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

Abstract

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 $\pi /2<\theta <2\pi$, $\theta \ne \pi$, 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).

13 pages

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