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We introduce a heat flow associated to half-harmonic maps, which have beenintroduced by Da Lio and Rivi\`ere. Those maps exhibit integrability bycompensation in one space dimension and are related to harmonic maps with freeboundary. We consider a new flow associated to these harmonic maps with freeboundary which is actually motivated by a rather unusual heat flow forhalf-harmonic maps. We construct then weak solutions and prove their partialregularity in space and time via a Ginzburg-Landau approximation. The presentpaper complements the study initiated by Struwe and Chen-Lin.

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