We consider the two-dimensional incompressible Euler equation \[\begin{ cases} \partial_t \omega + u\cdot \nabla \omega=0 \\ \omega(0,x)=\omega_0(x). \end{ cases}\] 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. First, 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 $t \in [0,T]$. Second, we give a quantitative description when the initial vorticity has 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 $\epsilon \to 0$.
Categories: math.AP