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

TL;DR

This paper proves the global well-posedness of the defocusing 3D quadratic NLS in the critical space for radial data without assuming boundedness of the solution's norm, using almost conservation of pseudo conformal energy.

Contribution

It removes the a priori boundedness assumption for global well-posedness in the radial case by employing pseudo conformal energy conservation.

Findings

Global well-posedness established without a priori norm bounds

Method based on almost conservation of supercritical pseudo conformal energy

Local solutions constructed in supercritical weighted space away from the origin

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.