Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state

TL;DR

This paper proves that the focusing mass-critical nonlinear Schrödinger equation is globally well-posed and exhibits scattering for initial data with mass below that of the ground state, using interaction Morawetz estimates.

Contribution

It introduces a positive definite interaction Morawetz estimate for sub-ground state mass initial data in the mass-critical NLS, advancing understanding of global behavior.

Findings

Established global well-posedness for sub-ground state mass

Proved scattering for solutions with initial mass below ground state

Developed frequency localized interaction Morawetz estimates

Abstract

In this paper we prove that the focusing, $d$-dimensional mass critical nonlinear Schr{\"o}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$, $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} < \| Q \|_{L^{2}(\mathbf{R}^{d})}$, where $Q$ is the ground state, and $d \geq 1$. We first establish an interaction Morawetz estimate that is positive definite when $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} < \| Q \|_{L^{2}(\mathbf{R}^{d})}$, and has the appropriate scaling. Next, we will prove a frequency localized interaction Morawetz estimate similar to the estimates made in \cite{D2}, \cite{D3}, \cite{D4}. See also \cite{CKSTT4} for the energy critical case. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.