Turning waves and breakdown for incompressible flows

Turning waves and breakdown for incompressible flows Angel Castroa, Diego Córdobaa, Charles L. Feffermanb,1, Francisco Gancedoc, and María López-Fernándezd aInstituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain; bDepartment of Mathematics, Princeton University, 1102 Fine Hall, Washington Road, Princeton, NJ 08544; cDepartment of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637; and dInstitut für Mathematik, Universität Zürich Winterthurerstrasse 190 CH-8057 Zurich, Switzerland Contributed by Charles L. Fefferman, January 29, 2011 (sent for review November 30, 2010) We consider the evolution of an interface generated between two immiscible, incompressible, and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by ðα,f0ðαÞÞ, the interface reaches a regime in finite time in which is no longer a graph. Therefore there exists a time twhere the solution of the free boundary problem parameterized as ðα,fðα,tÞÞ blows up: ‖∂αf‖L∞ðtÞ¼∞.In particular, for the Muskat problem, this result allows us to reach an unstable regime, for which the Rayleigh–Taylor condition changes sign and the solution breaks down. 1. Introduction Here we study two problems of fluids mechanics concerning the evolution of two incompressible fluids of different characteristics in 2D. We consider that both fluids are immiscible and of different constant densities ρ1and ρ2, modeling the dynamics of an interface that separates the domains Ω1ðtÞand Ω2ðtÞ. That is, the liquid density ρ¼ρðx;tÞ;ðx;tÞ∈R2×Rþ, is defined by ρðx;tÞ¼ρ1;x∈Ω1ðtÞ ρ2;x∈Ω2ðtÞ¼R2−Ω1ðtÞ;[1] and satisfies the conservation of mass equation ρtþv·∇ρ¼0;∇·v¼0;[2] where v¼ðv1ðx;tÞ;v2ðx;tÞÞ is the velocity field. With a free boundary parameterized by ∂ΩjðtÞ¼fzðα;tÞ¼ðz1ðα;tÞ;z2ðα;tÞÞ:α∈Pg; we consider open curves vanishing at infinity lim α→∞ðzðα;tÞ−ðα;0ÞÞ¼0; or periodic in the space variable zðαþ2kπ;tÞ¼zðα;tÞþ2kπð1;0Þ: The scalar vorticity, ∇⊥·v, has the form ∇⊥·vðx;tÞ¼ωðα;tÞδðx−zðα;tÞÞ;[3] i.e., the vorticity is a Dirac measure on zdefined by <∇⊥·v;η>¼ZR ωðα;tÞηðzðα;tÞÞdα; with ηðxÞa test function. The system is closed by using one of the following fundamental fluid motion equations: Darcy’s law μ κv¼−∇p−gρð0;1Þ;[4] or Euler equations ρðvtþv·∇vÞ¼−∇p−gρð0;1Þ:[5] Here p¼pðx;tÞis pressure, ggravity, μviscosity, and κpermeability of the isotropic medium. The Muskat problem (1) is given by Eqs. 1,2, and 4, which considers the dynamics of two incompressible fluids of different densities throughout porous media and Hele–Shaw cells (2, 3). In this last setting, the fluid is trapped between two fixed parallel plates that are close enough together so that the fluid essentially only moves in two directions (4). Taking ρ1¼0, Eqs. 1–3and 5are known as the water waves problem (see ref. 5 and references therein), modeling the dynamics of the contour between an inviscid fluid with density ρ2and vacuum (or air) under the influence of gravity. Condition 3(deduced by [4], assumed for [5]) allows us to write the evolution equation in terms of the free boundary as follows. One could recover the velocity field from [3] by means of Biot–Savart law vðx;tÞ¼∇⊥Δ−1ð∇⊥·vÞðx;tÞ¼ 1 2πZR ðx−zðα;tÞÞ⊥ jx−zðα;tÞj2ωðα;tÞdα; applying the Dirac measure with amplitude ω. Taking limits on the above equation approaching the boundary in the normal direction inside Ωj, the velocity is shown to be discontinuous in the tangential direction, but continuous in the normal, and given by the Birkhoff–Rott integral of the amplitude ωalong the interface curve: BRðz;ωÞðα;tÞ¼ 1 2πPV ZR ðzðα;tÞ−zðβ;tÞÞ⊥ jzðα;tÞ−zðβ;tÞj2ωðβ;tÞdβ; where PV denotes principal value. This fact yields the curve velocity from which one can subtract any term cin the tangential without modifying the geometry of the interface ztðα;tÞ¼BRðz;ωÞðα;tÞþcðα;tÞ∂αzðα;tÞ:[6] Understanding the problem as weak solutions of [1,2, and 4]or [1–3and 5], the continuity of the pressure on the free boundary follows. Therefore, taking limits in Darcy’s law from both sides and subtracting the results in the tangential direction, it is easy to close the system for Muskat (in this paper we consider two fluids with the same viscosity): ωðα;tÞ¼−ðρ2−ρ1Þκg μ∂αz2ðα;tÞ:[7] Author contributions: A.C., D.C., C.L.F., F.G., and M.L.-F. designed research, performed research, and wrote the paper. The authors declare no conflict of interest. 1To whom correspondence should be addressed. E-mail: [email protected]. 4754–4759 ∣PNAS ∣March 22, 2011 ∣vol. 108 ∣no. 12 www.pnas.org/cgi/doi/10.1073/pnas.1101518108 In a similar way for water waves, Euler equations yield ωtðα;tÞ¼−2∂tBRðz;ωÞðα;tÞ·∂αzðα;tÞ−∂ αjωj2 4j∂αzj2ðα;tÞ þ∂αðcωÞðα;tÞþ2cðα;tÞ∂αBRðz;ωÞðα;tÞ·∂αzðα;tÞ −2g∂αz2ðα;tÞ:[8] Then, the two contour equations are set by [6and 7] and [6and 8]. For these models, the well-posedness turns out to be false for some settings. Rayleigh (6) and Saffman and Taylor (2) gave a condition that must be satisfied for the linearized model in order to exist a solution locally in time: The normal component of the pressure gradient jump at the interface has to have a distinguished sign. This quantity is known as the Rayleigh–Taylor condition. It reads as σðα;tÞ¼−ð∇p2ðzðα;tÞ;tÞ−∇p1ðzðα;tÞ;tÞÞ ·∂⊥ αzðα;tÞ>0; where ∇pjðzðα;tÞ;tÞdenotes the limit gradient of the pressure obtained approaching the boundary in the normal direction inside ΩjðtÞ. An easy linearization around a flat contour ðα;fðα;tÞÞ, allows us to find ft¼1 2HðωÞ where His the Hilbert transform which symbol on the Fourier side is given by b H¼−isign ðξÞ. The equations ω¼−ðρ2−ρ1Þκg μ∂αf; ðlinear MuskatÞ ωt¼2g∂αf; ðlinear water wavesÞ show the parabolicity of the Muskat problem when the denser fluid is below (ρ2>ρ1) and the dispersive character of water waves. 1. There is a wide literature on the Muskat problem and the dynamics of two fluids in a Hele–Shaw cell. There are works considering the case of a viscosity jump neglecting the effect of gravity (7, 8). Local existence in a more general situation (with discontinuous viscosity and density) is shown in ref. 9 and also treated in ref. 10. A different approach to prove local existence can be found in ref. 11 for the setting we are considering in this paper. The Rayleigh–Taylor stability depends upon the sign of ðρ2−ρ1Þ∂αz1ðα;tÞ(11) indicating that the heavier fluid has to be below in the stable case. If the lighter fluid is below, the problem has been shown to be ill-posed (11). Global-existence results for small initial data can be found in refs. 7 and 11–14. For large initial curves and parameterized by ðα;fðα;tÞÞ, there are maximum principles for the L∞and L2norms of f, and decay rates, together with global existence for Lipschitz curves if ‖∂αf‖L∞ð0Þ<1(15, 16, 17). 2. The water waves problem has been extensively considered (see refs. 5 and 18 and references therein). For sufficiently smooth free boundary, the Rayleigh–Taylor condition remains positive with no bottom considerations (19), a fact that was used to prove local existence (19). The Rayleigh–Taylor stability can play a different role for the case of non-“almost”-flat bottom (20). Recently, for small initial data, exponential time of existence has been proven in two dimensions (21) and global existence in the three-dimensional case (two-dimensional interface) (22, 23). 2. Rayleigh–Taylor Breakdown for Muskat This section is devoted to show the main ingredients to prove the Theorem 2.1. We consider the function FðzÞðα;βÞ¼ jβj2 jzðαÞ−zðα−βÞj2;α;β∈R; and in the periodic setting FðzÞðα;βÞ¼ ‖β‖2 2ðcoshðz2ðαÞ−z2ðα−βÞÞ −cosðz1ðαÞ−z1ðα−βÞÞÞ ; α;β∈T;[9] where ‖x‖¼distðx;2πZÞ.IfFðzÞ∈L∞ðR2Þ, then the curve zsatisfies the arc-chord condition. We say that the Rayleigh–Taylor (R-T) of the solution of the Muskat problem breaks down in finite time if for initial data z0satisfying σðα;0Þ¼ðρ2−ρ1Þ∂αz1ðα;0Þ>0 there exists a time t>0for which σðα;tÞis strictly negative in a nonempty open interval. Theorem 2.1. There exists a nonempty open set of initial data in H4, satisfying Rayleigh–Taylor and arc-chord conditions, for which the Rayleigh–Taylor condition of the solution of the Muskat problem [1,2, and 4] breaks down in finite time. After choosing the appropriate tangential term and a integration by parts, the contour equation reads ztðα;tÞ¼ρ2−ρ1 2πPV ZR ðz1ðα;tÞ−z1ðβ;tÞÞ jzðα;tÞ−zðβ;tÞj2ð∂αzðα;tÞ−∂ αzðβ;tÞÞdβ: For a 2πperiodic interface, removing the principal value at infinity, the equation becomes ztðαÞ¼ðρ2−ρ1Þ 4π ×ZT sinðz1ðαÞ−z1ðα−βÞÞð∂αzðαÞ−∂ αzðα−βÞÞ coshðz2ðαÞ−z2ðα−βÞÞ −cosðz1ðαÞ−z1ðα−βÞÞ dβ: [10] From now on, we shall use the periodic configuration. The steps of the proof are as follows: 1. First, for any initial curve z0ðαÞ¼zðα;0Þin H4that satisfy R-T ðρ2−ρ1Þ∂αz1ðα;0Þ>0 and the arc-chord condition then the solution to the Muskat problem zðα;tÞbecomes analytic for 0<t<T. Moreover, zðα;tÞis real analytic in a strip SðtÞ¼fαþiζ:jζj<ctg for t∈ð0;TÞwhere cdepends only on infð0Þ¼inf α ∂αz1ðα;0Þ j∂αzðα;0Þj2: The proof follows by controlling the quantities extended on SðtÞ: FðzÞðαþiζ;β;tÞ and gðαþiζ;tÞby using [9] and formula gðα;tÞ¼ZT ½sinðz1ðα;tÞ−z1ðα−β;tÞÞ∕½coshðz2ðα;tÞ −z2ðα−β;tÞÞ −cosðz1ðα;tÞ−z1ðα−β;tÞÞdβ; respectively. The norms Castro et al. PNAS ∣March 22, 2011 ∣vol. 108 ∣no. 12 ∣4755 MATHEMATICS ‖FðzÞ‖L∞ðSÞðtÞ¼ sup αþiζ∈SðtÞ;β∈T jFðzÞðαþiζ;βÞj; ‖z‖2 L2ðSÞðtÞ¼∑ ZT jzðαict;tÞj2dα; ‖z‖2 HjðSÞðtÞ¼‖z‖2 L2ðSÞðtÞþ∑ ZT j∂j αzðαict;tÞj2dα; for j∈N; infðtÞ¼ inf αþiζ∈SðtÞ ℜ∂αz1ðαþiζ;tÞ j∂αzðαþiζ;tÞj2: Then the quantity ‖z‖2 RTðtÞ¼‖z‖2 H4ðSÞðtÞþ‖FðzÞ‖L∞ðSÞðtÞ þ1∕ðinfðtÞ−c−K∥ℑðgÞ‖H2ðSÞðtÞÞ satisfies d dt ‖z‖RTðtÞ≤C‖z‖k RTðtÞ; for C,K, and kuniversal constants. It yields ‖z‖RTðtÞ≤‖z‖RTð0Þ ð1−C‖z‖k RTð0ÞtÞ1∕k; providing control of the analyticity and T¼1∕ðC‖z‖k RTð0ÞÞ. 2. Second, there is a lower bound on the strip of analyticity, which does not collapse to the real axis as long as the Rayleigh–Taylor is greater than or equal to 0. Then there is a time Tand a solution of the Muskat problem zðα;tÞdefined for 0<t≤Tthat continues analytically into a complex strip if ðρ2−ρ1Þ∂αz1≥0, where Tis either a small constant or it is the first time a vertical tangent appears, whichever occurs first. We redefine the strip SðtÞ¼fαþiζ:jζj<hðtÞ;0<hð0Þg; and the quantity ‖z‖2 S¼‖z‖2 H4ðSÞþ‖FðzÞ‖L∞ðSÞwith this new SðtÞ. For an hðtÞdecreasing [the expression of hðtÞis chosen later], we consider the evolution of the most singular quantity ∑ Zj∂4 αzðαihðtÞ;tÞj2dα: Taking a derivative in t, one finds d dt ∑ Zj∂4 αzðαihðtÞÞj2dα≤h0ðtÞ 10 ∑ ZΛð∂4 αzÞðαihðtÞÞ ·∂4 αzðαihðtÞÞdα −10h0ðtÞZΛð∂4 αzÞðαÞ·∂4 αzðαÞdα þ2∑  ℜZ∂4 αztðαihðtÞÞ ·∂4 αzðαihðtÞÞdα: Estimating in a wise way, one obtains d dt ∑ Zj∂4 αzðαihðtÞÞj2dα≤C‖z‖k SðtÞ −10h0ðtÞZΛð∂4 αzÞðαÞ·∂4 αzðαÞdαþðC‖z‖k SðtÞhðtÞ þ1 10 h0ðtÞÞ ZΛð∂4 αzÞðαihðtÞÞ ·∂4 αzðαihðtÞÞdα: Therefore, choosing hðtÞ¼hð0Þexpð−10CZt 0 ‖z‖k SðrÞdrÞ eliminates the most dangerous term. The other terms are easily controlled, giving finally d dt ∑ Zj∂4 αzðαihðtÞÞj2dα≤C‖z‖kþ2 SðtÞ; which allows us to reach a regime for which the boundary z develops a vertical tangent at time T. 3. Third, it is shown the existence of a large class of analytic curves for which there exist a point where the tangent vector is vertical and the velocity indicates that the curve is going to turn up and reach the unstable regime. For the equation ztðα;tÞ¼uðα;tÞ¼ðu1ðα;tÞ;u2ðα;tÞÞ; that is, a: ∂αz1ðαÞ>0if α≠0;b:∂αz1ð0Þ¼0; c:∂αz2ð0Þ>0;d:∂αu1ð0Þ<0; for analytic functions z1ðαÞand z2ðαÞsuch that zðαÞsatisfies the arc-chord condition. Here we consider the periodic case (being analogous for an open curve vanishing at infinity). We assume that zðαÞis a smooth odd curve satisfying the properties a,b, and c. Differentiating the expression 10 for the horizontal component of the velocity, at α¼0, it yields ð∂αu1Þð0Þ¼Zπ −π ½cosðz1ðβÞÞð∂αz1ðβÞÞ2 þsinðz1ðβÞÞ∂2 αz1ðβÞ∕½coshðz2ðβÞÞ −cosðz1ðβÞÞdβ −Zπ −π sinðz1ðβÞÞ∂αz1ðβÞ½sinðz1ðβÞÞ∂αz1ðβÞ −sinhðz2ðβÞÞð∂αz2ð0Þ−∂ αz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2dβ: Integration by parts provides 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: Therefore, it is easy to obtain that ð∂αu1Þð0Þ¼∂αz2ð0ÞZπ −π ½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβ ¼2∂αz2ð0ÞZπ 0 ½sinðz1ðβÞÞ sinhðz2ðβÞÞ∕½ðcoshðz2ðβÞÞ −cosðz1ðβÞÞÞ2∂αz1ðβÞdβ [11] Expression 11 allows us to determine the sign of ð∂αu1Þð0Þ. One could take z1ðβÞ¼−sinðβÞþβ 4756 ∣www.pnas.org/cgi/doi/10.1073/pnas.1101518108 Castro et al. and construct the function z2ðβÞin the following way: Let β1, β2,β3, and β4be real increasing numbers less than π. We pick z2ðβÞ≤0for β2<β<π,z2ðβÞ<c<0for β2<β<β4, and z 2ðβÞa smooth function with the following properties a: z 2ðβÞis odd;b:ð∂βz 2Þð0Þ>0; c: z 2ðβÞ>0if β∈ð0;β1Þ;d:z  2ðβÞ<0if β∈ðβ1;β2: Also, z 2ðβÞis 2π-periodic. For z2ðβÞ¼bz 2ðβÞ,0≤β≤β2, and b>0, the velocity satisfies 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; where A<0. The constant blarge enough yields ð∂αu1Þ ð0Þ<0. Rectifying the curve on the interval ½β2;β3, it is easy to obtain a smooth curve. Finally, convolving with the heat kernel the vertical component, the curve zðαÞis approximated by an analytic one. 4. Fourth, with the initial data found in 3 and no assumption on the R-T condition, we use a modification of Cauchy– Kowalewski theorems (24, 25) to show that there exists an analytic solution for the Muskat problem in some interval ½−T;T for a small enough T>0. Here we are forced to change substantially the method in ref. 26 because, in this case, the curve cannot be parameterized as a graph, so we have to deal with the arc-chord condition. Then, with fXrgr>0, a scale of Banach spaces given by real functions that can be extended analytically on the complex strip Sr¼fαþiζ∈C:jζj<rgwith norm ‖f‖r¼∑ ZjfðαirÞj2dαþZj∂4 αfðαirÞj2dα; and z0ðαÞa curve satisfying the arc-chord condition and z0ðαÞ∈Xr0for some r0>0, we prove the existence of a time T>0and 0<r<r0so that there is a unique solution to the Muskat problem in Cð½−T;T;XrÞ. This result allows us to find solutions that do not satisfy the R-T but shrink the strip of analyticity. We extend Eq. 10 as follows: ztðαþiζ;tÞ¼Gðzðαþiζ;tÞÞ; with GðzÞðα;tÞ¼ðρ2−ρ1Þ 4π ×ZT sinðz1ðα;tÞ−z1ðα−β;tÞÞð∂αzðα;tÞ−∂ αzðα−β;tÞÞ coshðz2ðα;tÞ−z2ðα−β;tÞÞ −cosðz1ðα;tÞ−z1ðα−β;tÞÞ dβ: For 0<r0<rand the open set Oin Srgiven by O¼fz∈Xr:‖z‖r<R; ‖FðzÞ‖L∞ðSrÞ<R2g;[12] the function Gfor G:O→Xr0is a continuous mapping and there is a constant CR(depending on Ronly) such that ‖GðzÞ‖r≤CR r−r0‖z‖r;[13] ‖Gðz2Þ−Gðz1Þ‖r0≤CR r−r0‖z2−z1‖r;[14] and sup αþiζ∈Sr;β∈T jGðzÞðαþiζÞ−GðzÞðαþiζ−βÞj ≤CRjβj;[15] for z;zj∈O. For initial data z0∈Xr0satisfying arc-chord, we can find a 0<r0 0<r0and a constant R0such that ‖z0‖r0 0<R0and ½coshðz0 2ðαþiζÞ−z0 2ðαþiζ−βÞÞ −cosðz0 1ðαþiζÞ−z0 1ðαþiζ−βÞÞ∕ð‖β‖2Þ>1 R2 0 ;[16] for αþiζ∈Sr0 0. We take 0<r<r0 0and R0<Rto define the open set Oas in [12]. Therefore we can use the classical method of successive approximations: znþ1ðtÞ¼z0þZt 0 GðznðsÞÞds; for G:O→Xr0and 0<r0<r. We assume by induction that ‖zk‖rðtÞ<R; and ‖FðzkÞ‖L∞ðSrÞðtÞ<R for k≤nand 0<t<Twith T¼minðTA;TCK Þand TCK the time obtaining in the proofs in refs 24 and 25. We get ‖znþ1‖rðtÞ<Rthat follows using [13 and 14]. The time TAis to yield ‖Fðznþ1Þ‖L∞ðSrÞðtÞ<R. Then, using the induction hypothesisand[15],wecancontrolthequantitytaking0<TA< ðR−2 0−R−2ÞðC2 Rþ2R0CRÞ−1. 5. Fifth, all the results above allow us to prove that there is a nonempty set of initial data in H4satisfying the arc-chord and R-Tconditions, such that the solution of the Muskat problem reaches the unstable regime: The R-T becomes strictly negative on a nonempty interval. We pick initial data as in 3. We apply the local-existence result in 4 to get an analytic solution zðα;tÞon ½−T;T. Then we consider a time 0<δ<T and a curve ωε δðα;tÞ, solving the Muskat problem with initial datum zðα;−δÞþηϵ δðαÞ. The function ηϵ δhas a small H4 norm, i.e., ‖ωε δð·;−δÞ−zð·;−δÞ‖H4¼‖ηϵ δ‖H4≤ε: The time δis small enough so that ωε δðα;−δÞsatisfies R-T: ðρ2−ρ1Þ∂αðωε δÞ1ðα;−δÞ>0. Then we apply the local-existence result in 1 that ωε δðα;tÞbecomes analytic for some time −δ<t. With 2, we assure the existence and analyticity of the solution even if ∂αðωε δÞ1ðα;tÞ≤0for some time t. Then, we show that both solutions are close in the H4topology as time evolves. We can apply to ωε δthe local-existence result in 4 if it is needed. Then, with δand εsmall enough, we find the desired result. 3. Turning Water Waves In this section, we prove for the water wave problem (ρ1¼0and [1–3and 5]) that with initial data given by a graph ðα;f0ðαÞÞ, the interface reaches a regime in finite time where it only can be parameterized as zðα;tÞ¼ðz1ðα;tÞ;z2ðα;tÞÞ;for α∈R, with ∂αz1ðα;tÞ<0for α∈I, a nonempty interval. Therefore there exists a time twhere the solution of the free boundary problem reparameterized by ðα;fðα;tÞÞ satisfies ‖fα‖L∞ðtÞ¼∞. Theorem 3.1. There exists a nonempty open set of initial data ðα;f0ðαÞÞ, with f0∈H5, such that in finite time tthe solution of Castro et al. PNAS ∣March 22, 2011 ∣vol. 108 ∣no. 12 ∣4757 MATHEMATICS the water waves problem (ρ1¼0and [1–3and 5]) given by ðα;fðα;tÞÞ satisfies ‖fα‖L∞ðtÞ¼∞. The solution can be continued for t>t as zðα;tÞwith ∂αz1ðα;tÞ<0for α∈I, a nonempty interval. In order to prove this theorem, we consider a curve zðαÞ∈H5 with the same properties as in point 3 of the previous section. Then, we pick zðα;tÞ¼zðαÞand ωðα;tÞ¼−∂αz 2ðαÞas a datum for the initial value problem. It is easy to find the same properties for the velocity, because the tangential direction does not affect the evolution. Picking the appropriate cðα;tÞand applying the local-existence result in ref. 18 (note that in this case it is not necessary analyticity, just H5regularity), there exists a solution of the water waves problem with zðα;tÞ∈Cð½t−δ;tþδ;H5Þ, ωðα;tÞ∈Cð½t−δ;tþδ;H4Þ, and δ>0small enough. Then, the initial datum ðz0ðαÞ;ω0ðαÞÞ¼ðα;f0ðαÞ;ω0ðαÞÞ is given by ðzðα;t−δÞ;ωðα;t−δÞÞ. 4. Muskat Breakdown In this section, we show that there exists a smooth initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later it breaks down. The outline of the proof is to construct a curve in the unstable regime which is analytic except in a single point. We show that, as we evolve backward in time, the curve becomes analytic and is as close as we desired (in the Hktopology with klarge enough) to the curve from part 3 of Section 2. Here we will work in the periodic setting and will consider the equation ∂tzðζ;tÞ¼Zw∈ΓþðtÞ sinðz1ðζ;tÞ−z1ðw;tÞÞ coshðz2ðζ;tÞ−z2ðw;tÞÞ −cosðz1ðζ;tÞ−z1ðw;tÞÞ ×ð∂ζzðζ;tÞ−∂ ζzðw;tÞÞdw; [17] where ζ∈ΩðtÞ, ΩðtÞ¼fζ∈C∕2kπ:jℑζj<hðℜz;tÞg; hðx;tÞis a positive periodic function with period 2πand smooth for fixed time t, and ΓðtÞ¼fζ∈C∕2kπ:ζ¼xþihðx;tÞg: This equation is equivalent to [1,2, and 4] for holomorphic functions. In order to prove the result, we will need the following theorem: Theorem 4.1. Let hðx;tÞbe a positive, smooth, and periodic function with period 2πfor fixed time t∈½t0−δ;t0. Let zðx;t0Þbe a curve satisfying the following properties: •z1ðx;t0Þ−xand z2ðx;t0Þare periodic with period 2π; •zðζ;t0Þis real for ζreal; •zðζ;t0Þis analytic in ζ∈Ωðt0Þ; •zðζ;tÞ∈HkðΓðt0ÞÞ with ka large enough integer. •Complex arc-chord condition: jcoshðz2ðζ;t0Þ−z2ðw;t0ÞÞ −cosðz1ðζ;t0Þ−z1ðw;t0ÞÞj ≥½jjℜðζ−wÞjj þ jℑðζ−wÞj2; for ζ,w∈Ωðt0Þ, where ‖x‖¼distanceðx;2kπÞ: •Generalized Rayleigh–Taylor condition: RTðζ;t0Þ>0, where RTðζ;tÞ¼ℜ−2π∂ζz1ðζ;tÞ ð∂ζz1ðζ;tÞÞ2þð∂ζz2ðζ;tÞÞ2ð1þi∂xhðℜζ;tÞÞ−1 þℑPV ZwΓþðtÞ ½sinðz1ðζ;tÞ −z1ðw;tÞÞ∕½coshðz2ðζ;tÞ−z2ðw;tÞÞ −cosðz1ðζ;tÞ−z1ðw;tÞÞdw þi∂thðζ;tÞ ×ð1þi∂xhðℜζ;tÞÞ−1: Then, for small enough δ, there exists a solution for Eq. 17 in the time interval t∈½t0−δ;t0, satisfying •z1ðx;tÞ−xand z2ðx;tÞare periodic with period 2π; •zðζ;tÞis real for ζreal; •zðζ;tÞis analytic in ζ∈Ωðt0Þ; •zðζ;tÞ∈HkðΓðtÞÞ with ka large enough integer. Now, let zðx;tÞbe the solution of the Muskat problem with zðx;0Þ¼z0ðxÞ, where z0ðxÞis the particular initial data from part 3 of the Section 2. We shall define this solution as the unperturbed solution. Let us denote the Rayleigh–Taylor function σ0 1ðx;tÞ≡−2π∂xz1ðx;tÞ ð∂xz1ðx;tÞÞ2þð∂xz2ðx;tÞÞ2: Notice the minus sign in the right-hand side of the previous expression. One can check the following properties of this Rayleigh–Taylor function: 1. σ0 1ð·;tÞis analytic on fxþiy:x∈T;jyj≤cbgwith jσ0 1ðxþiy;tÞj ≤C, for all xþiy as above and for all t≤½0;τ; 2. σ0 1ð0;0Þis real for x∈T,t∈½0;τ; 3. σ0 1has a priori bounded Ck0norm as a function of ðx;tÞ∈T×½0;τ(k0large enough); 4. σ0 1ð0;0Þ¼0; 5. ∂xσ0 1ð0;0Þ¼0; 6. ∂2 xσ0 1ð0;0Þ¼−c2<0; 7. ∂tσ0 1ð0;0Þ¼c1>0. In this setting, we define the following weight functions hðx;tÞ¼A−1ðτ2−t2ÞþðA−1−ðτ−tÞÞ sin2x 2for t∈½τ2;τ: [18] ℏðx;tÞ¼1 4ðA−1τ2þA−1sinx 2ÞþA−2τtþAt sinx 2 t∈½0;τ2;[19] with x∈T. First we choose the parameters Alarge enough and then τsmall enough, then one can show that σ0 1ðx;tÞþ∂thðx;tÞ−A1 2hðx;tÞ≥cτ2for x∈T;t ∈½τ2;τ[20] and σ0 1ðx;tÞþ∂tℏðx;tÞ−A1 2ℏðx;tÞ≥1 2A−2τfor x∈T;t ∈½0;τ2:[21] The inequalities 20 and 21 are one of the main ingredients of the proof of the following results. Theorem 4.2. Let zðx;tÞbe a solution of the Muskat equation in the interval t∈½0;τ. Let hðx;tÞand ℏðx;tÞas in the expressions 18 and 19, and ka large enough integer. Assume that zðx;tÞsatisfies 4758 ∣www.pnas.org/cgi/doi/10.1073/pnas.1101518108 Castro et al. •z1ðx;tÞ−xand z2ðx;tÞare periodic with period 2π; •zðζ;tÞis real for ζreal; •zðζ;tÞis analytic in ζ∈ΩðtÞ; •zðζ;tÞ∈HkðΓðtÞÞ with ka large enough integer. •Complex arc-chord condition: jcoshðz2ðζ;tÞ−z2ðw;tÞÞ −cosðz1ðζ;tÞ−z1ðw;tÞÞj ≥½‖ℜðζ−wÞ‖þjℑðζ−wÞj2; for ζ,w∈ΩðtÞ. Here, in the definition of ΩðtÞand ΓðtÞ, we use hðx;tÞif t∈½τ2;τ and ℏðx;tÞif t∈½0;τ2. Then 1 2 d dt Zw∈ΓþðtÞ j∂k ζzðζ;tÞ−∂ k ζzðζ;tÞj2dℜζ≥−CðAÞλ2; if t∈½τ2;τ Zw∈ΓþðtÞ j∂k ζzðζ;tÞ−∂ k ζzðζ;tÞj2dℜζ≤λ2 and λ≤τ50. In addition, 1 2 d dt Zw∈ΓþðtÞ j∂k ζzðζ;tÞ−∂ k ζzðζ;tÞj2dℜζ≥−CðAÞτ−1λ2; if t∈½0;τ2 Zw∈ΓþðtÞ j∂k ζzðζ;tÞ−∂ k ζzðζ;tÞj2dℜζ≤λ2 and λ≤τ50. This theorem implies that for all γ>0there is ε>0such that Zw∈ΓþðtÞ j∂k ζzðζ;tÞ−∂ k ζzðζ;tÞj2dℜζ≤γ for t∈½0;τif Zw∈ΓþðtÞ j∂k ζzðζ;τÞ−∂ k ζzðζ;τÞj2dℜζ≤ε and zðx;tÞsatisfies the requirements of the theorem. Lemma 4.3. Let zðx;tÞbe a solution of the Muskat problem satisfying the requirements of Theorem 4.2 and close enough to the unperturbed solution in t∈½0;τ. Let hðx;tÞand ℏðx;tÞbe as in [18] and [19] with a suitable choice of Aand τ. Then zðx;tÞsatisfies the generalized Rayleigh–Taylor condition in t∈½0;τ. In particular, the unperturbed solution satisfies the generalized Rayleigh–Taylor condition in t∈½0;τ Theorems 4.1 and 4.2 and Lemma 4.3 allow us to achieve the desired result. Indeed we can choose a curve zðx;τÞsuch that Zζ∈Γ j∂k ζzðζ;τÞ−∂ k ζzðζ;τÞj2dℜζ≤ε; with 0<ε<ε0(ε0small enough), satisfying the generalized Rayleigh–Taylor condition by Lemma 4.3 and satisfying the rest of the hypothesis of Theorem 4.1. Because hð0;τÞ¼0,zðx;tÞis allowed to be nonanalytic at x¼0[maybe zðx;τÞ∈HkðTÞbut zðx;τÞ∉Hkþ1ðTÞ]. By Theorem 4.1, there is a solution zðx;tÞ, analytic in ΩðtÞ, for some interval t∈½τ−δ;τwith small enough δ and for all ε. By Theorem 4.2, we can choose εsmall enough in such a way that, by Lemma 4.3, zðx;τ−δÞsatisfies the generalized Rayleigh–Taylor condition. Then we can go further the time τ−δ. Iterating this argument, we find we can extend zðx;tÞto be a solution of the Muskat problem, analytic in ΩðtÞfor all t∈½0;τ and as close as we want to the unperturbed solution. ACKNOWLEDGEMENTS A.C., D.C., and F.G. were partially supported by Grant MTM2008-03754 of the Ministerio de Ciencia e Innovación (MCINN) (Spain) and Grant StG-203138CDSIF of the European Research Council. C.F. was partially supported by National Science Foundation (NSF) Grant DMS-0901040 and Office of Naval Research Grant ONR00014-08-1-0678. F.G. was partially supported by NSF Grant DMS-0901810. M.L.-F. was partially supported by Grants MTM2008-03541 and MTM2010-19510 of the MCINN (Spain). 1. Muskat M (1934) Two fluid systems in porous media. The encroachment of water into an oil sand. Physics 5:250–264. 2. Saffman P-G, Taylor G (1958) The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc R Soc London Ser A 245:312–329. 3. Howison S (2000) A note on the two-phase Hele-Shaw problem. J Fluid Mech 409:243–249. 4. Hele-Shaw HS (1898) On the motion of a viscous fluid between two parallel plates. Trans Royal Inst Nav Archit, London 40:21. 5. Bardos C, Lannes D (2010) Mathematics for 2D interfaces. ArXiv:1005.5329. 6. Rayleigh J-W-S (1878) On the instability of jets. Proc Lond Math Soc 10:4–13. 7. Siegel M, Caflisch R, Howison S (2004) Global existence, singular solutions, and ill-posedness for the muskat problem. Commun Pur Appl Math 57:1374–1411. 8. Yi F (1996) Local classical solution of Muskat free boundary problem. J Partial Diff Eqs 9:84–96. 9. Córdoba A, Córdoba D, Gancedo F (2011) Interface evolution: The Hele-Shaw and Muskat problems. Ann Math 173:477–542. 10. Ambrose D (2004) Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur J Appl Math 15:597–607. 11. Córdoba D, Gancedo F (2007) Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun Math Phys 273:445–471. 12. Constantin P, Pugh M (1993) Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6:393–415. 13. Yi F (2003) Global classical solution of Muskat free boundary problem. J Math Anal Appl 288:442–461. 14. Escher J, Matioc B-V (2010) On the parabolicity of the Muskat problem: Wellposedness, fingering, and stability results. ArXiv:1005.2512. 15. Córdoba D, Gancedo F (2009) A maximum principle for the Muskat problem for fluids with different densities. Comm Math Phys 286:681–696. 16. Constantin P, Córdoba D, Gancedo F, Strain R-M (2010) On the global existence for the for the Muskat problem. ArXiv:1007.3744v1. 17. Córdoba D, Gancedo F, Orive R (2008) A note on the interface dynamics for convection in porous media. Physica D 237:1488–1497. 18. Córdoba A, Córdoba D, Gancedo F (2010) Interface evolution: Water wave in 2D. Adv Math 223:120–173. 19. Wu S (1997) Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent math 130:39–72. 20. Lannes D (2005) Well-posedness of the water-waves equations. J Am Math Soc 18:605–654. 21. Wu S (2009) Almost global wellposedness of the 2-D full water wave problem. Invent Math 177:45135. 22. Germain P, Masmoudi N, Shatah J (2009) Global solutions for the gravity water waves equation in dimension 3. C R Math Acad Sci 347:897–902. 23. Wu S (2009) Global well-posedness of the 3-D full water wave problem. ArXiv:0910.2473. 24. Nirenberg L (1972) An abstract form of the nonlinear Cauchy-Kowalewski theorem. J Differ Geom 6:561–576. 25. Nishida T (1977) A note on a theorem of Nirenberg. J Differ Geom 12:629–633. 26. Sulem C, Sulem P-L, Bardos C, Frisch U (1981) Finite time analyticity for the twoand three-dimensional Kelvin-Helmholtz instability. Commun Math Phys 80:485–516. Castro et al. PNAS ∣March 22, 2011 ∣vol. 108 ∣no. 12 ∣4759 MATHEMATICS