Existence, Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in General Target Manifolds

TL;DR

This paper investigates the fractional harmonic gradient flow into general target manifolds, establishing global existence, uniqueness, and regularity results for solutions with small energy, extending previous work to a non-local fractional setting.

Contribution

It extends the analysis of fractional harmonic gradient flows to general target manifolds, providing new results on existence, uniqueness, and regularity in a non-local framework.

Findings

Proved global existence of solutions with small energy.

Established uniqueness of solutions in the fractional setting.

Demonstrated regularity properties using commutator estimates.

Abstract

In this paper, we continue to study the fractional harmonic gradient flow on $S^{n-1}$ taking values in a general closed manifold $N \subset \mathbb{R}^n$, addressing global existence and uniqueness of solutions of energy class with sufficiently small energy, adding to the existing body of knowledge pertaining to the half-harmonic gradient flow and expanding upon our previous work in [34]. We extend the techniques by Struwe in [30] and Rivi\`ere in [22] to the non-local framework analogous to [34] to derive uniqueness, employ commutator estimates as in [8] for regularity and follow [30] for a general existence result.