Harmonic map heat flow with rough boundary data

Abstract: Let $B_1$ be the unit open disk in $\Real^2$ and $M$ be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\times B_1,M)$ whose energy is non-increasing in time, given initial data $u_0\in H^1(B_1,M)$ and boundary data $\gamma=u_0|{\partial B_1}$. Previously, this uniqueness result was obtained by Rivi`{e}re (when $M$ is the round sphere and the energy of initial data is small) and Freire (when $M$ is an arbitrary closed Riemannian manifold), given that $u_0\in H^1(B_1,M)$ and $\gamma=u_0|{\partial B_1}\in H^{3/2}(\partial B_1)$. The point of our uniqueness result is that no boundary regularity assumption is needed. Second, we prove the exponential convergence of the harmonic map heat flow, assuming that energy is small at all times.