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 Rivière 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.

References in corpus (6)

• n/p-Harmonic maps: regularity for the sphere case
• Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps
• Free boundary minimal surfaces: a nonlocal approach
• Infinite time blow-up for half-harmonic map flow from $\mathbb{R}$ into ${\mathbb{S}}^1$
• Almost minimizers for certain fractional variational problems
• A Pohozaev-type formula and Quantization of Horizontal Half-Harmonic Maps