Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Riviere. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.