  We show an energy convexity along any harmonic map heat flow with small
initial energy and fixed boundary data on the unit 2-disk. In particular, this
gives an affirmative answer to a question raised by W. Minicozzi asking whether
such harmonic map heat flow converges uniformly in time strongly in the
W^{1,2}-topology, as time goes to infinity, to the unique limiting harmonic
map.
