A relative entropy for expanders of the Harmonic map flow

TL;DR

This paper introduces a relative entropy concept to study the uniqueness and existence of expanding solutions for the Harmonic map flow originating from 0-homogeneous maps into closed Riemannian manifolds.

Contribution

It develops a new relative entropy framework to establish existence and generic uniqueness of expanding solutions for the Harmonic map flow from 0-homogeneous initial maps.

Findings

Existence of two expanding solutions for given initial data.

Proof of generic uniqueness of solutions with zero relative entropy.

Application of blow-up and blow-down techniques in solution analysis.

Abstract

In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a $0$-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved.