Uniqueness for the harmonic map flow from surfaces to general targets

LetM be a two-dimensional compact Riemannian manifold with smooth (possibly empty) boundary,N an arbitrary compact manifold. Ifu andv are weak solutions of the harmonic map flow inH ¹(Mx0,T]; N) whose energy is non-increasing in time and having the same initial datau ₀∈H¹(M, N) (and same boundary values if ?M≠Ø) thenu=v. Combined with a result of M. Struwe, this shows any suchu is smooth in the complement of a finite subset ofM×(0,T)c.