  We consider the two-dimensional Euler equations in non-smooth domains with
corners. It is shown that if the angle of the corner $\theta$ is strictly less
than $\pi/2$, the Lipschitz estimate of the vorticity at the corner is at most
single exponential growth and the upper bound is sharp. %near the stagnation
point. For the corner with the larger angle $\pi/2 < \theta <2\pi$, $\theta
\neq \pi$, we construct an example of the vorticity which loses continuity
instantaneously. For the case $\theta \le \pi/2$, the vorticity remains
continuous inside the domain. We thus identify the threshold of the angle for
the vorticity maintaining the continuity. For the borderline angle
$\theta=\pi/2$, it is also shown that the growth rate of the Lipschitz constant
of the vorticity can be double exponential, which is the same as in
Kiselev-Sverak's result (Annals of Math., 2014).
