  Inspired by work of Colding-Minicozzi on mean curvature flow, Zhang
introduced a notion of entropy stability for harmonic map flow. We build
further upon this work in several directions. First we prove the equivalence of
entropy stability with a more computationally tractable $\mathcal F$-stability.
Then, focusing on the case of spherical targets, we prove a general instability
result for high-entropy solitons. Finally, we exploit results of Lin-Wang to
observe long time existence and convergence results for maps into certain
convex domains and how they relate to generic singularities of harmonic map
flow.
