a Xi :ma h/0701447 1 [ma h.AP] 16 Jan 2007
Exis ence o he α-pa ch model and he
QG sha p on in Sobole spaces
F ancisco Gancedo
Abs ac
We conside a amily o con ou dynamics equa ions depending on a pa ame e α
wi h 0 < α ≤1. The o ex pa ch p oblem o he 2-D Eule equa ion is ob ained aking
α→0, and he case α= 1 co esponds o a sha p on o he QG equa ion. We p o e
local-in- ime exis ence o he amily o equa ions in Sobole spaces.
1 In oduc ion
The 2-D QG equa ion p o ides pa icula solu ions o he e olu ion o he empe a u e om a
gene al quasi-geos ophic sys em o a mosphe ic and oceanic lows. This equa ion is de i ed
conside ing small Rossby and Ekman numbe s and cons an po en ial o ici y (see [12] o
mo e de ails). I eads
θ (x, ) + u(x, )· ∇θ(x, ) = 0, x ∈R2,
θ(x, 0) = θ0(x).(1)
He e θis he empe a u e o he luid, he incomp essible eloci y uis exp essed by means
o he s eam unc ion as ollows
u=∇⊥ψ= (−∂x2ψ, ∂x1ψ),
and he ela ion be ween he s eam unc ion and he empe a u e is gi en by
θ=−(−∆)1/2ψ.
This sys em ha e been conside ed in on ogenesis, whe e he dynamics o ho and cold luids
is s udied oge he wi h he o ma ion and he e olu ion o on s (see [4], [5], [8], [11]).
F om a ma hema ical poin o iew, his equa ion ha e been p esen ed as a wo dimen-
sional model o he 3-D Eule equa ion due o hei s ong analogies (see [4]), being he
o ma ion o singula i ies o a egula ini ial da a an open p oblem (see [4], [6], [7]). Ne e -
heless he QG equa ion has global in ime weak solu ions due o an ex a cancella ion (see
[13]). A ew spa se esul s a e known abou weak solu ions o he 2-D and 3-D Eule equa ion
in i s p imi i e- a iable o m.
1
An ou s anding kind o weak solu ions o he QG equa ion a e hose in which he em-
pe a u e akes wo di e en alues in complemen a y domains, modelling he e olu ion o a
sha p on as ollows
θ(x1, x2, ) = θ1,Ω( )
θ2,R2 Ω( ).(2)
In his wo k we s udy a p oblem simila o he 2-D o ex pa ch p oblem, whe e he
o ici y o he 2-D Eule equa ion is gi en by a cha ac e is ic unc ion o a domain, and i is
conside ed he egula i y o he ee bounda y o such domain. Fo his equa ion he o ici y
sa is ies
w (x, ) + u(x, )· ∇w(x, ) = 0, x ∈R2,
w(x, 0) = w0(x),(3)
in a weak sense, and he eloci y is gi en by he Bio -Sa a law o analogously
u=∇⊥ψ, and w= ∆ψ.
Chemin [3] p o ed global-in- ime egula i y o he ee bounda y using pa adi e en ial cal-
culus. A simple p oo can be ound in [1] due o Be ozzi and Cons an in.
We poin ou ha in he QG equa ion, he eloci y is de e mined om he empe a u e
by singula in eg al ope a o s (see [15]) as ollows
u= (−R2θ, R1θ),(4)
whe e R1and R2a e he Riesz ans o ms, making he sys em mo e singula han (3).
Rod igo [14] p oposed he p oblem o he e olu ion o a sha p on o he QG equa ion.
He de i ed he eloci y on he ee bounda y in he no mal di ec ion, and p o ed local-
exis ence and uniqueness o a pe iodic C∞ on , i.e.
θ(x1, x2, ) = θ1,{ (x1, )> x2}
θ2,{ (x1, )≤x2},
wi h (x1, ) pe iodic, using he Nash-Mose i e a ion.
In his pape we s udy a amily o con ou dynamics equa ion gi en by weak solu ions o
he ollowing sys em
θ +u· ∇θ= 0, x ∈R2,
u=∇⊥ψ, θ =−(−∆)1−α/2ψ, 0< α ≤1,
(5)
whe e he ac i e scala θ(x, ) sa is ies (2). We no ice ha he case α= 0 is he 2-D o ex
pa ch p oblem, and α= 1 co espond o he sha p on o he QG equa ion.
This sys em was in oduced by C´o doba, Fon elos, Mancho and Rod igo in [9], whe e hey
p esen a p oo o local-exis ence o a pe iodic C∞ on , and show e idence o singula i ies
in ini e ime. The singula scena io is due o wo pa ches collapse poin -wise.
2
He e we gi e a p oo o local-exis ence o he sys em (5) whe e he solu ion sa is ies (2),
wi h he bounda y ∂Ω( ) gi en by he cu e
∂Ω( ) = {x(γ, ) = (x1(γ, ), x2(γ, )) : γ∈[−π, π]},
and x(γ, ) belongs o a Sobole space. In he cases 0 < α < 1 we show uniqueness.
I is well-known (see [10] and [14]) ha in his kind o con ou dynamics equa ions, he
eloci y in he angen ial di ec ion only mo es he pa icles on he bounda y. The e o e we
do no al e he shape o he con ou i we change he angen ial componen o he eloci y;
i.e., we a e making a change on he pa ame iza ion. In he mos singula case, α= 1 o
he QG equa ion, we need o change he eloci y in he angen ial di ec ion in o de o ge
exis ence in he Sobole spaces. We ake a angen ial eloci y in such a way ha |∂γx(γ, )|
sa is ies
|∂γx(γ, )|2=A( ),
and does no depend on γ. We would like o ci e he wo k o Hou, Loweng ub and Shelley
[10] in which his idea was used o s udy a con ou dynamics p oblem.
We no ice ha in o de o ge a non-singula no mal eloci y o he cu e o 0 < α ≤1
(see [9] and [14]), we need a one o one cu e, and pa ame e ized in such a way ha
|∂γx(γ, )|2>0.
Rigo ously, we need ha
|x(γ, )−x(γ−η, )|
|η|>0,∀γ, η ∈[−π, π],(6)
he e o e we gi e an ini ial da a sa is ying his p ope y, and we p o e ha his condi ion is
sa is ied locally in ime. We poin ou he impo ance o ake in o accoun he e olu ion o
his quan i y due o he nume ical simula ions in [9].
Finally, I wish o hank An onio C´o doba and my hesis ad iso Diego C´o doba o hei
s ong in luence in his wo k, hei ad ices and sugges ions. The au ho was pa ially sup-
po ed by he g an s PAC-05-005-2 o he JCLM (Spain) and MTM2005-05980 o he MEC
(Spain).
2 The Con ou Equa ion
In his sec ion we deduce he amily o con ou equa ions in e m o he ee bounda y x(γ, ).
We conside he equa ions gi en by he sys em (1), wi h a eloci y sa is ying
u(x, ) = ∇⊥ψ(x, ),(7)
o he s eam unc ion i ollows
θ=−(−∆)1−α/2ψ, (8)
and he ac i e scala ul ills
3
θ(x1, x2, ) = θ1,Ω( )
θ2,R2 Ω( ).(9)
The bounda y o Ω( ) is gi en by he cu e
∂Ω( ) = {x(γ, ) = (x1(γ, ), x2(γ, )) : γ∈[−π, π] = T},
wi h x(γ, ) one o one. Due o he iden i y (9), we ind ha
∇⊥θ= (θ1−θ2)∂γx(γ, )δ(x−x(γ, )),
whe e δis he Di ac dis ibu ion. Using (7) and (8), we go ha
u=−(−∆)α/2−1∇⊥θ.
Due o he in eg al ope a o s −(−∆)α/2−1a e Riesz po en ials (see [15]), using he las o
iden i ies we ob ain ha
u(x, ) = −Θα
2πZT
∂γx(γ−η, )
|x−x(γ−η, )|αdη, (10)
o x6=x(γ, ), and Θα= (θ1−θ2)Γ(α/2)/21−αΓ(2 −α/2). We no ice ha o α= 1, i
x→x(γ, ) he in eg al in (10) is di e gen . As we ha e showed be o e, we a e in e es ed in
he no mal eloci y o he sys ems. Then we ha e ha using he iden i y (10), and aking
he limi as ollows
u(x, )·∂⊥
γx(γ, ), x →x(γ, ),(11)
we ob ain
u(x(γ, ), )·∂⊥
γx(γ, ) = −Θα
2πZT
∂γx(γ−η, )·∂⊥
γx(γ, )
|x(γ, )−x(γ−η, )|αdη. (12)
This iden i y is well de ined o 0 < α ≤1 and a one o one cu e x(γ, ). Due o he ac ha
angen ial eloci y does no change he shape o he bounda y, we ix he con ou α-pa ch
equa ions as ollows
x (γ, ) = Θα
2πZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|αdη, 0< α ≤1,
x(γ, 0) = x0(γ).
(13)
Seeing he equa ion (10), we show ha he eloci y in QG p esen s a loga i hmic di e gence
in he angen ial di ec ion on he bounda y. Ne e heless i belongs o Lp(R2) o 1 < p < ∞,
and o he bounded mean oscilla ion space (see [15] o he de ini ion o he BMO space). In
QG he eloci y is gi en by (4), and w i ing he empe a u e in he ollowing way
θ(x, ) = (θ1−θ2)XΩ( )(x) + θ2,
we ind ha
u(x, ) = (θ1−θ2)(−R2(XΩ( )), R1(XΩ( ))).
Using ha XΩ( )∈Lp(R2) o 1 ≤p≤ ∞, we conclude de a gumen . In pa icula he
ene gy o he sys em is conse ed due o kukL2( ) = |θ1−θ2| |Ω( )|1/2, and he a ea o Ω( )
is cons an in ime.
4
3 Weak solu ions o he α-sys em
In his sec ion we show ha i θ(x, ) is de ined by (9) and he cu e x(γ, ) is con ec ed by
he no mal eloci y (12), hen θ(x, ) is a weak solu ion o he sys em (5) and con e sely. We
gi e he de ini ion o weak solu ion below.
De ini ion 3.1 The ac i e scala θis a weak solu ion o he α-sys em i o any unc ion
ϕ∈C∞
c(R2×(0, T)), we ha e
ZT
0ZR2
θ(x, )(∂ ϕ(x, ) + u(x, )· ∇ϕ(x, ))dxd = 0,(14)
whe e he incomp essible eloci y uis gi en by (7), and he s eam unc ion sa is ies (8).
Then
P oposi ion 3.2 I θ(x, )is de ined by (9), and he cu e x(γ, )sa is ies (6) and (12),
hen θ(x, )is a weak solu ion o he α-sys em. Fu he mo e, i θ(x, )is a weak solu ion o
he α-sys em gi en by (9), and x(γ, )sa is ies (6), hen x(γ, ) e i ies (12).
P oo : Le θ(x, ) be a weak solu ion o he α-sys em de ined by (9). In eg a ing by pa s
we ha e
I=ZT
0ZR2
θ(x, )∂ ϕ(x, )dxd =θ1ZT
0ZΩ( )
∂ ϕ(x, )dxd +θ2ZT
0ZΩ( ) R2
∂ ϕ(x, )dxd
=−(θ1−θ2)ZT
0ZT
ϕ(x(γ, ), )x (γ, )·∂⊥
γx(γ, )dγd .
On he o he hand, we ob ain
J=ZT
0ZR2
θ u · ∇ϕ dxd =θ1ZT
0ZΩ
u· ∇ϕ dxd +θ2ZT
0ZR2 Ω
u· ∇ϕ dxd .
Taking
Ωε
1( ) = {x∈Ω : dis (x, Ω( )) ≥ε},
and
Ωε
2( ) = {x∈R2 Ω : dis (x, R2 Ω( )) ≥ε},
we ha e ha Jε→Ji ε→0, whe e Jεis gi en by
Jε=θ1ZT
0ZΩε
1( )
u· ∇ϕ dxd +θ2ZT
0ZΩε
2( )
u· ∇ϕ dxd .
In eg a ing by pa in Jε, using ha he eloci y is di e gence ee, and aking he limi as
in (11), we ob ain
J= (θ1−θ2)ZT
0ZT
ϕ(x(γ, ), )u(x(γ, ), )·∂⊥
γx(γ, )dγd
=−(θ1−θ2)Θα
2πZT
0ZT
ϕ(x(γ, ), )ZT
∂γx(γ−η, )·∂⊥
γx(γ, )
|x(γ, )−x(γ−η, )|αdηdγd .
5

We ha e ha I+J= 0 using (14), and i ollows
ZT
0ZT
(γ, )x (γ, )·∂⊥
γx(γ, ) + Θα
2πZT
∂γx(γ−η, )·∂⊥
γx(γ, )
|x(γ, )−x(γ−η, )|αdηdγd = 0,
o (γ, ) pe iodic in γ. We ind ha (12) is sa is ied. Following he same a gumen s i is
easy o check ha i x(γ, ) sa is ies (12), hen θis a weak solu ion gi en by (9).
4 Local well-posedness o 0< α < 1
In his sec ion we p o e exis ence and uniqueness o he con ou equa ion in he cases
0< α < 1. We deno e he Sobole spaces by Hk(T), wi h no ms
kxk2
Hk=kxk2
L2+k∂k
γxk2
L2,
and he spaces Ck(T) wi h
kxkCk= max
j≤kk∂j
γxkL∞.
We need ha he cu e sa is ies
|x(γ, )−x(γ−η, )|
|η|>0,∀γ, η ∈[−π, π],(15)
hen we de ine
F(x)(γ, η, ) = |η|
|x(γ, )−x(γ−η, )|∀γ, η ∈[−π, π],(16)
wi h
F(x)(γ, 0, ) = 1
|∂γx(γ, )|.
The main heo em in his sec ion is he ollowing
Theo em 4.1 Le x0(γ)∈Hk(T) o k≥3wi h F(x0)(γ, η)<∞. Then he e exis s a ime
T > 0so ha he e is a unique solu ion o (13) o 0< α < 1in C1([0, T]; Hk(T)) wi h
x(γ, 0) = x0(γ).
P oo : We can choose Θα= 2πwi hou loss o gene ali y, ob aining he ollowing equa ion
x (γ, ) = ZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|αdη, 0< α < 1,
x(γ, 0) = x0(γ).
(17)
We p esen he p oo o k= 3, being analogous o k > 3, using ene gy es ima es (see [2]
o mo e de ails). We igno e he ime dependence o simpli y he no a ion in some e ms.
Conside ing he quan i y
6
ZT
x(γ)·x (γ)dγ =ZTZT
x(γ)·∂γx(γ)−∂γx(η)
|x(γ)−x(η)|αdηdγ
=−ZTZT
x(η)·∂γx(γ)−∂γx(η)
|x(γ)−x(η)|αdηdγ
=1
2ZTZT
(x(γ)−x(η)) ·(∂γx(γ)−∂γx(η))
|x(γ)−x(η)|αdηdγ
=1
2(2 −α)ZTZT
∂γ|x(γ)−x(γ−η)|2−αdγdη
= 0,
(18)
we ob ain d
d kxkL2( ) = 0.(19)
We decompose as ollows
ZT
∂3
γx(γ)·∂3
γx (γ)dγ =I1+I2+I3+I4,
whe e
I1=ZTZT
∂3
γx(γ)·∂4
γx(γ)−∂4
γx(γ−η)
|x(γ)−x(γ−η)|αdηdγ,
I2= 3
ZTZT
∂3
γx(γ)·(∂3
γx(γ)−∂3
γx(γ−η))∂γ(|x(γ)−x(γ−η)|−α)dηdγ,
I3= 3
ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η))∂2
γ(|x(γ)−x(γ−η)|−α)dηdγ,
I4=ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))∂3
γ(|x(γ)−x(γ−η)|−α)dηdγ.
Ope a ing as in (18), he e m I1becomes
I1=1
2ZTZT
(∂3
γx(γ)−∂3
γx(γ−η)) ·∂4
γx(γ)−∂4
γx(γ−η)
|x(γ)−x(γ−η)|αdηdγ
=1
4ZTZT
∂γ|∂3
γx(γ)−∂3
γx(γ−η)|2
|x(γ)−x(γ−η)|αdηdγ
=α
4ZTZT
|∂3
γx(γ)−∂3
γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η))
|x(γ)−x(γ−η)|α+2 dηdγ.
7
One inds ha
I1≤α
4ZTZT
|∂3
γx(γ)−∂3
γx(γ−η)|2|∂γx(γ)−∂γx(γ−η)|
|x(γ)−x(γ−η)|α+1 dηdγ,
and due o he inequali y |∂γx(γ)−∂γx(γ−η)||η|−1≤ kxkC2,i ollows
I1≤α
4kxkC2ZTZT
|η|−α|F(x)(γ, η)|1+α|∂3
γx(γ)−∂3
γx(γ−η)|2dηdγ
≤1
2kF(x)k1+α
L∞kxkC2ZT
|η|−αZT
(|∂3
γx(γ)|2+|∂3
γx(γ−η)|2)dγdη
≤ kF(x)k1+α
L∞kxkC2k∂3
γxk2
L2ZT
|η|−αdη
≤CαkF(x)k1+α
L∞kxkC2k∂3
γxk2
L2.
(20)
As be o e, we can ob ain I2=−6I1, and i yields
I2≤CαkF(x)k1+α
L∞kxkC2k∂3
γxk2
L2.(21)
In o de o es ima e he e m I3, we conside I3=J1+J2+J3, whe e
J1=−3αZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η)) A(γ, η)
|x(γ)−x(γ−η)|α+2 dηdγ,
J2=−3α
ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2
|x(γ)−x(γ−η)|α+2 dηdγ,
J3= 3α(2 + α)
ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η)) (B(γ, η))2
|x(γ)−x(γ−η)|α+4 dηdγ,
wi h
A(γ, η) = (x(γ)−x(γ−η)) ·(∂2
γx(γ)−∂2
γx(γ−η)),
and
B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)).
The iden i y
∂2
γx(γ)−∂2
γx(γ−η) = ηZ1
0
∂3
γx(γ+ (s−1)η)ds, (22)
yields
J1≤3Z1
0ZTZT
|η|(|∂2
γx(γ)|+|∂2
γx(γ−η)|)|∂3
γx(γ)||∂3
γx(γ+ (s−1)η)|
|x(γ)−x(γ−η)|α+1 dγdηds
≤3kF(x)k1+α
L∞kxkC2Z1
0ZT
|η|−αZT
(|∂3
γx(γ)|2+|∂3
γx(γ+ (s−1)η)|2)dγdηds
≤CαkF(x)k1+α
L∞kxkC2k∂3
γxk2
L2.
8
Using (22), we ha e o J2
J2=−3α
Z1
0ZTZT
|F(x)(γ, η)|2+α|∂γx(γ)−∂γx(γ−η)|2
η
∂3
γx(γ)·∂3
γx(γ+(s−1)η)
|η|αdγdηds
≤3kF(x)k2+α
L∞kxk2
C2Z1
0ZT
|η|−αZT
(|∂3
γx(γ)|2+|∂3
γx(γ+ (s−1)η)|2)dγdηds
≤CαkF(x)k2+α
L∞kxk2
C2k∂3
γxk2
L2.
The e m J3is es ima ed by
J3≤9
Z1
0ZTZT
|η||∂γx(γ)−∂γx(γ−η)|2|∂3
γx(γ)||∂3
γx(γ+(s−1)η)|
|x(γ)−x(γ−η)|α+2 dγdηds
≤CαkF(x)k2+α
L∞kxk2
C2k∂3
γxk2
L2.
We ge inally
I3≤Cα(kF(x)k1+α
L∞kxkC2+kF(x)k2+α
L∞kxk2
C2)k∂3
γxk2
L2.(23)
We decompose he e m I4=J4+J5+J6+J7+J8as ollows
J4=−αZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) C(γ, η)
|x(γ)−x(γ−η)|α+2 dηdγ,
J5=−3α
ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) D(γ, η)
|x(γ)−x(γ−η)|α+2 dηdγ,
J6= 5α(α+ 2)
ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) A(γ, η)B(γ, η)
|x(γ)−x(γ−η)|α+4 dηdγ,
J7= 5α(α+ 2)
ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))B(γ, η)|∂γx(γ)−∂γx(γ−η)|2
|x(γ)−x(γ−η)|α+4 dηdγ,
J8=−2α(α+ 2)(α+ 4)
ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) (B(γ, η))3
|x(γ)−x(γ−η)|α+6 dηdγ,
wi h
C(γ, η) = (x(γ)−x(γ−η)) ·(∂3
γx(γ)−∂3
γx(γ−η)),
D(γ, η) = (∂γx(γ)−∂γx(γ−η)) ·(∂2
γx(γ)−∂2
γx(γ−η)).
The mos singula e m is J4, in such a way ha
J4≤ kF(x)k1+α
L∞kxkC2ZT
|η|−αZT
|∂3
γx(γ)||∂3
γx(γ)−∂3
γx(γ−η)|dγdη
≤CαkF(x)k1+α
L∞kxkC2k∂3
γxk2
L2.
Fo J5, we ha e
J5≤3kF(x)k2+α
L∞kxk2
C2ZT
|η|−αZT
|∂3
γx(γ)||∂2
γx(γ)−∂2
γx(γ−η)|dγdη
≤CαkF(x)k2+α
L∞kxk2
C2k∂2
γxkL2k∂3
γxkL2.
9
Theo em 5.1 Le x0(γ)∈Hk(T) o k≥3wi h F(x0)(γ, η)<∞. Then he e exis s a ime
T > 0so ha he e is a solu ion o (31) in C1([0, T]; Hk(T)) wi h x(γ, 0) = x0(γ)and λ(γ, )
gi en by (36).
P oo : Being analogous o k > 3, we gi e he p oo o k= 3. We ha e showed be o e
ha (33) is sa is ied i x(γ, ) is a solu ion o (31). Then we can ew i e λ(γ, ) as ollows
λ(γ, ) = γ+π
2πA( )ZT
∂γx(γ, )·∂γZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|dηdγ
−1
A( )Zγ
−π
∂γx(η, )·∂ηZT
∂γx(η, )−∂γx(η−ξ, )
|x(η, )−x(η−ξ, )|dξdη.
(37)
We ob ain
ZT
x(γ)·x (γ)dγ =ZTZT
x(γ)·∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ +ZT
λ(γ)x(γ)·∂γx(γ)dγ
=I1+I2,
One inds ha I1= 0, since
I1=ZTZT
x(γ)·∂γx(γ)−∂γx(η)
|x(γ)−x(η)|dηdγ =−ZTZT
x(η)·∂γx(γ)−∂γx(η)
|x(γ)−x(η)|dηdγ
=1
2ZTZT
(x(γ)−x(η)) ·(∂γx(γ)−∂γx(η))
|x(γ)−x(η)|dηdγ =1
2ZTZT
∂γ|x(γ)−x(γ−η)|dγdη
= 0.
Fo he e m I2, one ob ains ha I2≤ kλkL∞kxkL2k∂γxkL2, and
kλkL∞≤2
A( )ZT
|∂γx(γ)|∂γZT
∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
≤2
A( )ZT
|∂γx(γ)|ZT
|∂2
γx(γ)−∂2
γx(γ−η)|
|x(γ)−x(γ−η)|dηdγ
+2
A( )ZT
|∂γx(γ)|ZT
|∂γx(γ)−∂γx(γ−η)|2
|x(γ)−x(γ−η)|2dηdγ =J1+J2.
Due o 1/A( )≤ kF(x)k2
L∞( ), we ha e
J1≤2kF(x)k3
L∞Z1
0ZTZT
|∂3
γx(γ+ (s−1)η)||∂γx(γ)|dγdηds ≤2kF(x)k3
L∞kxk2
H3,
and
J2≤2kF(x)k4
L∞kxkC1Z1
0ZTZT
|∂2
γx(γ+ (s−1)η)|2dγdηds ≤2kF(x)k4
L∞kxk3
H3.
16

The e o e we ob ain ha
d
d kxk2
L2( )≤CkF(x)k4
L∞( )kxk5
H3( ).(38)
We decompose as ollows
ZT
∂3
γx(γ)·∂3
γx (γ)dγ =ZT
∂3
γx(γ)·∂3
γZT
∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
+ZT
∂3
γx(γ)·∂3
γ(λ(γ)∂γx(γ))dγ
=I3+I4.
We ake I3=J3+J4+J5+J6whe e
J3=ZTZT
∂3
γx(γ)·∂4
γx(γ)−∂4
γx(γ−η)
|x(γ)−x(γ−η)|dηdγ,
J4= 3 ZTZT
∂3
γx(γ)·(∂3
γx(γ)−∂3
γx(γ−η))∂γ(|x(γ)−x(γ−η)|−1)dηdγ,
J5= 3 ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η))∂2
γ(|x(γ)−x(γ−η)|−1)dηdγ,
J6=ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))∂3
γ(|x(γ)−x(γ−η)|−1)dηdγ.
The e m J3can be w i en as
J3=1
2ZTZT
(∂3
γx(γ)−∂3
γx(γ−η)) ·∂4
γx(γ)−∂4
γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
=1
4ZTZT
∂γ|∂3
γx(γ)−∂3
γx(γ−η)|2
|x(γ)−x(γ−η)|dηdγ
=1
4ZTZT
|∂3
γx(γ)−∂3
γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η))
|x(γ)−x(γ−η)|3dηdγ.
I we de ine
B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)),
due o (32), we ob ain ha
J3=1
4ZTZT
|F(x)(γ, η)|3|∂3
γx(γ)−∂3
γx(γ−η)|2B(γ, η)η−2−∂γx(γ)·∂2
γx(γ)
|η|dηdγ.
17
Using ha

B(γ, η)η−2−∂γx(γ)·∂2
γx(γ)
η≤2kxk2
C2,1
2|η|−1/2,
we ind
J3≤ kF(x)k3
L∞kxk2
C2,1
2ZT
|η|−1/2ZT
(|∂3
γx(γ)|2+|∂3
γx(γ−η)|2)dγdη
≤CkF(x)k3
L∞kxk2
C2,1
2k∂3
γxk2
L2
≤CkF(x)k3
L∞kxk4
H3.
(39)
We ob ain ha J4=−6J3, and i yields
J4≤CkF(x)k3
L∞kxk4
H3.(40)
In o de o es ima e he e m J5, we conside J5=K1+K2+K3, whe e
K1=−3ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η)) C(γ, η)
|x(γ)−x(γ−η)|3dηdγ,
K2=−3ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2
|x(γ)−x(γ−η)|3dηdγ,
K3= 9 ZTZT
∂3
γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η)) (B(γ, η))2
|x(γ)−x(γ−η)|5dηdγ,
wi h
C(γ, η) = (x(γ)−x(γ−η)) ·(∂2
γx(γ)−∂2
γx(γ−η)).
The inequali y
|∂2
γx(γ)−∂2
γx(γ−η)||η|−1/2≤ kxkC2,1
2,(41)
yields
K1≤3kF(x)k2
L∞kxkC2,1
2Z1
0ZT
|η|−1/2ZT
|∂3
γx(γ)||∂3
γx(γ+ (s−1)η)|dγdηds
≤CkF(x)k2
L∞kxk3
H3.
As be o e, we ha e o K2 ha
K2≤CkF(x)k3
L∞kxk2
C2k∂3
γxk2
L2≤CkF(x)k3
L∞kxk4
H3.
The e m K3is es ima ed by
K3≤CkF(x)k3
L∞kxk2
C2k∂3
γxk2
L2≤CkF(x)k3
L∞kxk4
H3.
We ge inally
J5≤CkF(x)k3
L∞kxk4
H3.(42)
18
We decompose he e m J6=K4+K5+K6+K7+K8as ollows
K4=−ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) D(γ, η)
|x(γ)−x(γ−η)|3dηdγ,
K5=−3ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) E(γ, η)
|x(γ)−x(γ−η)|3dηdγ,
K6= 15 ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) B(γ, η)C(γ, η)
|x(γ)−x(γ−η)|5dηdγ,
K7= 15 ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))B(γ, η)|∂γx(γ)−∂γx(γ−η)|2
|x(γ)−x(γ−η)|5dηdγ,
K8=−30 ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) (B(γ, η))3
|x(γ)−x(γ−η)|7dηdγ,
wi h
D(γ, η) = (x(γ)−x(γ−η)) ·(∂3
γx(γ)−∂3
γx(γ−η)),
E(γ, η) = (∂γx(γ)−∂γx(γ−η)) ·(∂2
γx(γ)−∂2
γx(γ−η)).
We ob ain
K5≤3kF(x)k3
L∞kxk2
C2k∂3
γxk2
L2≤3kF(x)k3
L∞kxk4
H3,
K6≤15kF(x)k3
L∞kxk2
C2k∂3
γxk2
L2≤15kF(x)k3
L∞kxk4
H3,
K7≤15kF(x)k4
L∞kxk3
C2k∂3
γxkL2k∂2
γxkL2≤15kF(x)k4
L∞kxk5
H3,
and
K8≤30kF(x)k4
L∞kxk3
C2k∂3
γxkL2k∂2
γxkL2≤30kF(x)k4
L∞kxk5
H3.
Fo he mos singula e m, we ha e
K4=ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))η ∂γx(γ)·(∂3
γx(γ)−∂3
γx(γ−η)) −D(γ, η)
|x(γ)−x(γ−η)|3dηdγ
−ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))η ∂γx(γ)·(∂3
γx(γ)−∂3
γx(γ−η))
|x(γ)−x(γ−η)|3dηdγ
=L1+L2.
One inds ha
L1≤ kF(x)k3
L∞kxk2
C2ZTZT
|∂3
γx(γ)||∂3
γx(γ)−∂3
γx(γ−η)|dγdη ≤CkF(x)k3
L∞kxk4
H3.
The e m L2is decomposed, and i yields
19
L2=ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η))η(∂γx(γ)−∂γx(γ−η)) ·∂3
γx(γ−η)
|x(γ)−x(γ−η)|3dηdγ
−ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) η∂γx(γ)·∂3
γx(γ)−∂γx(γ−η)·∂3
γx(γ−η)
|x(γ)−x(γ−η)|3dηdγ
=M1+M2.
We es ima e he e m M1as ollows
M1≤ kF(x)k3
L∞kxk2
C2ZTZT
|∂3
γx(γ)||∂3
γx(γ−η)|dγdη ≤ kF(x)k3
L∞kxk4
H3.
Taking he de i a i e in (32), we ind ha ∂γx(γ)·∂3
γx(γ) = −|∂2
γx(γ)|2, and we ew i e
M2=ZTZT
∂3
γx(γ)·(∂γx(γ)−∂γx(γ−η)) η|∂2
γx(γ)|2− |∂2
γx(γ−η)|2
|x(γ)−x(γ−η)|3dηdγ.
The inequali y
||∂2
γx(γ)|2− |∂2
γx(γ−η)|2| ≤ 2kxkC2|η|Z1
0
|∂3
γx(γ+ (s−1)η)|ds, (43)
yields
M2≤2kF(x)k3
L∞kxk2
C2Z1
0ZTZT
|∂3
γx(γ)||∂3
γx(γ+ (s−1)η)|dγdηds ≤CkF(x)k3
L∞kxk4
H3.
We ecall ha K4=L1+L2=L1+M1+M2≤CkF(x)k3
L∞kxk4
H3,and inally i ollows
J6≤CkF(x)k4
L∞kxk5
H3.(44)
Due o (39), (40), (42) and (44), we ob ain
I3≤CkF(x)k4
L∞kxk5
H3.(45)
We ake I4=J7+J8+J9+J10, whe e
J7=ZT
λ(γ)∂3
γx(γ)·∂4
γx(γ)dγ, J8= 3 ZT
∂γλ(γ)|∂3
γx(γ)|2dγ,
J9= 3 ZT
∂2
γλ(γ)∂3
γx(γ)·∂2
γx(γ)dγ, J10 =ZT
∂3
γλ(γ)∂3
γx(γ)·∂γx(γ)dγ.
We in eg a e by pa s in he e m J7, and we ge
20
J7=−1
2ZT
∂γλ(γ)|∂3
γx(γ)|2dγ ≤1
2k∂γλkL∞k∂3
γxk2
L2.
Using (37), we ind ha
∂γλ(γ, ) = 1
2πA( )ZT
∂γx(γ, )·∂γZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|dηdγ
−1
A( )∂γx(γ, )·∂γZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|dη
=K9+K10.
(46)
The e m K9is es ima ed as J1and J2, ob aining
K9≤ kF(x)k4
L∞kxk3
H3.
We ha e o K10 ha
K10 ≤kxkC2
A( )ZT|∂2
γx(γ, )−∂2
γx(γ−η, )|
|x(γ, )−x(γ−η, )|+|∂γx(γ, )−∂γx(γ−η, )|2
|x(γ, )−x(γ−η, )|2dη
≤2kF(x)k4
L∞kxk3
C2,1
2ZT
|η|−1/2dη
≤CkF(x)k4
L∞kxk3
H3,
and he e o e
J7≤CkF(x)k4
L∞kxk5
H3.(47)
Due o he iden i y J8=−6J7, one inds ha
J8≤CkF(x)k4
L∞kxk5
H3.(48)
Using ha
∂2
γλ(γ, ) = −1
A( )∂2
γx(γ, )·∂γZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|dη
−1
A( )∂γx(γ, )·∂2
γZT
∂γx(γ, )−∂γx(γ−η, )
|x(γ, )−x(γ−η, )|dη,
one ge s
J9=−1
A( )ZT
∂3
γx(γ)·∂2
γx(γ)∂2
γx(γ)·∂γZT
∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
−1
A( )ZT
∂3
γx(γ)·∂2
γx(γ)∂γx(γ)·∂2
γZT
∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
=L3+L4.
21

The e o e
L3≤kxk2
C2
A( )ZTZT
|∂3
γx(γ)||∂2
γx(γ, )−∂2
γx(γ−η, )|
|x(γ, )−x(γ−η, )|+|∂γx(γ, )−∂γx(γ−η, )|2
|x(γ, )−x(γ−η, )|2dηdγ
≤ kF(x)k4
L∞kxk3
C2Z1
0ZTZT
|∂3
γx(γ)|(|∂3
γx(γ+ ( −1)η)|+|∂2
γx(γ+ ( −1)η)|)dγdηds
≤CkF(x)k4
L∞kxk5
H3.
Mo eo e
L4=−1
A( )ZTZT
∂3
γx(γ)·∂2
γx(γ)∂γx(γ)·∂3
γx(γ)−∂3
γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
+2
A( )ZTZT
∂3
γx(γ)·∂2
γx(γ)∂γx(γ)·(∂2
γx(γ)−∂2
γx(γ−η))B(γ, η)
|x(γ)−x(γ−η)|3dηdγ
−1
A( )ZTZT
∂3
γx(γ)·∂2
γx(γ)∂γx(γ)·(∂γx(γ)−∂γx(γ−η))∂2
γ(|x(γ)−x(γ−η)|−1)dηdγ
=M3+M4+M5.
The e ms M4and M5a e es ima ed as be o e, and we ob ain
M4+M5≤CkF(x)k5
L∞kxk6
H3.
The mos singula e m is M3, bu we ind ha
M3=1
A( )ZTZT
∂3
γx(γ)·∂2
γx(γ)∂3
γx(γ−η)·∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
−1
A( )ZTZT
∂3
γx(γ)·∂2
γx(γ)∂3
γx(γ)·∂γx(γ)−∂3
γx(γ−η)·∂γx(γ−η)
|x(γ)−x(γ−η)|dηdγ
=N1+N2.
We ob ain
N1≤ kF(x)k3
L∞kxk2
C2k∂3
γxk2
L2≤ kF(x)k3
L∞kxk4
H3,
and using (32)
N2=1
A( )ZTZT
∂3
γx(γ)·∂2
γx(γ)|∂2
γx(γ)|2− |∂2
γx(γ−η)|2
|x(γ)−x(γ−η)|dηdγ.
Due o (43), we conclude ha
N2≤2kF(x)k3
L∞kxk2
C2k∂3
γxk2
L2≤2kF(x)k3
L∞kxk4
H3.
We ha e J9=L3+L4=L3+M3+M4+M5=L3+N1+N2+M4+M5, and he e o e
J9≤ kF(x)k5
L∞kxk6
H3.(49)
22
The iden i y (32) yields
J10 =−ZT
∂3
γλ(γ)|∂2
γx(γ)|2dγ = 2 ZT
∂2
γλ(γ)∂3
γx(γ)·∂2
γx(γ)dγ =2
3J9,
and he e o e
J10 ≤ kF(x)k5
L∞kxk6
H3.(50)
Due o he inequali ies (47), (48), (49), and (50), we ge
I4≤CkF(x)k5
L∞kxk6
H3.
Using (45) and he las es ima e, we ha e
d
d k∂3
γxk2
L2( )≤CkF(x)k5
L∞( )kxk6
H3( ).
This inequali y and (38) bound he e olu ion o he Sobole no ms o he cu e as ollows
d
d kxkH3( )≤CkF(x)k5
L∞( )kxk5
H3( ).(51)
We con inue he a gumen conside ing he e olu ion o he quan i y kF(x)kL∞( ). Taking
p > 2, i yields
d
d kF(x)kp
Lp( )≤pZTZT|η|
|x(γ, )−x(γ−η, )|p+1 |x (γ, )−x (γ−η, )|
|η|dγdη.
We ha e
x (γ)−x (γ−η) = ZT
(∂γx(γ)−∂γx(γ−ξ)
|x(γ)−x(γ−ξ)|−∂γx(γ)−∂γx(γ−ξ)
|x(γ−η)−x(γ−η−ξ)|)dξ
+ZT
∂γx(γ)−∂γx(γ−η) + ∂γx(γ−η−ξ)−∂γx(γ−ξ)
|x(γ−η)−x(γ−η−ξ)|dξ
+ (λ(γ)−λ(γ−η))∂γx(γ) + λ(γ−η)(∂γx(γ)−∂γx(γ−η))
=I5+I6+I7+I8.
The e m I5yields
I5≤ZT
|∂γx(γ)−∂γx(γ−ξ)||x(γ)−x(γ−ξ)| − |x(γ−η)−x(γ−η−ξ)|
|x(γ)−x(γ−ξ)||x(γ−η)−x(γ−η−ξ)|dξ
≤ kF(x)k2
L∞kxkC2ZT
|ξ|−1|x(γ)−x(γ−η)−(x(γ−ξ)−x(γ−η−ξ))|dξ
≤ kF(x)k2
L∞kxkC2|η|Z1
0ZT∂γx(γ+ (s−1)η)−∂γx(γ+ (s−1)η−ξ)
|ξ|dξds
≤2πkF(x)k2
L∞kxk2
C2|η|.
23
Fo I6we ake
I6≤ kF(x)kL∞|η|Z1
0ZT∂2
γx(γ+ (s−1)η)−∂2
γx(γ+ (s−1)η−ξ)
|ξ|dξds
≤ kF(x)kL∞kxkC2,1
2|η|Z1
0ZT
|ξ|−1/2dξds
≤CkF(x)kL∞kxkC2,1
2|η|
We ha e o I7
I7≤2kxkC2
A( )|η|max
γ|∂γx(γ)||∂γZT
∂γx(γ)−∂γx(γ−η)
|x(γ)−x(γ−η)|dη|
≤2kF(x)k2
L∞kxk2
C2|η|max
γZT
|∂2
γx(γ)−∂2
γx(γ−η)|
|x(γ)−x(γ−η)|dη+ZT
|∂γx(γ)−∂γx(γ−η)|2
|x(γ)−x(γ−η)|2dη
≤4kF(x)k4
L∞kxk4
H3|η|.
Es ima ing kλkL∞as be o e, easily we ge
I8≤ kλkL∞kxkC2|η| ≤ 4kF(x)k4
L∞kxk4
H3|η|.
The las ou es ima es show ha
d
d kF(x)kLp( )≤Ckxk4
H3( )kF(x)k5
L∞( )kF(x)kLp( ),
by in eg a ing in ime and aking p→ ∞, we ob ain
kF(x)kL∞( +h)≤ kF(x)kL∞( )exp CZ +h
kxk4
H3(s)kF(x)k5
L∞(s)ds.
As in he p e ious sec ion, i ollows
d
d kF(x)kL∞( )≤Ckxk4
H3( )kF(x)k6
L∞( ).
Then, due o (51) and he abo e es ima e, we ind inally ha
d
d (kxkH3( ) + kF(x)kL∞( )) ≤C(kxkH3( ) + kF(x)kL∞( ))10.
In eg a ing, we ha e
kxkH3( ) + kF(x)kL∞( )≤kx0kH3+kF(x0)kL∞
1− Ckx0kH3+kF(x0)kL∞91
9
,
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whe e Cis a cons an .
We ha e used he equali y (32) o ob ain he a p io i es ima es. In o de o ge he
solu ion o (31), we ha e o choose an app op ia e egula ized p oblem p ese ing (32). We
p opose he sys em
xε,δ
(γ, ) = φε∗ZT
∂γ(φε∗xε,δ(γ, )−φε∗xε,δ(γ−η, ))
|xε,δ(γ, )−xε,δ(γ−η, )|+δdη +λε,δ(γ, )∂γxε,δ(γ, ),
xε,δ(γ, 0) = x0(γ),
(52)
wi h
λε,δ(γ, ) = γ+π
2πZT
∂γxε,δ(γ, )
|∂γxε,δ(γ, )|2·∂γφε∗ZT
∂γ(φε∗xε,δ(γ, )−φε∗xε,δ(γ−η, ))
|xε,δ(γ, )−xε,δ(γ−η, )|+δdηdγ
−Zγ
−π
∂γxε,δ(η, )
|∂γxε,δ(η, )|2·∂ηφε∗ZT
∂γ(φε∗xε,δ(η, )−φε∗xε,δ(η−ξ, ))
|xε,δ(η, )−xε,δ(η−ξ, )|+δdξdη.
We can ob ain ene gy es ima es o he sys em (52) depending on εand δ, bu wi hou using
(32), and he e o e we ob ain exis ence o (52). As long as he solu ion exis s, we ha e ha
∂γxε,δ(γ, )·∂2
γxε,δ(γ, ) = 0.
Using his p ope y o he solu ion, we ob ain ene gy es ima es ha depend only on δ, and
aking ε→0 we ge a solu ion o he ollowing equa ion
xδ
(γ, ) = ZT
∂γxδ(γ, )−∂γxδ(γ−η, ))
|xδ(γ, )−xδ(γ−η, )|+δdη +λδ(γ, )∂γxδ(γ, ),
xδ(γ, 0) = x0(γ),
(53)
wi h
λδ(γ, ) = γ+π
2πZT
∂γxδ(γ, )
|∂γxδ(γ, )|2·∂γZT
∂γxδ(γ, )−∂γxδ(γ−η, )
|xδ(γ, )−xδ(γ−η, )|+δdηdγ
−Zγ
−π
∂γxδ(η, )
|∂γxδ(η, )|2·∂ηZT
∂γxδ(η, )−∂γxδ(η−ξ, ))
|xδ(η, )−xδ(η−ξ, )|+δdξdη.
Again we ha e ha he solu ions o his sys em sa is y
∂γxδ(γ, )·∂2
γxδ(γ, ) = 0,
and aking ad an age o his, we ind ene gy es ima es independen o δ. I we end δ o 0,
we conclude he exis ence esul .
Re e ences
[1] A. L. Be ozzi and P. Cons an in. Global egula i y o o ex pa ches. Comm. Ma h.
Phys. 152 (1): 19–28, 1993.
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