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380

As soon as surface tension is included in the model, the phase regions Q'(t) are in

general no longer coupled to a sign condition for the temperature distributions, and one

cannot resort to a comparison principle. As a consequence, many of the methods which

were successfully applied for the classical Stefan problem are not available for the Stefan

problem with surface tension.

Results concerning the regularity of the free boundary for weak solutions of the mul-

tidimensional one-phase Stefan problem (without surface tension) were established in

[4,

5, 27, 35, 36], and continuity of the temperature was proved in [7]. The regularity

results were derived by formulating the Stefan problem as a parabolic variational inequal-

ity, see [18, 27]. In order to obtain the smoothness results, the authors in [27, 36] had to

impose restrictive geometric assumptions on the initial data which assure that the melting

is rapid and free from breaking off. Under fairly weak assumptions on the data it was

shown in [40] that any weak solution eventually becomes smooth and that T(t) approaches

the shape of a (growing) sphere. If the data are sufficiently smooth and satisfy high order

(up to order 23!) compatibility conditions, classical solutions were obtained in [31]. The

approach relies on the Nash-Moser implicit function theorem.

Continuity of the temperature for weak solutions of the multidimensional two-phase

Stefan problem (without surface tension) was obtained in [6, 14, 15, 54]. More recently,

the regularity of the free boundary for weak (viscosity) solutions was studied in [1, 2, 47]

under a non-degeneracy condition. Local existence of classical solutions in a small time

interval was proved in [41], provided that the initial data satisfy high order compatibility

conditions. The continuity of the temperature distribution for an m-phase Stefan model

with m > 2 has been studied in [17].

Although the Stefan problem with surface tension (1) has been around for many

decades, only few analytical results concerning existence and the regularity of solutions

are known, see [19, 28, 39, 41]. In [28], the authors consider system (1) with small surface

tension 0<<rCl and linearize the problem about a = 0. Assuming the existence of

smooth solutions for the case a = 0, that is, for the classical Stefan problem, the authors

prove existence and uniqueness of a weak solution for the linearized problem and then

investigate the effect of small surface tension on the shape of T(t). Existence of global

weak solutions for the two-phase problem (1) is established in [39], using a discretized

problem and a capacity-type estimate for approximating solutions. The weak solutions

obtained in [39] have a sharp interface, but are highly non-unique. In [41], the way

in which a spherical ball of ice in a supercooled fluid might melt down is investigated.

A proof for the existence of classical solutions for (1), assuming restrictive high order

compatibility conditions for the initial data, is sketched in [48]. In [19], the existence and

uniqueness of analytic solutions in case that F(i) is the graph of a function over R"

obtained.

If the diffusion equation

u'—£yu?

= 0 is replaced by the elliptic equation Au* = 0, and

the initial condition for u

is dropped, then the resulting problem is the quasi-stationary

Stefan problem with surface tension, which has also been termed the Mullins-Sekerka

model (or the Hele-Shaw model with surface tension). Existence, uniqueness, and regu-

larity of solutions for the quasi-stationary approximation has recently been investigated

in [3, 11, 12, 21, 22, 23, 24].

381

Weak solutions to the modified Stefan problem with surface tension and kinetic un-

dercooling, that is, with the condition u

<JK

—

0V on the free interface r(t), were

obtained by Visintin [50]. For classical solutions for the Stefan problem with the condi-

tion

<TK-

PV see [13, 48].

In order to describe our main result, we first give a concise formulation of the boundary

operators B' occurring in (1). Let

€

{0,1},

i = 1,2, be fixed numbers. Then we set

BV := S'dvtf + (1 - <5V, i = l,2, (3)

where v is the outward pointing normal on <9fi. The case

= 0 corresponds to a Dirichlet

boundary condition on the fixed boundary J*, whereas

= 1 results in a Neumann (that

is,

no-flux) boundary condition on J'.

In the following, we denote by W£ the Sobolev-Slobodeskii spaces of fractional order,

where s > 0 and p 6

(1,

oo). Let I = [0,T] be a given interval. In order to deal with the

inhomogeneous boundary conditions B

we introduce the function spaces

G'(J) := W^V^

to(I,L

))nL

(I,W<?-

ti)

''(J

)), i = 1,2. (4)

Theorem 1.1 Letp£ (n+2, oo). Suppose

that{U\,YQ,V)

satisfy the regularity conditions

K,r

,&') e w?-

a/p

(n&) x w^

'" x

(I)

and the compatibility conditions

ui\

= *K

B'4

(o),

[cUo] e

6/p

(5)

Then there exists a positive number

G / such that the Stefan problem (1) has a uni-

que solution on [0,To] in the above-defined class (6). The manifold M. := U

(o,T

)(r(i) x

{£}) is real analytic and

eC" (J

($T

(t) u r(t)) x {t} J.

\t£(0,T

)

Proof.

This follows from Theorem and from parabolic regularity theory. •

Remark 1.2 (a) Suppose that (u'

) satisfy the regularity conditions

(uj.ro.b*) e

)

x c

x (tfiiMJ)) nc(/,cV)))

and tte compatibility conditions

<XK

B*t4 = 6*(0).

Then the conclusions of Theorem 1.1

hold.

(b) The first two compatibility conditions in (5) arise naturally for classical solutions.

Besides, observe that

[<9„HO]

3/p

(Yo)

since u

belongs to

2/p

(fl'

Hence the

third compatibility condition in (5) is a mild assumption on the spatial regularity of the

initial velocity of

[t t—*

T(t)] in normal direction.

for

eacht e (0,T

382

The

transformed equations

A major difficulty

in the

mathematical treatment

of the

Stefan problem

(1)

comes from

the fact that

the

boundary

T(t), and

thus

the

geometry,

unknown

and

ever changing.

A widely used method

overcome this inherent difficulty

is to

choose

fixed reference

surface

£ and

then

introduce

unknown diffeomorphism 6{-,t)

: £

—>

F(i)

which

tracks

the

position

and the

regularity

of the

moving surface T(i).

In a

next step

one can

try

extend this diffeomorphism

to a

diffeomorphism

fixed reference regions

into

the unknown domains £}*(£). This approach

usually called

the

direct mapping method.

The treatment

of the

moving boundary problem

(1)

then proceeds

transforming

the

equations into

new system

equations defined

on the

fixed domain

from which

both

the

solution

and the

diffeomorphism

have

to be

determined.

In the

context

of the

Stefan problem this method

has

first been used

Hanzawa

[31].

In order

implement this approach we choose

closed compact analytic hypersurface

Scil which

To- Such

surface always exists,

see

[20].

We can

assume that

E divides

into

two

disjoint regions

and D

such that

enclosed

by E, and

such

that

= J

E for i = 1,2.

Clearly, we could choose

E = r

However, since

our

main

objective

is to

impose only minimal regularity

and

compatibility conditions

for the

initial

data

and

then

prove that solutions regularize,

the

choice

E =

is not

appropriate.

Let

p, be the

outer unit normal field

on E

with respect

to D

Then

we can

find

number

ao > 0

such that

/Ex(-ao,«o)

-+ R- \

V (p.n

+ w) /

smooth diffeomorphism onto

its

image

:—

im(X), that

is,

XeDifP°(Ex (-a

),n).

It will

convenient

decompose

the

inverse

of X

into

-1

(P, A) such that

fer(K,E), AeC°°(TZ,(-a

)).

Of course,

1Z,

is the set of

those points with distance less than

ao to E. For

further

use,

introduce

the set of

admissible functions

:= 2l

:= {p e

(£);

HPHCHE)

where

a e

(0,

) is a

fixed number. Given

G 21

define

the

mapping

p >-

P +

P{P)KP)

(7)

and

we set F

im(^). Then

is a

-hypersurface

and 6

: E

—v

is a C

dif-

feomorphism between

the

manifolds

E and T

Moreover,

if a

small enough then

is contained

in fl and it

separates

into

two

domains

Oj and Q

just

as E

does.

particular,

diY

= J

for i = 1,2.

We will

now

introduce

extension

: R

—>

R" of 6

with

the

property that

GDiff

(R",R

)

and

Diff

a) n

Diff

%), e

(8)

383

Let a g (0, oo/4) be given and let tp be a C^-cutoff function with the properties that

0 <

< 1, V

(

) = 1 if \r\ < a, y(r) = 0 if \r\ > 3a and sup

|</?'(r)|

< \ja. Given any

p £

we introduce the Hanzawa transformation

ftM

._J*(J'(*).A(s) + ¥>(A(z))p(P(s))) if xGTl,

W _

\ x if a; g ft.

The function [r i-» r +

^WPW]

strictly increasing for any p e 21 and any pCE,

since |<p'(r)p(p)| < 1, and (8) follows after a short computation. It should be observed

that there exists an open neighborhood U of J

U J

such that

\u = idu. (9)

We now consider time-dependent functions p e C-([0,T],C(Y:)) n C([0,T],2l) with

T > 0 a fixed number. Each p defines a family T := {T(i) =

p(t)

;

t 6 [0,T]} of

hypersurfaces. Note that T(t) can also be described as the zero-level set of the function

•R

[0,

-> R

(x,t) .-» A(x)-p(P(x),t)

(10)

that is r(<) =

(-,t)

(0), for £ G

[0,T].

Consequently, the unit normal field v{-,i) on

T(t) at the point x = X{p, p(p, £)) can be expressed in coordinates of E as

v{p,

t) =

V«MM)

IW„(a:,i)|

, pGS (11)

x=X(p,p(p,t))

where V denotes the gradient with respect to the x-variable. Moreover, the normal

velocity V of T at time t and at the point x = X(p,p(p,t)) e T(t) is given by

v(p,t)=

dtp{p

|V^(x,i)|

, pGE. (12)

x=X(p,p(j,,t))

We can now readily infer from (11)-(12) that the Stefan condition [d„v\ = IV in (1)

translates into

{p,t)={Wu

{x,t)-Vu\x,t)\W<Pp{x,t))\

x=x{pp{pt)y

peE, (13)

where (• | •) denotes the Euclidean inner product of ".

In the following, we will assume that

pec

([o,nc(E))nc

([o,r|

)

ai) (

)

where 21:= {g G C

(E); HsHc^s) < °} with a e (0,a

/4). For each p we define the

cylindrical domains

:= (D

U D

) x

[0,T],

n„,

:= (J (fi^, U ft

(t)

) x {t}.

te[o,T]

384

Every p which satisfies (14) induces a diffeomorphism

: D

PiT

§„(y, T) := (0

(y), r).

We can now consider the pull-back operator $* given by

:= u o $„, for u

BUC (Q„,

Analogously, the push-forward operator $£ is defined by

<$>y

o $-!, for v G BUC(D

For a given function p which satisfies (14), we consider the transformed differential

operators

A(p)v := SJ (A (*>«)),

C (p) „ := <^

(V (*£«»)

V0„) - -^y (V (fcjt;

) | V^),

for ?; G C((0,T],C

((D

U -D

)), where u' := t>|z>

[o,T], and where 7* denotes the trace

operator from D' onto E. As before, V stands for the gradient with respect to the space

variables. The transformed mean curvature operator is given by

H(p):=%K

, (16)

where «

r (t)

denotes the mean curvature of T

(

) for t e [0,T]. Lastly, we consider the

mapping R(v,p) given by

R(v,p)(y,t) := { f(My))d

(P(y),t) (%(V(^v))(y,t)\p(P(y))) y e K,

(1?)

It follows from (9) and from (12)-(17) that the two-phase Stefan problem with surface

tension (1) can be transformed into the following system of equations for v := $*«

v = A(p)v + R(v,p) in D

Bv = 6 on JT,

v = aH{p) onE

, ,.

p = C(p)v

onE

{

v (0) = v

in D,

p{0) = po,

where J

:= (J

U J

) x

[0,T],

:= E x

[0,T],

and where

(B«)Ux[o,n ~ BV, 6|

Jix|0

,T]=6

, t = l,2.

Theorem 2.1 ILet p G (n + 2,00). Suppose that the functions {vo,po,b) satisfy the

regularity assumptions

,b) e Wl-**{D) x W^p) x (\I) x

(/))

385

and the compatibility conditions

«b|E = ff#(A>), Bv

= b(0), C(p

eW^

(D).

Then there exists a positive number T

6 I such that the

coupled

system of transformed

equations (18) admits a unique solution (v,p) on I

:= [0,T

] with

v e

W}{I

(D)) n

W2(D)),

p € W

3/2

1/2p

, L

(E)) n

W}-

1,2p

(E)) n L

, W

1/p

(E)).

Moreover, the solution (v, p) satisfies

eC-(Ex(0,T

)),

C^dD*

U E) x (0,T

)) n C^V* x (0,T

)),

w/tere V:={ie(]; dist(a;, E) < a} and V* =

Vnl>.

Proof.

Based on [19], the proof of this result will be given in [20].

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Epidemic Models with Compartmental Diffusion

W.E. Fitzgibbon

, M. Langlais

and J.J. Morgan

Department of Mathematics, University of Houston

Houston, Texas 77204-3476, USA

UMR CNRS 5466, case 26

Universite Victor Segalen Bordeaux 2

33076 Bordeaux Cedex, France

Department of Mathematics

Texas A & M University, College Station, Texas

77843-3368, USA

Abstract

Using a basic SI model we undertake the question of assessing the effects of a

spatial heterogeneity upon the geographic spread of disease.

1 Introduction

In this note we undertake the question of assessing the effects of a spatial heterogeneity,

a fragmented habitat, upon the geographic spread of disease. Although generalizations

to more complex and realistic epidemiological models are possible, we shall consider only

the basic SI model for disease transmission. To be more precise we consider a population

subdivided into two subclasses susceptible and infective. Susceptibles are individuals who

have yet to contract the disease. The infective class consists of those infected with the

disease who are capable of transmitting. Interactions between susceptible and infective

will produce a loss from the susceptible class and a gain in the infective class. We shall

assume that the susceptible and the infective populations disperse throughout a bounded

region or habitat and that they remain confined to this region for all time. We employ

Fickian diffusion as a model of Brownian dispersion in the standard manner. Heretofore

the dynamics and the qualitative behavior of what we have described are well known.

What distinguishes the work at hand is that we shall assume existence of a subregion

Q* C f2 where the transmission of disease takes place. In the region Q

—

CI*

we assume

that the disease spreads only via diffusive transport. The rates of diffusion are different

for the susceptible and infective populations and the diffusion rates are distinct in fl*

and Q

—

Q*. To conceptualize this model we may envision S¥ as an urban environment

and the surrounding region fl

—

fi* as a rural environment. Here we are saying that

the disease is transmitted and spreads through the urban area f2* and out into the sur-

rounding rural region f2

—

where there is only dispersion for the population and no

389