Half-harmonic gradient flow: aspects of a non-local geometric PDE

The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in ^[47] is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see ^[40], is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as $t\to +\mathrm{\infty }$, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity's sake, we restrict our attention to the case of spherical target manifolds $S^{n-1}$, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds $N$ (cf. ^[48]).