Blow up dynamics for equivariant critical Schr\"odinger maps

TL;DR

This paper proves the existence of finite-time blow-up solutions for equivariant Schr"odinger maps from 2+1 dimensions into the 2-sphere, with solutions closely resembling a rescaled harmonic map, characterized by a specific scaling law.

Contribution

It establishes the existence of finite-time blow-up solutions near a harmonic map with a precise scaling behavior, advancing understanding of singularity formation in Schr"odinger maps.

Findings

Existence of equivariant blow-up solutions close to harmonic maps

Blow-up rate characterized by $t^{- u}$ with $ u>3/2$

Solutions exhibit dynamic rescaling near singularity

Abstract

For the Schr\"odinger map problem from 2+1 dimensions into the 2-sphere we prove the existence of equivariant finite time blow up solutions that are close to a dynamically rescaled lowest energy harmonic map, the scaling parameter being given by $t^{-\nu}$ with $\nu>3/2$.