  In this paper, we study the fractional harmonic gradient flow on $S^1$ taking
values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular
addressing uniqueness and regularity of solutions in the so-called energy class
with sufficiently small energy, adding to the existing body of knowledge which
includes existence of solutions and certain bubbling phenomena.

Analysis of PDEs