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370

Corollary 2.1 Fori el, xe dPj\dtt, we have \x\ > Ri-{1 - h

•

)

1/2

Another technical result which will be useful is

Lemma 2.3 Let i el, I e {K

{

+ 1, ..., r,}, j e {1, 2}, rj

K<-

Then

| MvT

•

(w(i?)

•

(ri))

| < diamKt

•

(8/%/l5)

•

| D

fa) |.

Proof:

Abbreviate a<

> := a^(i,l) for it G {1, ..., 4}. Then Djtpi{j)) = a® - a<

'. Since

oP-\ a*

', a'

' €

it follows (o« + a<

•

{r]) = 0. Thus

\\Vi(r,)\-

-(w(r))-D

(

))\

= \{Wi{v)\-'• Vi{v) ~ \aM+

W\-

-{a®+

W)).D

{T))\

< 4

•

| (1/2)

•

(a« + o

(3)

) - Mv)

I • I

(j)

+ a

(3)

•

| D

(

) \.

But | (1/2)

•

(a«> + a<

>) -

(pi{q)

| < diamKi, and by Corollary 2.1

(1/2)

• |

aP +

a™ |

> Ri-(1 -

-S-

f'\

Since /ij < 5/4 (see (A3)), we obtain Lemma 2.3 by combining the preceding estimates,

Lemma 2.3 allows us to estimate n^

°<Pi —

^ o

and gf'

—

Lemma 2.4 There is C > 0 such that for i, I,

as in Lemma 2.3, we have

(ft)

(wfa)) -

»><**>(

ifcfo))

< <7-fc.

Proof:

Take i, /, r\ as in Lemma 2.3. Put v := I'M

?)!

-1

fii

))-,

d for j € {1, 2},

««:= |o«(<

)

0-aW(*,Z)|-

-(a05(i

Z)-oW(t,0) = I^Wr'-^^W-

Let n denote the unit vector in K

satisfying two properties: first, n-aW = 0 for j e {1, 2},

and second, there is some e > 0 with t-n+ipi(rj) $ Ki for t

(0,

e).

Then v =

i>{ipi{r]))

and n = n^

\tpi{rj)), and there are S\, 8

, S

e K with u =

<5i •

' + S

•

' + S

-n. We

have

= n

•

v > 0, \v - n\ = y/2

•

(1 -

5z)

112

and

1 _ 5\ = |J

(D + S

•

= (6

•

a^ + 5

W). „

< Cl^xl -I- |<5

•

(Vl5

•

S-)-

where the last inequality follows with Lemma 2.3 and (A3). We further find with Lemma

2.1:

1 = \

> S

+ 8

-2-\5

-S

\-a

+ 8

> (l-ff2)-(<5i + (5

)72 +

Lemma 2.4 follows from the preceding estimates. o

Lemma 2.5 Let i el, I G

{K*

+ 1, ..., rj,

G K^. Assume in addition that

hi < (y/Tb/16)

•

(1 - alfl

, (2.4)

with 02 from Lemma 2.1. Then

Ml) ~ 9i(v)

< (16

/15

)

•

9l(v)

•

S-

•

(1 - a

371

Proof:

Abbreviate 7 := |<ft(7?)|

-1

<Pi{v),

•= oP\i,l) for j e {1, 2, 3}. Put

«) _

(3) = £>

j(/3

,(r;) for j e {1, 2}. Then

DjMv) = l^iWI"

•

- (7

•

) -7) for j £ {1, 2}.

Using this equation, we get after some computations

«4fo)

= .lwfo)r

-tf-(ftfa)

-*fo)), (2.5)

where 17(77)

defined by a{rj) := | (7

•

) — (7

•

) |

. We conclude with

(2.5):

lflifa)-nfa)l = 9i(v)-\l ~ (Ri/\fim)

(2.6)

+ Wlwfa)l)

-(l - (1 " <r(v)-9,(v)-

)

1/2

)\-

Dn the other hand, we have by Lemma 2.1

p,(„)2 = |

«|

.[l - (|a^r

-|aWr

-(o

(1)

-a

(2)

))

]

> |

.|a

(2)

-(l-*

md we may deduce from Lemma 2.3 and (A3)

a{rj) < (16

/15)-i?-

-(diamA-,)

-|a

(1)

-|a

(2)

r < (16

/15)

•

(hi/S)

•

\a^\

•

|a<

Therefore assumption (2.4) on ft, implies

(1 - aW-gtiv)-

)

> 1 - a(

)

•

(v)-

> 1- (1 - a

•

(16

/15)

•

> 0.

Jsing this estimate, recalling that

\ipi(77)|

> 1 — h

•

> 15/16 (see Corollary

l.l),

and noting that Ri/\

tp(rf)

| > 1, we may now deduce from (2.6):

9i{v) ~ Ml)

< ftfo)

•

max{ (1 - h

•

- 1,

(1 - h\

•

(1 - a

•

(16

/15)

•

S-

}

< gi(r,) •/i

•

max{ 16/15, (16

/15

)

•

(1 - a

}

)-(16

/15

)-/

-5-

.(l-a

^et us state some consequences of the preceding lemma.

Corollary 2.2 Put S

:= (15/128)

•

(1 -

af)

Then,

if i € 1 with h

< S

, I 6

{K;

+ 1, ..., T;}, 77 € A"'

', we have Ui{rf)/2 <

»(«?) < 2-^C?)-

Uorollary 2.2 yields vi{rj) > gi(r])/2 > 0. This, in turn, implies

Dorollary 2.3 In the situation of Corollary 2.2, the vectors Diipiir)),

D2ipi(i]),

with

tpi

/>,

, are linearly independent.

372

Thus

ipi

is a local parameter of dB^, for I €

{K;

4-1, ..., r;}, i e 1, hence by (2.2),

hdo

= JT J

hdo

= £

(h°<l>i)(.ri)-Ml)dTi

(2.7)

for i e X with h, < S

, h e

V-idB^).

Corollary 2.4 There is a constant C > 0 with the ensuing properties:

Let G e Hlffi), H e Hlffi)

, £,£,£e (0,oo),tel with

hi<S

, \G{t-x)\ + \H{t-x)\<£, | G{x) - G{t

•

x) \ +

H{x) - H(t

•

x) | < £,

\G{x) + H{x)-n

{BR

)

\<£ forx£dB

, t e [(1 - h

•

)

1/2

1].

• n<«>.

Abbreviate a

:= G

dB^ + (H | dB^)

•

h:=G\ dP^dQ + {H

dP^dQ)

Then

|k||

Whh

C-((£-1)

(£ +

~£)

•

(£

•

+£ +

(£ +

£). ^)

•

Ri-

Proof:

Let I e {K

{

+ 1, ..., rj, n G K&\ and put t :=

|<A(??)|/.RJ.

Then

1^,(77)

| =

Ri,

ip,{ri)

= t

•

ip{n). Sincej>,(r)) e dPi\dQ, Corollary 2.1 yields te[(l-

)

1/2

1].

Moreover, for 9, g e E, /i, /i, n, n e E

, we have

\(g + h-n)

-(g +

h-n)

< 2

•

\g + h

•

\g + h

•

n - g - h

•

n\ + (g + h-n-g~h- n)

Starting from these observations, we may deduce Corollary 2.4 from (2.3), (2.7), Lemma

2.4, Lemma 2.5 and Corollary 2.2. o

Proof of Theorem 1.1

Let i el The space W* defined in Theorem 1.1 is a Hilbert space with respect to the

inner product

DiUk

•

DjVk dx + R7

•

We denote the corresponding norm by | |W, that is,

(u,v)

{i)

:= / Y\ Djiik-DjVkdx + ft

• / u-vdo

(u,veW

I"I

= ( / £

(•

^*)

da;

-IN

lis)

« e

Wi.

Theorem 3.1 For A C R

, Zef 7

i(A) denote the set of polynomials over A with degree

less than or equal to 1, and put |w|o,2 := HHI2 f

0T w e

^{Af, |w|i,2 := (H-Di^H

. +

\\D

w\\

\\D

w\\

)

1/2

for w e tf

(A)'

. Set

K:={»eC°(H)

wlif,

e^i(if,) /or 1 < I < n } for iel.

373

Suppose there is C\ > 0 and for each i el, a linear operator IIj : i7

(Pj)

i-» V

such that

(^(diamKi)

^- \(w -n

(w))\K

)

1/2

< C

•

|«,|

1>2

(3.1)

1=1

/or

e iJ^Pj)

{0,1}.

Then there is a constant C

> 0 with

1/2

-\\(w- Uiiw)) | dPi\dQ ||

< C-\w |

lj2

for i eT, we

{Pif.

Proof:

Let i e I, u; e

.H"

I e {«; + 1, ..., n}. We shall write C for constants

which do not depend on i, w or I. For brevity, we set v := w

—

IT,(iy). Recalling that

(

(?7,

— r]i — 772)

<Pi{n)

for

e AT

(2)

(see Section 2), we get

L\v(<Pi(ri))\

-9i(v)dr] < C

•

(dmmK

)

•

[ \v(i

(

))\

< C-(diamJif,)

-|l«

^i|^

(3>

ll?

where the last inequality follows by a standard trace theorem on K^. Using the transfor-

mation rule and recalling the standard estimate

detAf

C •

(diamAT; )~

, we conclude

\v(<Pi(.v))\

•

9i(v) dr)

< C-diamX

^((diamA'

(

-k(2/)|

+ \Vv{y)\

)dy.

But diumKi < h

(

•

Ri/S < Ri by (A3), so Theorem 3.1 is implied by (2.3) and (3.1). o

We need some further technical results related to the outer part <9Pj\9Q of the bound-

ary of P^

Theorem 3.2 There is a constant C > 0 with

|| w

< C

• I

w |

(i)

for i el with h

< S

, we

{Pif,

where S

was introduced in Corollary 2.2.

Proof:

Take i € K. with h

(

< So, we

^(Pi).

Recall the definition of the patches

Fi = Fi and T; = T, on dB^ and 9P,\9f2, respectively, and of their local parameters

fi =

<Pi

ipi

, respectively

(K;

+ 1 < I < Tj); see Section 2. Then R~

•

i*} is a

subset of dB\, and

(tp,

o ipf^iRi -y)eTtC dPj\dtt for y e R;

•

and for l 6 {m + 1, ..., n}.

For such I, we set

r, := {r

•

y : y e R;

•

, r e K with S < r <

(

o^f

•

j,) |}.

Then we have by (A5)

u{r7 :

Ki +

i<i<

} = Pi\Bs, r,nr

for k,ie

{«*

+ !,..., rj, k^i.

374

Thus,

in view of (2.2) and (2.3), it suffices to show for / G {«i + 1, ..., r,} :

/ \w\

< C-( f \Vw\

dx + R'

• f \w\

). (3.2)

Jis/RtyF, Wr, JT, I

Here and in the rest of this

proof,

the letter C denotes constants only depending on S. In

order to prove (3.2), we fix I e {/c* + 1, ...,

TJ}.

Then we find for y € R~

•

Ft, with the

abbreviation 7 := |

((pi

o •0

)(i?

•

y) | :

\w(S-y)\

= \w(S-y) - w(j

•

y) + w(-y-y)\ = | / Vw(r

•

y dr + w(^

•

y) |

< (J\-

dr)

1/2

-(f\

-\Vw(r-y)\

dr)

1/2

\w^-y)\,

hence

/ S

-\w(S-y)\

< (S-T7-1 r

-|V™(r-y)|

dr + (S/Ri)

-A),

with

A := / flM «/(I (w o ^

){Ri

•

I •

y) |

It follows

*>*

< c-(

/" |v™|

<fa

-A).

J{S/Ri)-F, Wr, • >

The mapping R^

•

ipi

is a local parameter of .R,"

•

, with

•

as element of surface

area. Thus, taking into account (2.3), Corollary 2.2 and the definition of

ipi,

we get

= /~,J™(<M

?))|

-^('?)*? <

- L,J™(<Pi(v))\

-9i{'n)dr]

JKW JKW

2- J

\w(y)\

This estimate completes the proof of (3.2), and hence of Theorem 3.2. o

Theorem 3.3 There is C > 0 such that for i € X with hi < So, w €

^/^)

we have

112

-\\W\ dP^dQ ||

< C

•

(|| Vw ||

+ i?r

1/2

•

IM dBn, ||

Proof:

Theorem 3.3 follows by arguments analogous to those used in the proof of

Theorem 3.2. We omit the details. o

We note two consequences of the preceding theorems.

Theorem 3.4 There is C > 0 with || w | B

\^ll

< C

•

w \

{i}

for i e 1 with h

{

So,

w e Wi.

Proof:

According to [1, Lemma 4.1], there is some C > 0 with

IMI2 < C

•

(|| Vt; ||

+ S-

1/2

•

|| v

) for v e ^(B^H)

with v\dQ = 0.

Theorem 3.4 follows from this result and Theorem 3.2. o

375

Theorem 3.5 There is a constant C > 0 and for any i € I with h

< So, an operator

Vi-. L

{Pi) ^W

{

such that divV^-K) = v, |X>i(7r) |

(i)

< C

•

||TT

for

{P{).

Proof:

According to [1, Theorem 4.1], there is a constant C > 0 and for any i G I, an

operator Vi : L

)

>->

^^)

such that we have for g

(Q^) :

div^(g) =

Vi(

) |

an =

VV^Q)

+ R'

l/2

• ||

Ha)

OB*

c-\\

Thus,

if we set Vi{g) :=

T>i{g)

| Pi for i G I, g G

(Pi),

where g denotes the trivial

extension of g to Q^, then Theorem 3.5 readily follows from Theorem 3.3. o

Now we are in a position to carry out the

Proof of Theorem 1.1: Take i e I with h

< So- Let C > 0 denote constants in

(0,

oo) which only depend on Q, 5 and the parameter a

from Lemma 2.1. We have

(i)

)

< C-

(v,v), \a{v,w)\ < C-M«-M

for v, we H

{Pi)

(3.3)

Moreover, by Theorem 3.5, the pair of spaces (Wi,Z/

(P;)) satisfies the Babushka-Brezzi

condition, uniformly in i G X. Due to these results, we may apply the theory of mixed

variational problems, as presented in [7, p. 57-61]. It follows there is a pair (vi,gi) €

)

x L

(Pj) satisfying (1.6). The first inequality in (3.3) and a partial integration

which exploits (1.1) yield

|| V(« | Pi - v)\\

+ R]

•

|| (u - v) | dPi\dtt ||

(3.4)

< C-P

1/2

-||£R

un)(«,7r)||

;

compare the proof of [1, Theorem 5.1]. Using Theorem 3.5, we may deduce from (3.4)

that the term ||

| P

—

g ||

is also bounded by the right-hand side of (3.4). Details of this

argument can again be found in the proof of

[1,

Theorem 5.1]. Due to (3.4) and Theorem

3.4, the term ||(u

—

| Q

, too, may be seen to be bounded by the right-hand side of

(3.4).

Now inequality (1.7) follows with Corollary 2.4. o

References

[1] Deuring, P.: Finite element methods for the Stokes system in three-dimensional

exterior domains. Math. Methods Appl. Sci. 20 (1997), 245-269.

[2] Deuring, P.: Numerical tests of an artificial boundary condition for exterior Stokes

flow. In: Feistauer, M., Kozel, K., Rannacher, R. (eds.): Proceedings of the 3rd

Summer Conference on Numerical Modelling in Continuum Mechanics, Prague, 1997.

Matfyspress, Univerzity Karlovy, Prague, S. 227-235 (1997).

[3] Deuring, P.: A stable mixed finite element method on truncated exterior domains.

RAIRO Math. Model. Anal. Numer. 32 (1998), 283-305.

[4] Deuring, P.: Calculating Stokes flow around a sphere: comparison of artificial bound-

ary conditions. In: Proceedings of the 7th Conference on Navier-Stokes and Related

Equations, Ferrara, 1999. Ann. Univ. Ferrara - Sez. VII - Sc. Mat. 46, 1-9 (2000).

376

[5] Deuring, P., Kracmar, S.: Artificial boundary conditions for the Oseen system in 3D

exterior domains. Analysis 20 (2000), 65-90

[6] Galdi, G. P.: An introduction to the mathematical theory of Navier-Stokes equations.

I. Linearized steady problems (Rev. Ed.). Springer, New York e.a., 1998.

[7] Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations.

Springer, Berlin e.a., 1986.

[8] Goldstein, C. I.: The finite element method with nonuniform mesh sizes for un-

bounded domains. Math. Comp. 36 (1981), 387-404.

[9] Goldstein, G. I.: Multigrid methods for elliptic problems in unbounded domains.

SIAM J. Numer. Anal. 30 (1993), 159-183.

[10] Nazarov, S. A., Specovius-Neugebauer, M.: Approximation of exterior boundary

value problems for the Stokes system. Asymptotic Anal. 14 (1997), 233-255.

On the Stefan problem with surface tension

Joachim Escher

, Jan Priiss

, Gieri Simonett

Institute for Applied Mathematics

University of Hannover, D-30167 Hannover, Germany

Fachbereich Mathematik und Informatik

Martin-Luther-Universitat Halle-Wittenberg

D-60120 Halle, Germany

Department of Mathematics, Vanderbilt University

Nashville, TN 37240, USA

Email : escher@ifam.uni-hannover.de ; anok@mathematik.uni-halle.de ;

simonett@math.vanderbilt.edu

Abstract

We present results on the uniqueness and the existence of smooth solutions to

the one and two-phase Stefan problem with surface tension.

1 Introduction

The classical Stefan problem is a model for phase transitions in solid-liquid systems and

accounts for heat diffusion and exchange of latent heat in a homogeneous medium. The

strong formulation of this model corresponds to a moving boundary problem involving

a parabolic diffusion equation for each phase and a transmission condition prescribed

at the interface separating the phases. Molecular considerations attempting to explain

supercooling and dendritic growth of crystals suggest to also include surface tension on

the interface separating the solid from the liquid region.

In order to formulate the Stefan problem, we introduce the following notations. Let

f2 be a smooth bounded domain in R

, whose boundary dQ consists of two disjoint com-

ponents, an 'interior' part J

and an 'exterior' part J

. We think of O as a homogeneous

medium which is occupied by a liquid and a solid phase, say water and ice, that initially

occupy the regions Qj

an(

i ^o>

d ^

nat

an<

i ^o

are

separated by a sharp interface

. More precisely, we assume that r

C fi is a compact closed hypersurface, that fij

and fip are disjoint open sets such that H = fij U f2§ and such that

d£T

= J' U F

for

i = 1,2. For the sake of definiteness, we consider the open set fij as the region occupied

by the liquid phase. Consequently, the component J

is in contact with the liquid phase

and J

is in contact with the solid phase. The boundaries J

and J

, corresponding for

377

378

instance to the walls of a container, are fixed, whereas r

will change as time evolves, due

to solidification or liquidation of the two different phases.

Given t > 0, let T(t) be the position of r

at time i, and let V(-,t) and «(-, t) be the

normal velocity and the mean curvature of T(t). Moreover, let f2

(i) and fi

(£) be the two

regions in Q separated by T(t). According to our assumption, Q

(i) is the region occupied

by the liquid phase, and F(t) is a sharp interface which separates the liquid from the solid

phase. Let v(-,t) be the outer unit normal field on T(t) with respect to 0

(i). We shall

use the convention that the normal velocity is positive if Q

^) is expanding, and that the

mean curvature is positive if the intersection of H

(t) with a small ball centered at T(t)

is convex. Consequently, the normal velocity is positive if the liquid region is growing,

v points into the solid phase, and K is positive for a water ball surrounded by ice, and

negative for an ice ball surrounded by water.

Let To and u'

: Q'

—>

R be given, where uj and u

, denote the initial temperatures

of the liquid and solid phase, respectively. The strong formulation of the two-phase

Stefan problem with surface tension consists of finding a family T := {F(t); t > 0} of

hypersurfaces and functions u

: U

>o (Q'{t) x {t})

—>

R, satisfying

dt^ - Au*

rf(0)

r(o)

= 0

= V

= OK

= IV

where I > 0 is the latent heat per unit volume absorbed or released for melting or solidi-

fying, and a > 0 is the surface tension. Moreover,

u] := d

- d

u\ (2)

denotes the jump of the normal derivatives of u

and u

across the boundary T(i). Finally,

denotes the Dirichlet or the Neumann boundary operator on the fixed boundary J', and

are given functions, where i = 1,2. For simplicity, we assume the conductivity and the

diffusion coefficients for the different phases to be the same, which we then normalize to

one.

Using distinct constants for the solid and liquid phase does not alter the mathematics

in a significant way.

The condition u' = an on the free interface is usually called the Gibbs-Thomson

condition, see [8, 10, 29, 30, 32, 38, 44, 46, 53].

If either u

or u

is replaced by 0, while all the other aspects of the problem are left

unchanged, then the modified problem is called the one-phase Stefan model with surface

tension.

If the Gibbs-Thomson condition on the free interface T(t) is replaced by u'

—

0, then

(1) is called the (classical) Stefan model.

It should be emphasized that the Stefan problem with Gibbs-Thomson correction (1)

differs from the classical Stefan problem in a much more fundamental way than just in

(t),

on J',

(t),

on F (t),

infft,

(1)

379

the modification of an interface condition. This becomes evident, for instance, by the fact

that in the classical Stefan model, the temperature completely determines the phases,

that is, the liquid region can be characterized by the condition u > 0, whereas u < 0

characterizes the solid region, where u = 0 is the melting temperature. The inclusion of

surface tension will no longer allow to determine the phases merely by the sign of u.

The main reason for introducing the Gibbs-Thomson correction u = an stems from

the need to account for so-called supercooling, in which a fluid supports temperatures

below its freezing point, or superheating, the analogous phenomena for solids; or dendrite

formation, in which simple shapes evolve into complicated tree-like structures. The effect

of supercooling can be in the order of hundreds of degrees for certain materials and is

required for nucleation, namely the forming of a new phase in a set previously occupied

by the parental phase, see [10, Chapter 1] and [52]. However, (1) does not account for

nucleation effects and can only be applied in the presence of impurities or of some other

mechanism inducing nucleation. Supercooling is an equilibrium phenomenon and is not

merely a transient effect. A simple argument for appreciating these equilibrium effects as

a first-order correction to the continuum mechanics is given in [8, pp. 209-210]: Suppose

that u = 0 is the equilibrium temperature between a sohd and liquid separated by a

planar interface. Then a certain amount of energy is required for an atom at the surface

to overcome the binding energy of the crystal lattice and become part of the the liquid

with lower binding energy. The amount of energy required to produce this transition

depends on the number of nearest neighbors in the crystal structure and on the number

of nearest neighbors of an atom on the surface. If the interface between the solid and

liquid is curved, then an atom on the interface has fewer nearest neighbors and one expects

that it will require less energy to produce a phase transition. If one considers a solid with

constant mean curvature (an ice ball) in equilibrium with its melt, then one expects the

prevailing temperature to be lower, and the Gibbs-Thomson condition indeed predicts

that u = —o/R, where R is the radius of the ball. Hence, the fluid is supercooled even at

equilibrium.

A more detailed account of this argument, which is well-known to scientists in solid

state physics and material science, can be found in [10, 32]. A more satisfactory argu-

ment leading to the same conclusion (and further generalizations) may be obtained from

statistical mechanics see [9, 33, 44]. In [29, 30], a theoretical framework is developed

starting from general thermodynamical laws which are appropriate to a continuum and

which include interfacial contributions for both energy and entropy.

The Stefan problem has been studied in the mathematical literature for over a century,

see [49, 42] and [53, pp. 117-120] for a historic account, and has attracted the attention

of many prominent mathematicians.

The (classical) Stefan problem is known to admit unique global weak solutions, pro-

vided the given functions u'

and b' have the physically correct signs; see for instance

[25,

26, 34] and [37, pp. 496-503]. It is important to point out that the existence of weak

solutions is closely tied to the maximum principle, see for instance the proofs in [25]. If

the natural sign conditions for uj and b' are obstructed, then the Stefan problem becomes

ill-posed, see [16] for instance.