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

Regularity and uniqueness of the heat ow of biharmonic maps