Finite time singularities for water waves with surface tension

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