Energy identity for approximate harmonic maps from surfaces to general targets

Let un be a sequence of mappings from a closed Riemannian surface M to a general Riemannian manifold N. If un satisfiessupn⁡(‖∇un‖L2(M)+‖τ(un)‖Lp(M))≤Λfor some p>1, where τ(un) is the tension field of un, then the so called energy identity and neckless property hold during blowing up. This result is sharp by Parker's example, where the tension fields of the mappings from Riemannian surface are bounded in L1(M) but the energy identity fails.