Measurable Semigroup Selection of the Heat Flow for Harmonic Maps

TL;DR

This paper demonstrates the existence of infinitely many measurable semigroups solving the heat flow for harmonic maps from a 3-ball to a sphere, highlighting non-uniqueness in solutions where previous work showed non-uniqueness.

Contribution

It introduces a measurable semigroup framework for the heat flow of harmonic maps, extending Coron’s non-uniqueness results using techniques by Cardona and Kapitanski.

Findings

Existence of infinitely many measurable semigroups for the heat flow.

Non-uniqueness of solutions in specific geometric settings.

Extension of previous non-uniqueness results to semigroup solutions.

Abstract

J.-M. Coron proved in [5] that the global weak solutions of the heat flow from $M$ to $N$, starting at non-stationary weakly harmonic maps, are not unique when $M = B^3$ and $N = S^2$. Hence, the semigroup property of the solution map does not hold in general. The present short paper uses the techniques developed by J. Cardona and L. Kapitanski to show the existence of infinitely many measurable semigroups solving the heat flow in the same cases where non-uniqueness was shown by J.-M. Coron.