In this article, we focus on the uniqueness question for (expanding) solutions of the harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a 0-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved.