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4Математические заметки СВФУ Январь—март, 2022. Том 29, № 1
UDC 517.95
ON PHASE-FIELD EQUATIONS OF PENROSE—FIFE TYPE WITH THE NON-CONSERVED ORDER PARAMETER UNDER FLUX BOUNDARY CONDITION. I: GLOBAL-IN-TIME SOLVABILITY A. Tani
Abstract. We study the initial-boundary value problem for the non-conserved phase-field model proposed by Penrose and Fife in 1990 [1] under the flux boundary condition for the temperature field in higher space dimensions, which is obliged to overcome additional difficulties in the mathematical treatment. In all the existing works concerning this problem, only one due to Horn et al. [2] was discussed under the correct form of the flux boundary condition. Here we prove that the same correctly formulated problem as theirs is well-posed globally-in-time in Sobolev—Slobodetskii spaces. Moreover, it is shown that the temperature keeps positive through the time evolution.
DOI: 10.25587/SVFU.2022.97.11.008 Keywords: non-conserved phase-field equations, Penrose—Fife type, flux boundary condition, strong solution in Sobolev—Slobodetskii spaces.
Dedicated to Professor Ivan E. Egorov on his seventieth birthday
1. Introduction
In this paper we are concerned with the non-conserved phase-field equation proposed by Penrose and Fife [1,3] which is a continuum model for the description of dynamics of order-disorder phase transition taking into account of both the relaxation and the balance laws based on the second law of thermodynamics:
where p is an order parameter, 0 is an absolute temperature, M1, M2 and K are positive constants, e = e(p, 0) is the internal energy defined by
e = f~0% (1.2)
with the free energy density f = f (p, 0), and g = g(x, t) is a heat supply. The free energy density f (p, 0) assumed in [1, 3] is of the form
f{ip, e) = -cv(e\ne + s)-^tp2-btp + c- eso(v) (1.3)
© 2022 Tani A.
with given constants CV > 0, a > 0, S > 0, b, c and a concave function s0(p). The most commonly used form of s0 is the double equal-well potential,
1 2
*>(¥>)=(l-V2) . (1-4)
Inserting (1.2)-(1.4) into (1.1), we have
% = + v + Cvft-(a<P+b)d^ = -M2(1.5)
We consider (1.5) in a bounded domain Q c RN (N = 2, 3), with a smooth boundary T. The boundary conditions imposed on T are:
where n = n(x) is an outward unit normal vector to T at x G T, fl is a heat conductivity on the boundary T and de(> 0) is an external temperature. The latter condition for 6 is the so-called flux boundary condition.
A phase transition in two-phase system has been theorized as a continuum in which two phases may coexist, so that the transition between them is considered to occur smoothly within an appropriate layer or diffuse interface. The use of diffuse interface models in describing the phase transition is traced back to van der Waals [4,5], Landau [6] for the second order phase transitions by introducing the notion of order parameter, and Devonshire [7] for the first order transitions. Then, the theory has been extended as continuum models for the description of dynamics of order-disorder phase transition by Cahn [8-10] (Models A and B in [11]; see also [12,13]), Caginalp [14], Penrose and Fife [1,3] and so on.
In the sequel, without loss of generality, we assume that CV = M1 = 1, M2 = M and fl is a positive constant. And it is more convenient to use u =1/6 instead of 6, so that our problem is formulated as follows:
" §f = KAip - (p3 + Lp + (atp + b)u, x G tt, t > 0,
|| + (aip + b)u2^ = Mu2Au-gu2, x G il, t > 0,
i£ = o, = x&r,t> o,
^ (tp, u)|t=o = (<^o,uo)(x), x G Q.
Concerning the mathematical results related to problem (1.6) with fl = 0 the existence of a unique strong solution was proved by Zheng [15] and Sprekels and Zheng [16] (see also [17]). In the framework of weak solutions of (1.6) and its generalization there are many papers (see, [18-22] and the references therein). However, for problem (1.6) with fl > 0, as was pointed out in [13], all results except [2] were reported under the physically incorrect flux conditions, for example, in [21]
and in [22]
d6
The aim of the present paper is to show the unique existence of a strong solution in Sobolev-Slobodetskii spaces in higher space dimensions, which are different from [2]. Moreover, a boundedness of the solution is shown up to an arbitrary finite time.
In what follows, we focus our study on the most important case, N = 3. Let us describe our results.
Theorem 1.1. Let T be any positive number, and ft be a bounded domain in R3 with a boundary r belonging to W23/2+l, l > 1/2. Suppose that g G W2'l/2(QT) (Qt = ft x (0,T)), (9e G W21/2+l,1/4+l/2(rT) (Ft = r x (0,T)), and po,«o G W2x+l(ft) satisfy inf uq(x) = Mq > 0, and the compatibility conditions up to order max{[Z —
3/2], 0}. Then problem (1.6) has a unique solution (p, u) G W22+l'(2+l)/2(QT») x W2+1,(2+0/2(Qt-) for some T* G (0,T] such that
\\(lP,u)\\w22+l,{2+l)'2(QT - )
+ \\^e\Wi/2+1,1/4+1/2 (pT)), (1.7)
and 1
u(x,t) > -u0 for any (x,t) G Qt*- (1-8)
Theorem 1.2. Assume that the hypotheses in Theorem 1.1 with I = 2, and
inf 0e(x,t)^0e>O, SUP ^GL2(r;^(0,T)) (1.9)
MerT te{o,T] Q.e lrl ot
hold, where |ft| is a volume of ft. Then the solution in Theorem 1.1 is extended on [0,T ]:
\(p,u)\w24.2(QT) < C(T), u(x,t) > C*(T) for any (x, t) G Qt.
Here C(T) and C*(T) are positive constants depending non-decreasingly on both the data and T.
Remark 1.3. Our proof is applicable to the similar problems with more general f, for example,
/(¥>, e) = -Cvidind + s1e + s2) + ^(d- rv - ^p3 + ^-V + ^V
2 3 4 6
with positive constants CV, a1, d* and nonnegative constants 51, ¿2, a3, a4, a6
satisfying a4 + a6 > 0 proposed by Alt and Pawlow [23]. The corresponding results
to Theorems 1.1 and 1.2 still hold for such an f.
Throughout this paper we use the standard Sobolev-Slobodetskii spaces WJ(ft) with norms \H\r(\\-\\ = ||-\\o) and the anisotropic ones Wj's(Qt) = L((0,T; W2r(ft))n L2(ft; W2s(0,T)) for non-negative real numbers r, s and T(< By W|(0,T;
WJ(ft)) we denote the function space of u(t) defined on (0, T) with values in WJ(ft) such that u(t) G W|(0, T). In particular, the case s = r/2,
W(J't/2(Qt) = L((0,T; WJ(ft)) n L((ft; W(J/2(0,T)),
is mainly used. For a smooth manifold dQ = T, the spaces WJ(T) of functions defined on T are introduced in a standard manner by means of the local coordinates and the partition of unity, and W2"'s(TT), TT = T x (0, T), can be defined in the same way as above. The spaces of vector fields whose components belong to, for example, W2's(QT) are denoted by the same notation as the scalar case, W;T's(QT), and their norms are supposed to be equal to the sum of norms of all its components. In detail, see [24-26].
In what follows, we also denote by c the universal positive constants which may vary in different places; by Ci and Ci(T) (i = 1,2, 3,...) the positive constants depending non-decreasingly on the data but not on T, and may depend non-decrea-singly on both the data and T, respectively; besides some constants depend on the indicated quantities.
We give proofs of Theorems 1.1 and 1.2 in Sections 2 and 3, respectively.
2. Local-in-time existence: Proof of Theorem 1.1
((Step 1)) For simplicity let us assume 1 > l > 1/2, and consider a linear problem
dv dt
d(x, t)Av + f (x, t) (x G n, t > 0),
f- = h(x, t) (ier,t> 0), v\t=0 = 0 (x G iî)
(2.1)
where d(x,t) G w1'1/2(Qt), f (x,t) G W1//2(Qt) and h(x,t) G W21/2+M/4+i/2(TT) are given functions satisfying d(x,t) > d = const > 0 and the compatibility condition.
The theory of linear partial differential equations of parabolic type [25, 27] implies that problem (2.1) has a unique solution v G W^+i'1+i/2(QT) satisfying
(2.2)
Let us denote the extension of (p0,u0) by (p0,u0) G w2,+1,1+1/2(Qt) satisfying ll(p0, "0)||w2+i,i+I/2(Qt ) < c|(P0 J u0)| W21+l (n).
When we choose in (2.1)
d = K, h(x, t) = —= hi, d n
dp
f(x, t) = -p3 + p + (ap + b)u--— + KApo = /i(p, u),
the unique solution v gives p — p0; we choose in (2.1)
d = Mul = =
M \u
f{x, t) = -(ap + b)u- gu2 - ^ + M(u2 - u20)Au + Mu2Au0 = /2(p, u),
dt * dt the unique solution v gives u — ïïo.
In order to estimate fi, f2, hi, h2, we rely on the following lemma:
Lemma 2.1 [26, Lemma 3.5]. If wi G W2m'm/2(Qr), G W2m''m'/2(QT) with m > 5/2, m > to' > 0, then wiw2 G W2" 'm /2(QT) and
||WiW2|
W2
/2(QT) - l|wi|lWTm/2(QT)^W2^W2m'-m/2(QT)•
By using Lemma 2.1 and interpolation inequalities it is easy to obtain
llf
ill™ rl<l/2t
- OM^+M + I^Q,) + a|u|W22+I'1 + I/2(Qt) +1) l^lW2-'/2
(Qt)
(Qt)
U|W2''/2(Qt) + (K + 1)l<^o|W2 + I'1 + I/2
W22
2(Qt)
- C^ ||p||
(Qt)'
,1 + I/2
(Qt)
+ |b|(e2 + CE2t)lulW22+I'1+I/2(Q) + c(K + 1)|^0|w21+i(O) ,
(2.3)
llf2lw2',/2(Qt) - ((al^lW22+'-1+'/2(Qt) + |b|)
dp
dt
+
/;,;/2(
Iw2,I/2(qt )
X |u|W2+l',1 + l'/2 <n.\ + M l|Au|lW2-'/2(Qt)Hu2 - MÜ|w.2+''1
2(Qt)
w2'I/2(Qt)l
n',1+; '/2
(Qt)
+ (M II + 1)ll u0y W2+i,1 + i/2(Qt)
< C3(|p|w22+l'1+l/ 2(Qt), |u|W22+l'1 + l/2(Qt), I|g|IWll/2(QT), ||u0|W21+l(n)) x (^3 + Ce3 i)|u|w22+l'1+l/2(Qt) + c(M I|u011 W2+l (n) + 1)ll u0|W21 + l(n)'
(2.4)
Here l' is any number, l' G (1/2, l); C2 and C3 are positive constants depending on each indicated argument monotonically non-decreasingly; ei is any positive number; CEi is a positive constant which increases monotonically as ei tends to 0 (i = 1,2, 3).
Trace theorem leads to
|h1^W 1/2+l,1/4+l/2(n) < C4|P?0|w2+l,1+l/2(Q ) < cC4 1P0 1 w1 +
(2.5)
|h2|W21/2+i,1/4+i/2(pt)
<
A
M
W21/2 + I,1/4+I/2(Pt)
ß
+ T7llöellw3/2+!'1/4+!/2(rT) + C4||mo||w2+m+V2
M1
(Qt)
1
^ C5 ( \\U\\W2+I,l + 1/2(0^V — ) (£4+Ce4t)\\u\\w2 + l,l + l/2
ÜO
(Qt)
(Qt ) ¡3
+ ]\^ll6'ellw21/2+M/4+!/2(rT) + cC-i\\uo\\w^+l(n)' (2-6)
where C5, e4 and CE4 have the same properties as C3, e1 and CE1, respectively. Let A1 and A2 be any positive constants satisfying
Ai > 2c(1 + Ci(K + 1 + C4)) ||po||Wi+I(n)
and
A2 > 2c(1 + Ci(M|uo|w21+i(o) +1+ C4))|uolw21+'(n) +
2ßCi M
II*
e|Wl/2+l,1/4+'/2
(Ft )'
respectively.
Now we define the successive approximate solution {(p(m), u(m))}^=0 as follows:
(p(0) ,u(0)) = (po,ïïo),
p(m) - po is a solution to (2.1) with d = K, h = hi, / = /x(p(m-1),u(m-1)), u(m) — ÏÏ0 is a solution to (2.1) with
d = h = h2(u(m-1)), / = /2(p(m-1),u(m-1)) (m =1, 2, 3,... ),
provided
(p(m-1),u(m-1)) G W2+M+l/2(QT) x W2+M+l/2(QT), IP(m-1)|w22 + - + "2(QT ) < A1, H«(m-1)Hw2 + ,.1 + ,/2(QT ) < A2. Estimates (2.2), (2.3), (2.5) imply
A
\\^m>\\w2+l,1+l/2(Qt) < -f+ClC2(Al,A2)Al(el+Ceit) + \b\ClA2(e2 + Ce2t), (2.7) and (2.2), (2.4), (2.6) imply
A2
Hw22+!'1+!/2(Qt) < —+CiC3(A1,A2, \\g\\w^/2(Qty llMo||H/2i+!(0))A2(e3+C'e3i)
+ A2CiC5(A2,l/w0)(e4+Ce4i). (2.8)
First we choose e1-e4 so small that
Ci (C2(A1; A2)Ai£i + |6|A2e2) < Ci(C3(AI,A2, ||ff||wM/2(QT), lko||ww(n))£3 +C5(A2, l/wo)e4) <
1
hold, and then choose T0 G (0,T] to satisfy
C1(C2(A1,A2)A1Cei + \b\A2Ce2)T0 <
Ci (c3(Ai, A2, ||ff||w.,./2(QT), ||«o||wi+.(n))Ce3 + C5 ^A2, To < i.
1
¿o,
Thus, from (2.7) and (2.8) we conclude by induction \\¥>(m)\\w22+'.i+i/2(QTo) <Ai, \u(m)\w22+i ,i+i/2(QTo) <A( (m = 0,1, 2,...). (2.9)
((Step 2)) We show that |(p(m),u(m))}~=0 is a Cauchy sequence in W22+1'1+1/2(Qt-) x W22+1'1+1/2(Qt-)
for some T* G (0,T0]. For that we begin with the estimates f1(p, u) — f1(p', u'), f2(p, u) — f2(p',u') and h2(u) — h2(u'), where p, p' and u, u' satisfy (2.9)1 and
(2.9)2, respectively. In the similar manner as (2.3), (2.4) and (2.6) by the help of Lemma 2.1 and (2.9) we have
u) - /l(p', u')IIW2'l/2(Qt) < (3A2 + oA2 + + C£5 t)||p - P'|W22+l'1 + l/2(Qt) + (0A1 + |b|)(e6 + Ceet)||u - u'|| (2.10)
|/2(p,u) - /2(p',u')|W2,l/2(Qt)
< A2(3aA1 + |b|)(e7 + CE7t)||p - ^'|wHi,1+i/2(Qt)
+ (A2(2aA1 + 2|b| + 2||g||w2,l/2(QT) + (2 + M + Mc||u,||
x (es + Ce8T)|u - u'1 1+l/2(Qt), (2.11)
1 h2 (u) - h2(u')|W_l/2 + l,1/4+l/2(rt)
<C5(a2,-^ (£9+CE9t)\\u-u'\\w2+lll+l/2(Qt) (2.12)
for any t G (0, T0]. Here again ei is any positive number; CEi is a positive constant increasing monotonically as ei tends to 0 (i = 5, 6,... , 9). As in ((Step 1)) p(m) - p(m-1) satisfies (2.1) with
d = K, h(x, t) = 0, /(x,t)= /1(p(m-1) ,u(m-1)) - /1(p(m-2) ,u(m-2)), and u(m) - u(m-1) satisfies (2.1) with
d = Mu0, h(x,t) = h2(u(m-1)) - h2(u(m-2)),
/(x,t) = /2(p(m-1),u(m-1)) - /2(p(m-2),u(m-2)). Hence, inequalities (2.2) and (2.10) yield
|p(m) - p(m-1)|w22+l,1+l/2(Qt)
< C1 (3A2 + aA2 + 1) (es + CE5t)|p(m-1) - p(m-2) ||W2+l,1+l/2(Qt)
+ C1(oA1 + |6|)(ee + Ceet)||u(m-1) - u^H^+m+i^, (2.13)
and inequalities (2.2), (2.11) and (2.12) yield
|u(m) - u(m-1)|w22+l,1 + l/2(Qt)
< C1A2(3oA1 + |b|)(e7 + CE7t)||p(m-1) - p(m-2) |w22+M+I/!(Qt) + C1 [(A2(2oA1 + 2|b| + 2|g| w i,I/2(„t ) + (2 + M )A2) + Mc|«clW 1+l(
'w2'1/2(qt) ' "" >^2> W21+'(0)j
X (e8 + Ceat) + C5 (a2, (s9 + Cegt)
x\\u^-r>-u^-2'>\\wr,1+l/2{Qt). (2.14)
2
2
Now we take £5-69 first and then T1 G (0, T0] as
1
4Ci:
(3A? + 0A2 + 1)£5 + A((3aA1 + |bj)eT <
(0A1 + jbj)£6 + (A((2aA1 +2jbj + 2\\g\\w2. '/2(qt ) + (2 + M )A() + Mc\\u0\\W2i+'(n)) 68
+ c5 £9 < 7i-;
TI((3A? + oA2 + 1)C£b +A2(3aAi + |6|)Ce7) <
4C1
«0/ 4Ci 1
T1
(0A1 + jbj)Cee + (A( (20A1 + 2jbj + 2\\g\\w2 , ./2(qt ) + (2 + M )A()
+ Mc||U0||^2l+,(o))Ce8 + C5 U2, - ) Ce
12
1
< 4Ci:
so that (2.13) and (2.14) lead to
\P(m) — P(m-1)\W22 + 1 ■ 1 + 1/2(qt1 ) + — u(m-1)\w22+1 ■ 1 + 1/2(qt1 )
2vmv^ r IIw2 (qt1 )
(m-1) — u(m-2))i ^
+ \\u(m-1) — u(m-2))\w22+i , i+i/2(QTi)). (2.15)
This means that there exists a limit function (p,u) of (p(m),u(m)) on [0,T1] in the topology of W2+l,1+l/2(QTl) x W2+l,1+l/2(QTl), which satisfies
wpww^1 , 1 + 1/2 (qt1 ) < A1, \u\w22+1 • 1+1/2(qt1 ) < A2. (2.16)
The uniqueness of the solution follows from inequality (2.15) with (p(m), u(m)) = (p,u) and (p(m-1),u(m-1)) = (p',u') by assuming that (p,u) and (p',u') are two solutions of problem (1.6).
As for (1.8) it is easy to see that
t
/'du 1
— (x,r) dr > u0 — VtA2 > —Uq (2-17)
0
holds if t (>)0 is smaller than T2 = (u0/(2A2))2.
Set T* = min{T1,T2}, so that the proof of Theorem 1.1 is complete.
3. Global-in-time existence: Proof of Theorem 1.2
Assume that solution (p, u ) to problem (1.6) belongs to W24'2(Qt) x W24'2(Qt) and has properties (1.7), (1.8) for any T > 0. First we derive a priori estimates of (p, u) on [0, T]. 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 3.1. For any t G [0, T]
l|Vp(i)||2 + Mi)||l4(0)+ J (¿+logu) (x,t)dx
n
t 2 t + /( +||V«(t)||2)dt + J dr J (oeu + ^j(x,T)dT <Ce(T). (3.1)
0 r
Proof. Add (1.6)1 multiplied by dp/dt and (1.6)2 multiplied by 1/u, and integrate it over Q. Then we have
Jt (f l|Vp(i)||2 + \Mmltm -\Mm2 + /logudx
+
dp ~dt
(t)
+ M || Vu(t) |2 + J gudx + /3 J ee u dr = / |r |. (3.2)
Integrating (3.2) over (0, t), we easily see that 1
K 1 r
Yl|Vp(i)||2 + -||p(i)||l4(0)iir + J logu(x,t)dx
n
t t + M || Vu(r )||2^ dr + J dr J gudx + / J dr J 0eudr 0 n 0 r
<fl|Vpo||2 + i||po||l4(o)+ y"log«o(a;)da; + ^|r|i + i|ii|. (3.3)
Poincare inequality
Kt)ll<c||Vu(t)||+^ Ju(x,t)dT
(3.4)
implies
J dr J gu dx 0n
< |g(T )||u(T )|| dT
< Y j l|V«(r)|| I \\g(r)rdr + ^ I \\g(r)\\dr J 6eu(x,T) dT.
2
t
t
t
t
t
Therefore, we have
K 1 t
Yl|Vp(i)||2 + -||p(i)||l4(0) + J logu(x,t)dx
t 2 t dp
' —1| vm^tj ||" | cit + p' i cit i t/eu i
+ yl|vu(r)||^jdr + fi' j dr j 6eudT
0 0 r
<fl|Vpo||2 + i||po||l4(o)+ J loguo(x) dx + p\T\t + ^\i
with /3'= /3- sup ||<7(t)||vW(2e|r|) >0.
0<t<T
Next, multiply (1.6)2 by u-2 and integrate it over ft. Then one get
Jtj (u ~ ~ dx + 13 J udT = f gdx + 13 J 9edT>
n r n r
from which it readily follows
t t u(x,t)dx + (3 / dr -{x,t)d,T < —\\<p(t)\\li{n) + I dr / \q{x
j\{x,t) dx + 13 j dr j\{x,r) dT < ^Mt)\\i4{n] + j dr j | g(x,T)\dx
n 0 r 0 n
t
+ /3 j dr j 6e(x, t) dT + ^(o + \b\)2 + ^ |ft| + j 1- dx. (3.6)
ye(x, t) ai + | [a - , ^
0 r n
Since logu + 1/u > 0 holds, adding (3.5) to (3.6) leads to (3.1). □ Lemma 3.2. For any t G [0, T]
1
CVCrr1 < f -(x,t)dx<C7(T). (3.7)
u
n
Proof. The upper bound is derived from (3.6) and (3.1) with the elementary inequality — log u < 1/u. Moreover, (3.1) leads to
u
j log u(x,t) dx + j —(x,t) dx<Ce{T),
nn so that the lower bound is derived from Jensen's inequality
"108(m / ludx) - -M/log ludx = MIlogutdx- D
n n n
Put
u(t) = minu(x,t), u(t) = max u(x,t).
x£Q x£Q
Then (3.7) implies
u(t) < C7(T)\il\, u(t)>
|Q|
c7(ty
It is easily seen from Poincare inequality that for any t G [0,T]
(3.8)
IKi) - w(i)H + \\u(t)-u(t)\\ < c(||Vw(i)|| + 1).
(3.9)
Lemma 3.3. For any t G [0, T]
dp ~dt
(t)
+ II Vu(t)|2 + J(u0e - log(u0e)) dr
+
dVp
dT
(T)
+
1 du. u dT
+
dp, ,
dT < Cg(T). (3.10)
Proof. Adding (d/dt)(1.6)1 multiplied by dp/dt and (1.6)2 multiplied by u-2du/dt, and integrating it over Q, we have
1 d Ydt
dp ~dt
(t)
+ MII Vu(t) I 2 + 3 J (u0e - log(u^e)) dr r
+ K
dVp
dt
(t)
+ 3j |p|2
dp
dt
dx +
1 du . . u dt
dX + i nn
< au(t)
dp
dt
+
dp
dt
+
1 du u dt
d0e
dt
+ (3.11)
L2(r) V ^e,
Insert the inequality
u(t) <
|Ki)-Tl(i)|| + |Ki)-M(i)||+2i(
<
3
(c| Vu(t)| + c + C7(T) | Q |3/2) = c| Vu(t)| + C7 (T),
which is derived from (3.8) and (3.9), into the right most hand side of (3.11), and
t
2
2
2
2
2
2
2
2
2
2
1
2
3
integrate the rewritten (3.11) over (0,t). We have
dp ~dt
(t)
+ M||Vu(t)||2 + ß j(u0e - log(u0e)) dr
t
+ 2/1 K
0
dVp
ör t
(t )
< c(o + 1)/(
dp
+ 3
(t )
P
dp
+
1 du u dt
(t)
dT
+ ||Vu(t)||2J dT + C9(T)
= c(o + 1)Y (t) + Cg(T). (3.12)
Solving the differential inequality
we obtain
c(o + 1) V min{1, M}2 Thanks to (3.13), (3.10) immediately follows from (3.12). □ From (3.10) it easily follows
(3.13)
du
(t)
dt
1 du u dT
Z(t)2 dT < Cs(T)u*(T)2
(3.14)
u(t) < u*(T) = c\JC$(T) + C'7(T).
From (1.6)1 and its differentiation with respect to x
KVAip = ^^ + 3p2Vp - Vp - cwVp - (ap + 6)Vu,
it is clear to derive by the help of (3.1), (3.10) and (3.14) t
l|p(t)|lw22(fi) + / (||P(
t )|3 +
dp dr
(t )
dT < Cio(T) for any t G [0,T]. (3.15)
Lemma 3.4.
> Cn(T) for any t G [0,T],
(3.16)
Proof. Multiply (1.5)2 with Cy = 1, M2 = M by 0p-1 (p > 4) and integrate it over ft. By integration by parts and Young's inequality, we get
d_
dt
||0(t)HLp(n) + Mp(p - 1)| 0p-4|V0|2 dx + ppßI dr
n r
= pß J OP-^edT+p J eP-^acp + b^dx+p J g&P~x dx
2
2
2
1
2
2
2
t
t
2
2
1
n
a<p + b)^ep/2y/5 dx^j5/3 + C12M(^Je3(-p-2) dx^j
4C12M
1/3
+
4C12M
5/3
Using Nirenberg-Gagliardo inequality, the embedding theorem from W2(Q) in L6(Q) with a constant C0, |v|L6(n) < |M|W_1(n)/C0, (3.10) and (3.15), we find that
0p-1 |V0|2 dx
p - 2
V0(P-2)/2
dx
>
p - 2
\ 1/3
CJ / 03(p-2) dx - 0p-2 dx
0x
nn
6/5 x 5/3
(j + *)
< (oC10(T) + |b|)5
dp
dt
6\ 1/3
<
(oC10(T) + |b|)2
C0
(/*Sp/1 dx)1/3
n
2 /f \ 1/3
( I 03p/1 dx
W1(n)
dp
dt
5/3
<
cHlffllw^n)
03p/1 dx
n
1/3
p(p - 1) (j^T^j J 0P~2 dx < PiP ~ !) (^2) ^ / dP l d'X
for any positive constant C12.
Therefore, choosing C12 < C0, we obtain
03p/1 dx
1/3
d_
dt
II0(t)ILp(n) + (C0 - c^M^ 03(p-2) d^ V3 +
p
Lp(r)
1/3
< 3INILp(r) + P2C13(t^ 03p/1 dxj , (3.17)
n
p
2
2
2
2
2
p
since p > 4 and p(p - 1)/(p - 2)2 > 1. Here
dVp
_ (oCio(r) + H)2
Cl3(t) - 4MC0C12
dt
w^n)
1 —2 + 9M'°|1/3^)2.
^ 4MC0C12lmV)nwiW ^ 4^7x2 First letting p = 4 in (3.17), we have
d \1/3
*)ll(n) < mil(r) +42C13(t^ 03 dx
< 3|0eH4(r) +42C13(t)|Q|1/3||0(t)Hl4(n),
from which by virtue of Gronwall's lemma, (3.1), (3.10) and (3.12) it follows
I|0(t)lL4(n) < C1i(T) for any t G [0,T]. (3.18)
Now let us introduce the sequence {pk}£=0 defined by p0 = 4, pk = 4pk-1/3 (k = 1, 2,3,...). Integrating (3.17) with p = pk over (0, t), we obtain
II№Pfc(n) < l«0ispk(n)
t t
+ // |0e(T)ip;fc(r) dT + plf C13(T)|0(T)ip;k_1 (n) dT. (3.19)
00
Set
Bfe(t) = max{1, |00|i„(n), II^||WrT), sup ||0(t)||l(n)}.
T e(0,t) k
Then (3.19) becomes
Bk (t)Pk < ^1 + |Q| + 3|r |t + J C13(T) dT^ p2efe-1(t)Pk < C15(T)pkBk-1(t)Pk, and thus by induction
Bfc(i) < c15(T)1/p^2/PfeBfc_1(i) < ... < C15(iy* npf^W. № = E--
j=1 j=1p
Since it is easily seen that
- v-< vT-V---
"¿i w PO ~4'
2 /2 2 1 2
n p2/Pi = exp —^exp [Y,—
j=1 \ j=1pj / \j=1p
~ exp\R) ^ 00 0io^+iogpo])<+o°'
one can conclude
This means
lim Bk(t) < Cia(T).
k
lim sup ||0(T)||L (n) < Cia(T), o<r<t
or equivalently, (3.16) with CU(T) = 1/Cia(T). □ Lemma 3.5. For any t G [0, T]
t
||VAp(t)||2 +
dVp
dt
(t)
+
9V
dr2
(t)
+
dAp
ÖT
(t)
+ ||A2P(T)||2j dT
< Cir(T). (3.20)
Proof. Differentiate (1.6)1 with respect to t, and multiply it by d2p/dt2. Then by integration by part we have
KT
dt
dVp
<
d2p di2
dt
(t)
(t)
+2
9V di2
(t)
"feir-s
= 2 J lv("3^2 +1 + au)% + +
n
dp ~dt
(t)
2
+2(o||p||w22(n)
du di
(t)
Hence, integrating this over (0, t), we have by virtue of (3.10), (3.12), (3.14) and (3.15)
dVp
dt
(t)
+
d2y dr2
(t )
dT < Cis(T).
From this it easily follows that
|| VAp(t)||2 +
dAp
dT
(t )
+ ||A2p(T)||2 dT < Ci8(T)
by using the equations of (1.6)1 operated by V, d/dt and A. For the last estimate in (3.20) we use the inequality
||Au(T)||2 dT < Ciq(T),
(3.21)
which results from integrating (1.6)2 multiplied by u 2Au over Qt by the help of (3.14)-(3.16). □
oo
2
2
2
2
2
2
t
2
2
t
2
t
Lemma 3.6. For any t G [0, T]
du ~dt
(t)
+
+
dVu
dt
(t)
dVu
dr
+
(r )
1 du d6>e . . .
L2(r)
+
dr2
(r )
2
+
dAu
dr
(t)
+ II A2u(r)
+
1 du u d0e .
+ 2 ' ^
L2(r)
dr < C20(T). (3.22)
Proof. Recall the following equation derived from differentiating (1.6)2 multiplied by u-2 with respect to t:
1 d 2u dAu 2 M—— + —
i2 dt2
dt
du 0t
-{aip + b)—-al-
dt'
(3.23)
First, multiply (3.23) by du/dt and then integrate it over ft. We get by integration by part and Young's inequality
1 d 2di
1 du u dt
(t)
+M
dVu
dt
(t)
+ 0
1 du u d0e ,
+ 2~dt ' (t)
1 du\ * (, , ^d2p , / dp\2 , d^du «atJ + + +
dt 7 dt
L2(r) 0 i. 2 / d0e
<
1 du\ 3 u*(T )2 u dr J 2
daH— u —— I dr
4 J V dt r
2
2
dt
+
2
1 du u dt
¡3u*(T)2 4~
+
d(9e
dt
L2(r)
From this, together with the interpolation inequality, Lemmas 3.3-3.6, (3.14) and (3.15), it follows
d_
dt
1 du . .
«at(t)
+ C21(T)
+ 2M
<
dVu
dt
(t)
Cu (T )3
du dt
+ 20
(t)
1 du u d0e .
+ 2~dt ' W
du dt
(t)
ay
<9t2
< 5
(t)
du 0t
+
dp at
(t)
(t)
+ C (T)
dp dt
du dt
(t)
(t)
+
+
+
1 du u dt
L2(r) 2
(t)
dg_ dt
(t)
+
1 du u dt
dt
2
(t)
L2 (r)
(t)
+ C21(T)
d2p
dt2
+ C22 (T) (3.24)
with a constant 5 > 0 determined later and C (T) = cC11(T) 6 5 1.
2
2
2
2
d2u
2
2
2
u
2
2
2
2
2
1
2
2
2
2
2
3
2
2
1
4
2
Second, multiply (3.23) by d2u/dt2 and then integrate it over Q. We have by integration by part and Young's inequality
M d ~2 dt
dVu
dt
(t)
+
du
1 d2u u dt2 2
(t)
+
/3 d_ 2 dt
1 du d6>e . , ,
«7*+uaT>(t)
¿2(r)
d2p
dp
dg\ \ d2
1 /du\3 _ „ f d20e du _ t ( dde
dt + —~r /
2 dt J V di
r
vA-dt dT + l3J dt2 dt
<
1 d2u u dt2
+ c£
(t)
+ C23 (T)
1 du u dt
+
L3 (r)
du dt
du dt
+
l4(o) 2
L2 (r)
(ap + b)
92p <9t2
2
+
dp at
+
d20e
dt2
L2 (r)
+
/3 d_ 2 dt,
dr
+
dtfe
'at
<9t
dr.
Like (3.24), due to trace theorem, the interpolation inequality, and Lemmas 3.3-3.6, this inequality yields
M
d_
dt
dVu
dt
(t)
+
< 5'
+ C24 (T)
+
1 d2u u dt2
du dt
+ 0
1 du d(9, u dt
(t)
+ c5'
Wa
du dt
(t)
d2p 2
dt2 +
dp
dt
4
+
1
4
+
1
du dt
dp
dt
4
(t)
(t)
+
dt
4
du dt 2
dt
\ 2
r (t)
t L2
du (t)
+ dt
2
№ +
1
+ dt2
du dt 2
(t)
+
/3 d_ 2 dt.
L2 (r)
d^e l~dt~
dr (3.25)
with 5' > 0 is a constant determined later. Here we used
||u|||3(r) < c||u3/2||i||u3/2|| < 5'||u||2 + c5'-1(||Vu||4 + ||u||4). Third, integrating (3.23) multiplied by dAu/dt over Q, one obtains
M
dAu
dt
(t)
<
MCii(T )2
+ C25 (T)
1 d2u u2 dt2
d2p
+
MCn(T )6
dt2
+
dp
dt
+
dVu . (t) 4 +
dp 4 + dg
dt dt
du dt
(t)
. (3.26)
Then, add (3.24), (3.25) and (3.26) multiplied by MCn(T)2/(2c), choose 5 = 5' = M2C11(T)2/(8c) and integrate the resultant inequality with respect to t. Con-
2
2
2
2
2
2
4
2
2
2
1
2
2
3
2
2
2
2
4
2
2
4
2
4
2
c
c
2
4
1
sequently, we conclude thanks to Lemmas 3.1-3.5
1 du u dt
(t)
+ M
Wu
t
+ J I 2M 0
dt
Wu
(t)
(r )
dr
+ 20
+ 0
+
1 du dé>e . . .
¿2(r )
d2u
(r )
2
+
M 2CU(T )2
4c
dAu
dr
(r)
1 du u dé>e \ û~îh + 21h)
¿2(r )
dr
< C26(T) y 0
Then, (3.27) leads to
du
dr
dr + C27(T ) = C26(T )X (t) + C27(T ). (3.27)
d_
dt
X(t) < 2C28(T)2X(t)2 + 2C29(T)2
(3.28)
where C28(T) = C26(T)/ min{u*(T)-2,M}, C2g(T) = C2r(T)/min{Cu(T)-2,M}. Solving (3.28), we find
X(t) < ^4^tan(2C28(T)C29(T)T) = CW(T). C28 (T )
Therefore the left hand of (3.27) is bounded from above by C26(T)C*0(T) + C2r(T). Finally, recalling
1 du
dp
u2 dt
dt
from (1.6)2, multiplying this by A2u and integrating it over Qt, one get by virtue of (3.21), (3.27) and Lemmas 3.1-3.5
t
J || A2u(r)||2 dr < C31 (T).
(3.29)
Estimates (3.27) and (3.29) lead to (3.22) with C20(T) = C26(T)C30(T) +
C27(T )+ C31(T ). □
Combining Lemmas 3.1-3.6 with Theorem 1.1 implies the existence and uniqueness of the solution in QT for any T > 0 to problem (1.6). Therefore, the assertion of Theorem 1.2 is proved.
2
2
2
2
2
2
2
4
1
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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
