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PureMathematicsnØêÆ,2021,11(8),1517-1534
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.118171
‘ÚÑQuasi-Geostrophic•§|N
·½5
MMM
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2021c710F¶¹^Fϵ2021c817F¶uÙFϵ2021c824F
Á‡
©ïĘa‘ÚÑquasi-geostrophic •§|Њ¯KN·½5"ÏLÚ?˜ap$ª
äkØÓK5•I·Ü.Besov˜m§¿ÏLïáƒAÚÑŒ+3Ùþ˜—k.5O§
y²T‘ÚÑquasi-geostrophic •§|'u·Ü..Besov˜m¥˜—ЊN·
½5"
'…c
‘ÚÑQuasi-Geostrophic •§|§·Ü.Besov˜m§N·½5
GlobalWell-PosednessofTwo-Dimensional
DispersiveQuasi-GeostrophicEquations
RongShao
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Jul.10
th
,2021;accepted:Aug.17
th
,2021;published:Aug.24
th
,2021
©ÙÚ^:M.‘ÚÑQuasi-Geostrophic•§|N·½5[J].nØêÆ,2021,11(8):1517-1534.
DOI:10.12677/pm.2021.118171
M
Abstract
Thispaperisdevotedtostudyingtheglobalwell-posednessofCauchyproblemfor
thetwo-dimensionaldispersivequasi-geostrophicequations.Byintroducingakind
ofHybrid-Besovspaceswithdifferentregularityindicesathighfrequencyandlow
frequency,andbyestablishingtheuniformlyboundedestimationsofthe corresponding
dispersive operator semigroup on these newfunction spaces,the global well-posedness
of the 2D dispersive quasi-geostrophic equations is obtained for uniformly small initial
valuesinthecriticalfunctionalframework.
Keywords
Two-DimensionalDispersiveQuasi-GeostrophicEquations,Hybrid-BesovSpaces,
GlobalWell-Posedness
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2021.1181711518nØêÆ
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DOI:10.12677/pm.2021.1181711519nØêÆ
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DOI:10.12677/pm.2021.1181711520nØêÆ
M
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L
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k
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DOI:10.12677/pm.2021.1181711521nØêÆ
M
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ξ
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|ξ|
)
×∂
α
2
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t
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≤C|λx|
−N
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|=|α|
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|α|

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φ)(λ
−1
ξ)


|ξ|
−|α
1
|−|α
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|
e
−ν|ξ|
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t
(1+At)
|α
1
|
.
DOI:10.12677/pm.2021.1181711522nØêÆ
M
N= 3,é?¿λ&A, ¤á


(L
∗
)
3
(φ(λ
−1
ξ)
ˆ
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A
(t,ξ))


≤C|λx|
−3
e
−ν|ξ|
α
t
.
Ïd,¤á
Z
|x|≥λ
−1
|g(x,t)|dx≤Ce
−cλ
α
t
λ
2
Z
|x|≥λ
−1
|λx|
−3
dx≤Ce
−cλ
α
t
.(2.6)
(Ü(2.5) Ú(2.6) =Œ(2.4)¤á.ùÒ¤(2.3) ªy².
3.‚5OÚ¦È{K
Ún3.1s,σ∈R, α∈[0,2],(p,q) ∈[1,∞].K¤ á
1)é?¿θ∈
˙
B
s−
α
q
,σ−
α
q
2,p
,kG
A
(t)θk
˜
L
q
(0,∞;
˙
B
s,σ
2,p
)
≤Ckθk
˙
B
s−
α
q
,σ−
α
q
2,p
,
2)é?¿f∈
˜
L
1
(0,∞;
˙
B
s,σ
2,p
),¤á



Z
t
0
G
A
(t−τ)f(τ)dτ



˜
L
q
(0,∞;
˙
B
s+
α
q
,σ+
α
q
2,p
)
≤Ckfk
˜
L
1
(0,∞;
˙
B
s,σ
2,p
)
.
y²1) d½Â2.2,Œ•
kG
A
(t)θk
˜
L
q
(0,∞;
˙
B
s,σ
2,p
)
=sup
2
j
≤A
2
js
k∆
j
(G
A
(t)θ)k
L
q
(0,∞;L
2
)
+sup
2
j
>A
2
jσ
k∆
j
(G
A
(t)θ)k
L
q
(0,∞;L
p
)
.(3.1)
Äk,$^Ún2.5, Œ


∆
j
(G
A
(t)(t)θ)


L
q
(0,∞;L
2
)
≤C


e
−cλ
α
t
k∆
j
θk
L
2


L
q
(0,∞)
≤C2
−
α
q
j
k∆
j
θk
L
2
.(3.2)
Ùg,ÓnŒ


∆
j
(G
A
(t)θ)


L
q
(0,∞;L
p
)
≤C


e
−cλ
α
t
k∆
j
θk
L
p


L
q
(0,∞)
≤C2
−
α
q
j
k∆
j
θk
L
p
.(3.3)
ò(3.2) Ú(3.3) “\(3.1),=


G
A
(t)θ


˜
L
q
(0,∞;B
s,σ
2,p
)
≤Ckθk
˙
B
s−
α
q
,σ−
α
q
2,p
.
DOI:10.12677/pm.2021.1181711523nØêÆ
M
2)d½Â2.2, Œ•



Z
t
0
G
A
(t−τ)f(τ)dτ



˜
L
q
(0,∞;
˙
B
s+
α
q
,σ+
α
q
2,p
)
=sup
2
j
≤A
2
j(s+
α
q
)



∆
j
(
Z
t
0
G
A
(t−τ)f(τ)dτ)



L
q
(0,∞;L
2
)
+sup
2
j
>A
2
j(σ+
α
q
)



∆
j
(
Z
t
0
G
A
(t−τ)f(τ)dτ)



L
q
(0,∞;L
p
)
.
(3.4)
Äk,$^MinkowskiØª, Ún2.5(2) ±9YoungØª, Œ



∆
j

Z
t
0
G
A
(t−τ)f(τ)dτ




L
q
(0,∞;L
2
)
≤C



Z
t
0


G
A
(t−τ)∆
j
f(τ)


L
2
dτ



L
q
(0,∞)
≤C



Z
t
0
e
−2
αj
(t−τ)


∆
j
f(τ)


L
2
dτ



L
q
(0,∞)
≤C



e
−2
αj
t



L
q
(0,∞)



∆
j
f(τ)



L
1
(0,∞;L
2
)
≤C2
−
α
q
j



∆
j
f



L
1
(0,∞;L
2
)
.(3.5)
Ùg,ÓnŒ



∆
j

Z
t
0
G
A
(t−τ)f(τ)dτ




L
q
(0,∞;L
p
)
≤C



Z
t
0


G
A
(t−τ)∆
j
f(τ)


L
p
dτ



L
q
(0,∞)
≤C



Z
t
0
e
−c2
αj
(t−τ)


∆
j
f(τ)


L
p
dτ



L
q
(0,∞)
≤C



e
−2
αj
t



L
q
(0,∞)



∆
j
f(τ)



L
1
(0,∞;L
p
)
≤C2
−
α
q
j



∆
j
f



L
1
(0,∞;L
p
)
.(3.6)
ò(3.5) Ú(3.6) “\(3.4),=y(ؤá.
Ú n3.2p∈[2,4] ,α∈(2 −
2
p
,2], R
⊥
θ,θ∈
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)∩
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
),
K•3†A,R
⊥
θ,θÃ'~êC, ¦
kR
⊥
θ·θk
˜
L
1
(0,∞;
˙
B
3−α,
2
p
+2−α
2,p
)
≤C

kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
+kθk
˜
L
∞
(0,∞;
˙
B
2,
2
p
+1
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)

.(3.7)
y²dBony•ÈŒ,
∆
j

(R
⊥
θ)θ

=
X
|k−j|≤4
∆
j
(S
k−1
(R
⊥
θ)∆
k
θ)+
X
|k−j|≤4
∆
j
(S
k−1
θ∆
k
(R
⊥
θ))+
X
k≥j−2
∆
j
(∆
k
(R
⊥
θ)
˜
∆
k
θ)
:= I
j
+II
j
+III
j
.
DOI:10.12677/pm.2021.1181711524nØêÆ
M
Ú\•I8J
j
:=

(k
0
,k) : |k−j|≤4,k
0
≤k−2

.Ké2
j
>A,N´wÑ



I
j



L
1
(0,∞;L
p
)
≤
X
J
j



∆
j
(∆
k
0
(R
⊥
θ)∆
k
θ)



L
1
(0,∞;L
p
)
≤

X
J
j,ll
+
X
J
j,lh
+
X
J
j,hh




∆
j
(∆
k
0
(R
⊥
θ)∆
k
θ)



L
1
(0,∞;L
p
)
:= I
j,1
+I
j,2
+I
j,3
,
Ù¥
J
j,ll
=

(k
0
,k) ∈J
j
: 2
k
0
≤A,2
k
≤A

,
J
j,lh
=

(k
0
,k) ∈J
j
: 2
k
0
≤A,2
k
>A

,
J
j,hh
=

(k
0
,k) ∈J
j
: 2
k
0
>A,2
k
>A

.
|^Bernstein Øª!H¨olderØª±9ŽfR
⊥
3L
p
(R
2
)(1 <p<∞)þk.Œ
I
j,1
≤C
X
(k
0
,k)∈J
j,ll
2
k
0


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
∞
)
2
2k(
1
2
−
1
p
)


∆
k
θ


L
1
(0,∞;L
2
)
≤C
X
(k
0
,k)∈J
j,ll
2
k
0
(2−α)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2
)
2
2k


∆
k
θ


L
1
(0,∞;L
2
)
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
X
k
0
≤k−2
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
2
−k(
2
p
+2−α)
≤C2
−j(
2
p
+2−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
Óín,Œ„
I
j,2
≤C
X
(k
0
,k)∈J
j,lh
2
k
0


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
∞
)


∆
k
θ


L
1
(0,∞;L
p
)
≤C
X
(k
0
,k)∈J
j,lh
2
k
0
(2−α)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2
)
2
k(
2
p
+1)


∆
k
θ


L
1
(0,∞;L
p
)
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
(k
0
,k)∈J
j,lh
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
X
k
0
≤k−2
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
2
−k(
2
p
+2−α)
≤C2
−j(
2
p
+2−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
DOI:10.12677/pm.2021.1181711525nØêÆ
M
Ú
I
j,3
≤C
X
(k
0
,k)∈J
j,hh
2
2
p
k
0


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
p
)


∆
k
θ


L
1
(0,∞;L
p
)
≤C
X
(k
0
,k)∈J
j,hh
2
k
0
(
2
p
+1−α)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
p
)
2
k(
2
p
+1)


∆
k
θ


L
1
(0,∞;L
p
)
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
(k
0
,k)∈J
j,hh
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
X
k
0
≤k−2
2
(α−1)(k
0
−k)
2
−k(
2
p
+2−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
2
−k(
2
p
+2−α)
≤C2
−j(
2
p
+2−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
éu2
j
≤Aœ/, |^5Œ„


I
j


L
1
(0,∞;L
2
)
≤
X
J
j


∆
j
(∆
k
0
(R
⊥
θ)∆
k
θ)


L
1
(0,∞;L
2
)
≤

X
J
j,ll
+
X
J
j,lh
+
X
J
j,hh



∆
j
(∆
k
0
(R
⊥
θ)∆
k
θ)


L
1
(0,∞;L
2
)
:= I
j,4
+I
j,5
+I
j,6
.
|^Bernstein Øª!H¨older Øª±9ŽfR
⊥
3L
p
(R
2
)(1 <p<∞)þk.Œ
I
j,4
≤C
X
(k
0
,k)∈J
j,ll
2
k
0


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2
)


∆
k
θ


L
1
(0,∞;L
2
)
≤C
X
(k
0
,k)∈J
j,ll
2
k
0
(2−α)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2
)
2
2k


∆
k
θ


L
1
(0,∞;L
2
)
2
(α−1)(k
0
−k)
2
−k(3−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
X
k
0
≤k−2
2
(α−1)(k
0
−k)
2
−k(3−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
2
−k(3−α)
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
DOI:10.12677/pm.2021.1181711526nØêÆ
M
5¿p≤4…
2
p
+2−α>0ž,¤á
I
j,5
≤C
X
(k
0
,k)∈J
j,lh


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2p
p−2
)


∆
k
θ


L
1
(0,∞;L
p
)
≤C
X
(k
0
,k)∈J
j,lh
2
2k
0
p


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2
)


∆
k
θ


L
1
(0,∞;L
2
)
≤C
X
(k
0
,k)∈J
j,lh
2
k
0
(2−α)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2
)
2
k(
2
p
+1)


∆
k
θ


L
1
(0,∞;L
p
)
2
(k
0
−k)(
2
p
+α−2)
2
−k(3−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
X
k
0
≤k−2
2
(k
0
−k)(
2
p
+α−2)
2
−k(3−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
2
−k(3−α)
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
Ú
I
j,6
≤C
X
(k
0
,k)∈J
j,hh


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
2p
p−2
)


∆
k
θ


L
1
(0,∞;L
p
)
≤C
X
(k
0
,k)∈J
j,hh
2
2k
0
(
2
p
−
1
2
)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
p
)


∆
k
θ


L
1
(0,∞;L
p
)
≤C
X
(k
0
,k)∈J
j,hh
2
k
0
(
2
p
+1−α)


∆
k
0
(R
⊥
θ)


L
∞
(0,∞;L
p
)
2
k(
2
p
+1)


∆
k
θ


L
1
(0,∞;L
p
)
2
(k
0
−k)(
2
p
+α−2)
2
−k(3−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
X
k
0
≤k−2
2
(k
0
−k)(
2
p
+α−2)
2
−k(3−α)
≤C


R
⊥
θ


˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)


θ


˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
|k−j|≤4
2
−k(3−α)
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
Ïd,(ÜI
j,1
∼I
j,6
O,Œ
sup
2
j
≤1
2
j(3−α)
kI
j
k
L
1
(0,∞;L
2
)
+sup
2
j
>1
2
(
2
p
+2−α)j
kI
j
k
L
1
(0,∞;L
p
)
≤Ckθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.(3.8)
aqŒ
sup
2
j
≤1
2
j(3−α)
kII
j
k
L
1
(0,∞;L
2
)
+sup
2
j
>1
2
(
2
p
+2−α)j
kII
j
k
L
1
(0,∞;L
p
)
≤Ckθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.(3.9)
DOI:10.12677/pm.2021.1181711527nØêÆ
M
-K
j
:= {(k,k
0
) : k≥j−3,|k
0
−k|≤1}.
III
j
=

X
K
j,ll
+
X
K
j,lh
+
X
K
j,hl
+
X
K
j,hh

∆
j
(∆
k
(R
⊥
θ)∆
k
0
θ)
:= III
j,1
+III
j,2
+III
j,3
+III
j,4
,
Ù¥
K
j,ll
=

(k,k
0
) ∈K
j
: 2
k
≤A,2
k
0
≤A

,
K
j,lh
=

(k,k
0
) ∈K
j
: 2
k
≤A,2
k
0
>A

,
K
j,hl
=

(k,k
0
) ∈K
j
: 2
k
>A,2
k
0
≤A

,
K
j,hh
=

(k,k
0
) ∈K
j
: 2
k
>A,2
k
0
>A

.
|^Bernstein Øª!H¨older Øª±9ŽfR
⊥
3L
p
(R
2
)(1 <p<∞)þk.Œ


III
j,1


L
1
(0,∞;L
p
)
≤C2
2j(1−
1
p
)
X
(k,k
0
)∈K
j,ll


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
1
)
≤C2
2j(1−
1
p
)
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
(k,k
0
)∈K
j,ll
2
−k(2−α)
2
−2k
0
≤C2
2j(1−
1
p
)
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
X
|k−j|≤1
2
−2k
0
≤C2
2j(1−
1
p
)
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
2
−2k
≤C2
−j(
2
p
+2−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
9


III
j,1


L
1
(0,∞;L
2
)
≤C2
2j(1−
1
2
)
X
(k,k
0
)∈K
j,ll


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
1
)
≤C2
j
X
(k,k
0
)∈K
j,ll
2
k(2−α)


∆
k
(R
⊥
θ)


L
∞
(0,∞;L
2
)
2
2k
0


∆
k
0
θ


L
1
(0,∞;L
2
)
2
−k(2−α)
2
−2k
0
≤C2
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
X
|k−k
0
|≤1
2
−2k
0
≤C2
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
2
−2k
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
DOI:10.12677/pm.2021.1181711528nØêÆ
M
aq/,†íÑ


III
j,2


L
1
(0,∞;L
p
)
≤C2
j
X
(k,k
0
)∈K
j,lh
∪K
j,hl


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
2p
2+p
)
≤C2
j
X
K
j,lh
2
k(2−α)


∆
k
(R
⊥
θ)


L
∞
(0,∞;L
2
)
2
k
0
(
2
p
+1)


∆
k
0
θ


L
1
(0,∞;L
p
)
2
−k(2−α)
2
−k
0
(
2
p
+1)
≤C2
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
X
|k−k
0
|≤1
2
−k
0
(
2
p
+1)
≤C2
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
2
−k(
2
p
+1)
≤C2
−j(
2
p
+2−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
9


III
j,2


L
1
(0,∞;L
2
)
≤C2
2
p
j
X
(k,k
0
)∈K
j,lh


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
2p
2+p
)
≤C2
2
p
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
X
|k−k
0
|≤1
2
−k
0
(
2
p
+1)
≤C2
2
p
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(2−α)
2
−k(
2
p
+1)
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
|^Bernstein Øª!H¨older Øª±9ŽfR
⊥
3L
p
(R
2
)(1 <p<∞)þk.Œ


III
j,3


L
1
(0,∞;L
p
)
≤C2
j
X
(k,k
0
)∈K
j,hl


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
2p
2+p
)
≤C2
j
X
K
j,hl
2
k(
2
p
+1−α)


∆
k
(R
⊥
θ)


L
∞
(0,∞;L
p
)
2
2k
0


∆
k
0
θ


L
1
(0,∞;L
2
)
2
−k(
2
p
+1−α)
2
−2k
0
≤C2
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(
2
p
+1−α)
X
|k−k
0
|≤1
2
−2k
0
≤C2
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−2k
2
−k(
2
p
+1−α)
≤C2
−j(
2
p
+2−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
DOI:10.12677/pm.2021.1181711529nØêÆ
M
Ú


III
j,3


L
1
(0,∞;L
2
)
≤C2
2
p
j
X
(k,k
0
)∈K
j,hl


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
2p
2+p
)
≤C2
2
p
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(
2
p
+1−α)
X
|k−k
0
|≤1
2
−2k
0
≤C2
2
p
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−2k
2
−k(
2
p
+1−α)
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
5¿2 ≤p≤4, Œ


III
j,4


L
1
(0,∞;L
p
)
≤C
X
(k,k
0
)∈K
j,hh


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
p
2
)
≤C2
2
p
j
X
(k,k
0
)∈K
j,hh


∆
k
(R
⊥
θ)


L
∞
(0,∞;L
p
)


∆
k
0
v


L
1
(0,∞;L
p
)
≤C2
2
p
j
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(
2
p
+1−α)
2
−k(
2
p
+1)
≤C2
−j(
2
p
+1−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.


III
j,4


L
1
(0,∞;L
2
)
≤C2
2j(
2
p
−
1
2
)
X
(k,k
0
)∈K
j,hh


∆
k
(R
⊥
θ)∆
k
0
θ


L
1
(0,∞;L
p
2
)
≤C2
2j(
2
p
−
1
2
)
X
(k,k
0
)∈K
j,hh


∆
k
(R
⊥
θ)


L
∞
(0,∞;L
p
)


∆
k
0
θ


L
1
(0,∞;L
p
)
≤C2
j(
4
p
−1)
kR
⊥
θk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
X
k≥j−3
2
−k(
2
p
+1−α)
2
−k(
2
p
+1)
≤C2
−j(3−α)
kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
nÜIII
j,1
∼III
j,4
O,Œ
sup
2
j
≤1
2
j(3−α)
kIII
j
k
L
1
(0,∞;L
2
)
+sup
2
j
>1
2
(
2
p
+2−α)j
kIII
j
k
L
1
(0,∞;L
p
)
≤Ckθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.(3.10)
DOI:10.12677/pm.2021.1181711530nØêÆ
M
(Ü(3.8) ∼(3.10) =(3.7)¤á.
4.½n1.1y²
Ún4.1([26]) (X,k·k
X
) •Banach ˜m.B:X×X→X•V‚5Žf,…÷vé
?¿x
1
,x
2
∈X,•3~êη>0,¦kB(x
1
,x
2
)k
X
≤ηkx
1
k
X
kx
2
k
X
. XJ0<ε<
1
4η
…y∈X
÷vkyk
X
≤ε, K•§x= y+B(x,x) 3X¥•3•˜), …÷vkxk
X
≤2ε.
½n1.1y².½ÂBanach˜mX•
X=
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
(R
2
))∩
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
(R
2
)),
¿3ÙþDƒ‰ê
kθk
X
:= kθk
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
)
+kθk
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
.
½Â
B(θ,θ)(t) :=
Z
t
0
G
A
(t−τ)∇[R
⊥
θ(τ)·θ(τ)]dτ.
dÚn3.1 (2)ÚÚn3.2Œ•,é?¿θ∈X, •3~êC
1
≥0,Œ


B(θ,θ)


X
=



Z
t
0
G
A
(t−τ)∇[R
⊥
θ(τ)·θ(τ)]dτ



X
≤C
1


[R
⊥
θ(τ)·θ(τ)]


˜
L
1
(0,∞;
˙
B
3−α,
2
p
+2−α
2,p
)
≤C
1
kθk
X
kθk
X
.
¿…,dÚn3.1(1) qŒ•,é?¿θ
0
∈
˙
B
2−α,
2
p
+1−α
2,p
,•3~êC
0
>0 ¦
kG
A
(t)θ
0
k
X
≤C
0
kθ
0
k
˙
B
2−α,
2
p
+1−α
2,p
.
Ïd,dÚn4.1Œ,é?¿0≤≤
1
4C
1
±9?¿÷vkθ
0
k
˙
B
2−α,
2
p
+1−α
2,p
≤

C
0
θ
0
∈
˙
B
2−α,
2
p
+1−α
2,p
,•§(2.1)3X¥•3˜‡•˜)θ∈
˜
L
∞
(0,∞;
˙
B
2−α,
2
p
+1−α
2,p
) ∩
˜
L
1
(0,∞;
˙
B
2,
2
p
+1
2,p
)
…kθk
X
≤2.d,dIOÈ—5?Ø, Œ?˜Úy²θ∈C

0,∞;
˙
B
2−α,
2
p
+1−α
2,p

.ùÒ¤½
n1.1y².
DOI:10.12677/pm.2021.1181711531nØêÆ
M
ë•©z
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