be estimated in some rough way. This rough estimate of the local energy leads to a small perturbation of global bubbling solutions. This perturbation on global solutions, albeit small, has a non-negligible effect on the approximation of blowup solutions. This difficulty is particularly evident when we study bubbling solutions with multiple blowup points in [@linzhang4]. Therefore our method to obtain those sharp results is different from ones in Chen-Lin [@ChenLin1] for the mean field equations and Lin-Wei-Zhao [@lwz] for the $SU(3)$ Toda system (The methods in [@ChenLin1] and [@lwz] are similar).

The organization of the paper is as follows. In section two we first prove a uniqueness theorem for globally defined linearized Liouville systems. This result plays a central role for the delicate blowup analysis in sections three to five. The main idea of the proof uses a monotonicity property of solutions and we introduce a way to use maximum principles suitable for Liouville systems. In the second part of section two, we study the asymptotic behavior of global solutions to the Liouville system on $\mathbb R^2$ and obtain some Pohozaev identities. In section three and section four we obtain a sharp expansion result for blowup solutions around a blowup 