FEN BİLİMLERİ DERGİSİ JOURNAL OF SCIENCE
CONTINUOUS DEPENDENCE OF SOLUTIONS ON THE COEFFICIENT THERMAL DIFFUSIVITY FOR PHASE FIELD
EQUATIONS Şevket GÜR1 1
Sakarya Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü ,54187, SAKARYA
ABSTRACT
In this paper, investigate the continuous dependence on the coefficient thermal diffusivity.
Keyword: Phase field equation, continuous dependence.
FAZ ALAN DENKLEMLERİNİN ÇÖZÜMLERİNİN ISI İLETKENLİK KATSAYISINA SÜREKLİ BAĞIMLILIĞI ÖZET
Bu çalışmada faz alan denklemlerinin çözümlerinin ısı iletkenlik katsayısına sürekli bağımlılığı incelenmiştir.
Anahtar Kelimeler:Faz alan denklemi, sürekli bağımlılık.
I. INTRODUCTION We consider the problem
2
( ,
)
t f x2
u
h
1(
x
,
t
)
TQ
t
x
,
)
(
(1))
,
(
2 2K
u
h
x
t
u
l t t
(
x
,
t
)
Q
T (2) ) , ( tx
, u u( tx, ) (
x
,
t
)
0
,
T
(3))
(
)
0
,
(
x
0x
,u
(
x
,
0
)
u
0(
x
)
x (4)Faz Alan Denklemlerin Çözümlerinin Ş. GÜR where
Q
T
0
,
T
,
R
n(
n
1
)
is a bounded domain with a sufficiently smooth boundary ;
0
,
T
,
,
,
l
andK are positive constants characterizing the length scale, the relaxation time, the latent heat and the thermal diffusivity respectively.0 0
, u
,
,u
,h
1, h
2 andf
(
x
,
)
are given functions.In [1], G.Çağınalp has considered, as a model describing the phase transitions with a seperation surface of finite thickness and proves a global existence theorem for the classical solution of problem such type. In [2], Brochet, Hilhorst ve Chen investigated problem (1)-(4) considering
2 l u v ,
1 2 0)
(
p j j js
b
s
f
, b2p1 0,p2,h
i(
x
,
t
)
0
,)
2
,
1
(
i
and homogeneous Neumann boundary condition and they proved that the problem is well posed if( ,
0u
0)
L
2( )
2. In [3], Kalantarov has proved that the initial boundary value problem for system (1)-(2), under some conditions onf
(
x
,
)
homogeneous boundary conditions, global uniqualy solvable inC
(
R
,
X
)
, ( ) ( ) 0 1 0 1 H HX and the existence of global attractor.
II. CONTINUOUS DEPENDENCE THEOREM: If 2 1 1 2 1 1 2 1
)
(
,
)
(
1
)
,
(
x
s
f
x
s
c
s
s
s
s
f
p
p
(5)then the solution of problem (1)-(4) from
V
(
Q
T)
V
(
Q
T)
( [4],[5],[6]) depends continuously on the thermal diffusivity coefficient. Where,
1 2 ( T) T V Q W Q
v x t( , ) : v L Q2( T)
and
,
1
p
ifn
1
,
2
,
2
,
1
n
n
p
if n3.PROOF:
Let
1,u
1
and
2,u
2
be the solutions fromV
(
Q
T)
V
(
Q
T)
of problem (1)-(4) for different coefficientK
1 andK
2 respectively. We define difference variables
,u and K by2
1
,u
u
1
u
2, andK
K
1
K
2 (K
1K
2). Then
,
u
satifies the initial boundary value problemu x f x f t ( , 1) ( , 2) 2 2
(
x
,
t
)
Q
T (6) 2 1 2K
u
K
u
u
l t t
(
x
,
t
)
Q
T (7) 0 u
(8)0
)
0
,
(
)
0
,
(
x
u
x
(9)If we take the inner product in
L
2(
)
of (6) by
t
and of (7) byu l u l t 4 2 2
and we add the obtained equations, then by inequalities from making use of (5) dx x f x f( ,
) ( ,
))
t ( 1 2
c
t + t L n nt
c
c
2 2)
(
1 (10) and dx x f x f( ,
) ( ,
))
( 1 2
c
2 +
2 2)
(
1 n n Lt
c
c
(11) we obtain 2 t
+
2
2+4 1 u 2 l K + 2 2 2 t u l
+ 2 2 2 21 2 2 2 2 2 l u K u l dt d
2
(
u
,
)
+4 ( u2,u) l K + ) , ( 2 2 2 u ut l K
+ ( t,ut) l
+c
t + t L n nt
c
c
2 2)
(
1 +c
2 +
2)
(
1t
L nc
c
(12)Faz Alan Denklemlerin Çözümlerinin Ş. GÜR where denotes the norm on
L
2(
)
andc
1(
t
)
is depens on the given functions. Making use of Cauchy-Schwarz and
-Young’s inequalities, right hand side of (12) can be estimated. If we select the number
0 sufficiently small and by the inequality
2 2 2c
n n L (12) rewrite as follows: 2 4
t
+4 1 u 2 l K + 2 2 4 3 t u l
+ 2 2 2 21 2 2 2 2 2 l u K u l dt d
a
1(
t
)
2+ +2
u
2l
l
+a
2(
t
)
2+ 2 2 2 23
4
6
u
K
l
l
(13) Wherea
1(
t
)
anda
2(
t
)
are depends on given function and the parameters. If we denote ,2 ( ),1 2 2 , ) ( 2 max ) ( ~ 2 2 1
t a l t a t c and 2 2 1 2 2 2 22
2
2
)
(
u
l
K
u
l
t
Y
then from (13)
0
)
0
(
3
4
6
)
(
)
(
~
)
(
2 2 2 2Y
u
K
l
l
t
Y
t
c
dt
t
dY
According to Gronwall’s lemma, we have
)
(t
Y
2 ) ( 2 2 ) ( ~ 2 2 03
4
6
t t Q L ds s cu
K
e
l
l
Since
i,
u
i
V
(
Q
T)
V
(
Q
T)
) ( 2 2QT Lu
) , , , , , , ~ , ~ , , ( 2 2 ) ( 0 2 ) ( 0 1 )) ( ; , 0 ( 2 )) ( ; , 0 ( 2 ) ( 2 2 ) ( 1 1 1 1 2 1 1 1 2 1 2 2 2 K l u u h h C H H p W T W W T W Q L Q L p T T
C
1
K
2
)
(t
Y
2 ) ( ~ 2 1 2 0)
(
3
4
6
K
e
K
C
l
l
t ds s c
Moreover,
(
)
2 ) ( 2 1 1 2t
u
u
W 2 2 1 ) ( ~ 2 1 2(
)
(
)
3
4
6
0K
K
e
K
C
l
l
t ds s c
and
(
)
2 ) ( 2 1 1 2t
W
2 2 1 ) ( ~ 2 1 2(
)
(
)
3
4
6
0K
K
e
K
C
l
l
t ds s c
Hence we have proved the theorem. REFERENCES
1. Çaginalp, G. An Analysis of a Phase Field Model of a Free Boundary Arch. Rat. Mech. Anal.92, 205-245, ( 1986 ).
2. Brochet, D., Hilhorst, D., Chen, X., Finite Dimensional Exponential Attractor for the Phase Field Model, Appl. Analysis, vol.49, 197-212, ( 1993 ).
3. Kalantarov, V.K. On the Minimal Global Attractor of a System of Phase Field Equations, Zap.Nauchn.Semin. LOMI, 188, 70-86, ( 1991 ). 4. Soltanov, K.N. On Nonlinear Equations of the Form:
F(x; u;Du; u) = 0 .Russian Acad. Sic. Sb. Math. 80.(1995) no:2
367-3923, ( 1995 ).
5. Soltanov, K.N. Some Imbedding Theorems and Nonlinear Differential Equations, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 19, 125-146, ( 1999 ).
6. Gür, Ş. Phd Thesis. Global Behavior of Solutions of Initial Boundary Value Problems for Phase Field Equations, Hacettepe University, (2004).
