Global Well-Posedness and Scattering of Besov Data for the Energy-Critical Nonlinear Schrödinger Equation

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, Staffilani, Takaoka and Tao (2008). \bigskip 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 the sum of a finite energy component in $\dot{H}^1$ and a small part in a Besov space, with the size of the Besov part small enough compared to the size of the energy part. The solution is shown to scatter for initial conditions u0 ∈ B˙12,q, for 2 ≤ q < ∞, while it is only globally well-posed for q = ∞. 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, namely the spaces 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 u0 ∈ B˙1 2,q for 2 ≤ q < ∞ evolve to a global solution that scatters and all data u0 ∈ B˙1 2,∞ evolve to a global solution.Ph.D