Nonlinear dispersive equations with random initial data

Abstract

In the first part of this thesis we consider the defocusing nonlinear wave equation of power-type on R3. 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 Klein- Gordon equation in three dimensions in the symplectic phase space H 1/2 (T 3) x H -1/2 (T 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.