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

By B. Dodson. 2011

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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.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d4de9fd68a84ae7d1f09f00013ca714b8b2b68ab