## Doctoral Dissertations

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#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

2018

September

Andrea R. Nahmod

#### Subject Categories

Analysis | Partial Differential Equations

#### Abstract

This thesis studies the cubic nonlinear Sch\"rodinger equation (NLS) on tori both from the deterministic and probabilistic viewpoints. In Part I of this thesis, we prove global-in-time well-posedness of the Cauchy initial value problem for the defocusing cubic NLS on 4-dimensional tori and with initial data in the energy-critical space $H^1$. Furthermore, in the focusing case we prove that if a maximal-lifespan solution of the cubic NLS \, $u: I\times\mathbb{T}^4\to \mathbb{C}$\, satisfies \$\sup_{t\in I}\|u(t)\|_{\dot{H}^1(\mathbb{T}^4)}

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