TY  - JOUR
AU  - Wettstein, Jerome D.
PY  - 2023
TI  - Half-harmonic gradient flow: aspects of a non-local geometric PDE
JO  - Mathematics in Engineering
SP  - 1
EP  - 38
VL  - 5
IS  - 3
AB  - 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  →  +  ∞, 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]).
UR  - https://doi.org/10.3934/mine.2023058
DO  - 10.3934/mine.2023058