Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map

TL;DR

This paper studies finite-time blow-up solutions for the energy-critical Schrödinger map with equivariant symmetry, revealing how solutions concentrate energy and form singularities near the ground state.

Contribution

It establishes the existence of a codimension one set of initial data leading to blow-up, providing a detailed description of the singularity formation process.

Findings

Existence of blow-up solutions close to the ground state

Finite-time singularity formation through energy concentration

Characterization of the blow-up profile as a universal bubble

Abstract

We consider the energy critical Schr\"odinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.