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

arXiv:1102.1902v2 [math.AP] 10 Jun 2011 Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves ´ Angel Castro, Diego C´ordoba, Charles Fefferman, Francisco Gancedo and Mar´ıa L´opez-Fern´andez. June 14, 2011 Abstract The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning. 1 Introduction The Muskat problem [26] models the evolution of an interface between two fluids of different characteristics in porous media by means of Darcy’s law: µ κu=−∇p−(0,gρ),(1) where (x, t)∈R2×R+,u= (u1(x, t), u2(x, t)) is the incompressible velocity (i.e. ∇ · u= 0), p=p(x, t) is the pressure, µ(x, t) is the dynamic viscosity, κis the permeability of the isotropic medium, ρ=ρ(x, t) is the liquid density, and g is the acceleration due to gravity. More precisely, the interface separates the domains Ω1and Ω2defined by (µ, ρ)(x1, x2, t) = (µ1, ρ1), x ∈Ω1(t) (µ2, ρ2), x ∈Ω2(t) = R2−Ω1(t), and µ1, µ2, ρ1, ρ2are constants. This physical situation is also related to the evolution of two fluids of different characteristics in a Hele-Shaw cell [22], due to the fact that the laws which model both phenomena are mathematically analogous [31]. This paper is concerned with the case µ1=µ2which provides weak solutions of the following transport equation ρt+u· ∇ρ= 0, ρ0=ρ(x, 0), x ∈R2,(2) 1 where initially the scalar ρ0is given by ρ0=ρ(x1, x2,0) = ρ1in Ω1(0) = {x2> f0(x1)} ρ2in Ω2(0) = {x2< f0(x1)}.(3) Let the free boundary be parametrized by ∂Ωj(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α∈R} where z(α, t)−(α, 0) is 2π-periodic in the space parameter αor, an open contour vanishing at infinity lim α→±∞(z(α, t)−(α, 0)) = 0 with initial data z(α, 0) = z0(α) = (α, f0(α)). From Darcy’s law, we find that the vorticity is concentrated on the free boundary z(α, t), and is given by a Dirac distribution as follows: ∇⊥·u(x, t) = ω(α, t)δ(x−z(α, t)), with ω(α, t) representing the vorticity strength i.e. ∇⊥·uis a measure defined by <∇⊥·u, η >=Zω(α, t)η(z(α, t))dα, with η(x) a test function. Then z(α, t) evolves with an incompressible velocity field coming from the Biot-Savart law: u(x, t) = ∇⊥∆−1∇⊥·u(x, t). As (x, t) approaches a point z(α, t) on the contour the velocity uagrees, modulo tangential terms, with the Birkhoff-Rott integral: BR(z, ω)(α, t) = 1 2πPV Z(z(α, t)−z(β, t))⊥ |z(α, t)−z(β, t)|2ω(β, t)dβ. This yields an appropriate contour dynamics system: zt(α, t) = BR(z, ω)(α, t) + c(α, t)∂αz(α, t),(4) where the term c represents the change of parametrization and does not modify the geometric evolution of the curve [24]. The well-posedness is not guaranteed in general, in fact such a result turns out to be false for some initial data. Rayleigh [30] and Saffman-Taylor [31] gave a condition that must be satisfied for the linearized model in order to have a solution locally in time, namely that the normal component of the pressure gradient jump at the interface has to have a distinguished sign. This is known as the Rayleigh-Taylor condition: σ(α, t) = −(∇p2(z(α, t), t)− ∇p1(z(α, t), t)) ·∂⊥ αz(α, t)>0, 2 where ∇pj(z(α, t), t) denotes the limit gradient of the pressure obtained approaching the boundary in the normal direction inside Ωj(t). We call σ(α, t) the Rayleigh-Taylor of the solution z(α, t). Understanding the problem as weak solutions of (1-2) plus the incompressibility of the velocity, we find that the continuity of the pressure (p2(z(α, t), t) = p1(z(α, t), t)) follows as a mathematical consequence, making unnecessary to impose it as a physical assumption (for more details see [13] and [11]). For the surface tension case, there is a jump discontinuity of the pressure across the interface which is modeled to be equal to the local curvature times the surface tension coefficient: p2(z(α, t), t)−p1(z(α, t), t)) = τκ(α, t). This is known as the Laplace-Young condition, which makes the initial value problem more regular. Then there are no instabilities [18] but fingering phenomena arise [29, 19]. By means of Darcy’s law, we can find the following formula for the difference of the gradients of the pressure in the normal direction and the strength of the vorticity: σ(α, t) = (ρ2−ρ1)∂αz1(α, t) ω(α, t) = −(ρ2−ρ1)∂αz2(α, t).(5) Above g is taken equal to 1 for the sake of simplicity. Then, if we choose an appropriate term c in equation (4) (see section 2 below), the dynamics of the interface satisfies zt(α, t) = ρ2−ρ1 2πPV Z(z1(α, t)−z1(β, t)) |z(α, t)−z(β, t)|2(∂αz(α, t)−∂αz(β, t))dβ. (6) A wise choice of parametrization of the curve is to have ∂αz1(α, t) = 1 (for more details see [13]). This yields the denser fluid below the less dense fluid if ρ2> ρ1and therefore the Rayleigh-Taylor condition holds as long as the interface is a graph. This fact has been used in [13] to show local existence in the stable case (ρ2> ρ1), together with ill-posedness in the unstable situation (ρ2< ρ1). Local existence for the general case (µ16=µ2) is shown in [11], which was also treated in [34, 1]. ¿From (6) it is easy to find the evolution equation for the graph: ft(α, t) = ρ2−ρ1 2πPV ZR (α−β) (α−β)2+ (f(α, t)−f(β, t))2(∂αf(α, t)−∂αf(β, t))dβ, f(α, 0) = f0(α). (7) The above equation can be linearized around the flat solution to find the following nonlocal partial differential equation ft(x, t) = −ρ2−ρ1 2Λf(x, t), f(x, 0) = f0(x), x ∈R, 3 where the operator Λ is the square root of the Laplacian. This linearization shows the parabolic character of the system. Furthermore the stable system gives a maximum principle kfkL∞(t)≤ kfkL∞(0) [14]; decay rates are obtained for the periodic case: kfkL∞(t)≤ kf0kL∞e−Ct, and also for the case on the real line (flat at infinity): kfkL∞(t)≤kf0kL∞ 1 + Ct . There are several results on global existence for small initial data (small compared to 1 in several norms more regular than Lipschitz [9, 35, 32, 13, 19]) taking advantage of the parabolic character of the equation for small initial data. In [8] it is shown in the stable case that global existence for solutions holds if the first derivative of the initial data is smaller than an explicitly computable constant greater than 1/5. Furthermore, if kf0kL∞<∞and k∂xf0kL∞<1, then there exists a global-in-time solution that satisfies f(x, t)∈C([0, T]×R)∩L∞([0, T]; W1,∞(R)), for each T > 0. In particular fis Lipschitz continuous. Moreover, equation (7) yields an L2decay: kfk2 L2(t) + ρ2−ρ1 2πZt 0 ds ZR dα ZR dx ln 1 + f(x, s)−f(α, s) x−α2=kf0k2 L2, which does not imply, for large initial data, a gain of derivatives in the system (see [8]). We will see below that the solutions to the Muskat problem with initial data in H4become real analytic immediately despite the weakness of the above decay formula. The main result we present here is: Theorem 1.1 There exists a nonempty open set of initial data in H4with Rayleigh-Taylor strictly positive σ > 0such that in finite time the Rayleigh-Taylor σ(α, t)of the solution of (6) is strictly negative for all αin a nonempty open interval. The geometry of this family of initial data is far from trivial: numerical simulations performed in [16] show that there exist initial data with large steepness for which a regularizing effect appears. In fact, as will be explained in Section 2, the first evidence of a change of sign in the Rayleigh-Taylor has been experimentally found in a model with two interfaces. We proceed as follows: First, in section 3, we assume initial conditions at time t=t0that satisfy the Rayleigh- Taylor (σ > 0) and the arc-chord condition, and for which the boundary zinitially belongs to H4. Let C1be the constant in the arc-chord condition, let C2be an upper bound for the H4 norm of the initial data and let c3be a lower bound for σ. Then there exists t1> t0, with t1 depending only on C1, C2, c3, such that the Muskat problem has a solution for time t∈[t0, t1], satisfying also the arc-chord and Rayleigh-Taylor conditions. Moreover, for t0< t ≤t1, the 4 solution z(α, t) is real analytic in a strip S(t) = {α+iζ :|ζ| ≤ c(t−t0)}, where cdepends only on C1, C2, c3. Our goal in section 4 is to show that the region of analyticity does not collapse to the real axis as long as the Rayleigh-Taylor is greater than or equal to 0. This allows us to reach a regime for which the boundary zdevelops a vertical tangent. Section 5 is devoted to showing the existence of a large class of analytic curves for which there exists a point where the tangent vector is vertical and the velocities indicate that the curves are going to turn over and reach the unstable regime for a small time. Plugging these initial data into a Cauchy-Kowalewski theorem indicates that the analytic curves turn over. Therefore the unstable regime is reached. Finally, in section 6, a perturbative argument allows us to conclude that we can find curves in H4close enough to the special class of analytic curves described in Section 5, which satisfy the arc-chord and Rayleigh-Taylor conditions. Then we can show the existence of the curves passing the critical time and actually turning over. Therefore the unstable regime is reached for an entire H4−neighborhood of initial data. Remark 1.2 In a forthcoming paper (see [5]) we will exhibit a particular initial datum for which we will show that once the curve reaches the unstable regime the strip of analyticity collapses in finite time and the solution breaks down. In section 8 we provide a very brief sketch of our proof of breakdown of smoothness for the Muskat equation. These results were announced in [6]. Remark 1.3 The same approach can be done for the water waves problem, which shows that, starting with some initial data given by (α, f0(α)), in finite time the interface reaches a regime in which it is no longer a graph. Therefore there exists a time t∗where the solution of the free boundary problem parametrized by (α, f(α, t)) satisfies kfαkL∞(t∗) = ∞(see section 7). This scenario is known in the literature as wave breaking [7] and there are numerical simulations showing this phenomenon [4]. Remark 1.4 We conjecture that a result analogous to Theorem 1.1 holds, in which surface tension is included. We may simply use the same initial data as in Theorem 1.1, and take the coefficient of surface tension to be very small. The solutions are presumably changed only slightly by the surface tension (although we do not have a proof of this plausible assertion). Consequently, we believe that Muskat solutions with small surface tension can turn over. A similar remark applies to water waves (see theorem 7.1). There exist initial data for which water waves with surface tension turn over. A rigorous proof may be easily supplied, since local existence (backwards and forward in time) is known for water waves with surface tension (see [3]). 2 The contour equation and numerical simulations Here we present the evolution equation in terms of the free boundary which is going to be used throughout the paper, and the numerical experiment that motivated the Theorem. 5 2.1 The equation of motion By Darcy’s law: ∇⊥·u=−(ρ2−ρ1)∂αz2(α)δ(x−z(α)), and Biot-Savart yields zt(α) = −(ρ2−ρ1) 2πPV ZR (z(α)−z(α−β))⊥ |z(α)−z(α−β)|2∂αz2(α−β)dβ. (8) For the first coordinate above one finds (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 the identity PV ZR ∂βln(|z(α)−z(α−β)|2)dβ = 0. Therefore zt(α) = −(ρ2−ρ1) 2πPV ZR (z1(α)−z1(α−β)) |z(α)−z(α−β)|2∂αz(α−β)dβ. Here we point out that in the Biot-Savart law the perpendicular direction appears, but after the above integration by parts, we only see the tangential direction. Adding the tangential term (ρ2−ρ1) 2πPV ZR (z1(α)−z1(α−β)) |z(α)−z(α−β)|2dβ∂αz(α), we find that the contour equation is given by zt(α) = (ρ2−ρ1) 2πPV ZR z1(α)−z1(α−β) |z(α)−z(α−β)|2(∂αz(α)−∂αz(α−β))dβ. For the 2πperiodic interface the equation becomes zt(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 order to see (9) we take z(α) = z1(α) + iz2(α); it is easy to rewrite (8) as follows; zt(α) = −(ρ2−ρ1) 2πi P V ZR ∂αz2(β) z(α)−z(β)dβ. The classical identity 1 z+X k≥1 z z2−(2πk)2=1 2 tan(z/2) 6 allows us to conclude that zt(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, where T=R/2πZ. Analogously, using the equality (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 the appropriate tangential term, we obtain equation (9). 2.2 The scenario motivated by the numerics Our investigations started with the idea that interesting new phenomena may arise if we study three fluids, separated by two interfaces. Careful numerical studies indicated that one of the interfaces may turn over. In attempting to prove analytically the turnover indicated by the numerics, we discovered that a turnover can occur also for a single interface, i.e., for the Muskat problem. This section describes one of our numerical experiments. Proceeding as in the preceding section, one can derive the equations modeling the evolution of two interfaces separating three fluids with different densities ρj(j= 1,2,3). More precisely, assume that both interfaces can be parametrized by graphs (α, f(α, t)) and (α, g(α, t)), with flying above g. These equations read in the periodic case, cf. [16, 15] (this scenario has been recently also considered in [20]), ft(α, t) = ¯ρ1I[f(·, t), f(·, t)] + ¯ρ2I[f(·, t), g(·, t)], f(α, 0) = f0(α), gt(α, t) = ¯ρ2I[g(·, t), g(·, t)] + ¯ρ1I[g(·, t), f(·, t)], g(α, 0) = g0(α),(10) where ¯ρj= (ρj+1 −ρj)/(4π), j= 1,2, and, for given functions u(α), v(α), I[u, v] := PV ZT (∂αu(α)−∂αv(α−β)) tan(β/2)(1 −tanh2((u(α)−v(α−β))/2)) tan2(β/2) + tanh2((u(α)−v(α−β))/2) dβ. (11) The first terms I[f(·, t), f(·, t)] and I[g(·, t), g(·, t)] in (10) give the velocity of a unique interface. The cross terms I[f(·, t), g(·, t)] and I[g(·, t), f(·, t)] take into account the interaction of the two interfaces, and their contribution is getting bigger when the curves are getting closer. This, together with the diffusive behavior reported in [16] for the equation ft(α, t) = ¯ρ1I[f(·, t), f(·, t)], f(α, 0) = f0(α),(12) and the mean conservation for fand g, motivate the choice of the following initial data, in the hope that some non regularizing effect arises from the interaction of the two interfaces; f0(α) =      0.1−sin3π(α−M1+r1) 2r1,if α∈[M1−r1, M1+r1], 0.1,otherwise (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 Figure 1: Left: Solutions to (10) with initial data (13)-(14) at times t= 0 (dashed blue), t= 3.46 ·10−4(red points) and t= 7.66 ·10−4(black). Right: Solutions at t= 1.04 ·10−3 (dashed red) and t= 1.84 ·10−3(black) and g0(α) =      sin3π(α−M2+r2) 2r23 −0.92,if α∈[M2−r2, M2+r2], g0(α) = −0.92,otherwise. (14) The choice of parameters M1=π+ 0.1, r1= 0.7, M2=π/1.2, r2= 0.3, ¯ρ1= 20πand ¯ρ2=π/20, yielded a strong growth of the derivative in the the lower interface as the two curves approach, as shown in Figure 1. Moreover, after introducing a small modification in the lower interface so that the tangent at a certain point becomes actually infinite, and evaluating the normal velocity relative to this point along the modified curve, we obtain the result plotted in Figure 2. This graphic clearly indicates that the velocity field is forcing the interface to turn over. The numerical approximation of (10) addresses as a main difficulty the absolute lack of knowledge about the behavior of the solutions to (10). Indeed, the goal of our experiments is precisely the search for some singular behavior. The nonlocal terms make the computations expensive and special care has to be taken in order to evaluate the integrands in a neighborhood of β= 0. For this, we used Taylor expansions locally and computed exactly the principal value. In this situation, adaptivity is strongly indicated, both in space and time, since a good indicator of a singular behavior will be given either by a sudden accumulation of spatial nodes or a sudden reduction of the time steps. In order to attain the highest resolution in the integration of (10) and compute the solutions shown in Figure 1, cubic spline interpolation of the curves f(·, t) and g(·, t) with periodic boundary conditions was used. This provides a C2interpolant of each interface at every time and allows, in particular, the evaluation of the convolution terms at any β∈[0,2π]. Then, adaptive quadrature can be applied to approximate the integrals and evaluate the derivative at any time. In the experiments reported, adaptive Lobatto quadrature was used, 8 2.8 2.9 3 3.1 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 2: Zoom of the interface, modified so that its tangent is vertical at a single point P; and the normal velocity along the curve, minus that at P, scaled by a factor of 100. by means of the MATLAB routine quadl. For the time integration, the embedded Runge– Kutta formula due to Dormand and Prince, DOPRI5(4), was implemented, since the problem was not found to be particularly stiff, see for instance [21]. The time stepping was combined with a spatial node redistribution after every successful step. For the redistribution of the spatial nodes an algorithm following [17] was implemented, with some modifications taking into account that both interfaces are graphs. For several tolerance requirements and different choices of the parameters involved in the full adaptive routine, the integration always failed at a certain critical time, suggesting the explosion of the derivative at a certain point of the lower interface and the lack of validity of (10), once this curve stops being a graph. The phenomenon described above and the explicit representations of the maximum of the solutions derived in [14], motivated the search for special initial data which allowed us to understand that this behavior also arises in the one-interface case. 3 Instant Analyticity Here we show the main estimates that provide local-existence and instant analyticity for a single curve that satisfies initially the arc-chord and Rayleigh-Taylor conditions. We consider the function F(z)(α, β) = β2 |z(α)−z(α−β)|2, α, β ∈R, and in the periodic setting F(z)(α, β) = ||β||2 2(cosh(z2(α)−z2(α−β)) −cos(z1(α)−z1(α−β))), α, β ∈T, where ||x|| =dist(x, 2πZ). If F(z)∈L∞then we say that the curve satisfies the arc-chord condition, and the L∞ norm of Fis called the arc-chord constant. 9 using (23), and therefore (see section 9 in [11] for more details) d dtkzkRT (t)≤C(kzkRT (t) + 1)k. It follows that kzkRT (t)≤kzkRT (0) + 1 (1 −C(kzkRT (0) + 1)kt)1/k −1, providing the a priori estimate with Cand kuniversal constants. We approximate the problem as follows zε t(α, t) = 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), where φε(x) = φ(α/ε)/ε,φis the heat kernel and ε > 0. Picard’s theorem yields the existence of a solution zε(α, t) in C[0, Tε); H4which is analytic in the whole space for z0satisfying the arc-chord condition and εsmall enough. Using the same techniques we have developed above we obtain a bound for zε(α, t) in H4in the strip S(t) for a small enough Twhich is independent of ε. We need arc-chord, R-T, z0∈H4and c−m(0) <0. Then we can pass to the limit. 4 Getting all the way to breakdown of Rayleigh-Taylor This section is devoted to proving the following theorem. Theorem 4.1 Let z(α, 0) = z0(α)be an analytic curve in the strip S={α+iζ ∈C:|ζ|< h(0)}, with h(0) >0and satisfying: •The arc-chord condition, F(z0)(α+iζ, β)∈L∞(S×R) •The Rayleigh-Taylor condition, ∂αz0 1(α)>0. •The curve z0(α)is real for real α. •The functions z0 1(α)−αand z0 2(α)are periodic with period 2π. •The functions z0 1(α)−αand z0 2(α)belong to H4(∂S). Then there exist a time Tand a solution of the Muskat problem z(α, t)defined for 0< t ≤T that continues analytically into some complex strip for each fixed t∈[0, T ]. Here Tis either a small constant depending only on ||z0||Sor it is the first time a vertical tangent appears, whichever occurs first. Thus our Muskat solution is analytic as long as ∂αz1(α, t)≥0. We will use the following: 16 Lemma 4.2 Let ϕ(α±iζ) = PN k=−NAkeikα∓kζ. Then, for ζ > 0, we have ∂ ∂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) where Λϕ(α±iζ) = PN k=−N|k|Akeikαe∓kζ. Proof: First we shall compute the left hand side in the frequency space: X ±ZT |ϕ(α±iζ)|2dα = 4π N X k=−N |Ak|2cosh(2|k|ζ). On the other hand we have that 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. Differentiating in ζwe obtain ∂ ∂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 for any ζ > 0. Corollary 4.3 Let ϕ(α±iζ, t) = PN k=−NAk(t)eikαe∓kζ and h(t)>0be a decreasing function of t. Then ∂ ∂t X ±ZT |ϕ(α±ih(t))|2dα ≤h′(t) 10 X ±ZT Λϕ(α±ih(t))ϕ(α±ih(t))dα −10h′(t)ZT Λϕ(α)ϕ(α)dα + 2ℜX ±ZT ϕt(α±ih(t))ϕ(α±ih(t))dα. This corollary allows us to prove Theorem 4.1. Proof (Theorem 4.1): The norms kzkHk(S)and kzkSare defined as before using the new strip S(t) defined by S(t) = {α+iζ ∈C:|ζ|< h(t)}, where h(t) is a positive decreasing function of t. 17 We use the Galerkin approximation of equation (15), i.e. ∂tz[N](ζ, t) = ΠN[J[z[N]]](ζ, t), where ζ∈S(t), ΠNwill be specified below, and J[z](α, t) = Zπ −π sin(z1(α)−z1(β))(∂αz(α)−∂αz(β)) cosh(z2(α)−z2(β)) −cos(z1(α)−z1(β))dβ. We impose the initial condition z[N](α, 0) = z[N](α). Here, for a large enough positive integer N, we define z[N](α, 0) from z0(α) by using the projection ΠN: ∞ X −∞ Akeikα 7→ N X −N Akeikα. We define z[N](α) by stipulating that z[N] 1(α)−α= ΠN[z0 1(α)−α] and z[N] 2(α) = ΠN[z0 2(α)]. For Nlarge enough, the functions z[N](α, 0) satisfy the arc-chord and Rayleigh-Taylor condition. We shall consider the evolution of the most singular quantity X ±ZT |∂4 αz[N](α±ihN(t), t)|2dα, where hN(t) is a smooth positive decreasing function on t, with hN(0) = h(0), which will be given below. Also we denote SN(t) = {α+iζ ∈C:|ζ|< hN(t)}. ¿From now on, we will drop the dependency on Nfrom z[N]and hN(t) in our notation. We will return to the previous notation in the discussion below at the end of the section. Taking the derivative with respect to tyields d dt Z α∈T∂4 αzµ(α±ih(t), t) 2dα = 2ℜZ α∈T ∂4 αzµ(α±ih(t), t)∂t∂4 αzµ(α±ih(t), t) + ih′(t)∂5 αzµ(α±ih(t), t)dα = 2ℜZ α∈T ∂4 αzµ(α±ih(t), t)∂4 αΠN[Jµ[z]](α±ih(t), t) + ih′(t)∂5 αzµ(α±ih(t), t)dα 18 = 2ℜZ α∈T ∂4 αzµ(α±ih(t), t)ΠN[∂4 αJµ[z]](α±ih(t), t) + ih′(t)∂5 αzµ(α±ih(t), t)dα = 2ℜZ α∈T ∂4 αzµ(α±ih(t), t)∂4 αJµ[z](α±ih(t), t) + ih′(t)∂5 αzµ(α±ih(t), t)dα, since ∂4 αzµ(α±ih(t), t) is a trigonometric polynomial in the range of ΠN. Here µ= 1, 2. Using the above corollary we have that d dt X ±Z α∈T∂4 αzµ(α±ih(t), t) 2dα ≤h′(t) 10 X ±ZT Λ(∂4 αzµ)(α±ih(t)) ·∂4 αzµ(α±ih(t))dα −10h′(t)ZT Λ(∂4 αzµ)(α)·∂4 αzµ(α)dα + 2 X ± ℜZT ∂4 αJµ[z](α, t)(α±ih(t)) ·∂4 αzµ(α±ih(t))dα. We shall study in detail the most singular term in ∂4J[z](α, t), i.e. ∂4 αJ[z](α±ih(t), t) = Zπ −π sin(z1(α±ih(t), t)−z1(β, t))(∂5 αz(α±ih(t), t)−∂5 βz(β, t)) cosh(z2(α±ih(t), t)−z2(β, t)) −cos(z1(α±ih(t), t)−z1(β, t))dβ + l.o.t ≡X+ l.o.t., where ||l.o.t||L2(T)≤C(||z||S(t) + 1)k(see [11] and our previous discussion of (18)). We split Xin to the following terms X=Zπ −π K(α±ih(t), β)(∂5 αz(α±ih(t), t)−∂5 βz(β, t))dβ +σ(α±ih(t), t)Zπ −π cot α±ih(t)−β 2(∂5 αz(α±ih(t), t)−∂5 βz(β, t))dβ ≡X1+X2, where K(α, β) = sin(z1(α, t)−z1(β, t)) cosh(z2(α, t)−z2(β, t)) −cos(z1(α, t)−z1(β, t)) −∂αz1(α, t) (∂αz2(α, t))2+ (∂αz1(α, t))2cot α−β 2 and ˜σ(α, t) = ∂αz1(α, t) (∂αz1(α, t))2+ (∂αz2(α, t))2. Let us denote Γ±(t) = {ζ∈C:ζ=α±ih(t), α ∈T}. 19 Since K(α, β) is a holomorphic function in αand β, with α, β ∈S(t), for fixed twe have that X1=Zπ π K(α±ih(t), β)∂5 αz(α±ih(t), t)dβ −Zπ π K(α±ih(t), β)∂5 αz(β, t)dβ ≡X11 +X12, and integration by parts shows that the term X12 satisfies ||X12||L2(T)≤C(||z||S+ 1)k. In addition, we can write X11 as follows X11 =Zw∈Γ±(t) K(α±ih(t), w)∂5z(α±ih(t), t)dw =P.V. Zw∈Γ±(t) sin(z1(α±ih(t), t)−z1(w, t))∂5z(α±ih(t), t) cosh(z2(α±ih(t), t)−z2(w, t)) −cos(z1(α±ih(t), t)−z1(w, t))dw −∂5z(α±ih(t), t)σ(α±ih(t), t)P.V. Zw∈Γ±(t) cot α±ih(t)−w 2dw =P.V. Zπ −π sin(z1(α±ih(t), t)−z1(β±ih(t), t))∂5z(α±ih(t), t) cosh(z2(α±ih(t), t)−z2(β±ih(t), t)) −cos(z1(α±ih(t), t)−z1(β±ih(t), t))dβ, As before we call f(α±ih(t), t) =P.V. Zπ −π sin(z1(α±ih(t), t)−z1(β±ih(t), t)) cosh(z2(α±ih(t), t)−z2(β±ih(t), t)) −cos(z1(α±ih(t), t)−z1(β±ih(t), t))dβ =P.V. Zπ −π sin(z1(α±ih(t), t)−z1(α±ih(t)−β, t)) cosh(z2(α±ih(t), t)−z2(α±ih(t)−β, t)) −cos(z1(α±ih(t), t)−z1(α±ih(t)−β, t))dβ. Thus X11 =∂5z(α±ih(t), t)f(α±ih(t), t). Also we can write X2in the following way; X2=˜σ(α±ih(t), t)Zπ −π cot α±ih(t)−β 2(∂5 αz(α±ih(t), t)−∂5 βz(β, t))dβ =˜σ(α±ih(t), t)Zw∈Γ±(t) cot α±ih(t)−w 2(∂5 αz(α±ih(t), t)−∂5 βz(w, t))dw =˜σ(α±ih(t), t)P.V. Zw∈Γ±(t) cot α±ih(t)−w 2∂5 αz(α±ih(t), t)dw −˜σ(α±ih(t), t)P.V. Zw∈Γ±(t) cot α±ih(t)−w 2∂5 αz(w, t)dw 20 =−˜σ(α±ih(t), t)P.V. Zπ −π cot α−β 2∂5 αz(β±ih(t), t)dβ =−˜σ(α±ih(t), t)P.V. Zπ −π 1 2csc2α−β 2(∂4 αz(α±ih(t), t)−∂4 αz(β±ih(t), t))dβ and finally X2=−2π˜σ(α±ih(t), t)(Λ∂4 αz)(α±ih(t), t). Then we find two dangerous terms I1= 2ℜZT f(α±ih(t), t)(∂4 αzµ)(α±ih(t)) ·(∂5 αzµ)(α±ih(t))dα and I2=−4πℜZT ˜σ(α±ih(t), t)Λ(∂4 αzµ)(α±ih(t)) ·∂4 αzµ(α±ih(t))dα. The rest can be bounded by C(kzkS+ 1)k(t) as in the previous section. In order to bound I1and I2we use the following commutator estimate: ||Λ1 2(fg)−fΛ1 2g||L2(T)≤C||Λ1+εf||L2(T)||g||L2(T),(28) for f(α) = PN −Nfkeikx and g(α) = PN −Ngkeikx, where ε > 0 and Cdoes not depend on N. The proof of (28) will be left to the reader. First we estimate I1. We denote γ=α+ih(t). I1=2ℜZπ −π f(γ, t)∂4 αzµ(γ, t)∂5 αzµ(γ, t)dα =2 Zπ −π ℜ(f(γ)) ℜ(∂4 αzµ(γ, t))∂α(ℜ(∂4 αzµ(γ, t))) + ℑ(∂4 αzµ(γ, t))∂α(ℑ(∂4 αzµ(γ, t)))dα −2Zπ −π ℑ(f(γ)) ℜ(∂4 αzµ(γ, t))∂α(ℑ(∂4 αzµ(γ, t))) + ℑ(∂4 αzµ(γ, t))∂α(ℜ(∂4 αzµ(γ, t)))dα ≡I11 +I12. Integrating by parts we have that ||I11||L2(T)≤C(||z||S+ 1)k. In order to estimate I12 we note that f(γ, t) is real for real γ. Then ℑ(f(α±ih(t), t)) = h(t)˜ f±(α, t), where || ˜ f±||H2(T)≤C(||z||S(t) + 1)k. 21 Then we can write Zπ −π ℑ(f(γ))ℜ(∂4 αzµ(γ, t))∂α(ℑ(∂4 αzµ(γ, t)))dα =h(t)Zπ −π ˜ f±(α, t)ℜ(∂4 αzµ(γ, t))∂α(ℑ(∂4 αzµ(γ, t)))dα =−h(t)Zπ −π ˜ f±(α, t)ℜ(∂4 αzµ(γ, t))ΛH(ℑ(∂4 αzµ(γ, t)))dα =−h(t)Zπ −π Λ1 2(˜ f±(α, t)ℜ(∂4 αzµ)(γ, t))Λ1 2H(ℑ(∂4 αzµ(γ, t)))dα =−h(t)Zπ −πnΛ1 2(˜ f±(α, t)ℜ(∂4 αzµ)(γ, t)) −˜ f±(α)Λ1 2ℜ(∂4 αzµ)oΛ1 2H(ℑ(∂4 αzµ(γ, t)))dα −h(t)Zπ −π ˜ f±(α)Λ1 2ℜ(∂4 αzµ)Λ1 2H(ℑ(∂4 αzµ(γ, t)))dα ≤h(t)||Λ1 2(˜ f±(·, t))ℜ(∂4 αzµ(· ± ih(t), t)) −˜ f±(·, t)Λ1 2ℜ(∂4 αzµ(· ± ih(t)))||L2(T) × ||Λ1 2H(ℑ(∂4 αzµ(· ± ih(t), t)))||L2(T) +h(t)|| ˜ f±||L∞(T)||Λ1 2ℜ(∂4 αzµ(· ± ih(t)))||L2(T)||Λ1 2Hℑ(∂4 αzµ(· ± ih(t)))||L2(T). Using the estimate (28) yields Zπ −π ℑ(f(γ))ℜ(∂4 αzµ(γ, t))∂α(ℑ(∂4 αzµ(γ, t)))dα ≤h(t)||Λ1+ε˜ f±||L2(T)||ℜ(∂4 αzµ(· ± ih(t), t))||L2(T)||Λ1 2(ℑ(∂4 αzµ(· ± ih(t), t)))||L2(T) +h(t)|| ˜ f±||L∞(T)||Λ1 2ℜ(∂4 αzµ(· ± ih(t)))||L2(T)||Λ1 2ℑ(∂4 αzµ(· ± ih(t))||L2(T) ≤Ch(t)(||z||S+ 1)k+Ch(t)(||z||S+ 1)k||Λ1 2∂4 αzµ(· ± ih(t), t)||2 L2(T) =Ch(t)(||z||S+ 1)k+Ch(t)(||z||S+ 1)kZπ −π ∂4 αzµ(γ, t)Λ∂4 αzµ(γ, t)dα. Now I1is equal to the integral to the left, plus a similar integral that can be bounded in a similar way. Thus we obtain that X ± I1≤C(kzkS+ 1)k+Ch(t)(kzkS+ 1)kkΛ1/2∂4 αzk2 L2(S).(29) By assumption the R-T ˜σis bigger than zero for real values. In order to avoid problems with the imaginary part we may write ∂αz1(α±ih(t), t) (∂αz1(α±ih(t)))2+ (∂αz2(α±ih(t)))2=∂αz1(α, t) |∂αz(α, t)|2+h(t)g±(α, t). where ||g±||H2(T)≤C(||z||S+ 1)k. 22 One finds, I2=−2ℜZT ∂αz1(α) |∂αz(α)|2Λ(∂4 αzµ)(α±ih(t)) ·∂4 αzµ(α±ih(t))dα −h(t)2ℜZT g±(α, t)Λ(∂4 αzµ)(α±ih(t)) ·∂4 αzµ(α±ih(t))dα. The first term above can be treated as in section 3 taking advantage of the inequality (26). Here we just need ∂αz1(α)≥0. The second term can be treated using the inequality (28) as with the term I1. We find that X ± I2≤C(||z||S+ 1)k+Ch(t)kg±kH2(S)kΛ1/2∂4 αzk2 L2(S), and therefore X ± I2≤C(||z||S+ 1)k+Ch(t)(kzkS+ 1)kkΛ1/2∂4 αzk2 L2(S).(30) Using (29) and (30) we have that d dt X ±ZT |∂4 αzµ(α±ih(t))|2dα ≤C(kzkS(t) + 1)k−10h′(t)ZT Λ(∂4 αzµ)(α)·∂4 αzµ(α)dα +(C(kzkS(t) + 1)kh(t) + 1 10h′(t)) ZT Λ(∂4 αzµ)(α±ih(t)) ·∂4 αzµ(α±ih(t))dα. Choosing h(t) = h(0) exp(−10CZt 0 (kzkS+ 1)k(r)dr) we eliminate the most dangerous term. The other term in the expression above involves with a function on the real line and it is easily controlled. Indeed ZT Λ∂4 αzµ(α)·∂4 αzµ(α)≤C h(t)X ±ZT |∂4 αzµ(α±ih(t))|2dα, as one sees by examining the Fourier expansion of ∂4 αzµ(α, t). Thus  10h′(t)ZT Λ(∂4 αzµ)(α)·∂4 αzµ(α)dα ≤C|h′(t)| h(t)||z||2 S≤C(||z||S+ 1)k+2. And we obtain finally d dt X ±ZT |∂4 αz(α±ih(t))|2dα ≤C(kzkS(t) + 1)k+2. 23 Recovering the dependency on Nin our notation we have that d dt X ±ZT |∂4 αz[N](α±ihN(t))|2dα ≤C(kz[N]kSN(t) + 1)k+2.(31) As in the previous section, we can obtain a bound of the evolution of the arc-chord condition that depends on C(kz[N]kSN(t) + 1)k+2. This estimate is true whenever t∈[0, TN], where TNis the maximal time of existence of the solution z[N]. In addition inequality (31) shows that we can extend these solutions in H4(S) up to a small enough time Tindependent of Nand depending on the initial data. The above calculation shows that the strip may shrink but does not collapse as long as ∂αz1(α, t)≥0. 5 From an analytic curve in the stable regime to an analytic curve in the unstable regime In this section we show that there exist some initial data which are analytic curves satisfying the arc-chord and R-T conditions such that the solution of the Muskat problem reaches the unstable regime. In order to do it we will prove the local existence of solutions for analytic initial data without assuming the R-T condition. Then we will construct some suitable initial data for our purpose. Theorem 5.1 Let z0be an analytic curve satisfying the arc-chord condition. Then there exists an analytic solution for the Muskat problem in some interval [−T, T]for a small enough T > 0. Remark 5.2 Notice that in theorem (5.1) there is no assumption on the R-T condition. The proof we use here is analogous to the one in [33] based on Cauchy-Kowalewski theorems [27, 28] (for an application to the Euler equation see [2]). Here we cannot parametrize the curve as a graph, so we have to change the argument substantially in the proof in order to deal with the arc-chord condition. Proof: We use the same notation as before. Let {Xr}r>0be a scale of Banach spaces given by R2−valued real functions fthat can be extended into the complex strip Sr={α+iζ ∈ C:|ζ|< r}such that the norm kfk2 r=X ±ZT |f(α±ir)−(α±ir, 0)|2dα +ZT |∂4 αf(α±ir)|2dα, is finite and f(α)−(α, 0) is 2π−periodic. Let z0(α) be a curve satisfying the arc-chord condition and z0(α)∈Xr0for some r0>0. Then, we will show that there exist a time T > 0 and 0 < r < r0so that there is a unique solution to (16) in C([0, T]; Xr). 24 It is easy to check that Xr⊂Xr′for r′≤rdue to the fact that kfkr′≤ kfkr. A simple application of the Cauchy formula gives k∂αfkr′≤C r−r′kfkr,(32) for r′< r. Next, we write equation (16) as follows: zt(α+iζ, t) = G(z(α+iζ, t)), with G(z(α+iζ, t)) = 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 take 0 ≤r′< r and we introduce the open set Oin Srgiven by O={z, ω ∈Xr:kzkr< R, kF(z)kL∞(Sr)< R2},(33) with F(z)(α+iζ, β, t) given by (17). Then the function Gfor G:O→Xr′is a continuous mapping. In addition, there is a constant CR(depending on Ronly) such that kG(z)kr′≤CR r−r′kzkr,(34) kG(z2)−G(z1)kr′≤CR r−r′kz2−z1kr,(35) and sup α+iζ∈Sr,β∈T |G(z)(α+iζ)−G(z)(α+iζ −β)| ≤ CR|β|,(36) for z, zj∈O. The above inequalities can be proved by estimating as in previous sections. Then they yield the proof of theorem 5.1. The argument is analogous to [27] and [28]. We have to deal with the arc-chord condition so we will point out the main differences. For initial data z0∈Xr0satisfying arc-chord, we can find a 0 < r′ 0< r0and a constant R0such that kz0kr′ 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) for α+iζ ∈Sr′ 0. We take 0 < r < r′ 0and R0< R to define the open set Oas in (33). Therefore we can use the classical method of successive approximations: zn+1(t) = z0+Zt 0 G(zn(s))ds, (38) for G:O→Xr′and 0 < r′< r. We assume by induction that kzkkr(t)< R, and kF(zk)kL∞(Sr)(t)< R 25 II. Tε≥0 for all ε. Then we can apply a Cauchy-Kowalewski theorem to the initial data wε(α, 0) −z(α, 0) satisfying ||wε−z||S(0) ≤Cε. For t > 0 small enough, z(α, t) is in the unstable regime. We achieve the conclusion of theorem 1.1 by continuity with respect to the initial data. The rest of the section is devoted to proving inequality (44). We shall denote γ=α+ih(t) and d(γ, t) = ∂4 α(w(γ, t)−z(γ, t)) (we omit the superscript εin the notation) and we recall that w(α, t) and z(α, t) are real for real α(therefore we obtain similar similar estimates for γ=α−ih(t)). In order to prove inequality (44) we have to compute the following quantity d dt Zπ −π |d(γ, t)|2dα = 2ℜZπ −π d(γ, t)dt(γ, t)dα+ 2ℜih′(t)Zπ −π d(γ, t)∂αd(γ, t)dα. Again we treat in detail the most singular term in dt(γ, t). Recall K(α, β) from section 3 and write Kwand Kzfor corresponding expressions arising from zand w. Then we have that dt(γ, t) = Zπ −π Kw(γ, γ −β)∂5 α(w(γ, t)−w(γ−β, t)dβ −Zπ −π Kz(γ, γ −β)∂5 α(z(γ, t)−z(γ−β, t))dβ + l.o.t(α, t), where 2ℜZπ −π d(γ, t)l.o.t(α)dα≤C(||d(·+ih(t), t)||2 L2(T)+||w(·+ih(t), t)−z(·+ih(t), t)||2 L2(T)). Here Cis a constant which just depends on ε0and δ. We can write Zπ −π Kw(γ, γ −β)∂5 α(w(γ, t)−w(γ−β, t))dβ −Zπ −π Kz(γ, γ −β)∂5 α(z(γ, t)−z(γ−β, t))dβ =Zπ −π Kw(γ, γ −β)∂5 α((w(γ, t)−z(γ, t)) −(w(γ−β, t)−z(γ−β, t)))dβ +Zπ −π {Kz(γ, γ −β)−Kw(γ, γ −β)}∂5 α(z(γ, t)−z(γ−β, t))dβ =Zπ −π Kw(γ, γ −β)∂α(d(γ, t)−d(γ−β, t))dβ +Zπ −π {Kz(γ, γ −β)−Kw(γ, γ −β)}∂5 α(z(γ, t)−z(γ−β, t))dβ ≡X1(α, t) + X2(α, t). 32 Therefore d dt Zπ −π |d(γ, t)|2dα ≤C||d(·+ih(t), t)||2 L2(T) + 2ℜZπ −π d(γ, t)X1(α, t)dα+ 2ℜZπ −π d(γ, t)X2(α, t)dα + 2ℜih′(t)Zπ −π d(γ, t)∂αd(γ, t)dα. Following the computations in section 3 when t∈[−δ, ta] and those in section 4 when t∈ [ta,˜ Tε] we have that d dt Zπ −π |d(γ, t)|2dα ≤C||d(·+ih(t), t)||2 L2(T)+ 2ℜZπ −π d(γ, t)X2(α, t)dα. In addition  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))2cot β 2 −Kz(γ, γ −β)−∂αz1(γ) (∂αz1(γ))2+ (∂αz1(γ))2cot β 2 +∂αw1(γ) (∂αw1(γ))2+ (∂αw1(γ))2−∂αz1(γ) (∂αz1(γ))2+ (∂αz1(γ))2cot β 2 ≤(||d(·+ih(t), t)||L2(T)+||w(·+ih(t), t)−z(·+ih(t), t)||L2(T))C+C cot β 2. Also, |∂5 αz(α±ih(t), t)−∂5 αz(α±ih(t)−β, t)| ≤ C tan β 2 since zis the analytic unperturbed solution. Therefore 2ℜZπ −π d(γ, t)X2(α, t)dα≤C(||d(·+ih(t), t)||2 L2(T)+||w(·+ih(t), t)−z(·+ih(t), t)||2 L2(T)). We are done. 7 Turning water waves Let us consider an incompressible irrotational flow satisfying the Euler equations ρ(vt+v· ∇v) = −∇p−gρ(0,1),(45) 33 where ρsatisfies (2,3) and ρ1= 0. This system of equations provides the motion of the interface for the water wave problem (see [3, 25] and references therein), whose contour equation is given by zt(α, t) = BR(z, ω)(α, t) + c(α, t)∂αz(α, t),(46) and ωt(α, t) = −2∂tBR(z, ω)(α, t)·∂αz(α, t)−∂α(|ω|2 4|∂αz|2)(α, t) + ∂α(c ω)(α, t) + 2c(α, t)∂αBR(z, ω)(α, t)·∂αz(α, t)−2g∂αz2(α, t). (47) The values of z(α, t) and w(α, t) are given at an initial time t0:z(α, t0) = z0(α) and w(α, t0) = w0(α). For more details see [12]. As an application of section 5, we can consider initial data given by a graph (α, f0(α)) and show that in finite time the interface evolution reaches a regime where the contour only can be parametrized as z(α, t) = (z1(α, t), z2(α, t)),for α∈R, with ∂αz1(α, t)<0 for α∈I, a non-empty interval. This implies that there exists a time t∗where the solution of the free boundary problem reparametrized by (α, f(α, t)) satisfies kfαkL∞(t∗) = ∞. Theorem 7.1 There exists a non-empty open set of initial data z0(α) = (α, f0(α)) and w0(α), with f0∈H5and w0∈H4, such that in finite time t∗the solution of the water wave problem (46,47) given by (α, f(α, t)) satisfies kfαkL∞(t∗) = ∞. The solution can be continued for t > t∗as z(α, t)with ∂αz1(α, t)<0for α∈I, a non-empty interval. Proof: Let us consider a curve z∗(α)∈H5satisfying 1., 2. and 3. of Lemma 5.3. We point out that analyticity is not required here. In order to find a velocity with property (39) we pick for water waves ω(α, t∗) = −∂αz∗ 2(α) and a suitable z(α, t∗) = z∗(α) as an initial datum. Notice that the tangential term does not affect the evolution. Then, with the appropriate c(α, t), we can apply the local existence result in [12]: There exists a solution of the water wave problem with z(α, t)∈C([t∗−δ, t∗+δ]; H5), ω(α, t)∈C([t∗−δ, t∗+δ]; H4) and δ > 0 small enough. The initial data promised by theorem 7.1 are any sufficiently small perturbations of z(α, t) and w(α, t) at time t=t∗−δ. 8 Breakdown of Smoothness In [5] we will exhibit a solution z(α, t) of the Muskat equation, with the following properties. 1. At time t0, the interface is real-analytic and satisfies the arc-chord and Rayleigh-Taylor conditions. 2. At time t1> t0, the interface turns over. 3. At time t2> t1, the interface no longer belongs to C4, although it is real-analytic for all times t∈[t0, t2). In this section we provide a brief sketch of our proof of the existence of such a Muskat solution. Our Muskat solution z(α, t) will be a small perturbation of a Muskat solution z00(α, t), with the following properties. 34 4. z00(α, t) is real analytic in α, for |ℑα|< ε00 and |τ| ≤ τ00. 5. For t∈[−τ00,0), z00(α, t) satisfies the Rayleigh-Taylor and arc-chord conditions. 6. For t= 0, the curve z00(α, t) has a vertical tangent at α= 0. 7. For t∈(0, τ00], the curve z00(α, t) fails to satisfy the Rayleigh-Taylor condition. This paper constructs Muskat solutions z00 satisfying 4., 5., 6. and 7. Our problem is to pass from z00 to a nearby Muskat solution zsatisfying 1., 2. and 3. The idea is as follows. So far, we have studied the analytic continuation of Muskat solutions to a time-varying strip S(t) = {|ℑα| ≤ h(t)}, in the complex plane. In our forthcoming paper [5], we will study the analytic continuation of a Muskat solution to a carefully chosen time-varying domain of the form Ω(t) = {|ℑα| ≤ h(ℜα, t)},(48) defined for t∈[−τ10, τ].Here, τis a small enough positive number. For t∈[−τ10, τ], we will work with the space H4(Ω(t)), consisting of all analytic functions F: Ω(t)7→ C2whose derivatives up to order 4 belong to L2(∂Ω(t)). We will pick our time-varying domain Ω(t) in (48) so that h(x, t)>0 for all (x, t)∈ R/2πZ×[−τ10, τ) and h(x, τ)>0 for all x∈R/2πZ\ {0}, but h(0, τ) = 0. Thus, the domain Ω(t) has ’thickness’ zero at the origin. Consequently, H4(Ω(τ)) is not contained in C4(R/2πZ). We will also take τ < τ00 and h(x, t)< ε00, so that the Muskat solution z00(α, t) continues analytically to Ω(t), for each t∈[−τ10, τ]. We can therefore pick an ’initial’ curve z0(α), such that 8. z0(α)−z00(α, τ) belongs to H4(Ω(τ)) and has small norm, yet 9. z0(α) does not belong to C4(R/2πZ). We solve the Muskat problem backwards in time, with the ’initial’ condition 10. z(α, τ) = z0(α). By a more elaborate version of the analytic continuation arguments used in this paper, we find that our Muskat solution exists and continues analytically into Ω(t), for all t∈[t∗, τ] (for a suitable time t∗); moreover, 11. z(α, t)−z00(α, t) has small norm in H4(Ω(τ)), for all t∈[t∗, τ]. Here, either 12. t∗=−τ10 or 13. a modified Rayleigh-Taylor condition, adapted to the time-varying domain, fails at time t∗. We can rule out 13., thanks to 11., together with our understanding of z00(t) and Ω(t). Thus, we obtain a Muskat solution z(α, t), satisfying 9., 10., 11. and 12. Properties 1., 2. and 3. of z(α, t) now follow easily. 35 Acknowledgements AC, DC and FG were partially supported by the grant MTM2008-03754 of the MCINN (Spain) and the grant StG-203138CDSIF of the ERC. CF was partially supported by NSF grant DMS-0901040 and ONR grant ONR00014-08-1-0678. FG was partially supported by NSF grant DMS-0901810. MLF was partially supported by the grants MTM2008-03541 and MTM2010-19510 of the MCINN (Spain). The authors thank Professor Rafael de la Llave for helpful discussions. References [1] D. Ambrose. Well-posedness of Two-phase Hele-Shaw Flow without Surface Tension. Euro. Jnl. of Applied Mathematics 15, (2004), 597-607. [2] C. Bardos and S. Benachour. Domaine d’analyticite des solutions de l’equation d’Euler dans un ouvert de Rn.Annal. Sc. Normale Sup. di Pisa, (1977), no. 4, 647-687. [3] C. 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Angel Castro Instituto de Ciencias Matem´aticas Consejo Superior de Investigaciones Cient´ıficas Serrano 123, 28006 Madrid, Spain Email: angel [email protected] Diego C´ordoba Charles Fefferman Instituto de Ciencias Matem´aticas Department of Mathematics Consejo Superior de Investigaciones Cient´ıficas Princeton University Serrano 123, 28006 Madrid, Spain 1102 Fine Hall, Washington Rd, Email: [email protected] Princeton, NJ 08544, USA Email: [email protected] Francisco Gancedo Mar´ıa L´opez-Fern´andez Department of Mathematics Institut f¨ur Mathematik University of Chicago Universit¨at Z¨urich 5734 University Avenue, Winterthurerstr. 190, CH-8057 Chicago, IL 60637, USA Z¨urich, Switzerland Email: [email protected] Email: maria.lop[email protected]h 38