A new conformal heat flow of harmonic maps

TL;DR

This paper introduces a conformal heat flow for harmonic maps that aims to delay singularities, demonstrating the existence of global weak solutions that are mostly smooth with finitely many singular points.

Contribution

It proposes a novel conformal heat flow for harmonic maps and establishes the existence of global weak solutions with controlled singularities.

Findings

Global weak solutions exist for the flow.

Solutions are smooth except at finitely many points.

Flow delays but does not prevent bubble formation.

Abstract

We introduce and study a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. This flow is designed to postpone finite time singularity but does not get rid of possibility of bubble forming. We show that Struwe type global weak solution exists, which is smooth except at most finitely many points.