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

2024/10/06 by Shen, Jia, Wu, Yifei
#35B40 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.04337

Abstract

In this paper, we prove the global well-posedness of defocusing 3D quadratic nonlinear Schrödinger equation i∂_t u + \frac12Δu = |u| u, in its sharp critical weighted space \mathcal F 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 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 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.

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