Global well-posedness and scattering for mass-critical inhomogeneous NLS when $d\ge3$

TL;DR

This paper establishes global well-posedness and scattering for the mass-critical inhomogeneous nonlinear Schrödinger equation in dimensions three and higher, overcoming challenges posed by the inhomogeneity and lack of symmetries.

Contribution

It extends the concentration compactness/rigidity method to inhomogeneous NLS with singularity, proving results under new conditions and in Lorentz spaces.

Findings

Proves global well-posedness and scattering for large initial data.

Handles singularity at the origin due to inhomogeneity.

Uses Lorentz space estimates to exclude almost periodic solutions.

Abstract

We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\pm |x|^{-b}|u|^{\frac{4-2b}{d}}u$ for large $L^2(\mathbb{R} ^d)$ initial data with $d\ge3,0<b<\min \left\{ 2,\frac{d}{2} \right\}$; in the focusing case, we require that the mass is strictly less than that of the ground state. Compared with the classical Schr\"odinger case ($b=0$, Dodson, J. Amer. Math. Soc. (2012), Adv. Math. (2015)), the main differences for the inhomogeneous case ($b>0$) are that the presence of the inhomogeneity $|x|^{-b}$ creates a nontrivial singularity at the origin, and breaks the translation symmetry as well as the Galilean invariance of the equation, which makes the establishment of the profile decomposition and long time Strichartz estimates more difficult. To overcome these difficulties, we perform the concentration compactness/rigidity methods of [Kenig and Merle, Invent. Math. (2006)] in the Lorentz space framework, and reduces the problem to the exclusion of almost periodic solutions. The exclusion of these solutions will utilize fractional estimates and long time Strichartz estimates in Lorentz spaces. In our study, we obseve that the decay of the inhomogeneity $|x|^{-b}$ at infinity prevents the concentration of the almost periodic solution at infinity in either physical or frequency space. Therefore, we can use classical Morawetz estimates, rather than interaction Morawetz estimates, to exclude the existence of the quasi-soliton.