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In this paper we focus on the uniqueness question for (expanding) solutionsof the Harmonic map flow coming out of smooth 0-homogeneous maps with valuesinto a closed Riemannian manifold. We introduce a relative entropy for twopurposes. On the one hand, we prove the existence of two expanding solutionsassociated to any suitable solution coming out of a $0$-homogeneous map by ablow-up and a blow-down process. On the other hand, generic uniqueness ofexpanding solutions coming out of the same 0-homogeneous map of 0 relativeentropy is proved.

A relative entropy for expanders of the Harmonic map flow