Global and dynamical aspects of nonlinear Schrodinger equations on compact manifolds

We address two questions concerning the global existence and long-time dynamical behavior of nonlinear Schrodinger equations when the spatial domain is a compact manifold. The first is the problem of low-regularity global well-posedness below energy norm of nonlinear Schrodinger equations on arbitrary closed (compact without boundary) Riemannian manifolds; and the second is that of the behavior of Sobolev norms, in particular pertaining to weakly turbulent behavior and existence of unbounded orbits in Sobolev spaces Hs.;In Part I of the dissertation, we address the first problem of global well-posedness below energy norm for the defocusing cubic nonlinear Schrodinger equation on arbitrary closed Riemannian 2-manifolds. To this end, we prove a sharp bilinear oscillatory integral bound on Rd , which yields sharp (at relevant scales) bilinear Strichartz estimates on d--dimensional compact manifolds without boundary. We also develop some multilinear spectral analysis techniques on compact manifolds, namely bounds on Coifman-Meyer-type multilinear spectral multipliers and estimates on the spectral localization of products of Laplacian eigenfunctions. As a result, we are able to apply the I-method machinery introduced by Colliander, Keel, Staffilani, Takaoka, and Tao to prove global well-posedness of the defocusing cubic NLS on closed surfaces in Hs(M) for all s > ⅔. This generalizes without any loss of regularity previous results in [13, 33] for the torus T2 where Strichartz estimates only lose less-than-any-positive-power of derivatives and the multilinear analysis is classical thanks to the Fourier structure.;In Part II, we address the second problem pertaining to the existence of unbounded Hs--orbits for cubic and almost cubic Schrodinger nonlinearities on Td . It is conjectured that such orbits exist and are generic, a behavior which signals, in the case when s > 1, the cascade of energy to higher and higher frequency modes. Using a soft analysis argument that can be described as a "nonlinear uniform boundedness principle", we observe that this conjecture is implied by the "long-time strong instability" of the flow near certain families of initial data in Hs( T2 ). Long-time strong instability near zero was proved in a pioneering work of Colliander, Keel, Staffilani, Takaoka, and Tao [31], wherein they delicately constructed a "norm inflating" solution. This suggests a program to prove the "existence and genericness" of unbounded orbits based on the idea of extending the latter result in [31] to neighborhoods of a sufficiently large family of initial data. We take the first step in this program for the cubic nonlinearity by proving long-time strong instability near all single-frequency initial data in the range 0 < s < 1. We also illustrate how it successfully implies existence (and genericness in a metric subspace) of unbounded orbits for certain Schrodinger nonlinearities that are close to cubic.