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17Математические заметки СВФУ Апрель—июнь, 2022. Том 29, № 2
UDC 517.95
ON PHASE-FIELD EQUATIONS OF PENROSE—FIFE TYPE WITH NON-CONSERVED ORDER PARAMETER UNDER FLUX BOUNDARY CONDITION. II: UNIFORM BOUNDEDNESS A. Tani
Abstract. In [1] we showed the global-in-time solvability of the initial-boundary value problem for the non-conserved phase-field model proposed by Penrose and Fife [2,3] under the correct form of flux boundary condition for the temperature field in higher space dimensions. In this paper we discuss the uniform boundedness up to the infinite time of its solution in Sobolev—Slobodetskiï spaces.
DOI: 10.25587/SVFU.2022.49.45.008
Keywords: non-conserved phase-field equations, Penrose—Fife type, flux boundary condition, uniform boundedness of strong solution in Sobolev—Slobodetskiï spaces.
1. Introduction
In [1] we discussed the global-in-time solvability of the initial-boundary value problem for the non-conserved phase-field model proposed by Penrose and Fife [2, 3] under the correct form of flux boundary condition for the temperature field in higher space dimensions, which is a continuum model for description of dynamics of order/disorder phase transition taking into account of both the relaxation and balance laws based on the second law of thermodynamics:
^ = KAcp - (f3 + tp + {ay + b)u, ieii, t > o,
du d
+ (atp + b)u2-^- = Mu2Au - gu2, x G O, t > 0, dt ' dt (1.1)
^ = M^ = l3(1--ee), xgt, t>o, on on \u J
((p,u)\t=0 = (tpo,uo)(x), x e Q.
Here p is the order parameter, u =1/0 with 0 being the absolute temperature, g = g(x, t) is the heat supply, K > 0, M > 0, a > 0, and b are given constants, fl > 0 is the heat conductivity (constant) on the boundary T and 0e (> 0) is the external temperature; Q C RN (N = 2, 3) is a bounded domain with smooth boundary T,
© 2022 A. Tani
n = n(x) is the outward unit normal vector to T at x £ T. The latter boundary condition for u is the so-called flux boundary condition.
There are many papers (see [4]) with the mathematical results related to problem (1.1) with ft = 0. In the case ft > 0, there are also many papers under the physically incorrect flux condition, as was pointed out in [1,4], while under the physically correct flux condition there are only two papers [1,5], as far as the author knows.
In [1] we proved the following theorems.
Theorem 1.1. Let T be a positive number and £ a bounded domain in Rn (N = 2, 3) with the boundary T belonging to W23/2+l, I > 1/2 (N = 3) or
I > 0 (N = 2). Suppose that g £ w2'1/2(Qt), where Qt = £ x (0,T), 6e £ wi/2+i,i/a+i/2(tt), tt = T x (0,T) and (p0,u0 £ W2,+1 (£) are such that inf uo(x) =
Mq > 0 and they satisfy the compatibility conditions up to order max{[Z — 3/2], 0}. Then problem (1.1) has a unique solution (p,u) £ W2+l'(2+l)/2(QT») for some T* £ (0,T] such that
< C(||(^o,uo)||
W21 + l(0)
+ ||g|w^/2(QT ) +11 1 w21/2+i,1/4+1/2 (rT )); (1-2)
u(x, t) > -u0 for any (x, t) £ Qt* •
Theorem 1.2. Assume that the hypotheses in Theorem 1.1 with l = 2 are valid and the conditions
inf 9e(x,t) = 9e> 0, sup Vl '</3, £L2(T-,Wi(0,T)) (1.3)
(x,t)erT te{o,T] —e II ot
hold, where |£| is the volume of Then the solution from Theorem 1.1 is extended on [0,T]:
|(^,u)!w24.2(qt) < C(T), u(x,t) > C,(T) for any (x,t) £ Qt. (1.4)
Here C(T) and C*(T) are positive constants depending non-decreasingly on both the data and T.
The focus of the present paper is the following theorem on the uniform bounded-ness of its solution up to an infinite time.
Theorem 1.3. Assume that the hypotheses in Theorem 1.1 with I = 2 and
any T e (0, ro) hold. In addition, we suppose that g e L2(0, ro; W22(Q)) n LTO(0, ro; W21(Q)),
d0e _ T„1
<9i
e L2(Q^),
dt
e W1(0, ro; L2(T)) n Li(0, ro; L2(T)) n L4(TTO),
h(t) = J g(x, t) dx + fl J 0e(x, t) dT e L1(0, ro) n L2(0, ro);
sup ll^)ll2"i# sup
M|l|o<t<oo |1 \Ve 0<t<oo
> 0,
inf &e(x, t) = 0e > 0, u, oup we
(x.tjeToo (x,t)6rc
u* sup 0e(x, t) < 1 (see (2.13) for u*),
\ Q \1/3 <C0*.
Then the solution (p,u) from Theorem 1.2 has the following properties: for N = 2, 3
u* > u(x, t) > u* for any (x, t) e Q
(1.5)
yp(t)yl4(n) + IIVp(t)H2 +
dp dt
(t)
+ II Vu(t) |2 +
+
for N = 2
ay
dr2
(t )
+ || Vu(t
+
(t )
lim
t^oo
dp ~dt
(t)
du
|Vu(t)n =0;
dT < C for any t> 0, (1.6)
du ~dt
(t)
+ |Au(t)|1 +
|| VAu(t )|1 + dS7u
d 2u d^2
(t )
+
ot
(t )
dT < C* for any t> 0, (1.7)
lim
t^oo
du ~dt
(t)
0.
Here u*, u*, C, and C* are positive constants depending on the data only.
Remark 1.4. At this moment we have not succeeded to prove (1.7) for N = 3, since Lemma 2.1 below is not applicable for proving Lemma 3.6 in [1] for N = 3, or Lemmas 2.7 for N =3. This is really caused by the embedding of W2(Q) in L3(Q) in the applicable form of Lemma 2.1 for N = 2, but not for N = 3.
As in [1], the standard Sobolev-Slobodetskii spaces are used: WJ(Q) with norms II ■ IIr (I -II = II ■ IIo) and W2r(T) and the anisotropic
W2's(Qt) = L2(0, T; W2r(Q)) n L2(Q; W2s(0, T))
t
2
2
i
2
2
2
1
t
2
2
i
2
i
1
and
W2r's(rT) = L2(0, T; W2 (r)) n L2(r; W2s(0, T))
for non-negative real numbers r, s, and T < By W2s(0,T; WJ(fi)) we denote the function space of u(t) defined on (0,T) with values in WJ(£) such that u(t) G W2s(0, T). The spaces of vector fields are denoted the same as in the scalar case, and their norms are supposed to be equal to the sum of norms of all their components. In what follows, we also denote by c universal positive constants which may vary from place to place; C0 is the constant in the Sobolev-Poincare inequality; CO is the constant in the embedding theorem from W2(fi) in Lq(£2) (1 < q < 6), IM|Lq(o) < IMIwKfi)/CO, and C (i = 1, 2,3,...) are positive constants depending non-decreasingly only on the data.
2. Uniform boundedness: Proof of Theorem 1.3
Recall the a priori estimates for u) from [1, § 3]. In the present case, as well, some calculations in the following are certainly formal, because the regularity of the solution is not sufficient. However, as usual, we can easily derive the rigorous results by using the arguments of mollifiers and passing to the limit.
Lemma 2.1 [6]. Let y(t) and H(t) be nonnegative functions, dy(t)/dt be locally integrable on (0, tx>), and y(t) and H(t) satisfy the inequalities
^ < Biy(t)2 + B2 +H(t), J y{r) d,t < B3, J H(t) d,t < B4 for t G (0, 00)
0 0
with B1-B4 being positive constants independent of t. Then, for any r > 0, y(t + r) < (Jy + B2r + B^j eBlBa for any t G (0, 00).
Moreover, lim y(t) = 0.
t^^
Lemma 2.2. For any t > 0, we have
\\vvm2 + Mt)\\Un) + j (¿+losu) Met*
+
dip
(t )
+ ||Vu(rdr < Cl (2.1)
Proof. Add (1.1)i multiplied by dp/dt and (1.1)2 multiplied by 1/u, and
t
t
t
2
integrate it over Qt. Then
§ \\vtpm2 + lymUn)dT+J logu(x,t)dx t
dp
(t )
+ M||Vw(r)||2^ dr + J dr J gudx + ^ J dr J 9eudT
o o n o r
<fl|V^o||2 + iyo||l4(o)+ y"log«o(a;)da; + ^|r|i + i|ii|. (2.2)
n
Substituting the inequality
t
+mt<f J nv«(r)ii2dr 0
sup llffWf + ^P sup \\g(r)\\] f dr [ 6eudT
<T<t |1 |£e 0<T<t J J
' n r
f 1
dr / -—(x,T)dT,
0eu
J dT J gu dx o n
+ V +
/3 , eg
4 + 4M|r| 0<^t'
C2 4M|r|
sup Hg(7
o<T <t
o r
which is derived from the Sobolev-Poincare inequality IK*)II <Co||Vu(t)||+^P [u(x,t)dT
\T\
<Co||V«(i)|| +^E.j(eeu)(x,t)dT (2.3)
and
t t /3|r| t<j J dr J(6eu)(x,T)dT+P J dr J _L^(a;,r)dr, (2.4)
or
or
into (2.2), we have
K1
l|V^(i)||2 + -y(i)||l4(0)+ /log u(x,t)dx
2
+
+ yl|Vii(r)||2^ dr + /3' J dr J 6eudT
o r
K 1 f
< yl|V^o||2 + ^yo||l4(o) + J \ogu0(x)dx
n
t
+ j drj^-){x,r)dT, (2.5)
or
t
t
t
t
where
^ = 7-7In SUP SUP ll5(t)l1'
4 4A2 |1 | 0<t<oo |I |£e 0<t<oo
C2
= + SUP
4M |1 | 0<t<TO
Next, multiply (1.1)2 by u-2 and integrate it over Qt. Then t
0 r
dx + 13 J dr J\{x,t) dT <e-\\ip(t)\\ii{n)+ J -^-dx
n
t
J (I g(x, t) dx + /3 J 6e(x, r) dr (2.6)
t
+ + ......
0 n r
for any e > 0.
Add (2.5) to (2.6) multiplied by a constant j(> ¡3"/{f39_e)) and choose e < 1/(4y). Then we derive (2.1), because logu + 1/u > 0. □
Lemma 2.3. For any t > 0, we have
C^1 < J i(a:,t)dx < C2. (2.7)
n
Proof. As in [1, Lemma 3.2], it is clear how to get (2.7) by using (2.1) and (2.6). Indeed, the upper bound is derived from (2.6) and (2.1) with the elementary inequality — log u < 1/u. On the other hand, the lower bound results from Jensen's inequality
-1os(m / «dx) - ~W\ /log«dx = M /log udx'
nnn
and
1
/log u(x,t)dx+ f—(x,t)dx<Ci u
derived from (2.1) with log u + 1/u > 0. □ Then (2.7) implies that
u(t) =mmu(x,t) <C2\£\, u(t) = max«(i,t) > j^. (2.8)
xen xEfi (s2
It is easily seen from the Sobolev-Poincare inequality that, for any t > 0,
IK*) - u(t)II + I\u(t) - u(t)II < Coll Vu{t)\\ + c. (2.9)
Lemma 2.4. For any t > 0, we have
d(p ~dt
(t)
+ l|Vu(t)||2 +
dV^
dr
(t )
+
1 du
dT < C3
(2.10)
and
lim
t^oo
d(p ~dt
(t)
+ l|Vu(t)||2 =0.
Proof. Adding differentiated with respect to t equation (1.1)i multiplied by d(p/dt and (1.1)2 multiplied by u-2du/dt, and integrating over Q, we get
1 d Ydt
d(p ~dt
(t)
+ M|| Vu(t)||2 + ß J (u9e - log(u0e))(x, t) dT r
< au(t)
~dt
(t)
+
+ K
dV(p
dt
dT
(t)
(t )
+ 3
d(p
+
1 du u dt
(t)
+ 7;m%m2+ßVW\(m + 7r
We dt
(t)
L2 (r)
< || Vu(t) |2
d(p ~dt
(t)
+ (a2C02 + aC4 + 1)
+
d(p ~dt
+
Ode dt
(t)
(t)
L2 (r)
, /I^U , V^ + I + —
|Vu(t)|2
C4
Ode dt
(t)
. (2.11)
Here we used the inequality
u(t) <
3
\u(t)-H(t)\\ + \\u(t) - u(t)\\ +u(
< Co||Vu(t)|| + C4 (2.12)
derived from (2.8) and (2.9).
Applying Lemma 2.1 with r = T* and Lemma 2.2 to (2.11) and invoking the boundedness of the solution on [0,T*] in Theorem 1.1, we arrive at (2.10).
The last assertion in Lemma 2.4 follows from the last claim in Lemma 2.1. □
From (2.10) and (2.12) it easily follows that
du
Wr[T)
dT <
1 du u dT
0
(t) < u* EE Co\fch + C4 for any t > 0.
u(t)2 dT < C3u
2
(2.13)
From (1.1)1, with the help of Lemmas 2.2, 2.4 and (2.12), it is clear how to get the inequality
||A<^(t)||2 < C5 for any t> 0. (2.14)
t
2
2
2
2
2
2
2
2
2
2
e
2
2
t
t
2
2
Lemma 2.5. For any t > 0, we have
u(t) > C6.
Proof. Rewrite (1.1)2 for 9 = 1/u:
,, A 1 , ,,dp Yt=~MA - + {tup + b)-g+9.
(2.15)
(2.16)
Multiply (2.16) by 9p 1 (p > 4) and integrate it over Integrating by parts and using Young's and the Sobolev-Poincare inequalities, we get d
l\\9(t)\\lpin)+Mp(p-l) J 9P-i\V9\2dx+pf3 j + 9edT j 9^ dx
n r r n
r n n n
<
ppu* sup 9e J 9pdT + ^^ ^ J (W2
+
9~W\Igdx
n
5/3
6/5 \ 5/3
dx
n n n
(J^ + + op (J
1/3
1/3
C7M\J \ 'r '''at' J ~~J 4
nn
for any constant C7 > 0. Here we used
jgdxj9P-1 dx = („ — dJ9. d^/9P-1 dx.
n n r n
In both sides of the above inequality, we use the following inequalities by virtue of the interpolation inequality, the embedding theorem, (2.10), and (2.14):
J 9P-4\V9\2dx= J \y9[p-2)/2\2 dx
nn
2 / \ 1/3
>
2
p—2
■^(J*™*) — f9P- dx)
n
'(cq2 — 2/3)(y 93(p-2) dx^j
>
J ((ap + b)^/2^ ' cbj ' <(aC5 + \b\)2^j
2
p — 2
1/3
dp
dt
n
6\ 1/3
\ 4/3
93p/4 dxj
<
(aC5 + |b|)2
n* 2 C0
dp
dt
wl( n)
93p/4 dx
4/3
2
ep/2
g dx
6/5 x 5/3
dx
g dx
03p/4 dx
4/3
< C2||Vg||2(| 03p/4 dx
4/3
Substituting these for the respective terms in both sides, discarding the fourth term in the left-hand side, and choosing C7 = C02 — |Q|2/3, we find
1/3
lll^ll^ + T^2"1"12/3)(/'3(p-2)<^
n
< p2Cs(t)^J 03p/4 d^ 7, (2.17)
since p > 4 and p(p — 1)/(p — 2)2 > 1. Here
cm- c° iiv^mii2 i h[t? (aCs +1&|)2
with C7 = C02 — |Q|2/3.
First, letting p = 4 in (2.17), we get
d^ at
(t)
W(n)
d_
dt1
H^H^n) < 42C8(t)(|03 d^ V3 < 42C8(t)M1/3||0(t)||L4(n),
from which by Gronwall's lemma, (2.1), and (2.10) it follows that
||0(t)||i4(n) < C9 for any t> 0.
(2.18)
Now, introduce the sequence {pk}£=0 defined by p0 = 4, pk = 4pk-1/3 (k 1, 2,3,...). Integrating (2.17) with p = pk over (0, t), we obtain
HW4 (n) <MP4 (n) + plf c8(t M
|Lpfe-i(n)
dr.
(2.19)
Set
6fc(t) = max{1, ||0o|U„(n), sup ||0(r)||L(n)}.
T £(0,t) k
Then (2.19) becomes
Bk(t)pk < + |Q| + JC8(t) drj pkBk-1(t)pk = C1opkBk-1(t)pk,
and thus, by induction,
ek(t) < cUPkPl/Pkek-i(t) < ... < cfi f[p2/Pse0(t), m* = £ -■
j=1 j=1pj
2
1
1
La(n)
2
r
Since it is easily seen that
Mk
pi Pi "pi V4 J Po ~ 4'
k ( k 2 \ 2
n p2/p3 = exp —logpj -exp —logpj
j=1
we can conclude which means that
Vj=1
Pj
j=1
Pj
/2 j
< exp Y^ ) ( 3 log i + logpo ) ) < +00,
lim Bk(t) < C11
k
lim sup |9(r^r (n) < C11, o<T<t k
or, equivalently, we get (2.15) with C6 = 1/C11. □ Lemma 2.6. For any t > 0, we have
||VAp(t)||2 +
OVp
dt
(t)
+
d2p dr2
(r)
+
dAp
dr
(r)
dr < C12. (2.20)
Proof. Inequality (2.20) is proved by following the proof of Lemma 3.5 in [1]. Indeed, differentiate (1.1)1 with respect to t, multiply it by d2p/dt2, and integrate over After integrating by parts, we get
K
<
d_
dt
d2p dt2
OVp
dt
(t)
(t)
+ 2
d2p dt2
(t)
+2{np\\2w?m+l+au(t))'
dp ~dt
(t)
+2(a||p||w22 (n) +
du ~dt
(t)
Whence, integrating the above over (0, t), by virtue of (2.10) and (2.12)-(2.14), we get
dVp
dt
(t)
+
d2p dr2
(r)
From this it easily follows that
||VAp(t)||2 +
dAp
dr
(r)
dr < C13.
dr < C14
by the equations derived from applying d/dt and V to (1.1) 1. □ Lemma 2.6 yields
t
f || Au(r)||2 dr < C15
(2.21)
by integrating (1.1)2 multiplied by u 2Au over Qt with the help of (2.13)-(2.15). For further uniform estimates we assume N = 2.
t
2
2
2
2
2
2
2
2
2
t
2
2
t
2
Lemma 2.7. Let N = 2. For any t > 0, we have t
du ~dt
(t)
2
+
Wa
dT
(t )
2
and
+
lim
t
1 du u dde ,
+ 2 97
L2(r)
dT < Ci6
(2.22)
du ~dt
(t)
= 0.
Proof. The assertion of this lemma is proved by following the first part of the proof of Lemma 3.6 in [1]. In fact, differentiation (1.1)1 multiplied by u2 with respect to t derived by differentiating (1.1)2 multiplied by u-2 with respect to t, we get
1 d 2u dAu 2 M—— + —
i2 dt2
dt
du ~dt
-{a<p + b)—-al-
dt'
(2.23)
Multiply (2.23) by du/dt and then integrate over il. Integrating by parts, due to Young's inequality and (2.13), we have
1 d Ydt
1 du . . u dt
<
+M
dVu
dt
3
(t)
+ß
1 du u d0e u~dt + 2 ~dt
1 du\ u
-— dx + —
uot ) 2
*2 / u\ d2{P (a<p + oJ-^ttt + a
dt2
+
(t)
dip ~dt
L2(r) 2
+
1 du u dt
+
dg_ dt
ßu*2
2
dde
dt
L2(r)
By the interpolation inequality, Lemmas 2.4-2.6, and (2.14), this yields that
d_
dt
1 du . . u dt
+ C17
+ 2M
dVu
dt
(t)
c
-0s
C6
d2<p dt2
(t)
+ Ci7
ay
dt2
<M 2
2
+
du ~dt
du ~dt
dp
dt
(t) (t)
+ 2ß du
1 du u dde u~dt + 2 ~dt
dt
dp
dt
(t) (t)
+
1 du u dt
(t)
(t)
L2(r) 2
dt
(t)
+
dde ~dt
(t)
L2(r)
(t)
(t)
dp ~dt
+
C66M
(t)
dp ~dt
du ~dt 2
(t)
(t)
+
dg_ dt
(t)
1 du u dt
2
+
(t)
dde dt
(t)
L2(r)
. (2.24)
By Lemma 2.1, due to Lemmas 2.2-2.6, the assertion of Lemma 2.7 follows. □
By Lemmas 2.2, 2.4, and 2.7, (1.1)2 and its differentiation with respect to x, it implies that
t
||Au(t)||2 + y ||VAu(t)||2 dT < Ci8 for any t> 0. (2.25)
2
2
2
2
u
2
2
2
2
2
2
1
2
2
2
2
2
2
2
2
l
4
2
2
l
2
2
l
Lemma 2.8. Let N = 2. For any t > 0, we have
dt
(t)
+
1 du d0e ., ,
+
and
lim
t^oo
¿2(r)
d2u d
Wu
(r )
+
dAu
dr
(r )
dr < Cv
(2.26)
dt
(t)
0.
Proof. Multiply (2.23) by d2u/dt2 and then integrate it over Q. By integration by parts and Young's inequality, we find
M d ~2 dt
Wu
<
1
dt
1 d2u u dt2
+ cft
(t)
+
1 d2u . u dt2
+
(t)
+ C
20
du ~dt
f3_d_
2 dt
+
1 du d6>e ■, , ulit^lH^
¿4(H)
(ap + b)
1 du 3 + ¿3(r) du 2 d2ee
u dt ~dt + ¿2(r) dt2
<9i2 2
¿2(r)
+
+
dp
¿2(r)
4
dt
ftd_ 2 dt
+
¿4(H)
d0e
'at
dr.
Similarly to (2.24), this inequality, trace theorem, the interpolation inequality, and Lemmas 2.4-2.7 yield
dt
< s
Wu
dt du
(t)
dt
(t)
+
+ cS-
1 d2u u dt2
+ ft
Wu
dt
(t)
1 du d6>e . , ,
du ~dt
+
d2p 2 + dp 4 dg 2 d2ee
dt2 dt + 1 dt + dt2
(t)
¿2(r)
+ C21
ft d
¿2(r) du
dt
+
2 dt
(t)
dtfe
'at
+
du
(t)
dt
dr, (2.27)
where S > 0 is a constant determined later. Here we used the inequality ||u|||3(r) < c||u3/2||i||u3/2|| < S||u||2 + cS-1(yVuy4 + ||u||4). Next, squaring (2.23) and integrating it over Q, we obtain
M2
dAu
dt
(t)
<
c62
1 d2u u dt2
+
C66
du ~dt
C22
(t)
d2p
dt2
+
dp
dt
+
dg
dt
(2.28)
Then, add (2.27) and (2.28) multiplied by MC|/(2c) and choose S = M2C|/(4c). This implies
dVu, , 2 11 d2u 2 M2C2 dAu
dt
dt
(t)
+ ■
1 d2u u dt2
+ ■
4c
dt
+ ft
dt
1 du d0e\/.
¿2(r)
2
2
t
2
2
2
2
2
2
2
2
2
2
2
2
4
4
2
2
1
2
1
2
2
4
2
2
c
c
1
2
4
2
1
2
< C23 du . . Tt{t) 4 + 1 du . . 2 + 1 d2tp dt2 2 + dp dt 4 + 1 dg dt 2 +
+ dde dt 2 + ¿2(r ) dde dt
d2de
dt2
¿4(r )
¿2(r )
(2.29)
Applying Lemma 2.1 to (2.29), due to to Lemmas 2.2-2.7, we conclude (2.26). Last assertion in this lemma follows from the last claim in Lemma 2.1. □
From Lemmas 2.2, 2.4, 2.6, and 2.8 and the equation of (1.1)2 differentiated with respect to x, we obtain
||VAu(t)||2 < C25 for any t> 0. Finally, recall the equation
A
(2.30)
MA2i
1 du dp
V2m+[aLp + b)m+g
derived from (1.1)2. Squaring this equation and integrating it over Qt, we arrive at the estimate
J ||A2u(T)
■ dr< C2,
(2.31)
by (2.21) and Lemmas 2.2-2.7.
Therefore, (1.6) and (1.7) are followed from Lemmas 2.2-2.6 and Lemmas 2.7, 2.8, respectively, which concludes the proof of Theorem 3.
REFERENCES
1. Tani A., "On phase-field equations of Penrose—Fife type with the non-conserved order parameter under flux boundary condition. I: Global-in-time solvability," Mat. Zametki SVFU, No. 1, 29, 101-120 (2022).
2. Penrose O. and Fife P., "Thermodynamically consistent model of phase field type for kinetics of phase transitions," Phys. D, 43, 44-62 (1990).
3. Penrose O. and Fife P., "On the relation between the standard phase field model and a 'thermodynamically consistent' phase field model," Phys. D, 69, 107-113 (1993).
4. Brokate M. and Sprekels J., Hysteresis and Phase Transition, Springer (1996). (Appl. Math. Sci.; vol. 121).
5. Horn W., Laurençot Ph., and Sprekels J., "Global solutions to Penrose—Fife phase-field model under flux boundary conditions for the inverse temperature," Math. Meth. Appl. Sci., 19, 1053-1072 (1996).
6. Shen W. and Zheng S., "On the coupled Cahn-Hilliard equations," Commun. Partial Differ. Equations, 18, 701-727 (1993).
2
4
Submitted April 4, 2021 Revised February 18, 2022 Accepted February 28, 2022
Atusi Tani
Department of Mathematics, Keio University,
3-14-1 Hiyoshi, Yokohama 223-8522, Japan tani@math.keio.ac.jp
