Analytical and Computer-Assisted Proofs in Incompressible Fluids

Transcription

1 Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid Analytical and Computer-Assisted Proofs in Incompressible Fluids Javier Gómez Serrano Tesis doctoral dirigida por Diego Córdoba Gazolaz Madrid, Mayo 013

2 ii

3 iii Resumen y Conclusiones En esta memoria mostramos resultados sobre ecuaciones que provienen de la mecánica de fluidos incompresibles. En particular, tratamos problemas de frontera libre, que modelizan la evolución de la interfase que separa dos fluidos inmiscibles de diferentes densidades. Nuestro interés se ha centrado en el análisis de la formación de singularidades en tiempo finito. Uno de los problemas estudiados es el movimiento de las olas water waves en inglés, esto es, cuando los fluidos son agua y aire, que en nuestro caso asumiremos que tienen densidades iguales a 1 y a 0 respectivamente. Asumiremos también que dichos fluidos son irrotacionales y que la vorticidad el rotacional de la velocidad está concentrada en la interfase. Estas ecuaciones también son conocidas como el problema de frontera libre para las ecuaciones de Euler incompresibles. El otro problema que presentaremos es el problema de Muskat, que modeliza el comportamiento de la interfase entre dos fluidos incompresibles en un medio poroso, donde la ecuación del movimiento viene dada por la ley de Darcy. La estructura de esta disertación presenta dos partes claramente diferenciadas: la primera capítulos 1 a 3 corresponde a las técnicas clásicas del análisis y de las ecuaciones en derivadas parciales mientras que la segunda capítulos 4 a 6 utiliza el ordenador para probar de forma rigurosa los teoremas. La primera parte está dividida en tres capítulos. En el primero se realiza una introducción y un repaso del estado de la cuestión del problema de las water waves. En el segundo se presenta el resultado central de la tesis: la demostración de la formación de singularidades de tipo splash y splat, que corresponden físicamente al momento en el que la ola rompe al chocar consigo misma en un único punto splash o en una curva splat. Los ingredientes esenciales de la prueba comprenden una desingularización mediante una transformación conforme del dominio en el que ocurre la singularidad y estimaciones de energía para el problema en el nuevo dominio desingularizado con el fin de obtener existencia local. Dicha existencia local se demuestra tanto en el espacio de funciones analíticas como en espacios de Sobolev. Los resultados de este capítulo han sido publicados en [19] y [0]. En el tercero se estudia la influencia de la tensión superficial en el modelo y si ésta es capaz o no de prevenir la aparición de singularidades splash o splat. Aquí demostramos que dichas singularidades pueden surgir incluso en el caso en que haya tensión superficial. El estudio de este caso se puede encontrar publicado en [18]. La segunda parte se fragmenta en tres capítulos también. En el primero se realiza una introducción de la aritmética de intervalos y las pruebas asistidas por ordenador, haciendo énfasis en su uso en el marco del análisis y de las ecuaciones en derivadas parciales. En el segundo se describe un posible esquema de demostración del siguiente resultado: existen condiciones iniciales que inicialmente son un grafo, en un tiempo finito desarrollan una singularidad de tipo turning esto es, que la interfase deja de ser un grafo y finalmente colapsan en una singularidad de tipo splash. La primera parte del resultado fue demostrada por Castro, Córdoba, Fefferman, Gancedo y López-Fernández en [] mientras que la segunda es el capítulo dos de la primera parte de esta memoria. No es evidente la conexión entre ambos resultados a priori puesto que no se sabe que los conjuntos de soluciones que verifican cada

4 iv uno de los teoremas compartan elementos. En este capítulo se presentan resultados parciales en esta dirección y se sugiere cómo se podría completar el resto de la demostración mediante el uso extensivo de técnicas en las que predomina el uso del ordenador como herramienta rigurosa de demostración. En el tercer capítulo se usan las técnicas anteriores para demostrar rigurosamente una serie de teoremas sobre la formación de singularidades turning para el problema de Muskat. Castro, Córdoba, Fefferman, Gancedo y López-Fernández probaron en [] que existen datos iniciales que giran, pasando al régimen inestable. En nuestro caso realizamos un estudio sobre las condiciones en las que se puede dar el giro comparando diversos modelos: el modelo confinado en el que los fluidos se encuentran situados entre dos tapas situadas a una altura finita y el no confinado, así como los casos en los que el medio presenta un salto de permeabilidades modelo no homogéneo. El resultado de dicho trabajo se encuentra en [5]. Por último, se adjuntan los códigos correspondientes a las simulaciones numéricas de la primera parte y la prueba asistida por ordenador de la segunda parte en los apéndices A y B respectivamente.

5 v Abstract and Conclusions This dissertation is devoted to the study of equations arising in the field of fluid mechanics, more precisely incompressible fluids. In particular, we consider free boundary problems, which model the evolution of an interface between two immiscible fluids with different densities. Attention is focused on the analysis of finite time singularity formation. One of the studied problems is the so-called water waves problem, which approximates the behaviour of the sea waves, i.e. when the fluids are water and air, which in this case we think of having densities one and zero respectively. We will also assume that these fluids are irrotational and that the vorticity the curl of the velocity is concentrated on the interface. These equations also receive the name of the free boundary incompressible Euler equations. The other one is the Muskat problem, which models the behaviour of the interface between two incompressible fluids in a porous medium. In this case, the equation of movement is given by Darcy s Law. The structure of this work has two highly differentiated parts: the first part chapters 1 to 3 corresponds to the classical techniques coming from analysis and partial differential equations while the second chapters 4 to 6 employs the computer to rigorously prove the theorems. The first part is divided into three chapters. The first one consists of an introduction and a brief survey about the state of the art concerning the water waves problem. In the second one the main result of this thesis is presented: the proof of the formation of splash and splat singularities, which physically correspond to the moment in which the wave turns and breaks down while self-intersecting, either in a single point or along an arc. The main ingredients of the proof are a desingularization of the domain in which the singularity occurs by means of a conformal map and energy estimates for the new problem in the desingularized domain in order to obtain a local existence theorem. The space in which we can prove local existence can be either the space of analytic functions or a Sobolev space. The results of this chapter have been published in [19] and [0]. In the third chapter the influence of surface tension in the model is studied. More specifically, an answer to the question whether surface tension can prevent the appearance of splash or splat singularities is given. Such singularities can occur even when surface tension is present. This study can be found in [18]. The second part is fragmented into three chapters. In the first one an introduction to interval arithmetics and computer-assisted proofs is made, emphasizing in their use in the framework of analysis and partial differential equations. In the second one, we describe a possible approach to a proof of the following result: there exist initial conditions that initially can be written as a graph, develop a turning singularity this means the interface stops being a graph in finite time and finally collapse into a splash singularity. The first part of the result was proved by Castro, Córdoba, Fefferman, Gancedo and López-Fernández in [], while the second can be found in the second chapter of the first part of this manuscript. The connection between these two results is not evident since a priori it is not known whether the sets of solutions to both theorems have common elements. In this

6 vi chapter, partial results both rigorous and non rigorous are presented and suggestions about how the full proof could be completed are made. In our case, this completion is based in techniques in which the use of the computer as a rigorous theorem prover tool predominates. In the third chapter an example illustrating how the previous techniques can be put into practice is made. We prove several theorems concerning the formation of turning singularities for the Muskat problem. Castro, Córdoba, Fefferman, Gancedo and López-Fernández proved in [] that there exists a class of initial data that develops turning singularities for the Muskat problem, moving into the unstable regime. In our case, we carry out a study about the conditions under which the turning can be created by comparing different models: the confined model in which the fluids are hold between two fixed boundaries situated at a finite height and the non-confined model, as well as cases in which the medium presents a jump on the permeabilities. The result of this work appears in [5]. Finally, the codes corresponding to the numerical simulations of the first part and the rigorous computer-assisted proof of the second part can be found in appendices A and B respectively.

7 vii Agradecimientos Sin duda alguna, esta tesis no habría sido posible sin mi director, Diego Córdoba. Me gustaría agradecerle todo el tiempo y la dedicación que ha puesto, pero en especial su capacidad de motivación para sacar lo mejor de mí, aún y sobre todo cuando uno piensa que las cosas no van a salir como espera. Le doy las gracias por todas las oportunidades que me ha dado, porque la puerta de su despacho siempre estuvo abierta para mí y por su filosofía de trabajo, que han hecho de estos cuatro últimos años una época memorable. La posibilidad de trabajar con Charles Fefferman ha sido enriquecedora en todos los sentidos. He aprendido muchísimo de él, de su entusiasmo y visión de las matemáticas. También he sido testigo de cómo transformar lo imposible en una realidad de forma espectacular. Rafael de la Llave ha sido una de las personas que más influencia han tenido en este trabajo. Aparte de abrirme la puerta al mundo de las pruebas asistidas por ordenador y de demostrarme todo su poder, ha sido una gran fuente de ideas y las discusiones con él siempre han sido muy provechosas. Además, agradezco el tiempo que ha invertido en repasar esta memoria y sus comentarios ya que él ha sido el lector. También le estoy agradecido puesto que a través de él conocí a Jordi-Lluis Figueras, quien me invitó una semana a Uppsala: una gran experiencia. Quiero agradecer al resto del tribunal: Antonio Córdoba, Alberto Enciso, Daniel Faraco, David Lannes, Fabricio Macià y Daniel Peralta el interés que han mostrado en mi trabajo y la posibilidad de discutir con ellos sobre él. En especial agradezco a Dani Peralta que me acercara en coche a casa muchas tardes y me ahorrara un montón de tiempo de viaje. Mención aparte merece Angel Castro: aparte de ser un gran colaborador, siempre ha tenido el tiempo que hiciera falta para sentarse conmigo desde a calcular varias veces grandes cantidades de derivadas a depurar código, pasando por establecer teorías sobre mediciones mediante rendijas y su validez como teoremas. Además le doy las gracias a Angel por haberse leido la mayoría de esta tesis. Paco Gancedo ha sido un estupendo colaborador y le agradezco la paciencia que ha tenido siempre conmigo a la hora de hacer cuentas y todas las horas que ha pasado en la pizarra y a través de Skype explicándomelas. Estos años en el ICMAT no habrían sido lo mismo de no ser por dos grandes amigos, Miguel y Jezabel, con quienes he compartido momentos de los buenos y de los no tan buenos. Y por Rafa Granero, que me ha aguantado en el despacho y de quien he aprendido trabajando con él. No puedo olvidarme del vicedirector del ICMAT y tutor de esta tesis, Rafa Orive, que me ha ayudado siempre con todo lo relacionado con la UAM y me ha facilitado un montón la vida en el mundo burocrático. Por supuesto, algo que me llevo y que no olvidaré de todos estos años son todas las comidas, los cafés, las conversaciones y el buen ambiente con el resto de gente que ha pasado por el ICMAT y el departamento de la UAM, tanto con los que siguen como con los que ya se fueron, matemáticos o no. Me gustaría agradecer también a todos los profesores que han contribuido en mi formación, no solo matemática, sino también personal, las horas que han invertido haciéndolo. En particular, Albrecht Hess impulsó activamente mi vocación por las matemáticas durante mi etapa en el instituto y Salvador Roura consiguió establecer un grupo permanente de programadores del que aprendí muchísimo. También le estoy agradecido a toda la gente de la Olimpiada Matemática por el trabajo incansable que hacen día a día, y muy especialmente a María Gaspar y Josep Grané, quienes me animaron en todo momento y me han ayudado siempre que he tenido algún problema, fuera del tipo que fuera. Gracias a mis amigos, con quienes he pasado este tiempo muchos ratos en Madrid, en Barcelona o donde fuera, y pocos en esta última fase cuando no tenía tiempo para nada. Gracias por todos los viajes que hicimos juntos. Y a mi madre, de la que estoy orgulloso, que siempre ha estado ahí.

8 viii

9 ix A mi familia

10 x

11 Contents 1 Introduction to the Water Waves problem Statement of the Problem Splash singularity for water waves 9.1 Introduction Elementary Potential Theory Main Results Further Results Splash curves: transformation to the tilde domain and back Proof of real-analytic short-time existence in tilde domain Proof of short-time existence in Sobolev spaces in the tilde domain The Rayleigh-Taylor function in the tilde domain Definition of c in the tilde domain Time evolution of the function ϕ in the tilde domain Definition and a priori estimates of the energy in the tilde domain Estimates for BR Estimates for z t Estimates for ω t Estimates for ω Estimates for BR t Estimates for the Rayleigh-Taylor function σ Estimates for σ t Energy estimates on the curve Energy estimates for ω Finding the Rayleigh-Taylor function in the equation for ϕ t Higher order derivatives of σ Energy estimates for ϕ Energy estimates for z mq σt + 4 l=0 1 mq l t Proof of short-time existence Theorem

12 CONTENTS 3 Splash singularities for water waves with surface tension Introduction Properties of the curvature in the tilde domain Initial data Energy without the Rayleigh-Taylor condition The energy The energy estimates K ω Calculations of the time derivative of the energy Development of the derivative in B Collection of the terms High Order Low Order Type I Low Order Type II Regularized system High Order Low Order Type I Low Order Type II Energy with the Rayleigh-Taylor condition The energy The energy estimates K ϕ Calculations of the time derivative of the energy Development of the derivative of the B term Helpful estimates for the Birkhoff-Rott operator Introduction to Computer-Assisted Proofs Computer-Assisted Proofs and Interval arithmetics Automatic Differentiation Integration From a graph to a Splash Singularity Introduction Bounds for ft and gt Representation of the functions and Interpolation Rigorous bounds for Singular integrals Estimates of the norm of the Operator I + T Bounds for Ct and k Writing the differential inequality as a differential system of equations Estimates for the linear terms with Q = Estimates for the linear terms with Q derivatives in Q: Linear terms

13 CONTENTS derivative in Q: Linear terms derivatives in Q: Linear terms derivatives in Q: Linear terms derivatives in Q: Linear terms derivatives in Q: Totals derivative in Q: Totals derivatives in Q: Totals derivatives in Q: Totals derivatives in Q: Totals Writing the linear system for D and its derivatives Future improvements Proof of Theorem Computing the difference z x and ω γ Computing the difference ϕ ψ A Computer-Assisted Proof for the Muskat problem Introduction Main Results Technical details concerning the proofs A Water Waves: Codes 185 A.1 waterwaves potato.m A. my sqrt.m A.3 my angle.m A.4 adams bashforth potato iterative.m A.5 fode ww bhl potato iterative.m A.6 fode ww bhl potato iterative with gamma.m A.7 dh.m A.8 zp.m A.9 ftrans.m A.10 iftrans.m A.11 compute gamma iterative.m A.1 dzphi potato.m A.13 inverse potato.m B Turning waves for Muskat: Codes 195 B.1 Additional functions for C-XSC B.1.1 Code added to itaylor.hpp B.1. Code added to itaylor.cpp B.1.3 Code added to dimtaylor.hpp B.1.4 Code added to dimtaylor.cpp B. Theorem B.3 Theorem B.4 Theorem

14 4 CONTENTS

15 Chapter 1 Introduction to the Water Waves problem 1.1 Statement of the Problem The water wave equations or D incompressible free boundary Euler equations describe a system consisting of a connected water region Ωt R and a vacuum region R \ Ωt, evolving as a function of time t, and separated by a smooth interface Ωt = {z, t : R. We write Ω 1 t = R \ Ωt, Ω t = Ωt. The fluid velocity vx, y, t R and the pressure px, y, t R are defined for x, y Ωt. The fluid is assumed to be incompressible and irrotational v = 0, curl v = 0 in Ωt, 1.1 and to satisfy the D Euler equation [ t + v x ]vx, y, t = px, y, t 0, g in Ωt, 1. where g > 0 accounts for the gravitational acceleration. Neglecting surface tension, we assume that the pressure satisfies p = p t at Ωt, where p t is a function of t alone. 1.3 Finally, we assume that the interface moves with the fluid, i.e., t z, t = vz, t, t + c #, t z, t, 1.4 where c #, t is an arbitrary smooth function of, t the choice of c # affects only the parametrization of Ωt and z, t = z 1, t, z, t. At an initial time t 0, we specify the fluid region Ωt 0 and the velocity vx, y, t 0 x, y Ωt 0, subject to the constraint 1.1. We then solve equations with the given initial conditions, and we ask whether a singularity can form in finite time from an initially smooth 5

16 6 CHAPTER 1. INTRODUCTION TO THE WATER WAVES PROBLEM velocity v, t 0 and fluid interface Ωt 0. In this part of the thesis, we prove that water waves in two space dimensions can form a singularity in finite time by either of two simple, natural scenarios, which we call a splash and a splat. The water wave problem comes in three flavors: Asymptotically Flat: We may demand that z, t, 0 0 as ±. Periodic: We may instead demand that z, t, 0 is a π-periodic function of. Compact: Finally, we may demand that z, t is a π-periodic function of. To obtain physically meaningful solutions in the Asymptotically Flat and Periodic flavors, we demand that px, y, t + gy = O1 in Ωt and that Ωt vx, y, t dxdy < finite energy, 1.5 where we abuse of notation and identify Ωt with Ωt T R, T = R/πZ, in the Periodic case. Moreover, in Chapter 3 we study the relevance of considering the Laplace-Young condition for which the pressure on the interface Ωt is proportional to its curvature, meaning that the surface tension effect is considered: pz, t, t = τ z, t z, t z, t 3 τ K. 1.6 Above τ > 0 is the surface tension coefficient. In this thesis, we restrict attention to periodic water waves, although our arguments can be easily modified to apply to the other flavors. See Remark.1.5 below. Let us summarize some of the previous work on water waves. The existence and Sobolev regularity of water waves for short time is due to S. Wu [88]. Her proof applies to smooth interfaces that need not be graphs of functions, but [88] assumes the arc-chord condition z, t zβ, t c AC β, all, β R. 1.7 The constant c AC > 0 is called the arc-chord constant, which may vary with time. The issue of long-time existence has been treated in Alvarez-Lannes [5], where wellposedness over large time scales is shown, and several asymptotic regimes are justified. By taking advantage of the dispersive properties of the water-wave system, Wu [90] proved exponentially large time of existence for small initial data. In three space dimensions, Wu [89] proved short-time existence; and Germain et al [50], [51] and Wu [91] proved existence for all time in the case of small initial data. We draw the attention of the reader to two recent preprints: the first one by Ionescu and Pusateri [63] and

17 1.1. STATEMENT OF THE PROBLEM 7 the second one by Alazard and Delort, [3] in which they prove existence for all time in the two-dimensional case for small initial data. There are several important variants of the water wave problem. One can drop the assumption that the fluid is irrotational. See Christodoulou-Lindblad [6], Lindblad [70], Coutand-Shkoller [34], Shatah-Zeng [83], Zhang-Zhang [95]. Lannes [68] considered the case in which water is moving over a fixed bottom. Ambrose-Masmoudi [8] considered the case where the equations include surface tension, and the limit where the coefficient of surface tension tends to zero. Lannes [69] discussed the problem of two fluids separated by an interface with small non-zero surface tension. Alazard et al. [] took advantage of the dispersive properties of the equations to lower the regularity of the initial data. See also the papers of Córdoba et al. [8] and Alazard-Metivier [4]. In the case of large data for the two-dimensional problem , Castro et al. in [], [1] showed that there exist initial data for which the interface is the graph of a function, but after a finite time the water wave turns over and the interface is no longer a graph. For previous numerical simulations showing this turning phenomenon, see Baker et al. [11] and Beale et al. [13]. Next, we describe a singularity that can form in water waves. We start by presenting what we believe based on numerical simulations; then, we explain what we can prove. a The initial water region Ωt 0. b The water region Ωt 1 at a later time t 1. c A splash forms at time t > t 1. Figure 1.1: Evolution of a splash singularity. Our simulations show an initially smooth water wave, for which the fluid interface is a graph as in Figure 1.1a. At a later time t 1, the water wave has turned over as described in [], [1], i.e., the interface is no longer a graph. Finally, in Figure 1.1c, the fluid interface self-intersects at a single point 1, but is otherwise smooth. We call this scenario a splash, 1 Here, we regard the fluid interface as sitting inside T R; recall that our water waves are π-periodic under horizontal translation.

18 8 CHAPTER 1. INTRODUCTION TO THE WATER WAVES PROBLEM and we call the single point at which the interface self-intersects, the splash point. Note that the arc-chord condition holds for times t < t, but the arc-chord constant tends to zero as t tends to t. Now let us explain what we can prove regarding the splash scenario. Recall that [], [1] already proved that a water wave may start as in Figure 1.1a and later evolve to look like Figure 1.1b. In this part of the thesis, we prove that a water wave may start as in Figure 1.1b, and later form a splash, as in Figure 1.1c. We would like to prove that an initially smooth water wave may start as in Figure 1.1a, then turn over as in Figure 1.1b, and finally produce a splash as in Figure 1.1c. To do so, our plan is to use interval arithmetic [73] to produce a rigorous computer-assisted proof that, close to the approximate solution arising from our numerics, there exists an exact solution of that ends in a splash. The stability result announced in [19, Theorem 4.1] is a first step in this direction. A more detailed version of the plan will be given in Chapter 5. A variant of the splash singularity is shown in Figures 1.a and 1.b. a The initial water region b The splat. Figure 1.: Evolution of a splat singularity. The water wave starts out smooth, as in Figure 1.a, although the interface is not a graph. At a later time, the interface self-intersects along an arc, but is otherwise smooth. In the next section, we prove that water waves can form a splat. The stability theorem announced in [19] and proved in Chapter 5, Section 5.4 shows that a sufficiently small perturbation of the splash will come from an initial condition which is close to the one that lead to the splash. This holds true also for the splat. We make no claim that the splash and the splat are the only singularities that can arise in solutions of the water wave equation.

19 Chapter Splash singularity for water waves.1 Introduction.1.1 Elementary Potential Theory To formulate precisely our main results, and to explain some ideas from their proofs, we recall some elementary potential theory for irrotational divergence-free vector fields vx, y, t defined on a region Ωt R with a smooth periodic boundary {z, t : R for fixed t. We assume that v is smooth up to the boundary and π-periodic with respect to horizontal translations. We suppose that v has finite energy. Such a velocity field v may be represented in several ways: We may write v = φ for a velocity potential φx, y, t defined on Ωt and smooth up to the boundary. We may also write v = ψ = y ψ, x ψ for a stream function ψ, defined on Ωt and smooth up to the boundary. The normal component of v at the boundary, given by u normal, t = vz, t, t z, t z, t uniquely specifies v on Ωt. Here, u = u, u 1 for u = u 1, u R, and we always orient Ωt so that the normal vector z, t points into the vacuum region R \ Ωt. The function u normal, t satisfies u normal, t z, t d = 0, but is otherwise arbitrary. T Note that, because v has finite energy, φ and ψ are π-periodic with respect to horizontal translations. Without the assumption of finite energy, φ and ψ could be periodic plus linear. The functions φ and ψ are conjugate harmonic functions. 9

20 10 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES There is another way to specify v, namely vx, y, t = 1 π P V x z 1 β, t, y z β, t ωβ, tdβ, x, y Ωt.1 R x z 1 β, t, y z β, t for a π-periodic function ωβ, t called the vorticity amplitude. See [11]. Formula.1 holds only in the interior of Ωt. Taking the limit as x, y z 1, t, z, t Ωt from the interior, we find that vz, t, t = BRz, ω, t + 1 ω, t z, t z, t,. where BR denotes the Birkhoff-Rott integral BRz, ω, t = 1 π P.V. z 1, t z 1 β, t, z, t z β, t ωβ, tdβ..3 R z 1, t z 1 β, t, z, t z β, t To see that v may be represented as in.1,., one applies the Biot-Savart law to a discontinuous extension of v from its initial domain Ωt to all of R ; to make the extension, one solves a Neumann problem in R \ Ωt. Thus, our velocity field v admits multiple descriptions. Note that the description in terms of ω is significantly different from the descriptions in terms of φ, ψ and u normal, because we bring in the Neumann problem on R \ Ωt to justify.1 and.. When Ωt is a splash curve as in Figure 1.1c, there is no problem defining φ and it is smooth up to the boundary, except that it can take two different values at the splash point, for obvious reasons. The same is true of ψ. Similarly, u normal, t continues to behave well. However, there is no reason to believe that ω, t will be well-defined and smooth for a splash curve, since R \ Ωt is a somewhat pathological domain. Our numerics suggest that max ω, t C t, where t s t s is the time of the splash. Let us apply the above potential theory to the water wave problem. A standard formulation of the problem [11] takes z, t and ω, t as unknowns. Standard computations see e.g. [8, Section ] show that the water wave problem is equivalent to the following equations and t z, t = BRz, ω, t + c, t z, t.4 t ω, t = z, t t BRz, ω, t ω 4 z, t + c, tω, t + c, t z, t BRz, ω, t g z, t..5 Here, c, t is a function that we may pick arbitrarily, since it influences only the parametrization of Ωt. For future reference, we write down several standard equations

21 .1. INTRODUCTION 11 that follow from by routine computation and elementary potential theory. x φx, y, t = x ψx, y, t = 0 in Ωt; φ and ψ are harmonic conjugates. px, y, t = t φx, y, t 1 φx, y, t gy n ψ z,t = Φ, t, where Φ, t = φz, t, t z, t and n is the outward-pointing unit normal to Ωt. ψx + π, y, t = ψx, y, t and φx + π, y, t = φx, y, t ψx, y, t = O1 as y v = ψ in Ωt t z, t = vz, t, t + c, t z, t t Φ, t = 1 vz, t, t + c, tvz, t, t z, t gy, t + p t..6 We may write u, t to denote vz, t, t..1. Main Results Our main result is the following theorem. For the definition of a splash curve see Definition in Section.. The interface shown in Figure 1.1c is an example of a splash curve. Theorem.1.1 Let z 0 be a splash curve, where the splash point is given by z 0 1 = z 0, 1. Let u 0 normal be a scalar function in H4 T, satisfying u 0 normal z 0 d = 0.7 and T u 0 normal 1, u 0 normal < 0..8 Then there exist a time T > 0; a time-varying domain Ωt defined for t [0, T ] and a velocity field vx, y, t defined for x, y Ωt, t [0, T ] such that the following hold: Ωt and vx, y, t solve the water wave equations for all t [0, T ]..9 Ωt is given as a parametrized curve {z, t : R, with z, t, 0 π-periodic in for fixed t..10 z, t, 0 C[0, T ], H 4 T and vz, t, t C[0, T ], H 3 T.11 z, 0 = z 0 and u normal, 0 = u 0 normal for all R..1 For each t [0, T ], the curve Ωt satisfies the arc-chord condition, but the arc-chord constant tends to zero as t 0..13

22 1 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES This result was announced in [19]. To prove that splash singularities can form, we note that the water wave equations are invariant under time reversal. Therefore, it is enough to exhibit a solution of the water wave equations that starts as a splash at time zero, but satisfies the arc-chord condition for each small positive time. Theorem.1.1 provides such solutions. Since the curve touches itself it is not clear if the vorticity amplitude is well defined, although the velocity potential remains nonsingular. In order to get around this issue we will apply a transformation from the original coordinates to new ones which we will denote with a tilde. The purpose of this transformation is to be able to deal with the failure of the arc-chord condition. Let us consider the scenario in the periodic setting and then the transformation defined by z, t P z, t where P is a conformal map that will be given as: P z = z 1/ tan and the branch of the root will be taken in such a way that it separates the self-intersecting points of the interface. We will also need that the interface passes below the points ±π, 0 or, equivalently, that those points belong to the vacuum region in order for the tilde region to lie inside a closed curve and the vacuum region to lie on the outer part. See Figures.1 and 3.1. Here P z will refer to a dimensional vector whose components are the real and imaginary parts of P z 1 + iz. Its inverse is given by 1 iw P 1 w = i log 1 + iw = arctanw for w C. In this setting, P 1 z will be well defined modulo multiples of π. Remark.1. Note that P z is periodic such that P z + kπ = P z. Moreover, P z is one-to-one in the water region and single-valued except at the splash point. Remark.1.3 Although the transformation to the tilde domain is convenient, the real reason for Theorem.1.1 is that the potential theory inside the water region does not go bad as we approach the splash even though it goes bad in the vacuum region. We define the following quantities: ψ x, ỹ, t ψp 1 x, ỹ, t, φ x, ỹ, t φp 1 x, ỹ, t, ṽ x, ỹ, t φ x, ỹ, t, Φ, t = φ z, t, t, Ψ, t = ψ z, t, t. Also we define Ωt = P Ωt. Let us note that since ψ and φ are π periodic, the resulting ψ and φ are well defined. We do not have problems with the harmonicity of ψ or φ at the point which is mapped from minus infinity times i which belongs to the water region by P since φ and ψ tend to finite limits at minus infinity times i. Also, the periodicity of φ and ψ causes φ and ψ to be continuous and harmonic at the interior of P Ω t.

23 .1. INTRODUCTION y x Figure.1: Splash singularity at times t = 0 Red - splash, t = Blue - turning and t = Black - graph. Let us assume that there exists a solution of.6 and that we take u normal = Ψ z such that u normal 1, u normal < 0 for all 0 < t < T, with T small enough, thus z, t satisfies the arc-chord condition and does not touch the removed branch from P w. The system.6 in the new coordinates reads ψ x, ỹ, t = 0 in P Ω t n ψ = Φ, t z,t z, t ṽ ψ in P Ω t z t, t = Q, tũ, t + c, t z, t Φ t, t = 1 Q, t ũ, t + c, tũ, t z, t gp 1 z, t z, 0 = z 0 Φ, 0 = Φ 0 = Φ 0,.14 where ũ is the limit of the velocity coming from the fluid region in the tilde domain and Q z, t, t = dp dw P 1 z, t, Q, t = dp z, t dw. We can solve the Neumann problem in the complement of Ωt. Therefore we can represent the velocity field ṽ in terms of a vorticity amplitude ω. We will see that z and ω satisfy the following equations z t, t = Q, tbr z, ω, t + c, t z, t..15

24 14 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES ω t, t = t BR z, ω, t z, t BR z, ω Q, t + c, t BR z, ω z, t Q, t 4 ω, t z, t + c, t ω, t g P 1 z, t..16 Remark.1.4 Equations are analogous to In fact, if we set Q 1 in we recover Our strategy will be the following: we will consider the evolution of the solutions in the tilde domain and then see that everything works fine in the original domain. We will have to obtain the normal velocity once given the tangential velocity, and viceversa. To do this, we just have to notice that Φ, t = ũ, t z, t = BR z, ω z, t + ω, t. From that, we can invert the equation see [8] and get ω. Equation. in the tilde domain then tells us ṽ on the boundary Ωt. We now note that a solution of the system.14 in the tilde domain gives rise to a solution of the system.6 in the non-tilde domain, by inverting the map P. In fact, this will be the implication used in Theorem.1.1 finding a solution in the tilde domain, and therefore in the non-tilde. Remark.1.5 It is likely that a similar argument works for the other two settings closed contour and asymptotic to horizontal by choosing an appropriate P w that separates the singularity. For example, for the closed contour we could consider P clo z = z, taking the branch so that it separates the singularity, and for the asymptotic to horizontal scenario, it is enough to move the interface such that the water region is entirely contained in the lower halfplane and the point i belongs to the vacuum region and apply the relation z+i z i = z+i z i. We now state the local existence results that lead to the proof of the existence of a splash singularity Theorem.1.1. To avoid the failure of the arc-chord condition, we will prove the local existence in the tilde domain. This can be done in two different settings, namely in the space of analytic functions and the Sobolev space H s. For the analytic version we define we consider the space S r = { + iη, η < r, f L S r = ± f ± ir d, f r = f L S r + 3 f L S r, H 3 S r = { f analytic in S r, f r <, f π-periodic

25 .1. INTRODUCTION 15 and we take z 1, z, Φ H 3 S r 3 X r. The first results concerning the Cauchy problem for small data in Sobolev spaces near the equilibrium point are due to Craig [36], Nalimov [75] and Yosihara [9]. Beale et al. [14] considered the Cauchy problem in the linearized version. For local existence with small analytic data see Sulem-Sulem [85]. Our main results regarding local existence in the tilde domain are the following theorems: Theorem.1.6 Local existence for analytic initial data in the tilde domain Let z 0 be a splash curve and let u 0 z0 z = Φ0 0 be the initial tangential velocity such that z 0 for some r 0 > 0, and satisfying: z 0 1, z 0,, Φ 0 X r0, 1. u 0 normal 1 = u normal 1, 0 < 0, u 0 normal = u normal, 0 < 0. u 0 normal z 0 d = 0. T Then there exist a finite time T > 0, 0 < r < r 0, a time-varying curve z, t and a function Φ, t satisfying: 1. P 1 z 1, t, P 1 z, t are π-periodic,. P 1 z, t satisfies the arc-chord condition for all t 0, T ], and ũ, t with z 1, t, z, t, Φ, t C[0, T ], X r which provides a solution of the water wave equations.14 with z 0 = P z 0 and ũ, 0 z, 0 = ũ 0 z 0. The main tool in the proof is an abstract Cauchy-Kowalewski theorem from [78] and [79]. For more details see [4]. For the proof of local existence in Sobolev spaces we will take the following c, t: c, t = + π π Q z β β, t BR z, ω β β, t z β β, t dβ z β β, t z β β, t dβ. Q BR z, ω β β, t This choice of c will ensure that z, t depends only on t. We will also define an auxiliary function ϕ, t analogous to the one introduced in [14] for the linear case and [8] nonlinear case which helps us to bound several of the terms that appear: ϕ, t = Q, t ω, t z, t Then, we can prove the following theorem: c, t z, t..17

26 16 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Theorem.1.7 Local existence for initial data in Sobolev spaces in the tilde domain In the setting of Section.1.1, let z 0 be the image of a splash curve by the map P parametrized in such a way that z 0 does not depend on, and such that z 0 1, z0 H 4 T. Let ϕ, 0 H 3+ 1 T be as in.17 and let ω, 0 H T. Then there exist a finite time T > 0, a time-varying curve z, t C[0, T ]; H 4, and functions ω, t C[0, T ]; H and ϕ C[0, T ]; H 3+ 1 providing a solution of the water wave equations The proof is based on the adaptation of the local existence proof in [8] to the tilde domain. Some of the relevant estimates from [8] obviously hold here as well, with essentially unchanged proofs. We state such results in Lemmas.4. and Lemmas.4.5,...,.4.9 below; and refer the reader to the relevant sections of [8] for the proofs. However, [8] contains several miracles, i.e., complicated calculations and estimates that lead to simple favorable results for no apparent reason. To see that analogous miracles occur in our present setting, we have to go through the arguments in detail; see Lemmas.4.10 and.4.1,...,.4.15 below. We have tried to make it possible to check the correctness of our arguments without extreme effort, and without undue repetitions from [8]. It would be very interesting to understand a-priori why the miracles in this paper and in [8], [8] occur. Presumably there is a simple, conceptual explanation, which at present we do not know. At the end of Section. we will define the notion of a splat curve. The curve depicted in Figure 1.b is an example of a splat curve. In the statement of Theorem 3.5.1, we may take z 0 to be the image of a splat curve under P rather than the image of a splash curve. The proof of Theorem goes through for this case with trivial changes. Consequently, we obtain an analogue of Theorem.1.6, with hypothesis 1 replaced by Hypothesis 1 : u 0 normal = u normal, 0 is negative for all I 1 I, where I 1, I are the intervals appearing in the definition of a splat curve in Section.. Just as Theorem.1.6 implies the formation of splash singularities for water waves, the above analogue of Theorem.1.6 for splat curves implies Corollary.1.8 Splat singularity There exist solutions of the water wave system that collapse along an arc in finite time, but remain otherwise smooth..1.3 Further Results Here we mention some immediate consequences of our results which are relevant: 1. Splash and Splat singularities for 3D water waves It is possible to extend our results to the periodic three dimensional setting by considering scenarios invariant under translation in one of the coordinate directions. While preparing the final revisions of this manuscript, we noticed that in a very recent arxiv posting [35], Coutand-Shkoller consider additional 3D splash singularities.

27 .. SPLASH CURVES: TRANSFORMATION TO THE TILDE DOMAIN AND BACK ỹ x Figure.: Tilde domain at times t = 0 Red - splash, t = Blue - turning and t = Black - graph.. No gravity The existence of a splash singularity can also be proved in the case where the gravity constant g is equal to zero, as long as the Rayleigh-Taylor condition holds.. Splash curves: transformation to the tilde domain and back In this section we will rewrite the equations by applying a transformation from the original coordinates to new ones which we will denote by tilde. The purpose of this transformation is to be able to deal with the failure of the arc-chord condition. 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..1 We say that z = z 1, z is a splash curve if 1. z 1, z are smooth functions and π-periodic.. z satisfies the arc-chord condition at every point except at 1 and, with 1 < where z 1 = z and z 1, z > 0. This means z 1 = z, but if we remove either a neighborhood of 1 or a neighborhood of in parameter space, then the arc-chord condition holds. 3. The curve z 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 = z, z 1 z points to the vacuum region. We regard the interface to be part of the water region.

28 18 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES y x Figure.3: Zoom of the splash singularity at times t = 0 Red - splash, t = Blue - turning and t = Black - graph. 4. We can choose a branch of the function P on the water region such that the curve z = z 1, z = P z satisfies: a z 1 and z are smooth and π-periodic. b z is a closed contour. c z satisfies the arc-chord condition. We will choose the branch of the root that produces that independently of x. lim P x + iy = e iπ/4 y 5. P w is analytic at w and dp dw w 0 if w belongs to the interior of the water region. Furthermore, ±π, 0 and 0, 0 belong to the vacuum region. 6. z q l for l = 0,..., 4, where q 0 = 0, 0, q 1 = 1, 1 1, q =, 1 1, q 3 =, 1 1, q 4 =, From now on, we will always work with splash curves as initial data unless we say otherwise. Condition 6 will be used in the local existence theorems and can be proved to hold for short enough time as long as the initial condition satisfies it. It is also immediate to check that the previous choice of P transforms any periodic interface into a closed curve. Here are two examples of curves which are not splash curves see Figure.4.

29 .. SPLASH CURVES: TRANSFORMATION y 0 y x x Figure.4: Two examples of non splash curves. Now we will show a careful deduction of the equations in the tilde domain. From the definition of z we have that and z, t = P z, t z, t.19 z t, t = P z, t z t, t = P z, t u, t + c, tz, t = P z, t u, t + c z, t..0 Since φ = φ P and v = φ = φ P, we obtain This implies that v i = i φ = i φ P = j j φ P P j x i = j ṽ j P i P j..1 Plugging this into.0 we get u, t = P z, t T ũ, t.. z t, t = P z, t P z, t T ũ, t + c z, t..3 From the Cauchy-Riemann equations P z, t P z, t T = Q, t Id, Q, t = dp z dz In this particular case, this means that Q, t = 1 + z, t 4 4 z, t, z, t = z 1, t + i z, t...4 Recall that Φ is the restriction of φ to the interface, i.e. Φ, t = φ z, t, t. Then

30 0 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Thus, Φ satisfies Φ, t = φ z, t, t = φp 1 z, t, t = φz, t, t = Φ, t.5 Φ t = 1 u, t + c, tu, t z, t gz, t = 1 P z, tt ũ, t + c, tũ, t z, t gp 1 z, t,.6 where the subscript in the gravity term of the last line denotes the second component. Thus the system.6 in the new coordinates reads ψx, y, t = 0 in P Ω t n ψ = Φ, t z,t z, t ṽ ψ in P Ω t z t, t = Q, tũ, t + c, t z, t Φ t, t = 1 Q, t ũ, t + c, tũ, t z, t gp 1 z, t + p t z, 0 = z 0 Φ, 0 = Φ 0 = Φ 0..7 We have seen that ṽ can be represented in the form ṽ x, ỹ, t = 1 π ψ x, ỹ, t = π P.V x z 1, t, ỹ z, t ω, td. x z 1, t, ỹ z, t Taking limits from the fluid region we obtain ũ, t = BR z, ω + ω z z. The evolution of ω is calculated in the following way. First, let us recall the equations z t, t = Q, tũ, t + c, t z, t Φ t, t = 1 Q, t ũ, t + c, tũ, t z, t gp 1 z, t Φ, t = ũ, t z, t z, 0 = z 0 Φ, 0 = Φ 0 = Φ 0..8 Substituting the expression for ũ, t and performing the change c, t = c, t + 1 Q, t ω,t z,t we obtain

31 .. SPLASH CURVES: TRANSFORMATION 1 z t, t = Q, tbr z, ω, t + c, t z, t Φ, t = BR z, ω, t z, t + 1 ω, t Φ t, t = 1 Q, t ũ, t + c, tũ, t z, t gp 1 z, t = 1 Q, t BR z, ω, t Q, t 8 ω, t z, t + c, tbr z, ω z, t c, t ω, t gp z, t..9 On the one hand, by taking derivatives with respect to t in the second equation follows Φ t, t = t BR z, ω, t z, t + BR z, ω, t z t, t + ω t, t = t BR z, ω z, t + BR z, ω Q, t + Q, tbr z, ω BR z, ω + c, tbr z, ω z, t + c, tbr z, ω z, t + ω t, t..30 On the other, taking derivatives with respect to in the third equation in.9 yields Φ t, t = 1 BR z, ω Q, t + Q, tbr z, ω BR z, ω 1 Q, t ω, t 4 z, t + c, tbr z, ω z, t + c, t BR z, ω z, t + c, tbr z, ω z, t + 1 c, t ω, t gp 1 z, t..31 Combining both equations, we find that ω t, t = t BR z, ω, t z, t BR z, ω Q, t + c, t BR z, ω z, t Q, t 4 ω, t z, t + c, t ω, t gp 1 z, t..3 We will proceed in the following way: we will consider the evolution of the solutions in the tilde domain and see that everything works fine in the original domain. For example, the sign condition on the normal vectors in the non-tilde domain has an equivalent form in the tilde domain i.e. the two normal components have negative sign. In the non-tilde domain, this implies that the interface moves away from the branch removed from the square root, and therefore the interface touches neither the branch cut nor the conflictive points q l see Condition 6 in Definition Hence P and P 1 will be well defined and one-to-one. See Figure.7.

32 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Let us note that getting φ = φ P is not a problem since φ is bounded and harmonic. Moreover, as ṽ = ψ and v = P T ṽ P and P has exponential decay at infinity, the velocity v belongs to L Ω t [, π] R. Remark.. Ψ, Φ, u and z have easy transformations to the tilde domain but ω has not. We would like to discuss what happens to the amplitude of the vorticity ω in the non-tilde domain as the curve approaches the splash. If the vorticity belongs to C[0, T splash ], C δ T, then the normal velocity should be continuous at the splash point and therefore the normal component of the restriction of the velocity to the curve from the water region cannot have the same sign at z 1 and z see Theorem.1.1. This means that the C δ norm of the amplitude of the vorticity becomes unbounded at the time of the splash. We illustrate this phenomenon by plotting 1/ max ω see Figure.5, where the blue curve is the calculated ω and the red curve is a potential fitting to the data as numerical instabilities don t allow us to compute ω with enough precision when we are in the regime which is close to the splash. Time has been reversed so that the splash occurs at time t = 0 and the interface separates from itself at t > 0. 7 x /max ω t x 10 4 Figure.5: Vorticity amplitude in the nontilde domain. The vorticity reaches infinity at a rate 1 1 of approximately T splash t T splash t. The fit is given by F = 3.7 t We also have performed numerical simulations in order to get a blowup rate for the arcchord condition. As in Figure.5, we plot the inverse of the arc-chord constant. The blue curve is made by the calculated points and the red curve is the interpolating one. We see a very good fitting. Time follows the same convention as before and the numerical evidence indicates a blowup of the arc-chord as 1 T splash t. The results can be seen in Figure.6. The numerics that led us to Figures 1.1a, 1.1b and 1.1c were performed using the method of Beale-Hou-Lowengrub [14], with special modifications to maintain accuracy up to the splash i.e. taking into account the impact of Q on the equation. The code was written first in Matlab, and then ported into C++ GSL [49] to optimize in terms of speed. We

33 .. SPLASH CURVES: TRANSFORMATION 3 1. x Arc Chord Constant t x 10 5 Figure.6: Arc-chord condition in the non-tilde domain. The arc-chord reaches infinity at a rate of approximately 1 T splash t. The fit is given by F = t enclose the Matlab code in Appendix for clarity reasons. Actual results from our simulations are shown in Figures.1, 3.1 and.3. Figures 1.1 and 1. are cartoons. Instead of having an evolution equation for ω, Beale-Hou-Lowengrub introduce a velocity potential φ and study its evolution through time subject to the constraint imposed by being a potential. This is the set of equations.8. The initial data on the non-tilde domain was given by: z1 0 = + 1 3π sin + 1 sin + 1 π sin3 z 0 = 1 10 cos 3 10 cos cos3 Note that z π = z π splash. Instead of prescribing an initial condition for ω, we prescribed the normal component of the velocity to ensure a more controlled direction of the fluid. From that we got the initial ω, 0 using the following relations. Let ψ be such that ψ = v and Ψ its restriction to the interface. Recall that we can transform the initial condition on the normal component of the velocity into an initial condition on the tangential component by applying the transformations described in section. The initial normal velocity is then prescribed by setting u 0 n z = Ψ = 3 cos 3.4 cos + cos cos4. The simulations were done using a spatial mesh of N = 048 nodes and a time step t = The time direction was set to run backwards from the splash to the graph and the graph was obtained at approximately T g = We also kept track of the energy conservation. If we consider the following energy not to be confused with the one in Section.4: E S t = 1 Ω f t vx, y, t dxdy + 1 gz, t z 1, td E k t + E p t.33

34 4 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES where z, t = z 1, t, z, t, u, t = vz, t, t, and Ω f t = Ω t [, π] R is a fundamental domain in the water region in a period, then we can see that the energy is conserved; this is a check of the accuracy of our numerics. de k t dt = vx, y, tv t x, y, t + vx, y, t vx, y, tdxdy Ω f t = vx, y, t px, y, t g0, 1dxdy Ω f t = vx, y, t px, y, t + gydxdy Ω f t = vx, y, t n gyds Ω f t = gz, tu, t z, td.34 where we have used the incompressibility of the fluid v = 0 and the continuity of the pressure on the interface p t Ω t = 0. Next de p t dt = = = gz, t t z, t z 1, td + 1 gz, t t z, t z 1, td gz, t t z 1, td gz, t z, t t z 1, td gz, tu, t z, td..35 This proves that the energy is constant. Note that: vx, y, t dxdy = φx, y, t dxdy Ω f t Ω f t = φx, y, t φx, y, tdxdy Ω f t + φx, y, t φx, y, t n dxdy Ω f t φ=0 = Ω f t φx, y, t φx, y, t n dxdy.36 so the numerical calculation is restricted to the values at the boundary. We observe that the energy of our system is conserved, as we have E S t , max t E S t min E S t t E S t min t We now give the proof of Theorem.1.1 using Theorem

35 .3. PROOF OF REAL-ANALYTIC SHORT-TIME EXISTENCE IN TILDE DOMAIN 5 Proof of Theorem.1.1: Using the fact that there is local existence to the initial data in the tilde domain and applying P 1 to the solution obtained there, we can get a curve z, t that solves the water wave equation in the non tilde domain. Details on the local existence in the tilde domain are shown below. Note that the sign condition.8 assumed in Theorem.1.1 guarantees that for positive time t the curve in the nontilde domain will separate as depicted in Figure.7a instead of crossing itself as depicted in Figure.7b. More precisely, we check that for small positive time t the curve z, t = z 1, t, z, t = P 1 z, t R/πZ R is a simple closed curve, i.e. that z, t is one-to-one. Indeed, if not, there exist a sequence of positive times t ν 0 and points ν, ν such that ν ν mod πz, but z ν, t ν = z ν, t ν. Since the initial splash curve z, 0 satisfies the modified chord-arc condition described in Condition of Definition 3.3.1, we may assume without loss of generality that ν 1 and ν with 1, as in Definition The sign condition.8 therefore guarantees that for large ν, z ν, t ν and z ν, t ν lie in the image of the open time-zero water region under the map P. Moreover for large ν, z ν, t ν z ν, t ν since z 1, 0 z, 0. Since P 1 is one-to-one on the image of the open time-zero water region under P, it follows that for large ν we have z ν, t ν z ν, t ν R/πZ R, with z, t P 1 z, t. This contradicts the defining condition z ν, t ν = z ν, t ν, completing the proof that z, t is a simple closed curve for small positive t. The proof of Theorem.1.1 is complete. We end this section by defining a splat curve, as promised in Section.1. To do so, we simply modify our Definition for a splash curve, by replacing Condition in that definition by the following Condition : We are given two disjoint closed non-degenerate intervals I 1, I [0, π whose images under z 1, z R/πZ R coincide. The map z 1, z R/πZ R satisfies the chord-arc condition when restricted to the complement of any open interval J such that J I 1 or J I. As promised, the curve depicted in Figure 1.b is a splat curve. Observe that the curve in Figure 1.b cannot be real-analytic..3 Proof of real-analytic short-time existence in tilde domain The main goal of this section is to prove Theorem.1.6. In order to accomplish this task we will prove local well-posedness for the system.37 below. In this section, we will drop the tildes from the notation. The system arises from.8 taking c = 0: z t = dp dw P 1 z u Φ t = 1 dp dw P 1 z u gp 1 z u = BRz, ω + ω z z Φ = ω + BRz, ω z dp dw P 1 z, t = z 1,t+iz,t 4 z 1,t+iz,t P 1 z, t = log i+z 1,t+iz,t i z 1,t+iz,t..37

36 6 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES We demand that z 0 0, 0 to find the function dp dw P 1 z, t well defined. This condition is going to remain true for short time. We also consider z 0 q l, l = 1,..., 4 in 3.8 to get P 1 z, t well defined. Again this is going to remain true for short time. The main tool in this section is a Cauchy-Kowalewski theorem see [, Section 5] for more details. We recall the following definitions S r = { + iη, η < r, f L S r = ± f ± ir d, f r = f L S r + 3 f L S r, the space H 3 S r = { f analytic in S r, f r <, f π-periodic and we now take z 1, z, Φ H 3 S r 3 X r. We have the following theorem: Theorem.3.1 Let z 0 be a curve satisfying the arc-chord condition z 0 z 0 β β > 1 M which doesn t touch the points q l, l = 0,..., 4 in 3.8,and z 0, Φ 0 X r0 for some r 0 > 0. Then, there exist a time T > 0 and 0 < r < r 0 such that there is a unique solution to the system.37 in C[0, T ], X r with initial conditions z, 0 = z 0, Φ, 0 = Φ 0, for all T. Equation.37 can be extended for complex variables: z t + iξ, t = F 1 z + iξ, t, Φ + iξ, t, Φ t + iξ, t = F z + iξ, t, Φ + iξ, t. Here where we abuse notation by writing dp dw P 1 z + iξ, t F 1 z, Φ = dp dw P 1 z u = 1 Π 4 l=1 [z 1 + iξ, t q1 l + z + iξ, t q l ] 16 z 1 + iξ, t + z + iξ, t and u + iξ, t = BRz + iξ, t, ω + iξ, t + 1 ω + iξ, t z + iξ, t z 1 + iξ, t + z + iξ, t with BRz + iξ, t, ω + iξ, t =

37 .3. LOCAL EXISTENCE IN ANALYTIC SPACES TILDE DOMAIN 7 1 π P V z + iξ β, t z + iξ, t, z 1 + iξ, t z 1 + iξ β, t T z 1 + iξ, t z 1 + iξ β, t ω+iξ β, tdβ + z + iξ, t z + iξ β, t and ω given implicitly by Φ = ω + BRz, ω z. We will also abuse notation by writing u for u 1 + u, even for complex u = u 1, u. The operator F is given by where P 1 z + iξ, t = 1 F z, Φ = 1 dp dw P 1 z u gp 1 z 4 1 l log[z 1 + iξ, t q1 l + z + iξ, t q l ]. l=1 Below we will use a strip of analyticity small enough so that the complex logarithm above is continuous. We use the following proposition: Proposition.3. Consider 0 r < r and the open set O X r given by: O = {z, Φ X r : z i r, Φ r < R, inf z 1 + iξ q1 l + z + iξ q l > R, +iξ S r with l = 0,..., 4, inf Gz + iξ, β > R +iξ Sr β [,π] Gz + iξ, β = z 1 + iξ z 1 + iξ β + z + iξ z + iξ β then the function F = F 1, F for F : O X r is a continuous mapping. In addition, there is a constant C R depending on R only such that F z, Φ r β C R r r z, Φ r.38 and F z, Φ F z 1, Φ 1 r C R r r z z 1, Φ Φ 1 r.39 sup F 1 z, Φ + iξ F 1 z, Φ + iξ β C R β.40 +iξ Sr β [,π] for z, z j, Φ, Φ j O.

38 8 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Proof: First we point out that ω is given in term of Φ and z by the implicit equation Φ = ω + BRz, ω z 1 I + Jω. It is well known that the operator I + J is invertible on L for real functions with mean zero see [8, Section 5] for more details. Writing ω ± ir = Φ ± ir 1 z ± ir zβ z ± ir π z ± ir zβ ωβdβ one can find that T ω L S r Φ L S r + C R ω L S 0 where C R depends on R for z, Φ O. The bound of I + J 1 for real functions yields Thus Analogously, one finds that ω L S 0 I + J 1 L L Φ L S 0 C R Φ L S r. ω L S r C R Φ L S r. ω L S r C R Φ r. This allows us to assert that ω is at the same level as Φ in terms of derivatives: ω L S r + ω L S r C R Φ r C R Φ r..41 Then, inequality.38 follows as in [, Section 6.3]. We will see how to deal with the most singular terms. For the first term in the norm, it is easy to find that F z, Φ L S r C R z, Φ r C R z, Φ r..4 In order to control the second one, we will show how to deal with F 1 as F is analogous. Here we point out that the functions dp dw P 1 z + iξ, t, P 1 z + iξ, t have no loss of derivatives and they are regular as long as z, Φ O. Therefore, in F 3 1 the most singular term is given by dp dw P 1 z + iξ, t u 3 + iξ, t as the rest can be estimated in an easier manner see [8, Section 6.1] as an example with more details. From the definition it is easy to bound dp dw P 1 z in L, it remains to control u 3 in L S r. To simplify the exposition we ignore the time dependence of the functions, we denote γ = ± ir, z 1 γ z 1 γ β + z γ z γ β zγ zγ β,

39 .3. LOCAL EXISTENCE IN ANALYTIC SPACES TILDE DOMAIN 9 and Next, we split as follows z 1 γ + z γ z γ, z γ β z γ, z 1 γ z 1 γ β zγ zγ β. 3 u = I 1 + I + I 3 + I 4 + I 5 + I 6 + l.o.t. where l.o.t. denotes lower order terms which can be estimated in an easier manner. We have and I = 1 π P V For I 6 we find I 1 = 1 π P V zγ 3 zγ 3 β zγ zγ β ωγ βdβ, zγ zγ β zγ zγ β zγ zγ β 3 zγ zγ βωγ βdβ, 3 I 3 = 1 π P V and since z, Φ O we get zγ zγ β zγ zγ β ωγ 3 βdβ, I 4 = 1 ωγ zγ 4 z γ, I 5 = 1 ωγ zγ z γ zγ zγ 4 I 6 = 1 I 6 L S r 1 z L S r ωγ 3 zγ z γ. inf γ Sr β [,π] Gzγ, β 1 3 ω L S r I 6 L S r C R 3 ω L S r by using Sobolev embedding. A simple application of the Cauchy formula gives which allows us to find The bound.41 gives finally f L S r I 6 L S r I 6 L S r C r r f L S r C R r r ω L S r. C R r r Φ r.

40 30 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES In a similar way we obtain I 4 L S r + I 5 L S r C R 4 z L S r In I 3 we decompose further: I 3 = I 3,1 + I 3, where I 3,1 = 1 π P V Kγ, β ωγ 3 βdβ, I 3, = 1 where H denotes the Hilbert transform and the kernel K is given by C R r r z r. z γ z γ H ωγ, 3 zγ zγ β zγ zγ β z γ z γ 1 tanβ/. We can integrate by parts β ωγ β in I 3,1 to find I 3,1 L S r C R ω L S r C R Φ r see [8, Section 3] for more details. The term I 3, can be estimated by I 3, L S r C R H 3 ω L S r = C R 3 ω L S r A similar splitting in I = I,1 + I, with I,1 = 1 π P V Lγ, β zγ 3 zγ βdβ, 3 I, = ωγz γ z γ z γ Λ 3 zγ, where Λ = H gives the kernel L as follows C R r r Φ r. Lγ, β 3 zγ 3 zγ β = ωγz γ z γ z γ 3 zγ 3 zγ β 4 sin β/ + ωγ βzγ zγ β zγ zγ β zγ zγ β zγ 3 zγ β. 3 Heuristically, we regard this operator as no better or no worse than a Hilbert transform of 3 z. It is easy to prove that I,1 L S r C R 3 z L S r C R Φ r see [8, Section 6.1] for more details. The term I, can be bounded as follows I, L S r C R Λ 3 z L S r = C R 4 z L S r C R r r z r.

41 .3. LOCAL EXISTENCE IN ANALYTIC SPACES TILDE DOMAIN 31 Analogously, for I 1 we find I 1 L S r C R r r z r. This strategy allows us to deal with 3 u and therefore with 3 F 1. The same applies to 3 F and we can get finally.38. To get.39 we write where for z j O and j = 1,. This implies Φ 1 = 1 I + J z 1ω1, Φ = 1 I + J z ω J z jω = BRz j, ω z j which yields Φ Φ 1 = ω ω 1 + BRz, ω ω 1 z + BRz, ω 1 z BRz 1, ω 1 z 1 ω ω 1 = I + J z 1 Φ Φ 1 I + J z 1 BRz, ω 1 z BRz 1, ω 1 z 1. This helps us to find ω ω 1 L S r + ω ω 1 L S r CR Φ Φ 1 r + z z 1 r. We use a decomposition similar to the one used to prove.38 which allows us to get finally.39. Inequality.40 follows in an easier manner. Proof of Theorem.3.1: We apply the following result of Nirenberg [78] and Nishida [79]. Abstract Cauchy-Kowalewski Theorem: Consider the equation dut = F ut for t < δ.43 dt with initial condition u0 = u 0 X r0.44 For some numbers Ĉ, ˆR > 0, assume the following hypothesis: For every pair of numbers r, r such that 0 < r < r < r 0, F is a Lipschitz map from {u X r : u u 0 Xr < ˆR Ĉ into X r, with Lipschitz constant at most. Then the r r equation.43 with initial condition.44 has a solution ut in C[ δ, δ], X r for small enough r, δ > 0. The above Abstract Cauchy-Kowalewski Theorem is obviously equivalent to a special case of Nishida s Theorem [79], although our notation differs from that of [78]. In place of.43, Nirenberg and Nishida treat the more general equation dut = F ut, t. dt

42 3 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES The proof of the Abstract Cauchy-Kowalewski Theorem in [78] proceeds by showing that the obvious iteration scheme k+1 u 0 Z t = u + t F uk sds 0 converges in Xr for small enough r depending on t. du Our system.37 has the form = F u for u = z, Φ. Proposition.3. tells us that dt the hypothesis of the Abstract Cauchy-Kowalewski Theorem holds for the system.37. In particular, for R > 0 small enough, we obtain the arc-chord condition for every u = z, Φ such that kz, Φ z 0, Φ0 kxr < R for any arbitrarily small r > 0. Hence, the conclusion of Theorem.3.1 follows from the Abstract Cauchy-Kowalewski Theorem. Proof of Theorem.1.6: Applying Theorem.3.1, we obtain a solution of the water wave equation, with the correct initial conditions, in the tilde domain. Passing from the tilde domain back to the original problem, we obtain a solution of the water wave equations as asserted in Theorem.1.6. We have to make sure that, for small positive time, the splash curve evolves as in Figure.7a, rather than Figure.7b a Good b Bad Figure.7: Two different evolutions of the interface. This is guaranteed by the hypothesis of Theorem.1.6 regarding the sign of the normal component of the initial velocity at the splash point..4 Proof of short-time existence in Sobolev spaces in the tilde domain In this section we will show how to obtain a local existence theorem for the water wave equations in the tilde domain. The proof is based on energy estimates and uses the fact that the Rayleigh-Taylor function is positive.

43 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN The Rayleigh-Taylor function in the tilde domain We begin by recalling the function ϕ, t, which will be studied in detail in Section.4.3 and in the definition of the Rayleigh-Taylor condition, by the expression ϕ, t = Q ω, t z, t c z, t..45 Next we introduce the R-T function: σ BR t z, ω + ϕ z BR z, ω + Q BR z, ω + ω z z z + ω z z t + ϕ z z z Q z z + g P 1 z z..46 This function σ coincides with the expression z, t p z, t, t, where p = p P 1. Indeed, it is easy to check that And taking the gradient on the equation.47 yields ṽ t + 1 In addition we know that and therefore Q t φ + ṽ = p gp 1 + p t..47 Q ṽ + Q ṽ ṽ = p g P ṽ z, t, t = BR z, ω, t + d dtṽ z, t, t = tbr z, ω, t + t On the other hand, by using.48 we have ω, t z, t z, t + ω, t z, t z, t.49 ω, t z, t t z, t..50 dtṽ z, d t, t = tṽ z, t, t + t z, t ṽ z, t, t = 1 Q ṽ z, t, t Q ṽ z, t, t ṽ z, t, t p z, t, t g P 1 z, t + t z, t ṽ z, t, t..51 Furthermore the equation.9 together with.49 gives rise to t z, t =Q ṽ z, t, t Q ω, t z, t z, t + c z, t =Q 1 Q ω, t ṽ z, t, t z, t z, t c z, t z, t..5

44 34 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Therefore by.45, we obtain By introducing.53 in.51 we have t z, t =Q ṽ z, t, t ϕ, t z, t z, t..53 dtṽ z, d t, t = 1 Q ṽ z, t, t Q ṽ z, t, t ṽ z, t, t p z, t, t g P 1 z, t Therefore + Q ṽ z, t, t ṽ z, t, t ϕ, t z, t ṽ z, t, t. z, t dtṽ z, d t, t = 1 Q ṽ z, t, t ϕ, t ṽ z, t, t z, t p z, t, t g P 1 z, t..54 Next we take a derivative with respect to in the equation.49 to get z, t + ṽ z, t, t = BR z, ω, t + ω, t z, t Multiplying equation.54 by z, t and using.55 we learn d t, t z dtṽ z,, t = Q Q z, t ṽ z, t, t ω, t z, t ϕ, t z, t BR z, ω, t z, t ϕ, t ω, t z, t z, t z, t z, t z, t..55 p z, t, t z, t g P 1 z, t z, t..56 On the other hand, by multiplying.50 by z, t we have d t, t z dtṽ z,, t = t BR z, ω z ω, t + z, t t z, t z, t..57 From.56 and.57 we find t BR z, ω z, t + = Q Q z, t ṽ z, t, t ϕ, t ω, t z, t z, t ω z, t t z, t z, t ϕ, t z, t BR z, ω, t z, t z, t z, t p z, t, t z, t g P 1 z, t z, t..58

45 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 35 Finally, rearranging the terms in.58 yields p z, t, t z, t = t BR z, ω, t + ω, t ϕ, t + z, t t z, t + z, t z + Q ω, t BR z, ω, t + z, t z, t ϕ, t z, t BR z, ω, t z, t z, t + g P 1 Q z, t, and then, comparing with 5., we obtain the desired result p z, t, t z, t = σ, t. z, t Note that for the tilde domain, the Rayleigh-Taylor condition is the same as in the first domain, i.e: where p = p P 1 and p, t z, t = p, t z, t z, t = P z, t z, t z, t = J P z, tj z, t 0 1 where J is the rotation matrix. Together with the Cauchy-Riemann equations 1 0 this implies that Moreover Hence J P z, tj = P z, t. p, t = P z, t T p, t. p, t, z, t = P z, t T p, t, P z, t 1 z, t.59 = p, t, z, t..60 By taking the divergence on the Euler equation and because the flow is irrotational in the interior of the regions Ω j t follows p = v 0 which, together with the fact that the pressure is zero on the interface and px, y, t + gy = O1 when y tends to,then follows by Hopf s lemma in Ω t that σ, t z, t n pz, t, t > 0,

46 36 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES except in the case v = 0. This argument was suggested by Hou and Caflisch see [89], although the proof of the positivity of the Rayleigh-Taylor condition in the nontilde domain for all time was first introduced by Wu in [88]. The above proof shows that σ > 0 provided our domain Ωt arises by applying the map P to a domain Ωt with smooth boundary. Here, Ωt may be a splash curve, but we cannot allow boundaries Ωt whose inverse images under P look like figure.7b. Nevertheless, since σ > 0 for the image of P applied to a splash curve, we know that σ > 0 at time t = 0 in the context of Theorem Our estimates below will guarantee that the condition σ > 0 persists for a short time. Thus, in proving Theorem 3.5.1, we may use the positivity of σ..4. Definition of c in the tilde domain From now on, we will drop the tildes from the notation for simplicity. We will choose the following tangential term: c = + π π Here and in.9 we find and Q BR β z β z β dβ Q BR β P 1 = P 1 i + z 1, t + iz, t z, t = log i z 1, t + iz, t Q = Qz, t = z 1, t + iz, t 4 4 z 1, t + iz, t. z β dβ..61 z β These functions are regular as long as z, t q l. We deal with initial data which satisfy the above condition and we will show that it s going to remain true for short time. In order to measure it we define mq l t = min T z, t ql for l = 0,..., 4. We also point out that, because of our choice of c, t, solutions of satisfy that as in [9, Equations. -.5]. z, t = At for any T.4.3 Time evolution of the function ϕ in the tilde domain Recall that we have defined an auxiliary function ϕ, t adapted to the tilde domain, which helps us to bound several of the terms that appear: ϕ, t = Q, tω, t z, t c, t z, t..6

47 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 37 We will show how to find the evolution equation for ϕ t. We have and therefore that yields The equation for ω t reads: ϕ = Q ω z c z ϕ Q = Q ω 4 z + c z Q cω ϕ Q ω Q = 4 z c z ϕ ω t = BR t z QQ BR +cbr z {{ Q 1a Q + cω. c z + Q {{ 1b For the quantity 1 = 1a + 1b we write c z 1 = 1a + 1b = cbr z + = cbr z + c z = c[br z + z πq = c[ z πq Q BR β Q BR β Q and then.63 becomes ϕ ω t = BR t z QQ BR + c z πq Furthermore Q BR β gp Q c z Q Q 3 z β z β dβ Q BR z Q c z Q Q 3 ] z β z β dβ Q BR z c z Q Q Q 3 ] Q ω ϕ t = QQ t z Q ω z 3 z z t + Q ω t z tc z z β z β dβ 4cQ BR z c z Q Q Q 3 ω = QQ t z Q ω 1 Q BR β z π z β dβ + Q ϕ [ BR t z QQ BR z Q + c z πq Q BR β z β z β z β dβ 4cQ BR z c z Q Q Q 3.64 gp 1. ] gp 1 t c z.

48 38 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES We should remark that we have used that For simplicity, we denote Computing We can write ω ϕ t = QQ t z Q ω z Bt {{ a +c z Bt {{ b z z t = 1 π Q BR β z β dβ. Bt = 1 Q z β BR β dβ..65 π z β z BR t z Q3 Q BR Q ϕ z z Q Q cqq BR z c z Q Q z Q z =a + b = Btϕ, gp 1 t c z and it yields ϕ t = ϕbt Q z ϕ Q Q BR t z z + gp 1 z + QQ t ω z cbr z z QQ Q Q c z Q3 z Q BR t c z..66 We will use the equation above to perform energy estimates..4.4 Definition and a priori estimates of the energy in the tilde domain Let us consider for k 4 the following definition of energy Et: Et = 1 + z Q zσ H t + k 1 z k z d + Fz L t + ω H t + ϕ z t + k H k 1 mq σt + 4 l=0 1 mq l t,.67 where Fz = β z z β,, β [, π], and mq σ = min T {Q z, tσ, t. In the next section we shall show a proof of the following lemma.

49 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 39 Lemma.4.1 Let z, t and ω, t be a solution of Then, the following a priori estimate holds: for k 4 and C and p constants depending only on k. d dt Et CEp t,.68 The following subsections are devoted to proving Lemma.4.1 by showing the regularity of the different elements involved in the problem: the Birkhoff-Rott integral, z t, t, ω t, t, ω, t; BR t, t, the R-T function σ, t and its time derivative σ t, t Estimates for BR In this section we show that the Birkhoff-Rott integral is as regular as z. Lemma.4. The following estimate holds BRz, ω H k C Fz L + z H k+1 + ω H k j,.69 for k, where C and j are constants independent of z and ω. Remark.4.3 Using this estimate for k = we find easily that BRz, ω L C Fz L + z H 3 + ω H j,.70 which shall be used throughout the paper, where C and j are universal constants. Proof: The proof can be done as in [8, Section 6.1] since the definition for the Birkhoff-Rott operator is independent of the domain Estimates for z t In this section we show that z t is as regular as z. Lemma.4.4 The following estimate holds z t H k C Fz L + z H k+1 + ω H k + for k, where C and j are constants that depend only on k. 4 l=0 j 1 mq l,.71 t Proof: It follows from [8, Section 6.]. The only additional thing we need to control is an L norm of Q, which we can easily bound by the mq l terms which control the distance from the curve to the q l points, more precisely, the one that controls the distance from the origin.

50 40 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Estimates for ω t This section is devoted to showing that ω t is as regular as ω. Lemma.4.5 The following estimate holds ω t H k C Fz L + z H k+ + ω H k+1 + ϕ H k+1 + for k 1, where C and j are constants that depend only on k. 4 l=0 j 1 mq l,.7 t Proof: We use formula.64 and proceed as in [8, Section 6.3]. Note that in [8] an exponential growth appears in the bound of the estimates for the nonlocal operator acting on ω t see equation.64. However, in a recent paper [30] the authors get a polynomial growth for the operator in both and 3 dimensions. Note that even the exponential growth is still good enough to prove Theorem Estimates for ω In this section we show that the amplitude of the vorticity ω lies at the same level as z. We shall consider z H k T, ϕ H k 1 T and ω H k T as part of the energy estimates. The inequality below yields ω H k 1 T. Lemma.4.6 The following estimate holds ω H k 1 C Fz L + z H k + ω H k + ϕ H k 1 + for k 3, where C and j are constants that depend only on k. 4 l=0 j 1 mq l,.73 t Proof: We can apply the same techniques as in [8, Section 6.4] since the most singular terms are treated there and the other terms are harmless and can be easily estimated. The impact of Q is now taken into account by the mq l terms which now cover all of the points q 0,..., q Estimates for BR t. Here we prove that the time derivative of the Birkhoff-Rott integral is at the same level as z. Lemma.4.7 The following estimate holds BR t H k C Fz L + z H k+ + ω H k+1 + ϕ H k+1 + for k, where C and j are constants that depend only on k. 4 l=0 j 1 mq l,.74 t Proof: We proceed as in [8, Section 6.5], where BR t appears in the formula 5.. We use.71 and.7 to bound z t and ω t in BR t respectively.

51 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN Estimates for the Rayleigh-Taylor function σ Here we prove that the Rayleigh-Taylor function is at the same level as z. Lemma.4.8 The following estimate holds σ H k C Fz L + z H k+ + ω H k+1 + ϕ H k+1 + for k, where C and j are constants that depend only on k. 4 l=0 j 1 mq l,.75 t Proof: We proceed as in [8, Section 6.5] using formula 5.. There is a new term in the definition of σ, namely Q BRz, ω + ω z z Qz z, but this term is less singular than BR t z, ω z. Hence, the new term causes no trouble Estimates for σ t In this section we obtain an upper bound for the L norm of σ t that will be used in the energy inequalities and in the treatment of the Rayleigh-Taylor condition. Lemma.4.9 The following estimate holds σ t L C where C and j are universal constants. Fz L + z H 4 + ω H 3 + ϕ H l=0 j 1 mq l,.76 t Proof: Again, as in the previous subsection, the new term is less singular than the terms treated in [8, Section 6.6]. Hence we deal with them with no problem Energy estimates on the curve In this section we give the proof of the following lemma when, again, k = 4. The case k > 4 is left to the reader. Regarding z 4 L let us remark that we have z 4 Q σ z L t = Q σ z 4 z z Q σ d mq σt z 4 z d..77 T Lemma.4.10 Let z, t and ω, t be a solution of priori estimate holds: for d z dt H + k 1 St = T Then, the following a Q σ z k z d + Fz L St + CE p t,.78 and k 4, where C and p are constants that depend only on k. Q σ k z z z 3 Λ k 1 ϕd,.79

52 4 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES The term St is uncontrolled but it will appear in the equation of the evolution of ϕ with the opposite sign. Proof: Using.71 and.77 one gets easily We obtain d dt z H 3 C CE p t. z z t + 3 z 3 z t d d dt Fz L CEp t in a similar manner as in [8, Section 7.]. It remains to deal with the quantity d π Q σ π Q dt z 4 z σ d = z t 4 z Q σ d + z 4 z z 4 t d =I 1 + I. The bounds.71,.77 and.76 give us I 1 CE p t. Next for I we write Q σ π I = z 4 z Q 4 Q σ BRd + z 4 z c 4 zd = J 1 + J. The most singular terms in J 1 are given by K 1, K, K 3 and K 4 : K 1 = 1 π P V Q 4 σ z 4 z 4 z z 4 z z ω dβd, and K = π P V K 3 = 1 π P V K 4 = π Q 4 σ z 4 z z z z z 4 z z z 4 z 4 ω dβd, Q 4 σ z 4 z z z z z 4 ω dβd, Q 3 σ z 4 z BR Qz 4 zd, where the prime denotes a function in the variable β, i.e. f = f β.

53 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 43 Then we write: K 1 = 1 π P V = 1 π z P V + 1 π z P V = L 1 + L. Q 4 σ z z 4 4 z zβ 4 z zβ ωβdβd z 4 4 z zβ 4 z zβ 4 z 4 z 4 zβ z zβ Q 4 σωβ + Q 4 βσβω dβd Q 4 σωβ Q 4 βσβω dβd That is, we have performed a manipulation in K 1, allowing us to show that L 1, its most singular term, vanishes: L 1 = = = 0. 1 π z P V 1 π z P V zβ 4 4 z zβ 4 z zβ 4 z 4 zβ 4 z 4 zβ z zβ Q 4 σωβ + Q 4 βσβω dβd Q 4 σωβ+q 4 βσβω dβd The term L involves a S.I.O. Singular Integral Operator acting on 4 z thanks to the minus sign between the two terms Q 4 σω. One can show that L C Fz L z k H 3 ω C 1,δ σ C 1,δ Q 4 C 1,δ 4 z L CE p t. Inside K we find that z z 4 z 4 z can be written as follows: z z 4 z 4 z = z z z β 4 z 4 z βz z 4 z + βz 4 z z 4 z,.80 then using that z 4 z = 3 z 3 z,.81 we can split K as a sum of S.I.O.s operating on 4 z, plus a kernel of the form η,β β acting on z 3 z with η C allowing us to obtain again the estimate K CE p t. Note that below we will also use a variant of.81, namely The term K 3 is a sum of z 4 z 4 z = z z 4 z 3 z 3 z z 3 z..8 L 3 = 1 π Q 4 σ z 4 z [ z z z z z ] z 4 tanβ/ ω dβd,

54 44 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES plus the following term: L 4 = Q 4 σ 4 z z z 4 H 4 ωd. We can integrate by parts on L 3 with respect to β since ω 4 = β ω 3. This calculation gives a S.I.O. acting on ω 3 which can be estimated as before. Next in L 4 we write and decompose further L 4 = + + L 4 = Q σ 4 z z [ Q z 3 Q 4 σ 4 z z z 4 Λ 3 ωd Q σ 4 z z z 3 Λ 3 ϕd z Λ 3 ω Λ 3 Q z ω Q σ 4 z z z 3 Λ 3 c z d = M 1 + S + M 1, ] d for St given by.79. In M 1 we find a commutator that allows us to obtain Using.61 for M 1 we have where M 1 = M 1 CE p t. H Q σ 4 z z z 3 4 c z d = N 1 + N + N 3 + N 4, N 1 = N = H Q σ 4 z z z 3 Q BR 4 z z d, H Q σ 4 z z z 3 Q 4 BR N 3 = 4 H Q σ 4 z z Q Q 4 z 3 zbr z z d, z z d, and N 4 is given by the rest of the terms which can be controlled easily by the estimates from Section for the Birkhoff-Rott integral. Regarding N 1 a straightforward calculation gives and analogously for N 3 N 1 CE p t, N 3 CE p t.

55 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 45 Again, in N we consider the most singular terms given by O 1 = O = H Q σ 4 z z z z 3 H Q σ 4 z z z π z 3 z Q π P V z 4 z 4 z z ω dβd, z z z z 4 z z 4 z 4 z ω ddβ, z Q π P V O 3 = H Q σ 4 z z Q z z 3 z BRz, 4 ωd. Using the decomposition.80 we can easily estimate O as in our discussion of K. In O 3 we find z BRz, ω 4 = z π z z z β z z ω 4 dβ. Above we can integrate by parts as in our discussion of L 3. We find that O 3 CE p t. Next we split O 1 into a S.I.O. acting on z 4, which can be estimated as before, plus the term P 1 = H Q σ 4 z z z 3 Q ω z z 3 Λ 4 z d. Then the following estimate for the commutator Q ω z z 3 Λ 4 z ΛQ ω z z 3 4 z L CE p t yields where Using that We can write P 1 CE p t + R R = Q σ 4 z z z 3 Q ω z z 3 4 z d. HfΛgd = f gd. R = Q σ 4 z z z 3 Q ω 4 z z z 3 d + Q σ 4 z z z 3 Q ω 4 z z z 3 d and a straightforward integration by parts let us control R. This calculation allows us to get P 1 CE p t.

56 46 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES We can easily show that K 4 CE p t because we can bound Q 3 σbr Q in L. So finally we have controlled J 1 in the following manner: J 1 CE p t + S. To finish the proof let us observe that the term J can be estimated integrating by parts, using the identity z 4 z = 3 z 3 z to treat its most singular component. We have obtained T Q σ z 4 z z cd 4 = 3 T and this yields the desired control Energy estimates for ω In this section we show the following result. 1 z Q σ 3 z z 3 cd Lemma.4.11 Let z, t and ω, t be a solution of priori estimate holds: for k 4, where C and p are constants that depend only on k. Then, the following a d dt ω H k t CE p t,.83 Proof: We will discuss the case k = 4, leaving the other cases to the reader. Formula.7 shows easily that d dt ω H t Fz L t + z H t + ω 4 H t + ϕ 3 H t + 3 which together with.73 yields d dt ω H t CE p t. 4 l=0 j 1 mq l t Finding the Rayleigh-Taylor function in the equation for ϕ t. In this section we get the R-T function in the evolution equation for ϕ t. Lemma.4.1 Let z, t and ω, t be a solution of Then, the following identity holds: ϕ t = NICE ϕ z ϕ Q σ z z z 3.84

57 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 47 where NICE satisfies and Λ k 1 ϕ k NICEd CE p t.85 NICE H k CE p t.86 for k 4, where C and p are constants that depend only on k. Proof: We will give the proof for k = 4. From now on, when we show that a term f satisfies Λ 3 ϕ fd CE p t and f H CE p t we say that this term is NICE. Then, f becomes part of NICE and by abuse of notation we denote f by NICE. Notice that, whenever we can estimate the L norm of Λ 1/ f by CE p t, then f is NICE. We use.66 to compute Q ϕ ϕ t = Btϕ z Q Q 1 z gp z BR t + z z {{ ω + QQ t z c z {{ t. 3b Expanding 3 = 3a + 3b: cbr 3 =3a + 3b = Q BR t 3a z z QQ c z t z = Q BR t z z z Q BR t z = Q z BR t z Bt z t + Q BR z Q Q c z z Bt Q BR Q 3 z BR Q z z t z + QQ t BR z t z z. We use that to find z = z z z z z z ; z = z t z z t z z z

58 48 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES ϕ t = Btϕ {{ 4 ϕ z {{ 5 13 Q + z Q ϕ Q BR t z {{ 6 + Q BR z z t z z z {{ 3 + QQ t BR z {{ cbr z z QQ {{ 10 Q 7 Q c z {{ 11 Q 3 z z z 3 z Bt t 1 gp z Q z {{ 8 z BR Q {{ 1 The term z Bt t depends only on t so it is going to be part of NICE. 4 = Btϕ is NICE at the level of ϕ. 5 = ϕ z = ϕ z ϕ z ϕ. ω QQ t z {{ The first term is at the level of ϕ so it is NICE. The second one is the transport term which appears in.98. Q 6 = z Q ϕ = Q ϕ z Q + Q ϕϕ z Q + ϕ Q. Q z Above we find the first term at the level of z so it is NICE. The second term is at the level of ϕ so it is NICE. We write the last one as ϕ Q Q = ϕ z Q z z Qz + ϕ z Q Q z z z z 3. The first term is at the level of z so it is NICE. For the second term we have used that Finally: 7 = QQ t BR z = z z z z z z. 6 = NICE + ϕ Q Q z z z z 3. z z = Q Q t BR z z + Q Qt BR z + QQ t BR z The first term is at the level of z, z t, BR z so it is NICE. We use that z z.

59 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 49 Q t z = Q t z = Qz z t z Using equation.65 = z 1 Qz Qz z. z t z t and we find that Q t z =z t Qz z z t z = Bt z = z t z z t z z z + Qz z + Qz z z t z z z 3.88 z z Bt. That yields Finally: z 7 = QQ t BR z = NICE + QBR z z t z Qz z {{ NICE at the level of z,z t,br + QBR z Qz z z t z z 3 + QBR z Qz + QQ t BR z z. z z Bt {{ NICE at the level of z,z t,br z 7 = QQ t BR z = NICE + QBR z Qz z z t z z z 3 + QQ t BR z. which means 1 gp z 8 = Q z = Q gp 1 z z = Q Q z gp 1 z {{ NICE at the level of z Q gp 1 z z z z z 3, z z z Q z gp 1 z z z {{ NICE at the level of z

60 50 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES 1 gp z 8 = Q z = NICE Q gp 1 z z z z z 3. Next ω 9 = QQ t z ω = Q Q t +Q Q t ω z z {{ ω + QQ t. z NICE at the level of z,z t We use 5.1 to deal with Qt z. We find that For the next term Therefore ω 9 = QQ t z 10 = z cbr z QQ cbr z Q Qz z 10 = cbr = NICE + Qω Qz z z = cbr z Q {{ NICE as before z z z t z ω z 3 + QQ t. z cbr z z QQ z z 3 cbr z Qz Qz z. {{ NICE as before z z QQ = NICE cbr cbr z Q Q z z QQ z z z z 3. Next Q 11 = Q c z = c z Q Q c z Q Qz z z z z 3 z Qz z c z + Q Q Q c z. The fact that the last two terms are NICE, allows us to find that Q 11 = Q c z = NICE c z Q Q c z Q Qz z z z z 3.

61 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 51 Finally: Q 3 1 = z BR Q which implies that = 3Q Q BR z {{ NICE Q3 Q3 z BR Q z BR z Qz z Q 3 BR Qz z {{ NICE z z z 3 Q 3 1 = z BR Q = NICE Q3 z BR Q Q 3 BR Qz z z z z 3. We gather all the formulas from 4 to 1, keeping term 13 unchanged. They yield: ϕ t =NICE ϕ z ϕ + ϕ z z z 3 Q Qz z {{ 16a z z z 3 Q BR t z {{ 15a Q gp 1 z z z z z 3 {{ 15b + Qω Qz z z t z ω z {{ 3 + QQ t + QBR z Qz z z {{ 18a z z + QQ t BR {{ 14b c z Q Q {{ 17b 14a z cbr z {{ 17a c z Q Qz z z z z {{ 3 16c Q 3 BR Qz z z z z {{ 3 +Q BR z 16d We compute ω 14 = 14a + 14b = QQ t + QQ t BR z = Q t ω Q Q + Q t z z Q Q BR z = Q t Q ϕ Q t ω Q Q z Q t Q Q BR z t z z 3 {{ 18b z z z 3 QQ cbr z Q Qz z {{ 16b Q3 z BR Q {{ 17c z z z t z z 3. z z + Q t Q z Bt.

62 5 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES The last formula allows us to conclude that 14=NICE. We reorganize gathering 15 = 15a + 15b, 16 = 16a + 16b + 16c + 16d, 17 = 17a + 17b + 17c and 18 = 18a + 18b as follows: ϕ t =NICE ϕ z ϕ Q BR t z + gp 1 z z z z z {{ 3 Q 3 BR + c z Q 4 + c BR z Q ϕ Q 4 Qz z z z z {{ Q BR z z t z z 3 + Qω + QBR z Qz z z t z z {{ Q 3 BR + c z z + cbr Q Q. z Q z {{ 17 We add and subtract terms in order to find the R-T condition. We recall here that σ BR t + ϕ z BR z + ω z + Q BR + ω z z z t + ϕ z z z Qz z + gp 1 z z. Then, we find

63 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 53 ϕ t =NICE ϕ z ϕ Q BR t + ϕ z BR z + ω z z t + ϕ z z z + gp 1 z z z z z 3 +Q BR z z t z ϕ z 3 + Q z BR z + ω z z t + ϕ z z z z z z {{ 3 Q 3 BR + c z Q 4 + c BR z Q 19 ϕ Q 4 Qz z z z z 3 + Qω + QBR z Qz z z t z z 3 Q 3 BR + c z z + cbr Q Q. z Q z Line 19 can be written as 19 =Q BR z =Q BR z QQ BR z ϕ z =Q BR z z t z z 3 + Q BR z z t z z 3 + Q BR z 1 z 3 QQ BR z = 1 z 3 z t z + QQ BR z We expand z t to find We denote z z z 3 z t z + ϕ z z z z 3 ϕ z z z ϕ z z z z = 1 z 3 ϕ z z z z 3 + Q ω z z t + ϕ z z z z 3 + Q ω z ϕ z z z z z z 3 z t z + ϕ z z z + Q ω 1 z z 3 z t z + Q BR z + Q ω z z z Q BR z + Q ω z z z QQ BR z ϕ z z z z 3. ϕ z z z z z z 3 ϕ z z z z z D = Q BR z + Q ω z z z..89

64 54 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES We claim that where That means Thus We write Therefore We have D = AN + z H ϕ.90 AN H 3 CE p t..91 D = NICE. 19 = NICE QQ BR z ϕ z z z z 3. D = QQ BR z + Q 1 {{ π P V z z z z z ω dβ part of AN, at the level of z {{ part of AN, we use.8 Q 1 π P V z z z z z 4 z z z z ω dβ {{ part of AN, we use.80 and.81 + Q BRz, ω z + Q ω {{ z z z. AN + Q Hω ω D = AN + z Q H z Q ω = AN + z H z + Q ω z z z + Q ω z z z = AN + z H ϕ + H c z + Q ω z z z = AN + z Hϕ H Q BR z + Q ω z z z. Q BR z = QQ BR z {{ +Q 1 π P V z z z z z ω dβ AN Q 1 π P V z z z z z 4 z z z z ω dβ {{ AN, we use that z z z =z z βz z + Q 1 z z π P V z z z ω dβ. {{ AN, we use that z z z =z z βz z

65 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 55 For the second term on the right one finds 3 Q π P V z z z z z ω dβ = Q 4 π P V z z 4 z z z ω dβ + Q π P V z z z z ω dβ z 4 + Q π P V Q π P V z z z z z 4 z z z 3 z 3 ω dβ + l.o.t., z z z z z 3 ω dβ where in l.o.t. we gather the terms of lower order. Then, all the terms above can be estimated in L but the first one on the right. That is equal to 1 H Q 5 z z z ω plus a commutator which can be estimated in L. This means that Taking Hilbert transforms: Q BR z = AN + 1 H Q z z z ω. H Q 1 BR z = AN H Q z z z ω = AN + 1 Q z z z ω. Using that z z = z z we complete the proof of.90. Thus 19 yields ϕ z ϕ Q BR t + ϕ t =NICE ϕ z BR Q 3 BR + c z Q 4 z + ω z z t + + c BR z Q ϕ Q 4 + Qω + QBR z Qz z z t z z 3 Q 3 BR + c z + cbr z Q z ϕ z z z + gp 1 z z z z z 3 Qz z z z z 3 Q Q {{ z 0 QQ BR z ϕ z z z z {{ 3. 1

66 56 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES For 0 we write z t = Q 4 BR + c z + Q cbr z z t Q z = Q3 BR + c z z Q + QcBR z z. Now which means We write 0 = NICE z t Q z Q, = NICE z t Q z Q QQ BR z ϕ z z z z 3. z z t = z t z {{ z + z t z z z only depends on t = Bt z {{ + Q BR z + cz z z z See.65 = Btz + D {{ as in 5.14 Writing z t = Q BR + cz we compute z z ϕ z z z z z. To simplify we write z t z t = Q BR z Bt + DQ BR {{ NICE ϕ z z z Q BR z t z t = NICE z z {{ NICE because D is nice z z + cbt z. {{ NICE ϕ z z z Q z BR z. Setting the above formula in the expression of 0+1 allows us to find = NICE.

67 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 57 This yields ϕ z ϕ Q BR t + ϕ t =NICE ϕ z BR Q 3 BR + c z Q 4 + ω z z t + + c BR z Q + Qω + QBR z Qz z ϕ Q 4 z t z z 3. We now complete the formula for σ in 5. to find ϕ t =NICE ϕ z ϕ Q σ z z z 3 + Q 3 BR + ω z z Q z z z z 3 {{ + gp 1 z z z z z 3 ϕ z z Qz z z z z 3 + Q 3 BR c z Q 4 c BR z Q + ϕ Q 4 Qz z z z z {{ Qω + QBR z Qz z z t z z {{ 3. 4 Expanding we find ϕ Q 4 = ω 4 z + c z Q 4 ωc Q ω + 3 = Q 3 z + BR z ω z cbr z Q ωc Q Qz z z z z 3. Writing z t z = Q BR z + cz z we obtain that 4 = Qω + QBR z Qz z Q BR z z 3 + Qω + QBR z Qz z c z z z 3.

68 58 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Thus ω = Q 3 z + BR z ω z Qz z z z z 3 + Qω + QBR z Qz z Q BR z z 3 Q = Q Qz z ω z z ω + BR z z z 3 + Q BR z z 3 = Q Qz z 1 ω + BR z z 3 D = NICE. Finally, we obtain ϕ t = NICE ϕ z ϕ Q σ z z z 3. Corollary.4.13 If we disregard the condition on the H k norm for the definition of the NICE terms, imposing only the first condition, then Higher order derivatives of σ ϕ t = NICE Q σ z z z 3. In this section we deal with the highest order derivative of the R-T function. We show that Lemma.4.14 Let z, t and ω, t be a solution of Then, the following identity holds: k 1 Q σ = ANN + z H k 1 ϕ t + ϕh ϕ k.9 where ANN satisfies ANN L CE p t.93 for k 4, where C and p are constants that depend only on k. Proof: We show the proof for k = 4. From now on, if a term f satisfies f L CE p t we say that this term becomes part of ANN. By abuse of notation we will denote f by ANN. We recall

69 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 59 We write Q ω z Q σ =Q BR t + ϕ z BR z + Q ω z + Q 3 BR + ω z t + z t + z z Q z {{ this term is in H 3 so its third derivative is in ANN ϕ z z z + Q gp 1 z z. {{ this term is also in H 3 ϕ z z z = Q ω z Q BR z + cz z + ϕ = Q ω z = Q ω z Above we use 5.14 and.90 to find z t + Q ω z = Q ω z D. ϕ z z Q BR z + c + where AN is as in.91. The remaining terms in Q σ are z z z ϕ z z z Q BR z + Q ω z z z z = AN + Q ω z Hϕ,.94 L = Q BR t z + Q ϕ z BR z. We take 3 derivatives and consider the most dangerous characters: 3 L = M 1 + M + M 3 + ANN, where M 1 = Q BRz, 3 ω t z + Q ϕ z BRz, 4 ω z, M 3 = Q π M =Q 1 z 3 t z 3 t z π z z ω dβ 1 π + Q ϕ z Q ϕ z π π 4 z 4 z z z z ω dβ, z z z z z 4 z z z 3 t z 3 tω dβ z z z z z 4 z z 4 z 4 z ω dβ.

70 60 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Here we point out that in order to deal with BR t in the less singular terms we proceed using estimate.74. In M we find M = Q ω z Λ 3 z t z + Q ϕω z 3 Λ 4 z z + ANN. For the second term we use the usual trick For the first term we recall that 4 z z = 3 3 z z. z = At z z t = 1 A t z z t = 0 z z t + z z t = 0 z z t + z z t + z z t = 0 z z t = z z t z z t. This allows us to control M. For M 3 we find M 3 = Q ω z Λz 3 z t Q ϕω z 3 Λz 4 z + ANN so it can be estimated as M. There remains M 1. Using that z z z = z z z we find M 1 = Q H 3 ω t + Q ϕ z H 4 ω + ANN..95 We compute Q Q H 3 ω t = H 3 ω + ANN We compute the most singular term in t Q BR z = Q π t = H 3 z ϕ t + H 3 z c t + ANN = z H 3 ϕ t + H t Q BR z + ANN Q 3 z t 3 z t z z z ω dβ z z z π z z 4 z z z 3 t z 3 tω dβ {{ Q extra cancelation in z z z =z z z β z z z z z z ω 3 tdβ +ANN. π P V {{ extra cancelation as above

71 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 61 This shows that t Q BR z = Q ω z Λ 3 z t z + ANN. That gives Q t Q ω BR z = Λ z 3 zt z + ANN, which implies Q H t Q ω BR z = z 3 zt z + ANN = Q ω z z 3 t z + ANN. Plugging the above formula in 5.17 we find that Q H 3 ω t = z H ϕ 3 t Q ω z z 3 t z + ANN = z H 3 ϕ t Q ω z + ANN. As we did before, we expand 3 Q BR z to find Q 3 BR z =Q Qz zbr 4 z + Q π P V Q z z π P V z + Q π P V Q 3 BR z Q ω z c z 4 z 4 z 4 z z z z ω dβ z z 4 z z z 4 z 4 ω dβ z z z z z ω 4 dβ + ANN. Therefore, we can use.80,.81 and.81 to show that the most dangerous term is given by Q 1 H 4 ω. It implies 3 Q BR z = Q 1 H 4 ω + ANN and therefore Q H 3 ω t = z H ϕ 3 t Q ω Q z H 4 ω Q ω z c z 4 z + ANN.

72 6 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES We use the above formula and expand ϕ to find M 1 = Q H 3 ω t + Q ϕ z H 4 ω + ANN = z H 3 ϕ t Q ch 4 ω Q ω = z H 3 ϕ t c z H 4 z c Q ω z z 4 z + ANN Q ω z c z 4 z + ANN = z H 3 ϕ t c z H 4 ϕ c z H 4 c z Q ω z c We will show that It yields z 4 z + ANN. c z H c z 4 Q ω z c z 4 z = ANN..97 Q H 3 ω t + Q ϕ z H 4 ω = z H 3 ϕ t c z H 4 ϕ + ANN that together with.94 allows us to obtain.9. We have c z H c z 4 Q ω z c z 4 z = ch c z 4 Q ω z c z 4 z = ch 3 Q BR z Q ω z c We repeat the calculation for dealing with the most dangerous terms in Q 3 BR z = Λ 4z ωq z z + ANN. z 4 z. We recognized as before terms in ANN using that z z z gives an extra cancellation. We find that ch Q 3 BR z Q ω z c z 4 z = ch Λ 4z ωq z z = c z 4 ωq z z Q ω z c Q ω z c z 4 z + ANN z 4 z + ANN. Using that 4 z z = 4 z z we are done proving.97.

73 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN Energy estimates for ϕ In this section we prove the following result. Lemma.4.15 Let z, t and ω, t be a solution of priori estimate holds: Then, the following a d dt ϕ t St + CE p t.98 H k 1 for k 4, where C and p are constants that depend only on k. Proof: We shall present the details in the case k = 4, leaving the other cases to the reader. Using the estimates obtained before one has where d dt ϕ L t CE p t. Developing the derivative using Lemma.4.1, we get that: d dt Λ1/ ϕ 3 L t = Λ ϕ 3 ϕ 3 t d.99 T =I 1 + I + I 3, I 1 = Λ ϕ 3 NICEd, T I = Λ ϕ 3 ϕ T z ϕ d, I 3 = Λ ϕ 3 Q σ z z T z 3 d. We use.85 to control I 1. The most singular term in I is the one given by 1 T z Λ 3 ϕ ϕϕd 4 = T + Using the commutator estimate 1 z Λ1/ ϕ 3 1 z ϕ Λ 1/ ϕ 3 d. T [ ] ϕλ 1/ ϕ 4 Λ 1/ ϕ ϕ 4 d gλ 1/ f Λ 1/ g f L g C f H 1/.100 we can bound I. In I 3 we split further considering the most singular terms J 1 = Λ ϕq 3 σz z z 3 d, T J = T Λ 3 ϕq σ 4 z z z 3 d,

74 64 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES J 3 = Λ ϕ 3 Q σ z z T z 3 d. The term J 1 can be estimated as before. Recalling.79 we see that J = St. It remains to control J 3 in order to find.98. We decompose J 3 = K 1 + K where and Inequality.75 for k = 4 allows us to obtain K 1 = H ϕ 3 Q σ z z T z 3 d K = H ϕ 3 Q 3 σ z z T z 3 d. K 1 CE p t. To finish the proof we use formula.9 for k = 4 to find K = L 1 + L + L 3 where L 1 = H ϕann 3 z z T z 3 d, L = H ϕ z 3 H ϕ 3 t z z T z 3 d, L 3 = H ϕϕh 3 ϕ 4 z z z 3 d. T The term L 1 can be easily estimated using.93. For L we substitute the expression.98 for 3 ϕ t to get L = M 1 + M + M 3 : M 1 = H ϕ z 3 H NICE z z T z 3 d. M = T H 3 ϕ z H ϕϕ z z z z 3 d. M 3 = H ϕ z 3 H Q σ z z T z 3 z z z 3 d. By equation.85, M 1 is bounded. M is bounded knowing that we have room for half derivative in the term which is not the third factor. Finally we can bound M 3 in virtue of Lemma.4.8. To finish, in L 3 we integrate by parts to find L 3 = H ϕ 3 ϕ z z T z 3 d CE p t using Sobolev embedding.

75 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN Energy estimates for z mq σt + 4 l=0 1 mq l t. Lemma.4.16 Let z, t and ω, t be a solution of priori estimate holds: Then, the following a d z dt mq σt + for k 4, where C and p are constants that depend only on k. 4 l=0 1 mq l CE p t.101 t Proof: Inequalities.71 and.76 show that Q σ C 1 [0, T ] [, π] for some T and therefore mq σt is a Lipschitz function differentiable almost everywhere by Rademacher s theorem. Let mq σt = min [,π] Q σ, t = Q σ t, t. We can calculate the derivative of mq σt, to obtain mq σ t = Q σ t t, t for almost every t. Then it follows that: d 1 dt mq t = Q σ t t, t σ mq σ t almost everywhere. By using the previous a priori estimates for the L bounds, we get to d z dt mq t CE p t. σ On the other hand, we can apply the same argument to point where the minimum is attained we have that: d 1 dt mq l t = z t t, t z t, t q l mq l 3 t 1 mq l t. Denoting again by t the which again can be easily bounded and we get.101, as desired..4.5 Proof of short-time existence Theorem To conclude the proof of the local existence, we shall use the previous a priori estimates. We now introduce a regularized version of the evolution equation which is well-posed for short time independently of the sign condition on σ, t at t = 0. But for σ, 0 > 0, we shall find a time of existence uniformly in the regularization, allowing us to take the limit.

76 66 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Now, let z ε,δ,µ, t be a solution of the following system compare with.64: z ε,δ,µ t, t = φ δ φ δ Q z ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ, t + φ µ c ε,δ,µ φ µ z ε,δ,µ, t,.10 ω ε,δ,µ t = z ε,δ,µ Q Q z ε,δ,µ φ z ε,δ,µ [ δ φ δ z ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ t z ε,δ,µ, t ϕ Qz ε,δ,µ Qz ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ ε,δ,µ + cε,δ,µ z ε,δ,µ πqz ε,δ,µ Qz ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ β Qz ε,δ,µ 4cε,δ,µ Qz ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ z ε,δ,µ Qz ε,δ,µ cε,δ,µ z ε,δ,µ Qz ε,δ,µ ] Qz ε,δ,µ 3 gp 1 z ε,δ,µ z ε,δ,µ, t Q z ε,δ,µ + z ε,δ,µ, t Q z ε,δ,µ φ δ φ δ ε z ε,δ,µ Q z ε,δ,µ Λφ µ φ µ ϕ ε,δ,µ, z ε,δ,µ β z ε,δ,µ β dβ Qz ε,δ,µ t Qz ε,δ,µ ω ε,δ,µ z ε,δ,µ Q z ε,δ,µ ω ε,δ,µ z ε,δ,µ 3 z ε,δ,µ t z ε,δ,µ Qz ε,δ,µ t Qz ε,δ,µ ω ε,δ,µ z ε,δ,µ Q z ε,δ,µ ω ε,δ,µ z ε,δ,µ 3 z ε,δ,µ t z ε,δ,µ.103 z ε,δ,µ, 0 = z 0 and ω ε,δ,µ, 0 = ω 0 for ε > 0, δ > 0, µ > 0, φ δ and φ µ even mollifiers, and c ε,δ,µ = + π π β z ε,δ,µ β β z ε,δ,µ β φ δ φ δ β Q z ε,δ,µ βbrz ε,δ,µ, ω ε,δ,µ βdβ β z ε,δ,µ β β z ε,δ,µ β φ δ φ δ β Q z ε,δ,µ βbrz ε,δ,µ, ω ε,δ,µ βdβ, B ε,δ,µ t = 1 π + π C ε,δ,µ = φ δ φ δ π φ δ φ δ ϕ ε,δ,µ = Q z ε,δ,µ ω ε,δ,µ z ε,δ,µ C ε,δ,µ, z ε,δ,µ, t z ε,δ,µ, t Q z ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ, td, β z ε,δ,µ β β z ε,δ,µ β βq z ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ βdβ β z ε,δ,µ β β z ε,δ,µ β βq z ε,δ,µ βbrz ε,δ,µ, ω ε,δ,µ βdβ.

77 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 67 We start proving the following lemma: Lemma.4.17 Let z ε,δ,µ, t H 4 T, ω ε,δ,µ, t H T, ϕ ε,δ,µ, t H 3 T. Then ω ε,δ,µ, t H 3 T. Proof: We can write ω ε,δ,µ as: Taking three derivatives yields ω ε,δ,µ = z ε,δ,µ Q z ε,δ,µ ϕ ε,δ,µ + C ε,δ,µ. ω 3 ε,δ,µ = SAFE + z ε,δ,µ Q z ε,δ,µ 3 C ε,δ,µ = SAFE z ε,δ,µ Q z ε,δ,µ φ δ φ δ z ε,δ,µ z ε,δ,µ Q z ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ = SAFE z ε,δ,µ Q z ε,δ,µ φ z ε,δ,µ δ φ δ z ε,δ,µ Q z ε,δ,µ BRz 3 ε,δ,µ, ω ε,δ,µ = SAFE z ε,δ,µ Q z ε,δ,µ φ z ε,δ,µ δ φ δ z ε,δ,µ Q z ε,δ,µ BRz ε,δ,µ, ω 3 ε,δ,µ where SAFE means bounded in L. Using the representation BRz ε,δ,µ, ω 3 ε,δ,µ = SAFE + 1 z ε,δ,µ z ε,δ,µ H 3 ω ε,δ,µ we get that ω 3 ε,δ,µ = SAFE z ε,δ,µ Q z ε,δ,µ φ δ φ δ Q z ε,δ,µ 1 z ε,δ,µ z ε,δ,µ H 3 ω ε,δ,µ z ε,δ,µ z ε,δ,µ {{ =0 and we are done. We should remark that the lemma holds independently of δ, µ and ε. We define a distance between data z, ω and z, ω by taking dz, ω, z, ω = z z H 4 + ω ω H + ϕ ϕ H 3 where ϕ and ϕ arise from z, ω and z, ω respectively by.6. Let XX denote the resulting metric space. The proof of Lemma.4.17 gives also the following Corollary.4.18 The map z, ω ω is Lipschitz from any ball in XX into H 3 T. We note that throughout this section we will repeatedly use the following commutator estimate for convolutions: φ δ fg gφ δ f L C g L f L,.104

78 68 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES where the constant C is independent of δ, f and g. We can now operate to get the following expression for ϕ ε,δ,µ : t ϕ ε,δ,µ = Qzε,δ,µ t Qz ε,δ,µ ω ε,δ,µ z ε,δ,µ Q z ε,δ,µ ω ε,δ,µ z ε,δ,µ 3 z ε,δ,µ t z ε,δ,µ + Q z ε,δ,µ t ω ε,δ,µ z ε,δ,µ Q z ε,δ,µ [ = φ δ φ δ z ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ t z ε,δ,µ, t ϕ Qz ε,δ,µ Qz ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ ε,δ,µ Qz ε,δ,µ + cε,δ,µ z ε,δ,µ πqz ε,δ,µ Qz ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ β 4cε,δ,µ Qz ε,δ,µ BRz ε,δ,µ, ω ε,δ,µ z ε,δ,µ Qz ε,δ,µ cε,δ,µ z ε,δ,µ Qz ε,δ,µ ] Qz ε,δ,µ 3 gp 1 z ε,δ,µ z ε,δ,µ β z ε,δ,µ β dβ + φ δ φ δ Qz ε,δ,µ t Qz ε,δ,µ ω ε,δ,µ z ε,δ,µ Q z ε,δ,µ ω ε,δ,µ ελφ µ φ µ ϕ ε,δ,µ t C ε,δ,µ. z ε,δ,µ 3 z ε,δ,µ t z ε,δ,µ t C ε,δ,µ The RHS of the evolution equations for z ε,δ,µ and ϕ ε,δ,µ are Lipschitz in the spaces H 4 T and H 3+ 1 T since they are mollified. For the case of ω ε,δ,µ Lipschitz in the space H T we use that for δ small enough φ δ φ δ is close to the identity and the a priori bounds. In all of the cases we have taken advantage of Lemma Therefore we can solve for short time, thanks to Picard s theorem. Now, we can perform energy estimates as in the a priori case to get uniform bounds in µ and we can let µ go to zero. The energy estimates that we can get are the following: d z ε,δ,µ H + Fz ε,δ,µ dt 4 L + ωε,δ,µ H + ϕ ε,δ,µ H 3+ 1 Cε, δ + 4 l=0 z ε,δ,µ H 4 + Fz ε,δ,µ L + ωε,δ,µ H + ϕ ε,δ,µ H m ε,δ,µ q l t j m ε,δ,µ q l t. We should note that for the new system without the φ µ mollifier, the length of the tangent vector z ε,δ is now constant in space and depends only on time. Lemma.4.17 still applies and we can still perform energy estimates as in the a priori case. The only difference relies on the fact that we should have to move the mollifiers and apply the estimate We should also remark that because of the dissipative term ελϕ ε,δ it is enough to use the following estimate l=0

79 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 69 1 d dt Λ1/ ϕ 3 ε,δ L = Λ 3 ϕ ε,δ t 3 ϕ ε,δ d ε 64 Λ 3 ϕ ε,δ L + Cε t 3 ϕ ε,δ L and hence require only that t ϕ ε,δ H 3 T instead of the H 3+ 1 T that was required before except for the transport term that can be estimated as in subsection The estimations are performed following exactly the same steps of subsection.4.4. More precisely, we can get the following energy estimates: d z ε,δ H + Fz ε,δ dt 4 L + ωε,δ H + ϕ ε,δ H 3+ 1 Cε z ε,δ H 4 + Fz ε,δ L + ωε,δ H + ϕ ε,δ H m ε,δ q l t j m ε,δ q l t. Under these conditions, we can let δ go to zero. Finally, let z ε, t be a solution of the following system compare with.64: l=0 l=0 z ε t, t = Q z ε, tbrz ε, ω ε, t + c ε, t z ε, t,.105 ϕ ωt ε = BRz ε, ω ε t z ε Qz ε Qz ε BRz ε, ω ε ε Qz ε + cε z ε πqz ε Qz ε BRz ε, ω ε zβ ε β zβ ε dβ 4cε Qz ε BRz ε, ω ε z ε Qz ε cε z ε Qz ε Qz ε 3 gp 1 z ε ε z ε Q z ε Λϕε,.106 z ε, 0 = z 0 and ω ε, 0 = ω 0 for ε > 0, where c ε = + π π β z ε β β z ε β βq z ε βbrz ε, ω ε βdβ β z ε β β z ε β βq z ε βbrz ε, ω ε βdβ, ϕ ε = Q z ε ω ε z ε z ε c ε, B ε t = 1 z ε, t π z ε, t Q z ε BRz ε, ω ε, td.

80 70 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES Proceeding as in section.4.3 compare with equation.87 we find ϕ ε t = B ε tϕ ε ϕ ε Qz ε z ε + z ε Qz ε ϕε Qz ε t BRz ε, ω ε z ε z ε z ε z ε 3 t z ε B ε t + Qz ε BRz ε, ω ε z ε t z ε z ε z ε 3 + Qz ε t Qz ε BRz ε, ω ε z ε z ε Qz ε gp 1 z ε z ε + Qz ε t Qz ε ω ε z ε c ε BRz ε, ω ε z ε z ε Qzε Qz ε Qz ε Qz ε cε z ε Qz ε 3 z ε BRzε, ω ε Qz ε ελ ϕ ε. We also define compare with equation σ ε = t BRz ε, ω ε + ϕε z ε BRz ε, ω ε z ε + 1 z ε zt ε + ϕε z ε z ε z ε + Qz ε BRzε, ω ε + ωε z ε z ε Qz ε z ε + g P 1 z ε z ε. Remark.4.19 The system is analogous to the system considered in [8, Section 8]. We point out an unfortunate typographical error in that section; the Laplacian should have been written as the square root of the Laplacian. For this ε-system we now know that there is local-existence for initial data satisfying Fz 0, β < even if σ ε, 0 does not have the proper sign. In the following we shall show briefly how to obtain a solution of the regularized system with z ε C[0, T ε ], H k, ϕ ε C[0, T ε ], H k 1, ω ε C[0, T ε ], H k for k 4. The next step is to integrate the system during a time T independent of ε. We will show that for this system we have ω ε d dt Et CEp t,.108 where Et is given by the analogous formula.67 for the ε-system, and C and p are constants independent of ε. In the following we shall see what is the impact of the ε system on the a priori estimates and check that there is no practical impact for sufficiently small ε. To do that, we will show the corresponding uniform estimates for k = 4 and leave to the reader the remaining easier cases. Let us consider the one corresponding to I 3 in section.4.4.1, we have I ε 3 = 1 z ε 3 Λ 3 ϕ ε Q z ε σ ε z ε z ε d.

81 .4. LOCAL EXISTENCE IN SOBOLEV SPACES TILDE DOMAIN 71 Proceeding in the same way as before, we can perform the same splittings and get uniform bounds such that I ε 3 = Sε + M ε 4 + bounded terms where Sε corresponds to S in.79, and bounded terms CE p t, M4 ε = ε z ε z ε z ε H ϕ 3 ε HΛ ϕ 3 ε d. Then we can write M4 ε as follows M4 ε = ε Λ 1 z ε z ε z ε H ϕ 3 ε Λ 1 H 3 ϕ ε d, and therefore, for small ε which gives This finally shows.108 and therefore M ε 4 Λ 1 3 ϕ ε L + bounded terms, d dt Et CEp t ε Λ 3 ϕ L. Et Ct1 p + E 1 p 0 1/1 p. Now we are in position to extend the time of existence T ε so long as the above estimate works and obtain a time T dependent only on the initial data arc-chord, Rayleigh-Taylor, distance to the points q 0,..., q 4, and Sobolev norms of z, ω, and ϕ. We can let ε tend to 0, and get a solution of the original system. This concludes the proof.

82 7 CHAPTER. SPLASH SINGULARITY FOR WATER WAVES

83 Chapter 3 Splash singularities for water waves with surface tension 3.1 Introduction We establish the main result in the chapter for the system and 1.6. Theorem Consider z 0, 0 H k T for k 5. Then there exist a family of initial data satisfying 1.5 and the arc-chord condition 1.7 and a time T s > 0 such that the interface z, t H k T from the unique smooth solution of the system on the time interval [0, T s ] touches itself at a single point splash singularity or along an arc splat singularity at time t = T s. These solutions can be extended to the periodic 3D setting considering scenarios invariant under translations in one coordinate direction. 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 In order to deal with these obstacles we use a conformal map w 1/, P w = tan 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 dimensional vector whose components are the real and imaginary parts of P z 1 + iz. We also make sure that Ωt Ωt 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 and the new amplitude ω: z, t = P z, t z t, t = Q, tbr z, ω, t + c, t z, t,

84 74 CHAPTER 3. WATER WAVES WITH SURFACE TENSION where Q ω t, t = BR t z, ω, t z, t BR z, ω Q, t ω, t, t 4 z, t + c, tbr z, ω z, t + c, t ω, t P 1 z, t Q 3 + τ z, t 3 zt HP 1 z P1 1 z z T HP1 1 z P 1 z + τ Q z, t z, t z, t 3 3. Q, t = dp dw P 1 z, t and HPi 1 denotes the Hessian matrix of Pi 1, which is the i-th i = {1, component of the transformation P 1. Here, we choose c, t in such a way that z, t = At. This particular choice of c was first introduced by Hou et al. in [6] and was later used by Ambrose [6] and Ambrose- Masmoudi [8]. The choice of c implies, c, t = + π π Q z β β, t BR z, ω β β, t z β β, t dβ z β β, t z β β, t dβ Q BR z, ω β β, t It is easy to check that if we take Q 1 in we recover We also define the function ϕ, t = Q, t ω, t z, t c, t z, t 3.3 introduced by Beale et al. for the linear case [14] and by Ambrose-Masmoudi for the nonlinear one [8]. 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 3.3 is devoted to provide the appropriate initial data for the splash and splat singularities. In Section 3.4 we choose an energy which does not need a precise sign on the Rayleigh-Taylor function. In Section 3.5 we choose a different energy that involves the sign of the Rayleigh- Taylor 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 [6] and Ambrose-Masmoudi [9]. The Rayleigh-Taylor function is given by the following formula σ BR t z, ω + ϕ z BR z, ω + Q BR z, ω + ω z z z + ω z z t + ϕ z z z Q z z + P 1 z z. 3.4

85 3.1. INTRODUCTION 75 All solutions that we will consider throughout the paper will have finite energy, as discussed in [0]. 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 3.4 and 3.5. E S t = 1 Ω f t vx, y, t dxdy + 1 E k t + E p t + E τ t, z, t z 1, td + τ z, t d where z, t = z 1, t, z, t, u, t = vz, t, t, and Ω f t = Ωt [, π] R is a fundamental domain in the water region in a period, then it follows that the energy is conserved. de k t dt = = Ω f t Ω f t = = = Ω f t vx, y, tv t x, y, t + vx, y, t vx, y, tdxdy vx, y, t px, y, t 0, 1dxdy Ω f t vx, y, t px, y, t + ydxdy vx, y, t n yds + vx, y, t n τ Ω f t Kds z, tu, t z, td + τ u, t z, t z, t z, t z, t 3 d 3.5 where we have used the incompressibility of the fluid v = 0 and Laplace-Young s condition for the pressure on the interface. Next de p t dt = = = z, t t z, t z 1, td + 1 z, t t z, t z 1, td z, t t z 1, td z, t z, t t z 1, td z, tu, t z, td. 3.6 de τ t dt = τ = τ z, t t z, t d = τ z, t z, t u, t d = τ z, t z, t t z, t d z, t z, t z, t z, t 3 u, t z, td 3.7 Adding all the derivatives we get the desired result.

86 76 CHAPTER 3. WATER WAVES WITH SURFACE TENSION 3. Properties of the curvature in the tilde domain In this section we will rewrite the term corresponding to the curvature Kz, t in the new tilde variables z, t. We will proceed step by step. Let us recall that the curvature is defined by K, t = z, t z, t z, t 3 We begin with the term z, t 3. We have that z, t = P z, t, P z, t = P z, t z, t, P z, t z, t Since P and P 1 are conformal, by the Cauchy-Riemann equations that implies that We move to the other term P z, t T P z, t = Q, tid, z, t 3 = Q 3, t z, t 3 z, t, z, t = P 1 z, t z, t, P 1 z, t z, t = P 1 z, t z, t, P 1 z, t z, t + P 1 z, t z, t, P 1 z, t z, t W + X Again, by the Cauchy-Riemann equations W = Developing the terms in X, we get 1 Q, t z, t, z, t P 1 z, t z T z, t =, t HP1 1 z, t z, t z T, t HP 1 z, t z, t where HPi 1 denotes the Hessian of the i-th component of P 1 i = 1,. Hence, we can write X as This means that X = z T, t HP1 1 z, t z, t P 1 z, t z, t + z T, t HP 1 z, t z, t P1 1 z, t z, t. K, t = Q, t z, t z, t Q, t3 z, t 3 + X, t z, t 3 Q, t K, t + M, t,

87 3.. PROPERTIES OF THE CURVATURE IN THE TILDE DOMAIN 77 We will now try to simplify further by exploiting the Cauchy-Riemann equations. We can calculate the Hessian and the gradient terms as: 4 z P1,x 1 z = R 1 + z 4 Ra 4i z P1,y 1 z = R 1 + z 4 Ia 4 z P,x 1 z = I 1 + z 4 Ia 4i z P,y 1 z = I 1 + z 4 Ra 41 3 z P1,x,x 1 4 z = R 1 + z 4 Rb 4i1 3 z P1,x,y 1 4 z = R 1 + z 4 Ib 41 3 z P,x,x 1 4 z = I 1 + z 4 Ib 4i1 3 z P,x,y 1 4 z = I 1 + z 4 Rb Therefore the Hessians are HP1 1 Rb Ib = Ib Rb Calculating further:, HP 1 Ib Rb = Rb Ib, z T HP 1 z = Rb z 1 z + Ib z 1 z z T HP 1 1 z = Rb z 1 z Ib z 1 z X 1 = RaRb z 1 z + RaIb z 1 z 1 z z 1 + IaRb z 1 z + IbIb z 1 z z z X = RbRb z 1 z z z + RaIb z 1 z + IaIb z 1 z + IaRb z 1 z 1 z z 1 This means X = X 1 X = z 1 + z z RaRb + IaIb + z 1 RaIb IaRb z 1 + z Gz, z.

88 78 CHAPTER 3. WATER WAVES WITH SURFACE TENSION We can see that Q Q 3 = 1 1 Q = Ra + Ia = RaRb z 1 RaIb z + IaIb z 1 + IaRb z = Gz, z by the Cauchy-Riemann equations. If we take one derivative in space of X, we obtain This implies X = z 1 + z G z z, z + z 1 + z G z, z = z 1 + z G z z, z + z 3 K G z, z = z 1 + z G z z, z z 3 K Q Q 3, K = Q K Q 3 X z 3 K = Q K + Q3 z G z z, z KQ = Q K + M 1 + M Later, we will see that the M 1 is a low order term and can be absorbed by the energy. 3.3 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 We say that z = z 1, z is a splash curve if 1. z 1, z are smooth functions and π-periodic.. z satisfies the arc-chord condition at every point except at 1 and, with 1 < where z 1 = z and z 1, z > 0. This means z 1 = z, but if we remove either a neighborhood of 1 or a neighborhood of in parameter space, then the arc-chord condition holds. 3. The curve z 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 = z, z 1 z 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 P on the water region such that the curve z = z 1, z = P z satisfies:

89 3.3. INITIAL DATA 79 a z 1 and z are smooth and π-periodic. b z is a closed contour. c z satisfies the arc-chord condition. We will choose the branch of the root that produces that independently of x. lim P x + iy = e iπ/4 y 5. P w is analytic at w and dp dw w 0 if w belongs to the interior of the water region. Furthermore, ±π, 0 and 0, 0 belong to the vacuum region. 6. z q l for l = 0,..., 4, where 1 q 0 = 0, 0, q 1 =, 1, q = 1, 1 1, q 3 =, 1 1, q 4 =, Moreover, we will define a splat curve as a splash curve but replacing condition 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 P is regular, we need to control the distance to the points q l. In order to do so, we introduce the function mq l, t z, t q l for l = 0,..., 4. We have performed numerical simulations, as explained in [19] with the following initial data on the non-tilde domain: z1 0 = + 1 3π sin + 1 sin + 1 π sin3 z 0 = 1 10 cos 3 10 cos cos3 Note that z π = z π splash. Instead of prescribing an initial condition for ω, we prescribed the normal component of the velocity to ensure a more controlled direction of the fluid. From that we got the initial ω, 0 using the following relations. Let ψ be such that ψ = v and Ψ its restriction to the interface. The initial normal velocity is then prescribed by setting u 0 n z = Ψ = 3 cos 3.4 cos + cos cos4. The reader may easily check that the above z1 0 and z0 yield a splash curve, i.e. the conditions in Definition are satisfied. See Figure 3.1. In order to get an initial data for the splat singularity, one only needs to perturb the splash curve so that it z0 1 = 0 on a neighbourhood of both = ± π. The normal velocity

90 80 CHAPTER 3. WATER WAVES WITH SURFACE TENSION y x Figure 3.1: Splash singularity. The interface self intersects in a point. can be the same since it has the right sign the one that separates the curve. By continuity, the Rayleigh-Taylor function should remain positive. For the case where the energy is independent on the surface tension coefficient see Section 3.5, we need the curve to satisfy the Rayleigh-Taylor condition initially. This is always the case when the surface tension coefficient is small enough. To illustrate this phenomenon, we have plotted in the next figure the Rayleigh-Taylor condition for different values of the surface tension coefficient and the initial condition described above. We can see that for small enough values of τ 0 and 0.1: the Rayleigh-Taylor condition σ is strictly positive. For bigger values of τ, the Rayleigh-Taylor condition σ has distinct sign. 3.4 Energy without the Rayleigh-Taylor condition In this section, we prove local existence in the tilde domain, where the time of existence depends on the surface tension coefficient. This theorem has the advantage that the initial data does not need to satisfy the Rayleigh-Taylor condition and it works for every τ > 0. Theorem Let k 3. Let z 0 be the image of a splash curve by the map P parametrized in such a way that z 0 = L π, where L is the length of the curve in a fundamental period, and such that z 1 0, z0 Hk+ T. Let ω, 0 H k+ 1 T. Then there exist a finite time T > 0, a time-varying curve z, t C[0, T ]; H k+, and a function ω, t C[0, T ]; H k+ 1 providing a solution of the water wave equations The proof below is based in the following energy estimates: The energy We will define the energy for k 3 as

91 ENERGY WITHOUT THE RAYLEIGH-TAYLOR CONDITION σ Figure 3.: Rayleigh-Taylor function for different values of τ : τ = 0 blue, τ = 0.1 red, τ = 0.5 green, τ = 1 black Ek t = EE t+ z 3 Z Q k+1 {z A 1 + τ k K Z Qk+ k ω Λ k ω + {z B EE t = kz kl + kω kl + kfzkl t + 4 X l=0 1 z τ Z Qk+3 k ω ω, {z C 1, mq l t where mq l t = min T q l, t for l = 0,..., 4 and Λ = 1/. From now on, we will denote the Hilbert transform of a function f by Hf, where Z P V π f β dβ. Hf = π tan β Recall that the operator Λ can also be written as Λf = Hf The energy estimates The energy estimates for EE were proved in [8] and in [0]. In this section we will focus on the new terms A, B and C.

92 8 CHAPTER 3. WATER WAVES WITH SURFACE TENSION K Proposition 3.4. K t = NICE3 + Q z 3 H ω + 1 z 3 Q H ω, where NICE3 means Q j k K k NICE3 CE p k t for some positive constants C, p and any j. Proof: We start writing K t K t = 3 z 5 z t z z z + 1 z 3 z t z + z z t = P 0 + P 1 + P Calculating further P 0 we get that P 0 = 3 z 5 Q BR + c z z z z = NICE3, by the estimates proved in the Appendix. On the one hand, developing P, we obtain P = 1 z 3 z z t = 1 z 3 z z t = = 1 Q z 3 BR + c z z = NICE3 1 z 3 Q z z H ω + c z z = NICE3, since c is as regular as ω, z and therefore bounded in H k. On the other, P 1 gives rise to P 1 = 1 z 3 z t z = 1 Q z 3 BR + c z z = P 1,1 + P 1, We can further develop P 1, to obtain P 1, = NICE3, since the terms vanish either by integrating by parts, by being a dot product between two orthogonal vectors or because c = NICE3. We also have that P 1,1 = NICE3 + 1 z 3 Q BR + Q BR z = NICE3 + P 1,1,1 + P 1,1,

93 3.4. ENERGY WITHOUT THE RAYLEIGH-TAYLOR CONDITION 83 The only term in BR which is not NICE3 is when we hit with the derivative in ω. Therefore P 1,1,1 = NICE3 + 1 z 3 Q 1 H ω Finally, regarding P 1,1, and keeping in mind that hitting with all the derivatives in z leads us to a term which has the factor z z = z, giving us the extra regularity we needed to integrate the term. P 1,1, = NICE3 + Q 1 z zβ z 3 z π z zβ ω βdβ = NICE3 + Q z 3 H ω. We should notice that there doesn t appear a term proportional to H ω since the kernel that results from subtracting the Hilbert transform has room for two derivatives instead of one. Adding all the previous estimates together we get the desired result ω We first notice that M 1 one of the terms in the curvature is of the order of z and therefore it can be absorbed by the energy. Hence K = KQ KQ + low order terms We will follow the proof done by Ambrose in [6]. Taking into account the estimates for the implicit operator done in [8], we are left to see the impact of the Q factor in the singular term c ω, since the impact into the others is either trivial the ones that come from the factor proportional to the curvature or is zero the rest of the terms. Lemma k c ω = NICE35 + Q ω z H k K, where NICE35 means Q j Λ k ωnice35 CE p k t for some positive constants C, p and any j. Proof: The most singular term is when we hit all the derivatives in c, since if we hit all of them in ω, that term would belong to NICE35. Developing the new terms

94 84 CHAPTER 3. WATER WAVES WITH SURFACE TENSION k c ω = NICE35 ω k Q BR = NICE35 Q ω z k z z z z z β z H z = NICE35 Q ω z 4 k = NICE35 + Q ω z H k K. z zβ ωβdβ Lemma where NICE35 means k c ω = NICE35, Q j Λ k ωnice35 CE p k t for some positive constants C, p and any j. Proof: The most singular term is when we hit all the derivatives in ω, since if we hit all of them in c, that term would belong to NICE35. Thus, we have to estimate Q j H k+1 ω k+1 ω c = = 1 k+1 ωh k+1 ωq j c k+1 ω [ H k+1 ω cq j H k+1 ω cq j ] CE p k t, and therefore it is NICE 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 + = OK Q k+1 k K k Q H ω + 4QQ H ω kq k+ Q k K k 1 H ω Q k+3 k K k H ω 4Q k+ Q k K k H ω = A 1 + A + A 3,

95 3.4. ENERGY WITHOUT THE RAYLEIGH-TAYLOR CONDITION 85 where we will say that a term is OK if it is controlled by the energy. We should be careful while estimating B t because db dt = OK + 1 τ = OK + 1 τ = OK + τ Q k+ k ω t Λ k ω + 1 τ Q k+ k ω t Λ k ω + 1 τ Q k+ k ω t Λ k ω + 1 τ Q k+ k ωλ k ω t ΛQ k+ k ω k ω t Q k+ H k ω k ω t Hence db dt = OK + Q k+ Λ k τ ω Q ω z H k K + Q k+ Λ ω k Q k K + Q3 z 3 X + k + Q k+1 Q H ω k k+1 KQ = OK + Q k+ Λ k τ ω Q ω z H k K + Q k+ Λ ω k Q k K Q k+ Q Λ ω k k K + k + Q k+ Q H ω k k+1 K = OK + B 1 + B + B 3 + B 4 dc dt = OK + 1 z τ Q k+4 ω k ω k+1 K = OK + C Development of the derivative in B We start from the development of B 1, B, B 3 and B 4. We trivially have: B 1 = 1 Q k+ Λ k τ ω Q ω z H k K B 3 = Q k+ Q Λ ω k k K B 4 = OK k + Q k+ Q H k+1 ω k K

96 86 CHAPTER 3. WATER WAVES WITH SURFACE TENSION We now look at B. We can decompose it in the following way B = Q k+ Λ ω k Q k K + Q K = OK + Q k+ Λ ωq k k K + Q k+1 K + kq k K = OK + B,1 + B, + B,3 We can write down the terms B,1 and B,3 in the form B,1 = Q k+ H k+1 ωq k K B,3 = k Q k+ H k+1 ωq k K Integrating by parts in B, we establish B, = k + 3 Q k+3 Λ k+1 ω k K Q k+ Q Λ ω k k K = B,,1 + B,, Again, B,, can easily be reduced to the canonical form B,, = k + 3 Q k+ Q H k+1 ω k K 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 High Order From A: Q k+3 k K k H ω A From B: From C: No terms from C. Q k+3 Λ k+1 ω k K B,,1

97 3.4. ENERGY WITHOUT THE RAYLEIGH-TAYLOR CONDITION Low Order Type I From A: kq k+ Q k K k 1 H ω A 1 4Q k+ Q k K k H ω A 3 From B: Q k+ Q Λ ω k k K B 3 k + Q k+ Q H k+1 ω k K B 4 Q k+ H k+1 ωq k K B,1 k Q k+ H k+1 ωq k K B,3 k + 3 Q k+ Q H k+1 ω k K B,, From C: No terms from C Low Order Type II From A: No terms from A. From B: 1 τ Q k+ Λ k ω Q ω z H k K B 1 From C: 1 z τ Q k+4 ω k ω k+1 K C Regularized system Now, let z ε,δ,µ, t be a solution of the following system compare with : z ε,δ,µ t, t = φ δ φ δ Q z ε,δ,µ BR z ε,δ,µ, ω ε,δ,µ, t + φ µ c ε,δ,µ φ µ z ε,δ,µ, t, 3.9

98 88 CHAPTER 3. WATER WAVES WITH SURFACE TENSION ω ε,δ,µ t = φ δ φ δ BR t z ε,δ,µ, ω ε,δ,µ z ε,δ,µ BR z ε,δ,µ, ω ε,δ,µ Q z εδ,µ Q ω ε,δ,µ + τ +τ 4 z ε,δ,µ z ε,δ,µ Q zε,δ,µ + cε,δ,µ BR z ε,δ,µ, ω ε,δ,µ z ε,δ,µ + Q 3 z ε,δ,µ, t 3 zε,δ,µ T HP 1 z ε,δ,µ z ε,δ,µ 3 z ε,δ,µ P 1 εφ µ φ µ 1 z ε,δ,µ Λ ω ε,δ,µ c ε,δ,µ ω ε,δ,µ P 1 z ε,δ,µ, t z ε,δ,µ T HP 1 1 Q k+3 1 z ε,δ,µ P 1 z ε,δ,µ 3.10 z ε,δ,µ, 0 = z 0 and ω ε,δ,µ, 0 = ω 0 for ε > 0, δ > 0, µ > 0. The functions φ δ and φ µ are even mollifiers, and c ε,δ,µ = + π π c ε,δ,µ = + π π β z ε,δ,µ β β z ε,δ,µ β φ δ φ δ β Q z ε,δ,µ βbr z ε,δ,µ, ω ε,δ,µ βdβ β z ε,δ,µ β β z ε,δ,µ β φ δ φ δ β Q z ε,δ,µ βbr z ε,δ,µ, ω ε,δ,µ βdβ, β z ε,δ,µ β β z ε,δ,µ β βq z ε,δ,µ βbr z ε,δ,µ, ω ε,δ,µ βdβ β z ε,δ,µ β β z ε,δ,µ β βq z ε,δ,µ βbr z ε,δ,µ, ω ε,δ,µ βdβ, The RHS of the evolution equations for z ε,δ,µ and ω ε,δ,µ are Lipschitz in the spaces H k+ T and H k+ 1 T since they are mollified. Therefore we can solve 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 µ go to zero. The energy estimates that we can get are the following: d z ε,δ,µ H + F z ε,δ,µ dt 5 L + ωε,δ,µ H 3+ 1 Cδ z ε,δ,µ H 5 + F z ε,δ,µ L + ωε,δ,µ H m ε,δ,µ q l t j m ε,δ,µ q l t. We should note that for the new system without the φ µ mollifier, the length of the tangent vector z δ 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 K δ from the curve z δ. Similarly, we get let us omit the superscript δ, ε in z δ,ε and ω δ,ε l=0 l=0

99 3.4. ENERGY WITHOUT THE RAYLEIGH-TAYLOR CONDITION 89 K t = NICE3 + Q z 3 φ δ φ δ H ω + 1 z 3 Q φ δ φ δ H ω, k c ω = NICE35 + Q ω k c ω = NICE35, z H k K, and the following collection of terms: High Order From A: Q k+3 k K φ k δ φ δ H ω A From B: Q k+3 Λ k+1 ωφ δ φ δ k K B,,1 ε τ k+1 ω L D From C: No terms from C Low Order Type I From A: kq k+ Q k K k 1 φ δ φ δ H ω A 1 4Q k+ Q k K φ δ φ δ k H ω A 3 From B: Q k+ Q Λ ωφ k δ φ δ k K B 3 k + Q k+ Q H k+1 ωφ δ φ δ k K B 4 Q k+ H k+1 ωq φ δ φ δ k K B,1 k Q k+ H k+1 ωq φ δ φ δ k K B,3 k + 3 Q k+ Q H k+1 ωφ δ φ δ k K B,, From C: No terms from C.

100 90 CHAPTER 3. WATER WAVES WITH SURFACE TENSION Low Order Type II From A: No terms from A. From B: From C: 1 τ Q k+ Λ k ω Q ω 1 z τ z φ δ φ δ H k K B 1 Q k+4 ω k ωφ δ φ δ k+1 K C 1 We note that throughout this section we have repeatedly used the following commutator estimate for convolutions: φ δ fg gφ δ f L C g L f L, 3.11 where the constant C is 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. A + B,,1 + D = = Q k+3 k K φ δ φ δ H k+ ω Q k+3 H k+ ωφ δ φ δ k K ε k+1 ω L k K Q k+3 φ δ φ δ H k+ ω φ δ φ δ Q k+3 H k+ k K L Q k+3 L k+1 ω L ε k+1 ω L CεE p 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 A + B,,1 = 0. ω ε k+1 ω L 3.5 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 3.3. This Rayleigh-Taylor condition will hold in particular if the surface tension coefficient is small enough.

101 3.5. ENERGY WITH THE RAYLEIGH-TAYLOR CONDITION 91 Theorem Let k 3. Let z 0 be the image of a splash curve by the map P parametrized in such a way that z 0 = L π, where L is the length of the curve in a fundamental period, and such that z 1 0, z0 Hk+ T. Let ϕ, 0 H k+ 1 T be as in 3.3 and let ω, 0 H k 1 T. Then there exist a finite time T > 0, a time-varying curve z, t C[0, T ]; H k+, and functions ω, t C[0, T ]; H k 1 and ϕ C[0, T ]; H k+ 1 providing a solution of the water wave equations 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 for τ > 0. We will perform energy estimates on these solutions The energy We will define the energy for k 3 as Ek t = EE t + τ z Q k+1 k K + Q k k ϕλ k ϕ {{ {{ A B + z τ C Kt H 1 + KQ k+1 k 1 k 1 KΛ K + z {{ + z σq k k 1 C K {{ E z + mq k σt, C Kt H 1Q k k ϕ {{ D where mq k σ = min T Q k z, tσ, t and C is a sufficiently large constant such that C is strictly positive. Remember that ϕ was introduced in Equation 3.3. At this point is important to notice the following. Lemma 3.5. The following sentences hold. 1. Let ϕ H 3+ 1, ω H and z H k with k 4. Then ω H 3.. Let ϕ H 3+ 1, ω H 3 and z H k with k 5. Then ω H Let ω H 3+ 1, and z H k with k 5. Then ϕ H 3.5. This lemma shows that for a fixed τ > 0 the energy of this section is equivalent to this one in section This allows us to use this energy to extend the solutions of the theorem up to a time T which does not depend on τ for a small enough τ The energy estimates Again, we will only focus on the new terms A E since the estimates for the other ones were proved in [8] and in [0].

102 9 CHAPTER 3. WATER WAVES WITH SURFACE TENSION K Proposition K t = NICE3B + Q z 3 H ω + 1 z 3 Q H ω = NICE3B + 1 z H ϕ 1 z K ϕ, where NICE3B means Q j k K k NICE3B CE p 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 We now prove the second one. We begin by using the relation 3.3 to get K t = NICE3B + Q ϕ z H Q + Q ϕ z H Q + Q H z + Q c z H Q c Q = I + J We can easily see that c = z z Q BR = NICE3B since it is at the level of ω, z but we gain one derivative by multiplying by the tangential direction. This proves that Looking now to c we can see that J = NICE3B + Q z H ϕ Q. c = z z Q BR z z Q BR = I 1 + I. Using the standard estimates, the only thing that causes trouble in I 1 is when all the derivatives hit ω and therefore I 1 = NICE3B K Q z H ω. Regarding I, again, we need all the derivatives to hit BR to get the most singular terms, which are

103 3.5. ENERGY WITH THE RAYLEIGH-TAYLOR CONDITION 93 I = NICE3B Q [ Q 1 z z π Q z z [ 1 π [ z z π = NICE3B + Q 1 z z z β z zβ z zβ ] z zβ ω βdβ z z β z zβ z z z H ω Q Collecting all the terms from I 1 and I, we obtain ] ω βdβ ] ω βdβ z z z z H ω Q 1 ω z z z H z c Q = NICE3B + 1 z H K ω = NICE3B + 1 Q H K ϕ. We can finally write the total contribution as as we wanted to prove. K t = NICE3B + Q z H ϕ Q Q z H 4Q ϕ Q 3 1 z K ϕ + Q z H ϕ Q = NICE3B + Q z H ϕ Q 1 z K ϕ = NICE3B + 1 z H ϕ 1 z K ϕ ϕ Throughout this section, we will use the following estimate which was proved in [0] for the case without surface tension. The proof is exactly the same for the case with it. ϕ t = NICEB + ϕ ϕ z Q σ K + τ Q z KQ + M, where NICEB means for some positive constants C, p and any j. Q j Λ k ϕ k 1 NICEB CE p k t

104 94 CHAPTER 3. WATER WAVES WITH SURFACE TENSION 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 Q k+1 k z K H k ϕ τ Q k+1 k K k K ϕ = OK + τ Q k+1 k z K H k ϕ τ Q k+1 k k+1 K ϕ K = OK + A 1 + A Again, we need to be careful while computing the derivative of B as in Section 3.4. We obtain db dt = Q k Λ k ϕ k 1 ϕ t + Q k H k ϕ k 1 ϕ t ϕ = OK Q k Λ k ϕ k 1 ϕ z Q k Λ k ϕ k 1 Q σ K + τ Q k Λ k z ϕ Q k Q K τ Q k Q Λ k z ϕ k K τ + z k 1Qk+ Q H k ϕ k+1 K = OK + B 1 + B + B 3 + B 4 + B Development of the derivative of the B term We begin noticing that B 1 = OK, as it was proved in [8]. Integrating by parts in B 5, we have that B 5 τ = z k 1Qk+ Q H k+1 ϕ k K Furthermore, the only singular terms arising from B are when all derivatives hit either K or σ, this gives us B = OK Q k Λ k ϕ k 1 σ K Q k Λ k ϕ k 1 Kσ = OK + B,1 + B,. However, the only singular term of the Rayleigh-Taylor condition that is not in H k 1 is the one belonging to BR t z, ω z when the time derivative hits ω, this means B,1 = OK τ Q k Λ k ϕ KH k KQ = τ Q k+1 Λ k ϕ KH k K

105 3.5. ENERGY WITH THE RAYLEIGH-TAYLOR CONDITION 95 Finally, developing B 3 we obtain B 3 = τ Q k Λ k z ϕ Q k 3 K + τ Q k Λ k z ϕ Q k Q K = B 3,1 + B 3, Modulo lower order terms we can see that B 3, = OK + τ Q k Q Λ k z ϕ k K We can continue splitting B 3,1 into B 3,1 = OK + τ z = OK τ z = OK + B 3,1,1 + B 3,1, Q k+1 Λ k ϕ k+1 K + τ z Q k+1 Λ k+1 ϕ k K + τ z 3kQ k Q Λ k ϕ k K k 1Q k Q Λ k ϕ k K where in the last equality we have performed an integration by parts. We can observe that B 3, + B 4 = B 3,1, + B 5 = 0, B 3,1,1 + A 1 = 0 We will now see that B, cancels with the term arising from the derivative of E. Taking into account the previous lemmas de dt = σq k k 1 K H k+1 ϕ = OK B, Finally, we will see that the contributions from the time derivatives of C and D cancel B,1 and A. We start by noticing that, modulo lower order terms A = B,1. Furthermore dc dt dd dt = OK + τ = OK + τ C Kt H 1 + KQ k+1 H k+1 C Kt H 1Q k+1 k ϕ k+1 K, ϕλ k 1 K which, by integration by parts results in dc dt + dd dt + A + B,1 = OK. Adding all the contributions, we can bound the derivative in time of the energy by a power of the energy.

106 96 CHAPTER 3. WATER WAVES WITH SURFACE TENSION 3.6 Helpful estimates for the Birkhoff-Rott operator In this Section we will prove some of the estimates used throughout the paper for the sake of clarity to the reader. We begin with a classical decomposition of the Birkhoff-Rott operator. We should notice that we can write it in the following ways. On one hand: BR z, ω = 1 π + 1 π = 1 π z z β z z β z z ω βdβ tanβ/ z z ω βdβ tanβ/ z z β z z β z z ω βdβ + tanβ/ = z H ω + l.o.t ω. z On the other hand: z z H ω BR z, ω = 1 π = 1 π + ω π z z β ω π z z β ω β ωdβ + z z β π z z β dβ z z β z z β + ω z Λ z ω β ωdβ z z β 1 z z β 1 dβ 4 z sin β = ω z Λ z + l.o.t z. See [8], [8] for more details concerning the lower order terms. We will now prove energy estimates for the Birkhoff-Rott integral, showing that it is as regular as z. The proof is taken from [8, Section 6]. Lemma The following estimate holds BR z, ω H k C F z L + z H k+1 + ω H k j, 3.1 for k, where C and j are constants independent of z and ω. Remark 3.6. Using this estimate for k = we find easily that BR z, ω L C F z L + z H 3 + ω H j. 3.13

107 3.6. HELPFUL ESTIMATES FOR THE BIRKHOFF-ROTT OPERATOR 97 Proof: We shall present the proof for k =. Let us write BR z, ω, t = 1 C 1, β ω βdβ + z π z H ω where C 1 is given by C 1, β = z z β z z β z z tanβ/, 3.14 We shall show that C 1 L C F z L z C. To do so we split C 1 = D 1 + D + D 3 where and D 1 = z z β zβ z z β, D = β z[ z z β 1 z β ], The inequality D 3 = yields easily D 1 z C F z L. Then we can rewrite D as follows: z z [ 1 β 1 tanβ/ ]. z z β zβ z C β 3.15 D = z[ zβ z z β zβ + z z β z z β z ], β and, in particular, we have D zβ z z β zβ + z z β z z β. z β Using 3.15 we find that D z C F z L. Next let us observe that since β [, π] gives D 3 C F z L. The boundedness of the term C 1 in L gives us easily BR z, ω L C F z L z C ω L In BR z, ω, the most singular terms are given by P 3 = 1 π P V P 1 = 1 π P V z z β ω β z z β dβ, P = 1 π P V ω β z z β z z β dβ, ω β z z β z z β 4 z z β z z β dβ.

108 98 CHAPTER 3. WATER WAVES WITH SURFACE TENSION Again we have the expression P 1 = 1 C 1, β ω βdβ + z π z H ωd, giving us P 1 C F z j L z j C ω L + H ω Next let us write P = Q 1 + Q + Q 3 where Q = ω π Q 3 = 1 π where Λ = H. Using that Q 1 = 1 π ω z ω β ω z z β z z β dβ, z z β 1 z z β 1 dβ, z β z z β ω β dβ+ 4 sin β/ z Λ z, z z β β δ z C,δ, we get Q 1 + Q ω C 1 F z j z j C,δ, while for Q 3 we have that is Q 3 C ω L F z L z C + Λ z, P 1 + Λ z ω C 1 F z j z j C,δ Let us now consider P 3 = Q 4 + Q 5 + Q 6 + Q 7 + Q 8 + Q 9, where Q 4 = 1 π Q 5 = ω π ω β ω z z β z z β 4 z z β z z β dβ, Q 6 = ω z π z z β zβ z z β z z β z 4 z β dβ, β z z β zβ z z β z z β 4 dβ, Q 7 = ω z π z β π z z β 1 and Q 8 = ω z π z 4 z z z β 4 1 dβ, z 4 β 4 z z β 1 β 1 dβ, 4 sin β/ Q 9 = ω z z 4 z Λ z.

109 3.6. HELPFUL ESTIMATES FOR THE BIRKHOFF-ROTT OPERATOR 99 Proceeding as before we get P 3 C1 + Λ z ω C 1 F z j L z j C,δ, which together with 3.17 and 3.18 gives us the estimate P 1 + P + P 3 C1 + Λ z + H ω ω C 1 F z j L + z j H 3. For the rest of the terms in BR z, ω we obtain analogous estimates allowing us to conclude the equality BR z, ω L C1 + 3 z L + ω L ω C 1 F z j L z j C,δ. Finally the Sobolev inequalities yield 3.0 for k =. Lemma The following estimate will also be helpful BR z, ω z H k C F z L + z H k+ + ω H k j, 3.19 for k, where C and j are constants independent of z and ω. Proof: In BR z, ω z, the most singular terms are given by R 3 = 1 π P V R 1 = 1 π P V R = 1 π P V ω β z z z β z z β dβ, ω β z z z β z z β dβ, ω β z z z β z z β 4 z z β z z β dβ. R can be estimated in the same way as P. Regarding R 1, one can write it as R 1 = 1 [ z π P V z z β ω β z z β z z ] z dβ. Now, since ω β = β ω β, one can integrate by parts and bound the resulting kernel which has order -1 giving Finally, R 3 can be written in the form R 3 = 1 π P V R 1 C F z L + z H k+ + ω H k j. [ z z z β ω β z z β z z z β 4 z β z z β z 4 z z ] dβ, and bound R 3 by the kernel which has order 0 in L norm and ω in L norm. This completes the proof.

110 100 CHAPTER 3. WATER WAVES WITH SURFACE TENSION Then, the following corollary is immediate Corollary c H k C F z L + z H + ω k+ H j, 3.0 k for k, where C and j are constants independent of z and ω.

111 Chapter 4 Introduction to Computer-Assisted Proofs 4.1 Computer-Assisted Proofs and Interval arithmetics In the last 50 years computing power has experienced an enormous development. According to Moore s Law [7], every two years the number of transistors has doubled since the 1970 s. This phenomenon has resulted in the blooming of new techniques located in the verge between pure mathematics and computational ones. However, even nowadays when we can perform computations at the speeds of the order of Petaflops a quatrillion floating point operations per second we can not avoid the following questions, still fundamental in the rigorous analysis of the output of a computer program: Q1: Is a computer result influenced by the way the individual operations are done? Q: Does the environment operating system, computer architecture, compiler, rounding modes,... have any impact on the result? Sadly, the answer to these questions is Yes, which can be easily illustrated by the following C++ codes see Listings 4.1 and 4.3. The first one computes the harmonic series up to a given N in two ways: the first way adds the different numbers from the bigger ones to the smaller and the second one does the sum in the opposite way. The results for N = 10 6 can be seen in Listing 4.. They are not the same and curiously, the real result is not any of the two of them. The second program uses the MPFR library [47] to add two numbers given by the user in two different ways: rounding down and rounding up the result. The output is done in binary. We can see that the results differ Listing 4.4. Listing 4.1: Computation of the truncated Harmonic Series in two different ways int mainint argc, char* argv[]{ int N; cin >> N; cout.setfios::fixed; cout.precision15; double res1, res; res1 = res = 0.0; for int i=1; i<=n; i++{ 101

112 10 CHAPTER 4. INTRODUCTION TO COMPUTER-ASSISTED PROOFS res1 = res /doublei; for int i=n; i>=1; i--{ res = res + 1.0/doublei; cout << res1 << endl; cout << res << endl; Listing 4.: Result of the previous computation. Listing 4.3: Sum of two numbers with different rounding int main int argc, char **argv{ mpfr_t x, y, d, u; mpfr_prec_t prec; prec = atoi argv[1]; int pprec = prec - 1; mpfr_inits prec, x, y, d, u, mpfr_ptr 0; mpfr_set_str x, argv[], 0, GMP_RNDN; mpfr_printf "x = %.*Rb\n", pprec, x; mpfr_set_str y, argv[3], 0, GMP_RNDN; mpfr_printf "y = %.*Rb\n", pprec, y; mpfr_add d, x, y, GMP_RNDD; mpfr_printf "d = %.*Rb\n", pprec, d; mpfr_add u, x, y, GMP_RNDU; mpfr_printf "u = %.*Rb\n", pprec, u; return 0; x = p-4 y = p+0 d = p+0 u = p+0 Listing 4.4: Program executed with arguments This shows that even the simplest algorithms need a careful analysis: only two operations suffice to give different results if executed in different order or with different rounding methods. Fortunately, the theory of interval analysis developed by R. Moore [73] is an example of a tool, which albeit being impractical due to inefficient resources at the time of its conception, is now being widely used. It belongs to the paradigm known as rigorous computing in

113 4.1. COMPUTER-ASSISTED PROOFS AND INTERVAL ARITHMETICS 103 some contexts also called validated computing, in which numerical computations are used to provide rigorous mathematical statements about a result. The philosophy behind the theory of interval analysis consists in working with and producing objects which are not numbers, but intervals in which we are sure that the true result lies. Therefore the answer to the second question is also Yes. Nevertheless, we should be precise enough since even with plenty of resources, overestimation might lead to too big intervals which might not guarantee the desired result. Lately, interval methods have become quite popular among mathematicians. Several highly non-trivial results have been established by the use of interval arithmetics, see for example [58, 48, 81] as a small sample. However, there is a fraction of the mathematical community for which there exist doubts whether one can rely on the fundamentals of the physics in the sense that computers do the right thing according to given physical laws to prove a rigorous mathematical theorem or not [1]. Even so, it is my belief that after having seen the successful outcomes, it is clear that the future of mathematics must include techniques for performing validated numerics. In analysis, the most celebrated result is the proof of the dynamics of the Lorenz attractor Smale s 14th Problem done by Tucker [87] in 00. However, the study of the dynamics of a system has been restricted almost always to typically low-dimensional ODEs. As an example, we can cite the following papers related to the N-body problem, Rössler equations and the Henon map [66, 67, 93], but there is a big literature on the topic. Another problems involving ODEs but an infinite dimensional system are the computation of the ground state energy of atoms or the relativistic stability of matter see [8, 4, 41]. Regarding PDEs infinite dimension problems, most of the work has been carried out for dissipative systems i.e. systems in which the L -norm of the function decreases with time. The most popular ones are the Kuramoto-Sivashinsky equations or Navier-Stokes in low dimensions typically 1. The main feature of these models is that one can study the first N modes of the Fourier expansion of the function and see the rest as an error. Since the system is dissipative, if N is large enough, one can get a control on the error throughout time. We should remark here that the linearization of the water waves equation shows that the system is not dissipative, but dispersive. Other techniques such as the proof of existence of periodic orbits, for example reduce the problem to compute the norm of an adequate operator between Banach spaces and apply Brouwer s fixed point theorem to show that the operator has a fixed point the orbit or compute the Conley index of a certain region. These methods have been applied for instance in [94] Conley index for Kuramoto-Sivashinsky, [10] Bifurcation diagram for stationary solutions of Kuramoto-Sivashinsky, [37] Global atractors for viscous 1D Burgers, [45] Stationary solutions of viscous 1D Burgers with boundary conditions, [44] Traveling wave solutions for 1D Burgers equation, [43] Newton scheme around an approximate solution following the spirit of the one described in the next section, for Kuramoto-Sivashinsky. Representing an abstract concept such as a real number by a finite number of zeros and ones has the advantage that the calculations are finite and the framework is practical. The drawback is naturally that the amount of numbers that can be written in this way is finite although of the same order of magnitude as the age of the universe in seconds and inaccuracies might arise while performing mathematical operations. We will now discuss the

114 104 CHAPTER 4. INTRODUCTION TO COMPUTER-ASSISTED PROOFS basics of interval arithmetics. From now on, unless otherwise stated, we will suppose that the numbers are represented using 64 bits. All the reasoning can be easily reinterpreted in the case of arbitrary high precision multiprecision. Let F be the set of representable numbers by a computer. We will work with the set of representable closed intervals IR = {[a, a] a a, a, a F. For every element [a] IR we will refer to it by either [a] or by [a, a], whenever we want to stress the importance of the endpoints of the interval. We can now define an arithmetic by the theoretic-set definition [x] [y] = {x y x [x], y [y], 4.1 for any operation {+,,,. We can easily define them by the following equations: [x] + [y] = [x + y, x + y] [x] [y] = [x + y, x + y] [x] [y] = [min{xy, xy, xy, xy, max{xy, xy, xy, xy] [ 1 [x] [y] = [x] y, 1 ], whenever 0 [y]. y Note that this interval-valued operators can be extended to other algebraic expressions involving exponential, trigonometric, inverse trigonometric functions, etc. This derivation is purely theoretical, and we should keep in mind that, if carried out on a computer, the results of an operation have to be rounded up or down according to whether we are calculating the left or right endpoint so that the true result is enclosed in the produced interval. The main feature of the arithmetic is that if x [x], y [y], then necessarily x y [x] [y] for any operator. This property is fundamental in order to ensure that the true result is always contained in the interval we get from the computer. We remark that this arithmetic is not distributive, but subdistributive, i.e: [a] [b] + [c] [a] [b] + [a] [c] [a] [b] + [c] [a] [b] + [a] [c] Example If we set [a] = [3, 4], [b] = [1, ], [c] = [ 1, 1], then: [a] [b] + [c] = [0, 1] [a] [b] + [a] [c] = [ 1, 1] This illustrates that the way in which operations are executed in the interval-based arithmetic matters much more than in the real-based. As an example, consider the function fx = 1 x and a domain D = [ 1, 1]. Over the reals, we can write f as any of the following functions: f 1 x = 1 x f x = 1 x x f 3 x = 1 + x 1 x

115 4.. AUTOMATIC DIFFERENTIATION 105 However, evaluating f i over D we get the enclosures: f 1 D = [0, 1] f D = [0, ] f 3 D = [0, 4] We observe that although f 3 is completely factored, if we expand it we get an expression of the form x x which in the interval-based arithmetic is equal to an interval of a width twice the width of the domain in which we are evaluating the expression: a price too high to pay compared with the width of the interval [0, 0], another form to write the same expression over the reals. 4. Automatic Differentiation One of the main tasks in which we will need the help of a computer is to calculate a massive amount of function evaluations and their derivatives up to a given order at several points and intervals. In order to perform it, one could first think of trying to differentiate the expressions symbolically. However, we don t need the expression of the derivative, just its evaluation at given points. This, together with the fact that the size of the derivative might grow exponentially, makes the use of symbolic calculus impractical. Instead of calculating the expression of every derivative, we will use the so-called automatic differentiation methods. Suppose fx is a sufficiently regular function and let x 0 be the point or interval of which we want to calculate its image by f. We define f 0 = fx 0 d k f k = 1 k! dx k fx 0, k = 1,,..., N, where N stands for the maximum number of derivatives of the function we want to evaluate. We can think about f as being the coefficients of the Taylor series around x 0 up to order N. We now show how to compute the coefficients f for some of the functions that will appear in our programs. The generalization of the missing functions is immediate. However, it is possible to derive similar formulas for any solution of a differential equation e.g. Bessel functions. u ± v k = u k ± v k k u v k = u j v k j j=0 u v k = 1/v u k k v j u v k j j=1 sinu k = 1 k 1 j + 1cosu k 1 j u j+1 k j=0

116 106 CHAPTER 4. INTRODUCTION TO COMPUTER-ASSISTED PROOFS cosu k = 1 k 1 j + 1sinu k 1 j u j+1 k j=0 Automatic differentiation has become a natural technique in the field of Dynamical Systems, since the cost for evaluating an expression up to order k is Ok, making it a fast and powerful tool to approximate accurately trajectories [84]. It has also been used for the computation of invariant tori and their associated invariant manifolds [57, 56] or the computation of normal forms of KAM tori [54]. For more applications in Dynamical Systems we refer the reader to the survey [55]. Automatic Differentiation is also an important element in the so-called Taylor models [76, 71, 64], in which functions are represented by couples P,, being P a polynomial and an interval bound on the absolute value of the difference between the function and P. Nowadays, there are several packages that implement it, for example [65, 1]. 4.3 Integration In this section we will discuss the basics of rigorous integration. A more detailed version concerning singular integrals of piecewise-defined functions can be found in the next Chapter. We will only give the details of the one-dimensional case, omitting the multidimensional one. A brief description of a two-dimensional rigorous integration method can be found in the last chapter. The main problem we address here is to calculate bounds for a given integral I = b a fxdx, < a < b <. Different strategies can be used for this purpose. For instance, we can extend the classical integration schemes: I = N i=1 xi x i 1 fxdx, a = x 0 < x 1 <... < x N = b. In every interval, we approximate fx by a polynomial px and an error term. We detail some typical examples in Table 4.1: It is now clear where the interval arithmetic takes place. In order to enclose the value of the integral, we need to compute rigorous bounds for some derivative of the function at the integration region. Another approach consists of taking the Taylor series of the integrand up to order n as the polynomial p i x. Centering the Taylor series in the midpoint of the interval makes us integrate only roughly over half of the terms since the other half are equal to zero. We can see that

117 4.3. INTEGRATION 107 Midpoint Rule Trapezoid Rule Simpson s Rule N I p i xdx p i x f xi +x i 1 Error i=1 fx i 1 1 x x i 1 h i x xi 1 +fx i h i fx i 1 x x ix x i+x i 1 h i / xi + x i 1 x xi x x i 1 f h i /4 +f x i x x i 1x x i+x i 1 h i / b a 4 h f [a, b] b a 1 h f [a, b] b a 880 h4 f 4 [a, b] Table 4.1: Different Schemes for the rigorous integration. b a fxdx = b a b a fa + x af a fa + x af a x an f n a + n! x an f n a + n! x an+1 f n+1 ξxdx n + 1! x an+1 f n+1 [a, b]dx n + 1! = b afa + 1 b a f x an+1 a f n x an+ a + f n+1 [a, b]. n + 1! n +! {{ {{ Real number thin interval Error thick interval We now compare the two methods in the following examples, in which we integrate 1 0 ex dx. Example If we take N = 4 and use a trapezoidal rule, we enclose the integral in Example e x dx = 1 e 0 + e 1/4 + e 1/ + e 3/4 + e [1.77, 1.773] [ , ] e x dx 1 + x + x 0 + x3 6 e[0,1] dx = x + x + x3 x=1 6 + [1, e] x4 x=0 4 = [1, e] 4 = [ , ] x=1 x= e[0,1]

118 108 CHAPTER 4. INTRODUCTION TO COMPUTER-ASSISTED PROOFS The exact result is e We can see that there is a tradeoff between function evaluations efficiency of the scheme and quality precision of the results, since the first method is more exact but requires more evaluations of the integrand, while for the second it is enough to compute the Taylor series of the integrand.

119 Chapter 5 From a graph to a Splash Singularity 5.1 Introduction In this Chapter we will give details on a possible proof of the following result: Conjecture There exist initial data z 0, ω 0 that are solutions of the water wave equations such that at time 0 they can be parametrized as a graph, then turn over at a finite time T 1 > 0, and finally produce a splash at a finite time T > T 1. We should remark that this conjecture is a combination of Theorem.1.1 and [, Theorem 7.1] and is supported by numerical evidence. The proof follows along this lines. First of all, we will move backwards in time, being 0 the time of the splash, T T 1 the time of the turning and T the time in which the solution can be parametrized as a graph. We start computing a numerical approximation of a solution x, γ, ψ to the water waves equation that starts as a splash, turns over and finally is a graph. Such a candidate is depicted in Figures.1,3.1. Another ingredient is the following stability result that was announced in [19], that will allow us to conclude the following: if x, γ approximately satisfies equation 5.1, then near to x, γ there exists an exact solution z, ω. Below is the theorem. Theorem 5.1. Let D, t z, t x, t, d, t ω, t γ, t, D, t ϕ, t ψ, t 109

120 110 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY where x, γ, ψ are the solutions of x t = Q xbrx, γ + bx + f b = + π Q x BRx, γ π x d Q x BRx, γ β x dβ {{ + + π π f x b s x β f β x d x β dβ {{ b e 5.1 γ t ψ, t +BR t x, γ x = Q x BRx, γ + bbr x, γ x + bγ Q xγ 4 x = Q x,tγ,t x,t b s, t x, t, P 1 x + g z, ω, ϕ are the solutions of 5.1 with f g 0 and E the following norm for the difference Et D Q σ z H + 3 z 4 D + d H + D. H 3+ 1 Then we have that where and d dt Et CtEt + Ek t + cδt Ct = CEt, x H 5+ 1 t, γ H 3+ 1 t, ζ H 4+ 1 t, F x L t δt = f H 5+ 1 t, g H 3+ 1 tk, k big enough depend on the norms of f and g, and Et is given by Et = z Q σ z H t + 3 z 4 z d + F z L t where the L norm of the function T + ω H t + ϕ z t + H 3+ 1 mq σ z t + F z β z, t z β, t, measures the arc-chord condition, σ z BR t z, ω + ψ z BR z, ω + Q BRz, ω + ω z z, β T 4 l=0 z + ω z z t + 1 mq l t ψ z z z Qz z + P 1 z z 5.

121 5.. BOUNDS FOR F T AND GT 111 is the Rayleigh-Taylor function, mq σ z t min T Q, tσ z, t, and finally for l = 0,..., 4. mq l t min T z, t ql Remark We can absorb the terms in Et by Et raised to an appropriate power and terms in x, γ, ψ by performing the splitting z = z x + x or the analogous one for a different variable for any norm or any quantity that appears in Et. We can construct a solution z, ω, ϕ that satisfies 5.1 with f = g = 0 and very similar same initial conditions z 0 x 0, ω 0 γ 0, ϕ 0 ψ 0. If we knew Ct, ft, gt, k or bounds on them, a priori, then we could provide bounds on Et at any time T. We point out here that Et controls the norm z 1 x 1 L. Let T g be a time in which the approximate solution is a graph, i.e. x 1, T g > 0. Now, if ET g < x 1, T g then z 1, T g > z 1 x 1 L x 1, T g > 0, and this shows that z is a graph. In other words, the possible set of solutions of the water waves equation is a ball centered at x, γ, ζ with the topology given by E. All of the elements of this ball are graphs, therefore the solution is necessarily a graph. Thus, the problem is reduced to study and find bounds for Ct, ft, gt, k. 5. Bounds for ft and gt 5..1 Representation of the functions and Interpolation The first thing one has to decide is how to represent the data and how to pass from the cloud of points in space-time obtained by non-rigorous simulation to a function defined everywhere in [, π] [0, T ]. We need to interpolate in some way. One of the first things that can come to one s mind is to use the first N Fourier modes. This approach has two disadvantages. The first one is that the linearized water waves equation is not dissipative, hence we will not have a control for the tails uniformly for all time, even in the case where the tails have very small norms at time zero. The second disadvantage is numerical. Suppose N 10 3, which is a reasonable guess. Since we need to take 5.5 derivatives in the curve, the high order coefficients will be multiplied by roughly a factor of If we work with a 64-bit representation, machine epsilon is of the order and we will run into trouble because the computer will not distinguish between zero and non-zero values. Of course, this problem

122 11 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY can be solved if we use high precision arithmetics, but in any case we would be multiplying and dividing by very big constants. In our case, we chose to represent the functions x and γ by piecewise polynomials splines of high degree 10 in space, and low degree 3 in time. To do so, we first interpolate in space for every node in the time mesh. The interpolation is made via B-Splines. Since the interpolation is reduced to solve a linear interval system Ac = y, where A is constant in time and space and y depends on the values of the function at time t since the mesh in space is constant, we precondition by multiplying by the non-rigorous inverse of the midpoints of the entries of A. We remark that the system is interval-based because we need to produce a curve that is a splash i.e. there have to be two points 1, such that we can guarantee x 0 1 = x 0. Finally, the system is solved using a rigorous Gauss-Seidel iterative method. We also remark that the need for interval-based calculations is only strictly necessary at time t = 0 since it is the only point in which we have to guarantee some equality. By working with multiprecision 104 bits we can get widths in the coefficients of the order of In order to perform interpolation in time, we fix the values of the function and its time derivative at the mesh points. This gives us lots of systems of 4 equations the values of the function and its derivative at both endpoints and 4 unknowns the 4 coefficients of the degree 3 polynomial but with an explicit formula for each of them. With this method, our spline will be C 1 in time but it might not be C. 5.. Rigorous bounds for Singular integrals In this section we will discuss the computational details of the rigorous calculation of some singular integrals. In particular we will focus on the Hilbert transform, but the methods apply to any singular operator of order -1. Parts of the computation the N part are slightly related to the Taylor models with relative remainder presented in M. Joldeş thesis [64]. Let us suppose that we have a function f given explicitly by a spline piecewise polynomial which is C k 1 everywhere and C k everywhere but in finite points the points in which the different pieces of the spline are glued together. We need to calculate rigorously the Hilbert Transform of f, that is Hfx = P V π T fx fy tan x y dy, and we want to approximate it by a piecewise polynomial function with less regularity, plus an error that can be bounded in H q, 0 q c < k and in L. Let us assume that the knots of the spline are i, i = 0,..., N 1 and that we fix x [ i, i+1 ] where the indices are taken modulo N and the distance between the indices is taken over Z N. We can split our integral in

123 5.. BOUNDS FOR F T AND GT 113 Hfx = P V π = P V π T fx fy tan x y dy = P V π j i >K j+1 j Hf F x + Hf N x. j j+1 j fx fy tan x y dy + P V π fx fy tan x y dy j i K j+1 j fx fy tan x y dy Now, if we want to express Hf F x as a polynomial, it is easy since the integrand does not have a singularity. Hence Hf F x = P V π = j i >K j+1 j i >K j j+1 j j+1 fx fy tan x y dy = P V F j x, ydy π j i >K j c nm x x i m y y j n + Ex, ydy P x + Ex, n,m where E accounts for the error and is a polynomial with interval coefficients. Typically, we will use as the points for the Taylor expansions x i = i since we will compare the resulting polynomial with another one of the form j b jx x i j and we will also choose y j = j+ j+1. This choice is twofold: first, we will only have to integrate half of the terms since the rest will integrate to zero; and second, the error estimates will be better for this choice of y j in the sense that the coefficients will be smaller. All the computations will be carried out using automatic differentiation. We should remark that we can get estimates for the error E in any of the above mentioned norms without having to recompute it since the relation q xhf F x q xp x = q xex holds for every q < k. Now, we move on the the term Hf N x. In this case, we perform a Taylor expansion in both the denominator x y tan = x y + cx y 3, c = small interval constant and the numerator fx = fy + x yf y + 1 x y f y n! x yk 1 f k 1 η, where η belongs to an intermediate point between x and y, which we can enclose in the convex hull of [ i, i+1 ] and [ j, j+1 ] where the convex hull is understood in the torus. Since typically K will be very small compared to N there is no ambiguity in the definition. Finally, we can factor out x y and divide both in the numerator and the denominator. Since we know fy explicitly, we can perform the explicit integration and get a piecewise polynomial as a result.

124 114 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY 5..3 Estimates of the norm of the Operator I + T In this subsection we will outline how to compute the norm of the operator I + T = I + BRz,, z. Since the operator T behaves like a Hilbert Transform plus smoothing terms, we will describe how to calculate rigorously with the help of a computer an estimate for the norm of its inverse. The procedure is more general and can be applied to a bigger family of kernels. Let T = R/πZ, and let Ax, Bx be real-valued functions on T. Also, let Ex, y be a real-valued function on T T. We assume A, B and E are given by explicit formulas such as as perhaps piecewise trigonometric polynomials or splines, and Ex, y is a trigonometric polynomial on each rectangle I J of some partition of T T. We suppose A, B, E are smooth enough. Let H be the Hilbert transform acting on functions on T, i.e. Hfx = P V π T y cot fx ydy. Assume that A and B have no common zeros on T. Let Sfx = Axfx + BxHfx + Ex, yfydy, T f L T. Thus, S is a singular integral operator. We hope that S 1 exists and has a not-so-big norm on L, but we don t know this yet. Our goal here is to find approximate solutions F of the equation SF = f for suitable given f L T, and to check that SF f L T < δ for suitable δ. Our computation of F will be based on heuristic ideas, but the computation of an upper bound for SF f L T will be rigorous. In our case, Ax = 1, Bx = 1. To carry this out, let H 0 H 1 L T be finite-dimensional subspaces, e.g. with H i consisting of the span of wavelets from a wavelet bases having lengthscale N i. Here N 1 N say. Let π i be the orthogonal projection from L T to H i, and let us solve the equation π 1 Sπ 1 F = π 0 f. 5.3 If f is given explicitly in a wavelet bases, then 5.3 is a linear algebra problem, since π 1 Sπ 1 is of finite rank, and its matrix in terms of some given basis for H 1 can be computed explicitly. If π 0 f Rangeπ 1 Sπ 1, then our heuristic procedure fails. If π 0 f Rangeπ 1 Sπ 1, then we find F H 1 such that π 1 Sπ 1 F = π 0 f, i.e. π 1 SF = π 0 f. We then have SF f L T I π 1 SF L T + I π 0 f L T,

125 5.. BOUNDS FOR F T AND GT 115 and both norms on the right-hand side may be estimated explicitly. Now, our goal is to make a heuristic computation of an operator of the form Sfx = Ãxff + BxHfx + Ẽx, yfydy such that S S I has small norm on L T. Here, we will make a heuristic computation of S; later we will give a rigorous upper bound for the norm of S S I on L T. By a heuristic computation of S we mean a heuristic computation of Ã, B and Ẽ. We first find à and B by setting A + ibã + i B Aà = 1 B B = 1 A B + Bà = 0 Then, this means that S S = Aà B B + A B + BÃH + Smoothing terms = I + Smoothing terms So, from now on, we suppose that à and B are known. For the operator I +T, this means à = 1/, B = 1/. We want to compute Ẽ. Now, let {φ ν be some orthonormal basis for L T, for example a wavelet basis. By the previous methods, we can try to find functions ψ ν L T such that Sψ ν φ ν has small norm. We carry this for ν = 1,..., N for a large N. We now try to make Ẽ satisfy Ãxφ ν x + BxHφ ν x + Ẽx, yφ ν y = ψ ν x for ν = 1,..., N. 5.4 T Thus, we want Ẽx, yφ ν ydy = T T ψ ν x Ãxφ νx BxHφ ν x ψ ν # x, ν = 1,..., N. 5.5 Note that ψ ν # can be computed explicitly. Since the φ ν all ν form an orthonormal basis for L T, it is natural to define Ẽx, y = N ψ ν # φ ν y. This can be computed explicitly, and it satisfies 5.5. Thus, we can the compute S S = A + BH + Eà + BH + Ẽ ν=1 = Aà + A BH + AẼ + BHà + BH BH + BHẼ + Eà + E BH + EẼ = Aà + A BH + AẼ + BÃH + B[H, Ã] B B + B[H, B]H + BHẼ + Eà + E BH + EẼ = Aà B B + A B + BÃH + {AẼ + B[H, Ã] + B[H, B]H + BHẼ + Eà + E BH + EẼ 5.6

126 116 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY We claim that all terms enclosed in curly brackets are integral operators of the form S # fx = E # x, yfydy, for an E # that we can calculate. Let us go term by term T AẼ has the form S#, with E # x, y = AxẼx, y. B[H, Ã] has the form S#, with E # x, y = 1 π Bx cot x y Ãx Ãy. Note that if à is a piecewise trigonometric polynomial and Ck, then E # can easily be computed modulo a small error in C k 1. B[H, B]H has the form S #, with E # x, y = 1 x z 4π BxP V cot Bx Bz z y cot dz. = 1 { x z 4π BxP V cot Bx Bz B z y x cot dz. BHẼ has the form S#, with E # x, y = 1 π BxP V = 1 π BxP V x z cot Ẽz, ydz. x z cot Ẽz, y Ẽx, y dz. Eà has the form S#, with E # x, y = Ẽx, yãy. E BH has the form S #, with E # x, y = 1 π P V = 1 π P V Ex, z Bz z y cot dz. { Ex, z Bz By Ex, y cot EẼ has the form S#, with E # x, y = Ex, zẽz, ydz. z y dz. This proves the claim. Letting E # fx = T E# x, yfydy be the operator in curly brackets in 5.6, we see that S S = Aà B B + A B + BÃH + E#, and that the function E # x, y can be computed modulo a small error in C 0 T T. Therefore, we obtain an upper boundfor the norm of S S I, namely max Aà B B 1 + max A B + {max Bà + max E # x, y dy, max E # x, y dx. x y

127 5.. BOUNDS FOR F T AND GT 117 Defining S err := S S I, we obtain an explicit upper bound δ for the norm of S err on L T. We hope that δ < 1. If not, then we fail. Suppose δ < 1. Then S S = I + S err S SI + S err 1 = I, so we obtain a right inverse for S, namely SI + S err 1, which has norm at most S 1 δ 1, 5.7 where S denotes the norm of S as an operator on L T. Recall Sfx = Ãxfx + BxHfx + Ẽx, yfydy. T Therefore, { S max Ãx + max Bx + max max x Ẽx, y dy, max y Ẽx, y dx. Plugging that bound into 5.7, we obtain an explicit upper bound for the norm on L of a right inverse for S. Similarly by looking at SS instead of S S, we obtain an upper bound for the norm on L of a left inverse for S. Remark 5..1 To estimate e.g. max x T E# x, y dy it may be enough just to use the trivial estimate max E # x, y dy π max x T x,y E# x, y Remark 5.. Time dependent solutions For t [t 0, t 1 ] a small time interval, let S t fx = Ax, tfx + Bx, thfx + Ex, y, tfydy, T where for each t,a, t, B, t, E,, t are as assumed above. If A, B, E depend in a reasonable way on t, then one shows easilly that S t S t0 < η for all t [t 0, t 1 ]. We can make η small by taking t 1 close enough to t 0. Suppose we prove that St 1 0 C 0 by the previous methods. Then, of course we obtain an upper bound for St 1 valid for all t [t 0, t 1 ].

128 118 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY 5.3 Bounds for Ct and k Writing the differential inequality as a differential system of equations The calculation of a bound for Ct requires more effort than the previous one since one needs to calculate the terms one by one and add all their contributions to Ct. For example, in order to calculate the evolution of the norm D H kt a systematic approach is to take k derivatives k ranging from 0 to 4 in the equation for the evolution of z 5.1 with f = g = 0, take another k derivatives in the equation for x 5.1 with arbitrary f, g and subtract them. Let us focus from now on in the term Qz BRz, ω Qx BRx, γ and its derivatives. One notices that in order to write a term in the variables z, ω, ϕ composed of a factors minus its counterpart in the variables x, γ, η in a suitable way i.e. as a sum of terms that only have factors x, γ, η, D, d, D then the number of terms is a 1. The way of writing it is the classical way of adding and subtracting the same term with the purpose of creating differences of terms and eliminate all the occurrences of the variables z, ω, ϕ. An example for the Birkhoff-Rott operator with Q = 1 is given next. We should remark that the computation and bounding of the Birkhoff-Rott is the most expensive one, being easier the rest of the terms. BRz, ω BRx, γ = 1 x xβ ωβ γβ dβ π x xβ + 1 z zβ x xβ π x xβ γβ dβ + 1 z zβ x xβ π x xβ ωβ γβ dβ π z zβ 1 x xβ x xβ γβdβ π z zβ 1 x xβ x xβ ωβ γβdβ π z zβ 1 x xβ z zβ x xβ γβdβ π z zβ 1 x xβ z zβ x xβ ωβ γβdβ After having seen this, it is clear that a tool that can perform symbolic calculations derivation and basic arithmetic at least and the correct grouping of the factors is required since the performance at this task by a human is not satisfactory. We developed a tool in 900 lines of C++ code that could do all this and output the collection of terms in Tex. We show an excerpt of the terms concerning the fourth derivative of BRz, ω BRx, γ. The total number of terms in that case is 841.

129 5.3. BOUNDS FOR CT AND K 119 π BRx, 4 γ BRz, 4 ω = + x 4 x 4 β 1 d β x x β d + x 4 x 4 β 1 d β x x β 1 z z β d + x 4 x 4 β 1 γ β x x β 1 z z β d + 4 x 3 x 3 β 1 d β x x β d + 4 x 3 x 3 β 1 d β x x β 1 z z β 8 x 3 x 3 β d β 1 x x β x x β D D βd 8 x 3 x 3 β 1 d β x x β x x β x x βd 8 x 3 x 3 β 1 d β x x β D D β D D βd 8 x 3 x 3 β 1 d β x x β d x x β D D βd 8 x 3 x 3 β 1 1 d β x x β z z β D D β D D βd more terms... However, there is a significant way to reduce the number of terms in the estimates: writing the equation in complex form instead of vector form. Thus, we can write the evolution for z in the following way: t z, t = 1 1 π T z, t zβ, t ωβ, tdβ + c, t z, t In this formulation, the amount of terms of the fourth derivative accounts for only 140 terms. We present the first 10 below.

130 10 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY π BRx, 4 γ BRz, 4 ω = 7 x x 1 β x x β x x β D D βd βd 7 x x 1 β x x β x x β D D βγ βd 7 x x β D D 1 β x x β D D βd βd 7 x x β D D 1 β x x β D D βγ βd 36 D D β D D β 1 4 d βd x x β 36 D D β D D β 1 4 γ βd x x β x x β D D 3 β D D βd βd x x β D D 3 β D D βγ βd x x β D D β D D β d βd x x β D D β D D β γ βd more terms The final observation is that if we consider Et as a scalar, we might not get suitable estimates. In order to get better estimates, we will modify the energy into a vectorized version E v t, which we will also denote by Et by abuse of notation. This new vectorized energy will be as follows

131 5.3. BOUNDS FOR CT AND K 11 D L D H 1 D H D H Et = 3, d L d H 1 d H. where the inhomogeneous spaces H k have their norm defined by f H = f k k L. With this vectorized system, we avoid both the bounding of any given norm by the full energy and any constant factor arising from interpolation between two Sobolev spaces. Thus, our constant Ct will roughly be of a size comparable to the largest eigenvalue of the linearized system Estimates for the linear terms with Q = 1 Since we expect Et to be small, the terms that affect more to the evolution of Et are the linear ones. We now report on the non-rigorous experiments over the linear terms to obtain an approximate bound of the behavior of the full system i.e. an approximation to the largest eigenvalue of the linearized system. We remark that a multiplication of the estimates by a constant, even a small factor for example, has a big impact on the system, rendering the estimates useless and the estimations not tight enough, because the type of estimates we are going to get are exponential on the product of the time elapsed between the splash and the graph and the constant. Therefore, we should be very careful and fine estimates have to be developed. First of all, we will work with Q = 1 and later move on to the case Q 1. We will adopt the following convention do denote the different Kernels integral operators that appear: Θ a 1,a,a 3,a 4 b 1,b, β = 1 x xβ b 1 x xβ a 1 x xβ a x 3 xβ 3 a 3 x 4 xβ 4 a 4 b γβ Θ a 1,a,a 3,a 4 1 b 1, 1, β = x xβ b x 1 xβ a 1 x xβ a 3 x 3 xβ a 3 4 x 4 xβ a 4. The operators for which b 1 will act on D or its derivatives whereas the operators for which b = 1 will act on d or its derivatives. We now describe how to split the Kernels

132 1 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY in such a way that they can be computed. For the case where b 1 we illustrate this by splitting Θ 0,0,0,0,0, but the technique can be applied to any Kernel. 1 π where Θ 0,0,0,0,0 D Dβdβ = 1 π D K, βγβdβ 1 K, βγβdβdβ π {{ {{ + 1 D Dβ π c 1 T 1 γβdβ 4 sin β {{ T 3 K, β = 1 x xβ c 1 = 1 x c = x x π c T D Dβ γβdβ, 5.8 tan β {{ T 4 c 1 4 sin β c tan β We can think of c 1 and c as the Taylor coefficients of Θ, β around β =. We can bound the terms in 5.8 in the following way: We have then the estimates T 4 = c [HDγ DHγ] T 3 = c 1 [ΛDγ DΛγ] T 4 L c L D L γ L + D L Hγ L T 3 L c 1 L D L γ L + D L γ L + D L Λγ L. We now move on to T 1. We will estimate it in the following way: T 1 Dd = 1 D K, βγβdβd 1 π π D L K, βγβdβ To estimate the kernel T we will use the Generalized Young s inequality [46]: Defining T D L = 1 4π K, βγβdβk, σγσdσdβdσd. Kβ, σ = K, βγβk, σγσd, L.

133 5.3. BOUNDS FOR CT AND K 13 we have that T D L = 1 4π = 1 4π 1 4π D L Kβ, σdβdσdβdσ Kβ, σdσdσ Dβ 1 4π C D L, K,σdσ L { C = max max β dβ Kβ, σ dσ, max σ Kβ, σ dβ We finally show how to estimate the Kernels with b = 1. We will do this by showing how to estimate Θ 0,0,0,0 1, 1 but the technique can be applied to any Kernel. where 1 π Θ 0,0,0,0 1, 1 dβdβ = 1 K, βdβdβ π {{ K, β = c 1 = 1 x. + 1 π c 1 T 1 1 β dβdβ, tan {{ T 1 x xβ tan c 1 β We can easily estimate these two terms applying to T 1 the same estimates Young s inequality as for T in the previous case and by noting that T is 1 c 1Hd Estimates for the linear terms with Q 1 To perform the real estimates, where Q 1 we will use the estimates from the previous sections. We will explain how to pass from the former ones to the latter ones. We will illustrate this by computing the linear terms of the Birkhoff-Rott operator. First of all, the total number of terms will increase by a factor, since we will have Q zbrz, ω Q xbrx, γ = Q z Q xbrz, ω BRx, γ {{ nonlinear + Q xbrz, ω BRx, γ {{ calculated before + Q z Q xbrx, γ {{ new terms

134 14 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY In order to calculate the old terms with Q 1, the only thing we have to do is to incorporate a factor of Q k x on the estimates. The new terms can easily be calculated using that, up to linear order Q z Q x = x 4, 3x 8 x 1 x D + OD. The following tables summarize the non-rigorous estimations of the kernels obtained for the numerical simulations at time t = 0. We will mean by acting on D Dβ or its derivatives whenever D is referred, and dβ or its derivatives whenever d is referred.

135 derivatives in Q: Linear terms Num Kernel Acts T 1 T T 3,1 T 3, T 3,3 T 4,1 T 4, 1 Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,1 D Θ 1,0,0,0 3,0 D Θ 1,0,0,0, 1 d Θ 1,0,0,0 3,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0,1 D Θ 0,1,0,0, 1 d Θ 1,0,0,0, 1 d Θ,0,0,0 3, 1 d Θ 0,0,0,0 1, 1 d Θ 0,1,0,0 3,0 D Θ 1,0,0,0 3,1 D Θ,0,0,0 4,0 D Θ 0,0,0,0, D Θ 0,1,0,0 3,0 D Θ 1,1,0,0 3, 1 d Θ 1,0,0,0 3,0 D Θ 1,0,0,0 3,1 D Θ,0,0,0 4,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0,1 D Θ 0,0,0,0, D Θ 0,0,1,0, 1 d Θ 0,1,0,0, 1 d BOUNDS FOR CT AND K 15

136 11 Θ 1,1,0,0 4,0 D Θ 1,0,0,0, 1 d Θ,0,0,0 3, 1 d Θ 3,0,0,0 4, 1 d Θ 0,0,0,0 1, 1 d Θ 0,0,1,0 3,0 D Θ 0,1,0,0 3,1 D Θ 1,0,0,0 3, D Θ,0,0,0 4,1 D Θ 3,0,0,0 5,0 D Θ 0,0,0,0,3 D Θ 1,1,0,0 4,0 D Θ 0,0,1,0 3,0 D Θ 1,0,1,0 3, 1 d Θ 0,1,0,0 3,0 D Θ 0,1,0,0 3,1 D Θ 1,1,0,0 3, 1 d Θ,1,0,0 4, 1 d Θ 1,0,0,0 3,0 D Θ 1,0,0,0 3,1 D Θ 1,0,0,0 3, D Θ,0,0,0 4,0 D Θ,0,0,0 4,1 D Θ 3,0,0,0 5,0 D Estimated using an extra cancelation 15 Θ 0,0,0,0,1 D Θ 0,0,0,0, D Θ 0,0,0,0,3 D Θ 0,0,0,1, 1 d Θ 0,0,1,0, 1 d CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY

137 0 Θ 1,0,1,0 4,0 D Θ 0,1,0,0, 1 d Θ 1,1,0,0 4,1 D Θ,1,0,0 5,0 D Θ 1,0,0,0, 1 d Θ 0,,0,0 3, 1 d Θ,0,0,0 3, 1 d Θ 3,0,0,0 4, 1 d Θ 4,0,0,0 5, 1 d Θ 0,0,0,0 1, 1 d Θ 0,0,0,1 3,0 D Θ 0,0,1,0 3,1 D Θ 0,1,0,0 3, D Θ 1,0,0,0 3,3 D Θ 0,,0,0 4,0 D Θ,0,0,0 4, D Θ 3,0,0,0 5,1 D Θ 4,0,0,0 6,0 D Estimated using an extra cancelation derivative in Q: Linear terms Num Kernel Acts T 1 T T 3,1 T 3, T 3,3 T 4,1 T 4, 1 Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,1 D Θ 1,0,0,0 3,0 D Θ 1,0,0,0, 1 d BOUNDS FOR CT AND K 17

138 1 Θ 1,0,0,0 3,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0,1 D Θ 0,1,0,0, 1 d Θ 1,0,0,0, 1 d Θ,0,0,0 3, 1 d Θ 0,0,0,0 1, 1 d Θ 0,1,0,0 3,0 D Θ 1,0,0,0 3,1 D Θ,0,0,0 4,0 D Θ 0,0,0,0, D Θ 0,1,0,0 3,0 D Θ 1,1,0,0 3, 1 d 3 Θ 1,0,0,0 3,0 D Θ 1,0,0,0 3,1 D Θ,0,0,0 4,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0,1 D Θ 0,0,0,0, D Θ 0,0,1,0, 1 d Θ 0,1,0,0, 1 d Θ 1,1,0,0 4,0 D Θ 1,0,0,0, 1 d Θ,0,0,0 3, 1 d Θ 3,0,0,0 4, 1 d Θ 0,0,0,0 1, 1 d Θ 0,0,1,0 3,0 D Θ 0,1,0,0 3,1 D Θ 1,0,0,0 3, D Θ,0,0,0 4,1 D CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY

139 0 Θ 3,0,0,0 5,0 D Θ 0,0,0,0,3 D derivatives in Q: Linear terms Num Kernel Acts T 1 T T 3,1 T 3, T 3,3 T 4,1 T 4, 1 Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,1 D Θ 1,0,0,0 3,0 D Θ 1,0,0,0, 1 d Θ 1,0,0,0 3,0 D Θ 0,0,0,0,0 D Θ 0,0,0,0,1 D Θ 0,1,0,0, 1 d Θ 1,0,0,0, 1 d Θ,0,0,0 3, 1 d Θ 0,0,0,0 1, 1 d Θ 0,1,0,0 3,0 D Θ 1,0,0,0 3,1 D Θ,0,0,0 4,0 D Θ 0,0,0,0, D derivatives in Q: Linear terms Num Kernel Acts T 1 T T 3,1 T 3, T 3,3 T 4,1 T 4, 1 Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,0 D BOUNDS FOR CT AND K 19

140 1 Θ 0,0,0,0,0 D Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,1 D Θ 1,0,0,0 3,0 D Θ 1,0,0,0, 1 d derivatives in Q: Linear terms Num Kernel Acts T 1 T T 3,1 T 3, T 3,3 T 4,1 T 4, 1 Θ 0,0,0,0 1, 1 d Θ 0,0,0,0,0 D CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY

141 derivatives in Q: Totals Num Kernel D D D 3 D 4 D d d d 3 d 4 d 1 Θ 0,0,0,0 1, Θ 0,0,0,0, Total Θ 0,0,0,0, Θ 0,0,0,0 1, Θ 0,0,0,0, Θ 1,0,0,0 3, Θ 1,0,0,0, Total Θ 1,0,0,0 3, Θ 0,0,0,0, Θ 0,0,0,0, Θ 0,1,0,0, Θ 1,0,0,0, Θ,0,0,0 3, Θ 0,0,0,0 1, Θ 0,1,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ 0,0,0,0, Total Θ 0,1,0,0 3, Θ 1,1,0,0 3, Θ 1,0,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ 0,0,0,0, Θ 0,0,0,0, BOUNDS FOR CT AND K 131

142 8 Θ 0,0,0,0, Θ 0,0,1,0, Θ 0,1,0,0, Θ 1,1,0,0 4, Θ 1,0,0,0, Θ,0,0,0 3, Θ 3,0,0,0 4, Θ 0,0,0,0 1, Θ 0,0,1,0 3, Θ 0,1,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ 3,0,0,0 5, Θ 0,0,0,0, Total Θ 1,1,0,0 4, Θ 0,0,1,0 3, Θ 1,0,1,0 3, Θ 0,1,0,0 3, Θ 0,1,0,0 3, Θ 1,1,0,0 3, Θ,1,0,0 4, Θ 1,0,0,0 3, Θ 1,0,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ,0,0,0 4, Θ 3,0,0,0 5, Estimated using an extra cancelation 15 Θ 0,0,0,0, CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY

143 16 Θ 0,0,0,0, Θ 0,0,0,0, Θ 0,0,0,1, Θ 0,0,1,0, Θ 1,0,1,0 4, Θ 0,1,0,0, Θ 1,1,0,0 4, Θ,1,0,0 5, Θ 1,0,0,0, Θ 0,,0,0 3, Θ,0,0,0 3, Θ 3,0,0,0 4, Θ 4,0,0,0 5, Θ 0,0,0,0 1, Θ 0,0,0,1 3, Θ 0,0,1,0 3, Θ 0,1,0,0 3, Θ 1,0,0,0 3, Θ 0,,0,0 4, Θ,0,0,0 4, Θ 3,0,0,0 5, Θ 4,0,0,0 6, Estimated using an extra cancelation Total derivative in Q: Totals Num Kernel D D D 3 D 4 D d d d 3 d 4 d 1 Θ 0,0,0,0 1, Θ 0,0,0,0, BOUNDS FOR CT AND K 133

144 Total Θ 0,0,0,0, Θ 0,0,0,0 1, Θ 0,0,0,0, Θ 1,0,0,0 3, Θ 1,0,0,0, Total Θ 1,0,0,0 3, Θ 0,0,0,0, Θ 0,0,0,0, Θ 0,1,0,0, Θ 1,0,0,0, Θ,0,0,0 3, Θ 0,0,0,0 1, Θ 0,1,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ 0,0,0,0, Total Θ 0,1,0,0 3, Θ 1,1,0,0 3, Θ 1,0,0,0 3, Θ 1,0,0,0 3,1 D Θ,0,0,0 4,0 D Θ 0,0,0,0,0 D 3 D D Θ 0,0,0,0,1 D D Θ 0,0,0,0, D Θ 0,0,1,0, Θ 0,1,0,0, 1 d Θ 1,1,0,0 4, CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY

145 1 Θ 1,0,0,0, 1 d d Θ,0,0,0 3, 1 d Θ 3,0,0,0 4, Θ 0,0,0,0 1, 1 d 3 d d Θ 0,0,1,0 3, Θ 0,1,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ 3,0,0,0 5, Θ 0,0,0,0, Total derivatives in Q: Totals Num Kernel D D D 3 D 4 D d d d 3 d 4 d 1 Θ 0,0,0,0 1, Θ 0,0,0,0, Total Θ 0,0,0,0, Θ 0,0,0,0 1, Θ 0,0,0,0, Θ 1,0,0,0 3, Θ 1,0,0,0, Total Θ 1,0,0,0 3, Θ 0,0,0,0, Θ 0,0,0,0, Θ 0,1,0,0, Θ 1,0,0,0, Θ,0,0,0 3, BOUNDS FOR CT AND K 135

146 7 Θ 0,0,0,0 1, Θ 0,1,0,0 3, Θ 1,0,0,0 3, Θ,0,0,0 4, Θ 0,0,0,0, Total derivatives in Q: Totals Num Kernel D D D 3 D 4 D d d d 3 d 4 d 1 Θ 0,0,0,0 1, Θ 0,0,0,0, Total Θ 0,0,0,0, Θ 0,0,0,0 1, Θ 0,0,0,0, Θ 1,0,0,0 3, Θ 1,0,0,0, Total derivatives in Q: Totals Num Kernel D D D 3 D 4 D d d d 3 d 4 d 1 Θ 0,0,0,0 1, Θ 0,0,0,0, Total CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY

147 5.3. BOUNDS FOR CT AND K Writing the linear system for D and its derivatives We end this section packing all the previous estimates and getting the final estimates for the linear part of the energy concerning D and its derivatives. The estimates coming from the previous tables are summarized in We are left thus with the estimates coming from the term Q zbrz, ω Q xbrx, γ, which at a linear order are of the form BRx, γ x 4, 3x 8 x 1 x {{ D + OD. F We show now the different L estimates depending on the number of derivatives we are taking in every term In BR \In F This means the following estimates for the derivatives: Derivatives in BRF Estimates We summarize the contribution of these terms to the derivative of Et: The total contribution is thus given by D L D L D L 3 D L 4 D L

148 138 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY D L D L D L 3 D L 4 D L Finally, one can compute the largest eigenvalue λ 1 of the matrix. We obtain that it equals Since the growth of the energy is given roughly by expλ 1 T g /λ 1 f an this has to be smaller than min x 1, T g, which in our case is approximately 0.1. The estimates prove to be insufficient, although promising, since just an order of magnitude less could give us enough room to prove the theorem we assume the norms of f can be bounded by a constant of the order of We expect to lower λ 1 in the near future. Some ideas about how this can be carried out are presented in the next section Future improvements We give here a list of possible optimizations that although at this moment untested, we believe may lead us to get better estimates. The first one concerns the splitting of the kernels for the computation of the constant Ct. An improvement of the splitting may consist in changing the approximation of the Kernel. Previously it was done by Cauchy transforms over straight lines tangent at the curve. We propose to approximate it by Cauchy transforms over other curves. The most simple example is circumferences. In this case, let r β be the osculating circle at x, parameterized by β. Thus, we can approximate the kernel Θ 0,0,0,0,0 = γβ x β β x β β x xβ rβ β r r β c rβ β r r, β where c = r ββ x β rββ x β rβ x. β For the case of the circle, it is easy to see that rβ β r r β = ei 1 + rβ β r r β = ei + 4 tan tan i β i β 1 sin β, which allows us to write the previous operator as the usual Hilbert Transform plus other terms which are harmless and easy to estimate.

149 5.4. PROOF OF THEOREM The drawback of this method is that flat regions with a very small curvature might lead to high numerical errors since the curvature is infinite at those points. Perhaps a mixed method integrating in some regions over straight lines and in other over circumferences might yield better results. Another option to try is to consider a family curves which have a circular part an arc of a circle and are prolonged by straight segments, taking the best element of the family to split the kernel. Other improvements can consist on a different representation of the function: instead of representing it by splines in space-time, one could try to represent it by wavelets or by Chebychev polynomials in space and splines in time. The objective is that a big enough finite dimensional subspace of these function spaces is mapped by the Birkhoff-Rott integral into itself, making the orthogonal projection over this subspace almost zero. This could lead us to a better control of the norm of the orthogonal projection in time. Finally, one could try to perform a higher order non-rigorous simulation in order to get a better approximation of the real solution. By higher order we mean an improvement in the following senses: multiple precision for the double representation and higher order approximation of the integral approximations. We believe a higher order scheme in time will not produce significantly better results. Of course, there is a tradeoff between the computation time and the precision we can get. Therefore, in order to achieve these results the code needs to be highly optimized and perhaps run in parallel. 5.4 Proof of Theorem 5.1. In this section, we will prove the stability Theorem The equations are: SPLASH APPROX z t c ω t x t b γ t = Q zbr + cz = + π Q z BR π z d Q z β BR β z β dβ +BR t z = Q BR + cbr z + cϖ Q ϖ 4 z P 1 z = Q xbrx, γ + bx + f = + π Q x BR π x d Q x BR β x dβ {{ + + π π f x b s f β x β x d x β dβ {{ b e +BR t x, γ x = Q x BRx, γ + bbr x, γ x + bγ Q xγ 4 x P 1 x + g

150 140 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY where BRz, ϖ = 1 π P V f will be the error for z and g will be the error for ω Computing the difference z x and ω γ where We define now: The energy Et 1 and the Rayleigh-Taylor condition z z β ϖ βdβ z z β D z x, d ω γ, D ϕ ψ D Q z L + z σ z D 4 + d H + D H 3+ 1 BR t + σ z ϕ z BR + Q BR + ω z z z + ω z z t + ϕ z z z Q z P 1 z z Note that σ z > 0. We shall show that d dt Et CtEt + Ek t + cδt Ct = C x H 5+ 1 t, γ H 3+ 1 t, ψ H 4+ 1 t, F x L t and δt = f H 5+ 1 t, g H 3+ 1 tk, k big enough depend on the norms of f and g. Remark From now on, we will denote Et + Et k by P Et. 1 d dt D L CP Et + δt is left to the reader. We compute 1 d dt Q z z σ z 4 D = 1 Q zσ z t z 4 D + Q z z σ z 4 D 4 D t The first integral is easy to bound by CP Et, we proceed as in the local existence theorem We split I = Q z z σ z 4 D 4 D t = I 1 + I + I 3

151 5.4. PROOF OF THEOREM where We have: I 3 1 I 1 = I = I 3 = Q z z σ z 4 D 4 Q zbrz, ω Q xbrx, γd Q z z σ z 4 D 4 cz bx d Q z z σ z 4 D 4 fd Q z z σ z D 4 d + 1 Thus, we are done with I 3. We now split I 1 = l.o.t + I 1,1 + I 1, + I 1,3 + I 1,4 I 1,1 = I 1, = I 1,3 = Q x I 1,4 = 1 π Q z + Q 1 x π Q z Q z x 4 x 4 β x x β γ βdβ z σ z f 4 d CP Et + Q zσ z L δt z σ z D 4 Q 4 zbrz, ω Q 4 xbrx, γd Q z z σ z D 4 Q 1 π z 4 z 4 β z π z z β ω βdβ d Q z z σ z 4 D 1 π z z β z z β 4 z z β 4 z 4 z βω βdβ x x β x x β 4 x x β 4 x 4 x βγ βdβ Q z z σ z 4 D 4 Q zbrz, 4 ω Q xbrx, 4 γd where l.o.t stands for low order terms, nice terms easier to deal with. I 1,1 = l.o.t + I 1,1,1 where I 1,1,1 = = + Q z z σ z 4 D Qz 4 zbrz, ω Qx 4 xbrx, γd Q z z σ z 4 D Qz 4 DBRz, ωd Q z z σ z 4 D Qz 4 xbrz, ω Qx 4 xbrx, γd

152 14 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY Q z z σ z D 4 d QzBRz, ω L {{ + Q z CP Et z σ z 4 D + which means I 1,1 is done. From now on we will denote bounded as for local existence Q z z σ z Qz 4 xbrz, ω Qx 4 xbrx, γ d {{ l.o.t in D and d β z = z z β I 1, = I 1,,1 + I 1,, + I 1,,3 + I 1,,4 where I 1,,1 = I 1,, = I 1,,3 = I 1,,4 = Q z z σ z DQ 4 1 z π Q z z σ z DQ 4 1 z π Q z z σ z DQ 4 1 z π Q z z σ z 4 DQ z Q x 1 π β D 4 ω βdβd β z β x 4 1 β z 1 β x β x 4 d βdβd β x β x 4 γ βdβd β x ω βdβd Q 4 z I 1,,1 = z σ z D 4 1 β D 4 ω βdβd π β z = 1 1 π z π D 4 β D 4 Q 4 z σ z ωβ Q 4 zβσ z βω β z I 1,,1 = 1 π + Q4 zσ z ωβ + Q 4 zβσ z βω {{ ddβ this is zero as in local existence 4 D 4 D = 0 4 D z I 1,,1 CP Et β D 4 Q 4 z σ z ωβ Q 4 zβσ z βω β z {{ Hilbert transform applied to 4 D

153 5.4. PROOF OF THEOREM For I 1,, we can make a trick to get less derivatives in x. I 1,, = I 1 1,, + I 1,, + I 3 1,, Λ 4 x Q 4 z 1 z σ z Dω 4 z 1 { { 1 π β x 4 x π β dβ d I1,, 3 = 1 I1,, = 1 Q 4 z π z σ z D 4 β x 4 1 β z 1 z β + =0 { { 1 β x 1 x β + x x x 4 ωdβd β I1,, 1 = 1 Q 4 z π z σ z D 4 β x 4 We use that 1 z 1 x x + z z x D to find that I 3 1,, 1 4 CP Et We can use that Q z z σ z 4 D + Q z 6 L σ z L ω L 1 β z 1 z β + z z z 4 β and that 1 β z 1 z β + z z to find z 4 β z k C x k C 1 β 1/ D C + 1 F z k L F x k L I 1,, 1 8π We ve used that Q z z σ z 4 D 1 β z 1 β x x + z z x =0 { { z z z 4 β ω β ω dβd Sobolev inequalities { { D L z k C 1 β 1/ z C + 1 F z k L 1 β x 1 x β + x x x 4 β + C Q z 6 L σ z L z k C x k C D C x L F z k L F x k L d x 4 1/ β β 1/ dβ C x 4 L. Control of x H 5 { { Λ x 4 L

154 144 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY We split further in I 1 1,, = I1,1 1,, + I1, 1,, : Q 4 z z σ z D 4 β x 4 ω β ω + ω β dβd I 1,1 1,, = 1 π I 1, 1,, = 1 π Q 4 z z σ z 4 Dω β x 4 1 β z 1 β x β β z β β x dβd Inside of the β integral in I 1,1 follows: 1,, there is no principal value, so the appropriate estimate I 1,1 1,, CP Et For I 1, 1,, se proceed as for I 1,,, se decompose adding and subtracting 1 z β 1 x β. Thus, we are done with I 1,,. We decompose I 1,,3 = I1,,3 1 + I 1,,3 + I3 1,,3. I1,,3 1 Q z = z σ z DQ 4 1 z β π x 4 1 β x 1 x β + x x x 4 d βdβd β I 1,,3 = I 3 1,,3 = It s easy to obtain: Q z Q z z σ z 4 DQ z x 4 1 x π z σ z DQ z x π β d β dβd β d x 4 β dβd I1,,3 1 1 Q z 4π z σ z D 4 d + C Q z 6 L σ z L d L x k C F x k L x C,δ 4 x L CP Et I 1,,3 CP Et analogously since Λd L C d H I 3 1,,3 CP Et using Λd 4 x L C d H x H 5. We are done with I 1,,3. To deal with I 1,,4 se use that Q z Q x = Q1 tz + tx Q1 tz + tx D for t 0, 1. Then it is easy to find I 1,,4 CP Et,

155 5.4. PROOF OF THEOREM and we are done with I 1,. We decompose I 1,3 as I 1,3 = I 1,3,1 + I 1,3, + I 1,3,3 + I 1,3,4 + I 1,3,5 + I 1,3,6 I 1,3,1 = I 1,3, = I 1,3,3 = I 1,3,4 = Q z z σ z DQ 4 1 z π Q z z σ z DQ 4 1 z π Q z z σ z DQ 4 1 z π Q z z σ z DQ 4 1 z π Q z I 1,3,5 = z σ z DQ 4 1 z π 1 β z 4 1 β x 4 I 1,3,6 = β z β z 4 βz β 4 Dω βdβd β z β z 4 βz β 4 xd βdβd β z β z 4 βd β 4 xγ βdβd β D β z 4 βx β 4 xγ βdβd β x β x β 4 xγ β dβd Q z z σ z DQ 4 z Q x 1 π β x β x 4 βx β 4 xγ βdβd I 1,3,j, j =, 3, 4, 5, 6 are easier to deal with It can be done as before. Therefore we focus on I 1,3,1. I 1,3,1 = I1,3,1 1 + I1,3,1 + I1,3,1 3 I1,3,1 1 Q z = z σ z DQ 4 1 π β z z π β z 4 βzω β z z 4 z βω 1 β β Ddβd 4 I 1,3,1 = I 3 1,3,1 = Q z z σ z 4 DQ z Q z z σ z 4 DQ z 1 π 1 π z z 4 ω 4 D z z 4 ω z β z β dβd β z D 4 β dβd In I1,3,1 1 we find a commutator, which can be handled as before. It is also easy to estimate I1,3,1. To deal with I1,3,1 3 we remember that z 4 D = z 4 z x 4 x D 4 x = 3 z 3 z + 3 x 3 x D 4 x

156 146 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY That allows us to decompose further z 4 D = 3 z 3 D 3 D 3 x D 4 x which yields I 3 1,3,1 = I 3,1 I 3,1 1,3,1 = 3 π I 3, 1,3,1 = 3 π I 3,3 1,3,1 = 3 π 1,3,1 + I3, 1,3,1 + I3,3 1,3,1 Q 4 z z σ z 4 D z z 4 ω Q 4 z z σ z 4 D z z 4 ω Q 4 z z σ z 4 D z z 4 ω β z D 3 β dβd β D x 3 β dβd β D x 4 β dβd We use that β z 3 D β dβ C z D 3 CP Et L L to control I 3,1 1,3,1. I3, 1,3,1 follows similarly. We control I3,3 1,3,1 using that β D 4 x β dβ L D 4 x L D L 5 x L + D L 4 x L CP Et This allows us to finish the estimates for I 3,3 1,3,1 and I3 1,3,1. We are done with I 1,3,1 and I 1,3. We now decompose I 1,4. I 1,4 = I 1,4,1 + I 1,4, + I 1,4,3 + I 1,4,4 I 1,4,1 = I 1,4, = I 1,4,3 = I 1,4,4 = Q z z σ z 4 D Q zbrz, 4 ddβd Q z z σ z D 4 Q 1 z π Q z z σ z D 4 Q 1 z π β D β z 4 γ βdβd β x γ 4 β Q z z σ z 4 D Q z Q xbrx, 4 γd 1 β z 1 β x dβd

157 5.4. PROOF OF THEOREM We control I 1,4,, I 1,4,3 and I 1,4,4 as before. We further split I 1,4,1 = I1,4,1 1 + I1,4,1 + I 1,4,3 + I 1,4,4 I1,4,1 1 Q z = z σ z D 4 Q z BRz, d 4 1 z z H 4 d d I1,4,1 Q z = z σ z D 4 z Q z Q z z H 4 d H z z 4 d d I1,4,1 3 Q z = z σ z D 4 z Q H z z z Q x 4 x γ d I1,4,1 4 Q z = z σ z D 4 z Q H z ω 4 Q x 4 γ d z z x There are commutators in I1,4,1 1 and I 1,4,1 so they are easy to estimate. To get the estimate for I1,4,1 3 we bound Q H z z Q x x γ 4 L We now remember the following formulas: Q z z Q x x L {{ at the level of D 4 γ L CE t ϕ = Q zω z c z ψ = Q xγ x b s x These yield where I 4 1,4,1 = S + I 4,1 1,4,1 + I4, 1,4,1 + l.o.t, S = I 4,1 1,4,1 = I 4, 1,4,1 = Q zσ z D 4 z z 3 H Dd 4 Q z z σ z D 4 z Qz Qz H z 4 z z Q z z σ z D 4 z H 4 c z b s x d z ω Q x Qx x 4 γ d x

158 148 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY S is going to appear later with a negative sign and therefore cancel out. I 4,1 bounded as before since it is low order. We show how to deal with I 4, 1,4,1. We compute c z 4 = 3 Q z zbr ; 4 z b s x = 3 Q x xbr x Then, in 4 c z 4 b s x we consider the most singular terms 1,4,1 can be c z 4 b 4 s x = J 1 + J + J 3 + J 4 + J 5 + l.o.t. J 1 = Q z Qz zbrz, 4 z ω z + Q x Qx xbrx, 4 x γ x J = Q z 4 zbr z + x 4 Q xbr x J 3 = Q 1 z π Q 1 x π J 4 = Q zbrz, 4 ω β z β z 4 z z βz β 4 zω βdβ β x β x 4 x x βx β xγ 4 βdβ z z + Q xbrx, γ 4 x x J 5 will be given later. In J 1 and J we find 4th order terms in derivatives in z and x so they are fine. In J 3 we find inside the integrals β z z = z z β βz z 5.9 β x x = x x β βx x 5.10 This implies that we find Hilbert transforms applied to four derivatives of x and z. We are done with J 3. In J 4 we also find them inside the integrals 5.9 and 5.10 so it is easy to check that we have kernels of degree 0 applied to four derivatives of ω 4 and γ. 4 This implies that we have a Hilbert transform applied to ω 3 and γ 3 so we are done with J 4. The most dangerous term is J 5 which is given by J 5 = Q 1 z π We split further β z 4 β z z z ω βdβ + 1 Q x π β x 4 β x x γ βdβ x J 5 = J 5,1 + J 5, J 5,1 = Q 1 z π ω β z π z β z ω z 4 sin β 4 β/ z dβ + Q 1 x π γ β x π x β x γ x 4 sin β 4 β/ x dβ J 5, = Q 1 z z z 3 ω Λ 4 z + Q 1 x x x 3 γ Λ 4 x

159 5.4. PROOF OF THEOREM In J 5,1 we find a Hilbert transform applied to z 4 and x 4 so it is fine. further: We split J 5, = J 5,,1 + J 5,, + J 5,,3 J 5,,1 = Q 1 x x x 3 γ 1 z Q z z 3 ω Λ x 4 J 5,, = Λ Q 1 z z z 3 ω 4 D Q 1 z z z 3 ωλ 4 D J 5,,3 = Λ Q 1 z z z 3 ω 4 D J 5,,1 can be estimated as before there are more derivatives: 5 in total, but they are in x. In J 5,, we find a commutator. Finally: I 4, 1,4,1 CP Et Q zσ z D 4 z H Λ z We use that HΛ = and z 4 D = z 4 D to obtain: I 4, 1,4,1 CP Et 1 Q z 1 z z 3 ω 4 D d Q zσ z D 4 z Q z ω z z 4 D z d z CP Et 1 Q zσ z D 4 z 4 z D z Q z ω d z z {{ 1 Easy to estimate by CP Et Q Q zω zσ z z 4 D z 4 z D z d z {{ Integration by parts Then we are done with I 4, 1,4,1, I4 1,4,1, I 1,4,1, I 1,4 and I 1. To finish with I it remains to control I. We split it as: I = I,1 + I, + l.o.t I,1 = I, = Q z z σ z 4 Dc 5 z b 5 xd Q z z σ z 4 D 4 cz 4 bx d

160 150 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY The low order terms are easier to deal with. We further split I,1. I,1 = I,1,1 + I,1, + I,1,3 I,1,1 = I,1, = Q z z σ zc D 4 D 5 d {{ Integration by parts Q z z σ z Dc 4 b s x 5 d {{ 5 derivatives, but in x Q z I,1,3 = z σ z Db 4 e x 5 d {{ Error term We find I,1 CP Et + cδt. We decompose I,. I, = I,,1 + I,, + I,,3 I,,1 = I,, = I,,3 = Q z z σ z 4 D 4 c 4 b s z d Q z z σ z 4 D 4 b s {{ 5 derivatives in x Q z z σ z 4 D 4 b e x d {{ Error term Dd We deal with I,,1 more carefully. We use that to obtain 4 D z = 4 z z 4 x x 4 x D = 3 3 z z x x 4 x D = 3 3 D z 3 3 x D 4 x D I,,1 = I 1,,1 + I,,1 + I 3,,1 I 1,,1 = 3 I,,1 = 3 I 3,,1 = Q z z σ z 3 D z 4 c b s d Q z z σ z 3 x D 4 c b s d Q z z σ z 4 x D 4 c b s d

161 5.4. PROOF OF THEOREM We can integrate by parts in all of the above terms to get low order terms. We are finally done with I Computing the difference ϕ ψ From the local existence proof we find the equation for ϕ t : ϕ t = ϕb z t Q z ϕ z al Q Q 1 z P z z BR t + z z z +Q z Q z ω t z cbr z z Q zq z c z Qz Q3 z Q z z BR Q z c z t B z t = 1 Q z π zbr z d We will show how to find the equation for ψ t. We start from 5.11 and therefore that yields The equation for γ t reads: Then ψ Q x ψ Q Q = x γ x 4 x ψ = Q xγ x b s x = Q xγ 4 x + b s x Q γb s, x b s x Q x + γb s γ t = BR t x Q x BR + b s BR x ψ b Q + s x x Q P 1 z + b e BR x + b e γ + g x γ ω t =Q x Q x t Q xγ x x 3 x x t + Q xγ t x b s x t γ =Q x Q x t x Q xγ x B xt Q xγ 1 x f x π x d + Q x BR t x Q x ψ BR + b s BR x x P 1 z + b e BR x + b e γ + g b s x t We should remark that we have used that x x t = 1 π Q xbr x d + 1 π Q x + f x d b s x Q x

162 15 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY and Computing B x t = 1 Q x π xbr x d γ ψ t =Q x Q x t x Q xγ x B xt {{ where Q x ψ x Q x 1 + Q x Q xbr t x x Q3 x x BR Q x + Q xb s BR b s x Q x {{ 1 E 1 = Q x x BR x b e + Q x x b eγ + Q x x g Q xγ x are the error terms. We consider It yields 1 = Q xγ x B xt + Q x b s x Q x = Q xγ x B xt + b s x b s Qx Q x b s x Q x 1 P z b s x t + E 1 x 1 π f x x d = Q xγ x B xt + b s x B x t b s Q x xbr x Qx b Q s x x = B x tψ b s Q xbr x x Qx Q x b s x γ ψ t =Q x Q x t x B xtψ b s Q xbr Q xbr t x x Q3 x It is easy to check that = b s Q xbr x x {{ x BR Q x + Q xb s BR Qx Q x b s x x x {{ Q x ψ x Q Q x 1 P z b s x t + E 1 x x x x + Q xb s BR x x = b sbr x x Q xq x, x x

163 5.4. PROOF OF THEOREM then ψ t = B x tψ Q x ψ x Q Q x x BR t x x 1 P z + x γ + Q x Q x t x b x sbr x Q xq x Qx b Q s x Q3 x x x BR Q x b s x t + E 1 With this formula it is easy to find that 1 d D dx CP Et + cδt dt In order to deal with II II = Λ 3 D 3 D t d we take a derivative in in the equation for ω and ψ to reorganize the most dangerous terms. If we find a term of low order, we will denote it by NICE. Since the equations for ϕ t and ψ t are analogous except for the E 1 term, the NICE terms are going to be easier to estimate in terms of CP Et + cδt. Q ψ t = B x tψ x ψ x Q x γ + Q x Q x t x b s x {{ t 3 Expanding 3: +E 1 b s BR 3 = Q x xbr t b s x t x = Q xbr t x x Q xbr t = x B x t t + Q xbr We use that x Q x BR x t + x {{ 3 x x Q xq x x x x = x x x x x x ; z x P 1 Q x b Q s x x x B x t Q xbr x + Q x Q x t BR t Q 3 x x BR Q x x x t x x Q xbr t x = x t x x x t x x x x

164 154 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY to find ψ t = B x tψ {{ 4 ψ x {{ 5 + Q xbr x x t x x {{ 3 13 γ + Q x Q x t x {{ 9 Qx + ψ x Q x {{ 6 Q 3 x x BR Q x {{ 1 Q xbr t x x + Q x Q x t BR x {{ 7 x b s BR x Q xq x {{ 10 +E 1 x x x 3 x B x t t Q P 1 z x x {{ Qx 8 b Q s x x {{ 11 The term x B x t t depends only on t so it is not going to appear computing II. 4 = B x tψ is NICE at the level of ψ 5 = ψ x is a transparent term which is NICE even if we have to deal with Λ 1/ Qx 6 = ψ = Q x x Q x x Q x + Q x ψψ + ψ x Q x Q x Qx x The first term is at the level of x so it is NICE. The second term is at the level of x or ψ so it is NICE. We write the last one as ψ Q x Qx = ψ x Qx x Q x x + ψ x Qx x x x x Q x x 3 The first term is at the level of x or ψ so it is NICE. For the second term we have used that Finally: x = x x x x x x 6 = NICE + ψ x Qx x x x Q x x 3

165 5.4. PROOF OF THEOREM = Q x Q x t BR x x = Q x Q x t BR + Q x Q x t BR x x x x + Q x Qx t BR x x The first term is at the level of x, x t, BR x so it is NICE. We use that Q x t x = Q x t x = Qx x t x = x 1 Qx Qx x x t x t Using that and we find that Q x t x x x t x = B x t + 1 x f π x d x = x t x x x t x x = x t x Qx + Qx x x t x x x 3 x + Qx x B x 1 π xt + Qx x π f x d 5.1 x That yields Finally: x 7 = Q x Q x t BR x = NICE + Q xbr x x t x Qx x {{ NICE at the level of x,x t,br + Q x BR x Qx x x t x x 3 + Q x BR x Qx x x x B xt {{ NICE at the level of x,x t,br 1 x + Q x BR x Qx f x π x d x +Q x Q x t BR x {{ part of error terms x 7 = Q x Q x t BR x = NICE + Q xbr x Qx x x t x x x 3 + Q x Q x t BR x

166 156 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY 8 = Q x z x P 1 Q xx P 1 x x x {{ NICE at the level of x which means 8 = Q x γ 9 = Q x Q x t x = Q x P 1 x x x z x P 1 Q x P 1 x x x x x 3 We use 5.1 to deal with Qxt x. We find that γ 9 = Q x Q x t x x 10 = b s BR x Q xq x Therefore x x = Q x Q x x x P 1 x {{ NICE at the level of x = NICE Q x P 1 x x x x x 3 γ Q x t γ = Q x Q x t +Q x γ + Q x Q x t x x {{ x NICE at the level of x,x t = NICE + Q x γ Qx x x = b s BR x Q x {{ NICE as before b s BR x Q x Q x x x x x x 3 b s BR 10 = b s BR x t x γ x 3 + Q x Q x t x b s BR x x Q xx Q x x x {{ NICE as before x x Q xq x = NICE b s BR b s BR x Q x Q x x x x Q x Q x x x Q x Q x x Qx 11 = b Q s x = b s x Q x b s x Qx x x Q x Q x x Qx x b Q s x + Q x x Q x b s x x x x 3 x x x 3

167 5.4. PROOF OF THEOREM The fact that the last two terms are NICE, allows us to find that Qx 11 = b Q s x = NICE b s x Q x b s x Qx x x Q x Q x Finally: Q 3 1 = x x BR Q x which implies that = 3Q x Q x BR {{ NICE Q3 x Q3 x x BR Q x x BR x Qx x Q 3 x BR Qx x {{ NICE x x x 3 x x x 3 Q 3 1 = x x BR Q x = NICE Q3 x x BR Q x Q 3 x BR Qx x x x x 3 We gather all the formulas for 4 to 1 using that the error terms will be collected by E 1. We will denote the new term by Ẽ1. It yields: ψ t = NICE + ψ Qx x Q x x x x 3 {{ 16 Q xbr t x x x x {{ 3 15 Q x P 1 x x x x x 3 {{ 15 + Q x γ Qx x x t x γ x {{ 3 + Q x Q x t + Q x BR x Qx x x {{ 18 x x + Q x Q x t BR {{ 14 b s x Q x Q x {{ x b x BR x {{ 17 b s x Qx x x x Q x x {{ 3 16 Q 3 x BR Qx x x x x {{ 3 +Q xbr x 16 x t x x 3 {{ 18 Q x Q x b s BR x Q x Qx x x x x 3 {{ 16 Q3 x x BR Q x {{ 17 x t x x 3 + Ẽ1

168 158 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY We compute γ 14 = Q x Q x t x = Q x t Q x Q x γ x + Q x Q x t BR x x + Q x t Q x Q x BR x x = Q x t ψ Q x t Q γ Q x Q x x x Q x t Q x Q x BR x x Q x t x B x t Q x The last formula allows us to conclude that 14=NICE. We reorganize using 15, 16, 17 and 18. ψ t = NICE Q xbr t x + P 1 x x x x x 3 Q 3 BR + b s x Q 4 x + b s BR x Q x ψ Q 4 x Qx x x x x 3 + Q xbr x x t x x 3 + Q x γ + Q x BR x Qx x Q 3 x BR + b s x x + b s BR x Q x x z z x t x x 3 Q x Q x + Ẽ1 We add and subtract terms in order to find the R-T condition. We remember here that σ z = BR t + ϕ z BR z + ω z z t + ϕ z z z + Q z BR + ω Qz z + P 1 z z σ x = BR t + ψ x BR + Q x BR + γ x x x + γ x x t + ψ x x x Qx x + P 1 x x 5.13 In σ x there are error terms but they are not dangerous. Then, we find ψ t = NICE Q x BR t + ψ x BR x + γ x x t + ψ x x x + P 1 x x x x x 3 +Q xbr x x t x ψ x 3 + Q x x BR x + γ x x t + ψ x x x x x x {{ 3 Q 3 BR + b s x Q 4 x + b s BR x Q x 19 ψ Q 4 Qx x x x x x 3

169 5.4. PROOF OF THEOREM x t x x 3 + Q x γ + Q x BR x Qx x Q 3 x BR + b s x + b s BR x Q x Line 19 can be written as x Q x Q x + x Ẽ1 19 = Q xbr x x t x x 3 + Q xbr x ψ x x x x 3 + Q xγ x x t + ψ x x x x x x 3 = Q xbr x x t x x 3 + Q xbr x ψ x x x x 3 + Q xγ x x t x + ψ x x x x x x 3 Q x Q x BR x = Q xbr x 1 x 3 x t x + ψ x x x x t x + ψ x x Q x Q x BR x + Q xγ x 1 = 1 x 3 x 3 x t x + Q x Q x BR x We expand x t to find x x x ψ x x x ψ x x x x 3 19 = 1 x 3 Q xbr x + Q xγ x x x + x x x 3 Q xbr x + Q xγ x x x b e {{ error term: we incorporate it as Ẽ Q xbr x + Q xγ x x x ψ x x x x 3 Q x Q x BR x ψ x x x x 3 ψ x x x x 3 We denote We claim that G x = Q xbr x + Q xγ x x x 5.14 G x = NICE + x H ψ that becomes G x = NICE

170 160 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY Then 19 = NICE Q x Q x BR x ψ x x x x 3 + Ẽ We write G x = Q x Q x BR x 1 x x β x + Qx {{ π x x β γ βdβ NICE, at the level of x {{ NICE, we use that x =A xt Q 1 x x β x x π x x β 4 x x βx x βγ βdβ {{ NICE, we use that x only depends on time + Q xbrx, γ x {{ Hilbert transform applied to γ + Q xγ x x x Therefore γ G x = NICE + x Q xh x Q = NICE + x H x γ x + Q xγ x x x + Q xγ x x x = NICE + x H ψ + H b s x + Q xγ x x x = NICE + x Hψ H Q xbr x + Q xγ x x x Q xbr x = Q x Q x BR x {{ NICE +Q x 1 x x β x π x x β γ βdβ = Q 1 x x β x x π x x β 4 x x βx x βγ βdβ {{ NICE, extra cancelation in x x β x + Q 1 x x β x x π x x β γ βdβ {{ NICE, extra cancelation in x x β x This means that Q xbr x = NICE + 1 Q H x x x x γ

171 5.4. PROOF OF THEOREM Taking Hilbert transforms: H Q 1 xbr x = NICE H Q x x x x γ = NICE + 1 x x Q x x γ Using that x x = x x we are done. Thus 19 yields ψ t = NICE Q x For 0 we write BR t + + γ x x t + ψ x x Q 3 x BR + b s x Q 4 x ψ x BR x x + P 1 x x + b s BR x Q x + Q x γ + Q x BR x Qx x Q 3 x BR + b s x + x Q x ψ Q 4 x x t x x 3 x x x 3 Qx x x x x 3 x b s BR Q x Q x x {{ 0 Q x Q x BR x ψ x x x x {{ 3 +E, where E = Ẽ1 + Ẽ 1 x t = Q 4 x BR + b s x + Q xb s BR x + b e x + f + Q xbr x b e + b s b e x + Q xbr f + b s x f + b e x f {{ error terms Ẽ3 x t Q x x = Q3 x BR + b s x x + Q x b s BR x Q x x + Ẽ3 Q x x Now which means 0 = NICE x t Q x x Q x + Ẽ3 Q x x Q x = NICE x t Q x x Q x Q x Q x BR x ψ x x x x 3 + Ẽ3 Q x x Q x

172 16 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY We write x t = x t x x x {{ +x t x x x = = only depends on t B x t + 1 π B x t + 1 π f β + b e x x + f x = B x t + 1 f β π ψ x x x x β x x β dβ x + Q xbr x + bx x + f x x x β f β x β dβ x x x β x β dβ x x + Q xbr x + b s x x x x + Q xbr x + Q xγ x x x b e x x + f x x x x x {{ G x as in 5.14 x x Writing x t = Q xbr + b s x + b e x + f we compute x t x = Q xbr x {{ B xt + 1 x β f β π x β dβ + G xq x xbr x NICE {{ {{ error NICE because G x is nice ψ x x x Q x xbr x + x Q xbr b x e x x + f x + b s B x t + 1 x β f β π x β dβ x +b e B x t {{ {{ error NICE where Ê is an error term. To simplify we write x t x = NICE {{ error + 1 f β π ψ x x x Q x xbr x + errors Setting the above formula in the expression of 0+1 allows us to find x x β x x β dβ x + Ê = NICE + errors

173 5.4. PROOF OF THEOREM This yields BR t + ψ t = NICE Q x Q 3 x BR + b s x Q 4 x ψ x BR + γ + b s BR x Q x + Q x γ + Q x BR x Qx x x ψ Q 4 x x t x x 3 + E 3 x t + ψ x x + P 1 x x x x x 3 Qx x x x x 3 being E 3 a new error term. We now complete the formula for σ x in 5.13 to find ψ t = NICE Q x x xσ x x 3 + Q x BR + γ x x Qx x x x x {{ 3 + Q 3 x BR b s x BR x Q 4 b s x Q + ψ x Q 4 Qx x x x x x {{ Q x γ + Q x BR x Qx x x t x x {{ 3 4 +E 3 Expanding we find ψ Q 4 x = γ 4 x + b s x Q 4 γb s x Q x γ + 3 = Q 3 x x + BR x γ x b BR x s Q x γb s Q Qx x x x x x 3 Writing x t x = Q xbr x + b s x x + errors we obtain that 4 = Q x γ + Q x BR x Qx x Q xbr x x 3 + Q x γ + Q x BR x Qx x b s x x x 3 + errors

174 164 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY Thus γ = Q 3 x x + BR x γ x Qx x x x x 3 + Q x γ + Q x BR x Qx x Q xbr x x 3 + errors Q = Q x Qx x γ + BR x x γ x x x x 3 + Q xbr x x 3 + errors = Q x Qx x 1 γ + BR x x 3 D x + errors Finally, we obtain = NICE + errors For ϕ t we find ψ t = NICEx, γ, ψ Q xσ x x x x 3 + E 4 ϕ t = NICEz, ω, ϕ Q zσ z z z z 3, since we can apply the same methods as before to the equations with f = g = 0, which are satisfied by z, ω, ϕ. Then: II = Λ 3 D 3 D t = Λ 3 D NICEz, ω, ϕ NICEx, γ, ψ d Λ D 3 Q z z zσ z z 3 Q x x xσ x x 3 Λ DE 3 d 4 II 1 + II + II 3 II 1 CP Et because we are dealing with the NICE term II 3 CP Et + cδt because of the errors It remains to estimate II. We consider the most singular terms II = II,1 + II, + II,3 + l.o.t II,1 = Λ D 3 Q z z z σ z z 3 Q x x x σ x x 3 II, = Λ D 3 Q z 4 z zσ z d II,3 = z 3 Q x 4 x xσ x x 3 Λ D 3 Q z σ z z z z 3 Q x σ x x x x 3 d d

175 5.4. PROOF OF THEOREM II,1 = H D 3 Q z z z σ z z 3 Q x x x σ x x 3 d CP Et + cδt as before For II, we decompose further We find that II, = S + ĨI,, ĨI, = S = Λ 3 D where Q x 4 z zσ z z 3 Q x 4 x xσ x x 3 d Q zσ z 4 D z z 3 H 4 Dd ĨI, CP Et + cδt and S cancels out with S. We are done with II,. We write II,3 = H D 3 Q z σ 3 z z z z 3 Q x σ 3 x x x x 3 d + l.o.t We claim that Q x 3 σ x = x H 3 ψ t b s x H 4 ψ + errors + NICEx, γ, ψ 5.15 In the local existence we get This implies Q z 3 σ z = z H 3 ϕ t c z H 4 ϕ + NICEz, ω, ϕ II,3 = II,3,1 + II,3, + II,3,3 + II,3,4 II,3,1 = H D 3 z H ϕ 3 t z z z 3 x H ψ 3 t x x x 3 II,3, = H D 3 c z H ϕ 4 z z z 3 b s x H ψ 4 x x x 3 II,3,3 = H D 3 NICEz, ω, ϕ z z z 3 NICEx, γ, ψ x x x 3 II,3,4 = H 3 D x x x 3 d + errors d d d It is easy to find II,3,4 CP Et + cδt, error terms

176 166 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY II,3,3 CP Et, l.o.t In II,3, we split further: II,3, = II 1,3, + II,3, II 1,3, = II,3, = H D z 3 H D 4 z z z 3 d H DH 3 ψ 4 c z z z z 3 b s x x x x 3 d Then For II,3, : II 1,3, = 1 H D 3 z z z z 3 d CP Et II,3, = Λ 1/ ψλ H 3 1/ D 3 c z z z z 3 b s x x x x 3 d CP Et It remains II,3,1 = II 1,3,1 + II,3,1 II 1,3,1 = II,3,1 = H DH 3 D 3 t z z z d H D 3 H ψ 3 z z t {{ z x x x approx. sol. d Then II 1,3,1 CP Et + cδt At this point we remember that we had to deal with II = Λ 3 D 3 D t d so in II,3,1 1 can bound we find one derivative less or 1/ derivatives less and this shows that we II 1,3,1 CP Et + cδt by brute force. It remains to show claim We remember

177 5.4. PROOF OF THEOREM We write Q xγ x x + Q xγ x x t + Q xσ x = Q x BR t + ψ x BR + Q 3 x BR + γ x x Qx x {{ this term is in H 3 so it is NICE x t + = Q xγ x = Q xγ x ψ x x x ψ x x x + Q x P 1 x x {{ this term is also in H 3 Q x BR x + b s x x + b e x x + f x + ψ Q x BR x + b s + + Q xγ x b e x x + f x = Q xγ x ψ x x x Q x BR x + Q xγ x x x + errors = Q xγ x G x + errors = NICE + errors Finally, the most singular terms in Q xσ x are L = Q xbr t x + Q xψ x BR x We take 3 derivatives and consider the most dangerous characters: In M we find L = M 1 + M + M 3 + l.o.t M 1 = Q xbrx, γ 3 t x + Q xψ x BRx, 4 γ x M = Q 1 x π + Q xψ x M 3 = Q x π ψq x x π M = 1 π x 3 t x 3 t β x x x β γ βdβ x 4 x 4 β x x x β γ βdβ β x x β x 4 β x β 3 x t γ βdβ β x x β x 4 β x β 4 xγ βdβ Q xγ x Λ 3 x t x + Q xψγ x 3 Λ 4 x x + l.o.t x x x

178 168 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY For the second term we use the usual trick For the first term we remember that 4 x x = 3 3 x x x = At x x t = 1 A t x x t = 0 x x t + x x t = 0 x x t + x x t + x x t = 0 x x t = x x t x x t This allows us to control M. For M 3 we find M 3 = Q xγ x Λx 3 x t Q xψγ x 3 Λx 4 x + l.o.t so it can be estimated as M. It remains M 1. M 1 = Q xbrx, 3 γ t x + Q xψ x BRx, 4 γ x Using that β x x = β x x we find M 1 = Q x H 3 γ t + Q xψ x H 4 γ + l.o.t 5.16 We compute Q x Q H 3 γ t = H 3 x γ + NICE We compute the most singular term in t Q xbr x = Q x π This shows that t = H 3 x ψ t + H 3 x b s t + NICE = x H 3 ψ t + H t Q xbr x + NICE Q x 3 x t 3 x t β x β x γ βdβ β x x π β x 4 β x β x 3 t γ βdβ {{ Q x extra cancelation β x x β x γ 3 t βdβ + l.o.t. + NICE π {{ extra cancelation

179 5.4. PROOF OF THEOREM That gives t Q xbr x = Q xγ x Λ 3 x t x + l.o.t. + NICE t Q xbr x = Λ Q x γ x 3 x t x + l.o.t. + NICE which implies Q H t Q xbr x = x γ x 3 x t x + l.o.t. + NICE Q x H 3 γ t = x H 3 ψ t Q xγ x = Q xγ x Plugging the above formula in 5.17 we find that x 3 t x = x H 3 ψ t Q xγ x + NICE + errors 3 x t x + NICE + NICE Q 3 xbr x Q xγ x b s x 4 x + l.o.t As we did before, in 3 Q xbr x, the most dangerous term is given by Q x 1 H 4 γ, the tangential terms appear, which implies Q 3 xbr x = Q 1 x H 4 γ + NICE and therefore Q x H 3 γ t = x H ψ 3 t Q xγ Q x x H 4 γ Q xγ x b s x 4 x + NICE + errors We use 5.16 to find Q x H 3 γ t + Q xψ x H 4 γ = x H ψ 3 t Q x b sh γ 4 Q xγ x b s = x H 3 ψ t b s x H 4 Q x γ x + NICE + errors x 4 x Q xγ x b s x 4 x + NICE + errors = x H 3 ψ t b s x H 4 ψ b s x H 4 b s x Q xγ x b s 4 x x + NICE + errors

180 170 CHAPTER 5. FROM A GRAPH TO A SPLASH SINGULARITY We will show that is NICE and then we are done. b s x H b 4 s x Q xγ x b s x 4 x b s x H b 4 s x Q xγ x b s x 4 x = b s H b 4 s x Q xγ x b s x 4 x = b s H Q 3 xbr x Q xγ x b s x 4 x We repeat the calculation for dealing with the most dangerous terms in Q 3 xbr x = Λ x 4 γq x x x + l.o.t In the l.o.t we use that β x x gives an extra cancelation. We find that b s H Q 3 xbr x Q xγ x b s x 4 x = b s HΛ x 4 γq x x x Q xγ = b s 4 x x γq x x Using that 4 x x = 4 x x we are done. x b s x 4 x Q xγ x b s x 4 x + NICE + l.o.t + NICE

181 Chapter 6 Turning waves for the inhomogeneous Muskat problem: a computer-assisted proof 6.1 Introduction The evolution of a fluid in a porous medium is an interesting problem in fluid mechanics [15, 77]. Darcy, in 1856 tried to formulate the laws of a water flow through vertical homogeneous sand filters a porous medium. Without taking gravity into account, he postulated the following relation see Figure 6.1: where µ Q A = κp l P r, 6.1 L Q is the total discharge Vol/t. A is the cross section of the pipe. K is the length between the measure points. µ is the viscosity of the fluid. κ is the permeability of the medium. P l and P r are the pressure at the left and right ends respectively. For a continuous medium, if we also reflect the effect of gravity, Darcy s law is given by where µ v = p 0, gρ, 6. κ 171

182 17 CHAPTER 6. A COMPUTER-ASSISTED PROOF FOR THE MUSKAT PROBLEM Figure 6.1: Darcy s device for his experiment. Water flows through a porous medium. g is the gravitational acceleration constant. ρ is the density of the fluid. v is the velocity of the fluid. Darcy s law has been verified many times experimentally and can be derived from Navier- Stokes equations using homogenization methods [86]. Muskat, in 1937 [74], studied the evolution of ground water and its interaction with oil in a sandy medium by looking at the interface that separated the two fluids. Therefore, this is nowadays known as the Muskat problem. Darcy s law presents many similarities with the movement of a fluid trapped in a Hele- Shaw cell, which was studied by Hele-Shaw in 1898 [59, 60]. A Hele-Shaw cell consists of two thin parallel vertical plates, situated at a distance b which we will assume to be very small compared to the area of the plates see Figure 6.. Starting from the Stokes equations and setting v = v 1, v, v 3 as the velocity of the fluid we get ρv v = p + µ v 0, 0, gρ, divv = z y b x Figure 6.: A Hele-Shaw cell.

183 6.1. INTRODUCTION 173 Assuming that b is very small and that the plates are situated orthogonally to the x-axis, the velocity will depend only on y and z. Substituting this into 6.3, we get 0 = x p, ρv y v + v 3 z v = y p + µ v, ρv y v 3 + v 3 z v 3 = z p + µ v 3 ρg. We can also assume that the derivatives of v and v 3 in the y, z directions are small compared to the ones in the x direction. We can therefore approximate the previous system by x p = 0, y p = µ xx v, ρg + z p = µ xx v 3. Since p does not depend on x, and v takes vales 0 at the boundary x = 0 and x = b we can integrate the previous equations to get µv = 1 x bx y p, 6.4 µv 3 = 1 x bx z p + ρg. 6.5 Finally, averaging over every x [0, b] the mean velocity v can be written as 1µ v = p 0, gρ, 6.6 b which is analogous to Darcy s law by setting the permeability of the medium κ equal to b 1. Saffman and Taylor [80] studied the evolution of the interface between water and oil on a Hele-Shaw cell, obtaining the same equations as in the Muskat problem. This is why the equations for the Muskat problem are also known as the two-phase Hele-Shaw equations. We refer the reader to the papers [7], [17], [], [7], [9], [31], [3] and specially the survey paper [3] for the most important results concerning the unconfined Muskat problem. From now on, we will work in the two dimensional case, although the generalization to the 3D one is immediate. The Muskat problem has also been studied in what is called the confined regime. In the confined regime the two incompressible fluids can not penetrate into a top and a bottom layers which we will assume are at height L and L see Figure 6.3. Moreover, we will consider that both fluids have the same viscosity but different densities ρ 1 the upper fluid, ρ the lower one and we will denote by z, t = z 1, t, z, t the interface of the free boundary, which can be either horizontally periodic or flat at infinity. Thus, our system of equations is

184 174 CHAPTER 6. A COMPUTER-ASSISTED PROOF FOR THE MUSKAT PROBLEM µ κ v = p ρ0, g, v = 0, ρ t + v ρ = 0, v x, L, t = v x, L, t = 0. Figure 6.3: The confined Muskat problem. Sketch of the situation. since we don t take into account the effects of surface tension. Manipulating the system, one can get an evolution equation for z, t setting L = π and ρ ρ 1 = 4π: [ z zη sinhz 1 z 1 η t z, t = P.V. R coshz 1 z 1 η cosz z η + ] z 1 z 1 η, z + z η sinhz 1 z 1 η dη, 6.7 coshz 1 z 1 η + cosz + z η or, if the interface is parametrized as a graph, f, [ t f, t = P.V. f f η R + f + f η sinh η cosh η cosf f η sinh η cosh η + cosf + f η ] dη. 6.8

185 6.1. INTRODUCTION 175 For the confined model, local existence, a maximum principle and global existence for a class of initial data were proved in [33] in the case without surface tension. For the case with surface tension, local existence in a certain Hölder space was proved in [40], in which the authors also study bifurcations of the stationary solutions in the unstable case with surface tension. Similar results for the case with three fluids and two interfaces were discussed in [39]. We could also think of a model for the Muskat problem that also incorporates a jump in the permeabilities of the medium. this means that we can write the permeability of the medium as κx = κ 1 1 {x Ω 1 + κ 1 {x Ω, where Ω 1 and Ω respectively denote the space below or above a given boundary. This model has gained importance since it has been used for many applications: for example the description of a geothermal reservoir, oil exploration, soil physics, ground water hydrology, etc. see [5], [38] and the references therein. Again, Darcy s law governs the movement of the velocity of the fluids, which also have a jump of densities across an interface. We will again denote by z, t = z 1, t, z, t the interface and by b = h 1, h the fixed boundary at which the permeability jump is placed. Moreover, we will assume that this boundary is given by h 1 =, h = h for a constant L > h > 0 see Figure 6.4. We will designate by K = κ 1 κ κ 1 +κ the adimensional parameter relating the different permeabilities. It is easy to see that by definition 1 < K < 1. Figure 6.4: The inhomogeneous Muskat problem. Sketch of the situation.