Remarks on blow-up criteria for the derivative nonlinear Schrodinger equation under the optimal threshold setting

TL;DR

This paper investigates blow-up criteria for the derivative nonlinear Schrödinger equation at the critical mass threshold, establishing conditions for global well-posedness and analyzing blow-up profiles using concentration compactness.

Contribution

It extends the understanding of blow-up behavior at the critical mass for the derivative nonlinear Schrödinger equation using concentration compactness methods.

Findings

Global well-posedness when mass < 4pi or mass = 4pi with negative momentum

Characterization of blow-up solutions at the critical mass

Identification of limiting blow-up profiles

Abstract

We study the Cauchy problem of the mass critical nonlinear Schrodinger equation with derivative with the 4pi mass. One has the global well-posedness in H^1 whenever "the mass is strictly less than 4pi" or whenever "the mass is equal to 4pi and the momentum is strictly less than zero". In this paper, by the concentration compact principle as originally done by Kenig-Merle, we obtain the limiting profile of blow up solutions with the critical 4pi mass.