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

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.