Uniqueness and regularity of the fractional harmonic gradient flow in Sn−1

Abstract

In this paper, we study the fractional harmonic gradient flow on S1 taking values in Sn−1⊂Rn for every n≥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, see Schikorra et al. (2017), and bubbling phenomena as studied by Sire et al. (0000). We extend the techniques by Struwe (1985) and Rivière (1993) to the non-local framework and exploit integrability by compensation properties due to fractional Wente-type inequalities as in Mazowiecka and Schikorra (2018). Moreover, we briefly discuss convergence properties for solutions to the fractional gradient flow as t→∞.