a Xi :1204.6633 2 [ma h.AP] 19 Oc 2012
Fini e ime singula i ies o wa e wa es wi h su ace ension
Angel Cas o, Diego C´o doba, Cha les Fe e man,
F ancisco Gancedo and Ja ie G´omez-Se ano
Dedica ed o Pe e Cons an in on his 60 h Bi hday
Oc obe 22, 2012
Abs ac
He e we conside he 2D ee bounda y incomp essible Eule equa ion wi h su ace
ension. We p o e ha he su ace ension does no p e en a ini e ime splash o spla
singula i y, i.e. ha he cu e ouches i sel ei he in a poin o along an a c. To do so,
he main ing edien s o he p oo a e a ans o ma ion o desingula ize he cu e and a
p io i ene gy es ima es.
Keywo ds: Eule , incomp essible, blow-up, wa e wa es, splash, spla , su ace ension.
I In oduc ion
In his pape we con inue he wo k in [8] and [9] whe e we show he o ma ion o singula i ies
o he ee bounda y incomp essible Eule equa ions. He e we p o e ha in wo space
dimensions he ee bounda y p oblem de elops ini e ime “splash” and “spla ” singula i ies
when su ace ension is aken in o accoun (see below, in Sec ion III, he p ecise de ini ion o
he splash and spla cu es).
In o de o desc ibe he e olu ion o a luid wi h a mo ing domain Ω( )⊂R2, he 2D
incomp essible Eule equa ions a e used:
( + ·∇ )(x, y, ) = −∇p(x, y, )−(0,1),(x, y)∈Ω( ) (I.1)
wi h he luid eloci y (x, y, )∈R2and he p essu e p(x, y, )∈R. The ec o −(0,1)
ep esen s he ex e nal g a i a ional o ce ( he accele a ion due o g a i y is aken equal o
one o he sake o simplici y). The ee bounda y
∂Ω( ) = {z(α, ) = (z1(α, ), z2(α, )) : α∈R}(I.2)
is smoo h and con ec ed by he eloci y ield
z (α, )·z⊥
α(α, ) = (z(α, ), )·z⊥
α(α, ),(I.3)
which is assumed o be incomp essible and i o a ional
∇· (x, y, ) = 0,∇⊥· (x, y, ) = 0,(x, y)∈Ω( ).(I.4)
1
He e we s udy he ele ance o conside ing he Laplace-Young condi ion o which he p essu e
on he in e ace ∂Ω( ) is p opo ional o i s cu a u e, meaning ha he su ace ension e ec
is conside ed:
−p(z(α, ), ) = τ
2
zαα(α, )·z⊥
α(α, )
|zα(α, )|3≡τ
2K. (I.5)
Abo e τ > 0 is he su ace ension coe icien .
The esul s in his pape can be shown o h ee di e en scena ios:
1. Ω( ) a compac domain: z(α, ) is a 2π-pe iodic unc ion in α.
2. Asymp o ically la case: z(α, )−(α, 0) →0 as α→ ∞.
3. Ω( ) pe iodic in he ho izon al a iable: z(α, )−(α, 0) is a 2π-pe iodic unc ion in α.
The p oblem o s udy he e is he po en ial o ma ion o singula i ies o he sys em (I.1-
I.5) wi h smoo h in e ace and smoo h eloci y ield wi h ini e ene gy as ini ial da a:
Ω(0) = Ω0, ∂Ω0={z0(α) : α∈R},
(x, y, 0) = 0(x, y),ZΩ0| 0(x, y)|2dxdy < +∞.(I.6)
The smoo h ini ial cu e z0(α) mus sa is y he a c-cho d condi ion:
|z0(α)−z0(β)| ≥ cAC|α−β|, o all α, β ∈R,(I.7)
whe e cAC >0 is he a c-cho d cons an . The s udy o his quan i y has been employed by
o he au ho s o p o e local exis ence (see o example [23], [24]). We will quan i y how ou
cu e z(α) sa is ies he a c-cho d condi ion h ough he ollowing quan i y
F(z) = |β|
|z(α)−z(α−β)|, α, β ∈[−π, π].
Th oughou he pape we will only ocus on scena io 3 o he sake o simplici y. F om
now on, we will deno e Ω0∩[−π, π]×Rby Ω0by abuse o no a ion (a undamen al domain
in he pe iod).
We es ablish he main esul in he pape o he sys em (I.1-I.5).
Theo em I.1 Conside z0(α)−(α, 0) ∈Hk(T) o k≥5. Then he e exis a amily o
ini ial da a sa is ying (I.6) and he a c-cho d condi ion (I.7) and a ime Ts>0such ha he
in e ace z(α, )∈Hk(T) om he unique smoo h solu ion o he sys em (I.1-I.7) on he ime
in e al [0, Ts] ouches i sel a a single poin (“splash” singula i y) o along an a c (“spla ”
singula i y) a ime =Ts.
These solu ions can be ex ended o he pe iodic 3Dse ing conside ing scena ios in a ian
unde ansla ions in one coo dina e di ec ion. In [14], Cou and-Shkolle conside addi ional
3Dsplash and spla singula i ies. The case wi h small ini ial da a was ea ed by Wu in
he wo dimensional case [25] and he h ee dimensional case was s udied by Wu [26] and
Ge main e al. [16].
2
Fo o he long ime beha iou esul s see Al a ez-Lannes [3], Cas o e al. [10] and he
e e ences he ein.
In o de o p o e his heo em we p oceed as in [8] and [9]. Using (I.4) i is easy o
decla e ha is ha monic in Ω( ). This ac allows us o in oduce he momen ω(α, ) by
elemen a y po en ial heo y as ollows:
(x, y, ) = PV
2πZR
(x−z1(β, ), y −z2(β, )))⊥
|(x, y)−z(β, )|2ω(β, )dβ, (I.8)
whe e PV deno es p incipal alue a in ini y. This momen is also known in he li e a u e
as he o ici y ampli ude. Then he sys em (I.1-I.5) is equi alen o he ollowing e olu ion
equa ions which a e only w i en in e ms o he ee bounda y z(α, ) and he ampli ude
ω(α, ):
z (α, ) = BR(z, ω)(α, ) + c(α, )zα(α, ),(I.9)
ω (α, ) = −2BR (z, ω)(α, )·zα(α, )−ω2
4|∂αz|2α(α, ) + (cω)α(α, )
+ 2c(α, )BRα(z, ω)(α, )·zα(α, )−2(z2)α(α, ) + τzαα(α, )·z⊥
α(α, )
|zα(α, )|3α
(I.10)
( o de ails see o example [12, Sec ion 2]). Abo e BR(z, ω) is he Bi kho -Ro in eg al
de ined by
BR(z, ω) = 1
2πPV ZR
(z(α, )−z(β, ))⊥
|z(α, )−z(β, )|2ω(β, )dβ, (I.11)
and c(α, ) is a bi a y since he bounda y is con ec ed by he no mal eloci y (I.3).
Local exis ence in Sobole spaces was i s achie ed by Wu [23] assuming ini ially he
a c-cho d condi ion. Fo o he a ia ions and esul s see [15, 21, 7, 27, 24, 11, 20, 13, 22, 28,
18, 6, 4, 19, 1, 2, 12].
The s a egy o he p oo o he main esul is o es ablish a local exis ence heo em om
he ini ial da a ha has a splash o a spla singula i y (no ice ha he equa ions a e ime
e e sible in a ian ). Since he cu e sel -in e sec s ( ailu e o he a c-cho d condi ion), i
is no clea i he ampli ude o he o ici y emains smoo h and he meaning o equa ions
(I.9-I.10). In o de o deal wi h hese obs acles we use a con o mal map
P(w) =  an w
21/2, w ∈C,
whose in en ion is o keep apa he sel -in e sec ing poin s aking he b anch o he squa e
oo abo e passing h ough hose c ucial poin s. He e P(z) will e e o a 2 dimensional
ec o whose componen s a e he eal and imagina y pa s o P(z1+iz2). We also make su e
ha Ω( )∪∂Ω( ) do no con ain any singula poin o he ans o ma ion P. Then po en ial
heo y helps us o ge he ollowing analogous e olu ion equa ions o he new cu e
˜z(α, ) = P(z(α, ))
3
and he new ampli ude ˜ω:
˜z (α, ) = Q2(α, )BR(˜z, ˜ω)(α, ) + ˜c(α, )˜zα(α, ),(I.12)
˜ω (α, ) = −2BR (˜z, ˜ω)(α, )·˜zα(α, )−|BR(˜z, ˜ω)|2(Q2)α(α, )−Q2(α, )˜ω(α, )2
4|˜zα(α, )|2α
+ 2˜c(α, )BRα(˜z, ˜ω)·˜zα(α, ) + (˜c(α, )˜ω(α, ))α−2P−1
2(˜z(α, ))α
+τQ3
|˜zα(α, )|3(˜zT
αHP −1
2˜zα∇P−1
1·˜zα−˜zT
αHP −1
1˜zα∇P−1
2·˜zα)α
+τQ˜zαα(α, )·˜z⊥
α(α, )
|˜zα(α, )|3α
(I.13)
whe e
Q2(α, ) = 
dP
dw (P−1(˜z(α, )))
2
,
and HP −1
ideno es he Hessian ma ix o P−1
i, which is he i- h (i={1,2}) componen
o he ans o ma ion P−1.
He e, we choose ˜c(α, ) in such a way ha |˜zα(α, )|=A( ). This pa icula choice o ˜c
was i s in oduced by Hou e al. in [17] and was la e used by Amb ose [4] and Amb ose-
Masmoudi [5]. The choice o ˜cimplies
˜c(α, ) = α+π
2πZπ
−π
(Q2BR(˜z, ˜ω))β(β, )·˜zβ(β, )
|˜zβ(β, )|2dβ
−Zα
−π
(Q2BR(˜z, ˜ω))β(β, )·˜zβ(β, )
|˜zβ(β, )|2dβ
I is easy o check ha i we ake Q≡1 in (I.12-I.13) we eco e (I.9-I.10).
We also de ine he unc ion
˜ϕ(α, ) = Q2(α, )˜ω(α, )
2|˜zα(α, )|−˜c(α, )|˜zα(α, )|(I.14)
in oduced by Beale e al. o he linea case [7] and by Amb ose-Masmoudi o he nonlinea
one [5]. This unc ion will be used o p o e local exis ence in Sobole spaces.
In he sec ions below, we show a local exis ence heo em based on ene gy es ima es. Sec-
ion III is de o ed o p o ide he app op ia e ini ial da a o he splash and spla singula i ies.
In Sec ion IV we choose an ene gy which does no need a p ecise sign on he Rayleigh-Taylo
unc ion. In Sec ion V we choose a di e en ene gy ha in ol es he sign o he Rayleigh-
Taylo unc ion and he es ima es a e uni o m wi h espec o he su ace ension coe icien .
These wo ene gies a e based on he ones ob ained in he non- ilde domain by Amb ose ([4])
and Amb ose-Masmoudi ([6]).
4
The Rayleigh-Taylo unc ion is gi en by he ollowing o mula
σ≡BR (˜z, ˜ω) + ˜ϕ
|˜zα|BRα(˜z, ˜ω)·˜z⊥
α+˜ω
2|˜zα|2˜zα +˜ϕ
|˜zα|˜zαα·˜z⊥
α
+QBR(˜z, ˜ω) + ˜ω
2|˜zα|2˜zα
2
(∇Q)(˜z)·˜z⊥
α+ (∇P−1
2)(˜z)·˜z⊥
α.
(I.15)
All solu ions ha we will conside h oughou he pape will ha e ini e ene gy, as dis-
cussed in [8]. The sys em sa is ies he conse a ion o he mechanical ene gy. We de ine i
his way: (no o be con used wi h he subsequen de ini ions o some o he ene gies, see
sec ions IV and V).
ES( ) = 1
2ZΩ ( )| (x, y, )|2dxdy +1
2Zπ
−π
(z2(α, ))2∂αz1(α, )dα +τ
2Zπ
−π|∂αz(α, )|dα
≡ Ek( ) + Ep( ) + Eτ( ),
whe e z(α, ) = (z1(α, ), z2(α, )), u(α, ) = (z(α, ), ), and Ω ( ) = Ω( )∩[−π, π]×R
is a undamen al domain in he wa e egion in a pe iod, hen i ollows ha he ene gy is
conse ed.
dEk( )
d =ZΩ ( )
(x, y, )( (x, y, ) + (x, y, )·∇ (x, y, ))dxdy
=ZΩ ( )
(x, y, )(−∇p(x, y, )−(0,1))dxdy
=−ZΩ ( )
(x, y, )(∇(p(x, y, ) + y))dxdy
=−Z∂(Ω ( ))
(x, y, )·−→
n yds +Z∂(Ω ( ))
(x, y, )·−→
nτ
2Kds
=−Zπ
−π
z2(α, )u(α, )·∂αz⊥(α, )dα +τ
2Zπ
−π
u(α, )·∂αz⊥(α, )∂2
αz(α, )·∂αz⊥(α, )
|∂αz(α, )|3dα
(I.16)
whe e we ha e used he incomp essibili y o he luid (∇· = 0) and Laplace-Young’s condi ion
o he p essu e on he in e ace. Nex
dEp( )
d =Zπ
−π
z2(α, )∂ z2(α, )∂αz1(α, )dα +1
2Zπ
−π
(z2(α, ))2∂ ∂αz1(α, )dα
=Zπ
−π
z2(α, )∂ z2(α, )∂αz1(α, )dα −Zπ
−π
z2(α, )∂αz2(α, )∂ z1(α, )dα
=Zπ
−π
z2(α, )u(α, )·∂αz⊥(α, )dα. (I.17)
5

dEτ( )
d =τ
2Zπ
−π
∂αz(α, )·∂α∂ z(α, )
|∂αz(α, )|dα =−τ
2Zπ
−π
∂2
αz(α, )·∂ z(α, )
|∂αz(α, )|dα
=−τ
2Zπ
−π
∂2
αz(α, )·u(α, )
|∂αz(α, )|dα =−τ
2Zπ
−π
∂2
αz(α, )·∂αz⊥(α, )
|∂αz(α, )|3u(α, )·∂⊥
αz(α, )dα
(I.18)
Adding all he de i a i es we ge he desi ed esul .
II P ope ies o he cu a u e in he ilde domain
In his sec ion we will ew i e he e m co esponding o he cu a u e K(z(α, )) in he new
ilde a iables ˜z(α, ).
We will p oceed s ep by s ep. Le us ecall ha he cu a u e is de ined by
K(α, ) = zαα(α, )·z⊥
α(α, )
|zα(α, )|3
We begin wi h he e m |zα(α, )|3. We ha e ha
|˜zα(α, )|2=h∂αP(z(α, )), ∂αP(z(α, ))i=h∇P(z(α, )) ·zα(α, ),∇P(z(α, )) ·zα(α, )i
Since Pand P−1a e con o mal, by he Cauchy-Riemann equa ions
∇P(z(α, ))T∇P(z(α, )) = Q2(α, )Id2,
ha implies ha
|˜zα(α, )|3=Q3(α, )|zα(α, )|3
We mo e o he o he e m
hzαα(α, ), z⊥
α(α, )i=h∂α∇P−1(˜z(α, )) ·˜zα(α, ),(∇P−1(˜z(α, )) ·˜zα(α, ))⊥i
=h∇P−1(˜z(α, )) ·˜zαα(α, ),(∇P−1(˜z(α, )) ·˜zα(α, ))⊥i
+h∂α∇P−1(˜z(α, ))·˜zα(α, ),(∇P−1(˜z(α, )) ·˜zα(α, ))⊥i ≡ W+X
Again, by he Cauchy-Riemann equa ions
W=1
Q2(α, )h˜zαα(α, ),˜zα(α, )⊥i
De eloping he e ms in X, we ge
∇P−1(˜z(α, ))·˜zα(α, ) = ˜zT
α(α, )·HP −1
1(˜z(α, )) ·˜zα(α, )
˜zT
α(α, )·HP −1
2(˜z(α, )) ·˜zα(α, ),
6
whe e HP−1
ideno es he Hessian o he i- h componen o P−1(i= 1,2). Hence, we can
w i e Xas
X=−˜zT
α(α, )·HP −1
1(˜z(α, )) ·˜zα(α, )∇P−1
2(˜z(α, )) ·˜z(α, )
+ ˜zT
α(α, )·HP −1
2(˜z(α, )) ·˜zα(α, )∇P−1
1(˜z(α, )) ·˜z(α, ).
This means ha
K(α, ) = Q(α, )˜zαα(α, )·˜z⊥
α(α, )
|˜z(α, )|3+X(α, )Q(α, )3
|˜z(α, )|3≡Q(α, )˜
K(α, ) + M(α, )
We will now y o simpli y u he by exploi ing he Cauchy-Riemann equa ions. We can
calcula e he Hessian and he g adien e ms as:
P−1
1,x (˜z) = ℜ4˜z
1 + ˜z4≡ ℜ(a)
P−1
1,y (˜z) = ℜ4i˜z
1 + ˜z4≡ −ℑ(a)
P−1
2,x (˜z) = ℑ4˜z
1 + ˜z4≡ ℑ(a)
P−1
2,y (˜z) = ℑ4i˜z
1 + ˜z4≡ ℜ(a)
P−1
1,x,x(˜z) = ℜ4(1 −3˜z4)
(1 + ˜z4)2≡ ℜ(b)
P−1
1,x,y(˜z) = ℜ4i(1 −3˜z4)
(1 + ˜z4)2≡ −ℑ(b)
P−1
2,x,x(˜z) = ℑ4(1 −3˜z4)
(1 + ˜z4)2≡ ℑ(b)
P−1
2,x,y(˜z) = ℑ4i(1 −3˜z4)
(1 + ˜z4)2≡ ℜ(b)
The e o e he Hessians a e
HP −1
1=ℜ(b)−ℑ(b)
−ℑ(b)−ℜ(b), HP−1
2=ℑ(b)ℜ(b)
ℜ(b)−ℑ(b),
Calcula ing u he :
˜zT
αHP −1
2˜zα=ℜ(b)(2˜z1
α˜z2
α) + ℑ(b)((˜z1
α)2−(˜z2
α)2)
˜zT
αHP −1
1˜zα=ℜ(b)((˜z1
α)2−(˜z2
α)2)−ℑ(b)(2˜z1
α˜z2
α)
7
X1=ℜ(a)ℜ(b)(2(˜z1
α)2˜z2
α) + ℜ(a)ℑ(b)((˜z1
α)2˜z1
α−(˜z2
α)2˜z1
α)
+ℑ(a)ℜ(b)(−2˜z1
α(˜z2
α)2) + ℑ(b)ℑ(b)((˜z1
α)2˜z2
α−(˜z2
α)2˜z2
α)
X2=ℜ(b)ℜ(b)((˜z1
α)2˜z2
α−(˜z2
α)2˜z2
α) + ℜ(a)ℑ(b)(−2˜z1
α(˜z2
α)2)
+ℑ(a)ℑ(b)(2(˜z1
α)2˜z2
α) + ℑ(a)ℜ(b)((˜z1
α)2˜z1
α−(˜z2
α)2˜z1
α)
This means
X=X1−X2= ((˜z1
α)2+ (˜z2
α)2)(˜z2
α(ℜ(a)ℜ(b) + ℑ(a)ℑ(b)) + ˜z1
α(ℜ(a)ℑ(b)−ℑ(a)ℜ(b)))
≡((˜z1
α)2+ (˜z2
α)2)hG(z),˜zαi.
We can see ha
−Qα
Q3=1
2∂α1
Q2=∂α(ℜ(a)2+ℑ(a)2)
=ℜ(a)ℜ(b)˜z1
α−ℜ(a)ℑ(b)˜z2
α+ℑ(a)ℑ(b)˜z1
α+ℑ(a)ℜ(b)˜z2
α
=hG(z),˜z⊥
αi
by he Cauchy-Riemann equa ions.
I we ake one de i a i e in space o X, we ob ain
∂αX= ((˜z1
α)2+ (˜z2
α)2)h∇G(˜z)·˜zα,˜zαi+ ((˜z1
α)2+ (˜z2
α)2)hG(˜z),˜zααi
= ((˜z1
α)2+ (˜z2
α)2)h∇G(˜z)·˜zα,˜zαi+|˜zα|3˜
KhG(˜z),˜z⊥
αi
= ((˜z1
α)2+ (˜z2
α)2)h∇G(˜z)·˜zα,˜zαi−|˜zα|3˜
KQα
Q3,
This implies
K=Q˜
K−Q3X
|˜z|3⇒Kα= (Q˜
K)α+Q3
|˜zα|h∇G(˜z)·˜zα,˜zαi− ˜
KQα= (Q˜
K)α+M1+M2
La e , we will see ha he M1is a low o de e m and can be abso bed by he ene gy.
III Ini ial da a
Fo ini ial da a we a e in e es ed in conside ing a sel -in e sec ing cu e in one poin . Mo e
p ecisely, we will use as ini ial da a splash cu es which a e de ined his way:
De ini ion III.1 We say ha z(α) = (z1(α), z2(α)) is a splash cu e i
8
1. z1(α)−α, z2(α)a e smoo h unc ions and 2π-pe iodic.
2. z(α)sa is ies he a c-cho d condi ion a e e y poin excep a α1and α2, wi h α1< α2
whe e z(α1) = z(α2)and |zα(α1)|,|zα(α2)|>0. This means z(α1) = z(α2), bu i we
emo e ei he a neighbo hood o α1o a neighbo hood o α2in pa ame e space, hen
he a c-cho d condi ion holds.
3. The cu e z(α)sepa a es he complex plane in o wo egions; a connec ed wa e egion
and a acuum egion (no necessa ily connec ed). The wa e egion con ains each poin
x+iy o which y is la ge nega i e. We choose he pa ame iza ion such ha he no mal
ec o n=(−∂αz2(α),∂αz1(α))
|∂αz(α)|poin s o he acuum egion. We ega d he in e ace o be
pa o he wa e egion.
4. We can choose a b anch o he unc ion Pon he wa e egion such ha he cu e
˜z(α) = (˜z1(α),˜z2(α)) = P(z(α)) sa is ies:
(a) ˜z1(α)and ˜z2(α)a e smoo h and 2π-pe iodic.
(b) ˜zis a closed con ou .
(c) ˜zsa is ies he a c-cho d condi ion.
We will choose he b anch o he oo ha p oduces ha
lim
y→−∞ P(x+iy) = −e−iπ/4
independen ly o x.
5. P(w)is analy ic a wand dP
dw (w)6= 0 i wbelongs o he in e io o he wa e egion.
Fu he mo e, (±π, 0) and (0,0) belong o he acuum egion.
6. ˜z(α)6=ql o l= 0, ..., 4, whe e
q0= (0,0) , q1=1
√2,1
√2, q2=−1
√2,1
√2, q3=−1
√2,−1
√2, q4=1
√2,−1
√2.
(III.1)
Mo eo e , we will de ine a spla cu e as a splash cu e bu eplacing condi ion (2) by
he ac ha he cu e ouches i sel along an a c, ins ead o a poin .
Le us no e ha in o de o measu e when he ans o ma ion Pis egula , we need o
con ol he dis ance o he poin s ql. In o de o do so, we in oduce he unc ion
m(ql)(α, )≡ |˜z(α, )−ql|
o l= 0,...,4.
We ha e pe o med nume ical simula ions, as explained in [9] wi h he ollowing ini ial
da a on he non- ilde domain:
z0
1(α) = α+1
4−3π
2−1.9sin(α) + 1
2sin(2α) + 1
4π
2−1.9sin(3α)
9
dC
d = OK + 1
|˜zα|τZQ2k+4 ˜ω2∂k
α(˜ω)∂k+1
α(˜
K) = OK + C1
IV.D De elopmen o he de i a i e in B
We s a om he de elopmen o B1,B2,B3and B4. We i ially ha e:
B1=1
τZQ2k+2Λ(∂k
α(˜ω))Q2˜ω2
|˜zα|H(∂k
α(˜
K))
B3=−2ZQ2k+2QαΛ(∂k
α(˜ω))∂k
α(˜
K)
B4= OK −Z(2k+ 2)Q2k+2QαH(∂k+1
α(˜ω))∂k
α(˜
K)
We now look a B2. We can decompose i in he ollowing way
B2= 2 ZQ2k+2Λ(∂k
α(˜ω))∂k
α(Qα˜
K+Q˜
Kα)
= OK + 2 ZQ2k+2Λ(∂k
α(˜ω))(Qα∂k
α(˜
K) + Q∂k+1
α(˜
K) + kQα∂k
α(˜
K))
= OK + B2,1+B2,2+B2,3
We can w i e down he e ms B2,1and B2,3in he o m
B2,1= 2 ZQ2k+2H(∂k+1
α(˜ω))Qα∂k
α(˜
K)
B2,3= 2kZQ2k+2H(∂k+1
α(˜ω))Qα∂k
α(˜
K)
In eg a ing by pa s in B2,2we es ablish
B2,2=−2ZQ2k+3Λ(∂k+1
α(˜ω))∂k
α(˜
K)
−2(2k+ 3) ZQ2k+2QαΛ(∂k
α(˜ω))∂k
α(˜
K)
=B2,2,1+B2,2,2
Again, B2,2,2can easily be educed o he canonical o m
B2,2,2=−2(2k+ 3) ZQ2k+2QαH(∂k+1
α(˜ω))∂k
α(˜
K)
IV.E Collec ion o he e ms
We will spli all he uncon olled e ms in o h ee ca ego ies: high o de and low o de ypes
I and II and we will see ha he sum o he e ms in each ca ego y adds up o low enough
o de e ms, deno ed by OK.
16

IV.E.1 High O de
F om A:
2ZQ2k+3 ∂k
α(˜
K)∂k
α(H(˜ωαα)) (A2)
F om B:
−2ZQ2k+3Λ(∂k+1
α(˜ω))∂k
α(˜
K) (B2,2,1)
F om C:
No e ms om C.
IV.E.2 Low O de Type I
F om A:
2Z2kQ2k+2Qα∂k
α(˜
K)∂k−1
α(H(˜ωαα)) (A1)
2Z4Q2k+2Qα∂k
α(˜
K)∂k
α(H(˜ωα)) (A3)
F om B:
−2ZQ2k+2QαΛ(∂k
α(˜ω))∂k
α(˜
K) (B3)
−Z(2k+ 2)Q2k+2QαH(∂k+1
α(˜ω))∂k
α(˜
K) (B4)
2ZQ2k+2H(∂k+1
α(˜ω))Qα∂k
α(˜
K)) (B2,1)
2kZQ2k+2H(∂k+1
α(˜ω))Qα∂k
α(˜
K) (B2,3)
−2(2k+ 3) ZQ2k+2QαH(∂k+1
α(˜ω))∂k
α(˜
K) (B2,2,2)
F om C:
No e ms om C.
IV.E.3 Low O de Type II
F om A:
No e ms om A.
F om B:
1
τZQ2k+2Λ(∂k
α(˜ω))Q2˜ω2
|˜zα|H(∂k
α(˜
K)) (B1)
F om C:
1
|˜zα|τZQ2k+4 ˜ω2∂k
α(˜ω)∂k+1
α(˜
K) (C1)
17
IV.F Regula ized sys em
Now, le ˜zε,δ,µ(α, ) be a solu ion o he ollowing sys em (compa e wi h (I.12 - I.13)):
˜zε,δ,µ
(α, ) = φδ∗φδ∗Q2(˜zε,δ,µ)BR(˜zε,δ,µ,˜ωε,δ,µ)(α, ) + φµ∗˜cε,δ,µ φµ∗∂α˜zε,δ,µ(α, ),
(IV.1)
˜ωε,δ,µ
=φδ∗φδ∗−2BR (˜zε,δ,µ,˜ωε,δ,µ)·˜zε,δ,µ
α−|BR(˜zε,δ,µ,˜ωε,δ,µ)|2(Q2(˜zεδ,µ))α
−Q2˜
(ωε,δ,µ)2
4|˜zε,δ,µ
α|2α+ 2cε,δ,µBRα(˜zε,δ,µ, ωε,δ,µ)·˜zε,δ,µ
α+cε,δ,µ ˜ωε,δ,µα−2P−1
2(˜zε,δ,µ(α, ))α
+τ Q3(˜zε,δ,µ)
|˜zε,δ,µ
α(α, )|3(˜zε,δ,µ
α)THP−1
2˜zε,δ,µ
α∇P−1
1·˜zε,δ,µ
α−(˜zε,δ,µ
α)THP−1
1˜zε,δ,µ
α∇P−1
2·˜zε,δ,µ
α)!α
+τ Q˜zε,δ,µ
αα ·(˜zε,δ,µ
α)⊥
|˜zε,δ,µ
α|3!α!−εφµ∗φµ∗Λ(˜ωε,δ,µ)1
Q2k+3 (IV.2)
˜zε,δ,µ(α, 0) = ˜z0(α) and ˜ωε,δ,µ(α, 0) = ˜ω0(α) o ε > 0, δ > 0, µ > 0. The unc ions φδand φµ
a e e en molli ie s,
˜cε,δ,µ(α) =α+π
2πZπ
−π
∂β˜zε,δ,µ(β))
|∂β˜zε,δ,µ(β)|2·φδ∗φδ∗(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ
−Zα
−π
∂β˜zε,δ,µ(β)
|∂β˜zε,δ,µ(β)|2·φδ∗φδ∗(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ,
and
cε,δ,µ(α) =α+π
2πZπ
−π
∂β˜zε,δ,µ(β))
|∂β˜zε,δ,µ(β)|2·(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ
−Zα
−π
∂β˜zε,δ,µ(β)
|∂β˜zε,δ,µ(β)|2·(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ,
The RHS o he e olu ion equa ions o ˜zε,δ,µ and ˜ωε,δ,µ a e Lipschi z in he spaces Hk+2(T)
and Hk+1
2(T) since hey a e molli ied. The e o e we can sol e (IV.1-IV.2) o sho ime,
hanks o Pica d’s heo em.
Now, we can pe o m ene gy es ima es o ge uni o m bounds in µ(we jus deal wi h a
anspo e m and a dissipa i e) and we can le µgo o ze o. The ene gy es ima es ha we
can ge a e he ollowing:
d
d k˜zε,δ,µk2
H5+kF(˜zε,δ,µ)k2
L∞+k˜ωε,δ,µk2
H3+ 1
2+
4
X
l=0
1
mε,δ,µ(ql)!( )
≤C(δ) k˜zε,δ,µk2
H5+kF(˜zε,δ,µ)k2
L∞+k˜ωε,δ,µk2
H3+ 1
2+
4
X
l=0
1
mε,δ,µ(ql)!j
( ).
18
We should no e ha o he new sys em wi hou he φµmolli ie , he leng h o he angen
ec o |∂α˜zδ|is now cons an in space and depends only on ime. Nex we will pe o m ene gy
es ima es as in he p e ious case by using he cu a u e ˜
Kδ om he cu e ˜zδ.
Simila ly, we ge (le us omi he supe sc ip δ, ε in ˜zδ,ε and ˜ωδ,ε)
•˜
K = NICE3 + Q2
2|˜zα|3φδ∗φδ∗H(˜ωαα) + 1
|˜zα|3(Q2)αφδ∗φδ∗H(˜ωα),
•∂k
α(cα˜ω) = NICE35 + Q2˜ω2
2|˜zα|H(∂k
α(˜
K)),
•∂k
α(˜c˜ωα) = NICE35 ,
and he ollowing collec ion o e ms:
IV.F.1 High O de
F om A:
2ZQ2k+3 ∂k
α(˜
K)∂k
αφδ∗φδ∗(H(˜ωαα)) (A2)
F om B:
−2ZQ2k+3Λ(∂k+1
α(˜ω))φδ∗φδ∗∂k
α(˜
K) (B2,2,1)
−2ε
τk∂k+1
α˜ωk2
L2(D)
F om C:
No e ms om C.
IV.F.2 Low O de Type I
F om A:
2Z2kQ2k+2Qα∂k
α(˜
K)∂k−1
αφδ∗φδ∗(H(˜ωαα)) (A1)
2Z4Q2k+2Qα∂k
α(˜
K)φδ∗φδ∗∂k
α(H(˜ωα)) (A3)
F om B:
−2ZQ2k+2QαΛ(∂k
α(˜ω))φδ∗φδ∗∂k
α(˜
K) (B3)
−Z(2k+ 2)Q2k+2QαH(∂k+1
α(˜ω))φδ∗φδ∗∂k
α(˜
K) (B4)
2ZQ2k+2H(∂k+1
α(˜ω))Qαφδ∗φδ∗∂k
α(˜
K)) (B2,1)
2kZQ2k+2H(∂k+1
α(˜ω))Qαφδ∗φδ∗∂k
α(˜
K) (B2,3)
−2(2k+ 3) ZQ2k+2QαH(∂k+1
α(˜ω))φδ∗φδ∗∂k
α(˜
K) (B2,2,2)
19
F om C:
No e ms om C.
IV.F.3 Low O de Type II
F om A:
No e ms om A.
F om B:
1
τZQ2k+2Λ(∂k
α(˜ω))Q2˜ω2
|˜zα|φδ∗φδ∗H(∂k
α(˜
K)) (B1)
F om C:
1
|˜zα|τZQ2k+4 ˜ω2∂k
α(˜ω)φδ∗φδ∗∂k+1
α(˜
K) (C1)
We no e ha h oughou his sec ion we ha e epea edly used he ollowing commu a o
es ima e o con olu ions:
kφδ∗(∂α g)−gφδ∗(∂α )kL2≤Ck∂αgkL∞k kL2,(IV.3)
whe e he cons an Cis independen o δ, and g.
Also using his commu a o es ima e we can ind all he cancela ions we need in he
p e ious collec ion o e ms o low o de ype I and II o ob ain a sui able ene gy es ima e.
Rega ding he high o de e ms, we will do he es ima es in de ail. We will see he need
o he dissipa i e e m since he e a e e ms ha escape o hal o a de i a i e.
A2+B2,2,1+D= 2 ZQ2k+3 ∂k
α(˜
K)φδ∗φδ∗H(∂k+2
α˜ω)
−2ZQ2k+3H(∂k+2
α(˜ω))φδ∗φδ∗∂k
α(˜
K)−2εk∂k+1
α˜ωk2
L2
= 2 Z∂k
α(˜
K)Q2k+3φδ∗φδ∗H(∂k+2
α˜ω)−φδ∗φδ∗Q2k+3H(∂k+2
α˜ω)−2εk∂k+1
α˜ωk2
L2
≤ k∂k
α˜
KkL2k∂αQ2k+3kL∞k∂k+1
α˜ωkL2−2εk∂k+1
α˜ωk2
L2≤C(ε)Ep( ),
which is uni o m in δ. This p o es ha we can pass o he limi δ→0.
Finally, by applying he a p io i ene gy es ima es o he new sys em (which only depend
on ε) we can pass o he limi ε→0 since now we don’ ha e he p e ious p oblems and
A2+B2,2,1= 0.
V Ene gy wi h he Rayleigh-Taylo condi ion
In his sec ion, we p o e local exis ence in he ilde domain, whe e he ime o exis ence does
no depend on he su ace ension coe icien . In his heo em, we need ini ial da a o sa is y
he Rayleigh-Taylo condi ion as we explain in Sec ion III. This Rayleigh-Taylo condi ion
will hold in pa icula i he su ace ension coe icien is small enough.
20
Theo em V.1 Le k≥3. Le ˜z0(α)be he image o a splash cu e by he map Ppa ame ized
in such a way ha |∂α˜z0(α)|=L
2π, whe e Lis he leng h o he cu e in a undamen-
al pe iod, and such ha ˜z0
1(α),˜z0
2(α)∈Hk+2(T). Le ˜ϕ(α, 0) ∈Hk+1
2(T)be as in (I.14)
and le ˜ω(α, 0) ∈Hk−1(T). Then he e exis a ini e ime T > 0, a ime- a ying cu e
˜z(α, )∈C([0, T]; Hk+2), and unc ions ˜ω(α, )∈C([0, T]; Hk−1)and ˜ϕ∈C([0, T ]; Hk+1
2)
p o iding a solu ion o he wa e wa e equa ions (I.12 - I.13). Assume ha ini ially, he
Rayleigh-Taylo condi ion is s ic ly posi i e.
In o de o p o e his heo em we will use he solu ions we ha e ob ained in heo em IV.1
o τ > 0. We will pe o m ene gy es ima es on hese solu ions.
V.A The ene gy
We will de ine he ene gy o k≥3 as
E2
k( ) = EE2( ) + τ|˜zα|
2ZQ2k+1 ∂k
α(˜
K)2
|{z }
A
+ZQ2k−2∂k
α( ˜ϕ)Λ(∂k
α( ˜ϕ))
| {z }
B
+|˜zα|2τZ(Ck ˜
K( )kH1+˜
K)Q2k+1∂k−1
α(˜
K)Λ(∂k−1
α(˜
K))
|{z }
C
+ 2|˜zα|ZCk ˜
K( )kH1Q2k−2∂k
α( ˜ϕ)2
|{z }
D
+|˜zα|2ZσQ2k∂k−1
α(˜
K)2
|{z }
E
+|˜zα|2
m(Q2kσ)( ),
whe e m(Q2kσ) = minα∈TQ2k(˜z(α, ))σ(α, ) and Cis a su icien ly la ge cons an such ha
Cis s ic ly posi i e. Remembe ha ˜ϕwas in oduced in Equa ion I.14.
A his poin is impo an o no ice he ollowing.
Lemma V.2 The ollowing sen ences hold.
1. Le ˜ϕ∈H3+ 1
2,˜ω∈H2and z∈Hkwi h k≥4. Then ˜ω∈H3.
2. Le ˜ϕ∈H3+ 1
2,˜ω∈H3and z∈Hkwi h k≥5. Then ˜ω∈H3.5.
3. Le ˜ω∈H3+ 1
2, and ˜z∈Hkwi h k≥5. Then ˜ϕ∈H3.5.
This lemma shows ha o a ixed τ > 0 he ene gy o his sec ion is equi alen o his one in
sec ion IV.A. This allows us o use his ene gy o ex end he solu ions o he heo em IV.1
up o a ime Twhich does no depend on τ( o a small enough τ).
V.B The ene gy es ima es
Again, we will only ocus on he new e ms (A−E) since he es ima es o he o he ones
we e p o ed in [12] and in [8].
21

V.B.1 ˜
K
P oposi ion V.3
˜
K =NICE3B +Q2
2|˜zα|3H(˜ωαα) + 1
|˜zα|3(Q2)αH(˜ωα)
=NICE3B +1
|˜zα|2H( ˜ϕαα)−1
|˜zα|(˜
K˜ϕ)α,
whe e NICE3B means ZQj∂k
α(˜
K)∂k
α(NICE3B)≤CEp
k( )
o some posi i e cons an s C, p and any j.
P oo : The i s equali y ollows om he p oo om he las sec ion since he ene gies a e
equi alen (see Lemma V.2). We now p o e he second one. We begin by using he ela ion
(I.14) o ge
˜
K = NICE3B + Q2
|˜zα|2H ˜ϕ
Q2αα+Q2
|˜zα|H ˜c
Q2αα
+2(Q2)α
|˜zα|2H ˜ϕ
Q2α+2(Q2)α
|˜zα|H ˜c
Q2α=I+J
We can easily see ha
˜cα=−˜zα
|˜zα|2·(Q2BR)α= NICE3B
since i is a he le el o ˜ωα,˜zαα bu we gain one de i a i e by mul iplying by he angen ial
di ec ion. This p o es ha
J= NICE3B + 2(Q2)α
|˜zα|2H˜ϕα
Q2.
Looking now o ˜cαα we can see ha
˜cαα =−˜zαα
|˜zα|2·(Q2BR)α−˜zα
|˜zα|2·(Q2BR)αα =I1+I2.
Using he s anda d es ima es, he only hing ha causes ouble in I1is when all he de i a-
i es hi ˜ωand he e o e
I1= NICE3B −KQ2
2|˜zα|H(˜ωα).
Rega ding I2, again, we need all he de i a i es o hi BR o ge he mos singula e ms,
which a e
22
I2= NICE3B −Q2
|˜zα|2˜zα·2
2πZπ
−π
(˜zα(α)−˜zα(β))⊥
|˜z(α)−˜z(β)|2˜ωα(α−β)dβ
−Q2
|˜zα|2˜zα·1
2πZπ
−π
(˜z(α)−˜z(β))⊥
|˜z(α)−˜z(β)|2˜ωαα(α−β)dβ
−Q2
|˜zα|2˜zα·1
2πZπ
−π
(˜zαα(α)−˜zαα(β))⊥
|˜z(α)−˜z(β)|2˜ωα(α−β)dβ
= NICE3B + 2Q2
|˜zα|2
1
2
˜zα·˜z⊥
αα
|˜zα|2H(˜ωα)−Q2
|˜zα|2
˜zα·˜z⊥
αα
|˜zα|2H(˜ωα)−Q2
|˜zα|2
1
2
˜ω
|˜zα|2˜zα·H(˜z⊥
ααα)
Collec ing all he e ms om I1and I2, we ob ain
˜cαα
Q2= NICE3B + 1
2|˜zα|H(( ˜
K˜ω)α)
= NICE3B + 1
Q2H(( ˜
K˜ϕ)α).
We can inally w i e he o al con ibu ion as
˜
K = NICE3B + Q2
|˜zα|2H˜ϕαα
Q2−Q2
|˜zα|2H4Qα˜ϕα
Q3
−1
|˜zα|(˜
K˜ϕ)α+2(Q2)α
|˜zα|2H˜ϕα
Q2
= NICE3B + Q2
|˜zα|2H˜ϕαα
Q2−1
|˜zα|(˜
K˜ϕ)α
= NICE3B + 1
|˜zα|2H( ˜ϕαα)−1
|˜zα|(˜
K˜ϕ)α
as we wan ed o p o e.

V.B.2 ˜ϕ
Th oughou his sec ion, we will use he ollowing es ima e which was p o ed in [8] o he
case wi hou su ace ension. The p oo is exac ly he same o he case wi h i .
ϕα = NICE2B + ˜ϕ˜ϕαα
|˜zα|−Q2σ˜
K+τQ2
2|˜zα|(˜
KQ)α+Mα,
whe e NICE2B means
ZQjΛ(∂k
α( ˜ϕ))∂k−1
α(NICE2B)≤CEp
k( )
o some posi i e cons an s C, p and any j.
23
V.C Calcula ions o he ime de i a i e o he ene gy
Using he p e ious lemmas and p oposi ions, we can ge he ollowing es ima es o he
de i a i e o he ene gy:
dA
d = OK + τ
|˜zα|ZQ2k+1∂k
α(˜
K)∂k
α(H( ˜ϕαα)) −τZQ2k+1∂k
α(˜
K)∂k
α(( ˜
K˜ϕ)α)
= OK + τ
|˜zα|ZQ2k+1∂k
α(˜
K)∂k
α(H( ˜ϕαα)) −τZQ2k+1∂k
α(˜
K)∂k+1
α( ˜ϕ)˜
K= OK + A1+A2
Again, we need o be ca e ul while compu ing he de i a i e o Bas in Sec ion IV. We ob ain
dB
d = 2 ZQ2k−2Λ(∂k
α( ˜ϕ))∂k−1
α( ˜ϕα ) + Z(Q2k−2)αH(∂k
α( ˜ϕ))∂k−1
α( ˜ϕα )
= OK −2ZQ2k−2Λ(∂k
α( ˜ϕ))∂k−1
α˜ϕ˜ϕαα
|˜zα|
−2ZQ2k−2Λ(∂k
α( ˜ϕ))∂k−1
α(Q2σ˜
K)
+τ
|˜zα|ZQ2k−2Λ(∂k
α( ˜ϕ))∂k
α(Q2(Q˜
K)α)
−τ
|˜zα|ZQ2kQαΛ(∂k
α( ˜ϕ))∂k
α(˜
K)
+Zτ
|˜zα|(k−1)Q2k+2QαH(∂k
α( ˜ϕ))∂k+1
α(˜
K)
= OK + B1+B2+B3+B4+B5
V.D De elopmen o he de i a i e o he B e m
We begin no icing ha B1= OK, as i was p o ed in [12]. In eg a ing by pa s in B5, we
ha e ha
B5=−Zτ
|˜zα|(k−1)Q2k+2QαH(∂k+1
α( ˜ϕ))∂k
α(˜
K)
Fu he mo e, he only singula e ms a ising om B2a e when all de i a i es hi ei he
˜
Ko σ, his gi es us
B2= OK −2ZQ2kΛ(∂k
α( ˜ϕ))∂k−1
α(σ)˜
K−2ZQ2kΛ(∂k
α( ˜ϕ))∂k−1
α(˜
K)σ= OK + B2,1+B2,2.
Howe e , he only singula e m o he Rayleigh-Taylo condi ion ha is no in Hk−1is
he one belonging o BR (˜z, ˜ω)·˜zαwhen he ime de i a i e hi s ω, his means
B2,1= OK −τZQ2kΛ(∂k
α( ˜ϕ)) ˜
KH(∂k
α(˜
KQ))
=−τZQ2k+1Λ(∂k
α( ˜ϕ)) ˜
KH(∂k
α(˜
K))
24
Finally, de eloping B3we ob ain
B3=τ
|˜zα|ZQ2k−2Λ(∂k
α( ˜ϕ))∂k
α(Q3˜
Kα)
+τ
|˜zα|ZQ2k−2Λ(∂k
α( ˜ϕ))∂k
α(Q2Qα˜
K)
=B3,1+B3,2
Modulo lowe o de e ms we can see ha
B3,2= OK + τ
|˜zα|ZQ2kQαΛ(∂k
α( ˜ϕ))∂k
α(˜
K)
We can con inue spli ing B3,1in o
B3,1= OK + τ
|˜zα|ZQ2k+1Λ(∂k
α( ˜ϕ))∂k+1
α(˜
K) + τ
|˜zα|Z3kQ2kQαΛ(∂k
α( ˜ϕ))∂k
α(˜
K)
= OK −τ
|˜zα|ZQ2k+1Λ(∂k+1
α( ˜ϕ))∂k
α(˜
K) + τ
|˜zα|Z(k−1)Q2kQαΛ(∂k
α( ˜ϕ))∂k
α(˜
K)
= OK + B3,1,1+B3,1,2
whe e in he las equali y we ha e pe o med an in eg a ion by pa s. We can obse e ha
B3,2+B4=B3,1,2+B5= 0, B3,1,1+A1= 0
We will now see ha B2,2cancels wi h he e m a ising om he de i a i e o E. Taking
in o accoun he p e ious lemmas
dE
d = 2 ZσQ2k∂k−1
α(˜
K)H(∂k+1
α( ˜ϕ)) = OK −B2,2
Finally, we will see ha he con ibu ions om he ime de i a i es o Cand Dcancel
B2,1and A2. We s a by no icing ha , modulo lowe o de e ms A2=B2,1. Fu he mo e
dC
d = OK + 2τZ(Ck ˜
K( )kH1+˜
K)Q2k+1H(∂k+1
α( ˜ϕ))Λ(∂k−1
α(˜
K))
dD
d = OK + 2τZCk ˜
K( )kH1Q2k+1 ∂k
α( ˜ϕ)∂k+1
α(˜
K),
which, by in eg a ion by pa s esul s in
dC
d +dD
d +A2+B2,1= OK.
Adding all he con ibu ions, we can bound he de i a i e in ime o he ene gy by a
powe o he ene gy.
25
[26] S. Wu. Global wellposedness o he 3-D ull wa e wa e p oblem. In en . Ma h.,
184(1):125-220, 2011.
[27] H. Yosiha a. G a i y wa es on he ee su ace o an incomp essible pe ec luid o ini e
dep h. Publ. Res. Ins . Ma h. Sci., 18(1):49-96, 1982.
[28] P. Zhang and Z. Zhang. On he ee bounda y p oblem o h ee-dimensional incomp ess-
ible Eule equa ions. Comm. Pu e Appl. Ma h., 61(7):877-940, 2008.
Angel Cas o
D´epa emen de Ma h´ema iques e Applica ions
´
Ecole No male Sup´e ieu e
45, Rue d’Ulm, 75005 Pa is
Email: cas [email protected].
Diego C´o doba Cha les Fe e man
Ins i u o de Ciencias Ma em´a icas Depa men o Ma hema ics
Consejo Supe io de In es igaciones Cien ´ı icas P ince on Uni e si y
C/ Nicol´as Cab e a, 13-15 1102 Fine Hall, Washing on Rd,
Campus Can oblanco UAM, 28049 Mad id P ince on, NJ 08544, USA
Email: dcg@icma .es Email: c @ma h.p ince on.edu
F ancisco Gancedo Ja ie G´omez-Se ano
Depa amen o de An´alisis Ma em´a ico Ins i u o de Ciencias Ma em´a icas
Uni e sidad de Se illa Consejo Supe io de In es igaciones Cien ´ı icas
C/ Ta ia, s/n C/ Nicol´as Cab e a, 13-15
Campus Reina Me cedes, 41012 Se illa Campus Can oblanco UAM, 28049 Mad id
Email: ga[email p o ec ed] Email: ja ie .gomez@icma .es
32