Nonlinear dispersive equations with random initial data

In the first part of this thesis we consider the defocusing nonlinear wave equation of power-type on R 3. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of initial data of supercritical regularity which lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposition and improved averaging effects for the free evolution of the randomized initial data. In the second part of this thesis, we consider the periodic defocusing cubic nonlinear KleinGordon equation in three dimensions in the symplectic phase space H2 (T3) x H--i 3). This space is at the critical regularity for this equation, and in this setting there is no global well-posedness nor any uniform control on the local time of existence for arbitrary initial data. We prove several non-squeezing results: a local in time result and a conditional result which states that uniform bounds on the Strichartz norms of solutions for initial data in bounded subsets of the phase space implies global-in-time non-squeezing. As a consequence of the conditional result, we conclude nonsqueezing for certain subsets of the phase space and, in particular, we obtain deterministic small data non-squeezing for long times. To prove non-squeezing, we employ a combination of probabilistic and deterministic techniques. Analogously to the work of Burq and Tzvetkov, we first define a set of full measure with respect to a suitable randomization of the initial data on which the flow of this equation is globally defined. The proofs then rely on several approximation results for the flow, one which uses probabilistic estimates for the nonlinear component of the flow map and deterministic stability theory, and another which uses multilinear estimates in adapted function spaces built on UP and VP spaces. We prove non-squeezing using a combination of these approximation results, Gromov's finite dimensional non-squeezing theorem and the infinite dimensional symplectic capacity defined by Kuksin. Thesis Supervisor: Gigliola Staffilani Title: Abby Rockefeller Mauz6 Professor of Mathematics Acknowledgments I would like to thank my advisor Gigliola Staffilani, who has made this thesis possible and given me so much of her time over these past five years. She has taught me an incredible amount about mathematics and being a mathematician, all with an immense amount of kindness and patience. I owe much gratitude to Andrea Nahmod for many mathematical conversations and plentiful sound advice, and to Michael Eichmair for support and guidance during some stressful times. I would also like to thank David Jerison and Jared Speck for serving on my thesis committee, and Jonas Liihrmann for our extremely enjoyable collaboration (which has, in particular, given rise to the contents of Chapter 2 of this thesis). My time at MIT would have been nowhere near as enjoyable without the friendship of my fellow classmates, particularly Michael Andrews, Nate Bottman, Saul Glasman, Jiayong Li and Alex Moll, and I am grateful to them for making these past years at the math department as colorful as they have been. I would like to thank my family and friends for their support, including my grandparents for having set the best possible example for us all. My parents, Beverley and Morton, have given me so much throughout my life. I would like to specifically thank them for the confidence they have always had in me, and for believing that I was capable of finishing this thesis, even at times when I did not. Finally, thank you, Michael, for being my family and my home.