  In this paper, we investigate the persistency of loglog singular structure
for 2D Euler equation under the perturbation of continuous function. First, we
prove that when loglog vortex is perturbed by the function with modulus of
continuity $\mu(r)=\left( \log \frac{1}{r}\right)^{-\alpha} ~~ (0<\alpha<1)$,
the loglog vortex is propagated while the perturbation term maintaining same
modulus of continuity. We then show that this result is sharp: There exists a
smooth function such that when the loglog vortex is perturbed by the smooth
function, the norm of perturbation term corresponding to modulus of continuity
$\mu(r)=\left( \log \frac{1}{r}\right)^{-\alpha} $ instantly blows up for all
$\alpha>1$.
