On uniqueness for the Harmonic Map Heat Flow in supercritical dimensions

TL;DR

This paper investigates the uniqueness of solutions to the harmonic map heat flow in supercritical dimensions, revealing that singular initial data can lead to multiple stable solutions, and discusses methods to restore uniqueness.

Contribution

It demonstrates that, in supercritical dimensions, singular data can produce multiple stable solutions, challenging previous assumptions of uniqueness, and explores ways to recover uniqueness.

Findings

Singular data can lead to multiple stable solutions in supercritical dimensions.

Both solutions satisfy the local energy inequality.

Discussion on methods to recover uniqueness.

Abstract

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved.