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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(5),2049-2066
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.125209
˜až¢šg£‡46N3‘k.•þ
·½5
ééé
-Ÿ“‰ŒÆêƉÆÆ§-Ÿ
ÂvFϵ2023c47F¶¹^Fϵ2023c429F¶uÙFϵ2023c56F
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©ïÄ˜ašg£ž¢‡46N•§3‘k.•þN·½5"Äk^Garlekin•{
ïá)•35§,|^UþO•{)•˜5Ú-½5"
'…c
‡46N§Ã•ž¢§·½5
TheWell-PosednessofaDelayed
Non-AutonomousMicropolarFluid
on2DBoundedDomains
QilingWang
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:Apr.7
th
,2023;accepted:Apr.29
th
,2023;published:May6
th
,2023
Abstract
Inthispaper,westudythewell-posenessofanon-autonomousdelayedmicropolar
©ÙÚ^:é .˜až¢šg£‡46N3‘k.•þ·½5[J].A^êÆ?Ð,2023,12(5):2049-2066.
DOI:10.12677/aam.2023.125209
é
fluidon2Dboundeddomains.Weprovetheexistenceofsolutionsbythemethodof
theGarlerinapproximation.Then we usetheenergymethodtoprovetheuniqueness
andthestabilityofsolutions.
Keywords
MicropolarFluid,Delay,Well-Posedness
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.1252092051A^êÆ?Ð
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DOI:10.12677/aam.2023.1252092052A^êÆ?Ð
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DOI:10.12677/aam.2023.1252092053A^êÆ?Ð
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DOI:10.12677/aam.2023.1252092054A^êÆ?Ð
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(
b
H) 7→G(t,ξ) ∈(L
2
(Ω))
3
÷vµ
(i)é∀ξ∈BCL
−∞
(
b
H),N[0,T] 3t7→G(t,ξ) ∈(L
2
(Ω))
3
´Œÿ¶
(ii)G(·,0) = (0,0,0);
(iii)•3˜‡~êL
G
>0 ¦éu?¿t∈[0,T],ξ,η∈BCL
−∞
(
b
H),
kG(t,ξ)−G(t,η)k6L
G
kξ−ηk
BCL
−∞
(
b
H)
.
^‡(ii) Ú(iii) L²
kG(t,ξ)k6L
G
kξk
BCL
−∞
(
b
H)
,∀ξ∈BCL
−∞
(
b
H).(22)
e5y²•§(20)-(21) f)•35!•˜5!-½5"
½n3.1.(f)•35)é?¿T>0,bφ∈BCL
−∞
(
b
H) ´‰½§¿…b(I)Ú(II)
¤á§K¯K(20)-(21) 3«m(−∞,T]¥–•3˜‡f)"
y ·‚ò^n‡Ú½5y²½n3.1.
Ú½1.³7Cq)ÛÜ•35
·‚Äk£½ÂŽfA˜5Ÿ"¯¢þ§ŠâýŽf²;ÌnØ(„©z[24])§
•3˜‡AŠS{λ
n
}
∞
n=1
÷v
0 <λ
1
6λ
2
6···6λ
n
6···,λ
n
→+∞n→∞,
ÚÙéAA•þx{v
n
}
∞
n=1
⊆D(A),§´
b
H˜mFËAħ…d{v
1
,v
2
,···,v
n
,···}Ü
DOI:10.12677/aam.2023.1252092055A^êÆ?Ð
é
¤§3
b
V˜m¥È—§¦
Av
n
= λ
n
v
n
,∀n∈N.
b8ÜV
m
:= span{v
1
,···,v
m
},¿•ÄNP
m
w:=
m
P
j=1
(w,v
j
)v
j
,w∈
b
H½w∈
b
V§éuz‡
T>0,½Â¯K(20)-(21) Galerkin Cq)•
w
m
(t) :=
m
X
j=1
γ
m,j
(t)v
j
,
ùpXêγ
m,j
(t)÷ve~‡©…ܯK
d
dt

w
m
(t),v
j

+hAw
m
(t),v
j
i+hB(u
m
(t)),w
m
(t)),v
j
i+hN(w
m
(t)),v
j
i
= hF(t),v
j
i+(G(t,w
m
t
),v
j
), 1 6j6m,t∈(0,T),(23)
w
m
(s) = P
m
φ(s), s∈(−∞,0],(24)
3D
0
(0,T)¥•Ä•§(23).
þãអ~•¼‡©•§|÷vÛÜ)•3•˜5^‡(ë•©z[25])"Ïd§·‚
Ñ(23)-(24)k˜‡½Â3[0,t
m
)…0 ≤t
m
≤T•˜ÛÜ)"e5§·‚ò¼˜‡k
O§¿()w
m
(¢•3u‡«m[0,T].
Ú½2.kO
ò(23)z‡•§¦±γ
m,j
(t),j= 1,···,m¿¦Ú§|^(18)ÚhB(u,ψ),ψi= 0(„©z[22]
)k
1
2
d
dt
kw
m
(t)k
2
+δ
1
kw
m
(t)k
2
b
V
6hF(t),w
m
(t)i+(G(t,w
m
),w
m
(t)), t∈(0,T).(25)
qϕ
kw
m
t
k
BCL
−∞
(
b
H)
= sup
θ60
kw
m
(t+θ)k>kw
m
(t)k,0 <t<T,(26)
·‚(Ü(22),(25),(26) ÚCauchy Øªk
1
2
d
dt
kw
m
(t)k
2
+δ
1
kw
m
(t)k
2
b
V
6hF(t),w
m
(t)i+(G(t,w
m
),w
m
(t))
6kF(t)k
b
V
∗
kw
m
(t)k
b
V
+L
G
kw
m
t
k
BCL
−∞
(
b
H)
kw
m
(t)k
6
δ
1
2
kw
m
(t)k
2
b
V
+
1
2δ
1
kF(t)k
2
b
V
∗
+L
G
kw
m
t
k
2
BCL
−∞
(
b
H)
,(27)
ª(27) L²
DOI:10.12677/aam.2023.1252092056A^êÆ?Ð
é
d
dt
kw
m
(t)k
2
+δ
1
kw
m
(t)k
2
b
V
6
1
δ
1
kF(t)k
2
b
V
∗
+2L
G
kw
m
t
k
2
BCL
−∞
(
b
H)
, t∈(0,T).(28)
0 6s6t6T¿ò•§(28) •
d
dt
kw
m
(s)k
2
+δ
1
kw
m
(s)k
2
b
V
6
1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
, s∈(0,T).(29)
é(29) ¥s3[0,t] þÈ©k
kw
m
(t)k
2
+δ
1
t
Z
0
kw
m
(s)k
2
b
V
ds6kw
m
(0)k
2
+
t
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds.(30)
˜•¡§Šâ˜mBCL
−∞
(
b
b
H)‰ê½Â§·‚k
ku
m
t
k
2
BCL
−∞
(
b
H)
= sup
θ60
kw
m
(t+θ)k
2
6max
n
sup
θ≤−t
kw
m
(t+θ)k
2
,sup
−t6θ60
kw
m
(t+θ)k
2
o
,(31)
Ú
sup
θ≤−t
kw
m
(t+θ)k
2
=sup
θ≤−t
kφ(t+θ)k
2
.(32)
,˜•¡§·‚bt>t+θ>0,,é(29) 'us3«m[0,t+θ] þÈ©k
kw
m
(t+θ)k
2
+δ
1
t+θ
Z
0
kw
m
(s)k
2
b
V
ds
6kw
m
(0)k
2
+
t+θ
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds.(33)
d(33) ·‚k
kw
m
(t+θ)k
2
6kw
m
(0)k
2
+
t+θ
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds.(34)
é(34) 'uθ∈[−t,0] þ(.k
sup
−t6θ60
kw
m
(t+θ)k
2
6sup
−t6θ60

kw
m
(0)k
2
+
t+θ
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds

.(35)
DOI:10.12677/aam.2023.1252092057A^êÆ?Ð
é
l(31)-(35) k
kw
m
t
k
2
BCL
−∞
(
b
H)
6max
n
sup
θ≤−t
kφ(t+θ)k
2
,sup
−t6θ60

kw
m
(0)k
2
+
t+θ
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds

o
.(36)
du
sup
θ∈(−∞,−t]
kφ(t+θ)k= sup
θ60
kφ(θ)k= kφk
BCL
−∞
(
b
H)
,(37)
l(36)-(37) ·‚
kw
m
t
k
2
BCL
−∞
(
b
H)
6kφk
2
BCL
−∞
(
b
H)
+
t
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds.(38)
l(38) ÚGronwall Øªk
kw
m
t
k
2
BCL
−∞
(
b
H)
6(kφk
2
BCL
−∞
(
b
H)
+
1
δ
1
t
Z
0
kF(s)k
2
b
V
∗
ds

e
2L
G
t
,∀t∈[0,T].(39)
éuR>0, kφk
BCL
−∞
(
b
H)
6R,K(39) L²ùp•3˜‡'uδ
1
,L
G
,F,TÚR~ êc,
¦
kw
m
t
k
2
BCL
−∞
(
b
H)
6c(T,R),∀m>1.(40)
·‚l(26) Ú(40) Œ±
{w
m
}3L
∞
(0,T;
b
H)´k."(41)
·‚l(26),(30) Ú(40)•Œ±
δ
1
t
Z
0
kw
m
(s)k
2
b
V
ds6kw
m
(0)k
2
+
t
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
kw
m
s
k
2
BCL
−∞
(
b
H)
)ds.
6R
2
+
t
Z
0

1
δ
1
kF(s)k
2
b
V
∗
+2L
G
c(T,R)

ds.(42)
aqu(40),·‚Œ±l(42)ùp•3,˜‡~êc(T,R) >0¦
kw
m
k
2
L
2
(0,T;
b
V)
6c(T,R),∀m>1.(43)
DOI:10.12677/aam.2023.1252092058A^êÆ?Ð
é
d(23) ·‚Œ±§é∀v∈Vk
|h(w
m
(t))
0
,vi|6|hAw
m
(t),vi|+|hB(u
m
(t),w
m
(t)),vi|
+|hN(w
m
(t)),vi|+|hF(t),vi|+|(G(t,w
m
t
),v)|
6c(λ,ν
r
)(kw
m
(t)k
b
V
+ku
m
(t)k
1
2
k∇u
m
(t)k
1
2
kw
m
(t)k
1
2
k∇w
m
(t)k
1
2
+kF(t)k
b
V
∗
+kG(t,w
m
t
)k)kvk
b
V
,
6c(λ,ν
r
)(kw
m
(t)k
b
V
+kw
m
(t)kk∇w
m
(t)k+kF(t)k
b
V
∗
+kG(t,w
m
t
)k)kvk
b
V
.(44)
ùp~êc(λ,ν
r
)ûuλÚÚn2.1 ~êc(ν
r
),l(44) k
k(w
m
(t))
0
k
b
V
∗
6c(λ,ν
r
)(kw
m
(t)k
b
V
+kw
m
(t)kk∇w
m
(t)k+kF(t)k
b
V
∗
+kG(t,w
m
t
)k).(45)
(22)Ú(41),(43)Ú(45) L²
{(w
m
(t))
0
}3L
2
(0,T;
b
V
∗
)´k."(46)
l(40)-(41),(43),(46) †Ú½1 ¥ÛÜ•35ƒ(ܧ·‚3¤kžmt∈[0,T] 
GalerkinCq)Û•35"
Ú½3.Ûf)•35
·‚òy²³7Cq)4•¼ê´•§(20)-(21)f)"•Ä(40),(41),(43)Ú
(46),ÏLé‚À{§·‚Œ±˜‡fS(E,^ƒÓÎÒL«){w
m
},,‡ƒ
w∈L
∞
(0,T;
b
H)∩L
2
(0,T;
b
V) Úw
0
∈L
2
(0,T;
b
V
∗
),¿…ξ(t) ∈L
2
(0,T;(L
2
(Ω))
3
)¦
w
m
*w3L
∞
(0,T;
b
H)f(喤(47)
w
m
*w3L
2
(0,T;
b
V)f喤(48)
(w
m
)
0
*w
0
3L
2
(0,T;
b
V
∗
)f喤(49)
w
m
→w3L
2
(0,T;(L
2
(Ω))
3
)r喤(50)
G(·,w
m
) *ξ(t)3L
2
(0,T;(L
2
(Ω))
3
)fÂñ"(51)
y3§d(48)-(49) Ú;i\½n(ë„©z[26])§
u
m
∈C
0
([0,T];H),u∈C
0
([0,T];H).
Ó§l(48)-(50),·‚Œ±¼(é¤kfS¤á)
u
m
(t) →u(t)3Ha.e.t∈(0,T).
••§(20)-(21) f)§·‚I‡éª(23) ¥m4•m→∞,e5y²š‚5‘
DOI:10.12677/aam.2023.1252092059A^êÆ?Ð
é
eÂñ'Xµ
lim
m→∞
T
Z
0
hB(u
m
(t),w
m
(t)),vidt=
T
Z
0
hB(u(t),w(t)),vidt,∀v∈
b
V,(52)
lim
m→∞
T
Z
0

G(t,w
m
t
),v

=
T
Z
0

G(t,w
t
),v

dt,∀v∈
b
V.(53)
du(52) y²Ú©z[22] y²aq§¤±3ùpŽÑy²§y3·‚y²(53).
e5·‚Äky²BCL
−∞
(
b
H)¥Ð©ŠCq(„©z[27,(11)]).
P
m
φ→φ, 3BCL
−∞
(
b
H).(54)
XJ(54) ؤá§K•3>0 Ú˜‡Sθ
m
⊂(−∞,0] ¦
|P
m
φ(θ
m
)−φ(θ
m
)|>,∀m.(55)
·‚bθ
m
→−∞.XJθ
m
→θ,K
P
m
φ(θ
m
) →φ(θ
m
),
m→∞ž§
|P
m
φ(θ
m
)−φ(θ
m
)|≤|P
m
φ(θ
m
)−P
m
φ(θ)|+|P
m
φ(θ)−φ(θ)|→0,
½Âx=lim
θ→−∞
φ(θ),·‚k
|P
m
φ(θ
m
)−φ(θ
m
)|≤|P
m
φ(θ
m
)−P
m
x)|+|P
m
x−x|+|x−φ(θ
m
)|→0,
ù†(55) gñ§¤±(54) ¤á"|^(54),·‚k
sup
θ60
ku
m
(t+θ)−u(t+θ)k
= max

sup
θ∈(−∞,−t]
kP
m
φ(θ+t)−φ(θ+t)k,sup
θ∈[−t,0]
ku
m
(t+θ)−u(t+θ)k

6max

kP
m
φ−φk
BCL
−∞
(H)
,max
θ∈[0,t]
ku
m
(θ)−u(θ)k

→0,
ùL²
u
m
t
→u
t
3BCL
−∞
(
b
H)rÂñ§∀t6T.(56)
lb(II)(iii)Ú(56) Œ±Ñ
g(t,u
m
t
) →g(t,u
t
)3L
2
(0,T;H)rÂñ"(57)
DOI:10.12677/aam.2023.1252092060A^êÆ?Ð
é
d(57) ·‚Œ±(53).
3ù«œ¹e§·‚Œ±|^(48)-(50) Ú(52)-(53)=z(23)¥4•§Ñu´¯K
(20)-(21)f)"ddy²½n3.1 f)•35"
e5·‚ïÄf)•˜5"
½n3.2.(•˜5) ½n3.1^‡¤á§@oéuz‡Ð©Šu
0
= φ(s) ∈BCL
−∞
(
b
H),∀T>0,
•§(20)-(21) äk•˜f)"
yw
1
=(w
1
,ω
1
) Úw
2
=(w
2
,ω
2
) ´•§(20)-(21) 3«m(−∞,T] ü ‡)§ kÓ
Њw
1
0
= w
2
0
= φ(s).-w= w
1
−w
2
,Két∈(0,T] ·‚k
dw(t)
dt
+Aw(t)+B(u
1
,w
1
)−B(u
2
,w
2
)+N(w)
= G(t,w
1
t
)−G(t,w
2
t
).(58)
ò(58) Úw(t) ŠSÈk
1
2
d
dt
kw(t)k
2
+hAw(t),w(t)i+hB(u
1
(t),w
1
(t))−B(u
2
(t),w
2
(t)),w(t)i+hN(w),w(t)i
= (G(t,w
1
t
)−G(t,w
2
t
),w(t)),t∈(0,T].(59)
l©z[22],·‚k
|hB(u
1
(t),w
1
(t))−B(u
2
(t),w
2
(t)),w(t)i|
= |hB(u
1
(t)−u
2
(t),w
1
(t)w(t))−B(u
2
(t),w(t)),w(t)i|
= |hB(u(t),w
1
(t))w(t)i|
6λku(t)k
1
2
k∇u(t)k
1
2
kw(t)k
1
2
k∇w(t)k
1
2
k∇w
1
(t)k
1
2
6λkw(t)kkw(t)k
b
V
kw
1
(t)k
b
V
.(60)
(Ü(18),(60) Úb(II)(iii)§l(59)·‚Œ±
1
2
d
dt
kw(t)k
2
+δ
1
kw(t)k
2
b
V
6|hB(u
1
(t),w
1
(t))−B(u
2
(t),w
2
(t)),w(t)i|+(G(t,w
1
t
)−G(t,w
2
t
),w(t))
6λkw(t)kkw(t)k
b
V
kw
1
(t)k
b
V
+L
G
kw
t
k
BCL
−∞
(
b
H)
kw(t)k.(61)
du
w(θ) = 0, ∀θ60.
Ú
kw
s
k
BCL
−∞
(
b
H)
= sup
θ60
kw(s+θ)k6sup
θ∈[−s,0]
kw(s+θ)k=sup
r∈[0,s]
kw(r)k, 0 6s6T.
DOI:10.12677/aam.2023.1252092061A^êÆ?Ð
é
é(61) 3[0,t] þÈ©k
kw(t)k
2
+2δ
1
t
Z
0
kw(s)k
2
b
V
ds
62L
G
t
Z
0
kw
s
k
BCL
−∞
(
b
H)
kw(s)kds+2λ
t
Z
0
kw(s)kkw(s)k
b
V
kw
1
(s)k
b
V
ds
62L
G
t
Z
0
sup
r∈[0,s]
kwrkkw(s)kds+2λ
t
Z
0
kw(s)kkw(s)k
b
V
kw
1
(s)k
b
V
ds
62L
G
t
Z
0
sup
r∈[0,s]
kwrk
2
ds+2δ
1
t
Z
0
kw(s)k
2
b
V
ds+
λ
2
2δ
1
t
Z
0
kw(s)k
2
kw
1
(s)k
2
b
V
ds
6(2L
G
+
λ
2
2δ
1
)
t
Z
0
(1+kw
1
(s)k
2
b
V
)sup
r∈[0,s]
kw(r)k
2
ds+2δ
1
t
Z
0
kw(s)k
2
b
V
ds.(62)
l(62) k
sup
r∈[0,t]
kw(t)k
2
6(2L
G
+
λ
2
2δ
1
)
t
Z
0
(1+kw
1
(s)k
2
b
V
)sup
r∈[0,s]
kw(r)k
2
ds.(63)
é(63) ¦^Gronwall Øªk
sup
r∈[0,t]
kw(t)k= 0,∀t∈[0,T].
l·‚y²)•˜5"
•§·‚yf)ƒéuЩŠ-½5"
½n3.3.()'uЊ-½5)½n3.1 ^‡¤á§…u
(i)
´¯K(20)-(21)3i=1,2 ž
éAuЊφ
(i)
∈BCL
−∞
(
b
H) )§K
max
r∈[0,t]
kw
(1)
(r)−w
(2)
(r)k
2
6c

kφ
(1)
(0)−φ
(2)
(0)k
2
+kφ
(1)
(s)−φ
(2)
(s)k
2
BCL
−∞
(
b
H)

×exp

c
t
Z
0
(1+kw
(1)
(s)k
2
b
V
)ds

,(64)
kw
(1)
t
−w
(2)
t
k
2
BCL
−∞
(
b
H)
6ckφ
(1)
−φ
(2)
k
2
BCL
−∞
(
b
H)
exp

c
t
Z
0
(1+kw
(1)
(s)k
2
b
V
)ds

.(65)
yw
(i)
= (u
(i)
,ω
(i)
)(i= 1,2) ´¯K(20)-(21) éAuЊφ
(i)
∈BCL
−∞
(
b
H) )§·‚
½Â
DOI:10.12677/aam.2023.1252092062A^êÆ?Ð
é
u= u
(1)
−u
(2)
,ω= ω
(1)
−ω
(2)
,w= (u,ω) = w
(1)
−w
(2)
,φ(.) = φ
(1)
(.)−φ
(2)
(.).
Kk
1
2
d
dt
kw(t)k
2
+hAw(t),w(t)i+hB(u
(1)
(t),w
(1)
(t))−B(u
(2)
(t),w
(2)
(t)),w(t)i
+hN(w),w(t)i= (G(t,w
(1)
t
)−G(t,w
(2)
t
),w(t)),t∈(0,T].(66)
aqu(60) k
|hB(u
(1)
(t),w
(1)
(t))−B(u
(2)
(t),w
(2)
(t)),w(t)i|
= |hB(u(t),w
(1)
(t))w(t)i|
6λkw(t)kkw(t)k
b
V
kw
1
(t)k
b
V
6δ
1
kw(t)k
2
b
V
+
λ
2
4δ
1
kw(t)k
2
kw
1
(t)k
2
b
V
.(67)
éus∈[0,t],·‚k
kw
s
k
BCL
−∞
(
b
H)
= sup
θ60
kw(s+θ)k
= max

sup
θ∈(−∞,−s]
kφ(s+θ)k,sup
θ∈[−s,0]
kw(s+θ)k

6max

kφ(s)k
BCL
−∞
(
b
H)
,max
θ∈[0,s]
kw(s)k

.(68)
db(II)(iii)Ú(68) 
t
Z
0

G(s,w
(1)
s
)−G(s,w
(2)
s
),w(s)

ds6
t
Z
0
L
G
kw
s
k
BCL
−∞
(
b
H)
kw(s)kds
6
t
Z
0
L
G
kφ(s)k
BCL
−∞
(
b
H)
kw(s)kds+
t
Z
0
L
G
kw(s)k·max
θ∈[0,s]
kw(θ)kds.(69)
nÜ(18),(67) Ú(69),é(66) È©
1
2
kw(t)k
2
+δ
1
Z
t
0
kw(s)k
2
b
V
ds
6
1
2
kφ(0)k
2
+δ
1
Z
t
0
kw(s)k
2
b
V
ds+
λ
2
4δ
1
Z
t
0
kw(s)k
2
kw
1
(s)k
2
b
V
ds
+
t
Z
0
L
G
kφ(s)k
BCL
−∞
(
b
H)
kw(s)kds+
t
Z
0
L
G
kw(s)k·max
θ∈[0,s]
kw(θ)kds.
DOI:10.12677/aam.2023.1252092063A^êÆ?Ð
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=
kw(t)k
2
6kφ(0)k
2
+
λ
2
2δ
1
Z
t
0
kw(s)k
2
kw
1
(s)k
2
b
V
ds+
t
Z
0
2L
G
kφ(s)k
BCL
−∞
(
b
H)
kw(s)kds
+
t
Z
0
2L
G
kw(s)k·max
θ∈[0,s]
kw(θ)kds.(70)
du
2L
G
kφ(s)k
BCL
−∞
(
b
H)
t
Z
0
kw(s)kds6
L
G
2
kφ(s)k
2
BCL
−∞
(
b
H)
+2L
G

t
Z
0
kw(s)kds

2
6
L
G
2
kφ(s)k
2
BCL
−∞
(
b
H)
+2L
G
t
t
Z
0
kw(s)k
2
ds
6ckφ(s)k
2
BCL
−∞
(
b
H)
+c
t
Z
0
max
r∈[0,s]
kw(r)k
2
ds.(71)
òtO†•r∈[0,t],é(70) ¥r(r∈[0,t])•ŒŠ§¿$^(71) ·‚k
max
r∈[0,t]
kw(r)k
2
6kφ(0)k
2
+ckφ(s)k
2
BCL
−∞
(
b
H)
+c
t
Z
0

1+kw
(1)
(s)k
2
b
V

max
r∈[0,s]
kw(r)k
2
ds.(72)
éª(72) A^Gronwall Øª§ª(64).dª(72)Úª(68) ÚѪ(65).ddy²f)
-½5"
ë•©z
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