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The Role of Smoothing Effect in Some Dispersive Equations Ricardo The role of smoothing effect in some dispersive equations by Ricardo Grande Izquierdo Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 © Massachusetts Institute of Technology 2020. All rights reserved. Author............................................................................ Department of Mathematics May 1, 2020 Certified by........................................................................ Professor Gigliola Staffilani Abby Rockefeller Mauze Professor Thesis Supervisor Accepted by....................................................................... Davesh Maulik Chairman, Department Committee on Graduate Studies The role of smoothing effect in some dispersive equations by Ricardo Grande Izquierdo Submitted to the Department of Mathematics on May 1, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, we study the role of smoothing effect in the local well-posedness theory of dispersive partial differential equations in three different contexts. First, we use it to overcome a loss of derivatives in a family of nonlocal dispersive equations. Second, we exploit a discrete version of the smoothing effect to study a discrete system of particles and how to approximate it by a continuous dispersive equation. Third, we use an anisotropic version of the smoothing effect to establish the local well-posedness theory of the two-dimensional Dysthe equation, which is used to model oceanic rogue waves. Thesis Supervisor: Professor Gigliola Staffilani Title: Abby Rockefeller Mauze Professor Acknowledgements I would like to thank my advisor, Gigliola Staffilani, for her support, advice and encour- agement over the past five years. I also want to thank her for her generosity and wisdom, for creating a fantastic atmosphere to study mathematics at MIT, and for being the only one to help me out that one time at Sulmona. I thank David Jerison and Andrew Lawrie for serving on my thesis committee and for their useful suggestions. I am grateful to David Jerison, Yannick Sire and Luis Vega for taking an interest in this project and for many useful conversations. I also thank Paolo An- tonelli, Michele Dolce, Pierangelo Marcati and Raffaele Scandone for inviting me to L’Aquila and for being such wonderful hosts. I would like to single out some of my professors and mentors back at home, and thank them for their support and for their commitment to quality teaching from an early stage: Pedro Alegría, Javi Duoandikoetxea, Luis Escauriaza, Raúl Ibáñez, Miren Iriondo, Marta Macho-Stadler, Juan Luis Mañes and Luis Martínez. I am immensely grateful to Vishal, Sylvain, Hood, Robin, Ethan, Kristin, Gwen, Tudor, Chris, Kevin and many other friends at MIT for keeping me sane and making this experience so enjoyable. I would especially like to thank Ethan for putting up with me both as an officemate and as a roommate, for not giving up on my vowels and for teaching me lots of mathematics despite his own inability to do arithmetic. I also thank Kevin for sharing his love for margaritas and for those ridiculous walks in search of a nonexistent tree. I am fortunate to have had many good friends in Boston, particularly Iñigo, Jordi, Laura, Pablo, Javi, Adrià, Ferran, Sabine, and Nathaniel. I cannot thank Miguel enough for his endless phone calls, his sharp arguments and for making me fat. I am grateful to Laura for sharing the best part of this journey, for all her support and for introducing me to the best cacio e pepe in the world. I want to thank Vilar and Lekue for keeping in touch all these years and for making sure I feel welcome every time I visit. I also thank Josué for showing me the importance of collaboration, and for sending me at least nineteen incomprehensible postcards. Finalmente, quisiera dedicar esta tesis a mis padres, Miguel Ángel y Yolanda, y agrade- cerles su apoyo, confianza y paciencia; y a mi hermano Diego, por su templanza y por aguantarme con excelente sentido del humor. Some of the research in this thesis was supported by the National Science Foundation grants DMS-1500771, DMS-1462401, DMS-1764403, and DMS-1362509. 5 Contents Chapter 1. Introduction 9 1. Background and main results 9 2. Notation 19 Chapter 2. Space-time fractional Nonlinear Schrödinger equation 21 2.1. Introduction 21 2.2. Linear estimates 29 2.3. Proof of Theorem 2.1.2 50 2.4. General proof of Theorem 2.1.3 54 2.5. Ill-posedness 59 Chapter 3. Continuum limit for discrete NLS with memory effect 71 3.1. Introduction 71 3.2. The linear equation 76 3.3. Lwp of the discrete model 86 3.4. Continuum limit 92 Chapter 4. Dysthe equation and oceanic rogue waves 111 4.1. Introduction 111 4.2. Strichartz estimates 114 4.3. Smoothing effect 120 4.4. Maximal function estimates 124 4.5. Contraction mapping argument 136 4.6. Ill-posedness 141 Appendix A. Harmonic Analysis 149 Bibliography 151 7 CHAPTER 1 Introduction 1. Background and main results In this thesis, we study various dispersive equations and their local well-posedness theory. We say that an equation is of dispersive nature when wave solutions of different frequencies propagate at different speeds when no boundary conditions are imposed. There are many important natural phenomena that are described using dispersive equations: they often appear in models in nonlinear optics, they describe the propagation of water waves, and they play a fundamental role in quantum field theory. One important question in the theory of dispersive equations is whether there exists a solution to the PDEs used to model these physical phenomena. Informally, we say that a PDE is well-posed if a solution exists, it is unique and it depends continuously on the initial conditions of the system. In the case of evolution equations (i.e. involving time), local well-posedness means that such conditions are met for some, possibly short, time. Much of the work in this thesis concerns the problem of establishing well-posedness for a variety of dispersive equations. Another important aspect of the study of dispersive equations is their connection with the physical phenomena that they should model. One way of connecting the model to the equation is via the description of a discrete system of particles whose interactions follow a particular set of laws dictated by the physical phenomenon, and then letting the number of particles go to infinity. This limit is described by a continuous PDE, and it often constitutes a good approximation to the discrete system when there is a large number of particles. Chapter 3 in this thesis will deal with this problem. From the mathematical point of view, many of the dispersive equations that we have mentioned are nonlinear. The nonlinear term is often some power of the unknown function, but it sometimes involves derivatives of the unknown function itself. When that happens, the study of these equations becomes more intricate and we say that there is a loss of derivatives. We will elaborate on this later, but one way to handle this problem is to find ways to recover these derivatives. The smoothing effect is a technique that allows us to do just that, and in this thesis we explore how to use it in different contexts. 9 10 1. INTRODUCTION In order to set the stage, let us write a general form of a one-dimensional dispersive equation: ( @ u + L(u) = N(u); (t; x) 2 2 (1.1) t R ujt=0 = u0; where L(u) is the linear part of the equation and N(u) is the nonlinear term. To simplify the discussion, suppose that N(u) = 0 and that our initial data u0 lives in the Sobolev space s H (R) with enough regularity to make the following computations rigorous. We will also assume that the linear term L is a skew-adjoint, constant-coefficient differential operator: n X j L(u)(x) := cj @xu(x); j=0 where the coefficients cj 2 C do not depend on x, and n 2 N0 is the order of the differential operator L. By taking the Fourier transform of (1.1), we find that 0 n 1 X j (1.2) @tub(t; ξ) + @ cj (−iξ) A ub(t; ξ) = 0: j=0 Because L is skew-adjoint and has constant coefficients, we can write n X j cj (−iξ) = i h(ξ); j=0 for some real valued polynomial h : R ! R. The fact that h is real-valued is fundamental, as this is what makes it constitute a dispersive equation. One can then solve the ODE (1.2) and find the following ansatz for the solution: −it h(ξ)+ix ξ u(t; x) = e uc0(ξ) dξ =: W (t)u0: ˆR 2 Here fW (t)gt2R is the one-parameter unitary group (in L (R)) describing the solution to the linear problem associated to (1.1), and deriving linear estimates for W (t) is one of the main ways to study dispersive equations. Let us now consider the full nonlinear problem (1.1) including N(u). There are several ways to study the local well-posedness of this problem, but the techniques involved may be roughly classified as perturbative or non-perturbative. Examples of non-perturbative methods include conservation laws and monotonicity formulae, but we will not elaborate on these here. Instead, we will rely on perturbative methods which consist in approximating the nonlinear equation by more tractable equations. 1. BACKGROUND AND MAIN RESULTS 11 Because the formulation in (1.1) requires n derivatives, we often prefer to work with an integral reformulation of the problem given by the Duhamel formula: t 0 0 0 (1.3) u(t) = W (t)u0 + W (t − t ) N(u(t )) dt : ˆ0 This is not the only way to reformulate the problem (1.1), but it is convenient in order to study the nonlinear problem as a perturbation of the linear problem, i.e.
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