Abstract
I will report some recent progress of the global well-posedness and scattering for the defocusing nonlinear Schrodinger equations (NLS), which is based on the joint work with Prof. Yifei Wu. We will first give the global well-posedness of 3D quadratic NLS in the critical weighted space, namely |x|^{1/2}u_0 \in L^2. Previously, Killip, Masaki, Murphy, and Visan proved its conditional global well-posedness and scattering in such space. Our result removes the a priori assumption for the global well-posedness part. Next, we will consider the scattering of mass subcritical NLS. Previously, it is shown by Tsutsumi-Ogawa that the scattering holds in the first-order weighted space, and by Lee that the continuity of the scattering operator breaks down in L^2. We extend the scattering result below the first-order weight. Furthermore, we also give the scattering with a large class of L^2-data based on probabilistic method.