Long time dynamics for nonlinear Schr{\"o}dinger equations at critical regularity

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Date
2019-07-02
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Johns Hopkins University
Abstract
In this thesis, we study the long time dynamics for nonlinear Schr{\"o}dinger equations (NLS) with power-type nonlinearity at critical regularity. There are five problems studied by the author. On one hand, as for Euclidean case, we discuss problems including the \emph{Critical norm conjecture for high dimensional inter-critical NLS} and the \emph{Dynamics of threshold solutions for focusing energy-critical NLS}. Techniques including concentration compactness/Rigidity method, double Duhamel method and Morawetz-type arguments have been applied. On the other hand, we discuss global well-posedness and scattering theory for NLS on product spaces $\mathbb{R}^m \times \mathbb{T}^n$ (also known as waveguide manifolds) and corresponding resonant systems. Function spaces, Stricharz estimates and profile decompositions in the setting of waveguide manifolds, Virial identities and long time Stricharz estimate are used. My works can be found in \cite{CGZ,GZ,SZ,ZZ1,ZZ2}. \cite{ZZ1} is published in the \emph{Journal of Hyperbolic Differential Equations}, \cite{GZ} is to appear in the \emph{Journal of Differential Equations} and the other three papers are currently under review.
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NLS, well-posedness, scattering, waveguide manifold
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