The growth of the vorticity gradient for the two-dimensional Euler flows on domains with corners
The growth of the vorticity gradient for the two-dimensional Euler flows on domains with corners
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发表时间：
2016-02
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通讯作者：
Tsubasa Itoh;H. Miura;Tsuyoshi Yoneda
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作者：
Tsubasa Itoh;H. Miura;Tsuyoshi Yoneda
我们考虑有角的非光滑域中的二维欧拉方程。结果表明，如果角点角度$\theta$严格小于$\pi/2$，则角点处涡量的Lipschitz估计至多为单指数增长，且上界尖锐。 %接近停滞点。对于角度较大的角$\pi/2 < \theta <2\pi$、$\theta \neq \pi$，我们构建了一个瞬时失去连续性的涡量示例。对于 $\theta \le \pi/2$ 的情况，涡度在域内保持连续。因此，我们确定了维持连续性的涡度的角度阈值。对于边界角$\theta=\pi/2$，还表明涡度的Lipschitz常数的增长率可以是双指数的，这与Kiselev-Sverak的结果相同(Annals of Math., 2014)。
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).