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

Conservation and Breakdown of Modulus of Continuity in the Propagation of Loglog Vortex for 2D Euler equation