Finite time singularities for water waves with surface tension

arXiv:1204.6633v2 [math.AP] 19 Oct 2012 Finite time singularities for water waves with surface tension Angel Castro, Diego C´ordoba, Charles Fefferman, Francisco Gancedo and Javier G´omez-Serrano Dedicated to Peter Constantin on his 60th Birthday October 22, 2012 Abstract Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates. Keywords: Euler, incompressible, blow-up, water waves, splash, splat, surface tension. I Introduction In this paper we continue the work in [8] and [9] where we show the formation of singularities for the free boundary incompressible Euler equations. Here we prove that in two space dimensions the free boundary problem develops finite time “splash” and “splat” singularities when surface tension is taken into account (see below, in Section III, the precise definition of the splash and splat curves). In order to describe the evolution of a fluid with a moving domain Ω(t)⊂R2, the 2D incompressible Euler equations are used: (vt+v·∇v)(x, y, t) = −∇p(x, y, t)−(0,1),(x, y)∈Ω(t) (I.1) with the fluid velocity v(x, y, t)∈R2and the pressure p(x, y, t)∈R. The vector −(0,1) represents the external gravitational force (the acceleration due to gravity is taken equal to one for the sake of simplicity). The free boundary ∂Ω(t) = {z(α, t) = (z1(α, t), z2(α, t)) : α∈R}(I.2) is smooth and convected by the velocity field zt(α, t)·z⊥ α(α, t) = v(z(α, t), t)·z⊥ α(α, t),(I.3) which is assumed to be incompressible and irrotational ∇·v(x, y, t) = 0,∇⊥·v(x, y, t) = 0,(x, y)∈Ω(t).(I.4) 1 Here we study the relevance of considering the Laplace-Young condition for which the pressure on the interface ∂Ω(t) is proportional to its curvature, meaning that the surface tension effect is considered: −p(z(α, t), t) = τ 2 zαα(α, t)·z⊥ α(α, t) |zα(α, t)|3≡τ 2K. (I.5) Above τ > 0 is the surface tension coefficient. The results in this paper can be shown for three different scenarios: 1. Ω(t) a compact domain: z(α, t) is a 2π-periodic function in α. 2. Asymptotically flat case: z(α, t)−(α, 0) →0 as α→ ∞. 3. Ω(t) periodic in the horizontal variable: z(α, t)−(α, 0) is a 2π-periodic function in α. The problem to study here is the potential formation of singularities for the system (I.1I.5) with smooth interface and smooth velocity field with finite energy as initial data: Ω(0) = Ω0, ∂Ω0={z0(α) : α∈R}, v(x, y, 0) = v0(x, y),ZΩ0|v0(x, y)|2dxdy < +∞.(I.6) The smooth initial curve z0(α) must satisfy the arc-chord condition: |z0(α)−z0(β)| ≥ cAC|α−β|,for all α, β ∈R,(I.7) where cAC >0 is the arc-chord constant. The study of this quantity has been employed by other authors to prove local existence (see for example [23], [24]). We will quantify how our curve z(α) satisfies the arc-chord condition through the following quantity F(z) = |β| |z(α)−z(α−β)|, α, β ∈[−π, π]. Throughout the paper we will only focus on scenario 3 for the sake of simplicity. From now on, we will denote Ω0∩[−π, π]×Rby Ω0by abuse of notation (a fundamental domain in the period). We establish the main result in the paper for the system (I.1-I.5). Theorem I.1 Consider z0(α)−(α, 0) ∈Hk(T)for k≥5. Then there exist a family of initial data satisfying (I.6) and the arc-chord condition (I.7) and a time Ts>0such that the interface z(α, t)∈Hk(T)from the unique smooth solution of the system (I.1-I.7) on the time interval [0, Ts]touches itself at a single point (“splash” singularity) or along an arc (“splat” singularity) at time t=Ts. These solutions can be extended to the periodic 3Dsetting considering scenarios invariant under translations in one coordinate direction. In [14], Coutand-Shkoller consider additional 3Dsplash and splat singularities. The case with small initial data was treated by Wu in the two dimensional case [25] and the three dimensional case was studied by Wu [26] and Germain et al. [16]. 2 For other long time behaviour results see Alvarez-Lannes [3], Castro et al. [10] and the references therein. In order to prove this theorem we proceed as in [8] and [9]. Using (I.4) it is easy to declare that vis harmonic in Ω(t). This fact allows us to introduce the moment ω(α, t) by elementary potential theory as follows: v(x, y, t) = PV 2πZR (x−z1(β, t), y −z2(β, t)))⊥ |(x, y)−z(β, t)|2ω(β, t)dβ, (I.8) where PV denotes principal value at infinity. This moment is also known in the literature as the vorticity amplitude. Then the system (I.1-I.5) is equivalent to the following evolution equations which are only written in terms of the free boundary z(α, t) and the amplitude ω(α, t): zt(α, t) = BR(z, ω)(α, t) + c(α, t)zα(α, t),(I.9) ωt(α, t) = −2BRt(z, ω)(α, t)·zα(α, t)−ω2 4|∂αz|2α(α, t) + (cω)α(α, t) + 2c(α, t)BRα(z, ω)(α, t)·zα(α, t)−2(z2)α(α, t) + τzαα(α, t)·z⊥ α(α, t) |zα(α, t)|3α (I.10) (for details see for example [12, Section 2]). Above BR(z, ω) is the Birkhoff-Rott integral defined by BR(z, ω) = 1 2πPV ZR (z(α, t)−z(β, t))⊥ |z(α, t)−z(β, t)|2ω(β, t)dβ, (I.11) and c(α, t) is arbitrary since the boundary is convected by the normal velocity (I.3). Local existence in Sobolev spaces was first achieved by Wu [23] assuming initially the arc-chord condition. For other variations and results see [15, 21, 7, 27, 24, 11, 20, 13, 22, 28, 18, 6, 4, 19, 1, 2, 12]. The strategy of the proof of the main result is to establish a local existence theorem from the initial data that has a splash or a splat singularity (notice that the equations are time reversible invariant). Since the curve self-intersects (failure of the arc-chord condition), it is not clear if the amplitude of the vorticity remains smooth and the meaning of equations (I.9-I.10). In order to deal with these obstacles we use a conformal map P(w) = tan w 21/2, w ∈C, whose intention is to keep apart the self-intersecting points taking the branch of the square root above passing through those crucial points. Here P(z) will refer to a 2 dimensional vector whose components are the real and imaginary parts of P(z1+iz2). We also make sure that Ω(t)∪∂Ω(t) do not contain any singular point of the transformation P. Then potential theory helps us to get the following analogous evolution equations for the new curve ˜z(α, t) = P(z(α, t)) 3 and the new amplitude ˜ω: ˜zt(α, t) = Q2(α, t)BR(˜z, ˜ω)(α, t) + ˜c(α, t)˜zα(α, t),(I.12) ˜ωt(α, t) = −2BRt(˜z, ˜ω)(α, t)·˜zα(α, t)−|BR(˜z, ˜ω)|2(Q2)α(α, t)−Q2(α, t)˜ω(α, t)2 4|˜zα(α, t)|2α + 2˜c(α, t)BRα(˜z, ˜ω)·˜zα(α, t) + (˜c(α, t)˜ω(α, t))α−2P−1 2(˜z(α, t))α +τQ3 |˜zα(α, t)|3(˜zT αHP −1 2˜zα∇P−1 1·˜zα−˜zT αHP −1 1˜zα∇P−1 2·˜zα)α +τQ˜zαα(α, t)·˜z⊥ α(α, t) |˜zα(α, t)|3α (I.13) where Q2(α, t) =  dP dw (P−1(˜z(α, t))) 2 , and HP −1 idenotes the Hessian matrix of P−1 i, which is the i-th (i={1,2}) component of the transformation P−1. Here, we choose ˜c(α, t) in such a way that |˜zα(α, t)|=A(t). This particular choice of ˜c was first introduced by Hou et al. in [17] and was later used by Ambrose [4] and AmbroseMasmoudi [5]. The choice of ˜cimplies ˜c(α, t) = α+π 2πZπ −π (Q2BR(˜z, ˜ω))β(β, t)·˜zβ(β, t) |˜zβ(β, t)|2dβ −Zα −π (Q2BR(˜z, ˜ω))β(β, t)·˜zβ(β, t) |˜zβ(β, t)|2dβ It is easy to check that if we take Q≡1 in (I.12-I.13) we recover (I.9-I.10). We also define the function ˜ϕ(α, t) = Q2(α, t)˜ω(α, t) 2|˜zα(α, t)|−˜c(α, t)|˜zα(α, t)|(I.14) introduced by Beale et al. for the linear case [7] and by Ambrose-Masmoudi for the nonlinear one [5]. This function will be used to prove local existence in Sobolev spaces. In the sections below, we show a local existence theorem based on energy estimates. Section III is devoted to provide the appropriate initial data for the splash and splat singularities. In Section IV we choose an energy which does not need a precise sign on the Rayleigh-Taylor function. In Section V we choose a different energy that involves the sign of the RayleighTaylor function and the estimates are uniform with respect to the surface tension coefficient. These two energies are based on the ones obtained in the non-tilde domain by Ambrose ([4]) and Ambrose-Masmoudi ([6]). 4 The Rayleigh-Taylor function is given by the following formula σ≡BRt(˜z, ˜ω) + ˜ϕ |˜zα|BRα(˜z, ˜ω)·˜z⊥ α+˜ω 2|˜zα|2˜zαt +˜ϕ |˜zα|˜zαα·˜z⊥ α +QBR(˜z, ˜ω) + ˜ω 2|˜zα|2˜zα 2 (∇Q)(˜z)·˜z⊥ α+ (∇P−1 2)(˜z)·˜z⊥ α. (I.15) All solutions that we will consider throughout the paper will have finite energy, as discussed in [8]. The system satisfies the conservation of the mechanical energy. We define it this way: (not to be confused with the subsequent definitions of some other energies, see sections IV and V). ES(t) = 1 2ZΩf(t)|v(x, y, t)|2dxdy +1 2Zπ −π (z2(α, t))2∂αz1(α, t)dα +τ 2Zπ −π|∂αz(α, t)|dα ≡ Ek(t) + Ep(t) + Eτ(t), where z(α, t) = (z1(α, t), z2(α, t)), u(α, t) = v(z(α, t), t), and Ωf(t) = Ω(t)∩[−π, π]×R is a fundamental domain in the water region in a period, then it follows that the energy is conserved. dEk(t) dt =ZΩf(t) v(x, y, t)(vt(x, y, t) + v(x, y, t)·∇v(x, y, t))dxdy =ZΩf(t) v(x, y, t)(−∇p(x, y, t)−(0,1))dxdy =−ZΩf(t) v(x, y, t)(∇(p(x, y, t) + y))dxdy =−Z∂(Ωf(t)) v(x, y, t)·−→ n yds +Z∂(Ωf(t)) v(x, y, t)·−→ nτ 2Kds =−Zπ −π z2(α, t)u(α, t)·∂αz⊥(α, t)dα +τ 2Zπ −π u(α, t)·∂αz⊥(α, t)∂2 αz(α, t)·∂αz⊥(α, t) |∂αz(α, t)|3dα (I.16) where we have used the incompressibility of the fluid (∇·v= 0) and Laplace-Young’s condition for the pressure on the interface. Next dEp(t) dt =Zπ −π z2(α, t)∂tz2(α, t)∂αz1(α, t)dα +1 2Zπ −π (z2(α, t))2∂t∂αz1(α, t)dα =Zπ −π z2(α, t)∂tz2(α, t)∂αz1(α, t)dα −Zπ −π z2(α, t)∂αz2(α, t)∂tz1(α, t)dα =Zπ −π z2(α, t)u(α, t)·∂αz⊥(α, t)dα. (I.17) 5 dEτ(t) dt =τ 2Zπ −π ∂αz(α, t)·∂α∂tz(α, t) |∂αz(α, t)|dα =−τ 2Zπ −π ∂2 αz(α, t)·∂tz(α, t) |∂αz(α, t)|dα =−τ 2Zπ −π ∂2 αz(α, t)·u(α, t) |∂αz(α, t)|dα =−τ 2Zπ −π ∂2 αz(α, t)·∂αz⊥(α, t) |∂αz(α, t)|3u(α, t)·∂⊥ αz(α, t)dα (I.18) Adding all the derivatives we get the desired result. II Properties of the curvature in the tilde domain In this section we will rewrite the term corresponding to the curvature K(z(α, t)) in the new tilde variables ˜z(α, t). We will proceed step by step. Let us recall that the curvature is defined by K(α, t) = zαα(α, t)·z⊥ α(α, t) |zα(α, t)|3 We begin with the term |zα(α, t)|3. We have that |˜zα(α, t)|2=h∂αP(z(α, t)), ∂αP(z(α, t))i=h∇P(z(α, t)) ·zα(α, t),∇P(z(α, t)) ·zα(α, t)i Since Pand P−1are conformal, by the Cauchy-Riemann equations ∇P(z(α, t))T∇P(z(α, t)) = Q2(α, t)Id2, that implies that |˜zα(α, t)|3=Q3(α, t)|zα(α, t)|3 We move to the other term hzαα(α, t), z⊥ α(α, t)i=h∂α∇P−1(˜z(α, t)) ·˜zα(α, t),(∇P−1(˜z(α, t)) ·˜zα(α, t))⊥i =h∇P−1(˜z(α, t)) ·˜zαα(α, t),(∇P−1(˜z(α, t)) ·˜zα(α, t))⊥i +h∂α∇P−1(˜z(α, t))·˜zα(α, t),(∇P−1(˜z(α, t)) ·˜zα(α, t))⊥i ≡ W+X Again, by the Cauchy-Riemann equations W=1 Q2(α, t)h˜zαα(α, t),˜zα(α, t)⊥i Developing the terms in X, we get ∇P−1(˜z(α, t))·˜zα(α, t) = ˜zT α(α, t)·HP −1 1(˜z(α, t)) ·˜zα(α, t) ˜zT α(α, t)·HP −1 2(˜z(α, t)) ·˜zα(α, t), 6 where HP−1 idenotes the Hessian of the i-th component of P−1(i= 1,2). Hence, we can write Xas X=−˜zT α(α, t)·HP −1 1(˜z(α, t)) ·˜zα(α, t)∇P−1 2(˜z(α, t)) ·˜z(α, t) + ˜zT α(α, t)·HP −1 2(˜z(α, t)) ·˜zα(α, t)∇P−1 1(˜z(α, t)) ·˜z(α, t). This means that K(α, t) = Q(α, t)˜zαα(α, t)·˜z⊥ α(α, t) |˜z(α, t)|3+X(α, t)Q(α, t)3 |˜z(α, t)|3≡Q(α, t)˜ K(α, t) + M(α, t) We will now try to simplify further by exploiting the Cauchy-Riemann equations. We can calculate the Hessian and the gradient terms as: P−1 1,x (˜z) = ℜ4˜z 1 + ˜z4≡ ℜ(a) P−1 1,y (˜z) = ℜ4i˜z 1 + ˜z4≡ −ℑ(a) P−1 2,x (˜z) = ℑ4˜z 1 + ˜z4≡ ℑ(a) P−1 2,y (˜z) = ℑ4i˜z 1 + ˜z4≡ ℜ(a) P−1 1,x,x(˜z) = ℜ4(1 −3˜z4) (1 + ˜z4)2≡ ℜ(b) P−1 1,x,y(˜z) = ℜ4i(1 −3˜z4) (1 + ˜z4)2≡ −ℑ(b) P−1 2,x,x(˜z) = ℑ4(1 −3˜z4) (1 + ˜z4)2≡ ℑ(b) P−1 2,x,y(˜z) = ℑ4i(1 −3˜z4) (1 + ˜z4)2≡ ℜ(b) Therefore the Hessians are HP −1 1=ℜ(b)−ℑ(b) −ℑ(b)−ℜ(b), HP−1 2=ℑ(b)ℜ(b) ℜ(b)−ℑ(b), Calculating further: ˜zT αHP −1 2˜zα=ℜ(b)(2˜z1 α˜z2 α) + ℑ(b)((˜z1 α)2−(˜z2 α)2) ˜zT αHP −1 1˜zα=ℜ(b)((˜z1 α)2−(˜z2 α)2)−ℑ(b)(2˜z1 α˜z2 α) 7 X1=ℜ(a)ℜ(b)(2(˜z1 α)2˜z2 α) + ℜ(a)ℑ(b)((˜z1 α)2˜z1 α−(˜z2 α)2˜z1 α) +ℑ(a)ℜ(b)(−2˜z1 α(˜z2 α)2) + ℑ(b)ℑ(b)((˜z1 α)2˜z2 α−(˜z2 α)2˜z2 α) X2=ℜ(b)ℜ(b)((˜z1 α)2˜z2 α−(˜z2 α)2˜z2 α) + ℜ(a)ℑ(b)(−2˜z1 α(˜z2 α)2) +ℑ(a)ℑ(b)(2(˜z1 α)2˜z2 α) + ℑ(a)ℜ(b)((˜z1 α)2˜z1 α−(˜z2 α)2˜z1 α) This means X=X1−X2= ((˜z1 α)2+ (˜z2 α)2)(˜z2 α(ℜ(a)ℜ(b) + ℑ(a)ℑ(b)) + ˜z1 α(ℜ(a)ℑ(b)−ℑ(a)ℜ(b))) ≡((˜z1 α)2+ (˜z2 α)2)hG(z),˜zαi. We can see that −Qα Q3=1 2∂α1 Q2=∂α(ℜ(a)2+ℑ(a)2) =ℜ(a)ℜ(b)˜z1 α−ℜ(a)ℑ(b)˜z2 α+ℑ(a)ℑ(b)˜z1 α+ℑ(a)ℜ(b)˜z2 α =hG(z),˜z⊥ αi by the Cauchy-Riemann equations. If we take one derivative in space of X, we obtain ∂αX= ((˜z1 α)2+ (˜z2 α)2)h∇G(˜z)·˜zα,˜zαi+ ((˜z1 α)2+ (˜z2 α)2)hG(˜z),˜zααi = ((˜z1 α)2+ (˜z2 α)2)h∇G(˜z)·˜zα,˜zαi+|˜zα|3˜ KhG(˜z),˜z⊥ αi = ((˜z1 α)2+ (˜z2 α)2)h∇G(˜z)·˜zα,˜zαi−|˜zα|3˜ KQα Q3, This implies K=Q˜ K−Q3X |˜z|3⇒Kα= (Q˜ K)α+Q3 |˜zα|h∇G(˜z)·˜zα,˜zαi− ˜ KQα= (Q˜ K)α+M1+M2 Later, we will see that the M1is a low order term and can be absorbed by the energy. III Initial data For initial data we are interested in considering a self-intersecting curve in one point. More precisely, we will use as initial data splash curves which are defined this way: Definition III.1 We say that z(α) = (z1(α), z2(α)) is a splash curve if 8 1. z1(α)−α, z2(α)are smooth functions and 2π-periodic. 2. z(α)satisfies the arc-chord condition at every point except at α1and α2, with α1< α2 where z(α1) = z(α2)and |zα(α1)|,|zα(α2)|>0. This means z(α1) = z(α2), but if we remove either a neighborhood of α1or a neighborhood of α2in parameter space, then the arc-chord condition holds. 3. The curve z(α)separates the complex plane into two regions; a connected water region and a vacuum region (not necessarily connected). The water region contains each point x+iy for which y is large negative. We choose the parametrization such that the normal vector n=(−∂αz2(α),∂αz1(α)) |∂αz(α)|points to the vacuum region. We regard the interface to be part of the water region. 4. We can choose a branch of the function Pon the water region such that the curve ˜z(α) = (˜z1(α),˜z2(α)) = P(z(α)) satisfies: (a) ˜z1(α)and ˜z2(α)are smooth and 2π-periodic. (b) ˜zis a closed contour. (c) ˜zsatisfies the arc-chord condition. We will choose the branch of the root that produces that lim y→−∞ P(x+iy) = −e−iπ/4 independently of x. 5. P(w)is analytic at wand dP dw (w)6= 0 if wbelongs to the interior of the water region. Furthermore, (±π, 0) and (0,0) belong to the vacuum region. 6. ˜z(α)6=qlfor l= 0, ..., 4, where q0= (0,0) , q1=1 √2,1 √2, q2=−1 √2,1 √2, q3=−1 √2,−1 √2, q4=1 √2,−1 √2. (III.1) Moreover, we will define a splat curve as a splash curve but replacing condition (2) by the fact that the curve touches itself along an arc, instead of a point. Let us note that in order to measure when the transformation Pis regular, we need to control the distance to the points ql. In order to do so, we introduce the function m(ql)(α, t)≡ |˜z(α, t)−ql| for l= 0,...,4. We have performed numerical simulations, as explained in [9] with the following initial data on the non-tilde domain: z0 1(α) = α+1 4−3π 2−1.9sin(α) + 1 2sin(2α) + 1 4π 2−1.9sin(3α) 9 dC dt = OK + 1 |˜zα|τZQ2k+4 ˜ω2∂k α(˜ω)∂k+1 α(˜ K) = OK + C1 IV.D Development of the derivative in B We start from the development of B1,B2,B3and B4. We trivially have: B1=1 τZQ2k+2Λ(∂k α(˜ω))Q2˜ω2 |˜zα|H(∂k α(˜ K)) B3=−2ZQ2k+2QαΛ(∂k α(˜ω))∂k α(˜ K) B4= OK −Z(2k+ 2)Q2k+2QαH(∂k+1 α(˜ω))∂k α(˜ K) We now look at B2. We can decompose it in the following way B2= 2 ZQ2k+2Λ(∂k α(˜ω))∂k α(Qα˜ K+Q˜ Kα) = OK + 2 ZQ2k+2Λ(∂k α(˜ω))(Qα∂k α(˜ K) + Q∂k+1 α(˜ K) + kQα∂k α(˜ K)) = OK + B2,1+B2,2+B2,3 We can write down the terms B2,1and B2,3in the form B2,1= 2 ZQ2k+2H(∂k+1 α(˜ω))Qα∂k α(˜ K) B2,3= 2kZQ2k+2H(∂k+1 α(˜ω))Qα∂k α(˜ K) Integrating by parts in B2,2we establish B2,2=−2ZQ2k+3Λ(∂k+1 α(˜ω))∂k α(˜ K) −2(2k+ 3) ZQ2k+2QαΛ(∂k α(˜ω))∂k α(˜ K) =B2,2,1+B2,2,2 Again, B2,2,2can easily be reduced to the canonical form B2,2,2=−2(2k+ 3) ZQ2k+2QαH(∂k+1 α(˜ω))∂k α(˜ K) IV.E Collection of the terms We will split all the uncontrolled terms into three categories: high order and low order types I and II and we will see that the sum of the terms in each category adds up to low enough order terms, denoted by OK. 16 IV.E.1 High Order From A: 2ZQ2k+3 ∂k α(˜ K)∂k α(H(˜ωαα)) (A2) From B: −2ZQ2k+3Λ(∂k+1 α(˜ω))∂k α(˜ K) (B2,2,1) From C: No terms from C. IV.E.2 Low Order Type I From A: 2Z2kQ2k+2Qα∂k α(˜ K)∂k−1 α(H(˜ωαα)) (A1) 2Z4Q2k+2Qα∂k α(˜ K)∂k α(H(˜ωα)) (A3) From B: −2ZQ2k+2QαΛ(∂k α(˜ω))∂k α(˜ K) (B3) −Z(2k+ 2)Q2k+2QαH(∂k+1 α(˜ω))∂k α(˜ K) (B4) 2ZQ2k+2H(∂k+1 α(˜ω))Qα∂k α(˜ K)) (B2,1) 2kZQ2k+2H(∂k+1 α(˜ω))Qα∂k α(˜ K) (B2,3) −2(2k+ 3) ZQ2k+2QαH(∂k+1 α(˜ω))∂k α(˜ K) (B2,2,2) From C: No terms from C. IV.E.3 Low Order Type II From A: No terms from A. From B: 1 τZQ2k+2Λ(∂k α(˜ω))Q2˜ω2 |˜zα|H(∂k α(˜ K)) (B1) From C: 1 |˜zα|τZQ2k+4 ˜ω2∂k α(˜ω)∂k+1 α(˜ K) (C1) 17 IV.F Regularized system Now, let ˜zε,δ,µ(α, t) be a solution of the following system (compare with (I.12 - I.13)): ˜zε,δ,µ t(α, t) = φδ∗φδ∗Q2(˜zε,δ,µ)BR(˜zε,δ,µ,˜ωε,δ,µ)(α, t) + φµ∗˜cε,δ,µ φµ∗∂α˜zε,δ,µ(α, t), (IV.1) ˜ωε,δ,µ t=φδ∗φδ∗−2BRt(˜zε,δ,µ,˜ωε,δ,µ)·˜zε,δ,µ α−|BR(˜zε,δ,µ,˜ωε,δ,µ)|2(Q2(˜zεδ,µ))α −Q2˜ (ωε,δ,µ)2 4|˜zε,δ,µ α|2α+ 2cε,δ,µBRα(˜zε,δ,µ, ωε,δ,µ)·˜zε,δ,µ α+cε,δ,µ ˜ωε,δ,µα−2P−1 2(˜zε,δ,µ(α, t))α +τ Q3(˜zε,δ,µ) |˜zε,δ,µ α(α, t)|3(˜zε,δ,µ α)THP−1 2˜zε,δ,µ α∇P−1 1·˜zε,δ,µ α−(˜zε,δ,µ α)THP−1 1˜zε,δ,µ α∇P−1 2·˜zε,δ,µ α)!α +τ Q˜zε,δ,µ αα ·(˜zε,δ,µ α)⊥ |˜zε,δ,µ α|3!α!−εφµ∗φµ∗Λ(˜ωε,δ,µ)1 Q2k+3 (IV.2) ˜zε,δ,µ(α, 0) = ˜z0(α) and ˜ωε,δ,µ(α, 0) = ˜ω0(α) for ε > 0, δ > 0, µ > 0. The functions φδand φµ are even mollifiers, ˜cε,δ,µ(α) =α+π 2πZπ −π ∂β˜zε,δ,µ(β)) |∂β˜zε,δ,µ(β)|2·φδ∗φδ∗(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ −Zα −π ∂β˜zε,δ,µ(β) |∂β˜zε,δ,µ(β)|2·φδ∗φδ∗(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ, and cε,δ,µ(α) =α+π 2πZπ −π ∂β˜zε,δ,µ(β)) |∂β˜zε,δ,µ(β)|2·(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ −Zα −π ∂β˜zε,δ,µ(β) |∂β˜zε,δ,µ(β)|2·(∂β(Q2(˜zε,δ,µ)(β)BR(˜zε,δ,µ,˜ωε,δ,µ))(β))dβ, The RHS of the evolution equations for ˜zε,δ,µ and ˜ωε,δ,µ are Lipschitz in the spaces Hk+2(T) and Hk+1 2(T) since they are mollified. Therefore we can solve (IV.1-IV.2) for short time, thanks to Picard’s theorem. Now, we can perform energy estimates to get uniform bounds in µ(we just deal with a transport term and a dissipative) and we can let µgo to zero. The energy estimates that we can get are the following: d dt k˜zε,δ,µk2 H5+kF(˜zε,δ,µ)k2 L∞+k˜ωε,δ,µk2 H3+ 1 2+ 4 X l=0 1 mε,δ,µ(ql)!(t) ≤C(δ) k˜zε,δ,µk2 H5+kF(˜zε,δ,µ)k2 L∞+k˜ωε,δ,µk2 H3+ 1 2+ 4 X l=0 1 mε,δ,µ(ql)!j (t). 18 We should note that for the new system without the φµmollifier, the length of the tangent vector |∂α˜zδ|is now constant in space and depends only on time. Next we will perform energy estimates as in the previous case by using the curvature ˜ Kδfrom the curve ˜zδ. Similarly, we get (let us omit the superscript δ, ε in ˜zδ,ε and ˜ωδ,ε) •˜ Kt= NICE3 + Q2 2|˜zα|3φδ∗φδ∗H(˜ωαα) + 1 |˜zα|3(Q2)αφδ∗φδ∗H(˜ωα), •∂k α(cα˜ω) = NICE35 + Q2˜ω2 2|˜zα|H(∂k α(˜ K)), •∂k α(˜c˜ωα) = NICE35 , and the following collection of terms: IV.F.1 High Order From A: 2ZQ2k+3 ∂k α(˜ K)∂k αφδ∗φδ∗(H(˜ωαα)) (A2) From B: −2ZQ2k+3Λ(∂k+1 α(˜ω))φδ∗φδ∗∂k α(˜ K) (B2,2,1) −2ε τk∂k+1 α˜ωk2 L2(D) From C: No terms from C. IV.F.2 Low Order Type I From A: 2Z2kQ2k+2Qα∂k α(˜ K)∂k−1 αφδ∗φδ∗(H(˜ωαα)) (A1) 2Z4Q2k+2Qα∂k α(˜ K)φδ∗φδ∗∂k α(H(˜ωα)) (A3) From B: −2ZQ2k+2QαΛ(∂k α(˜ω))φδ∗φδ∗∂k α(˜ K) (B3) −Z(2k+ 2)Q2k+2QαH(∂k+1 α(˜ω))φδ∗φδ∗∂k α(˜ K) (B4) 2ZQ2k+2H(∂k+1 α(˜ω))Qαφδ∗φδ∗∂k α(˜ K)) (B2,1) 2kZQ2k+2H(∂k+1 α(˜ω))Qαφδ∗φδ∗∂k α(˜ K) (B2,3) −2(2k+ 3) ZQ2k+2QαH(∂k+1 α(˜ω))φδ∗φδ∗∂k α(˜ K) (B2,2,2) 19 From C: No terms from C. IV.F.3 Low Order Type II From A: No terms from A. From B: 1 τZQ2k+2Λ(∂k α(˜ω))Q2˜ω2 |˜zα|φδ∗φδ∗H(∂k α(˜ K)) (B1) From C: 1 |˜zα|τZQ2k+4 ˜ω2∂k α(˜ω)φδ∗φδ∗∂k+1 α(˜ K) (C1) We note that throughout this section we have repeatedly used the following commutator estimate for convolutions: kφδ∗(∂αfg)−gφδ∗(∂αf)kL2≤Ck∂αgkL∞kfkL2,(IV.3) where the constant Cis independent of δ, f and g. Also using this commutator estimate we can find all the cancelations we need in the previous collection of terms of low order type I and II to obtain a suitable energy estimate. Regarding the high order terms, we will do the estimates in detail. We will see the need for the dissipative term since there are terms that escape for half of a derivative. A2+B2,2,1+D= 2 ZQ2k+3 ∂k α(˜ K)φδ∗φδ∗H(∂k+2 α˜ω) −2ZQ2k+3H(∂k+2 α(˜ω))φδ∗φδ∗∂k α(˜ K)−2εk∂k+1 α˜ωk2 L2 = 2 Z∂k α(˜ K)Q2k+3φδ∗φδ∗H(∂k+2 α˜ω)−φδ∗φδ∗Q2k+3H(∂k+2 α˜ω)−2εk∂k+1 α˜ωk2 L2 ≤ k∂k α˜ KkL2k∂αQ2k+3kL∞k∂k+1 α˜ωkL2−2εk∂k+1 α˜ωk2 L2≤C(ε)Ep(t), which is uniform in δ. This proves that we can pass to the limit δ→0. Finally, by applying the a priori energy estimates to the new system (which only depend on ε) we can pass to the limit ε→0 since now we don’t have the previous problems and A2+B2,2,1= 0. V Energy with the Rayleigh-Taylor condition In this section, we prove local existence in the tilde domain, where the time of existence does not depend on the surface tension coefficient. In this theorem, we need initial data to satisfy the Rayleigh-Taylor condition as we explain in Section III. This Rayleigh-Taylor condition will hold in particular if the surface tension coefficient is small enough. 20 Theorem V.1 Let k≥3. Let ˜z0(α)be the image of a splash curve by the map Pparametrized in such a way that |∂α˜z0(α)|=L 2π, where Lis the length of the curve in a fundamental period, and such that ˜z0 1(α),˜z0 2(α)∈Hk+2(T). Let ˜ϕ(α, 0) ∈Hk+1 2(T)be as in (I.14) and let ˜ω(α, 0) ∈Hk−1(T). Then there exist a finite time T > 0, a time-varying curve ˜z(α, t)∈C([0, T]; Hk+2), and functions ˜ω(α, t)∈C([0, T]; Hk−1)and ˜ϕ∈C([0, T ]; Hk+1 2) providing a solution of the water wave equations (I.12 - I.13). Assume that initially, the Rayleigh-Taylor condition is strictly positive. In order to prove this theorem we will use the solutions we have obtained in theorem IV.1 for τ > 0. We will perform energy estimates on these solutions. V.A The energy We will define the energy for k≥3 as E2 k(t) = EE2(t) + τ|˜zα| 2ZQ2k+1 ∂k α(˜ K)2 |{z } A +ZQ2k−2∂k α( ˜ϕ)Λ(∂k α( ˜ϕ)) | {z } B +|˜zα|2τZ(Ck ˜ K(t)kH1+˜ K)Q2k+1∂k−1 α(˜ K)Λ(∂k−1 α(˜ K)) |{z } C + 2|˜zα|ZCk ˜ K(t)kH1Q2k−2∂k α( ˜ϕ)2 |{z } D +|˜zα|2ZσQ2k∂k−1 α(˜ K)2 |{z } E +|˜zα|2 m(Q2kσ)(t), where m(Q2kσ) = minα∈TQ2k(˜z(α, t))σ(α, t) and Cis a sufficiently large constant such that Cis strictly positive. Remember that ˜ϕwas introduced in Equation I.14. At this point is important to notice the following. Lemma V.2 The following sentences hold. 1. Let ˜ϕ∈H3+ 1 2,˜ω∈H2and z∈Hkwith k≥4. Then ˜ω∈H3. 2. Let ˜ϕ∈H3+ 1 2,˜ω∈H3and z∈Hkwith k≥5. Then ˜ω∈H3.5. 3. Let ˜ω∈H3+ 1 2, and ˜z∈Hkwith k≥5. Then ˜ϕ∈H3.5. This lemma shows that for a fixed τ > 0 the energy of this section is equivalent to this one in section IV.A. This allows us to use this energy to extend the solutions of the theorem IV.1 up to a time Twhich does not depend on τ(for a small enough τ). V.B The energy estimates Again, we will only focus on the new terms (A−E) since the estimates for the other ones were proved in [12] and in [8]. 21 V.B.1 ˜ K Proposition V.3 ˜ Kt=NICE3B +Q2 2|˜zα|3H(˜ωαα) + 1 |˜zα|3(Q2)αH(˜ωα) =NICE3B +1 |˜zα|2H( ˜ϕαα)−1 |˜zα|(˜ K˜ϕ)α, where NICE3B means ZQj∂k α(˜ K)∂k α(NICE3B)≤CEp k(t) for some positive constants C, p and any j. Proof: The first equality follows from the proof from the last section since the energies are equivalent (see Lemma V.2). We now prove the second one. We begin by using the relation (I.14) to get ˜ Kt= NICE3B + Q2 |˜zα|2H ˜ϕ Q2αα+Q2 |˜zα|H ˜c Q2αα +2(Q2)α |˜zα|2H ˜ϕ Q2α+2(Q2)α |˜zα|H ˜c Q2α=I+J We can easily see that ˜cα=−˜zα |˜zα|2·(Q2BR)α= NICE3B since it is at the level of ˜ωα,˜zαα but we gain one derivative by multiplying by the tangential direction. This proves that J= NICE3B + 2(Q2)α |˜zα|2H˜ϕα Q2. Looking now to ˜cαα we can see that ˜cαα =−˜zαα |˜zα|2·(Q2BR)α−˜zα |˜zα|2·(Q2BR)αα =I1+I2. Using the standard estimates, the only thing that causes trouble in I1is when all the derivatives hit ˜ωand therefore I1= NICE3B −KQ2 2|˜zα|H(˜ωα). Regarding I2, again, we need all the derivatives to hit BR to get the most singular terms, which are 22 I2= NICE3B −Q2 |˜zα|2˜zα·2 2πZπ −π (˜zα(α)−˜zα(β))⊥ |˜z(α)−˜z(β)|2˜ωα(α−β)dβ −Q2 |˜zα|2˜zα·1 2πZπ −π (˜z(α)−˜z(β))⊥ |˜z(α)−˜z(β)|2˜ωαα(α−β)dβ −Q2 |˜zα|2˜zα·1 2πZπ −π (˜zαα(α)−˜zαα(β))⊥ |˜z(α)−˜z(β)|2˜ωα(α−β)dβ = NICE3B + 2Q2 |˜zα|2 1 2 ˜zα·˜z⊥ αα |˜zα|2H(˜ωα)−Q2 |˜zα|2 ˜zα·˜z⊥ αα |˜zα|2H(˜ωα)−Q2 |˜zα|2 1 2 ˜ω |˜zα|2˜zα·H(˜z⊥ ααα) Collecting all the terms from I1and I2, we obtain ˜cαα Q2= NICE3B + 1 2|˜zα|H(( ˜ K˜ω)α) = NICE3B + 1 Q2H(( ˜ K˜ϕ)α). We can finally write the total contribution as ˜ Kt= NICE3B + Q2 |˜zα|2H˜ϕαα Q2−Q2 |˜zα|2H4Qα˜ϕα Q3 −1 |˜zα|(˜ K˜ϕ)α+2(Q2)α |˜zα|2H˜ϕα Q2 = NICE3B + Q2 |˜zα|2H˜ϕαα Q2−1 |˜zα|(˜ K˜ϕ)α = NICE3B + 1 |˜zα|2H( ˜ϕαα)−1 |˜zα|(˜ K˜ϕ)α as we wanted to prove.  V.B.2 ˜ϕ Throughout this section, we will use the following estimate which was proved in [8] for the case without surface tension. The proof is exactly the same for the case with it. ϕαt = NICE2B + ˜ϕ˜ϕαα |˜zα|−Q2σ˜ K+τQ2 2|˜zα|(˜ KQ)α+Mα, where NICE2B means ZQjΛ(∂k α( ˜ϕ))∂k−1 α(NICE2B)≤CEp k(t) for some positive constants C, p and any j. 23 V.C Calculations of the time derivative of the energy Using the previous lemmas and propositions, we can get the following estimates for the derivative of the energy: dA dt = OK + τ |˜zα|ZQ2k+1∂k α(˜ K)∂k α(H( ˜ϕαα)) −τZQ2k+1∂k α(˜ K)∂k α(( ˜ K˜ϕ)α) = OK + τ |˜zα|ZQ2k+1∂k α(˜ K)∂k α(H( ˜ϕαα)) −τZQ2k+1∂k α(˜ K)∂k+1 α( ˜ϕ)˜ K= OK + A1+A2 Again, we need to be careful while computing the derivative of Bas in Section IV. We obtain dB dt = 2 ZQ2k−2Λ(∂k α( ˜ϕ))∂k−1 α( ˜ϕαt) + Z(Q2k−2)αH(∂k α( ˜ϕ))∂k−1 α( ˜ϕαt) = OK −2ZQ2k−2Λ(∂k α( ˜ϕ))∂k−1 α˜ϕ˜ϕαα |˜zα| −2ZQ2k−2Λ(∂k α( ˜ϕ))∂k−1 α(Q2σ˜ K) +τ |˜zα|ZQ2k−2Λ(∂k α( ˜ϕ))∂k α(Q2(Q˜ K)α) −τ |˜zα|ZQ2kQαΛ(∂k α( ˜ϕ))∂k α(˜ K) +Zτ |˜zα|(k−1)Q2k+2QαH(∂k α( ˜ϕ))∂k+1 α(˜ K) = OK + B1+B2+B3+B4+B5 V.D Development of the derivative of the Bterm We begin noticing that B1= OK, as it was proved in [12]. Integrating by parts in B5, we have that B5=−Zτ |˜zα|(k−1)Q2k+2QαH(∂k+1 α( ˜ϕ))∂k α(˜ K) Furthermore, the only singular terms arising from B2are when all derivatives hit either ˜ Kor σ, this gives us B2= OK −2ZQ2kΛ(∂k α( ˜ϕ))∂k−1 α(σ)˜ K−2ZQ2kΛ(∂k α( ˜ϕ))∂k−1 α(˜ K)σ= OK + B2,1+B2,2. However, the only singular term of the Rayleigh-Taylor condition that is not in Hk−1is the one belonging to BRt(˜z, ˜ω)·˜zαwhen the time derivative hits ω, this means B2,1= OK −τZQ2kΛ(∂k α( ˜ϕ)) ˜ KH(∂k α(˜ KQ)) =−τZQ2k+1Λ(∂k α( ˜ϕ)) ˜ KH(∂k α(˜ K)) 24 Finally, developing B3we obtain B3=τ |˜zα|ZQ2k−2Λ(∂k α( ˜ϕ))∂k α(Q3˜ Kα) +τ |˜zα|ZQ2k−2Λ(∂k α( ˜ϕ))∂k α(Q2Qα˜ K) =B3,1+B3,2 Modulo lower order terms we can see that B3,2= OK + τ |˜zα|ZQ2kQαΛ(∂k α( ˜ϕ))∂k α(˜ K) We can continue splitting B3,1into B3,1= OK + τ |˜zα|ZQ2k+1Λ(∂k α( ˜ϕ))∂k+1 α(˜ K) + τ |˜zα|Z3kQ2kQαΛ(∂k α( ˜ϕ))∂k α(˜ K) = OK −τ |˜zα|ZQ2k+1Λ(∂k+1 α( ˜ϕ))∂k α(˜ K) + τ |˜zα|Z(k−1)Q2kQαΛ(∂k α( ˜ϕ))∂k α(˜ K) = OK + B3,1,1+B3,1,2 where in the last equality we have performed an integration by parts. We can observe that B3,2+B4=B3,1,2+B5= 0, B3,1,1+A1= 0 We will now see that B2,2cancels with the term arising from the derivative of E. Taking into account the previous lemmas dE dt = 2 ZσQ2k∂k−1 α(˜ K)H(∂k+1 α( ˜ϕ)) = OK −B2,2 Finally, we will see that the contributions from the time derivatives of Cand Dcancel B2,1and A2. We start by noticing that, modulo lower order terms A2=B2,1. Furthermore dC dt = OK + 2τZ(Ck ˜ K(t)kH1+˜ K)Q2k+1H(∂k+1 α( ˜ϕ))Λ(∂k−1 α(˜ K)) dD dt = OK + 2τZCk ˜ K(t)kH1Q2k+1 ∂k α( ˜ϕ)∂k+1 α(˜ K), which, by integration by parts results in dC dt +dD dt +A2+B2,1= OK. Adding all the contributions, we can bound the derivative in time of the energy by a power of the energy. 25 [26] S. Wu. Global wellposedness of the 3-D full water wave problem. Invent. Math., 184(1):125-220, 2011. [27] H. Yosihara. Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci., 18(1):49-96, 1982. [28] P. Zhang and Z. Zhang. On the free boundary problem of three-dimensional incompressible Euler equations. Comm. Pure Appl. Math., 61(7):877-940, 2008. Angel Castro D´epartement de Math´ematiques et Applications ´ Ecole Normale Sup´erieure 45, Rue d’Ulm, 75005 Paris Email: [email protected] Diego C´ordoba Charles Fefferman Instituto de Ciencias Matem´aticas Department of Mathematics Consejo Superior de Investigaciones Cient´ıficas Princeton University C/ Nicol´as Cabrera, 13-15 1102 Fine Hall, Washington Rd, Campus Cantoblanco UAM, 28049 Madrid Princeton, NJ 08544, USA Email: [email protected] Email: [email protected] Francisco Gancedo Javier G´omez-Serrano Departamento de An´alisis Matem´atico Instituto de Ciencias Matem´aticas Universidad de Sevilla Consejo Superior de Investigaciones Cient´ıficas C/ Tarfia, s/n C/ Nicol´as Cabrera, 13-15 Campus Reina Mercedes, 41012 Sevilla Campus Cantoblanco UAM, 28049 Madrid Email: fga[email protected] Email: [email protected] 32