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Multi-dimensional stefan problems with dynamicboundary conditionsAiki Toyohiko aa Department of Mathemtiacs Faculty of Education , Gifu University , 1–1,Yangagido, Gifu, 501–11, JapanPublished online: 02 May 2007.
To cite this article: Aiki Toyohiko (1995) Multi-dimensional stefan problems with dynamic boundary conditions,Applicable Analysis: An International Journal, 56:1-2, 71-94, DOI: 10.1080/00036819508840311
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Multi-dimensional Stefan Problems with Dynamic Boundary Conditions Communicated by J.R. Ockendon
TOYOHIKO AIKI Department of Mathematics Faculty of Education, Gifu University,
1-1, Yanagido, Gifu, 501-11, Japan
AMS: 35K85, 49A29, 49A2 1
ABSTRACT - We consider multi-phase Stefan problems for a class of nonlin- ear parabolic equations with dynamic boundary conditions formulated in bounded domains in RN, N 2 2. Our dynamic boundary condition is described by a non- linear parabolic (pseudo) differential equation for the boundary temperature. The existence and uniqueness for a weak solution as well as the monotone dependence of the solution upon data will be shown. Our approach to this problem is based on the abstract theory of nonlinear evolution equations governed by time-dependent subdifferentials in Hilbert spaces.
KEY WORDS: nonlinear degenerate parabolic equations, dynamic boundary conditions, subdifferentials
(Received for Publication 7 August 1993; in final form 14 November 1993)
0 INTRODUCTION
In the author's previous paper [I], one-dimensional two-phase Stefan problems for nonlinear parabolic equations with dynamic boundary conditions were studied in the classical formulation. In this work we deal with the enthalpy formulation of multi-dimensional Stefan problems with dynamic boundary conditions.
We consider a medium which occupies a bounded domain R c RN (N > 2), with smooth boundary r := dfl, and in which a solid and liquid coexist; we postulate that the medium is surrounded by a sufficiently thin material having large thermal conductivity. We neglect the thickness of the boundary material. For instance we
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can consider an iron ball in which water and ice coexist; they are separated by a hypersurface S (interface).
Let T be any positive number, Q := (0,T) x R and C := (0, T) x r. In the classical formulation, the Stefan problem is to find the interior temper-
ature distribution 0 = O(t,x) in Q and the boundary temperature V = V(t,x) on C determined, and the location of the interface S, which satisfy the following (2) (2v):
(2) (Heat equation) In Q /S := Ql U Q, where QI and Q, denote the liquid and , , \ - . solid region, respectively, the heat equations are satisfied:
where f is an internal heat source and Cl(resp. C,) is a positive constant determined by the thermal conductivity and specific heat of the liquid (resp. solid).
(ii) (Free boundary condition) On S, the temperature vanishes and the energy conservation law holds, that is,
where b is a positive constant (the latent heat of change of phase), Z is the normal unit vector to S and [.Is indicates the jump across S along Z.
(iii) (Boundary conditions) We quote Langer [7] for the physical interpretation of boundary conditions. First, by thermal contact of the medium in R and the boundary material,
O = V onI'.
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Next, the heat flow from the inside of fl to the boundary I. is -%, where v is the outward normal unit vector to r , and the accumulation rate (in time) of heat
dV on the boundary is x. (by neglecting the thickness of the boundary materials). Then, denoting by g(t ,x ,V) the heat flow from the outside of R to the boundary r, we have as the lateral boundary condition the following relation:
(iv) (Initial conditions) At t = 0, the initial conditions are prescribed for 6, V and S :
O(0, .) = 80,
V(0, .) = Vo, S(0) := S n {t = 0) = SO (initial interface),
where the obvious compatibility conditions are satisfied, namely, Oo = 0 on So, = Vo, 00 5 0 on one side of So, whereas 00 2 0 on the other side.
The weak formulation for problem (i) - (iv) is given by the following system:
where p : R R is the function given by
Cl(r - b) for r _> b,
P(r) = for 0 < r < b, C,r f o r r 5 0 ,
and uo = P-*(Bo). In this formulation, u represents the enthalpy and P(u) = 8 the temperature. We denote by SP = SP(P; g; f ; uo, Vo) the above system '(0.1) - (0.5).
By many authors initial-boundary value problems for (0.1) with usual boundary conditions have been studied. In particular, in case the flux condition is of the form -a,B(u)/dv = g(t, x, P(u)), the problem was uniquely solved in the variational sense by Visintin [ l l ] , Niezgodka, Pawlow & Visintin [9] and Niezgodka & Pawlow [8]. Also, some interesting results dealing with the boundary condition (0.2) are found in Cannon [3], Hintermann [5] and Grobbelaar & Dalsen [4]. Recently, boundary con- ditions similar to {(0.2), (0.3)) were discussed by Primicerio & Rodrigues [lo] and in Aiki [I] for one-dimensional Stefan problems with dynamic boundary conditions the local in time existence and uniqueness of classical solutions were shown.
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The purpose of the present paper is to establish an existence and a uniqueness theorem and a comparison theorem for S P . The idea for construction of the solution is as follows. We approximate P, g, uo, by smooth functions PC, g,, uo,,, and solve the approximate problem SP, := SP(PE;gE; f ; u o , , , P , ( ~ ~ , ~ ) ) by using some results Kenmochi [6] in the theory of nonlinear evolution equations involving time- dependent subdifferential operators in Hilbert spaces. Denoting by u, and V, the solution of SP,, we then show that u, and V, converge to some functions u and V, respectively, and the pair {u, V) is a solution of SP(P; g; f ; uo, Vo) The proofs of uniqueness and comparison of solutions are quite similar to those of Niezgodka & Pawlow [8]. The main results of the present paper were already announced in Aiki 121 without proofs; the complete proofs are given in this paper.
In this paper, for a general (real) Banach space V we denote by I . ( v the norm in V. When V is a Hilbert space, we denote (., the inner product in V. For a Hilbert space V and 0 < T < +m, we denote by C,,,([O,T]; V) the space of all weakly continuous functions from [0, T] into V. We,mean by u, 4 u in C,([O, TI; V) that (u,(t), z)v --+ (u(t), Z ) V uniformly in t E [O,T] as n 4 ca for each z E V.
1 STATEMENTS OF MAIN RESULTS
We begin with the precise assumptions on P, g, f , uo and under which SP is discussed.
(P l ) P is non-decreasing and Lipschitz continuous on R with Lipschitz constant Cp, P(0) = 0 and
IP(r)l 2 Plrl - f f o r any r E R
where p, ,f3 are positive constants; (P2) fGr some positive constant ro, p is bi-Lipschitz continuous both on
(-m, -ro] and [TO, m ) . (gl) g = g(t, x, () : (0, T) x F x R -t R be aiunction satisfying the Carathkodory
condition, i.e. g(t, x, () is continuous and non-decreasing in ( E R for a.e. (t, x) E C and is measurable in (t, x) E C for all ( E R;
(92) g(t, x, () is non-decreasing in ( E R for a.e. (t, x) E C; (93) for any E R, g(. , . ,<) E LZ(C); (94) g(., ., 0) E Lm(C) and g(t, x, () is Lipschitz continuous in ( uniformly with
respect to (t, x) E C, that is, there is a positive constant Cg such that
for all ( , ( I E R and a.e. ( t , x) E C; condition (94) can be replaced by the following (95);
(95) g(t, x,() is locally Lipschitz continuous in ( uniformly with respect to (t, x) E C, that is, for each M > 0 there is a positive constant Cg(M) such that
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for all t , t' with 5 M , )J'1 5 M and a.e. (t, x) E C and there are constants ml, m2 with ml 5 m2 such that g(t, x, P(rn1)) 5 O,g(t, x, P(m2)) > 0 for a.e. (t, x) E C.
( A l l f E L Y Q h
(A2) uo E LO"(R) and Vo E Lm(I').
We give a weak formulation for SP(& g; f ; uo, Vo) in the variational sense. Definition A couple {u, V} of functions u : [O,T] t L2(fl) and V : [O,T] t L2(I') is a weak solution of SP(P;g; f ; UO, Vo), if it satisfies the following (Sl) - (S3): (Sl) u E Cw([O, TI; L2(R))nLm(Q), P(u) E L2([o, TI; W1*2(R)), V E C,(O, T; L2(r ) ) nLm(C) and P(u) = V for a.e. on C;
+ / g(t, x, V)qdI'dt = /p fqdxdt for any q E W, C
where dI' denotes the usual surface element on r and W = {q E C1([O, TI; W1t2(R)); 9(T) = ~ ( 0 ) = 01; (S3) u(0, x) = uo(x) for a.e. x E R and V(O,x) = Vo(x) for a.e. x E I?.
We now state an existence result for SP
Next we mention a uniqueness result for SP.
Theorem 1.2 Under the same assumptions as in Theorem 1.1, SP has at most one weak solution {u, V).
Finally, we show a comparison result for SP.
Theorem 1.3 Suppose that the same assumptions as in Theorem 1.3 hold and a pair { ~ ~ , v ~ ) satisfies (A2). Let (u, V) (resp. {E,V)) be a weak solution of SP(P; g; f ; uo, Vo) (ESP. SP(P;g; f; zo,'if,)). Then for any t E [O, TI,
The construction of a solution to SP is done in the following way. For each E > 0 we define an approximations P, and g, of P and g, respectively,
by
&(r) = p ( r ) + ~r for r E R (1.2)
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and
where g(T, x, r ) for t > T, x E T and r E R,
a t , 2, = g t x r) for O 5 t 5 T, x E F and r E R.
Furthermore for each M > 0 and (t , x) E C, we put
g , ( t ,x ,M) f o r r > M ,
g , ( t ,x , r ) f o r - M < r < M ,
gE( t ,x l -M) for r < -M.
At first, we consider the initial-boundary value problem SpjM) := SP(P,; gLM); f ; uo., @,(uO,,)) where uo,, is a smooth approximation of uo. This problem can be uniquely solved as a direct application of the theory of nonlinear evolution equations generated by time-dependent subdifferentials and a solution {uLM), @M)) of sPjM) has the following regularities;
The next step is to show that there exists a positive number Mo independent of E, M such that
( t , x ) < M - 1 for a.e. ( t ,x) E C, M > 0 and E E (O,l].
Therefore { u $ ~ Q ) , VJMO)) is a solution of SP, := SP(,&; g,; f ; UO,,, P E ( ~ O , E ) ) . Finally, by using the uniform estimates for solutions {u,, V, ) of SP, with respect
to E , we show that there are functions u, V and a subsequence {E,) of { E ) such that u,,, converges to u , V,, converges to V as n tends to infinity and {u, V) is a weak solution of SP.
2 INITIAL-BOUNDARY VALUE PROBLEM SpjM)
In this section, we fix positive numbers E , M, use the same notations as in the previous section and show that for E , M there exists a solution {uLM), of SpjM) satisfying (1.3).
Throughout this section we make the same assumptions as in Theorem 1.1. Then there exists a positive number CM such that
for all [,(' E R and a.e. (t, a ) E C.
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In the sequel, for simplicity we put
and
( .f11)~ = / R fgdx + f&dr for .f= (f , f r ) , a = (g,gr) E H,
and denote by X* the dual space of X. Also, we define an operator P : X -+ H by putting
P v = (v ,vJr) for v E X.
Clearly, H is a Hilbert space with inner product (f:;)H, and the range R ( P ) of P is a dense subspace of H and P is continuous.
In order to solve SpjM) by applying the subdifferential operator theory, we introduce a family {4:,M(.))tE(~,Tl of functions f$f,M on H formulated by
J, IVzJ2dx + J, ~ ! ~ ) ( t , x, z l r )dr if z'= (2, zlr) E R ( P ) , ~ , M ( Z ) = otherwise ,
where GLM)(tl x l [) = J: g!M)(t, x, r)dr.
As to the family {44,M) we have:
Lemma 2.1 (1) For each t E [O,T], 44,# is a proper 1.s.c. convex function on H. (2) There are positive constants R1, R2 and Rg such that
(3) There is a positive constant R4 such that 4:,, has the following property (*): (*) For any 0 5 s < t < T , z' E R(P),
where 4 t ) = S,(IZ?(T, ., M I l ~ y r ) + I ~ ( T , ., - - M ) l ~ q r ) ) d ~ . (4) For each t E [O,T], the subdiflerential of in H is single valued and char- acterized as follouw > = dd:,M(Z) if and only if > = (z', z;) E H, i? = P z , z E X and
Since we can prove this lemma by elementary calculations, we omit the proof. Furthermore, by B, we denote the operator from D(B,) = H into itself given
by BE(;) = ((P,(z), ~ r ) for t = (2, zr) E H.
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Clearly, BE is bi-Lipschitz continuous on H , and is the subdifferential of the convex function j, on H given by
1 j.(;) = /oZ(') ,&(r)drdz + - 2 / r z;dr for i = ( 1 , ~ ~ ) E H.
Also, we put
yo(;) = rtdz + $dl. for i= ( I , i r ) E H.
Lemma 2.2 (1) yo(z') + T O ( - ~ = l z l~ l (n ) + ( .zr l~l(r) for 2 = (2, ~ y ) E H . (2) For each t E [O,T], 84fVM o B, is yo-accretive in H , i.e, if 6 and .?2 are any elements in H , B,G and BEG E D(b'4:,M), then
where dyo is the subdiflerential of yo in H . (3) For any r 2 0, t E [O,T], the set {z' E H; (qH < r, )$4,M(q( 5 r ) is compact in H .
This lemma is quite standard, so we omit the proof (cf. Chapter 3 in Ken- mochi [6]) .
Since R ( P ) is a dense subspace of H and P, is bi-Lipschitz continuous on R, we can choose a sequence { u o , E C 1 ( a ) ; 0 < E 5 1 ) such that
uo, -, 210 in L2(R) as E 1 0,
P,(uo,~) - VO in L 2 ( r ) as E 1 0 ,
{uo,,} is bounded in LW(R).
Now, we consider the following Cauchy problem CPE(4:,M; B,; f:, & ) :
2 + &:,,(BEZ(t)) = f for a.e. t E [O, TI,
v'(0) = GogE := B;lPP,(~O,s) ,
where f is a given H-valued function. The above Lemmas 2.1 and 2.2 allow us to apply the abstract_results in Ken-
mochi 161 to CPE(4E,M; Be; f; GO,,) , and we see that for each f E L2(0, T ; H ) , CPE(4ETM; Be; f:, G,,) has a unique solution v' in W'12(0, T; H) such that t -+
4 f , ~ ( & G ( t ) ) is bounded on [0, TI, hence, BEG E R(P) for all t E [0, TI. Let 6 , ~ be a solution of CPE($4,M; BE; f:, Go,=) for f = ( f , 0 ) and put uiM) = /?;l(P-l BE;.,M) and V j M ) = / ~ , ( Z L $ ~ ) ) I C . Then by Lemma 2.1(4) it is easy to see that {u iM) , GM)} is a unique solution of SPjM) and satisfies (1.3).
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STEFAN PROBLEM 7 9
3 UNIFORM ESTIMATES FOR APPROXIMATE SOLUTIONS
In this section, let us use the same notations as in the previous section. We now give uniform estimates for approximate solutions. For simplicity, we put
If p satisfies ( p l ) and (@2), then by (1.2) there exists a positive number To independent of E E ( O , l ] such that
@:(r) 2 To > 0 for a.e. r E R with J r J > ro. (3.1)
Lemma 3.1 Under the same assumptions as in Theorem 1.1. Let (uLM), vJM)) be the solution of sP(P,; g!M); f ; uo., @ c ( ~ O , c ) ) . Then there ezists a positive number Mo independent of E , M depending only on I f l ~ - ( ~ ) , Cg, 1g(., ., O)l~m(c), ro, To, Cp, po and T such that
JuIM)(t, x)I 5 MO - 1 for a.e. (t, x) E Q, all E E (O,1] and all M > 0,
M ( t , x ) 5 - 1 f0ra.e. ( t , x ) E El a l l € € ( O , l ] and ail M > 0 . (3.2)
Proof At first consider the case when ( D l ) is satisfied. Put
1 for r > 0,
-1 f o r r < O ,
and take a sequence {an) of smooth functions on R such that ah 2 0, -1 5 a, 5 1 on R , a,(O) = 0 and
a, --t uo pointwise on R as n -t CQ.
Let M1 and 5 be any positive numbers. For simplicity we put z(t, x) = Ml(t + 6) for (t, x) E Q, and u and V for uiM) and V = GM), respectively.
For a.e. t E [0, T]
On the other hand, since f - = 8m:,M(BcZ) for C = (u ,V) , it follows from t Lemma 2.1 (4) that for a.e. t E [0, TI D
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By the definition of on,
Now, put fig = Ig(., .,O)Jtm(c) and choose a positive number 6 so that
Moreover, choose a positive number M1 so that
Then using (3.1),
where - ro t -
- 4Cg(Cp + 1) '
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It follows from (3.3) (3.5) that for a.e. t E [O,tl],
Furthermore, if Ml 2 maxi( f lLm(g), 9, Mz), then
and hence
Since 6 and tl is independent of E , po and M, there exists a positive number MA such that
u(t, x) < MA - 1 for a.e. (t, x) E Q,
V(t, x) 5 MA - 1 for a.e. (t, x) E C.
Similarly, we obtain that for a positive number M: independent of E and M
-M: + 1 _< u(t, x ) for a.e. (t, x) E Q,
-M{ + 1 5 V(t, x) for a.e. (t ,x) E C.
Thus (3.2) holds for Mo = max{MA, Mt) . Next, consider the case when ( 0 2 ) is satisfied. With the same notations as
above, we take 6 = 1 and Ml satisfying MI 2 max{ml + 1, Cp). Then following inequalities hold:
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and
- Pc(z)t(t)o0([~(t) - PE(z)(t)1+)dr 5 0. (3.7)
By (3.6) and (3.7), (3.2) can be obtained in a way similar to the previous case. O
Under all the conditions of Theorem 1.1 Lemma 3.1 implies that {u!~"), vJMo)) is a solution of SP,. Therefore we write u,, V, for uiM0), viM0), respectively.
Next, we show a lemma on uniform estimates for the solutions {u,, V , ) by using the energy inequalities.
Lemma 3.2 Suppose that the same assumptions as in Theorem 1.1 are satisfied. Then there exists a positive constant M3 independent of a such that
h Iu.(t)I2dx 5 M3 for all t E [O,T] and all E E (O,l],
d ix( t ) i2dr 5 M3 for all t E [O,T] and all r E (0, 11,
Proof We put p,(r) = &'P,(s)ds for r E R. By (pl) and (1.2),
1- 2 p.(r) 2 aPr - for r E R.
For simplicity, we write u, V for u,, V,, respectively. Then for t E [0, TI,
On the other hand, for t E [O,T]
Now, note that
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STEFAN PROBLEM
and
1 - JtJ V,VdFdr = -- J Iv(t)I2dl' + 1 J lB . (~o ,~) /~dI ' . o r 2 r 2 r (3.11)
It follows from (3.8) (3.11) that for t E [0, TI,
By the Gronwall's inequality, there exists a positive number M4 such that
A lu(t)12dx + 1v(t)l2dI' 5 M4 for t E [0, TI.
Combining this inequality with (3.12), we get the assertions of the lemma. o
Now, for t E [0, TI and s E (O,1] we define Ec(t) E X* by putting
By (1.3), E, E W112(0,T; X*) for each E E (O,l]. Finally in this section, we give an uniform estimate for E,.
Lemma 3.3 Under the same assumptions as in Lemma 9.2, there exists a positive number Me independent of E such that
Proof By Lemma 3.1, there exists a positive constant M6 such that
Igr(t, ~ ~ 0 ) - g,(t,x, K)I 5 M61VcI for (42) E C and E E ( O , l ] .
Hence,
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Therefore for a.e. t E [0, T I ,
By this inequality and Lemma 3.2,
so that the lemma holds. 0
4 PROOF OF THEOREM 1.1
In this section, we assume that all the conditions of Theorem 1.1 and use the same notations as in the previous section, too. First, note that for each E the following variational identity holds:
= f q d s d t for all 7 E W. (4.1)
By Lemmas 3.2 and 3.3
{u , ) is bounded in L m ( Q ) , (4.2) {P,(u,)} is bounded in L 2 ( 0 , T ; X), (4.3) {V,) is bounded in Lw(O, T ; L 2 ( I ' ) ) , (4.4) {E,) is bounded in W 1 3 2 ( 0 , T ; X * ) . (4.5)
By (4 .2) N (4.5) it is possible to extract a sequence {E , ) with E, J 0 (as n --+ m ) such that
u,, =: u , --+ C weakly* in Lm(Q), (4.6)
,BE" ( u E n ) =: V, -+ 6 weakly in ~ ~ ( 0 , T ; x ) , (4.7) Xn =: V -+ V weakly* in L ~ ( o , T ; L ~ ( I ' ) ) , (4.8)
E,, =: E, -+ E weakly in W19'(0, T ; X*). (4.9)
Also, wn 1 -+ 5 weakly in ~ ~ ( 0 , T ; L'(I ' ) ) .
Hence, GIr = V a.e, on C. Since H is compactly imbedded in X*, the facts (4.2), (4.4) and (4.9) imply that
En --+ E in C([O, T I ; X * ) . (4.10)
According to (4 .6) , (4.8) and (4.10),
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STEFAN PROBLEM
for any 71 E L2(0, T; X).
Furthermore, (4.10) and (4.11) imply that
un + ii in c,([o, TI; ~ ' ( 0 ) ) ~
Vn + in c,([o, TI; L2(r)) .
In order to prove Theorem 1.1 we prepare three lemmas.
Lemma 4.1 G = p(G) a.e. on Q .
Proof Since /3 is maximal monotone as a mapping in L2(0, T; L2(R)), it is sufficient to show the following inequality:
l T ( i i - W , i. - P(~ i ) )~z (n )d t 2 0 for any w E LZ(O,T; L2(R)). (4.14)
In order to prove (4.14), let w be any function L2(0, T; L2(R)). Then
I + + O a s n - i c o f o r i = 2 , 3 .
Also, it is clear from the monotonicity of P,, that
1 4 + 2 0 f o r n = 1 , 2 , . . . .
Since
= i T \ ~ ( t ) - ~n(t)l(vn(t))dt - JT(p - vn>vn)~2(r)dtl 0
(4.10) implies that
Il,n + 0 as n --t m. (4.18)
It follows from (4.15) (4.18) that the inequality (4.14) holds. Thus Lemma 4.1 is proved. 0
Next we investigate the regularity of (2, p) in time. Define a function j on H by
1 j (Z) = /o ,o(z(z))dz + - / $dl? for i = ( i , i r ) E H,
2 r where p ( r ) = J,T P(()d(. Then we have:
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Lemma 4.2 Put Z( t ) = ( i ( t ) , p ( t ) ) for t E [O,T]. Then j (Z( t ) ) is absolutely continuous on [0, T ] and
d . - j (Z( t ) ) dt = [ p ( t ) ] ( p ( i ) ( t ) ) f o ~ a.e. t E [0, TI.
Proof For 0 I to I t l I T we put
{ ( ~ ( t ) , ,(t)) for to 5 t I T , (u*( t ) , V*(t ) ) =
( i ( t o ) , V(t0)) for t I
and
[ E * ( t ) ] ( 7 ) = J, .*(t))?dX + ~ * ( t ) ~ d T . for all 7 E X and t E [O, T I ,
[E.(t)](v) = ~ * ( t ) ~ d x + / r V.(t)qdr for all 7 E X and t E 10, TI.
Since E E W112 (0, T ; X * ) ,
E*( . + h ) - E*(.) _ g , E*(') - - h, + E' in ~ ' ( t ~ , t l ; x*) as h J, 0. h h
Now we observe that for h > 0
1
J , (p (u*( t + h ) ) - p ( ~ * ( t ) ) ) d x d t + % 1;' L ( v * ( t h)' - V*(t)2)drdt
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STEFAN PROBLEM
because Q E C,([O, TI; L2(R)). SimiIarly,
limsup h ~ o 2h lrth l(G'(t0) ' - G'(t)2)d~dt 5 0.
Therefore,
which implies
We can prove the following inequality in a way similar as above:
and
1; [e'(t)](P(ii)(t))dt = j(G(tl)) - j($to)) for all 0 5 to < tl < T .
Thus the lemma has been proved.
Lemma 4.3 P,,(un) -t P( l ) in L2(0, T ; X).
proof. For simplicity, we put
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On the other hand,
Also, by (4.12) and (4.13) we have
liminf Jn( t ) 2 J ( t ) for all t E [0, TI. n-m
Therefore, on account of Lemma 4.2,
Besides, by the monotonicity of g,,
It follows from (4.21) - (4.25) that
This implies that V&(un) -+ VP( i ) in L2(0,T; L2(S1)) as n -+ co, whence the assertion of the lemma holds.
From Lemma 4.1, Lemma 4.3 and (4.1) we immediately see that
9€,(., ., PE,(u~)) -+ d . , ., V ) ) in L2(0,T; L 2 ( W ,
and (6, v ) is a weak solution of SP. Thus the proof of Theorem 1.1 is complete.
5 PROOFS OF THEOREMS 1.2 AND 1.3
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STEFAN PROBLEM 8 9
Throughout this section assume (D l ) or (02) . Suppose that {ul, 1/11, (242, Vz} are solutions of SP(P; g; f; uo, Vo). Observe first
that the following variational identity holds:
where w = {v E C2*'(Q);v(T) = 0). In order to avoid surplus confusion for notations we introduce the following
functions:
By the definition of solutions to SP and the assumptions (Pl) , (g3)(resp. (g4)),
where and cg = Cg (resp. Cg(M)) for M = max{(V1(~w(c), ( h ( ~ - ( c ) ) . Using the above notations we can rewrite (5.1) in the form
By virtue of (5.3), (5.4) we can choose the following sequences {a,) c Cm(B) and {G,) c Cm(E) such that properties (Cl), (C2), (C3), (C4) and (C5) are satisfied:
(Cl ) la, - a l ~ z ( ~ ) L Con-'; (C2) a, 2 n-' in Q; (C3) a, = n-' on C, (C4) G, + G in L2(C);
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90 T. AIKI
( C 5 ) n-I 1 G, lCg+l on C, where Co is a positive constant independent of n.
Making use of a, we formulate the regularized parabolic problem RP, for each n:
z ( T , x ) = 0 for x E 0,
Lemma 5.1 For each n the regularized problem RP, has one and only one solution zn E W112(0 ,T ; L 2 ( R ) ) n Lm(O,T; W1g2(R) ) with znlc E W 1 j 2 ( 0 , T ; L 2 ( r ) ) ) .
We shall show the proof of Lemma 5.1 in Appendix.A. Remark 5.1 We can take the solution z,,n = 1 , 2 , . . . of RPn as a test function in (5.5).
As a consequence of assumption (C) we can prove:
Lemma 5.2 For n = 1,2, let z, be the solution of RP,. Then there exist positive constants lil, I i2 independent of n , such that
Proof (5.6) can be proved in a way similar to that of Lemma 3.1. Next we show (5.7). For simplicity we write z for r,.
Let us take the solution z,, n = 1,2 , ... of the regularized problem RP, as a test functions in (5.5). Then,
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STEFAN PROBLEM 9 1
By Lemma 5.2, assumption (Cl) and the definition of solution to SP, that is, u E Lw(Q) and U E Lw(C),
Hence IQ fou dxdt = 0 for any fo E D(Q).
This implies that u = 0 a.e. in Q, i.e. ul = u2 a.e. in Q and Vl = V2 a.e. on C. Thus Theorem 1.3 has been proved.
Finally, we show Theorem 1.4. Let {u, V) (resp. { ~ , v ) ) be the solution of SP(P; g; f ; uo, Vo) (resp. SP(P; g; f ; Eo, VO)) and {ue, V, ) (resp. {Z,, vc)) be the solution of SP(Pc; g,; f ; UO,C, Pc(u~,c)) (resp. SP(Pe; gc; f ; ZO,,, P,(ZO,~)) where P,, g,, U O , ~ , ti^,^ are same functions as in section 1.
Then, by the standard L1-space technique, we have
It follows from the proof of Theorem 1.1 and the uniqueness of the solution to SP that
Therefore, letting E 1 0 gives (1.1). Thus Theorem 1.4 is proved.
APPENDIX. PROOF OF LEMMA 5.1
In this section, we assume that all conditions of Lemma 5.1 hold and use same notations as in section 5. In order to prove Lemma 5.1 it is sufficient to show that for each n there is a unique solution z of the following initial-boundary value problem (IBP),:
where G,(t,x) = an(T - t ,x) for ( t ,x) E Q, j ( t ,x ) = f ( T - t ,x ) for (t ,x) E Q, e n ( t , x ) = Gn(T - t ,x) for ( t ,x) E C.
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92 T. AIKI
We give a definition of solutions to ( IBP) , for each n. Definition A We say that a function z on Q is a solution of ( IBP) , , if the follow- ing properties (i), (ii) are fulfilled: (i) z E W1,2(0, T ; L2(fl)) n Lm(O,T;X) and zlc E W1,'(O,T; L2( r ) ) ;
= (f*, V ) L Z ( Q ) for a.e. t E [0, TI and all 7 E X.
1 We note that i, = on C by (C3). Let n = 1 ,2 , . . ., fixed. In order to show existence of the solution to ( I B P ) , we
define an inner product of H, := L2(R) x L2(I') by putting
define an operator P, : X + H, by
Pnv = (v,vlr) for v E X ,
and introduce a family {$t)tE[o,Tl of functions $t on H, formulated by
Clearly, as to the family {@) we prove:
Lemma A . l ( 1 ) For each t E [O,T], @ is a proper 1.s.c. convex function on H,. (2) For each t E [O,T], the subdifferential of $t in H, is single valued and charac- terized as follows: 2 = d?lt(G) if and only if u" E H,, ii = Pnul u E X and
1 (2, Pn~)Hn = 8,Vu V u d z + - / G,uvdr for any v E X .
n r
We now consider the evolution equation
2 ( t ) + d@(z'(t)) = f in H, for a.e. t E [O,T],
q o ) = 20,
where f E L2(0, T; H,) and E H,. Under (a. l) , the following result follows easily from Lemma A . l (2) and Kenmochi (61.
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STEFAN PROBLEM
Lemma A.2 For any y E L2(0, T ; X ) there exists a unique function z' in W112(0, T ; H,) which satisjies (a.2) for f = (f - V$, Vy, 0) and Zo = (0,O). In addition, put z := P;'z' and z satisfies z E W1,'(O, T; L2(0)) n Lm(O,T; X ) and
= (f - Vc?, . v y , 7 1 ) ~ 2 ( ~ ) for a.e. t E [O, T] and all 7 E X.
Consider the mapping A, : L2(0, T ; X ) -t L2(0, T ; X ) with assigns to each y E LZ(O, T ; X) z := P;'z' where z' is a solution of (a.2) with f = (f - VZL, . Vy, 0) and z'o = (0,O). By elementary calculations and the (Banach's) fixed point theorem we show that the mapping A, has a fixed point z,, that is, A,z, = z,. Obviously, z, is a solution of (IBP), .
References
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[lo] M. Prirnicerio and J. F. Fbdrigues, The Hele-Shaw problem with nonlocal injec- tion condition, to appear in Proceeding of the meetnig "Nolinear Mathematical Problems in Industry,", Iwaki, Japan, 1992.
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