Large global solutions for energy-critical nonlinear Schr\"odinger equation

TL;DR

This paper proves global well-posedness and scattering for radial solutions of the 3D energy-critical nonlinear Schrödinger equation using outgoing and incoming decompositions, extending understanding of solution behavior in critical regimes.

Contribution

It introduces a method to decompose initial data into outgoing and incoming parts, establishing global solutions and scattering for each component in the energy-critical setting.

Findings

Radial initial data can be decomposed into outgoing and incoming parts.

Solutions with outgoing data scatter forward in time.

Solutions with incoming data scatter backward in time.

Abstract

In this work, we consider the 3D defocusing energy-critical nonlinear Schr\"odinger equation $i\partial_t u+\Delta u =|u|^4 u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^3$. Applying the outgoing and incoming decomposition presented in the recent work \cite{BECEANU-DENG-SOFFER-WU-2021}, we prove that any radial function $f$ with $\chi_{\leq1}f\in H^1$ and $\chi_{\geq1}f\in H^{s_0}$ with $\frac{5}{6}<s_0<1$, there exists an outgoing component $f_+$ (or incoming component $f_-$) of $f$, such that when the initial data is $f_+$, then the corresponding solution is globally well-posed and scatters forward in time; when the initial data is $f_-$, then the corresponding solution is globally well-posed and scatters backward in time.