Energy identity of approximate biharmonic maps to Riemannian manifolds and its application

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac{4}{3}$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic maps on ${\mathbb{R}}^4$. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps at time infinity.