AI 中文总结
研究具有毛细作用的三维液滴自由边界欧拉方程恒定涡度解，先介绍变分法得出的刚性结果，再对赤道截面接近圆盘的流体域进行摄动分析，证明阈值之上也有刚性结果。
AI 中文摘要
我们考虑具有毛细作用的三维液滴的自由边界欧拉方程的与时间无关的恒定涡度解。最近用变分法证明了该问题解的一个刚性结果：如果涉及涡度参数、毛细系数和液滴赤道截面面积的某个量低于某个值，那么解必然具有柱对称性，液滴形状为扁球体，两极扁平赤道鼓起，且每个流体粒子以恒定角速度沿水平圆周轨迹运动。本文针对赤道截面在\(C^2\)范数下接近圆盘的流体域对该问题进行摄动分析，并表明在变分法得到的阈值之上也成立刚性结果。
英文摘要
We consider time-independent solutions with constant vorticity of the free boundary Euler equations for a 3D liquid drop with capillarity. A rigidity result for the solutions of this problem has been recently proved with variational methods: if a certain quantity involving the vorticity parameter, the capillarity coefficient and the area of the equatorial section of the drop is below a certain value, then the solution has necessarily cylindrical symmetry, the shape of the drop is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity. In this paper we develop a perturbation analysis of the problem for fluid domains whose equatorial section is close in C2 norm to a disc, and we show that a rigidity result holds also above the threshold obtained with variational methods.
