a Xi :1102.1902 2 [ma h.AP] 10 Jun 2011
Rayleigh-Taylo b eakdown o he Muska p oblem
wi h applica ions o wa e wa es
´
Angel Cas o, Diego C´o doba, Cha les Fe e man,
F ancisco Gancedo and Ma ´ıa L´opez-Fe n´andez.
June 14, 2011
Abs ac
The Muska p oblem models he e olu ion o he in e ace be ween wo di e en luids
in po ous media. The Rayleigh-Taylo condi ion is na u al o each linea s abili y o he
Muska p oblem. We show ha he Rayleigh-Taylo condi ion may hold ini ially bu
b eak down in ini e ime. As a consequence o he me hod used, we p o e he exis ence
o wa e wa es u ning.
1 In oduc ion
The Muska p oblem [26] models he e olu ion o an in e ace be ween wo luids o di e en
cha ac e is ics in po ous media by means o Da cy’s law:
µ
κu=−∇p−(0,gρ),(1)
whe e (x, )∈R2×R+,u= (u1(x, ), u2(x, )) is he incomp essible eloci y (i.e. ∇ · u= 0),
p=p(x, ) is he p essu e, µ(x, ) is he dynamic iscosi y, κis he pe meabili y o he
iso opic medium, ρ=ρ(x, ) is he liquid densi y, and g is he accele a ion due o g a i y.
Mo e p ecisely, he in e ace sepa a es he domains Ω1and Ω2de ined by
(µ, ρ)(x1, x2, ) = (µ1, ρ1), x ∈Ω1( )
(µ2, ρ2), x ∈Ω2( ) = R2−Ω1( ),
and µ1, µ2, ρ1, ρ2a e cons an s. This physical si ua ion is also ela ed o he e olu ion o wo
luids o di e en cha ac e is ics in a Hele-Shaw cell [22], due o he ac ha he laws which
model bo h phenomena a e ma hema ically analogous [31].
This pape is conce ned wi h he case µ1=µ2which p o ides weak solu ions o he
ollowing anspo equa ion
ρ +u· ∇ρ= 0,
ρ0=ρ(x, 0), x ∈R2,(2)
1
whe e ini ially he scala ρ0is gi en by
ρ0=ρ(x1, x2,0) = ρ1in Ω1(0) = {x2> 0(x1)}
ρ2in Ω2(0) = {x2< 0(x1)}.(3)
Le he ee bounda y be pa ame ized by
∂Ωj( ) = {z(α, ) = (z1(α, ), z2(α, )) : α∈R}
whe e
z(α, )−(α, 0)
is 2π-pe iodic in he space pa ame e αo , an open con ou anishing a in ini y
lim
α→±∞(z(α, )−(α, 0)) = 0
wi h ini ial da a z(α, 0) = z0(α) = (α, 0(α)). F om Da cy’s law, we ind ha he o ici y
is concen a ed on he ee bounda y z(α, ), and is gi en by a Di ac dis ibu ion as ollows:
∇⊥·u(x, ) = ω(α, )δ(x−z(α, )),
wi h ω(α, ) ep esen ing he o ici y s eng h i.e. ∇⊥·uis a measu e de ined by
<∇⊥·u, η >=Zω(α, )η(z(α, ))dα,
wi h η(x) a es unc ion.
Then z(α, ) e ol es wi h an incomp essible eloci y ield coming om he Bio -Sa a
law:
u(x, ) = ∇⊥∆−1∇⊥·u(x, ).
As (x, ) app oaches a poin z(α, ) on he con ou he eloci y uag ees, modulo angen ial
e ms, wi h he Bi kho -Ro in eg al:
BR(z, ω)(α, ) = 1
2πPV Z(z(α, )−z(β, ))⊥
|z(α, )−z(β, )|2ω(β, )dβ.
This yields an app op ia e con ou dynamics sys em:
z (α, ) = BR(z, ω)(α, ) + c(α, )∂αz(α, ),(4)
whe e he e m c ep esen s he change o pa ame iza ion and does no modi y he geome ic
e olu ion o he cu e [24].
The well-posedness is no gua an eed in gene al, in ac such a esul u ns ou o be alse
o some ini ial da a. Rayleigh [30] and Sa man-Taylo [31] ga e a condi ion ha mus be
sa is ied o he linea ized model in o de o ha e a solu ion locally in ime, namely ha he
no mal componen o he p essu e g adien jump a he in e ace has o ha e a dis inguished
sign. This is known as he Rayleigh-Taylo condi ion:
σ(α, ) = −(∇p2(z(α, ), )− ∇p1(z(α, ), )) ·∂⊥
αz(α, )>0,
2
whe e ∇pj(z(α, ), ) deno es he limi g adien o he p essu e ob ained app oaching he
bounda y in he no mal di ec ion inside Ωj( ). We call σ(α, ) he Rayleigh-Taylo o he
solu ion z(α, ).
Unde s anding he p oblem as weak solu ions o (1-2) plus he incomp essibili y o he
eloci y, we ind ha he con inui y o he p essu e (p2(z(α, ), ) = p1(z(α, ), )) ollows as
a ma hema ical consequence, making unnecessa y o impose i as a physical assump ion ( o
mo e de ails see [13] and [11]). Fo he su ace ension case, he e is a jump discon inui y o
he p essu e ac oss he in e ace which is modeled o be equal o he local cu a u e imes
he su ace ension coe icien :
p2(z(α, ), )−p1(z(α, ), )) = τκ(α, ).
This is known as he Laplace-Young condi ion, which makes he ini ial alue p oblem mo e
egula . Then he e a e no ins abili ies [18] bu inge ing phenomena a ise [29, 19].
By means o Da cy’s law, we can ind he ollowing o mula o he di e ence o he
g adien s o he p essu e in he no mal di ec ion and he s eng h o he o ici y:
σ(α, ) = (ρ2−ρ1)∂αz1(α, )
ω(α, ) = −(ρ2−ρ1)∂αz2(α, ).(5)
Abo e g is aken equal o 1 o he sake o simplici y.
Then, i we choose an app op ia e e m c in equa ion (4) (see sec ion 2 below), he
dynamics o he in e ace sa is ies
z (α, ) = ρ2−ρ1
2πPV Z(z1(α, )−z1(β, ))
|z(α, )−z(β, )|2(∂αz(α, )−∂αz(β, ))dβ. (6)
A wise choice o pa ame iza ion o he cu e is o ha e ∂αz1(α, ) = 1 ( o mo e de ails
see [13]). This yields he dense luid below he less dense luid i ρ2> ρ1and he e o e he
Rayleigh-Taylo condi ion holds as long as he in e ace is a g aph. This ac has been used
in [13] o show local exis ence in he s able case (ρ2> ρ1), oge he wi h ill-posedness in he
uns able si ua ion (ρ2< ρ1). Local exis ence o he gene al case (µ16=µ2) is shown in [11],
which was also ea ed in [34, 1].
¿F om (6) i is easy o ind he e olu ion equa ion o he g aph:
(α, ) = ρ2−ρ1
2πPV ZR
(α−β)
(α−β)2+ ( (α, )− (β, ))2(∂α (α, )−∂α (β, ))dβ,
(α, 0) = 0(α).
(7)
The abo e equa ion can be linea ized a ound he la solu ion o ind he ollowing nonlocal
pa ial di e en ial equa ion
(x, ) = −ρ2−ρ1
2Λ (x, ),
(x, 0) = 0(x), x ∈R,
3
whe e he ope a o Λ is he squa e oo o he Laplacian. This linea iza ion shows he
pa abolic cha ac e o he sys em.
Fu he mo e he s able sys em gi es a maximum p inciple k kL∞( )≤ k kL∞(0) [14];
decay a es a e ob ained o he pe iodic case:
k kL∞( )≤ k 0kL∞e−C ,
and also o he case on he eal line ( la a in ini y):
k kL∞( )≤k 0kL∞
1 + C .
The e a e se e al esul s on global exis ence o small ini ial da a (small compa ed o 1
in se e al no ms mo e egula han Lipschi z [9, 35, 32, 13, 19]) aking ad an age o he
pa abolic cha ac e o he equa ion o small ini ial da a. In [8] i is shown in he s able case
ha global exis ence o solu ions holds i he i s de i a i e o he ini ial da a is smalle
han an explici ly compu able cons an g ea e han 1/5. Fu he mo e, i k 0kL∞<∞and
k∂x 0kL∞<1, hen he e exis s a global-in- ime solu ion ha sa is ies
(x, )∈C([0, T]×R)∩L∞([0, T]; W1,∞(R)),
o each T > 0. In pa icula is Lipschi z con inuous.
Mo eo e , equa ion (7) yields an L2decay:
k k2
L2( ) + ρ2−ρ1
2πZ
0
ds ZR
dα ZR
dx ln 1 +  (x, s)− (α, s)
x−α2=k 0k2
L2,
which does no imply, o la ge ini ial da a, a gain o de i a i es in he sys em (see [8]). We
will see below ha he solu ions o he Muska p oblem wi h ini ial da a in H4become eal
analy ic immedia ely despi e he weakness o he abo e decay o mula.
The main esul we p esen he e is:
Theo em 1.1 The e exis s a nonemp y open se o ini ial da a in H4wi h Rayleigh-Taylo
s ic ly posi i e σ > 0such ha in ini e ime he Rayleigh-Taylo σ(α, )o he solu ion o
(6) is s ic ly nega i e o all αin a nonemp y open in e al.
The geome y o his amily o ini ial da a is a om i ial: nume ical simula ions pe o med
in [16] show ha he e exis ini ial da a wi h la ge s eepness o which a egula izing e ec
appea s. In ac , as will be explained in Sec ion 2, he i s e idence o a change o sign in
he Rayleigh-Taylo has been expe imen ally ound in a model wi h wo in e aces.
We p oceed as ollows:
Fi s , in sec ion 3, we assume ini ial condi ions a ime = 0 ha sa is y he Rayleigh-
Taylo (σ > 0) and he a c-cho d condi ion, and o which he bounda y zini ially belongs o
H4. Le C1be he cons an in he a c-cho d condi ion, le C2be an uppe bound o he H4
no m o he ini ial da a and le c3be a lowe bound o σ. Then he e exis s 1> 0, wi h 1
depending only on C1, C2, c3, such ha he Muska p oblem has a solu ion o ime ∈[ 0, 1],
sa is ying also he a c-cho d and Rayleigh-Taylo condi ions. Mo eo e , o 0< ≤ 1, he
4
solu ion z(α, ) is eal analy ic in a s ip S( ) = {α+iζ :|ζ| ≤ c( − 0)}, whe e cdepends
only on C1, C2, c3.
Ou goal in sec ion 4 is o show ha he egion o analy ici y does no collapse o he
eal axis as long as he Rayleigh-Taylo is g ea e han o equal o 0. This allows us o each
a egime o which he bounda y zde elops a e ical angen .
Sec ion 5 is de o ed o showing he exis ence o a la ge class o analy ic cu es o which
he e exis s a poin whe e he angen ec o is e ical and he eloci ies indica e ha he
cu es a e going o u n o e and each he uns able egime o a small ime. Plugging hese
ini ial da a in o a Cauchy-Kowalewski heo em indica es ha he analy ic cu es u n o e .
The e o e he uns able egime is eached.
Finally, in sec ion 6, a pe u ba i e a gumen allows us o conclude ha we can ind
cu es in H4close enough o he special class o analy ic cu es desc ibed in Sec ion 5, which
sa is y he a c-cho d and Rayleigh-Taylo condi ions. Then we can show he exis ence o he
cu es passing he c i ical ime and ac ually u ning o e . The e o e he uns able egime is
eached o an en i e H4−neighbo hood o ini ial da a.
Rema k 1.2 In a o hcoming pape (see [5]) we will exhibi a pa icula ini ial da um o
which we will show ha once he cu e eaches he uns able egime he s ip o analy ici y
collapses in ini e ime and he solu ion b eaks down. In sec ion 8 we p o ide a e y b ie
ske ch o ou p oo o b eakdown o smoo hness o he Muska equa ion. These esul s we e
announced in [6].
Rema k 1.3 The same app oach can be done o he wa e wa es p oblem, which shows
ha , s a ing wi h some ini ial da a gi en by (α, 0(α)), in ini e ime he in e ace eaches a
egime in which i is no longe a g aph. The e o e he e exis s a ime ∗whe e he solu ion o
he ee bounda y p oblem pa ame ized by (α, (α, )) sa is ies k αkL∞( ∗) = ∞(see sec ion
7). This scena io is known in he li e a u e as wa e b eaking [7] and he e a e nume ical
simula ions showing his phenomenon [4].
Rema k 1.4 We conjec u e ha a esul analogous o Theo em 1.1 holds, in which su ace
ension is included. We may simply use he same ini ial da a as in Theo em 1.1, and ake
he coe icien o su ace ension o be e y small. The solu ions a e p esumably changed only
sligh ly by he su ace ension (al hough we do no ha e a p oo o his plausible asse ion).
Consequen ly, we belie e ha Muska solu ions wi h small su ace ension can u n o e .
A simila ema k applies o wa e wa es (see heo em 7.1). The e exis ini ial da a o
which wa e wa es wi h su ace ension u n o e . A igo ous p oo may be easily supplied,
since local exis ence (backwa ds and o wa d in ime) is known o wa e wa es wi h su ace
ension (see [3]).
2 The con ou equa ion and nume ical simula ions
He e we p esen he e olu ion equa ion in e ms o he ee bounda y which is going o be
used h oughou he pape , and he nume ical expe imen ha mo i a ed he Theo em.
5

2.1 The equa ion o mo ion
By Da cy’s law:
∇⊥·u=−(ρ2−ρ1)∂αz2(α)δ(x−z(α)),
and Bio -Sa a yields
z (α) = −(ρ2−ρ1)
2πPV ZR
(z(α)−z(α−β))⊥
|z(α)−z(α−β)|2∂αz2(α−β)dβ. (8)
Fo he i s coo dina e abo e one inds
(ρ2−ρ1)
2πPV ZR
(z2(α)−z2(α−β))
|z(α)−z(α−β)|2∂αz2(α−β)dβ
=−(ρ2−ρ1)
2πPV ZR
(z1(α)−z1(α−β))
|z(α)−z(α−β)|2∂αz1(α−β)dβ
using he iden i y
PV ZR
∂βln(|z(α)−z(α−β)|2)dβ = 0.
The e o e
z (α) = −(ρ2−ρ1)
2πPV ZR
(z1(α)−z1(α−β))
|z(α)−z(α−β)|2∂αz(α−β)dβ.
He e we poin ou ha in he Bio -Sa a law he pe pendicula di ec ion appea s, bu a e
he abo e in eg a ion by pa s, we only see he angen ial di ec ion.
Adding he angen ial e m
(ρ2−ρ1)
2πPV ZR
(z1(α)−z1(α−β))
|z(α)−z(α−β)|2dβ∂αz(α),
we ind ha he con ou equa ion is gi en by
z (α) = (ρ2−ρ1)
2πPV ZR
z1(α)−z1(α−β)
|z(α)−z(α−β)|2(∂αz(α)−∂αz(α−β))dβ.
Fo he 2πpe iodic in e ace he equa ion becomes
z (α) = (ρ2−ρ1)
4πZπ
−π
sin(z1(α)−z1(α−β))(∂αz(α)−∂αz(α−β))
cosh(z2(α)−z2(α−β)) −cos(z1(α)−z1(α−β))dβ. (9)
In o de o see (9) we ake z(α) = z1(α) + iz2(α); i is easy o ew i e (8) as ollows;
z (α) = −(ρ2−ρ1)
2πi P V ZR
∂αz2(β)
z(α)−z(β)dβ.
The classical iden i y
1
z+X
k≥1
z
z2−(2πk)2=1
2 an(z/2)
6
allows us o conclude ha
z (α) = (ρ2−ρ1)
4πZT
(sinh(z2(α)−z2(β)),−sin(z1(α)−z1(β)))
cosh(z2(α)−z2(β)) −cos(z1(α)−z1(β)) ∂αz2(β)dβ,
whe e T=R/2πZ.
Analogously, using he equali y
(ρ2−ρ1)
4πPV ZR
sinh(z2(α)−z2(β))
cosh(z2(α)−z2(β)) −cos(z1(α)−z1(β))∂αz2(β)dβ
=−(ρ2−ρ1)
4πPV ZR
sin(z1(α)−z1(β))
cosh(z2(α)−z2(β)) −cos(z1(α)−z1(β))∂αz1(β)dβ
and adding he app op ia e angen ial e m, we ob ain equa ion (9).
2.2 The scena io mo i a ed by he nume ics
Ou in es iga ions s a ed wi h he idea ha in e es ing new phenomena may a ise i we
s udy h ee luids, sepa a ed by wo in e aces. Ca e ul nume ical s udies indica ed ha one
o he in e aces may u n o e . In a emp ing o p o e analy ically he u no e indica ed
by he nume ics, we disco e ed ha a u no e can occu also o a single in e ace, i.e., o
he Muska p oblem. This sec ion desc ibes one o ou nume ical expe imen s.
P oceeding as in he p eceding sec ion, one can de i e he equa ions modeling he e olu-
ion o wo in e aces sepa a ing h ee luids wi h di e en densi ies ρj(j= 1,2,3). Mo e p e-
cisely, assume ha bo h in e aces can be pa ame ized by g aphs (α, (α, )) and (α, g(α, )),
wi h lying abo e g. These equa ions ead in he pe iodic case, c . [16, 15] ( his scena io
has been ecen ly also conside ed in [20]),
(α, ) = ¯ρ1I[ (·, ), (·, )] + ¯ρ2I[ (·, ), g(·, )], (α, 0) = 0(α),
g (α, ) = ¯ρ2I[g(·, ), g(·, )] + ¯ρ1I[g(·, ), (·, )], g(α, 0) = g0(α),(10)
whe e ¯ρj= (ρj+1 −ρj)/(4π), j= 1,2, and, o gi en unc ions u(α), (α),
I[u, ] := PV ZT
(∂αu(α)−∂α (α−β)) an(β/2)(1 − anh2((u(α)− (α−β))/2))
an2(β/2) + anh2((u(α)− (α−β))/2) dβ.
(11)
The i s e ms I[ (·, ), (·, )] and I[g(·, ), g(·, )] in (10) gi e he eloci y o a unique in e -
ace. The c oss e ms I[ (·, ), g(·, )] and I[g(·, ), (·, )] ake in o accoun he in e ac ion o
he wo in e aces, and hei con ibu ion is ge ing bigge when he cu es a e ge ing close .
This, oge he wi h he di usi e beha io epo ed in [16] o he equa ion
(α, ) = ¯ρ1I[ (·, ), (·, )], (α, 0) = 0(α),(12)
and he mean conse a ion o and g, mo i a e he choice o he ollowing ini ial da a, in
he hope ha some non egula izing e ec a ises om he in e ac ion o he wo in e aces;
0(α) = 




0.1−sin3π(α−M1+ 1)
2 1,i α∈[M1− 1, M1+ 1],
0.1,o he wise
(13)
7
0 2 4 6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0 2 4 6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Figu e 1: Le : Solu ions o (10) wi h ini ial da a (13)-(14) a imes = 0 (dashed blue),
= 3.46 ·10−4( ed poin s) and = 7.66 ·10−4(black). Righ : Solu ions a = 1.04 ·10−3
(dashed ed) and = 1.84 ·10−3(black)
and
g0(α) = 




sin3π(α−M2+ 2)
2 23
−0.92,i α∈[M2− 2, M2+ 2],
g0(α) = −0.92,o he wise.
(14)
The choice o pa ame e s M1=π+ 0.1, 1= 0.7, M2=π/1.2, 2= 0.3, ¯ρ1= 20πand
¯ρ2=π/20, yielded a s ong g ow h o he de i a i e in he he lowe in e ace as he wo
cu es app oach, as shown in Figu e 1. Mo eo e , a e in oducing a small modi ica ion
in he lowe in e ace so ha he angen a a ce ain poin becomes ac ually in ini e, and
e alua ing he no mal eloci y ela i e o his poin along he modi ied cu e, we ob ain he
esul plo ed in Figu e 2. This g aphic clea ly indica es ha he eloci y ield is o cing he
in e ace o u n o e .
The nume ical app oxima ion o (10) add esses as a main di icul y he absolu e lack o
knowledge abou he beha io o he solu ions o (10). Indeed, he goal o ou expe imen s is
p ecisely he sea ch o some singula beha io . The nonlocal e ms make he compu a ions
expensi e and special ca e has o be aken in o de o e alua e he in eg ands in a neigh-
bo hood o β= 0. Fo his, we used Taylo expansions locally and compu ed exac ly he
p incipal alue. In his si ua ion, adap i i y is s ongly indica ed, bo h in space and ime,
since a good indica o o a singula beha io will be gi en ei he by a sudden accumula ion
o spa ial nodes o a sudden educ ion o he ime s eps.
In o de o a ain he highes esolu ion in he in eg a ion o (10) and compu e he
solu ions shown in Figu e 1, cubic spline in e pola ion o he cu es (·, ) and g(·, ) wi h
pe iodic bounda y condi ions was used. This p o ides a C2in e polan o each in e ace a
e e y ime and allows, in pa icula , he e alua ion o he con olu ion e ms a any β∈[0,2π].
Then, adap i e quad a u e can be applied o app oxima e he in eg als and e alua e he
de i a i e a any ime. In he expe imen s epo ed, adap i e Loba o quad a u e was used,
8
2.8 2.9 3 3.1
−1
−0.8
−0.6
−0.4
−0.2
0
Figu e 2: Zoom o he in e ace, modi ied so ha i s angen is e ical a a single poin P;
and he no mal eloci y along he cu e, minus ha a P, scaled by a ac o o 100.
by means o he MATLAB ou ine quadl. Fo he ime in eg a ion, he embedded Runge–
Ku a o mula due o Do mand and P ince, DOPRI5(4), was implemen ed, since he p oblem
was no ound o be pa icula ly s i , see o ins ance [21]. The ime s epping was combined
wi h a spa ial node edis ibu ion a e e e y success ul s ep. Fo he edis ibu ion o he
spa ial nodes an algo i hm ollowing [17] was implemen ed, wi h some modi ica ions aking
in o accoun ha bo h in e aces a e g aphs. Fo se e al ole ance equi emen s and di e en
choices o he pa ame e s in ol ed in he ull adap i e ou ine, he in eg a ion always ailed
a a ce ain c i ical ime, sugges ing he explosion o he de i a i e a a ce ain poin o he
lowe in e ace and he lack o alidi y o (10), once his cu e s ops being a g aph.
The phenomenon desc ibed abo e and he explici ep esen a ions o he maximum o he
solu ions de i ed in [14], mo i a ed he sea ch o special ini ial da a which allowed us o
unde s and ha his beha io also a ises in he one-in e ace case.
3 Ins an Analy ici y
He e we show he main es ima es ha p o ide local-exis ence and ins an analy ici y o a
single cu e ha sa is ies ini ially he a c-cho d and Rayleigh-Taylo condi ions. We conside
he unc ion
F(z)(α, β) = β2
|z(α)−z(α−β)|2, α, β ∈R,
and in he pe iodic se ing
F(z)(α, β) = ||β||2
2(cosh(z2(α)−z2(α−β)) −cos(z1(α)−z1(α−β))), α, β ∈T,
whe e ||x|| =dis (x, 2πZ).
I F(z)∈L∞ hen we say ha he cu e sa is ies he a c-cho d condi ion, and he L∞
no m o Fis called he a c-cho d cons an .
9
using (23), and he e o e (see sec ion 9 in [11] o mo e de ails)
d
d kzkRT ( )≤C(kzkRT ( ) + 1)k.
I ollows ha
kzkRT ( )≤kzkRT (0) + 1
(1 −C(kzkRT (0) + 1)k )1/k −1,
p o iding he a p io i es ima e wi h Cand kuni e sal cons an s.
We app oxima e he p oblem as ollows
zε
(α, ) = φε∗Zsin(φε∗zε
1(α)−φε∗zε
1(β))(∂α(φε∗zε)(α)−∂β(φε∗zε)(β))
cosh(zε
2(α)−zε
2(β)) −cos(zε
1(α)−zε
1(β)) dβ
zε(α, 0) = φε∗z0(α),
whe e φε(x) = φ(α/ε)/ε,φis he hea ke nel and ε > 0. Pica d’s heo em yields he exis ence
o a solu ion zε(α, ) in C[0, Tε); H4which is analy ic in he whole space o z0sa is ying
he a c-cho d condi ion and εsmall enough. Using he same echniques we ha e de eloped
abo e we ob ain a bound o zε(α, ) in H4in he s ip S( ) o a small enough Twhich is
independen o ε. We need a c-cho d, R-T, z0∈H4and c−m(0) <0. Then we can pass o
he limi .
4 Ge ing all he way o b eakdown o Rayleigh-Taylo
This sec ion is de o ed o p o ing he ollowing heo em.
Theo em 4.1 Le z(α, 0) = z0(α)be an analy ic cu e in he s ip
S={α+iζ ∈C:|ζ|< h(0)},
wi h h(0) >0and sa is ying:
•The a c-cho d condi ion, F(z0)(α+iζ, β)∈L∞(S×R)
•The Rayleigh-Taylo condi ion, ∂αz0
1(α)>0.
•The cu e z0(α)is eal o eal α.
•The unc ions z0
1(α)−αand z0
2(α)a e pe iodic wi h pe iod 2π.
•The unc ions z0
1(α)−αand z0
2(α)belong o H4(∂S).
Then he e exis a ime Tand a solu ion o he Muska p oblem z(α, )de ined o 0< ≤T
ha con inues analy ically in o some complex s ip o each ixed ∈[0, T ]. He e Tis ei he
a small cons an depending only on ||z0||So i is he i s ime a e ical angen appea s,
whiche e occu s i s .
Thus ou Muska solu ion is analy ic as long as ∂αz1(α, )≥0.
We will use he ollowing:
16

Lemma 4.2 Le ϕ(α±iζ) = PN
k=−NAkeikα∓kζ. Then, o ζ > 0, we ha e
∂
∂ζ X
±ZT
|ϕ(α±iζ)|2dα ≥1
10 X
±ZT
Λϕ(α±iζ)ϕ(α±iζ)dα −10 ZT
Λϕ(α)ϕ(α)dα, (27)
whe e Λϕ(α±iζ) = PN
k=−N|k|Akeikαe∓kζ.
P oo : Fi s we shall compu e he le hand side in he equency space:
X
±ZT
|ϕ(α±iζ)|2dα = 4π
N
X
k=−N
|Ak|2cosh(2|k|ζ).
On he o he hand we ha e ha
X
±ZT
Λϕ(α±iζ)ϕ(α±iζ)dα = 4π
N
X
k=−N
|k||Ak|2cosh(2|k|ζ),
while
ZT
Λϕ(α)ϕ(α)dα = 2π
N
X
k=−N
|k||Ak|2.
Di e en ia ing in ζwe ob ain
∂
∂ζ ZT
|ϕ(α±iζ)|2dα = 8π
N
X
k=−N
|k||Ak|2sinh(2|k|ζ).
The lemma holds since sinh(ζ)≥cosh(ζ)−1 o any ζ > 0.
Co olla y 4.3 Le ϕ(α±iζ, ) = PN
k=−NAk( )eikαe∓kζ and h( )>0be a dec easing unc ion
o . Then
∂
∂ X
±ZT
|ϕ(α±ih( ))|2dα ≤h′( )
10 X
±ZT
Λϕ(α±ih( ))ϕ(α±ih( ))dα
−10h′( )ZT
Λϕ(α)ϕ(α)dα + 2ℜX
±ZT
ϕ (α±ih( ))ϕ(α±ih( ))dα.
This co olla y allows us o p o e Theo em 4.1.
P oo (Theo em 4.1): The no ms kzkHk(S)and kzkSa e de ined as be o e using he new
s ip S( ) de ined by
S( ) = {α+iζ ∈C:|ζ|< h( )},
whe e h( ) is a posi i e dec easing unc ion o .
17
We use he Gale kin app oxima ion o equa ion (15), i.e.
∂ z[N](ζ, ) = ΠN[J[z[N]]](ζ, ),
whe e ζ∈S( ), ΠNwill be speci ied below, and
J[z](α, ) = Zπ
−π
sin(z1(α)−z1(β))(∂αz(α)−∂αz(β))
cosh(z2(α)−z2(β)) −cos(z1(α)−z1(β))dβ.
We impose he ini ial condi ion
z[N](α, 0) = z[N](α).
He e, o a la ge enough posi i e in ege N, we de ine z[N](α, 0) om z0(α) by using he
p ojec ion
ΠN:
∞
X
−∞
Akeikα 7→
N
X
−N
Akeikα.
We de ine z[N](α) by s ipula ing ha
z[N]
1(α)−α= ΠN[z0
1(α)−α]
and
z[N]
2(α) = ΠN[z0
2(α)].
Fo Nla ge enough, he unc ions z[N](α, 0) sa is y he a c-cho d and Rayleigh-Taylo con-
di ion.
We shall conside he e olu ion o he mos singula quan i y
X
±ZT
|∂4
αz[N](α±ihN( ), )|2dα,
whe e hN( ) is a smoo h posi i e dec easing unc ion on , wi h hN(0) = h(0), which will be
gi en below. Also we deno e
SN( ) = {α+iζ ∈C:|ζ|< hN( )}.
¿F om now on, we will d op he dependency on N om z[N]and hN( ) in ou no a ion.
We will e u n o he p e ious no a ion in he discussion below a he end o he sec ion.
Taking he de i a i e wi h espec o yields
d
d Z
α∈T∂4
αzµ(α±ih( ), )
2dα
= 2ℜZ
α∈T
∂4
αzµ(α±ih( ), )∂ ∂4
αzµ(α±ih( ), ) + ih′( )∂5
αzµ(α±ih( ), )dα
= 2ℜZ
α∈T
∂4
αzµ(α±ih( ), )∂4
αΠN[Jµ[z]](α±ih( ), ) + ih′( )∂5
αzµ(α±ih( ), )dα
18
= 2ℜZ
α∈T
∂4
αzµ(α±ih( ), )ΠN[∂4
αJµ[z]](α±ih( ), ) + ih′( )∂5
αzµ(α±ih( ), )dα
= 2ℜZ
α∈T
∂4
αzµ(α±ih( ), )∂4
αJµ[z](α±ih( ), ) + ih′( )∂5
αzµ(α±ih( ), )dα,
since ∂4
αzµ(α±ih( ), ) is a igonome ic polynomial in he ange o ΠN. He e µ= 1, 2.
Using he abo e co olla y we ha e ha
d
d X
±Z
α∈T∂4
αzµ(α±ih( ), )
2dα ≤h′( )
10 X
±ZT
Λ(∂4
αzµ)(α±ih( )) ·∂4
αzµ(α±ih( ))dα
−10h′( )ZT
Λ(∂4
αzµ)(α)·∂4
αzµ(α)dα + 2 X
±
ℜZT
∂4
αJµ[z](α, )(α±ih( )) ·∂4
αzµ(α±ih( ))dα.
We shall s udy in de ail he mos singula e m in ∂4J[z](α, ), i.e.
∂4
αJ[z](α±ih( ), ) = Zπ
−π
sin(z1(α±ih( ), )−z1(β, ))(∂5
αz(α±ih( ), )−∂5
βz(β, ))
cosh(z2(α±ih( ), )−z2(β, )) −cos(z1(α±ih( ), )−z1(β, ))dβ
+ l.o. ≡X+ l.o. .,
whe e ||l.o. ||L2(T)≤C(||z||S( ) + 1)k(see [11] and ou p e ious discussion o (18)). We spli
Xin o he ollowing e ms
X=Zπ
−π
K(α±ih( ), β)(∂5
αz(α±ih( ), )−∂5
βz(β, ))dβ
+σ(α±ih( ), )Zπ
−π
co α±ih( )−β
2(∂5
αz(α±ih( ), )−∂5
βz(β, ))dβ
≡X1+X2,
whe e
K(α, β) = sin(z1(α, )−z1(β, ))
cosh(z2(α, )−z2(β, )) −cos(z1(α, )−z1(β, ))
−∂αz1(α, )
(∂αz2(α, ))2+ (∂αz1(α, ))2co α−β
2
and
˜σ(α, ) = ∂αz1(α, )
(∂αz1(α, ))2+ (∂αz2(α, ))2.
Le us deno e
Γ±( ) = {ζ∈C:ζ=α±ih( ), α ∈T}.
19
Since K(α, β) is a holomo phic unc ion in αand β, wi h α, β ∈S( ), o ixed we ha e ha
X1=Zπ
π
K(α±ih( ), β)∂5
αz(α±ih( ), )dβ
−Zπ
π
K(α±ih( ), β)∂5
αz(β, )dβ
≡X11 +X12,
and in eg a ion by pa s shows ha he e m X12 sa is ies ||X12||L2(T)≤C(||z||S+ 1)k. In
addi ion, we can w i e X11 as ollows
X11 =Zw∈Γ±( )
K(α±ih( ), w)∂5z(α±ih( ), )dw
=P.V. Zw∈Γ±( )
sin(z1(α±ih( ), )−z1(w, ))∂5z(α±ih( ), )
cosh(z2(α±ih( ), )−z2(w, )) −cos(z1(α±ih( ), )−z1(w, ))dw
−∂5z(α±ih( ), )σ(α±ih( ), )P.V. Zw∈Γ±( )
co α±ih( )−w
2dw
=P.V. Zπ
−π
sin(z1(α±ih( ), )−z1(β±ih( ), ))∂5z(α±ih( ), )
cosh(z2(α±ih( ), )−z2(β±ih( ), )) −cos(z1(α±ih( ), )−z1(β±ih( ), ))dβ,
As be o e we call
(α±ih( ), )
=P.V. Zπ
−π
sin(z1(α±ih( ), )−z1(β±ih( ), ))
cosh(z2(α±ih( ), )−z2(β±ih( ), )) −cos(z1(α±ih( ), )−z1(β±ih( ), ))dβ
=P.V. Zπ
−π
sin(z1(α±ih( ), )−z1(α±ih( )−β, ))
cosh(z2(α±ih( ), )−z2(α±ih( )−β, )) −cos(z1(α±ih( ), )−z1(α±ih( )−β, ))dβ.
Thus
X11 =∂5z(α±ih( ), ) (α±ih( ), ).
Also we can w i e X2in he ollowing way;
X2=˜σ(α±ih( ), )Zπ
−π
co α±ih( )−β
2(∂5
αz(α±ih( ), )−∂5
βz(β, ))dβ
=˜σ(α±ih( ), )Zw∈Γ±( )
co α±ih( )−w
2(∂5
αz(α±ih( ), )−∂5
βz(w, ))dw
=˜σ(α±ih( ), )P.V. Zw∈Γ±( )
co α±ih( )−w
2∂5
αz(α±ih( ), )dw
−˜σ(α±ih( ), )P.V. Zw∈Γ±( )
co α±ih( )−w
2∂5
αz(w, )dw
20
=−˜σ(α±ih( ), )P.V. Zπ
−π
co α−β
2∂5
αz(β±ih( ), )dβ
=−˜σ(α±ih( ), )P.V. Zπ
−π
1
2csc2α−β
2(∂4
αz(α±ih( ), )−∂4
αz(β±ih( ), ))dβ
and inally
X2=−2π˜σ(α±ih( ), )(Λ∂4
αz)(α±ih( ), ).
Then we ind wo dange ous e ms
I1= 2ℜZT
(α±ih( ), )(∂4
αzµ)(α±ih( )) ·(∂5
αzµ)(α±ih( ))dα
and
I2=−4πℜZT
˜σ(α±ih( ), )Λ(∂4
αzµ)(α±ih( )) ·∂4
αzµ(α±ih( ))dα.
The es can be bounded by C(kzkS+ 1)k( ) as in he p e ious sec ion. In o de o bound
I1and I2we use he ollowing commu a o es ima e:
||Λ1
2( g)− Λ1
2g||L2(T)≤C||Λ1+ε ||L2(T)||g||L2(T),(28)
o (α) = PN
−N keikx and g(α) = PN
−Ngkeikx, whe e ε > 0 and Cdoes no depend on N.
The p oo o (28) will be le o he eade .
Fi s we es ima e I1. We deno e γ=α+ih( ).
I1=2ℜZπ
−π
(γ, )∂4
αzµ(γ, )∂5
αzµ(γ, )dα
=2 Zπ
−π
ℜ( (γ)) ℜ(∂4
αzµ(γ, ))∂α(ℜ(∂4
αzµ(γ, ))) + ℑ(∂4
αzµ(γ, ))∂α(ℑ(∂4
αzµ(γ, )))dα
−2Zπ
−π
ℑ( (γ)) ℜ(∂4
αzµ(γ, ))∂α(ℑ(∂4
αzµ(γ, ))) + ℑ(∂4
αzµ(γ, ))∂α(ℜ(∂4
αzµ(γ, )))dα
≡I11 +I12.
In eg a ing by pa s we ha e ha ||I11||L2(T)≤C(||z||S+ 1)k. In o de o es ima e I12 we
no e ha (γ, ) is eal o eal γ. Then
ℑ( (α±ih( ), )) = h( )˜
±(α, ),
whe e
|| ˜
±||H2(T)≤C(||z||S( ) + 1)k.
21

Then we can w i e
Zπ
−π
ℑ( (γ))ℜ(∂4
αzµ(γ, ))∂α(ℑ(∂4
αzµ(γ, )))dα
=h( )Zπ
−π
˜
±(α, )ℜ(∂4
αzµ(γ, ))∂α(ℑ(∂4
αzµ(γ, )))dα
=−h( )Zπ
−π
˜
±(α, )ℜ(∂4
αzµ(γ, ))ΛH(ℑ(∂4
αzµ(γ, )))dα
=−h( )Zπ
−π
Λ1
2(˜
±(α, )ℜ(∂4
αzµ)(γ, ))Λ1
2H(ℑ(∂4
αzµ(γ, )))dα
=−h( )Zπ
−πnΛ1
2(˜
±(α, )ℜ(∂4
αzµ)(γ, )) −˜
±(α)Λ1
2ℜ(∂4
αzµ)oΛ1
2H(ℑ(∂4
αzµ(γ, )))dα
−h( )Zπ
−π
˜
±(α)Λ1
2ℜ(∂4
αzµ)Λ1
2H(ℑ(∂4
αzµ(γ, )))dα
≤h( )||Λ1
2(˜
±(·, ))ℜ(∂4
αzµ(· ± ih( ), )) −˜
±(·, )Λ1
2ℜ(∂4
αzµ(· ± ih( )))||L2(T)
× ||Λ1
2H(ℑ(∂4
αzµ(· ± ih( ), )))||L2(T)
+h( )|| ˜
±||L∞(T)||Λ1
2ℜ(∂4
αzµ(· ± ih( )))||L2(T)||Λ1
2Hℑ(∂4
αzµ(· ± ih( )))||L2(T).
Using he es ima e (28) yields
Zπ
−π
ℑ( (γ))ℜ(∂4
αzµ(γ, ))∂α(ℑ(∂4
αzµ(γ, )))dα
≤h( )||Λ1+ε˜
±||L2(T)||ℜ(∂4
αzµ(· ± ih( ), ))||L2(T)||Λ1
2(ℑ(∂4
αzµ(· ± ih( ), )))||L2(T)
+h( )|| ˜
±||L∞(T)||Λ1
2ℜ(∂4
αzµ(· ± ih( )))||L2(T)||Λ1
2ℑ(∂4
αzµ(· ± ih( ))||L2(T)
≤Ch( )(||z||S+ 1)k+Ch( )(||z||S+ 1)k||Λ1
2∂4
αzµ(· ± ih( ), )||2
L2(T)
=Ch( )(||z||S+ 1)k+Ch( )(||z||S+ 1)kZπ
−π
∂4
αzµ(γ, )Λ∂4
αzµ(γ, )dα.
Now I1is equal o he in eg al o he le , plus a simila in eg al ha can be bounded in a
simila way.
Thus we ob ain ha
X
±
I1≤C(kzkS+ 1)k+Ch( )(kzkS+ 1)kkΛ1/2∂4
αzk2
L2(S).(29)
By assump ion he R-T ˜σis bigge han ze o o eal alues. In o de o a oid p oblems
wi h he imagina y pa we may w i e
∂αz1(α±ih( ), )
(∂αz1(α±ih( )))2+ (∂αz2(α±ih( )))2=∂αz1(α, )
|∂αz(α, )|2+h( )g±(α, ).
whe e
||g±||H2(T)≤C(||z||S+ 1)k.
22
One inds,
I2=−2ℜZT
∂αz1(α)
|∂αz(α)|2Λ(∂4
αzµ)(α±ih( )) ·∂4
αzµ(α±ih( ))dα
−h( )2ℜZT
g±(α, )Λ(∂4
αzµ)(α±ih( )) ·∂4
αzµ(α±ih( ))dα.
The i s e m abo e can be ea ed as in sec ion 3 aking ad an age o he inequali y
(26). He e we jus need ∂αz1(α)≥0. The second e m can be ea ed using he inequali y
(28) as wi h he e m I1. We ind ha
X
±
I2≤C(||z||S+ 1)k+Ch( )kg±kH2(S)kΛ1/2∂4
αzk2
L2(S),
and he e o e
X
±
I2≤C(||z||S+ 1)k+Ch( )(kzkS+ 1)kkΛ1/2∂4
αzk2
L2(S).(30)
Using (29) and (30) we ha e ha
d
d X
±ZT
|∂4
αzµ(α±ih( ))|2dα ≤C(kzkS( ) + 1)k−10h′( )ZT
Λ(∂4
αzµ)(α)·∂4
αzµ(α)dα
+(C(kzkS( ) + 1)kh( ) + 1
10h′( )) ZT
Λ(∂4
αzµ)(α±ih( )) ·∂4
αzµ(α±ih( ))dα.
Choosing
h( ) = h(0) exp(−10CZ
0
(kzkS+ 1)k( )d )
we elimina e he mos dange ous e m. The o he e m in he exp ession abo e in ol es wi h
a unc ion on he eal line and i is easily con olled. Indeed
ZT
Λ∂4
αzµ(α)·∂4
αzµ(α)≤C
h( )X
±ZT
|∂4
αzµ(α±ih( ))|2dα,
as one sees by examining he Fou ie expansion o ∂4
αzµ(α, ).
Thus

10h′( )ZT
Λ(∂4
αzµ)(α)·∂4
αzµ(α)dα
≤C|h′( )|
h( )||z||2
S≤C(||z||S+ 1)k+2.
And we ob ain inally
d
d X
±ZT
|∂4
αz(α±ih( ))|2dα ≤C(kzkS( ) + 1)k+2.
23
Reco e ing he dependency on Nin ou no a ion we ha e ha
d
d X
±ZT
|∂4
αz[N](α±ihN( ))|2dα ≤C(kz[N]kSN( ) + 1)k+2.(31)
As in he p e ious sec ion, we can ob ain a bound o he e olu ion o he a c-cho d
condi ion ha depends on C(kz[N]kSN( ) + 1)k+2.
This es ima e is ue whene e ∈[0, TN], whe e TNis he maximal ime o exis ence
o he solu ion z[N]. In addi ion inequali y (31) shows ha we can ex end hese solu ions in
H4(S) up o a small enough ime Tindependen o Nand depending on he ini ial da a.
The abo e calcula ion shows ha he s ip may sh ink bu does no collapse as long as
∂αz1(α, )≥0.
5 F om an analy ic cu e in he s able egime o an analy ic
cu e in he uns able egime
In his sec ion we show ha he e exis some ini ial da a which a e analy ic cu es sa is ying
he a c-cho d and R-T condi ions such ha he solu ion o he Muska p oblem eaches he
uns able egime. In o de o do i we will p o e he local exis ence o solu ions o analy ic
ini ial da a wi hou assuming he R-T condi ion. Then we will cons uc some sui able ini ial
da a o ou pu pose.
Theo em 5.1 Le z0be an analy ic cu e sa is ying he a c-cho d condi ion. Then he e
exis s an analy ic solu ion o he Muska p oblem in some in e al [−T, T] o a small enough
T > 0.
Rema k 5.2 No ice ha in heo em (5.1) he e is no assump ion on he R-T condi ion.
The p oo we use he e is analogous o he one in [33] based on Cauchy-Kowalewski heo ems
[27, 28] ( o an applica ion o he Eule equa ion see [2]). He e we canno pa ame ize he
cu e as a g aph, so we ha e o change he a gumen subs an ially in he p oo in o de o
deal wi h he a c-cho d condi ion.
P oo : We use he same no a ion as be o e. Le {X } >0be a scale o Banach spaces gi en
by R2− alued eal unc ions ha can be ex ended in o he complex s ip S ={α+iζ ∈
C:|ζ|< }such ha he no m
k k2
=X
±ZT
| (α±i )−(α±i , 0)|2dα +ZT
|∂4
α (α±i )|2dα,
is ini e and (α)−(α, 0) is 2π−pe iodic.
Le z0(α) be a cu e sa is ying he a c-cho d condi ion and z0(α)∈X 0 o some 0>0.
Then, we will show ha he e exis a ime T > 0 and 0 < < 0so ha he e is a unique
solu ion o (16) in C([0, T]; X ).
24
I is easy o check ha X ⊂X ′ o ′≤ due o he ac ha k k ′≤ k k . A simple
applica ion o he Cauchy o mula gi es
k∂α k ′≤C
− ′k k ,(32)
o ′< . Nex , we w i e equa ion (16) as ollows:
z (α+iζ, ) = G(z(α+iζ, )),
wi h
G(z(α+iζ, )) = Zπ
−π
sin(z1(α+iζ)−z1(α+iζ −β))(∂αz(α+iζ)−∂αz(α+iζ −β))
cosh(z2(α+iζ)−z2(α+iζ −β)) −cos(z1(α+iζ)−z1(α+iζ −β))dβ.
We ake 0 ≤ ′< and we in oduce he open se Oin S gi en by
O={z, ω ∈X :kzk < R, kF(z)kL∞(S )< R2},(33)
wi h F(z)(α+iζ, β, ) gi en by (17). Then he unc ion G o G:O→X ′is a con inuous
mapping. In addi ion, he e is a cons an CR(depending on Ronly) such ha
kG(z)k ′≤CR
− ′kzk ,(34)
kG(z2)−G(z1)k ′≤CR
− ′kz2−z1k ,(35)
and
sup
α+iζ∈S ,β∈T
|G(z)(α+iζ)−G(z)(α+iζ −β)| ≤ CR|β|,(36)
o z, zj∈O. The abo e inequali ies can be p o ed by es ima ing as in p e ious sec ions.
Then hey yield he p oo o heo em 5.1. The a gumen is analogous o [27] and [28]. We
ha e o deal wi h he a c-cho d condi ion so we will poin ou he main di e ences. Fo ini ial
da a z0∈X 0sa is ying a c-cho d, we can ind a 0 < ′
0< 0and a cons an R0such ha
kz0k ′
0< R0and
2cosh(z0
2(α+iζ)−z0
2(α+iζ −β)) −cos(z0
1(α+iζ)−z0
1(α+iζ −β))
||β||2>1
R2
0
,(37)
o α+iζ ∈S ′
0. We ake 0 < < ′
0and R0< R o de ine he open se Oas in (33).
The e o e we can use he classical me hod o successi e app oxima ions:
zn+1( ) = z0+Z
0
G(zn(s))ds, (38)
o G:O→X ′and 0 < ′< . We assume by induc ion ha
kzkk ( )< R, and kF(zk)kL∞(S )( )< R
25
II. Tε≥0 o all ε. Then we can apply a Cauchy-Kowalewski heo em o he ini ial da a
wε(α, 0) −z(α, 0)
sa is ying
||wε−z||S(0) ≤Cε.
Fo > 0 small enough, z(α, ) is in he uns able egime. We achie e he conclusion o
heo em 1.1 by con inui y wi h espec o he ini ial da a.
The es o he sec ion is de o ed o p o ing inequali y (44). We shall deno e γ=α+ih( )
and d(γ, ) = ∂4
α(w(γ, )−z(γ, )) (we omi he supe sc ip εin he no a ion) and we ecall
ha w(α, ) and z(α, ) a e eal o eal α( he e o e we ob ain simila simila es ima es o
γ=α−ih( )). In o de o p o e inequali y (44) we ha e o compu e he ollowing quan i y
d
d Zπ
−π
|d(γ, )|2dα = 2ℜZπ
−π
d(γ, )d (γ, )dα+ 2ℜih′( )Zπ
−π
d(γ, )∂αd(γ, )dα.
Again we ea in de ail he mos singula e m in d (γ, ). Recall K(α, β) om sec ion 3 and
w i e Kwand Kz o co esponding exp essions a ising om zand w. Then we ha e ha
d (γ, ) = Zπ
−π
Kw(γ, γ −β)∂5
α(w(γ, )−w(γ−β, )dβ
−Zπ
−π
Kz(γ, γ −β)∂5
α(z(γ, )−z(γ−β, ))dβ + l.o. (α, ),
whe e
2ℜZπ
−π
d(γ, )l.o. (α)dα≤C(||d(·+ih( ), )||2
L2(T)+||w(·+ih( ), )−z(·+ih( ), )||2
L2(T)).
He e Cis a cons an which jus depends on ε0and δ.
We can w i e
Zπ
−π
Kw(γ, γ −β)∂5
α(w(γ, )−w(γ−β, ))dβ
−Zπ
−π
Kz(γ, γ −β)∂5
α(z(γ, )−z(γ−β, ))dβ
=Zπ
−π
Kw(γ, γ −β)∂5
α((w(γ, )−z(γ, )) −(w(γ−β, )−z(γ−β, )))dβ
+Zπ
−π
{Kz(γ, γ −β)−Kw(γ, γ −β)}∂5
α(z(γ, )−z(γ−β, ))dβ
=Zπ
−π
Kw(γ, γ −β)∂α(d(γ, )−d(γ−β, ))dβ
+Zπ
−π
{Kz(γ, γ −β)−Kw(γ, γ −β)}∂5
α(z(γ, )−z(γ−β, ))dβ
≡X1(α, ) + X2(α, ).
32

The e o e
d
d Zπ
−π
|d(γ, )|2dα ≤C||d(·+ih( ), )||2
L2(T)
+ 2ℜZπ
−π
d(γ, )X1(α, )dα+ 2ℜZπ
−π
d(γ, )X2(α, )dα
+ 2ℜih′( )Zπ
−π
d(γ, )∂αd(γ, )dα.
Following he compu a ions in sec ion 3 when ∈[−δ, a] and hose in sec ion 4 when ∈
[ a,˜
Tε] we ha e ha
d
d Zπ
−π
|d(γ, )|2dα ≤C||d(·+ih( ), )||2
L2(T)+ 2ℜZπ
−π
d(γ, )X2(α, )dα.
In addi ion

sin(w1(γ)−w1(γ−β))
cosh(w2(γ)−w2(γ−β)) −cos(w1(γ)−w1(γ−β))
−sin(z1(γ)−z1(γ−β))
cosh(z2(γ)−z2(γ−β)) −cos(z1(γ)−z1(γ−β))
=Kw(γ, γ −β)−∂αw1(γ)
(∂αw1(γ))2+ (∂αw1(γ))2co β
2
−Kz(γ, γ −β)−∂αz1(γ)
(∂αz1(γ))2+ (∂αz1(γ))2co β
2
+∂αw1(γ)
(∂αw1(γ))2+ (∂αw1(γ))2−∂αz1(γ)
(∂αz1(γ))2+ (∂αz1(γ))2co β
2
≤(||d(·+ih( ), )||L2(T)+||w(·+ih( ), )−z(·+ih( ), )||L2(T))C+C
co β
2.
Also,
|∂5
αz(α±ih( ), )−∂5
αz(α±ih( )−β, )| ≤ C
an β
2
since zis he analy ic unpe u bed solu ion. The e o e
2ℜZπ
−π
d(γ, )X2(α, )dα≤C(||d(·+ih( ), )||2
L2(T)+||w(·+ih( ), )−z(·+ih( ), )||2
L2(T)).
We a e done.
7 Tu ning wa e wa es
Le us conside an incomp essible i o a ional low sa is ying he Eule equa ions
ρ( + · ∇ ) = −∇p−gρ(0,1),(45)
33
whe e ρsa is ies (2,3) and ρ1= 0. This sys em o equa ions p o ides he mo ion o he
in e ace o he wa e wa e p oblem (see [3, 25] and e e ences he ein), whose con ou
equa ion is gi en by
z (α, ) = BR(z, ω)(α, ) + c(α, )∂αz(α, ),(46)
and
ω (α, ) = −2∂ BR(z, ω)(α, )·∂αz(α, )−∂α(|ω|2
4|∂αz|2)(α, ) + ∂α(c ω)(α, )
+ 2c(α, )∂αBR(z, ω)(α, )·∂αz(α, )−2g∂αz2(α, ).
(47)
The alues o z(α, ) and w(α, ) a e gi en a an ini ial ime 0:z(α, 0) = z0(α) and w(α, 0) =
w0(α). Fo mo e de ails see [12].
As an applica ion o sec ion 5, we can conside ini ial da a gi en by a g aph (α, 0(α))
and show ha in ini e ime he in e ace e olu ion eaches a egime whe e he con ou only
can be pa ame ized as z(α, ) = (z1(α, ), z2(α, )), o α∈R, wi h ∂αz1(α, )<0 o α∈I,
a non-emp y in e al. This implies ha he e exis s a ime ∗whe e he solu ion o he ee
bounda y p oblem epa ame ized by (α, (α, )) sa is ies k αkL∞( ∗) = ∞.
Theo em 7.1 The e exis s a non-emp y open se o ini ial da a z0(α) = (α, 0(α)) and
w0(α), wi h 0∈H5and w0∈H4, such ha in ini e ime ∗ he solu ion o he wa e wa e
p oblem (46,47) gi en by (α, (α, )) sa is ies k αkL∞( ∗) = ∞. The solu ion can be con inued
o > ∗as z(α, )wi h ∂αz1(α, )<0 o α∈I, a non-emp y in e al.
P oo : Le us conside a cu e z∗(α)∈H5sa is ying 1., 2. and 3. o Lemma 5.3. We
poin ou ha analy ici y is no equi ed he e. In o de o ind a eloci y wi h p ope y
(39) we pick o wa e wa es ω(α, ∗) = −∂αz∗
2(α) and a sui able z(α, ∗) = z∗(α) as an
ini ial da um. No ice ha he angen ial e m does no a ec he e olu ion. Then, wi h he
app op ia e c(α, ), we can apply he local exis ence esul in [12]: The e exis s a solu ion o
he wa e wa e p oblem wi h z(α, )∈C([ ∗−δ, ∗+δ]; H5), ω(α, )∈C([ ∗−δ, ∗+δ]; H4)
and δ > 0 small enough. The ini ial da a p omised by heo em 7.1 a e any su icien ly small
pe u ba ions o z(α, ) and w(α, ) a ime = ∗−δ.
8 B eakdown o Smoo hness
In [5] we will exhibi a solu ion z(α, ) o he Muska equa ion, wi h he ollowing p ope ies.
1. A ime 0, he in e ace is eal-analy ic and sa is ies he a c-cho d and Rayleigh-Taylo
condi ions.
2. A ime 1> 0, he in e ace u ns o e .
3. A ime 2> 1, he in e ace no longe belongs o C4, al hough i is eal-analy ic o
all imes ∈[ 0, 2).
In his sec ion we p o ide a b ie ske ch o ou p oo o he exis ence o such a Muska solu ion.
Ou Muska solu ion z(α, ) will be a small pe u ba ion o a Muska solu ion z00(α, ),
wi h he ollowing p ope ies.
34
4. z00(α, ) is eal analy ic in α, o |ℑα|< ε00 and |τ| ≤ τ00.
5. Fo ∈[−τ00,0), z00(α, ) sa is ies he Rayleigh-Taylo and a c-cho d condi ions.
6. Fo = 0, he cu e z00(α, ) has a e ical angen a α= 0.
7. Fo ∈(0, τ00], he cu e z00(α, ) ails o sa is y he Rayleigh-Taylo condi ion.
This pape cons uc s Muska solu ions z00 sa is ying 4., 5., 6. and 7. Ou p oblem is o pass
om z00 o a nea by Muska solu ion zsa is ying 1., 2. and 3. The idea is as ollows.
So a , we ha e s udied he analy ic con inua ion o Muska solu ions o a ime- a ying
s ip
S( ) = {|ℑα| ≤ h( )},
in he complex plane. In ou o hcoming pape [5], we will s udy he analy ic con inua ion
o a Muska solu ion o a ca e ully chosen ime- a ying domain o he o m
Ω( ) = {|ℑα| ≤ h(ℜα, )},(48)
de ined o ∈[−τ10, τ].He e, τis a small enough posi i e numbe .
Fo ∈[−τ10, τ], we will wo k wi h he space H4(Ω( )), consis ing o all analy ic unc ions
F: Ω( )7→ C2whose de i a i es up o o de 4 belong o L2(∂Ω( )).
We will pick ou ime- a ying domain Ω( ) in (48) so ha h(x, )>0 o all (x, )∈
R/2πZ×[−τ10, τ) and h(x, τ)>0 o all x∈R/2πZ {0}, bu h(0, τ) = 0. Thus, he
domain Ω( ) has ’ hickness’ ze o a he o igin. Consequen ly, H4(Ω(τ)) is no con ained in
C4(R/2πZ).
We will also ake τ < τ00 and h(x, )< ε00, so ha he Muska solu ion z00(α, ) con inues
analy ically o Ω( ), o each ∈[−τ10, τ].
We can he e o e pick an ’ini ial’ cu e z0(α), such ha
8. z0(α)−z00(α, τ) belongs o H4(Ω(τ)) and has small no m, ye
9. z0(α) does no belong o C4(R/2πZ).
We sol e he Muska p oblem backwa ds in ime, wi h he ’ini ial’ condi ion
10. z(α, τ) = z0(α).
By a mo e elabo a e e sion o he analy ic con inua ion a gumen s used in his pape , we
ind ha ou Muska solu ion exis s and con inues analy ically in o Ω( ), o all ∈[ ∗, τ]
( o a sui able ime ∗); mo eo e ,
11. z(α, )−z00(α, ) has small no m in H4(Ω(τ)), o all ∈[ ∗, τ].
He e, ei he
12. ∗=−τ10 o
13. a modi ied Rayleigh-Taylo condi ion, adap ed o he ime- a ying domain, ails a ime
∗.
We can ule ou 13., hanks o 11., oge he wi h ou unde s anding o z00( ) and Ω( ).
Thus, we ob ain a Muska solu ion z(α, ), sa is ying 9., 10., 11. and 12. P ope ies 1., 2.
and 3. o z(α, ) now ollow easily.
35
Acknowledgemen s
AC, DC and FG we e pa ially suppo ed by he g an MTM2008-03754 o he MCINN
(Spain) and he g an S G-203138CDSIF o he ERC. CF was pa ially suppo ed by NSF
g an DMS-0901040 and ONR g an ONR00014-08-1-0678. FG was pa ially suppo ed by
NSF g an DMS-0901810. MLF was pa ially suppo ed by he g an s MTM2008-03541
and MTM2010-19510 o he MCINN (Spain). The au ho s hank P o esso Ra ael de la
Lla e o help ul discussions.
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Angel Cas o
Ins i u o de Ciencias Ma em´a icas
Consejo Supe io de In es igaciones Cien ´ı icas
Se ano 123, 28006 Mad id, Spain
Email: angel cas o@icma .es
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
Se ano 123, 28006 Mad id, Spain 1102 Fine Hall, Washing on Rd,
Email: dcg@icma .es P ince on, NJ 08544, USA
Email: c @ma h.p ince on.edu
F ancisco Gancedo Ma ´ıa L´opez-Fe n´andez
Depa men o Ma hema ics Ins i u ¨u Ma hema ik
Uni e si y o Chicago Uni e si ¨a Z¨u ich
5734 Uni e si y A enue, Win e hu e s . 190, CH-8057
Chicago, IL 60637, USA Z¨u ich, Swi ze land
Email: gancedo@ma h.uchicago.edu Email: ma ia.lop[email p o ec ed]h
38