## Departmental PhD Thesis Exam – David Reiss

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, June 28, 2017
3:10 p.m.
BA6183

PhD Candidate:  David Reiss
Co-Supervisors:  Jim Colliander, Catherine Sulem
Thesis title: Global Well-Posedness and Scattering of Besov Data for the Energy-Critical Nonlinear Schr\”{o}dinger Equation

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Abstract:

We examine the Defocusing Energy-Critical Nonlinear Schr\”{o}dinger Equation in dimension 3.  This equation has been studied extensively when the initial data is in the critical homogeneous Sobolev space $\dot{H}^1,$ and a satisfactory theory is given in the work of Colliander, Keel, Sataffilani, Takaoka and Tao.  We extend the analysis of this equation to include infinite energy data $u_0 \in \dot{B}^1_{2,q}$ ($2 \leq q \leq \infty$) that can be decomposed as a finite energy component (a part  in $\dot{H}^1$) and a small Besov part, with the size of the energy part depending on the size of the Besov part.  If $2 \leq q < \infty,$ the solution is shown to scatter.  For $q = \infty$, the solution is shown to be globally well-posed.  Traditionally, the well-posedness theory has been studied in Strichartz spaces, but we use more subtle spaces to deal with the high frequencies that arise from the Besov data, $X^q(I)$.  These spaces are variants of bounded variation spaces and satisfy a duality that allows us to recover the traditional multilinear estimate along with a Strichartz variant that allows for extracting smallness by shrinking the time interval.

We also discuss a conjecture that all data $u_0 \in \dot{B}^1_{2,q}$ for $2 \leq q < \infty$ evolve to a global solution that scatters and we discuss the next steps to proving this.

A copy of the thesis can be found here:  ut-thesis-DR