A universal bound on the blow-up rate for the focusing mass-critical nonlinear Schr\"odinger equation

TL;DR

This paper establishes a universal bound on the blow-up rate for the focusing mass-critical nonlinear Schrödinger equation, proving the nonexistence of self-similar blow-up solutions and deriving a sharp log-log correction under radial symmetry.

Contribution

It extends previous results by providing a universal blow-up bound for general initial data and introduces a new analysis method that does not depend on ansatz or variational techniques.

Findings

No self-similar blow-up solutions exist.

Sharp log-log correction to blow-up rate under radial symmetry.

Universal blow-up bound for general initial data.

Abstract

In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schr\"odinger equation for general initial data in $L^2(\mathbb R^d)$, extending previous knowledge for mass near the ground-state threshold due to Merle and Rapha\"el. The main results are twofold. First, we show the nonexistence of self-similar rate blow-up solutions. Second, under radial symmetry, we establish the sharp log--log correction to the self-similar bound on the blow-up rate. The proofs are based on a new analysis of general blow-up solutions, which does not rely on any ansatz or variational structure.