  We introduce a heat flow associated to half-harmonic maps, which have been
introduced by Da Lio and Rivi\`ere. 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.
