Harmonic Map Heat Flow with Rough Boundary Data

Abstract. Let B1 be the unit open disk in R2 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 H1([0, T ]×B1,M) whose energy is non-increasing in time, given initial data u0 ∈ H(B1,M) and boundary data γ = u0|∂B1 . Previously, this uniqueness result was obtained by Rivière (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 u0 ∈ H(B1,M) and γ = u0|∂B1 ∈ H (∂B1). 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.