Remarks on approximate harmonic maps in dimension two

Abstract: For the class of approximate harmonic maps $u\in W^{1,2}(\Sigma,N)$ from a closed Riemmanian surface $(\Sigma,g)$ to a compact Riemannian manifold $(N, h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps ${u_n}:\Sigma\to N$, with tension fields $\tau(u_n)$ bounded in the Morrey space $M^{1,\delta}(\Sigma)$ for some $0\le\delta<2$; and (ii) if an approximate harmonic map $u$ has tension field $\tau(u)\in L\log L(\Sigma)\cap M^{1,\delta}(\Sigma)$ for some $0\le\delta<2$, then $u\in W^{2,1}(\Sigma, N)$. 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 $\tau(u_n)$ uniformly bounded in $M^{1,\delta}(\Sigma)$ for some $0\le\delta<2$ and uniformly integrable in $L\log L(\Sigma)$.