Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

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. Re e ences [1] D. Amb ose. Well-posedness o Two-phase Hele-Shaw Flow wi hou Su ace Tension. Eu o. Jnl. o Applied Ma hema ics 15, (2004), 597-607. [2] C. Ba dos and S. Benachou . Domaine d’analy ici e des solu ions de l’equa ion d’Eule dans un ou e de Rn.Annal. Sc. No male Sup. di Pisa, (1977), no. 4, 647-687. [3] C. Ba dos and D. Lannes. Ma hema ics o 2d In e aces. a Xi :1005.5329. To appea in Pano ama e Syn heses (2010). [4] J. Beale, T. Y. Hou and J. Loweng ub. 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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