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Remarks on approximate harmonic maps in dimension two

For the class of approximate harmonic maps $u\in W^{1,2}(Σ,N)$ from a closed Riemmanian surface $(Σ,g)$ to a compact Riemannian manifold $(N, h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps $\{u_n\}:Σ\to N$, with tension fields $τ(u_n)$ bounded in the Morrey space $M^{1,δ}(Σ)$ for some $0\leδ<2$; and (ii) if an approximate harmonic map $u$ has tension field $τ(u)\in L\log L(Σ)\cap M^{1,δ}(Σ)$ for some $0\leδ<2$, then $u\in W^{2,1}(Σ, N)$. Based on these estimates, we further establish the bubble tree convergence, referring to energy identity both $L^{2,1}$ of gradients and $L^1$-norm of hessians and the oscillation convergence, for a weakly convergent sequence of approximate harmonic maps $\{u_n\}$, with tension fields $τ(u_n)$ uniformly bounded in $M^{1,δ}(Σ)$ for some $0\leδ<2$ and uniformly integrable in $L\log L(Σ)$.