Global well-posedness and scattering of the defocusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation with radial data

TL;DR

This paper proves global well-posedness and scattering for the defocusing energy-critical inhomogeneous nonlinear Schrödinger equation with radial data, using Lorentz spaces and quantitative bounds, extending previous methods beyond concentration compactness.

Contribution

It introduces a Lorentz space-based approach to establish global solutions and scattering, providing explicit exponential bounds, differing from traditional concentration compactness techniques.

Findings

Global well-posedness and scattering for most dimensions and parameters.

Quantitative exponential bounds on the spacetime norm.

Extension of Tao's method with Lorentz spaces for INLS.

Abstract

We consider the defocusing energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation (INLS) $iu_t + \Delta u = |x|^{-b}|u|^{k}u$ in $\mathbb{R} \times \mathbb{R}^{n}$ where $n \geq 3$, $0<b<\min(2, n/2)$, and $k=(4-2b)/(n-2)$. We show that for every spherically symmetric initial data $\phi \in H^1(\mathbb{R}^n)$, or preferably $\dot{H}^1(\mathbb{R}^n)$, the solution is globally well-posed and scatters for every such $n$ and $b$ except for $n=4$ with $1\leq b<2$ and $n=5$ with $1/2\leq b\leq 5/4$. We mainly apply the arguments of Tao (2005), but inspired by the work of Aloui and Tayachi (2021), we utilize Lorentz spaces to define spacetime norms. This method is distinct from the widespread concentration compactness principle and establishes a quantitative bound for the solution's spacetime norm. The bound has an exponential form $C\exp(CE[\phi]^C)$ in terms of the energy $E[\phi]$, similar to Tao's work.