains unchanged, while (3.21) has,
on the right-hand side, the term c{ut in the solid and the term c2ut in the liquid.
4. A weak formulation of the general problem. The general problem associated with
(1.2), (1.3) is extremely difficult,11 chiefly because of the presence of curvature and
normal velocity in the free-boundary condition. In this section we suggest a possible
weak formulation, assuming, for convenience, that Q = to avoid discussing
boundary conditions.
Guided by the physics, we divide the basic equations into two parts:
(i) the energy equation:
c,Mr = -divq, q = -K,Vm in the solid,
c2ut — - divq, q =-K2Vm in the liquid,
Iv = [q] • n on the interface; (1.4b)
(ii) the interface condition:
u = B(n) ■ L - 0(n)v on the interface. (4.2)
The energy equations (4.1) are consequences of balance of energy and the bulk consti-
tutive equations, the jump condition (4.1b) arising because of constitutive differences
between the bulk phases; the interface condition (4.2) is a consequence of the con-
stitutive equations for the interface in conjunction with thermodynamic arguments
(cf. [10]).
In the classical Stefan problem, the solid and liquid are defined by u < 0 and
u > 0, respectively, and the bulk energy is given by c,w in the solid and I + c2u in
the liquid. In fact, defining a global energy e(u) and a global heat flux q(Vw) by
e(u) = clu + h{u)[l + {c2-cl)u],
q(Vn) = -[K1VM + A(w)(K2-K1)VM],
where h is the heaviside function, the energy equations (4.1) are formally equivalent
to the single equation
e(u)t = - divq(Vw) (4.4)
to be interpreted in the sense of distributions. The standard weak formulation of the
classical Stefan problem is based on (4.4).
The presence of supercooling and/or superheating—which is necessary if the tem-
perature at the interface is to satisfy (4.2) rather than u = 0—render (4.3) inappli-
cable. In this case, motivated by (4.3), we use the phase distribution / to define the
" Luckhaus [16] establishes global existence and lack of uniqueness for certain weak solutions of the
isotropic problem with equal heat capacities. Duchon and Robert [7] establish local existence and unique-
ness for the isotropic one-phase problem with fast diffusion.
302 M. E. GURTIN and H. M. SONER
global energy e(u, x) and the global heat flux q(Vw, x) through
(4.5)
e(u,x) = cxu + x[l + (c2 -cx)u\,
q(Vu,x) = -[KlVu + x(K2-Kl)Vu]
and this formally reduces the energy equations to the distributional equation
e(u, x), = -divq(Vw, x)- (4.6)
Our next step is to write a weak form for the interface condition. Here we use an
idea due to Sethian [9] and Osher and Sethian [11], who note that equations of the
form (4.2) (with u interpreted as data) arise automatically as equations satisfied by
the level surface of solutions of certain partial differential equations defined on all
of K3.
To apply the Osher-Sethian procedure to (4.2), we define b{\) and B(v) e lin(R3)
for any vector v / 0 by
b(\) = p( w), B(v) = I(w)B(w)P(w), w = |vf'v, (4.7)
3 _L _j_
where P(w) is the projection of K onto w , while I(w) is the inclusion of w
into R3. Next, we note that, given a function <p
n = V?>/|V?»|, v = -<pJ\V(p\ (4.8)
give the unit normal and normal velocity of level surfaces of (p , with n pointing in
the direction of increasing <p , and further
L= |Vfl»|-1P(n)VVpI(n) (4.9)
is the corresponding curvature tensor. Thus, formally, the level surfaces of any func-
tion (p{x, t) consistent with the equation
b{S7(p)(pt = B(Vip) Wtp + \Vtp\u (4.10)
will evolve according to the interface condition (4.2). This allows us to replace the
interface condition (4.2) by (4.10), with the interface the zero-level surface of (p and
the region occupied by the solid the set on which (p < 0 . An advantage of considering
the interface in this manner is that (4.10) admits a weak interpretation in the viscosity
12
sense;
We art
X(x, t):
sense-,12 that is, as a viscosity solution in the sense of Crandall and Lions [6],
We are therefore led to formally consider the following system 13 for u(x, t) and
e(u, x), = -divq(Vw, x) the sense of distributions,
b{Vtp)(pt — B(Vip) • W<p + \Vcp\u in the viscosity sense,
12 Cf. Chen, Giga, and Goto [13], Evans and Spruck [14], and Soner [17], who use viscosity solution of
equations of the form (4.12) to study surfaces that evolve according to equations of the form (4.2) ( u
prescribed).
13 This formulation was discovered independently by Sethian and Strain and presented by Strain at the
SIAM meeting, Chicago, July 1990. The formulation was also noted independently by Noel Walkington
(private communication).
THE STEFAN PROBLEM WITH SURFACE STRUCTURE 303
with e(u, x) and q(Vw, x) defined by (4.5) and
0 whenever <p(x, t) < 0,
X(x, t) {? whenever <p{x, t) > 0.
Acknowledgment. We gratefully acknowledge the Army Research Office and the
National Science Foundation for their support of this research.
References
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containing a more viscous liquid, Proc. Roy. Soc. London Ser. A 245, 312-329 (1958)
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capillary balance law, Arch. Rational Mech. Anal. 104, 195-221 (1988)
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on Hamilton-J acobi formulations, J. Comput. Phys. 79, 12-49 (1988)
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of an isothermal interface, Arch. Rational Mech. Anal. 108, 323-391 (1989)
[13] Y. -G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized
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temperature, European J. Appl. Math. 1, 101-111 (1990)
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... We base on a well–known generalization of the classical Stefan problem, cf. [2, 21, 23, 25, 38] . In order to fix notations, we recall the two–dimensional case of this model. ...
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