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100

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Transmission problems for degenerate parabolic

equations

Takeshi Fukao

Department of Mathematics

Graduate School of Science and Technology

Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 Japan

Email : t-fukao@math.e.chiba-u.ac.jp

Abstract

In this paper, we consider a transmission problem for nonlinear two degenerate

parabolic PDEs, in such a setting that a fixed bounded domain in R

is separated

by a moving interface into two subdomains. The degenerate parabolic PDEs are

formulated on each of these subdomains and moreover the so-called transmission

condition is imposed on the moving interface. The purpose of this paper is to prove

the existence and the uniqueness of a solution.

1 Introduction

We consider the following initial-boundary value problem for degenerate parabolic equa-

tions in a cylindrical domain Q := (0, T) x Q, where 0 < T < oo and fl is a bounded

domain in R

with smooth boundary T. Moreover, we suppose that 0 is separated by a

time dependent smooth interface T

(t) into two subdomains fii(£) and Ch{t), that is :

Q, := n-i.(t) U n

(t) U T

(t) for each t € (0,T):

«,-Aft(a)+v-Vu = / in Q; := U

te(or)

{i} x f2i(t), i = l,2,

ft(«)=ft(«), ^r + ^i^ = ° on E

:= U

(6(0

,T)W x r

(t),

^^ + noA(«) = p onS, := U

o,r)W * WW \

IW*)},

» = 1,2,

(1)

u(0) =

on Q.

Here u

:= du/dt, Au :=

ELI^V'^?)

and Vu := {du/dx

du/dx

,du/dx

). We

denote by u

:= u

(t,x) the 3-dimensional unit vector normal to T

(i) at x e T

(t),

pointing to fi2(i), by v~ := i/

(£,:r) the vector — v

and by v := ^(a;) the 3-dimensional

unit vector normal to V at x € T. no is a positive constant, / and p are given functions

prescribed on Q and E, respectively. u

is the initial datum on Q. fa and fa are functions

defined on R and which satisfy the following conditions:

103

104

• Pi

•

—•

K is a Lipschitz continuous and non-decreasing function for i = 1,2.

• A(0) = 0 and

\k(r)\

|rf

- c^, Vr e R and i = 1,2,

where /3;(r) := J

f3i(s)ds, i = 1,2 and c

ftl

c^ are positive constants.

In the above problem, (1)

describes a relation between Pi(u) and feW) on the moving

interface T

(i). It is called a transmission condition. In our geometric setting, the

following two cases are allowed, the first one is that <9fi;(*) = Fur

(i) and dQ,j{t) = T

(t)

for (i,j) = (I, 2) or (2,1), and the second one is that a part of <9f2;(i) is included in T for

i = 1,2. For a typical example of the latter case, see [7].

In this paper, we assume that the following conditions are satisfied:

(Al) For each t 6

[0,T],

there exists a C

-diffeomorphism © := ©(£, •) on fi, such that

6 £ C

(Q) and

Q(t,Q

(t))=Q

{0), i = l,2, O(i,r

(t)) = r

(0), Vt€[0,T].

This assumption (Al) says that the motion of T

(t) is prescribed and smooth in time

and the C

-regularity of 6 is actually used in the existence and uniqueness proofs of

solutions (Lemma 3.3, Lemma 4.2).

(A2) We assume that the convection vector field v belongs to C

(Q)

and that the

following properties are satisfied:

div (v) = 0 in Q, (2)

•

= vx

on E

, (3)

vi/

= 0 on£:=(0,T)xr, (4)

where u

:= v-E

(t,x) is the normal speed of T

(t) at (t,x) € S

We define the time-dependent function

(3 :

Q x R

—*

P(t,x,r) := Pi(r)xQ

(t) + P2{r)xn

(t),

where

x<U(t)

is the characteristic function of Qi(t) for i = 1,2 and t €

[0,T].

Then the

problem (1) is reformulated as the following form:

' in- AP{t,x,u) +v- Vu = f in Q,

dP(t x,u)

= onE; (5)

«(0) = UQ in O.

For the solvability of (5), it seems that the time-dependent subdifferential operator

theory is useful (cf. [1, 4, 7, 8, 9, 10, 16]). In sections 3 and 4, we prove the existence and

uniqueness of a solution.

105

2 Definition of the weak solution and the main result

We begin with the definition of a weak solution of (5) and then formulate our main result.

To this end we introduce some function spaces. We denote by V the Sobolev space iJ

(Q)

with norm j

•

|y given by

1/2

\u\

{|Vu|£

2(n)

+ n

|u||

2(r)

| Vu € V,

which is an equivalent norm to the standard one of H

(f2).

The dual space of V is denoted

by V* and the duality pairing between V* and V is denoted by

{-,-)

.y.

The duality

mapping F

—>

V* is defined by

(Fu

)v«,v := / Vui

•

\7u

dx + n

/ uiu

aT Vuj,u

€ V.

Jn Jv

Moreover, we define an inner product (•, -)y in V* by

(u\,u*

)v •= «,

)

\ful,u*

6 V*.

Then V* is a Hilbert space with the following dense and compact imbeddings:

V ^ L

(Q) ^ V*.

Now we give the definition of a weak solution of (5).

Definition 2.1 u :

[0,

-* L

{0.) is called a weak solution of (5) ifue

(0,

T; V*) n

L°°(0,T;L

(f2)), /3(u) := f3(t,x,u) G L

(0,T;V), and u satisfies (5) in the variational

sense :

- J

ur]

dxdt + J

Vp(u)

•

Vvdxdt 4 n

f}{u)rjdTdt

—

u(v

•

Vrfidxdt

= J

fndxdt + /

prjdTdt + J

ri(0)dx

V77

e W, ' ^

where W := {77 € H

(Q) | rj(T) = 0 on fi}.

Our main result is stated as follows.

Theorem 2.2 Assume that (Al) and (A2)

hold.

Let f €

{Q),

p e C

{Y,), u

e L

{Q)

and f3(u

) € if

(f7). Then there exists one and only one weak solution u of (5).

The existence proof for (5) will be given in section 3, using the abstract theory of

time-dependent subdifferential operators. The uniqueness proof will be given in section

106

3 Existence proof

Let us prove the existence of a weak solution of (5), applying the existence result in

[16] (or [8]) for a nonlinear evolution equations including time-dependent subdifferential

operators and a perturbation of the form:

f ut(t) + d(t>\u{i)) + g(t,u(t)) B f*(t) in V* for a.e. t £

[0,T],

\ w(0) = u

in V,

where the functions (p

, g(t, •) and f*(t) for each t £

[0,

T] are defined as follows:

• cf/ : L

(Q)

—>

(—oo,

oo]

is defined by

(z(x))dx iizeL

(Q),

(?)

/ /

(z(x))dx+

\ J fii(t) J f

ftiz)

'•=

\ J fii(t) J n

(t)

if ze V*\L

(fi).

Clearly 0* is a proper, lower semi-continuous and convex function on V.

• g(t,

•)

: L

(tt) -> V* is defined by

(ff(t,z),ri)vy •=- I z(x)(v{t)

•

Vn)(x)dx

V77 &V,VZ£

(Q).

• /* £ L

(0, T; V*) is defined by

(f(t),ri}v*,v := { f(t,x)

(x)dx+ f p(t,x)n(x)oT

Vr,eV.

Jn Jr

It is easy to see that (5) is equivalent to (7). Therefore, it is enough to solve (7) with

the above-defined functions $, g(t, •) and

/*(*).

The characterization of

dcj/

is given by the following lemma.

Lemma 3.1 Let z £ D(90') and j3(z) := /3(t,x,z(x)). Then z* £ d<jt{z) in V* if and

only if

(F-

z*,w)^

i) = {P{z),w)^

(

)

V»6l

(fi), (8)

where

(V)L

(S))

denotes the inner product in

(Q).

Moreover, this is equivalent to the

statement that j3(z) £ V and

{z*,T))

= /

V/3{z)

•

+ n

J P(z)ndT Vr? £ V. (9)

Jn Jr

The next lemma is concerned with the demi-closedness for g(t, •).

Lemma 3.2 The operator g(t, •) : D(g(t, •)) = L

{Q.) -*V*,te [0,T], satisfies :

\g(t,z)\

<j>\z)

+ Ci Vt £ [0,T], Vz £ L

(Q), (10)

where C$ and C\ are positive constants, and if {4>

)} is bounded and z

—>

z in V*

and t

—>

t, when n

—>

oo for {z

} c L

(Q) and {t

} C

[0,

T], then g(t

)

—+

g(t,z)

weakly in V*. Furthermore, the set L

:= (J

0<t<T

{z £ V* \ (^{z) + \z\y, < r} is bounded

in L

(Q) and hence relatively compact in V* for each r > 0.

107

The proofs of these lemmas are quite standard, so we omit them.

In the following lemma, we check the time-dependence of ft and the conditions required

in order to apply the abstract result in [16] (or [8]) to (7).

Lemma 3.3 There exists a positive constant C

with the following property : for every

t € [0,T], s <t and z e D(ft) there exists Zi £ D(ft) such that

\zi-z\v<C

\t-s\{l + \ft{z)\

), (11)

\4>

{zi)-ft{z)\<C

\t-s\{l + \ft{z)\). (12)

Proof.

For every z G D(ft) :=

(Q),

we take

zx(x):=z(Q-

(s,e{t,x))) Viefi.

Then, by the geometric assumption (Al), z

e L

(fl) = D{ft) and the mapping

&t,s '•= Q(s,

-)~

° ©(*,-) i

s a

-diffeomorphism on fi for every s, t G [0,T] such that

©M (£k{t)) = &i{s), i = 1,2. Moreover there exist two positive constants C

and C4 such

that

|det(j

,(z)) -1| <C

\t-s\ Vs,t €

[0,T],

Vz G fi,

k ° e

, -

v\LHn)

< C

\t -

s\\n\

Vs,t e

[0,71,

V77

G V,

(cf. Kenmochi [9, Lemma 3.2.2]), where Je

is the Jacobian of Q

and det

(^e

,„)

is its

determinant. Using these properties, we immediately obtain (11) and (12). •

Using Lemmas 3.1, 3.3 and an existence result of [16] (or [8]), we see that there exists

at least one solution u of (7) in

{0,

T; V*) n L°°(0, T; L

{9)) and this solution u is a

weak solution of (5) in the sense of Definition 2.1. •

4 Uniqueness proof

In this section, we prove the uniqueness of a weak solution of (5) obtained in the previous

section. Let u\ and u

be two weak solutions. Then taking the difference of their weak

variational forms (6) we have

- J

(ui - u

)rj

dxdt + J

(V/9(wi) - V/?(u

))

•

Vndxdt -

(ui

- u

)(v

•

Vn)dxdt

(/?K) - P{u

))r,dYdt = 0

\fr,eW.

Moreover, we define the function b

—>

R as

( /?M*,s))-/?Mts)) jf

(t }

b{t,x):=i

(t,x)-u

(t,x)

lr A

[ 0 if «i(t,a;) = «

(i,:r),

which is non-negative and bounded on Q. With this function b, Green's formula implies

- f (u

-u

){n

+ bA

+v-Vn}dxdt+ f {PiuJ-Piui)) l^+n

ri\ dTdt = 0

\fneW.

Q E

" (13)

108

Now we take smooth and positive approximations b

of b such that

b <b

a.e. on Q, e <b

< M

a.e. on Q, . .

—•

b a.e. on Q as £

—*

where M

is a positive constant. In order to prove the uniqueness, we consider the

following backward problem for the auxiliary linear parabolic equation for any given

ieD(Q):

Ar)

+ v

•

Vr?

= I in Q,

^+n

= 0 on£, (15)

(T,-) = 0 onO.

Using Ladyzenskaja, Solonnikov and Ural'ceva [11, Chapter 4, Theorem 5.3], problem

(15) has a unique solution

€ H

2+a

1+a

(Q) for a certain 0 < a < 1.

In the next two lemmas, we give the L°°-estimate of {r)

(t)} in V and the boundedness

of {bl

} in

{Q).

Lemma 4.1 There exists a positive constant Mi such that

sup \r,e{s)\l+ ( [ b

\An

dxdt < M

f j v •V{rj

)dTdt +M^^.y)

0<s<T

Js Ja Js JT

for alls e [0,T] and e G

(0,1].

Proof.

Multiplying the first equation in (15) by An

and integrating over Q with respect

to x, with the help of Green's formula we have

-|l»felv + 21 %\\AVe\

dx < (6|v|

cl(5)3

+ l)|ife|», +

\l\*

+ 2n

J v

•

V(^)dT,

Vie [0,T].

(16)

In fact, in order to get (16) we use the following equations and inequalities

- / (v

•

Vr)

)Ar)

dx = V(v

•

Vr/

)

•

Vj)

dx - f (v

•

Vr]

)^-dY

J si J si \

3|v|

cl(g)

| V^dz + /

•

^|VT?

J dx+^J^v V(

)dT

and by virtue of (2) and (4)

jf v

•

(\\^vA

dx = Jdiv Q|V%|

v) dx = J i|V^|

•

)dT = 0.

Finally we integrate (16) over

[s,

in time and apply Gronwall's inequality to get the

conclusion. •

109

Lemma 4.2 There exists a positive constant M

such that:

v(<) ' V(7

(i))dr < M

(t)\

L2{T)

Vt e

[0,71,

Ve €

(0,1].

Proof.

Our geometric condition (Al) ensures that without loss of generality we may

assume by taking an approximate local coordinate transformation and a partition of unity

on r that:

• Qc{x=(ii,i

, x

); x

< 0} and T contains a relatively open set T

in K

= {x |

= 0}.

• supp (r/

) is compact infiU r

, Ve

(0,1] and

In the above situation, we observe for any t £

[0,

T] that

/ v-V(7?

)dr = / vV(

= -

{

i,x!

+ v

)ri

+ / 2v

^-dx

Jtf J R

= -/ (vi,

2iX2

+2n

)r]

dxidx

J R

where v := (^i,

•02,^3)-

Therefore, for any t €

[0,T],

' v(?i

)ai ^ V^oi-^;|v|c.i

(

Q)3|'/

(r)

. v-V(r;

)dr<(2n

+ 2)|v|

cl(

)

Thus we have the conclusion of the lemma. •

Now, on account of Lemmas 4.1 and 4.2 we have the uniform estimate

sup

0<t<T

h(t)\

v+ f b

\An

dxdt < M

, (17)

where M

is a positive constant independent of e €

(0,1].

By (17) there exists a sequence

} with e„

—*

0 (n

—>

00) and a function r\ 6 V such that

—»

77, weakly in £

(Q),

Vrj

—>

V77, weakly in

(Q)

Take r/

as a test-function in (14), and pass to the limit

to obtain that

[ (uj - u

)£dxdt = 0 WeV{Q).

Thus Uj = u

a.e. on

<Q.

•

Remark 4.3 TTie idea of our uniqueness proof is due to Ladyzenskaja, Solonnikov and

Ural'ceva [11, Chapter 3, Section 3], and this was also extensively used in Niezgodka

and Pawlow

[12],

Rodrigues and Yi [15] and Rodrigues

[14]

for the uniqueness proof of

generalized Stefan problems and continuous casting problems. A similar technique was

recently applied for the uniqueness proof of Stefan problems arising in the Czochralski

process of crystal growth (cf. Fukao, Kenmochi and Pawlow [6, 7]).