  In this paper, we first establish regularity of the heat flow of biharmonic
maps into the unit sphere $S^L\subset\mathbb R^{L+1}$ under a smallness
condition of renormalized total energy. For the class of such solutions to the
heat flow of biharmonic maps, we prove the properties of uniqueness, convexity
of hessian energy, and unique limit at time infinity. We establish both
regularity and uniqueness for the class of weak solutions $u$ to the heat flow
of biharmonic maps into any compact Riemannian manifold $N$ without boundary
such that $\nabla^2 u\in L^q_tL^p_x$ for some $p>n/2$ and $q>2$ satisfying
(1.13).
