Uniqueness for the harmonic map flow in two dimensions (Q1343746)

The following uniqueness theorem is proved: Let \(M\) be a 2-dimensional Riemannian manifold with smooth boundary. If \(u,v \in H^ 1 (M \times [0,T]\); \(S^ N)\) are weak heat flows for harmonic maps, with nonincreasing energy in time, and with the same initial data and boundary value, then \(u=v\). This theorem differs from a similar result due to T. Rivière, in which a small energy assumption was made but without the nonincreasing energy assumption.