We consider the two-dimensional incompressible Euler equation \(\begin{aligned}{\left\{ \begin{array}{ll} \partial _t \omega + u\cdot \nabla \omega =0, \\ \omega (x,0)=\omega _0(x). \end{array}\right. }\end{aligned}\) We are interested in the cases when the initial vorticity has the form \(\omega _0=\omega _{0,\epsilon }+\omega _{0p,\epsilon }\) , where \(\omega _{0,\epsilon }\) is concentrated near M disjoint points \(p_m^0\) and \(\omega _{0p,\epsilon }\) is a small perturbation term. We prove that for such initial vorticities, the solution \(\omega (x,t)\) admits a decomposition \(\omega (x,t)=\omega _{\epsilon }(x,t)+\omega _{p,\epsilon }(x,t)\) , where \(\omega _{\epsilon }(x,t)\) remains concentrated near M points \(p_m(t)\) and \(\omega _{p,\epsilon }(x,t)\) remains small for \(t \in [0,T]\) . As a consequence of such decomposition, we are able to consider the initial vorticity of the form \(\omega _0(x)=\sum _{m=1}^M \frac{\gamma _m}{\epsilon ^2}\eta (\frac{x-p_m^0}{\epsilon })\) , where we do not assume \(\eta \) to have compact support. Finally, we prove that if \(p_m(t)\) remains separated for all \(t\in [0,+\infty )\) , then \(\omega (x,t)\) remains concentrated near M points at least for \(t \le c_0 |\log A_{\epsilon }|\) , where \(A_{\epsilon }\) is small and converges to 0 as \(\epsilon \rightarrow 0\) .
