Global well-posedness for the defocusing 3D quadratic NLS in the sharp critical space

Abstract

In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schr\"odinger equation \begin{align*} i\partial_t u + \frac12\Delta u = |u| u, \end{align*} in its sharp critical weighted space $\mathcal F \dot H_x^{1/2}$ for radial data. Killip, Masaki, Murphy, and Visan [2017, NoDEA] have proved its global well-posedness and scattering, if the $\mathcal F \dot H_x^{1/2}$-norm of the solution is bounded in the maximal lifespan. Now, we remove this a priori assumption for the global well-posedness statement in the radial case. Our method is based on the almost conservation of pseudo conformal energy. This energy scales like $\dot H_x^{-1}$, which is supercritical. We are still able to derive the global well-posedness using this monotone quantity. The main observation is that we can establish the local solution in supercritical weighted space when the initial time is away from the origin.