Local and global well-posedness in $L^{2}(\mathbb R^{n})$ for the inhomogeneous nonlinear Schr\"{o}dinger equation

Abstract

This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in L^{2}(\mathbb R^{n})$, where $\lambda \in \mathbb C$, $0<b<\min \left\{2,{\rm \; }n\right\}$ and $0<\sigma \le \frac{4-2b}{n} $. We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $\sigma =\frac{4-2b}{n} $, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $\sigma <\frac{4-2b}{n} $. In order to obtain the results above, we establish the Strichartz estimates in Lorentz spaces and use the contraction mapping principle based on Strichartz estimates.