The growth of the vorticity gradient for the two-dimensional Euler flows on domains with corners

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 $\pi \mathrm{/}2$, 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 $\pi \mathrm{/}2<\theta <2\pi$, $\theta \neq \pi$, we construct an example of the vorticity which loses continuity instantaneously. For the case $\theta \le \pi \mathrm{/}2$, 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 $\theta =\pi \mathrm{/}2$, 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).