SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 3366-3384, January 2021.
We prove a continuation criterion for incompressible liquids with free surface boundary when the liquid occupies a bounded region. We combine the energy estimates of Christodoulou and Lindblad [Comm. Pure Appl. Math., 53 (2000), pp. 1536--1602] with an analogue of the estimate due to Beale, Kato, and Majda [Comm. Math. Phys., 94 (1984), pp. 61--66] for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary as well as the material derivative of the normal derivative of the pressure remain bounded, assuming that the Taylor sign condition holds.

中文翻译：

---

关于具有自由表面边界的不可压缩欧拉方程解的分解

SIAM 数学分析杂志，第 53 卷，第 3 期，第 3366-3384 页，2021 年 1 月。
当液体占据有界区域时，我们证明了具有自由表面边界的不可压缩液体的连续准则。我们结合了 Christodoulou 和 Lindblad [Comm. 纯应用 Math., 53 (2000), pp. 1536--1602] 与 Beale、Kato 和 Majda [Comm. 数学。Phys., 94 (1984), pp. 61--66] 以涡度表示速度的梯度，并用它来说明只要自由边界的第二基本形式和注入半径可以继续求解，涡度和自由边界上速度的一个导数以及压力的法向导数的材料导数保持有界，假设泰勒符号条件成立。