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Nash's Implicit Function Theorem and The Stefan Problem
Lecture Notes i n N u m . Appl. Anal., 5 , 55-60 (1982) Nonlineur PDE in Applied Science. U.S.-JLlpan Semincir, To!iyo. 1982
iJash's Implicit Function Theorem and. The Stefan Problem
Ei-Lchi Hanzawa Department of Mathematics Faculty of Science Hokkaido University SAPPORO 060, JAPAN
W e sketch out a proof of t h e l o c a l existence of the
c l a s s i c a l solutions f o r the mltidimensional Stefan problem and i t s relevance t o Nash's implicit function theorem.
!the Stefan problem i s a mthematical model of t h e melting of a body of i c e ,
where we suppose that a body of i c e melts, at each point of the surface, with velocity i n proportion t o t h e n o w 1 gradient of t h e thermal d i s t r i b u t i o n i n W e show the
water and that t h e thermal d i s t r i b u t i o n satsfies t h e heat equation.
l o c a l existence of t h e c l a s s i c a l s o l u t i o s for t h e i n i t i a l value problem of t h e Stefan problem, by using Nash's i m p l i c i t function theorem, which enables u s t o reduce the s o l v a b i l i t y of a nonlinear problem t o that of t h e l i n e a r i z e d problem even i f a loss of r e g u l a r i t y of t h e solution f o r given data occurs. Here we suppose that t h e initial surface
C"
hypersurface i n Rn, the e x t e r i o r of
i n t e r i o r of
To
the domin Ro
of a body of i c e i s a closed
To i s the i c e part, a heater i s i n the m
whose surface J i s a l s o a closed bounded by
r0
and J
C
hypersurface i n Rn
i s t h e Hater part.
and
(Except the
conditions of regularity, these assumptions, e.g., th2t t h e water p a r t i s i n the i c e p a r t , are not e s s e n t i a l . )
The locus of a surface of a body of ice, which i s
t h e free boundary t o be determined, i s denoted by T
@,T
= {(x,t)
E
R"
X
[O, TI;
56
Ei-Ichi H A N Z A W A
0(x, t) = 0}, where t is the time variable. We denote the free domin with
water by RO,T.
See Figure 1.
Figure 1.
0
?T
t
Cur unknowns are the definlng function
0
and the t h e m 1 distribution u
in water in QQ,T" Our equations and result are as follows.
muat ions, (1) QltZO = a0,
~
(3) u = b on J (4)
u
=
0 on
l
x
=
a, ~ where = ~ {x; 0 0(x) = 01 = r 0'
[0, TI.
rO,T.
Nash's Theorem and Stefan Problem
(5) at@
n - kCiZl(ax d(ax i
@) = 0
i
51
on rQYT, where k i s a positive (from a
physical reason) constant.
Theorem.
Suppose t h a t
a and b
m
are nonnegative C
s a t i s f y the compatibility conditions up t o
m
(which are necessary conditions of the existence of a
( 5 ) ) . Then, for sufficiently small T > m
sense) e x i s t a C
ro
order on m
C
b e t i o n s and and J
at
solution f o r
t = 0
(1)-
0, there uniquely ( i n the e s s e n t i a l
function 0 and a Cm
function u on fl
@YT
which satsfy
(1)-(5). 1. For the one-dimensional Stefan problem, it i s well known t h a t
Remk
the unique global c l a s s i c a l solutions a r e obtained (see Rubinstein
[51). For t h e
mltidimensional problem, the unique global weak solutions a r e obtained by Kamenomostskaja [2]. m
R e m k 2. Tanigawa [6].
The C -ness of the solutions i n Theorem i s r e w k e d by M. The author's original theorem i s t h a t for solutions with f i n i t e
d i f f e r e n t i a b i l i t y of any order. technical.
The reason f o r giving this limitation was purely
That i s , the author dld not know whether there a r e smoothing
operators up t o
O3
order on a scale of Banach spaces which is used i n [I].
'They
a r e constracted by Tanigawa.
Remk
3. For the general existence theorem of t h e c l a s s i c a l solutions f o r
the multidimensional Stefan problem, it seems that Nash's implicit f'unction theorem i s necessary, i . e . , we encounter an e s s e n t i a l loss of regularity. occurs because t o solve the single first order equation (5) f o r
@
on
It OYT
does not cover the l o s s of regularity of t h e normal derivative of t h e t h e m 1 distribution u, that is, the former gains the regularity only along the
Ei-lchi H A N Z A W A
58
characteristic curves although t,he l a t t e r loses the regularity i n every direction.
See Figure
The situation of t h i s phenomenon is clearly
2.
recognized when we linearize the problem (1)-(5). Note that t h i s d i f f i c u l t y does not occur i n the one-dimensional Stefan problem, because the f r e e boundary
i s one-dimensional so that it i s covered by the characteristic curve.
Figure
2: The reason why the loss of regularity occurs.
the f r e e boundary
I
r
@YT
the characteristic curves
t h e directions of loss of regularity of t h e normal derivative of
(at -
Remark 4.
on
ro
When a body of i c e melts rapidly, e .g.
A)U =
u with
o
, when I grad
at t = 0, we do not need Nash's implicit f'unction theorem.
Kinderlehrer and Nirenberg [3] and M e i m o v
[41.
a1
2
E > 0
See
G . Komtsu suggests t o t h e
author that the essential reason why we can get around the d i f f i c u l t y of the l o s s
of regularity i n this case i s i n the f a c t that a heat potential cover losses of regularity i n the time direction when t h e melting i s rapid.
Remark 5.
The assumption that t h e i n i t i a l data
a and b a r e nonnegative
(which i s natural in physics) enables us t o solve t h e linearized problem of (1)-
(5). We resolve the linearized problem i n t o a parabolic mlxed problem and an i n i t i a l value problem i n
R
@
,T
for a first order operator which has t h e form
59
Nash's Theorem and Stefan Problem
3,
-
k f2(avu)av on Yo,T,, where v
{x; O(x, t)
= 01 in
liearize the problem.
is an outward unit normal to the surface
Rn and u is the t h e m 1 distribution at which we We can solve the latter problem if the' characteristic
curves starting f r o m Ro at t = 0 cover the domain RQYT. This requirement is satisfied because u = a on
3.
2
0 on Qo at t = O , u = b -> O on Jx[O,T],u=O
r
and we have the mximum principle for the heat equation. See Figure QYT This fact is the core of the present work.
Figure 3: The reason why we can solve the linearized Stefan problem.
!he characteristic curves
,/"
of the operato r
Y,l
->
0 and
on %,T*
References.
[l] E. Hanzawa, Classical solutions of the Stefan problem, T6hoku Math. J.
33 (1981), 297-335. [2] S. L. Kamenomostskaja, On Stefan's problem (in Russian), Nau&
Dokl.
Vysg. skoly 1 (1958), No.1, 60-62.
[3] D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Corn. Pure Appl. Math. 31 (1978), 257-282.
[4] A. M. Meymnov, On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations, Math. USSR Sbornik
Ei-Ichi HANZAWA
60
40 (1981), 157-178.
[51 L. I. Rubinstein, Monographs, Vol.
[61 M. Tanigawa, The elsewhere.
The Stefan problem, Translations of Mathematical
27, h e r . m
Math. Soc., Providence, R . I . ,
1972.
C -ness of solutions of t h e Stefan problem, t o appear
iJash's Implicit Function Theorem and. The Stefan Problem
Ei-Lchi Hanzawa Department of Mathematics Faculty of Science Hokkaido University SAPPORO 060, JAPAN
W e sketch out a proof of t h e l o c a l existence of the
c l a s s i c a l solutions f o r the mltidimensional Stefan problem and i t s relevance t o Nash's implicit function theorem.
!the Stefan problem i s a mthematical model of t h e melting of a body of i c e ,
where we suppose that a body of i c e melts, at each point of the surface, with velocity i n proportion t o t h e n o w 1 gradient of t h e thermal d i s t r i b u t i o n i n W e show the
water and that t h e thermal d i s t r i b u t i o n satsfies t h e heat equation.
l o c a l existence of t h e c l a s s i c a l s o l u t i o s for t h e i n i t i a l value problem of t h e Stefan problem, by using Nash's i m p l i c i t function theorem, which enables u s t o reduce the s o l v a b i l i t y of a nonlinear problem t o that of t h e l i n e a r i z e d problem even i f a loss of r e g u l a r i t y of t h e solution f o r given data occurs. Here we suppose that t h e initial surface
C"
hypersurface i n Rn, the e x t e r i o r of
i n t e r i o r of
To
the domin Ro
of a body of i c e i s a closed
To i s the i c e part, a heater i s i n the m
whose surface J i s a l s o a closed bounded by
r0
and J
C
hypersurface i n Rn
i s t h e Hater part.
and
(Except the
conditions of regularity, these assumptions, e.g., th2t t h e water p a r t i s i n the i c e p a r t , are not e s s e n t i a l . )
The locus of a surface of a body of ice, which i s
t h e free boundary t o be determined, i s denoted by T
@,T
= {(x,t)
E
R"
X
[O, TI;
56
Ei-Ichi H A N Z A W A
0(x, t) = 0}, where t is the time variable. We denote the free domin with
water by RO,T.
See Figure 1.
Figure 1.
0
?T
t
Cur unknowns are the definlng function
0
and the t h e m 1 distribution u
in water in QQ,T" Our equations and result are as follows.
muat ions, (1) QltZO = a0,
~
(3) u = b on J (4)
u
=
0 on
l
x
=
a, ~ where = ~ {x; 0 0(x) = 01 = r 0'
[0, TI.
rO,T.
Nash's Theorem and Stefan Problem
(5) at@
n - kCiZl(ax d(ax i
@) = 0
i
51
on rQYT, where k i s a positive (from a
physical reason) constant.
Theorem.
Suppose t h a t
a and b
m
are nonnegative C
s a t i s f y the compatibility conditions up t o
m
(which are necessary conditions of the existence of a
( 5 ) ) . Then, for sufficiently small T > m
sense) e x i s t a C
ro
order on m
C
b e t i o n s and and J
at
solution f o r
t = 0
(1)-
0, there uniquely ( i n the e s s e n t i a l
function 0 and a Cm
function u on fl
@YT
which satsfy
(1)-(5). 1. For the one-dimensional Stefan problem, it i s well known t h a t
Remk
the unique global c l a s s i c a l solutions a r e obtained (see Rubinstein
[51). For t h e
mltidimensional problem, the unique global weak solutions a r e obtained by Kamenomostskaja [2]. m
R e m k 2. Tanigawa [6].
The C -ness of the solutions i n Theorem i s r e w k e d by M. The author's original theorem i s t h a t for solutions with f i n i t e
d i f f e r e n t i a b i l i t y of any order. technical.
The reason f o r giving this limitation was purely
That i s , the author dld not know whether there a r e smoothing
operators up t o
O3
order on a scale of Banach spaces which is used i n [I].
'They
a r e constracted by Tanigawa.
Remk
3. For the general existence theorem of t h e c l a s s i c a l solutions f o r
the multidimensional Stefan problem, it seems that Nash's implicit f'unction theorem i s necessary, i . e . , we encounter an e s s e n t i a l loss of regularity. occurs because t o solve the single first order equation (5) f o r
@
on
It OYT
does not cover the l o s s of regularity of t h e normal derivative of t h e t h e m 1 distribution u, that is, the former gains the regularity only along the
Ei-lchi H A N Z A W A
58
characteristic curves although t,he l a t t e r loses the regularity i n every direction.
See Figure
The situation of t h i s phenomenon is clearly
2.
recognized when we linearize the problem (1)-(5). Note that t h i s d i f f i c u l t y does not occur i n the one-dimensional Stefan problem, because the f r e e boundary
i s one-dimensional so that it i s covered by the characteristic curve.
Figure
2: The reason why the loss of regularity occurs.
the f r e e boundary
I
r
@YT
the characteristic curves
t h e directions of loss of regularity of t h e normal derivative of
(at -
Remark 4.
on
ro
When a body of i c e melts rapidly, e .g.
A)U =
u with
o
, when I grad
at t = 0, we do not need Nash's implicit f'unction theorem.
Kinderlehrer and Nirenberg [3] and M e i m o v
[41.
a1
2
E > 0
See
G . Komtsu suggests t o t h e
author that the essential reason why we can get around the d i f f i c u l t y of the l o s s
of regularity i n this case i s i n the f a c t that a heat potential cover losses of regularity i n the time direction when t h e melting i s rapid.
Remark 5.
The assumption that t h e i n i t i a l data
a and b a r e nonnegative
(which i s natural in physics) enables us t o solve t h e linearized problem of (1)-
(5). We resolve the linearized problem i n t o a parabolic mlxed problem and an i n i t i a l value problem i n
R
@
,T
for a first order operator which has t h e form
59
Nash's Theorem and Stefan Problem
3,
-
k f2(avu)av on Yo,T,, where v
{x; O(x, t)
= 01 in
liearize the problem.
is an outward unit normal to the surface
Rn and u is the t h e m 1 distribution at which we We can solve the latter problem if the' characteristic
curves starting f r o m Ro at t = 0 cover the domain RQYT. This requirement is satisfied because u = a on
3.
2
0 on Qo at t = O , u = b -> O on Jx[O,T],u=O
r
and we have the mximum principle for the heat equation. See Figure QYT This fact is the core of the present work.
Figure 3: The reason why we can solve the linearized Stefan problem.
!he characteristic curves
,/"
of the operato r
Y,l
->
0 and
on %,T*
References.
[l] E. Hanzawa, Classical solutions of the Stefan problem, T6hoku Math. J.
33 (1981), 297-335. [2] S. L. Kamenomostskaja, On Stefan's problem (in Russian), Nau&
Dokl.
Vysg. skoly 1 (1958), No.1, 60-62.
[3] D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Corn. Pure Appl. Math. 31 (1978), 257-282.
[4] A. M. Meymnov, On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations, Math. USSR Sbornik
Ei-Ichi HANZAWA
60
40 (1981), 157-178.
[51 L. I. Rubinstein, Monographs, Vol.
[61 M. Tanigawa, The elsewhere.
The Stefan problem, Translations of Mathematical
27, h e r . m
Math. Soc., Providence, R . I . ,
1972.
C -ness of solutions of t h e Stefan problem, t o appear