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Global Well-Posedness And Scattering For The Defocusing Quintic Nonlinear Schrödinger Equation In Two Dimensions, Xueying Yu
Doctoral Dissertations
In this thesis we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in two dimensions. We take general data in the critical homogeneous Sobolev space dot H1/2. We show that if a solution remains bounded in dot H1/2 in its maximal time interval of existence, then the time interval is infinite and the solution scatters.
Well-Posedness For The Cubic Nonlinear Schrödinger Equations On Tori, Haitian Yue
Well-Posedness For The Cubic Nonlinear Schrödinger Equations On Tori, Haitian Yue
Doctoral Dissertations
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)}