Probabilistic and Deterministic Wellposedness for Low Regularity Dispersive Equations

In this thesis we will present two results on global existence for nonlinear dispersive equations with data at or below the scaling regularity. In chapter 1 we take a probabilistic perspective to study the energy-critical nonlinear Schrödinger equation in dimensions d>6. We prove that the Cauchy problem is almost surely globally well-posed with scattering for a range of super-critical initial data. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results in dimension 4 and the main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity. Chapter 2 concerns the half-wave maps equation, a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. We will prove that in three dimensions the equation is "weakly" globally well-posed for angularly regular data which is small in a critical Besov space, partially generalising known results in dimensions d=4. The main difficulty in moving to three dimensions is the loss of a key endpoint Strichartz estimate. We overcome this by using Sterbenz's improved Strichartz estimates in conjunction with commuting vector fields to develop trilinear estimates in weighted Strichartz spaces which avoid the use of the Strichartz endpoint.