Turning waves and breakdown for incompressible flows

Tu ning wa es and b eakdown o incomp essible lows Angel Cas oa, Diego Có dobaa, Cha les L. Fe e manb,1, F ancisco Gancedoc, and Ma ía López-Fe nándezd aIns i u o de Ciencias Ma emá icas, Consejo Supe io de In es igaciones Cien í icas, Se ano 123, 28006 Mad id, Spain; bDepa men o Ma hema ics, P ince on Uni e si y, 1102 Fine Hall, Washing on Road, P ince on, NJ 08544; cDepa men o Ma hema ics, Uni e si y o Chicago, 5734 Uni e si y A enue, Chicago, IL 60637; and dIns i u ü Ma hema ik, Uni e si ä Zü ich Win e hu e s asse 190 CH-8057 Zu ich, Swi ze land Con ibu ed by Cha les L. Fe e man, Janua y 29, 2011 (sen o e iew No embe 30, 2010) We conside he e olu ion o an in e ace gene a ed be ween wo immiscible, incomp essible, and i o a ional luids. Speci ically we s udy he Muska and wa e wa e p oblems. We show ha s a - ing wi h a amily o ini ial da a gi en by ðα, 0ðαÞÞ, he in e ace eaches a egime in ini e ime in which 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 e ized as ðα, ðα, ÞÞ blows up: ‖∂α ‖L∞ð Þ¼∞.In pa icula , o he Muska p oblem, his esul allows us o each an uns able egime, o which he Rayleigh–Taylo condi ion changes sign and he solu ion b eaks down. 1. In oduc ion He e we s udy wo p oblems o luids mechanics conce ning he e olu ion o wo incomp essible luids o di e en cha ac e is ics in 2D. We conside ha bo h luids a e immiscible and o di e - en cons an densi ies ρ1and ρ2, modeling he dynamics o an in e ace ha sepa a es he domains Ω1ð Þand Ω2ð Þ. Tha is, he liquid densi y ρ¼ρðx; Þ;ðx; Þ∈R2×Rþ, is de ined by ρðx; Þ¼ρ1;x∈Ω1ð Þ ρ2;x∈Ω2ð Þ¼R2−Ω1ð Þ;[1] and sa is ies he conse a ion o mass equa ion ρ þ ·∇ρ¼0;∇· ¼0;[2] whe e ¼ð 1ðx; Þ; 2ðx; ÞÞ is he eloci y ield. Wi h a ee bound- a y pa ame e ized by ∂Ωjð Þ¼ zðα; Þ¼ðz1ðα; Þ;z2ðα; ÞÞ:α∈Pg; we conside open cu es anishing a in ini y lim α→∞ðzðα; Þ−ðα;0ÞÞ¼0; o pe iodic in he space a iable zðαþ2kπ; Þ¼zðα; Þþ2kπð1;0Þ: The scala o ici y, ∇⊥· , has he o m ∇⊥· ðx; Þ¼ωðα; Þδðx−zðα; ÞÞ;[3] i.e., he o ici y is a Di ac measu e on zde ined by <∇⊥· ;η>¼ZR ωðα; Þηðzðα; ÞÞdα; wi h ηðxÞa es unc ion. The sys em is closed by using one o he ollowing undamen al luid mo ion equa ions: Da cy’s law μ κ ¼−∇p−gρð0;1Þ;[4] o Eule equa ions ρð þ ·∇ Þ¼−∇p−gρð0;1Þ:[5] He e p¼pðx; Þis p essu e, gg a i y, μ iscosi y, and κpe meabil- i y o he iso opic medium. The Muska p oblem (1) is gi en by Eqs. 1,2, and 4, which conside s he dynamics o wo incomp essible luids o di e en densi ies h oughou po ous media and Hele–Shaw cells (2, 3). In his las se ing, he luid is apped be ween wo ixed pa allel pla es ha a e close enough oge he so ha he luid essen ially only mo es in wo di ec ions (4). Taking ρ1¼0, Eqs. 1–3and 5a e known as he wa e wa es p oblem (see e . 5 and e e ences he ein), modeling he dynamics o he con ou be ween an in iscid luid wi h densi y ρ2and acuum (o ai ) unde he in luence o g a i y. Condi ion 3(deduced by [4], assumed o [5]) allows us o w i e he e olu ion equa ion in e ms o he ee bounda y as ol- lows. One could eco e he eloci y ield om [3] by means o Bio –Sa a law ðx; Þ¼∇⊥Δ−1ð∇⊥· Þðx; Þ¼ 1 2πZR ðx−zðα; ÞÞ⊥ jx−zðα; Þj2ωðα; Þdα; applying he Di ac measu e wi h ampli ude ω. Taking limi s on he abo e equa ion app oaching he bounda y in he no mal di ec ion inside Ωj, he eloci y is shown o be discon inuous in he angen ial di ec ion, bu con inuous in he no mal, and gi en by he Bi kho –Ro in eg al o he ampli ude ωalong he in e ace cu e: BRðz;ωÞðα; Þ¼ 1 2πPV ZR ðzðα; Þ−zðβ; ÞÞ⊥ jzðα; Þ−zðβ; Þj2ωðβ; Þdβ; whe e PV deno es p incipal alue. This ac yields he cu e eloci y om which one can sub ac any e m cin he angen ial wi hou modi ying he geome y o he in e ace z ðα; Þ¼BRðz;ωÞðα; Þþcðα; Þ∂αzðα; Þ:[6] Unde s anding he p oblem as weak solu ions o [1,2, and 4]o [1–3and 5], he con inui y o he p essu e on he ee bounda y ollows. The e o e, aking limi s in Da cy’s law om bo h sides and sub ac ing he esul s in he angen ial di ec ion, i is easy o close he sys em o Muska (in his pape we conside wo luids wi h he same iscosi y): ωðα; Þ¼−ðρ2−ρ1Þκg μ∂αz2ðα; Þ:[7] Au ho con ibu ions: A.C., D.C., C.L.F., F.G., and M.L.-F. designed esea ch, pe o med esea ch, and w o e he pape . The au ho s decla e no con lic o in e es . 1To whom co espondence should be add essed. E-mail: [email p o ec ed]. 4754–4759 ∣PNAS ∣Ma ch 22, 2011 ∣ ol. 108 ∣no. 12 www.pnas.o g/cgi/doi/10.1073/pnas.1101518108 In a simila way o wa e wa es, Eule equa ions yield ω ðα; Þ¼−2∂ BRðz;ωÞðα; Þ·∂αzðα; Þ−∂ αjωj2 4j∂αzj2ðα; Þ þ∂αðcωÞðα; Þþ2cðα; Þ∂αBRðz;ωÞðα; Þ·∂αzðα; Þ −2g∂αz2ðα; Þ:[8] Then, he wo con ou equa ions a e se by [6and 7] and [6and 8]. Fo hese models, he well-posedness u ns ou o be alse o some se ings. Rayleigh (6) and Sa man and Taylo (2) ga e a condi ion ha mus be sa is ied o he linea ized model in o de o exis a solu ion locally in ime: The 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 quan i y is known as he Rayleigh–Taylo condi ion. I eads as σðα; Þ¼−ð∇p2ðzðα; Þ; Þ−∇p1ðzðα; Þ; ÞÞ ·∂⊥ αzðα; Þ>0; whe e ∇pjðzðα; Þ; Þdeno 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 Ωjð Þ. An easy linea iza ion a ound a la con ou ðα; ðα; ÞÞ, allows us o ind ¼1 2HðωÞ whe e His he Hilbe ans o m which symbol on he Fou ie side is gi en by b H¼−isign ðξÞ. The equa ions ω¼−ðρ2−ρ1Þκg μ∂α ; ðlinea Muska Þ ω ¼2g∂α ; ðlinea wa e wa esÞ show he pa abolici y o he Muska p oblem when he dense luid is below (ρ2>ρ1) and he dispe si e cha ac e o wa e wa es. 1. The e is a wide li e a u e on he Muska p oblem and he dynamics o wo luids in a Hele–Shaw cell. The e a e wo ks conside ing he case o a iscosi y jump neglec ing he e ec o g a i y (7, 8). Local exis ence in a mo e gene al si ua ion (wi h discon inuous iscosi y and densi y) is shown in e . 9 and also ea ed in e . 10. A di e en app oach o p o e local exis ence can be ound in e . 11 o he se ing we a e con- side ing in his pape . The Rayleigh–Taylo s abili y depends upon he sign o ðρ2−ρ1Þ∂αz1ðα; Þ(11) indica ing ha he hea ie luid has o be below in he s able case. I he ligh e luid is below, he p oblem has been shown o be ill-posed (11). Global-exis ence esul s o small ini ial da a can be ound in e s. 7 and 11–14. Fo la ge ini ial cu es and pa a- me e ized by ðα; ðα; ÞÞ, he e a e maximum p inciples o he L∞and L2no ms o , and decay a es, oge he wi h global exis ence o Lipschi z cu es i ‖∂α ‖L∞ð0Þ<1(15, 16, 17). 2. The wa e wa es p oblem has been ex ensi ely conside ed (see e s. 5 and 18 and e e ences he ein). Fo su icien ly smoo h ee bounda y, he Rayleigh–Taylo condi ion emains posi i e wi h no bo om conside a ions (19), a ac ha was used o p o e local exis ence (19). The Rayleigh–Taylo s abili y can play a di e en ole o he case o non-“almos ”- la bo om (20). Recen ly, o small ini ial da a, exponen ial ime o exis ence has been p o en in wo dimensions (21) and global exis ence in he h ee-dimensional case ( wo-dimensional in- e ace) (22, 23). 2. Rayleigh–Taylo B eakdown o Muska This sec ion is de o ed o show he main ing edien s o p o e he Theo em 2.1. We conside he unc ion FðzÞðα;βÞ¼ jβj2 jzðαÞ−zðα−βÞj2;α;β∈R; and in he pe iodic se ing FðzÞðα;βÞ¼ ‖β‖2 2ðcoshðz2ðαÞ−z2ðα−βÞÞ −cosðz1ðαÞ−z1ðα−βÞÞÞ ; α;β∈T;[9] whe e ‖x‖¼dis ðx;2πZÞ.I FðzÞ∈L∞ðR2Þ, hen he cu e zsa- is ies he a c-cho d condi ion. We say ha he Rayleigh–Taylo (R-T) o he solu ion o he Muska p oblem b eaks down in ini e ime i o ini ial da a z0sa is ying σðα;0Þ¼ðρ2−ρ1Þ∂αz1ðα;0Þ>0 he e exis s a ime >0 o which σðα; Þis s ic ly nega i e in a nonemp y open in e al. Theo em 2.1. The e exis s a nonemp y open se o ini ial da a in H4, sa is ying Rayleigh–Taylo and a c-cho d condi ions, o which he Rayleigh–Taylo condi ion o he solu ion o he Muska p oblem [1,2, and 4] b eaks down in ini e ime. A e choosing he app op ia e angen ial e m and a in eg a- ion by pa s, he con ou equa ion eads z ðα; Þ¼ρ2−ρ1 2πPV ZR ðz1ðα; Þ−z1ðβ; ÞÞ jzðα; Þ−zðβ; Þj2ð∂αzðα; Þ−∂ αzðβ; ÞÞdβ: Fo a 2πpe iodic in e ace, emo ing he p incipal alue a in ini y, he equa ion becomes z ðαÞ¼ðρ2−ρ1Þ 4π ×ZT sinðz1ðαÞ−z1ðα−βÞÞð∂αzðαÞ−∂ αzðα−βÞÞ coshðz2ðαÞ−z2ðα−βÞÞ −cosðz1ðαÞ−z1ðα−βÞÞ dβ: [10] F om now on, we shall use he pe iodic con igu a ion. The s eps o he p oo a e as ollows: 1. Fi s , o any ini ial cu e z0ðαÞ¼zðα;0Þin H4 ha sa is y R-T ðρ2−ρ1Þ∂αz1ðα;0Þ>0 and he a c-cho d condi ion hen he solu ion o he Muska p oblem zðα; Þbecomes analy ic o 0< <T. Mo eo e , zðα; Þis eal analy ic in a s ip Sð Þ¼ αþiζ:jζj<c g o ∈ð0;TÞwhe e cdepends only on in ð0Þ¼in α ∂αz1ðα;0Þ j∂αzðα;0Þj2: The p oo ollows by con olling he quan i ies ex ended on Sð Þ: FðzÞðαþiζ;β; Þ and gðαþiζ; Þby using [9] and o mula gðα; Þ¼ZT ½sinðz1ðα; Þ−z1ðα−β; ÞÞ∕½coshðz2ðα; Þ −z2ðα−β; ÞÞ −cosðz1ðα; Þ−z1ðα−β; ÞÞdβ; espec i ely. The no ms Cas o e al. PNAS ∣Ma ch 22, 2011 ∣ ol. 108 ∣no. 12 ∣4755 MATHEMATICS ‖FðzÞ‖L∞ðSÞð Þ¼ sup αþiζ∈Sð Þ;β∈T jFðzÞðαþiζ;βÞj; ‖z‖2 L2ðSÞð Þ¼∑ ZT jzðαic ; Þj2dα; ‖z‖2 HjðSÞð Þ¼‖z‖2 L2ðSÞð Þþ∑ ZT j∂j αzðαic ; Þj2dα; o j∈N; in ð Þ¼ in αþiζ∈Sð Þ ℜ∂αz1ðαþiζ; Þ j∂αzðαþiζ; Þj2: Then he quan i y ‖z‖2 RTð Þ¼‖z‖2 H4ðSÞð Þþ‖FðzÞ‖L∞ðSÞð Þ þ1∕ðin ð Þ−c−K∥ℑðgÞ‖H2ðSÞð ÞÞ sa is ies d d ‖z‖RTð Þ≤C‖z‖k RTð Þ; o C,K, and kuni e sal cons an s. I yields ‖z‖RTð Þ≤‖z‖RTð0Þ ð1−C‖z‖k RTð0Þ Þ1∕k; p o iding con ol o he analy ici y and T¼1∕ðC‖z‖k RTð0ÞÞ. 2. Second, he e is a lowe bound on he s ip o analy ici y, which does no collapse o he eal axis as long as he Rayleigh–Taylo is g ea e han o equal o 0. Then he e is a ime Tand a solu ion o he Muska p oblem zðα; Þde ined o 0< ≤T ha con inues analy ically in o a complex s ip i ðρ2−ρ1Þ∂αz1≥0, whe e Tis ei he a small cons an o i is he i s ime a e ical angen appea s, whiche e occu s i s . We ede ine he s ip Sð Þ¼ αþiζ:jζj<hð Þ;0<hð0Þg; and he quan i y ‖z‖2 S¼‖z‖2 H4ðSÞþ‖FðzÞ‖L∞ðSÞwi h his new Sð Þ. Fo an hð Þdec easing [ he exp ession o hð Þis chosen la e ], we conside he e olu ion o he mos singula quan i y ∑ Zj∂4 αzðαihð Þ; Þj2dα: Taking a de i a i e in , one inds d d ∑ Zj∂4 αzðαihð ÞÞj2dα≤h0ð Þ 10 ∑ ZΛð∂4 αzÞðαihð ÞÞ ·∂4 αzðαihð ÞÞdα −10h0ð ÞZΛð∂4 αzÞðαÞ·∂4 αzðαÞdα þ2∑  ℜZ∂4 αz ðαihð ÞÞ ·∂4 αzðαihð ÞÞdα: Es ima ing in a wise way, one ob ains d d ∑ Zj∂4 αzðαihð ÞÞj2dα≤C‖z‖k Sð Þ −10h0ð ÞZΛð∂4 αzÞðαÞ·∂4 αzðαÞdαþðC‖z‖k Sð Þhð Þ þ1 10 h0ð ÞÞ ZΛð∂4 αzÞðαihð ÞÞ ·∂4 αzðαihð ÞÞdα: The e o e, choosing hð Þ¼hð0Þexpð−10CZ 0 ‖z‖k Sð Þd Þ elimina es he mos dange ous e m. The o he e ms a e ea- sily con olled, gi ing inally d d ∑ Zj∂4 αzðαihð ÞÞj2dα≤C‖z‖kþ2 Sð Þ; which allows us o each a egime o which he bounda y z de elops a e ical angen a ime T. 3. Thi d, i is shown he exis ence o a la ge class o analy ic cu es o which he e exis a poin whe e he angen ec o is e ical and he eloci y indica es ha he cu e is going o u n up and each he uns able egime. Fo he equa ion z ðα; Þ¼uðα; Þ¼ðu1ðα; Þ;u2ðα; ÞÞ; ha is, a: ∂αz1ðαÞ>0i α≠0;b:∂αz1ð0Þ¼0; c:∂αz2ð0Þ>0;d:∂αu1ð0Þ<0; o analy ic unc ions z1ðαÞand z2ðαÞsuch ha zðαÞsa is ies he a c-cho d condi ion. He e we conside he pe iodic case (being analogous o an open cu e anishing a in ini y). We assume ha zðαÞis a smoo h odd cu e sa is ying he p ope ies a,b, and c. Di e en ia ing he exp ession 10 o he ho izon al componen o he eloci y, a α¼0, i yields ð∂αu1Þð0Þ¼Zπ −π ½cosðz1ðβÞÞð∂αz1ðβÞÞ2 þsinðz1ðβÞÞ∂2 αz1ðβÞ∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ −Zπ −π sinðz1ðβÞÞ∂αz1ðβÞ½sinðz1ðβÞÞ∂αz1ðβÞ −sinhðz2ðβÞÞð∂αz2ð0Þ−∂ αz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2dβ: In eg a ion by pa s p o ides Zπ −π ½sinðz1ðβÞÞ∂2 αz1ðβÞ∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ ¼−Zπ −π cosðz1ðβÞÞ½ð∂αz1ðβÞÞ2∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ þZπ −π sinðz1ðβÞÞ∂αz1ðβÞ½sinðz1ðβÞÞ∂αz1ðβÞ þsinhðz2ðβÞÞ∂αz2ðβÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2dβ: The e o e, i is easy o ob ain ha ð∂αu1Þð0Þ¼∂αz2ð0ÞZπ −π ½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβ ¼2∂αz2ð0ÞZπ 0 ½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβ [11] Exp ession 11 allows us o de e mine he sign o ð∂αu1Þð0Þ. One could ake z1ðβÞ¼−sinðβÞþβ 4756 ∣www.pnas.o g/cgi/doi/10.1073/pnas.1101518108 Cas o e al. and cons uc he unc ion z2ðβÞin he ollowing way: Le β1, β2,β3, and β4be eal inc easing numbe s less han π. We pick z2ðβÞ≤0 o β2<β<π,z2ðβÞ<c<0 o β2<β<β4, and z 2ðβÞa smoo h unc ion wi h he ollowing p ope ies a: z 2ðβÞis odd;b:ð∂βz 2Þð0Þ>0; c: z 2ðβÞ>0i β∈ð0;β1Þ;d:z  2ðβÞ<0i β∈ðβ1;β2: Also, z 2ðβÞis 2π-pe iodic. Fo z2ðβÞ¼bz 2ðβÞ,0≤β≤β2, and b>0, he eloci y sa is ies ð∂αu1Þð0Þ<2ð∂αz2Þð0Þ ×Zβ1 0 ½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβ þZπ β3 ½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβ ¼2ð∂αz2Þð0ÞZβ1 0 ½sinðz1ðβÞÞ sinhðbz 2ðβÞÞ∕½ðcoshðbz 2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβþA; whe e A<0. The cons an bla ge enough yields ð∂αu1Þ ð0Þ<0. Rec i ying he cu e on he in e al ½β2;β3, i is easy o ob ain a smoo h cu e. Finally, con ol ing wi h he hea ke nel he e ical componen , he cu e zðαÞis app oxima ed by an analy ic one. 4. Fou h, wi h he ini ial da a ound in 3 and no assump ion on he R-T condi ion, we use a modi ica ion o Cauchy– Kowalewski heo ems (24, 25) o show ha he e exis s an ana- ly ic solu ion o he Muska p oblem in some in e al ½−T;T o a small enough T>0. He e we a e o ced o change sub- s an ially he me hod in e . 26 because, in his case, he cu e canno be pa ame e ized as a g aph, so we ha e o deal wi h he a c-cho d condi ion. Then, wi h X g >0, a scale o Banach spaces gi en by eal unc ions ha can be ex ended analy ically on he complex s ip S ¼ αþiζ∈C:jζj< gwi h no m ‖ ‖ ¼∑ Zj ðαi Þj2dαþZj∂4 α ðαi Þj2dα; and z0ðαÞa cu e sa is ying he a c-cho d condi ion and z0ðαÞ∈X 0 o some 0>0, we p o e he exis ence o a ime T>0and 0< < 0so ha he e is a unique solu ion o he Muska p oblem in Cð½−T;T;X Þ. This esul allows us o ind solu ions ha do no sa is y he R-T bu sh ink he s ip o analy ici y. We ex end Eq. 10 as ollows: z ðαþiζ; Þ¼Gðzðαþiζ; ÞÞ; wi h GðzÞðα; Þ¼ðρ2−ρ1Þ 4π ×ZT sinðz1ðα; Þ−z1ðα−β; ÞÞð∂αzðα; Þ−∂ αzðα−β; ÞÞ coshðz2ðα; Þ−z2ðα−β; ÞÞ −cosðz1ðα; Þ−z1ðα−β; ÞÞ dβ: Fo 0< 0< and he open se Oin S gi en by O¼ z∈X :‖z‖ <R; ‖FðzÞ‖L∞ðS Þ<R2g;[12] he unc ion G o G:O→X 0is a con inuous mapping and he e is a cons an CR(depending on Ronly) such ha ‖GðzÞ‖ ≤CR − 0‖z‖ ;[13] ‖Gðz2Þ−Gðz1Þ‖ 0≤CR − 0‖z2−z1‖ ;[14] and sup αþiζ∈S ;β∈T jGðzÞðαþiζÞ−GðzÞðαþiζ−βÞj ≤CRjβj;[15] o z;zj∈O. Fo ini ial da a z0∈X 0sa is ying a c-cho d, we can ind a 0< 0 0< 0and a cons an R0such ha ‖z0‖ 0 0<R0and ½coshðz0 2ðαþiζÞ−z0 2ðαþiζ−βÞÞ −cosðz0 1ðαþiζÞ−z0 1ðαþiζ−βÞÞ∕ð‖β‖2Þ>1 R2 0 ;[16] o αþiζ∈S 0 0. We ake 0< < 0 0and R0<R o de ine he open se Oas in [12]. The e o e we can use he classical me hod o successi e app oxima ions: znþ1ð Þ¼z0þZ 0 GðznðsÞÞds; o G:O→X 0and 0< 0< . We assume by induc ion ha ‖zk‖ ð Þ<R; and ‖FðzkÞ‖L∞ðS Þð Þ<R o k≤nand 0< <Twi h T¼minðTA;TCK Þand TCK he ime ob aining in he p oo s in e s 24 and 25. We ge ‖znþ1‖ ð Þ<R ha ollows using [13 and 14]. The ime TAis o yield ‖Fðznþ1Þ‖L∞ðS Þð Þ<R. Then, using he induc ion hypo hesisand[15],wecancon ol hequan i y aking0<TA< ðR−2 0−R−2ÞðC2 Rþ2R0CRÞ−1. 5. Fi h, all he esul s abo e allow us o p o e ha he e is a nonemp y se o ini ial da a in H4sa is ying he a c-cho d and R-Tcondi ions, such ha he solu ion o he Muska p o- blem eaches he uns able egime: The R-T becomes s ic ly nega i e on a nonemp y in e al. We pick ini ial da a as in 3. We apply he local-exis ence esul in 4 o ge an analy ic solu ion zðα; Þon ½−T;T. Then we conside a ime 0<δ<T and a cu e ωε δðα; Þ, sol ing he Muska p oblem wi h ini ial da um zðα;−δÞþηϵ δðαÞ. The unc ion ηϵ δhas a small H4 no m, i.e., ‖ωε δð·;−δÞ−zð·;−δÞ‖H4¼‖ηϵ δ‖H4≤ε: The ime δis small enough so ha ωε δðα;−δÞsa is ies R-T: ðρ2−ρ1Þ∂αðωε δÞ1ðα;−δÞ>0. Then we apply he local-exis ence esul in 1 ha ωε δðα; Þbecomes analy ic o some ime −δ< . Wi h 2, we assu e he exis ence and analy ici y o he solu ion e en i ∂αðωε δÞ1ðα; Þ≤0 o some ime . Then, we show ha bo h solu ions a e close in he H4 opology as ime e ol es. We can apply o ωε δ he local-exis ence esul in 4 i i is needed. Then, wi h δand εsmall enough, we ind he desi ed esul . 3. Tu ning Wa e Wa es In his sec ion, we p o e o he wa e wa e p oblem (ρ1¼0and [1–3and 5]) ha wi h ini ial da a gi en by a g aph ðα; 0ðαÞÞ, he in e ace eaches a egime in ini e ime whe e i only can be pa ame e ized as zðα; Þ¼ðz1ðα; Þ;z2ðα; ÞÞ; o α∈R, wi h ∂αz1ðα; Þ<0 o α∈I, a nonemp y in e al. The e o e he e exis s a ime whe e he solu ion o he ee bounda y p oblem epa ame e ized by ðα; ðα; ÞÞ sa is ies ‖ α‖L∞ð Þ¼∞. Theo em 3.1. The e exis s a nonemp y open se o ini ial da a ðα; 0ðαÞÞ, wi h 0∈H5, such ha in ini e ime  he solu ion o Cas o e al. PNAS ∣Ma ch 22, 2011 ∣ ol. 108 ∣no. 12 ∣4757 MATHEMATICS he wa e wa es p oblem (ρ1¼0and [1–3and 5]) gi en by ðα; ðα; ÞÞ sa is ies ‖ α‖L∞ð Þ¼∞. The solu ion can be con inued o > as zðα; Þwi h ∂αz1ðα; Þ<0 o α∈I, a nonemp y in e al. In o de o p o e his heo em, we conside a cu e zðαÞ∈H5 wi h he same p ope ies as in poin 3 o he p e ious sec ion. Then, we pick zðα; Þ¼zðαÞand ωðα; Þ¼−∂αz 2ðαÞas a da um o he ini ial alue p oblem. I is easy o ind he same p ope ies o he eloci y, because he angen ial di ec ion does no a ec he e olu ion. Picking he app op ia e cðα; Þand applying he local-exis ence esul in e . 18 (no e ha in his case i is no necessa y analy ici y, jus H5 egula i y), he e exis s a solu ion o he wa e wa es p oblem wi h zðα; Þ∈Cð½ −δ; þδ;H5Þ, ωðα; Þ∈Cð½ −δ; þδ;H4Þ, and δ>0small enough. Then, he ini ial da um ðz0ðαÞ;ω0ðαÞÞ¼ðα; 0ðαÞ;ω0ðαÞÞ is gi en by ðzðα; −δÞ;ωðα; −δÞÞ. 4. Muska B eakdown In his sec ion, we show ha he e exis s a smoo h ini ial da a in he s able egime o he Muska p oblem such ha he solu ion u ns o he uns able egime and la e i b eaks down. The ou line o he p oo is o cons uc a cu e in he uns able egime which is analy ic excep in a single poin . We show ha , as we e ol e back- wa d in ime, he cu e becomes analy ic and is as close as we desi ed (in he Hk opology wi h kla ge enough) o he cu e om pa 3 o Sec ion 2. He e we will wo k in he pe iodic se ing and will conside he equa ion ∂ zðζ; Þ¼Zw∈Γþð Þ sinðz1ðζ; Þ−z1ðw; ÞÞ coshðz2ðζ; Þ−z2ðw; ÞÞ −cosðz1ðζ; Þ−z1ðw; ÞÞ ×ð∂ζzðζ; Þ−∂ ζzðw; ÞÞdw; [17] whe e ζ∈Ωð Þ, Ωð Þ¼ ζ∈C∕2kπ:jℑζj<hðℜz; Þg; hðx; Þis a posi i e pe iodic unc ion wi h pe iod 2πand smoo h o ixed ime , and Γð Þ¼ ζ∈C∕2kπ:ζ¼xþihðx; Þg: This equa ion is equi alen o [1,2, and 4] o holomo phic unc ions. In o de o p o e he esul , we will need he ollowing heo- em: Theo em 4.1. Le hðx; Þbe a posi i e, smoo h, and pe iodic unc ion wi h pe iod 2π o ixed ime ∈½ 0−δ; 0. Le zðx; 0Þbe a cu e sa is ying he ollowing p ope ies: •z1ðx; 0Þ−xand z2ðx; 0Þa e pe iodic wi h pe iod 2π; •zðζ; 0Þis eal o ζ eal; •zðζ; 0Þis analy ic in ζ∈Ωð 0Þ; •zðζ; Þ∈HkðΓð 0ÞÞ wi h ka la ge enough in ege . •Complex a c-cho d condi ion: jcoshðz2ðζ; 0Þ−z2ðw; 0ÞÞ −cosðz1ðζ; 0Þ−z1ðw; 0ÞÞj ≥½jjℜðζ−wÞjj þ jℑðζ−wÞj2; o ζ,w∈Ωð 0Þ, whe e ‖x‖¼dis anceðx;2kπÞ: •Gene alized Rayleigh–Taylo condi ion: RTðζ; 0Þ>0, whe e RTðζ; Þ¼ℜ−2π∂ζz1ðζ; Þ ð∂ζz1ðζ; ÞÞ2þð∂ζz2ðζ; ÞÞ2ð1þi∂xhðℜζ; ÞÞ−1 þℑPV ZwΓþð Þ ½sinðz1ðζ; Þ −z1ðw; ÞÞ∕½coshðz2ðζ; Þ−z2ðw; ÞÞ −cosðz1ðζ; Þ−z1ðw; ÞÞdw þi∂ hðζ; Þ ×ð1þi∂xhðℜζ; ÞÞ−1: Then, o small enough δ, he e exis s a solu ion o Eq. 17 in he ime in e al ∈½ 0−δ; 0, sa is ying •z1ðx; Þ−xand z2ðx; Þa e pe iodic wi h pe iod 2π; •zðζ; Þis eal o ζ eal; •zðζ; Þis analy ic in ζ∈Ωð 0Þ; •zðζ; Þ∈HkðΓð ÞÞ wi h ka la ge enough in ege . Now, le zðx; Þbe he solu ion o he Muska p oblem wi h zðx;0Þ¼z0ðxÞ, whe e z0ðxÞis he pa icula ini ial da a om pa 3 o he Sec ion 2. We shall de ine his solu ion as he unpe u bed solu ion. Le us deno e he Rayleigh–Taylo unc ion σ0 1ðx; Þ≡−2π∂xz1ðx; Þ ð∂xz1ðx; ÞÞ2þð∂xz2ðx; ÞÞ2: No ice he minus sign in he igh -hand side o he p e ious exp ession. One can check he ollowing p ope ies o his Rayleigh–Taylo unc ion: 1. σ0 1ð·; Þis analy ic on xþiy:x∈T;jyj≤cbgwi h jσ0 1ðxþiy; Þj ≤C, o all xþiy as abo e and o all ≤½0;τ; 2. σ0 1ð0;0Þis eal o x∈T, ∈½0;τ; 3. σ0 1has a p io i bounded Ck0no m as a unc ion o ðx; Þ∈T×½0;τ(k0la ge enough); 4. σ0 1ð0;0Þ¼0; 5. ∂xσ0 1ð0;0Þ¼0; 6. ∂2 xσ0 1ð0;0Þ¼−c2<0; 7. ∂ σ0 1ð0;0Þ¼c1>0. In his se ing, we de ine he ollowing weigh unc ions hðx; Þ¼A−1ðτ2− 2ÞþðA−1−ðτ− ÞÞ sin2x 2 o ∈½τ2;τ: [18] ℏðx; Þ¼1 4ðA−1τ2þA−1sinx 2ÞþA−2τ þA sinx 2 ∈½0;τ2;[19] wi h x∈T. Fi s we choose he pa ame e s Ala ge enough and hen τsmall enough, hen one can show ha σ0 1ðx; Þþ∂ hðx; Þ−A1 2hðx; Þ≥cτ2 o x∈T; ∈½τ2;τ[20] and σ0 1ðx; Þþ∂ ℏðx; Þ−A1 2ℏðx; Þ≥1 2A−2τ o x∈T; ∈½0;τ2:[21] The inequali ies 20 and 21 a e one o he main ing edien s o he p oo o he ollowing esul s. Theo em 4.2. Le zðx; Þbe a solu ion o he Muska equa ion in he in e al ∈½0;τ. Le hðx; Þand ℏðx; Þas in he exp essions 18 and 19, and ka la ge enough in ege . Assume ha zðx; Þsa is ies 4758 ∣www.pnas.o g/cgi/doi/10.1073/pnas.1101518108 Cas o e al. •z1ðx; Þ−xand z2ðx; Þa e pe iodic wi h pe iod 2π; •zðζ; Þis eal o ζ eal; •zðζ; Þis analy ic in ζ∈Ωð Þ; •zðζ; Þ∈HkðΓð ÞÞ wi h ka la ge enough in ege . •Complex a c-cho d condi ion: jcoshðz2ðζ; Þ−z2ðw; ÞÞ −cosðz1ðζ; Þ−z1ðw; ÞÞj ≥½‖ℜðζ−wÞ‖þjℑðζ−wÞj2; o ζ,w∈Ωð Þ. He e, in he de ini ion o Ωð Þand Γð Þ, we use hðx; Þi ∈½τ2;τ and ℏðx; Þi ∈½0;τ2. Then 1 2 d d Zw∈Γþð Þ j∂k ζzðζ; Þ−∂ k ζzðζ; Þj2dℜζ≥−CðAÞλ2; i ∈½τ2;τ Zw∈Γþð Þ j∂k ζzðζ; Þ−∂ k ζzðζ; Þj2dℜζ≤λ2 and λ≤τ50. In addi ion, 1 2 d d Zw∈Γþð Þ j∂k ζzðζ; Þ−∂ k ζzðζ; Þj2dℜζ≥−CðAÞτ−1λ2; i ∈½0;τ2 Zw∈Γþð Þ j∂k ζzðζ; Þ−∂ k ζzðζ; Þj2dℜζ≤λ2 and λ≤τ50. This heo em implies ha o all γ>0 he e is ε>0such ha Zw∈Γþð Þ j∂k ζzðζ; Þ−∂ k ζzðζ; Þj2dℜζ≤γ o ∈½0;τi Zw∈Γþð Þ j∂k ζzðζ;τÞ−∂ k ζzðζ;τÞj2dℜζ≤ε and zðx; Þsa is ies he equi emen s o he heo em. Lemma 4.3. Le zðx; Þbe a solu ion o he Muska p oblem sa is ying he equi emen s o Theo em 4.2 and close enough o he unpe - u bed solu ion in ∈½0;τ. Le hðx; Þand ℏðx; Þbe as in [18] and [19] wi h a sui able choice o Aand τ. Then zðx; Þsa is ies he gene alized Rayleigh–Taylo condi ion in ∈½0;τ. In pa icula , he unpe u bed solu ion sa is ies he gene alized Rayleigh–Taylo condi ion in ∈½0;τ Theo ems 4.1 and 4.2 and Lemma 4.3 allow us o achie e he desi ed esul . Indeed we can choose a cu e zðx;τÞsuch ha Zζ∈Γ j∂k ζzðζ;τÞ−∂ k ζzðζ;τÞj2dℜζ≤ε; wi h 0<ε<ε0(ε0small enough), sa is ying he gene alized Rayleigh–Taylo condi ion by Lemma 4.3 and sa is ying he es o he hypo hesis o Theo em 4.1. Because hð0;τÞ¼0,zðx; Þis allowed o be nonanaly ic a x¼0[maybe zðx;τÞ∈HkðTÞbu zðx;τÞ∉Hkþ1ðTÞ]. By Theo em 4.1, he e is a solu ion zðx; Þ, ana- ly ic in Ωð Þ, o some in e al ∈½τ−δ;τwi h small enough δ and o all ε. By Theo em 4.2, we can choose εsmall enough in such a way ha , by Lemma 4.3, zðx;τ−δÞsa is ies he gene al- ized Rayleigh–Taylo condi ion. Then we can go u he he ime τ−δ. I e a ing his a gumen , we ind we can ex end zðx; Þ o be a solu ion o he Muska p oblem, analy ic in Ωð Þ o all ∈½0;τ and as close as we wan o he unpe u bed solu ion. ACKNOWLEDGEMENTS A.C., D.C., and F.G. we e pa ially suppo ed by G an MTM2008-03754 o he Minis e io de Ciencia e Inno ación (MCINN) (Spain) and G an S G-203138CDSIF o he Eu opean Resea ch Council. 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