  For the $2D$ Euler equation in vorticity formulation, we construct localized
smooth solutions whose critical Sobolev norms become large in a short period of
time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes
$H^1$ immediately for $t>0$. Our main observation is that a localized chunk of
vorticity bounded in $L^\infty \cap H^1$ with odd-odd symmetry is able to
generate a hyperbolic flow with large velocity gradient at least for a short
period of time, which stretches the vorticity gradient.
